R-Parity-violating supersymmetry

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R-PARITY-VIOLATING SUPERSYMMETRY

R. Barbier1, C. Berat2, M. Besancon3, M. Chemtob4,A. Deandrea1, E. Dudas5,6, P. Fayet7, S. Lavignac4,8,

G. Moreau9, E. Perez3 and Y. Sirois10

1 IPNL, Universite Claude Bernard, IN2P3-CNRS, 69622 Villeurbanne, France2 LPSC, Universite de Grenoble 1, IN2P3-CNRS, 38026 Grenoble, France

3 DAPNIA/Service de Physique des Particules, CEA-Saclay, 91191 Gif-sur-Yvette, France4 Service de Physique Theorique, CEA-Saclay, 91191 Gif-sur-Yvette, France

5 Laboratoire de Physique Theorique, Universite de Paris-Sud, 91405 Orsay, France6 Centre de Physique Theorique, Ecole Polytechnique, 91128 Palaiseau, France

7 Laboratoire de Physique Theorique, Ecole Normale Superieure, 75005 Paris, France8 CERN Theory Division, CH-1211 Geneve, Suisse

9 Service de Physique Theorique, Universite Libre de Bruxelles, 1050 Brussels, Belgium10 Laboratoire Leprince-Ringuet, Ecole Polytechnique, IN2P3-CNRS, 91128 Palaiseau, France

Abstract

Theoretical and phenomenological implications ofR-parity violation in supersymmetric the-ories are discussed in the context of particle physics and cosmology. Fundamental aspectsinclude the relation with continuous and discrete symmetries and the various allowed patternsof R-parity breaking. Recent developments on the generation ofneutrino masses and mixingswithin different scenarios ofR-parity violation are discussed. The possible contribution ofR-parity-violating Yukawa couplings in processes involvingvirtual supersymmetric particles andthe resulting constraints are reviewed. Finally, direct production of supersymmetric particlesand their decays in the presence ofR-parity-violating couplings is discussed together with asurvey of existing constraints from collider experiments.

To be submitted to Physics Reports

Contents

INTRODUCTORY REMARKS . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1 WHAT IS R-PARITY ? 7

1.1 What IsR-Parity, and How Was It Introduced ? . . . . . . . . . . . . . . . . . 7

1.2 Nature Does Not Seem To Be Supersymmetric ! . . . . . . . . . . . .. . . . . 11

1.3 ContinuousR-Invariance, and Electroweak Breaking . . . . . . . . . . . . . . 14

1.4 R-Invariance andR-Parity in the Supersymmetric Standard Model . . . . . . . 15

1.5 Gravitino and Gluino Masses: FromR-Invariance toR-Parity . . . . . . . . . 19

2 HOW CAN R-PARITY BE VIOLATED? 22

2.1 R-Parity-Violating Couplings . . . . . . . . . . . . . . . . . . . . . . . . .. 22

2.1.1 Superpotential Couplings . . . . . . . . . . . . . . . . . . . . . . .. . 23

2.1.2 Lagrangian Terms Associated with the SuperpotentialCouplings . . . . 24

2.1.3 Soft Supersymmetry-Breaking Terms . . . . . . . . . . . . . . .. . . 26

2.1.4 Choice of the Weak Interaction Basis . . . . . . . . . . . . . . .. . . 27

2.1.5 Constraints on the Size of6Rp Couplings . . . . . . . . . . . . . . . . 28

2.2 Patterns ofR-Parity Breaking . . . . . . . . . . . . . . . . . . . . . . . . . . 29

2.3 Effects of BilinearR-Parity violation . . . . . . . . . . . . . . . . . . . . . . 31

2.3.1 Distinguishing Between Higgs and Lepton Doublet Superfields . . . . 32

2.3.2 Trilinear Couplings Induced by Bilinear6Rp Terms . . . . . . . . . . . 35

2.3.3 Higgsino-Lepton Mixing . . . . . . . . . . . . . . . . . . . . . . . . .36

2.3.4 Experimental Signatures of BilinearR-Parity Violation . . . . . . . . . 38

2.4 Spontaneous Breaking ofR-Parity . . . . . . . . . . . . . . . . . . . . . . . . 39

2.5 Constraining6Rp Couplings from Flavour Symmetries . . . . . . . . . . . . . . 41

2.6 R-Parity Violation in Grand Unified Theories . . . . . . . . . . . . . .. . . . 44

2.7 Restrictions onR-Parity Violations from Generalized Matter, Baryon or LeptonParities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

3 RENORMALIZATION GROUP SCALE EVOLUTION OF 6Rp COUPLINGS 51

3.1 Renormalization Group Equations . . . . . . . . . . . . . . . . . . .. . . . . 52

3.1.1 Evolution of the Bilinearµ Terms . . . . . . . . . . . . . . . . . . . . 52

3.1.2 Evolution of the Trilinear6Rp Yukawa Couplings . . . . . . . . . . . . 54

3.1.3 Evolution of the Gauge Couplings . . . . . . . . . . . . . . . . . .. . 55

3.2 Perturbative Unitarity Constraints . . . . . . . . . . . . . . . .. . . . . . . . 56

3.3 Quasi-fixed points analysis for6Rp couplings . . . . . . . . . . . . . . . . . . . 56

3.4 Supersymmetry Breaking . . . . . . . . . . . . . . . . . . . . . . . . . . .. . 59

4 COSMOLOGY AND ASTROPHYSICS 60

4.1 Constraints from the lifetime of the Lightest Supersymmetric Particle . . . . . 61

4.1.1 Decays of the Lightest Supersymmetric Particle . . . . .. . . . . . . . 61

4.1.2 Gravitino Relics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

4.2 Cosmological Baryon Asymmetry . . . . . . . . . . . . . . . . . . . . .. . . 65

4.2.1 Baryogenesis fromR-Parity-Violating Interactions . . . . . . . . . . . 65

4.2.2 Survival of a Baryon Asymmetry in the Presence of6Rp Interactions . . 68

5 NEUTRINO MASSES AND MIXINGS 72

5.1 6Rp Contributions to Neutrino Masses and Mixings . . . . . . . . . . . .. . . 72

5.1.1 R-Parity Violation as a Source of Neutrino Masses . . . . . . . . . .. 72

5.1.2 Tree-Level Contribution Generated by Neutrino-Neutralino Mixing . . 73

5.1.3 One-Loop Contributions Generated by Trilinear6Rp Couplings . . . . . 75

5.1.4 One-Loop Contributions Generated by both Bilinear and Trilinear 6Rp

Couplings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

5.2 Explicit Models of Neutrino Masses . . . . . . . . . . . . . . . . . .. . . . . 80

5.2.1 Experimental Constraints on Neutrino Masses and Mixings . . . . . . . 80

5.2.2 Classification of Models . . . . . . . . . . . . . . . . . . . . . . . . .81

a) Models with Trilinear Couplings only . . . . . . . . . . . . . . . . .81

b) Models with both Bilinear and Trilinear Couplings . . . . . .. . . . 82

c) Models with Bilinear Couplings only . . . . . . . . . . . . . . . . . 83

5.3 Neutrino Transition Magnetic Moments . . . . . . . . . . . . . . .. . . . . . 85

5.4 Neutrino Flavour Transitions in Matter Induced by6Rp Interactions . . . . . . . 86

5.5 ∆L = 2 Sneutrino Masses and Sneutrino-Antisneutrino Mixing . . . .. . . . 87

6 INDIRECT BOUNDS ON R-parity ODD INTERACTIONS 89

6.1 Assumptions and Framework . . . . . . . . . . . . . . . . . . . . . . . . .. . 90

6.1.1 The Single Coupling Dominance Hypothesis . . . . . . . . . .. . . . 90

6.1.2 Choice of the Lepton and Quark Superfield Bases . . . . . . .. . . . . 90

6.1.3 A Basis-Independent Parametrization ofR-Parity Violation . . . . . . 92

6.1.4 Specific Conventions Used in this Chapter . . . . . . . . . . .. . . . . 93

6.2 Constraints on Bilinear6Rp Terms and on Spontaneously BrokenR-Parity . . . 94

6.2.1 Models with ExplicitR-Parity Breaking . . . . . . . . . . . . . . . . . 94

6.2.2 Models with SpontaneousR-Parity Breaking . . . . . . . . . . . . . . 95

6.3 Constraints on the Trilinear6Rp Interactions . . . . . . . . . . . . . . . . . . . 97

6.3.1 Charged Current Interactions . . . . . . . . . . . . . . . . . . . .. . . 97

6.3.2 Neutral Current Interactions . . . . . . . . . . . . . . . . . . . .. . . 102

6.3.3 Anomalous Magnetic Dipole Moments . . . . . . . . . . . . . . . .. 109

6.3.4 CP Violation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

6.4 Trilinear6Rp Interactions in Flavour Violating Processes and in6B and6L Processes120

6.4.1 Hadron Flavour Violating Processes . . . . . . . . . . . . . . .. . . . 121

6.4.2 Lepton Flavour Violating Processes . . . . . . . . . . . . . . .. . . . 129

6.4.3 Lepton Number Non-Conserving Processes . . . . . . . . . . .. . . . 132

6.4.4 Baryon Number Non-Conserving Processes . . . . . . . . . . .. . . . 135

6.5 General Discussion of Indirect Trilinear Bounds . . . . . .. . . . . . . . . . . 140

6.5.1 Summary of Main Experimental Bounds . . . . . . . . . . . . . . .. 140

6.5.2 Observations on the Bound Robustness and Validity . . .. . . . . . . 147

a) Natural Order of Magnitude for6Rp Couplings . . . . . . . . . . . . 147

b) Impact of the SUSY Masses on the Bounds . . . . . . . . . . . . . . 147

c) Validity of the Assumption of one or two Dominant Couplings . . . . 148

d) Bound Robustness in Regards to Model Dependence . . . . . . . .. 149

6.5.3 Phenomenological Implications of Bounds . . . . . . . . . .. . . . . 149

7 PHENOMENOLOGY AND SEARCHES AT COLLIDERS 152

7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .152

7.2 Interaction Strength and Search Strategies . . . . . . . . . .. . . . . . . . . . 153

7.3 Decay of Sparticles Involving6Rp Couplings . . . . . . . . . . . . . . . . . . . 155

7.3.1 Direct6Rp Decays of Sfermions . . . . . . . . . . . . . . . . . . . . . 155

7.3.2 Direct6Rp Decays of Gauginos-Higgsinos . . . . . . . . . . . . . . . . 156

7.3.3 Cascade Decays Initiated by Gauge Couplings . . . . . . . .. . . . . 159

7.3.4 Decays Through Mixing Involving Bilinear Interactions . . . . . . . . 161

7.4 6Rp Phenomenology from Pair Produced Sparticles . . . . . . . . . . . .. . . . 162

7.4.1 Gaugino-Higgsino Pair Production . . . . . . . . . . . . . . . .. . . . 162

a) Production and Final States . . . . . . . . . . . . . . . . . . . . . . 162

b) Searches for Gaugino Pair Production ate+e− Colliders . . . . . . . 164

c) Searches for Gaugino Pair Production at Hadron Colliders. . . . . . 168

7.4.2 Sfermion Pair Production . . . . . . . . . . . . . . . . . . . . . . . .. 168

a) Production and final states . . . . . . . . . . . . . . . . . . . . . . . 168

b) Results on Slepton Searches . . . . . . . . . . . . . . . . . . . . . . 169

c) Results on Squark Searches . . . . . . . . . . . . . . . . . . . . . . 176

d) Sfermion and Gluino Pair Production at Hadron Colliders .. . . . . 178

7.4.3 Effects of Bilinear6Rp Interactions . . . . . . . . . . . . . . . . . . . . 181

7.5 Single Sparticle Production . . . . . . . . . . . . . . . . . . . . . . .. . . . . 183

7.5.1 Single Sparticle Production at Leptonic Colliders . .. . . . . . . . . . 184

7.5.2 Single Sparticle Production at Lepton-Hadron Colliders . . . . . . . . 190

7.5.3 Single Sparticle Production at Hadron-Hadron Colliders . . . . . . . . 194

7.6 Virtual Effects involving6Rp Couplings . . . . . . . . . . . . . . . . . . . . . 204

7.6.1 Fermion Pair Production . . . . . . . . . . . . . . . . . . . . . . . . .205

7.6.2 6Rp Contributions to FCNC . . . . . . . . . . . . . . . . . . . . . . . . 212

7.6.3 6Rp Contributions toCP Violation . . . . . . . . . . . . . . . . . . . 216

8 CONCLUSIONS AND PROSPECTS 219

A Notations and Conventions 225

B Yukawa-like 6Rp Interactions Associated with the Trilinear 6Rp Superpotential 228

C Production and Decay Formulae 230

C.1 Sfermions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231

C.2 Neutralinos . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233

C.3 Charginos . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235

INTRODUCTORY REMARKS

The possible appearance ofR-parity-violating couplings, and hence the question of theconservation or non-conservation of baryon and lepton numbers (B andL) in supersymmetrictheories, has been emphasized for a long time. The rich phenomenology implied byR-parityviolation has gained full attention in the search for supersymmetry. We shall discuss here thetheoretical as well as phenomenological aspects of6Rp supersymmetry in particle and astropar-ticle physics.

In chapter 1 we introduce fundamental aspects of supersymmetry, having in mind the ques-tion of the definition of conserved baryon and lepton numbersin supersymmetric theories. Insupersymmetric extensions of the Standard ModelR-parity has emerged as a discrete rem-nant of a group of continuousU(1) R-symmetry transformations acting on the supersymmetrygenerator.R-parity is intimately connected with baryon and lepton numbers, its conservationnaturally allowing for conserved baryon and lepton numbersin supersymmetric theories. Con-versely, the violation ofR-parity requires violations ofB and/orL conservation laws. Thisgenerally leads to important phenomenological difficulties, unlessR-parity-violating interac-tions are sufficiently small. How small they have to be, and how these difficulties may beturned into opportunities in some specific cases, concerning for example neutrino masses andmixings, constitute important aspects of this review.

Chapter 2 is devoted to the discussion ofhowR-parity may be broken. The correspondingsuperpotential couplings (and resulting Lagrangian terms) and soft supersymmetry-breakingterms are recalled. Various possible patterns ofR-parity breaking are discussed, includingbilinear breaking as well as spontaneous breaking. Furthertheoretical insights on the possibleorigin of such terms violatingB and/orL as well as theR-parity symmetry are reviewed.This includes more recent developments on abelian family symmetries, grand-unified gaugesymmetries, and other discrete symmetries, and what they could tell us about possible6Rp terms.

The high-energy convergence of the gauge couplings obtained by renormalization-groupevolution of low-energy measurements gets remarkably improved once supersymmetry is in-troduced. More generally, the renormalization group equations governing the evolutions of thecoupling and mass parameters between two energy scales provide a way to test, at lower ener-gies, physical assumptions postulated at a much higher energy scale; or conversely to translateavailable experimental data into quantities at a higher energy scale. In chapter 3 we considerthe effects of the renormalization group equations, in the presence ofR-parity-violating inter-actions. We focus in particular, within the supergravity framework, on the evolution of the con-straints associated with perturbative unitarity, the existence of infrared fixed points and the testsof grand-unification schemes. The additional effects of thenew soft supersymmetry-breakingterms associated withRp-violations are also discussed.

Supersymmetric theories with conservedR-parity naturally provide a (color and electricallyneutral) stable lightest supersymmetric particle (LSP), i.e. a weakly-interacting massive particlewhich turns out to be a very good Dark Matter candidate. In contrast, one of the most strikingfeatures of supersymmetric theories withR-parity-violating interactions stems from the fact thatthe LSP can now decay into Standard Model particles only. We discuss in chapter 4 how suchan unstable LSP might still remain (if its lifetime is sufficiently long) a possible Dark Mattercandidate. We also discuss the gravitino relic issue, and the origin of the cosmological baryonasymmetry, reviewing several attempts at generating this asymmetry, as well as how it couldsurvive in the presence ofR-parity-violating interactions.

The most dramatic implication ofL-violating interactions fromR-parity violations is theautomatic generation ofneutrino masses and mixings. The possibility that the results of atmo-spheric and solar neutrino experiments be explained by neutrino masses and mixings originat-ing fromR-parity-violating interactions has motivated a large number of studies and models.R-parity violation in the lepton sector also leads to many newphenomena related to neutrinoand sneutrino physics. These aspects of neutrino physics related toL-violating interactions arereviewed in chapter 5.

In chapter 6 we discuss the possible contribution ofR-parity violating couplings to pro-cesses involving thevirtual effectsof supersymmetric particles. IndeedR-parity-violating cou-plings in the Supersymmetric Standard Model introduce new interactions between ordinary andsupersymmetric particles which can contribute to a large variety of low, intermediate and high-energy processes, not involving the direct production of supersymmetric particles in the finalstate. The requirement that theR-parity-violating contribution to a given observable avoids con-flicting with actual experimental measurements, leads to upper bounds on theR-parity-violatingcouplings possibly involved. These bounds are extensivelydiscussed, the main ones being sum-marized at the end of the chapter, in section 6.5. Their robustness as well as phenomenologicalimplications are also discussed at the end of this chapter.

The search for6Rp -supersymmetry processes has been a major analysis activity at high-energy colliders over the past 15 years, and is likely to be pursued at existing and future col-liders. Chapter 7.1 is dedicated to the phenomenology anddirect searches, at colliders, forsupersymmetric particles involvingR-parity-violating couplings. The essential ingredients ofthe corresponding phenomenology at colliders, including discussions on the magnitude ofR-parity-violating couplings and the subsequent decay of supersymmetric particles, are reviewed.

We then discuss the main and generic features of theR-parity-violating phenomenology forgaugino-higgsino pair productionandsfermion pair production, both at leptonic and hadroniccolliders. Furthermore, a remarkable specificity of the phenomenology of6Rp at colliders comesfrom possibility of producinga single supersymmetric particle. (This is also discussed in chap-ter 7 for leptonic, lepton-hadron and hadronic colliders.)The phenomenology of6Rp at collidersalso covers virtual effects such as those concerning fermion pair production, contributions toflavor-changing neutral currents and toCP violation. These aspects are also met in chapter 7.1.

Altogether, many direct experimental limits have accumulated during the last 15 years ofsearches for6Rp processes at colliders. We do not aim here at an exhaustive (and possiblytedious) catalog of all these searches with the corresponding limits. We rather choose to refer thereader interested in specific limits and details of experimental analyses to the relevant literatureand emphasize only the description of generic features of the phenomenology ofR-parity-violating processes at colliders, illustrated by examplesfrom the literature.

Conclusions and prospects for supersymmetry withRp-violating couplings are given inchapter 8. Finally, notations and conventions are summarized in appendix A. The Yukawa-like 6Rp interactions associated with the trilinear6Rp superpotential couplings (given in appendixA) are derived in appendix B. Useful formulae for the production and decays of sfermions,neutralinos and charginos are given in appendix C.

Chapter 1

WHAT IS R-PARITY ?

In this chapter, we recall howR-parity emerged, in supersymmetric extensions of the Stan-dard Model, as a discrete remnant of a continuousU(1) R-symmetry group acting on thesupersymmetry generator, necessarily broken so as to allowfor the gravitino and gluinos toacquire masses.R-parity naturally forbids unwanted squark and slepton exchanges, allowingfor conserved baryon (B) and lepton (L) numbers in supersymmetric theories. It guarantees thestability of the “Lightest Supersymmetric Particle”, which is, also, a very good candidate forthe non-baryonic Dark Matter of the universe.A contrario, R-parity violations are necessarilyaccompanied byB and/orL violations. This is, usually, a source of phenomenologicaldif-ficulties, unlessR-parity-violating (6Rp) interactions are sufficiently small, as we shall discussin this review article.R-parity violations, on the other hand, could also appear as adesiredfeature, since they may provide a source of Majorana masses for neutrinos. WhetherR-parityturns out to be absolutely conserved, or not, it plays an essential role in the phenomenology ofsupersymmetric theories, and the experimental searches for the new sparticles.

1.1 What Is R-Parity, and How Was It Introduced ?

Among the problems one had to solve before thinking of applying supersymmetry to the realworld, was the question of the definition of conserved quantum numbers, like baryon numberB and lepton numberL . These are carried by Dirac fermions, the spin-1

2quarks and lep-

tons. But supersymmetric theories make a systematic use ofMajorana fermions, in particularthe fermionic partners of the spin-1 gauge bosons (now called gauginos). This makes it verydifficult, even in general practically impossible, for themto carry additive conserved quantumnumbers likeB andL, in a supersymmetric theory.

Still, even Majorana fermions may be arranged into (chiral or non-chiral) Dirac fermionsso as to carry non-zero values of a new additive quantum number, called R . In an earlySU(2) × U(1) supersymmetric electroweak model with two chiral doublet Higgs superfields,now calledHd and Hu (or H1 and H2 ), the definition of a continuousR-symmetry actingon the supersymmetry generator allowed for an additive conserved quantum number,R, oneunit of which is carried by the supersymmetry generator [1].The values ofR for bosons andfermions differ by± 1 unit inside the multiplets of supersymmetry, the photon, for example,havingR = 0 while its spin-1

2partner, constrained from the continuousR-invariance to remain

massless, carriesR = ± 1 . Such a quantum number might tentatively have been identified as

a lepton number, despite the Majorana nature of the spin-12

partner of the photon, if the lattercould have been identified as one of the neutrinos. This, however, is not the case. The fermionicpartner of the photon should be considered as a neutrino of a new type, a “photonic neutrino”,called in 1977 thephotino.

This still leaves us with the question of how to define, in suchtheories, Dirac spinorscarrying conserved quantum numbers likeB and L . Furthermore, these quantum numbers,presently known to be carried by fundamental fermions only,not by bosons, seem to appearas intrinsically-fermionic numbers. Such a feature cannot be maintained in a supersymmetrictheory (in the usual framework of the “linear realizations”of supersymmetry), and one had toaccept the (then rather heretic) idea of attributing baryonand lepton numbers to fundamentalbosons, as well as to fermions. These new bosons carryingB or L are the superpartners of thespin-1

2quarks and leptons, namely the now-familiar, although still unobserved, spin-0squarks

and sleptons. Altogether, all known particles should be associated withnew superpartners[2].

This introduction of squarks and sleptons now makes the definition of baryon and leptonnumbers in supersymmetric theories a quasi-triviality – these new spin-0 particles carryingBand L , respectively, almost by definition – to the point that this old problem is now hardlyremembered, since we are so used to its solution. This does not mean, however, that thesenewly-definedB andL shouldnecessarilybe conserved, since new interactions that could bepresent in supersymmetric theories might spoil our familiar baryon and lepton-number conser-vation laws, even without taking into account the possibility of Grand Unification !

In fact the introduction of a large number of new bosons has a price, and carries with itthe risk of potential difficulties. Could these new bosons beexchanged between ordinary par-ticles, concurrently with the gauge bosons of electroweak and strong interactions ? But knowninteractions are due to the exchanges of spin-1 gauge bosons, not spin-0 particles ! Can wethen construct supersymmetric theories of weak, electromagnetic and strong interactions, whichwould be free of this potential problem posed by unwanted interactions mediated by spin-0 par-ticles ? Fortunately, the answer is yes. As a matter of fact the above problem, related with theconservation or non-conservation ofB and L , comes with its own natural solution, namelyR-invariance or, more precisely, a discrete version of it, known asR-parity. This one is closelyrelated, of course, with the definitions ofB and L , once we have decided, and accepted,to attributeB and L to the new squarks and sleptons, as well as to the ordinary quarks andleptons.

R-parity is associated with aZ2 subgroup of the group of continuousU(1) R-symmetrytransformations – often referred to asU(1)R – acting on the gauge superfields and the two chi-ral doublet Higgs superfieldsHd andHu responsible for electroweak symmetry breaking [1],with their definition extended to quark and lepton superfields so that quarks and leptons carryR = 0 , and squarks and sleptons,R = ± 1 [2]. As we shall see later,R-parity appears in factas the discrete remnant of this continuousU(1) R-invariance when gravitational interactionsare introduced [3], in the framework of local supersymmetry(supergravity), in which the grav-itino must at some point acquire a massm3/2 (which breaks the continuousR-invariance). Inaddition, either the continuousR-invariance, or simply its discrete version ofR-parity, if im-posed, naturally forbid the unwanted direct exchanges of the new squarks and sleptons betweenordinary quarks and leptons. It is, therefore, no surprise if the re-introduction of (unnecessary)6Rp terms in the Lagrangian density generally introduces again, most of the time, the problemsthat were elegantly solved byR-parity.

The precise definition ofR-invariance, which acts chirally on the anticommuting Grass-mann coordinateθ appearing in the definition of superspace and superfields, will be given later

(see Table 1.3 in subsection 1.4).R-transformations are defined so as not to act on ordinary par-ticles, which all haveR = 0 , their superpartners having, therefore,R = ±1 . This allows oneto distinguish between two separate sectors ofR-even andR-odd particles.R-even particles(having R-parity Rp = + 1 ) include the gluons, the photon, theW± and Z gauge bosons,the quarks and leptons, the Higgs bosons originating from the two Higgs doublets (required insupersymmetry to trigger the electroweak breaking and to generate quark and lepton masses)– and the graviton.R-odd particles (havingR-parity Rp = − 1 ) include their superpartners,i.e. the gluinos and the various neutralinos and charginos,squarks and sleptons – and the grav-itino. According to this first definition,R-parity simply corresponds to the parity of the additivequantum numberR associated with the above continuousU(1) R-invariance, as given by theexpression [4]:

R-parity Rp = (− 1 )R =

+ 1 for ordinary particles,

− 1 for their superpartners.(1.1)

But should we limit ourselves to the discreteR-parity symmetry, rather than consideringits full continuous parentR-invariance ? Thiscontinuous U(1) R-invariance, from whichR-parity has emerged, is indeed a symmetry of all four necessary basic building blocks of theSupersymmetric Standard Model [2]:

1) the Lagrangian density for theSU(3)×SU(2)×U(1) gauge superfields responsible forstrong and electroweak interactions;

2) the SU(3) × SU(2) × U(1) gauge interactions of the quark and lepton superfields;3) the SU(2) × U(1) gauge interactions of the two chiral doublet Higgs superfields Hd

andHu responsible for the electroweak breaking;4) and the “super-Yukawa” interactions responsible for quark and lepton masses, through

the trilinear superpotential couplings of quark and leptonsuperfields with the Higgs superfieldsHd andHu ,

W = λeij HdLiE

cj + λd

ij HdQiDcj − λu

ij HuQiUcj , (1.2)

in which chiral quark and lepton superfields are all taken as left-handed and denoted byQi, Uci , D

ci

andLi, Eci respectively (withi = 1, 2 or 3 being the generation index).

Since all the corresponding contributions to the Lagrangian density are invariant under thiscontinuousR-symmetry, why not simply keep it instead of abandoning it infavour of its dis-crete version,R-parity ? But an unbroken continuousR-invariance, which acts chirally ongluinos, would constrain them to remain massless, even after a spontaneous breaking of the su-persymmetry. We would then expect the existence of relatively light “ R-hadrons” [5, 6] madeof quarks, antiquarks and gluinos, which have not been observed. Once the continuousR-invariance is abandoned, and supersymmetry is spontaneously broken, radiative corrections doindeed allow for the generation of gluino masses [7], a pointto which we shall return later. Fur-thermore, the necessity of generating a mass for the Majorana spin-3

2gravitino, once local su-

persymmetry is spontaneously broken, also forces us to abandon the continuousR-invariance infavour of the discreteR-parity symmetry, thereby automatically allowing for gravitino, gluino,and other gaugino masses [3].

Once we drop the continuousR-invariance in favour of its discreteR-parity version, it islegitimate to look back and ask: how general is this notion ofR-parity, and, correlatively, are weforced to have thisR-parity conserved ? As a matter of fact, there is from the beginning a close

connection betweenR-parity and baryon and lepton-number conservation laws, which has itsorigin in our desire to get supersymmetric theories in whichB andL could be conserved, and,at the same time, to avoid unwanted exchanges of spin-0 particles.

Actually the superpotential of the supersymmetric extensions of the Standard Model dis-cussed in Ref. [2] was constrained from the beginning, for that purpose, to be aneven func-tion of the quark and lepton superfields. In other terms,odd gauge-invariant superpotentialterms (W ′, also denotedW6Rp ), which would have violated the “matter-parity” symmetry(−1)(3B+L), were then excluded from the beginning, to be able to recoverB andL conserva-tion laws, and avoid direct Yukawa exchanges of spin-0 squarks and sleptons between ordinaryquarks and leptons.

Tolerating unnecessary superpotential terms which areodd functions of the quark and lep-ton superfields (i.e.6Rp terms, precisely those that we are going to discuss in this review), doesindeed create, in general, immediate problems with baryon-and lepton-number conservationlaws [8]. Most notably, a squark-induced proton instability with a much too fast decay rate, ifboth B and L violations are simultaneously allowed; or neutrino masses(and other effects)that could be too large, ifL violations are allowed so that ordinary neutrinos can mix with neu-tral higgsinos and gauginos. The aim of this review is to discuss in detail how much of these6Rp contributions – parametrized by sets of coefficientsλijk, λ

′ijk, λ

′′ijk (and possiblyµi, etc.) –

may be tolerated in the superpotential and in the various supersymmetry-breaking terms.

The above intimate connection betweenR-parity andB and L conservation laws can bemade explicit by re–expressing theR-parity (1.1) in terms of the spinS and a matter-parity(−1) 3B+L , as follows [5]:

R-parity = (−1) 2S (−1) 3B+L . (1.3)

To understand the origin of this formula we note that, for allordinary particles,(−1) 2S coin-cides with (−1) 3B+L, expressing that among Standard Model fundamental particles, leptonsand quarks, and only them, are fermions,i.e. thatB and L normally appear as intrinsically-fermionic numbers. The quantity(−1) 2S (−1) 3B+L is always, trivially, identical to unity forall known particles (whether fundamental or composite) andfor Higgs bosons as well, all ofthem previously defined as havingR-parity +1 . (Indeed expression (1.1) ofR-parity comesfrom the fact that the (additive) quantum numberR was defined so as to vanish for ordinaryparticles, which then haveR-parity + 1 , their superpartners having, therefore,R-parity − 1 .)This immediately translates into the equivalent expression (1.3) ofR-parity.

R-parity may also be rewritten as(−1)2S (−1) 3 (B−L) , showing that this discrete symme-try (now allowing for gravitino and gluino masses) may stillbe conserved even if baryon andlepton numbers are separately violated, as long as their difference (B −L ) remains conserved,even only modulo 2. Again, it should be emphasized that the conservation (or non-conservation)of R-parity is closely related with the conservation (or non-conservation) of baryon and leptonnumbers,B andL . AbandoningR-parity by tolerating bothB andL violations, simultane-ously, would allow for the proton to decay, with a very short lifetime !

TheR-parity operator plays an essential role in the construction of supersymmetric theoriesof interactions, and the discussion of the experimental signatures of the new particles.R-invariance, or simply its discrete version ofR-parity, guarantees thatthe new spin-0 squarksand sleptons cannot be directly exchangedbetween ordinary quarks and leptons. It ensures that

the newR-odd sparticles can only be pair-produced, and that the decay of an R-odd sparticleshould always lead to another one (or an odd number of them). ConservedR-parity also ensuresthe stability of the “Lightest Supersymmetric Particle” (or LSP), a neutralino for example (orconceivably a sneutrino, or gravitino)1, which appears as an almost ideal candidate to constitutethe non-baryonic Dark Matter that seems to be present in our universe.

Expression (1.3) ofR-parity in terms ofB and L makes very apparent that imposingR-parity is equivalent to imposing a matter-parity symmetry. Still the definition ofR-parityoffers the additional advantage of identifying the two separate sectors ofRp = +1 particlesandRp = −1 sparticles, making apparent the pair-production law of thenew R-odd sparticles,and the stability of the LSP, ifR-parity is conserved. Considering “matter-parity” alone wouldonly imply directly the stability of the lightest “matter-odd” particle, not a very useful result !

Obviously, in the presence ofR-parity violations, the LSP is no longer required to be stable,superpartners being allowed to decay into ordinary particles.

1.2 Nature Does Not Seem To Be Supersymmetric !

The algebraic structure of supersymmetry involves a spin-12

fermionic symmetry generatorQsatisfying the (anti) commutation relations in four dimensions [9, 10, 11]:

Q, Q = − 2 γµPµ ,

[Q,P µ ] = 0 .(1.4)

This spin-12

supersymmetry generatorQ , here written as a 4-component Majorana spinor,was originally introduced as relating fermionic with bosonic fields, in relativistic quantum fieldtheories. The presence of the generator of space-time translationsP µ on the right-hand side ofthe anticommutation relations (1.4) is at the origin of the relation of supersymmetry with generalrelativity and gravitation, since a locally supersymmetric theory must be invariant under localcoordinate transformations [12].

The supersymmetry algebra (1.4) was introduced with quite different motivations: in con-nection with parity violation, with the hope of understanding parity violation in weak interac-tions as a consequence of a (misidentified) intrinsically parity-violating nature of the supersym-metry algebra [9]; in an attempt to explain the masslessnessof the neutrino from a possibleinterpretation as a spin-1

2Goldstone particle [10]; or by extending to four dimensionsthe su-

persymmetry transformations acting on the two-dimensional string worldsheet [11]. However,the mathematical existence of an algebraic structure does not mean that it could play a role asan invariance of the fundamental laws of Nature2.

Indeed many obstacles seemed, long ago, to prevent supersymmetry from possibly beinga fundamental symmetry of Nature. Is spontaneous supersymmetry breaking possible at all ?Where is the spin-1

2Goldstone fermion of supersymmetry, if not a neutrino ? Can we use

1The possibility of acharged or colored LSP may also be considered, although it seems rather stronglydisfavoured, as it could lead to new heavy isotopes of hydrogen and other elements, which have not been observed(cf. subsection 4.1.1 in chapter 4).

2Incidentally while supersymmetry is commonly referred to as “relating fermions with bosons”, its algebra(1.4) does not even require the existence of fundamental bosons ! With non-linear realizations of supersymmetrya fermionic field can be transformed into acompositebosonic field made of fermionic ones [10].

supersymmetry to relate directly known bosons and fermions? And, if not, why ? If knownbosons and fermions cannot be directly related by supersymmetry, do we have to accept thisas the sign that supersymmetry isnot a symmetry of the fundamental laws of Nature ? Canone define conserved baryon and lepton numbers in such theories, although they systematicallyinvolve self-conjugateMajorana fermions, (so far) unknown in Nature ? And finally, if wehave to postulate the existence of new bosons carryingB andL – the new spin-0 squarks andsleptons – can we prevent them from mediating new unwanted interactions ?

While bosons and fermions should have equal masses in a supersymmetric theory, this iscertainly not the case in Nature. Supersymmetry should clearly be broken. But it is a specialsymmetry, since the Hamiltonian, which appears on the right-hand side of the anticommutationrelations (1.4), can be expressed proportionally to the sumof the squares of the componentsof the supersymmetry generator, asH = 1

4

∑α Q

2α . This implies that a supersymmetry-

preserving vacuum state must have vanishing energy, while astate which is not invariant undersupersymmetry could naıvely be expected to have a larger, positive, energy. As a result, poten-tial candidates for supersymmetry-breaking vacuum statesseemed to be necessarily unstable.This led to the question

Q1 : Is spontaneous supersymmetry breaking possible at all ? (1.5)

As it turned out, and despite the above argument, several ways of breaking spontaneouslyglobal or local supersymmetry have been found [13, 14]. But spontaneous supersymmetrybreaking remains in general rather difficult to obtain, at least in global supersymmetry (andeven without adressing yet the issue of how this breaking could lead to a realistic theory), sincetheories tend to prefer systematically, for energy reasons, supersymmetric vacuum states. Onlyin very exceptional situations can the existence of such states be avoided ! In local supersym-metry, which includes gravity, one also has to arrange, at the price of a very severe fine-tuning,for the energy density of the vacuum to vanish exactly, or almost exactly, to an extremely goodaccuracy, so as not to generate an unacceptably large value of the cosmological constantΛ .

We still have to break supersymmetry in an acceptable way, soas to get – if this is indeedpossible – a physical world which looks like the one we know ! Of course just acceptingexplicit supersymmetry-breaking terms without worrying too much about their possible originwould make things much easier (unfortunately also at the price of introducing a large number ofarbitrary parameters). But such terms must have their origin in a spontaneous supersymmetry-breaking mechanism, if we want supersymmetry to play a fundamental role, especially if it is tobe realized as a local fermionic gauge symmetry, as it shouldin the framework of supergravitytheories.

But the spontaneous breaking of the global supersymmetry must generate a massless spin-12

Goldstone particle, leading to the next question,

Q2 : Where is the spin-12

Goldstone fermion of supersymmetry ? (1.6)

Could it be one of the neutrinos [10] ? A first attempt at implementing this idea within aSU(2) × U(1) electroweak model of “leptons” [1] quickly illustrated that it could not be pur-sued very far. The “leptons” of this model were soon to be reinterpreted as the “charginos” and“neutralinos” of the Supersymmetric Standard Model.

If the Goldstone fermion of supersymmetry is not one of the neutrinos, why hasn’t it beenobserved ? Today we tend not to think at all about the question, since: 1) the generalized use

of soft terms breakingexplicitly the supersymmetry seems to render this question irrelevant;2) since supersymmetry has to be realized locally anyway, within the framework of supergrav-ity [12], the massless spin-1

2Goldstone fermion (“goldstino”) should in any case be eliminated

in favour of extra degrees of freedom for a massive spin-32

gravitino [3, 14].

But where is the gravitino, and why has no one ever seen a fundamental spin-32

particle ?To discuss this properly we need to know which bosons and fermions could be associated undersupersymmetry. Still, even before adressing this crucial question we might already anticipatethat the interactions of the gravitino, with amplitudes proportional to the square root of theNewton constant

√GN ≃ 10−19 GeV−1, should in any case be absolutely negligible in particle

physics experiments, so that we don’t have to worry about thefact that no gravitino has beenobserved.

This simple but naıve answer is, however, not true in all circumstances ! It could be that thegravitino is light, possibly even extremely light, so that it would still interact very much like themassless Goldstone fermion of global supersymmetry, according to the “equivalence theorem”of supersymmetry [3]. Its interaction amplitudes are then determined by the ratio

√GN/m3/2

(i.e. are inversely proportional to the square of the “supersymmetry-breaking scale”Λss). Asa result a sufficiently light gravitino could have non-negligible interactions, which might evenmake it observable in particle physics experiments, provided that the supersymmetry-breakingscale parameter fixing the value of its massm3/2 is not too large [3, 15] ! Because of theconservation ofR-parity, at least to a good approximation, in the Supersymmetric StandardModel, theR-odd gravitino should normally be produced in association with anotherR-oddsuperpartner, provided the available energy is sufficient.Gravitinos could also be pair-produced,although these processes are normally suppressed at lower energies. But they would remainessentially “invisible” in particle physics, as soon as thesupersymmetry-breaking scale is largeenough (compared to the electroweak scale), which is in factthe most plausible and widelyconsidered situation.

In any case, much before getting to the Supersymmetric Standard Model, the crucial ques-tion to ask, if supersymmetry is to be relevant in particle physics, is:

Q3 : Which bosons and fermions could be related by supersymmetry? (1.7)

But there seems to be no answer since known bosons and fermions do not appear to havemuch in common (excepted, maybe, for the photon and the neutrino). In addition the numberof (known) degrees of freedom is significantly larger for fermions than for bosons. And thesefermions and bosons have very different gauge symmetry properties ! Furthermore, as discussedin subsection 1.1, the question

Q4 :How could one define (conserved)

baryon and lepton numbers, in a supersymmetric theory ?(1.8)

once appeared as a serious difficulty, owing in particular tothe presence ofself-conjugateMa-jorana fermions in supersymmetric theories. Of course nowadays we are so used to dealing withspin-0 squarks and sleptons, carrying baryon and lepton numbers almost by definition, that wecan hardly imagine this could once have appeared as a problem. Its solution required acceptingthe idea of attributing baryon or lepton numbers to a large number of new fundamental bosons.Even then, if such new spin-0 squarks and sleptons are introduced, their direct (Yukawa) ex-changes between ordinary quarks and leptons, if allowed, could lead to an immediate disaster,preventing us from getting a theory of electroweak and strong interactions mediated by spin-1

gauge bosons, and not spin-0 particles, with conservedB andL quantum numbers ! This maybe expressed by the question

Q5 :How can we avoid unwanted interactions

mediated by spin-0 squark and slepton exchanges ?(1.9)

Fortunately, we can naturally avoid such unwanted interactions, thanks toR-parity, which, ifpresent, guarantees that squarks and sleptons cannot be directly exchanged between ordinaryquarks and leptons, allowing for conserved baryon and lepton numbers in supersymmetric the-ories.

1.3 ContinuousR-Invariance, and Electroweak Breaking

The definition of the continuousR-invariance we are using arose from an early attempt at relat-ing known bosons and fermions together, in particular the spin-1 photon with a spin-1

2neutrino.

If we want to try to identify the companion of the photon as being a “neutrino”, although it ini-tially appears as a self-conjugate Majorana fermion, we need to understand how it could carrya conserved quantum number that we might attempt to interpret as a “lepton” number. This ledto the definition ofa continuousU(1) R-invariance[1], which also guaranteed the massless-ness of this “neutrino” (“νL”, carrying +1 unit of R ), by acting chirally on the Grassmanncoordinateθ which appears in the expression of the various gauge and chiral superfields3.

Attempting to relate the photon with one of the neutrinos could only be an exercise of limitedvalidity. The would-be “neutrino”, in particular, while having in this model aV −A couplingto its associated “lepton” and the chargedW± boson, was in fact what we would now call a“photino”, not directly coupled to theZ ! Still this first attempt, which became a part of theSupersymmetric Standard Model, illustrated how one can break spontaneously aSU(2)×U(1)electroweak gauge symmetry in a supersymmetric theory, usinga pair of chiral doublet Higgssuperfields, now known asHd and Hu ! Using only a single doublet Higgs superfield wouldhave left us witha massless charged chiral fermion, which is, evidently, unacceptable. Ourprevious charged “leptons” were in fact what we now call two winos, or charginos, obtainedfrom the mixing of charged gaugino and higgsino components,as given by the mass matrix

MC =

(M2 = 0 )g vu√

2= MW

√2 sin β

g vd√2

= MW

√2 cos β µ = 0

(1.10)

in the absence of a direct higgsino mass originating from aµ HuHd mass term in the superpo-tential4. The whole construction showed that one could deal elegantly with elementary spin-0

3This R-invariance itself originates from an analogousQ-invariance used, in a two-Higgs-doublet presuper-symmetry model, to restrict the allowed Yukawa andϕ4 interactions, in a way which prepared for the two Higgsdoublets and chiral fermion doublets to be related by supersymmetry [16].Q-transformations were then modifiedinto R-symmetry transformations, which survive the electroweakbreaking and allow for massive Dirac fermionscarrying the new quantum numberR. Transformations similar toR-transformations were also considered in [17],but acted differently on two disconnected sets of chiral superfieldsφ+ and φ− , with no mutual interactions; i.e.they acted differently on the Grassmann coordinateθ, depending on whether the superfields considered belongedto the first or second set.

4This µ term initially written in [1] (which would have broken explicitly the continuousU(1) R-invariance)was immediately replaced by aλ HuHd N trilinear coupling involving anextra neutral singlet chiral superfieldN : µ HuHd 7−→ λ HuHd N , as in the so-called Next-to-Minimal Supersymmetric Standard Model (NMSSM).

Higgs fields (not a very popular ingredient at the time), in the framework of spontaneously-broken supersymmetric theories. Quartic Higgs couplings are no longer completely arbitrary,but fixed by the values of the electroweak gauge couplingsg and g′ through the following“D-terms” in the scalar potential given5 in [1]:

VHiggs =g2

8(h†d ~τ hd + h†u ~τ hu)

2 +g′2

8(h†dhd − h†uhu)

2 + ...

=g2 + g′2

8(h†dhd − h†uhu)

2 +g2

2|h†dhu|2 + ... .

(1.11)

This is precisely the quartic Higgs potential of the “minimal” version of the SupersymmetricStandard Model, the so-called Minimal Supersymmetric Standard Model (MSSM). Further con-tributions to this quartic Higgs potential also appear in the presence of additional superfields,such as the neutral singlet chiral superfieldN already introduced in the previous model, whichplays an important role in the NMSSM, i.e. in “next-to-minimal” or “non-minimal” versions ofthe Supersymmetric Standard Model. Charged Higgs bosons (called H±) are always presentin this framework, as well as several neutral ones, three of them at least. Their mass spectrumdepends on the details of the supersymmetry-breaking mechanism considered: soft-breakingterms, possibly “derived from supergravity”, presence or absence of extraU(1) gauge fieldsand/or additional chiral superfields, role of radiative corrections, etc..

1.4 R-Invariance and R-Parity in the Supersymmetric Stan-dard Model

These two Higgs doubletsHd and Hu are precisely those used to generate the masses ofcharged leptons and down quarks, and of up quarks, in supersymmetric extensions of the Stan-dard Model [2]. Note that at the time having to introduce Higgs fields was generally consideredas rather unpleasant. While one Higgs doublet was taken as probably unavoidable to get to theStandard Model or at least simulate the effects of the spontaneous breaking of the electroweaksymmetry, having to consider two doublets, necessitating charged Higgs bosons as well as sev-eral neutral ones, in addition to the “doubling of the numberof particles”, was usually consid-ered as further indication of the irrelevance of supersymmetry. As a matter of fact, considerablework was devoted for a while on attempts to avoid fundamentalspin-0 Higgs fields (and ex-tra sparticles), before returning to fundamental Higgs bosons, precisely in this framework ofsupersymmetry.

In the previousSU(2)×U(1) model [1], it was impossible to view seriously for very long“gaugino” and “higgsino” fields as possible building blocksfor our familiar lepton fields. Thisled to consider that all quarks, and leptons, ought to be associated with new bosonic partners,the spin-0 squarks and sleptons. Gauginos and higgsinos, mixed together by the spontaneousbreaking of the electroweak symmetry, correspond to a new class of fermions, now knownas “charginos” and “neutralinos”. In particular, the partner of the photon under supersymme-try, which cannot be identified with any of the known neutrinos, should be viewed as a new“photonic neutrino”, thephotino; the fermionic partner of the gluon octet is an octet of self-conjugate Majorana fermions calledgluinos (although at the timecolored fermionsbelonging

5With a different denomination for the two Higgs doublets, such that ϕ′′ 7→ Higgs doublethd,(ϕ′)c 7→ Higgs doublethu, tan δ = v′/v′′ 7→ tan β = vu/vd .

1) the threeSU(3) × SU(2) × U(1) gauge superfields;

2) chiral superfields for the three quark and lepton families;

3) the two doublet Higgs superfieldsHd and Hu responsiblefor the spontaneous electroweak breaking,and the generation of quark and lepton masses;

4) the trilinear superpotential of Eq. (1.2).

Table 1.1: The basic ingredients of the Supersymmetric Standard Model.

to octet representations of the colourSU(3) gauge group were generally believed not to exist),etc..

The two doublet Higgs superfields6 Hd and Hu generate quark and lepton masses [2], inthe usual way, through the familiar trilinear superpotential of Eq. (1.2). The vacuum expec-tation values of the two corresponding spin-0 Higgs doublets hd and hu generate charged-lepton and down-quark masses, and up-quark masses, with mass matrices given byme

ij =

λeij vd/

√2 , md

ij = λdij vd/

√2 , and mu

ij = λuij vu/

√2 , respectively. This constitutes the

basic structure of theSupersymmetric Standard Model, which involves, at least, the basicingredients shown in Table 1.1. Other ingredients, such as aµ HuHd direct Higgs superfieldmass term in the superpotential, or an extra singlet chiral superfieldN with a trilinear superpo-tential couplingλ HuHdN + ... possibly acting as a replacement for a directµ HuHd massterm [1], and/or extraU(1) factors in the gauge group, may or may not be present, dependingon the particular version of the Supersymmetric Standard Model considered.

In any case, independently of the details of the supersymmetry-breaking mechanism ulti-mately considered and of the absence or presence of6Rp interactions, we obtain the followingminimal particle content of the Supersymmetric Standard Model, as summarized in Table 1.2.Each spin-1

2quark q or charged leptonl− is associated withtwo spin-0 partners collectively

denoted byq or l− , while a left-handed neutrinoνL is associated with asingle spin-0 sneu-trino νL. We have ignored for simplicity, in this table, further mixing between the various “neu-tralinos” described by neutral gaugino and higgsino fields,schematically denoted byγ, Z1,2,andh0. More precisely, all such models include four neutral Majorana fermions at least, mix-ings of the fermionic partners of the two neutralSU(2)×U(1) gauge bosons (usually denotedby Z and γ, or W3 and B ) and of the two neutral higgsino components (h 0

d and h 0u ).

Non-minimal models also involve additional gauginos and/or higgsinos.

Let us return to the definition of the continuousU(1) R-symmetry, and discreteR-parity,transformations. As explained earlier, the newadditive quantum numberR associated withthis continuousU(1) R-symmetry is carried by the supersymmetry generator, and distinguishesbetween bosons and fermions within the multiplets of supersymmetry [1]. Gauge bosons and

6The correspondence between earlier notations and modern ones is as follows:

S =

(S0

S−

)and T =

(T 0

T−

)7−→ Hd =

(H 0

d

H −

d

)and Hu =

(H +

u

H 0u

).

(left-handed) (right-handed) (both left-handed)

Spin 1 Spin 1/2 Spin 0

gluons g gluinos gphoton γ photino γ

—————— −−−−−−−−−− —————————W±

Z

winos W ±1,2

zinos Z1,2

higgsino h0

H±

H

h, A

Higgsbosons

leptons l sleptonslquarks q squarksq

Table 1.2: Minimal particle content of the Supersymmetric Standard Model.

V ( x, θ, θ ) → V ( x, θ e−iα, θ eiα ) for the SU(3) × SU(2) × U(1)

gauge superfields

Hd,u ( x, θ ) → Hd,u ( x, θ e−iα ) for the left-handed doubletHiggs superfieldsHd and Hu

S( x, θ ) → eiα S( x, θ e−iα ) for the left-handed(anti)quark and (anti)lepton

superfieldsQi, U ci , Dc

i , Li, Eci

Table 1.3: Action of a continuousU(1) R-symmetry transformation on the gauge and chiralsuperfields of the Supersymmetric Standard Model.

Higgs bosons haveR = 0 while their partners under supersymmetry, now interpretedas gaug-inos and higgsinos, haveR = ±1 . This definition is extended to the chiral quark and leptonsuperfields, spin-1

2quarks and leptons havingR = 0, and their spin-0 superpartners,R = + 1

(for qL, lL ) or R = − 1 (for qR, lR ) [2]. The action of theseR-symmetry transformations,which survive the spontaneous breaking of the electroweak symmetry (see also footnote 3 insection 1.3), is given in Table 1.3.

This continuousU(1) R-symmetry (U(1)R) is indeed a symmetry of the four basic build-ing blocks of the Supersymmetric Standard Model (cf. Table 1.1). This includes the self-interactions of theSU(3) × SU(2) × U(1) gauge superfields, and their interactions with thechiral quark and lepton superfields, and the two doublet Higgs superfieldsHd and Hu . Alsoinvariant under the continuousU(1) R-symmetry are the super-Yukawa interactions ofHd

andHu , responsible for the generation of quark and lepton masses through the superpotential(1.2). Indeed it follows from Table 1.3 that this trilinear superpotentialW transforms under thecontinuousR-symmetry with “R-weight” nW =

∑i ni = 2 , i.e. according to

W ( x, θ ) → e2 iα W ( x, θ e−iα ) . (1.12)

Bosons Fermions

gauge and Higgs bosons

graviton(R = 0 )

gauginos and higgsinos

gravitino(R = ± 1 )

R-parity + R-parity −

sleptons and squarks (R = ± 1 ) leptons and quarks (R = 0 )

R-parity − R-parity +

Table 1.4:R-parities in the Supersymmetric Standard Model.

Its auxiliary “F -component”, obtained from the coefficient of the bilinearθ θ term in the ex-pansion of this superpotentialW , is thereforeR-invariant, generatingR-invariant interac-tion terms in the Lagrangian density. Note, however, that a direct Higgs superfield mass termµ HuHd in the superpotential, which hasR-weight n = 0 , does not lead to interactionsinvariant under the continuousR-symmetry (see also footnote 4 in section 1.3 and [1] for areplacement of theµ term by a trilinear coupling with an extra singlet chiral superfield, as in theNMSSM). But it gets in general re-allowed as soon as the continuousR-symmetry gets reducedto its discrete version ofR-parity.

This R-invariance led us to distinguish between a sector ofR-even particles, which in-cludes all those of the Standard Model, withR = 0 (and thereforeRp = (−1)R = +1 ); andtheir R-odd superpartners, gauginos and higgsinos, sleptons and squarks, withR = ± 1 (andRp = −1 ), as indicated in Table 1.4.

More precisely the necessity of generating masses for the (Majorana) spin-32

gravitino [3]and the spin-1

2gluinos did not allow us to keep a distinction betweenR = + 1 andR = − 1

particles, forcing us to abandon the continuousR-invariance in favour of its discrete version,R-parity. The – even or odd – parity character of the (additive)R quantum number correspondsto the well-knownR -parity, first defined as+ 1 for the ordinary particles and− 1 for theirsuperpartners, simply written as(−1 )R in (1.1) [4], then re-expressed as(−1)2S(−1)3B+L

in (1.3) as an effect of the close connection betweenR-parity and baryon and lepton-numberconservation laws.

This R-parity symmetry operator may also be viewed as a non-trivial geometrical discretesymmetry associated with a reflection of the anticommuting fermionic Grassmann coordinate,θ → − θ , in superspace [18]. ThisR-parity operator plays an essential role in the constructionof supersymmetric theories of interactions, and in the discussion of the experimental signaturesof the new particles. A conservedR-parity guarantees thatthe new spin-0 squarks and sleptonscannot be directly exchangedbetween ordinary quarks and leptons, as well as the absolutestability of the LSP. But let us discuss more precisely the reasons which led to discarding thecontinuousR-invariance in favour of its discrete version,R-parity.

1.5 Gravitino and Gluino Masses: FromR-Invariance to R-Parity

There are two strong reasons, at least, to abandon the continuous R-invariance in favour ofits discreteZ2 subgroup generated by theR-parity transformation. One is theoretical, thenecessity – once gravitation is introduced – of generating amass for the (Majorana) spin-3

2

gravitino in the framework of spontaneously-broken locally supersymmetric theories [3]. Theother is phenomenological, the non-observation of massless (or even light)gluinos. Both par-ticles would have to stay massless in the absence of a breaking of the continuousU(1) R-invariance, thereby preventing, in the case of the gravitino, supersymmetry from being sponta-neously broken. (A third reason could now be the non-observation at LEP of a chargedwino –also calledchargino – lighter than theW±, that would exist in the case of a continuousU(1)R-invariance [1, 2], as shown by the mass matrixMC given in Eq. (1.10).)

All this is, therefore, also connected with the mechanism bywhich the supersymmetryshould get spontaneously broken, in the Supersymmetric Standard Model. The question hasnot received a definitive answer yet. The inclusion of universal soft supersymmetry-breakingterms for all squarks and sleptons,

−∑

q, l

m 20 ( q† q + l† l ) , (1.13)

was already considered in 1976, for lack of a better solution. But such terms should in fact begenerated by some spontaneous supersymmetry-breaking mechanism, if supersymmetry is tobe realized locally. In any case one now considers in generalall soft supersymmetry-breakingterms [19] (possibly “induced by supergravity”), which essentially serve as a parametrisation ofour ignorance about the true mechanism of supersymmetry breaking chosen by Nature to makesuperpartners heavy.

But let us return to gluino masses. SinceR-transformations actchirally on the Majoranaoctet of gluinos,

g → e γ5 α g (1.14)

a continuousR-invariance would require the gluinos to remain massless, even after a spon-taneous breaking of the supersymmetry ! As mentioned before(see section 1.1), one wouldthen expect the existence of“R-hadrons” which have not been observed [5, 6]. Present exper-imental constraints indicate that gluinos, if they exist, must be very massive [20], requiring asignificant breaking of the continuousR-invariance, in addition to the necessary breaking ofthe supersymmetry.

In the framework of global supersymmetry it is not so easy to generate large gluino masses.Even if global supersymmetry is spontaneously broken, and if the continuousR-symmetry isnot present, it is still in general rather difficult to obtainlarge masses for gluinos, since: i)no direct gluino mass term is present in the Lagrangian density; and ii) no such term may begenerated spontaneously, at the tree approximation, via gluino couplings involvingcolouredspin-0 fields.

A gluino mass may then be generated by radiative correctionsinvolving a new sector ofquarks sensitive to the source of supersymmetry breaking, that would now be called “mes-senger quarks” [7], but iii) this can only be through diagrams which “know” both about: a)the spontaneous breaking of the global supersymmetry, through some appropriately generated

VEVs for auxiliary gauge or chiral components,< D >, < F > or < G > ’s; b) the exis-tence of superpotential interactions which do not preservethe continuousU(1) R-symmetry.Such radiatively generated gluino masses, however, generally tend to be rather small, unless oneintroduces, in some often rather complicated “hidden sector”, very large mass scales≫ MW .

Fortunately gluino masses may also result directly from supergravity, as observed longago [3]. Gravitational interactions require, within localsupersymmetry, that the spin-2 gravitonbe associated with a spin-3/2 partner [12], the gravitino. Since the gravitino is the fermionicgauge particle of supersymmetry it must acquire a massm3/2 as soon as the local supersym-metry gets spontaneously broken7. Since the gravitino is a self-conjugate Majorana fermionits mass breaks the continuousR-invariance which acts chirally on it, just as for the gluinos,forcing us to abandon the continuousU(1) R-invariance, in favour of its discreteZ2 subgroupgenerated by theR-parity transformation. We can no longer distinguish between the values+1 and−1 of the (additive) quantum numberR; but only between “R-odd” particles (havingR = ±1 ) and “R-even” ones, i.e. between particles havingR-paritiesRp = (−1)R = − 1,and + 1, respectively (cf. Table 1.4).

In particular, when the spin-32

gravitino mass termm3/2 , which corresponds to a changein R ∆R = ± 2 , is introduced, the “left-handed sfermions”fL, which carryR = + 1,can mix with the “right-handed” onesfR, carryingR = − 1, through mixing terms having∆R = ± 2 , which may naturally (but not necessarily) be of orderm3/2 mf (so that thelightest of the squarks and sleptons may well turn out to be one of the two stop quarkst ).Supergravity theories offer a natural framework in which toinclude, in addition, direct gauginoMajorana mass terms

− 1

2M3

¯gaga − 1

2M2

¯W iW i − 1

2M1

¯BB , (1.15)

which also correspond to∆R = ± 2 , just as for the gravitino mass itself. The mass parame-tersM3, M2 andM1 associated with theSU(3)×SU(2)×U(1) gauginos may then naturally(but not necessarily) be of the same order as the gravitino massm3/2 .

Furthermore, once the continuousR-invariance is reduced to its discreteR-parity subgroup,a direct Higgs superfield mass termµ HuHd which was not allowed by the continuousU(1)R-symmetry (but could be replaced by a trilinearλ HuHdN superpotential term), gets real-lowed in the superpotential, as for example in the MSSM. The size of this supersymmetricµparameter might conceivably have been a source of difficulty, in case this parameter, presenteven if there is no supersymmetry breaking, turned out to be large. But since theµ term breaksexplicitly the continuousR-invariance of Table 1.3 (and, also, another “extraU(1)” symmetryacting axially on quark and lepton superfields [2]), its sizemay be controlled by considering oneor the other of these two symmetries. Even better, sinceµ got reallowed just as we abandonedthe continuousR-invariance so as to allow for gluino and gravitino masses, the size ofµ maynaturally be of the same order as the supersymmetry-breaking gaugino mass parametersMi, orthe gravitino massm3/2, since they all appear in violation of the continuousR-symmetry ofTable 1.3 [21]. Altogether there is here no specific hierarchy problem associated with the sizeof µ .

In general, irrespective of the supersymmetry-breaking mechanism considered, still un-known (and parametrized using a variety of possible soft supersymmetry-breaking terms), one

7Depending on the notation this mass may be expressed asm3/2 = κ d/√

6 , or m3/2 = κ FS/√

3 =√8 π/3 FS/MP , whereFS (or d) is the supersymmetry-breaking scale parameter,κ2 = 8π GN , and MP

is the Planck mass. Supersymmetry is often said to be broken “at the scale”√

FS (or√

d) = Λss ≈√m3/2 MP .

normally expects the various superpartners not to be too heavy. Otherwise the correspondingnew mass scale introduced in the game would tend to contaminate the electroweak scale, therebycreating a hierarchy problem in the Supersymmetric Standard Model. Superpartner masses arethen normally expected to be naturally of the order ofMW , or at most in the∼ TeV range.

Beyond that, in a more ambitious framework involving extra spacetime dimensions,R-parity may also be responsible for an elegant way of implementing supersymmetry breaking bydimensional reduction, by demandingdiscrete – periodic or antiperiodic – boundary condi-tions for ordinaryR-even particles and theirR-odd superpartners, respectively. The masses ofthe lowest-lying superpartners would now get fixed by the compactification scale – e.g. in thesimplest case and up to radiative correction effects, by

m3/2 = Mi =π ~

L c=

~

2Rc, (1.16)

in terms of the sizeL of the extra dimension responsible for the supersymmetry breaking [22].This led to consider, already a long time ago, the possibility of relatively “large” extra dimen-sions (as compared for example to the Planck length of10−33 cm), associated with a com-pactification scale that could then be as “low” as∼ TeV scale. If this is true, the discovery ofsuperpartners in the<∼ TeV scale should be followed by the discovery of series of heavy copiesfor all particles, corresponding to the Kaluza-Klein excitations of the extra dimensions – quitea spectacular discovery !

Landing back on four dimensions, the Supersymmetric Standard Model (whether “minimal”or not), with itsR-parity symmetry (whether it turns out to be absolutely conserved or not),provided the basis for the experimental searches for the newsuperpartners and Higgs bosons,starting with the first searches for gluinos and photinos, selectrons and smuons, in 1978-1980.But how the supersymmetry should actually be broken – if indeed it is a symmetry of Nature !– is not known yet, and this concentrates a large part of the remaining uncertainties in the Su-persymmetric Standard Model. Furthermore, it is worth to discuss more precisely the questionof the conservation, or non-conservation, ofR-parity. In other terms, how much violation ofthe R-parity symmetry may be tolerated, without running in conflict with existing experimen-tal results on proton decay, neutrino masses, and various accelerator or astrophysical data ? Or,conversely, could6Rp effects be responsible for the generation of small Majoranamasses forneutrinos ? This is the subject of the present review article.

Chapter 2

HOW CAN R-PARITY BE VIOLATED?

The non-conservation of baryon (B) and lepton (L) numbers is a generic feature of numerousextensions of the Standard Model. In theories involving newsymmetries valid at some highenergy scale such as Grand Unified Theories (GUTs), suchB- andL-violating effects are usu-ally suppressed by powers of the high scale, and therefore generally small - even though someof them, like proton decay, may be observable. By constrast,in supersymmetric extensions ofthe Standard Model, the scale of possible baryon and lepton-number violations is associatedwith the masses of the superpartners (squarks and leptons) responsible for the violations, whichmay lead to unacceptably large effects. Avoiding this, was the main interest of the introductionof R-parity, as discussed in the previous chapter. In view of theimportant phenomenologicaldifferences between supersymmetric models with and without R-parity, it is worth studying theextent to whichR-parity can be broken. Furthermore, there are in principle other symmetries(discrete or continuous, global or local) that can forbid proton decay while still allowing for thepresence of someR-parity-violating couplings. Their classification will allow to explore whichkind ofR-parity-violating terms can possibly appear.

2.1 R-Parity-Violating Couplings

In the Standard Model (assuming two-component, massless neutrino fields) it is impossibleto write down renormalizable, gauge-invariant interactions that violate baryon or lepton num-bers. This is no longer the case in supersymmetric extensions of the Standard Model where,for each ordinary fermion (boson), the introduction of a scalar (fermionic) partner allows fornew interactions that do not preserve baryon or lepton number. As explained in chapter 1, theseinteractions can be forbidden by introducingR-parity. This leads in particular to the popu-lar “Minimal Supersymmetric Standard Model” (MSSM), the supersymmetric extension of theStandard Model with gauge symmetrySU(3)C ×SU(2)L ×U(1)Y , with minimal particle con-tent and for whichR-parity invariance is generally assumed. Throughout most of this review,the MSSM will normally be our reference model, although the discussion ofR-parity violationdoes not in general depend much on the specific version of the Supersymmetric Standard Modelwhich is considered.

6Rp couplings originate either from the superpotential itself, or from soft supersymmetry-breaking terms. There are various kinds of such couplings, of dimensions 4, 3 or 2 only, with apotentially rich flavour structure. In this section, we shall write down explicitly all possible6Rp

terms in the framework of the MSSM, assuming the most generalbreaking ofR-parity. We shallthen consider particular scenarios which allow to reduce the number of independent couplingsused to parametrizeR-parity violation.

2.1.1 Superpotential Couplings

AssumingR-parity invariance, the superpotential of the Supersymmetric Standard Model withminimal particle content contains only one supersymmetricHiggs mass term, theµ-term, andthe supersymmetric Yukawa interactions generating massesfor the quarks and charged leptons(see section 1.1, and Appendix A for the definition of superfields),

WRp ≡ WMSSM = µHuHd + λeij HdLiE

cj + λd

ij HdQiDcj − λu

ij HuQiUcj . (2.1)

Other versions of the Supersymmetric Standard Model, with an extended Higgs sector and/oradditionalU(1) gauge factors, may have a slightly differentR-parity conserving superpotential,especially since they involve in general additional chiralsuperfields. This is for instance the casein the NMSSM where an extra neutral singlet is coupled to the two doublet Higgs superfieldsHd andHu.

In the absence ofR-parity, however,R-parity odd terms allowed by renormalizability andgauge invariance must also, in principle, be included in thesuperpotential. The ones that violatelepton-number conservation can be easily found by noting that the lepton superfieldsLi and theHiggs superfieldHd have exactly the same gauge quantum numbers. Thus, gauge invarianceallows for bilinear and trilinear lepton-number-violating superpotential couplings obtained byreplacingHd byLi in Eq. (2.1). The only other renormalizable6Rp superpotential term allowedby gauge invariance,U c

i DcjD

ck , breaks baryon-number conservation. Therefore the most gen-

eral renormalizable,R-parity odd superpotential consistent with the gauge symmetry and fieldcontent of the MSSM is1 [8] (see also [23]),

W6Rp= µiHuLi +

1

2λijk LiLjE

ck + λ′ijk LiQjD

ck +

1

2λ′′ijk U

ci D

cjD

ck , (2.2)

where, like in Eq. (2.1), there is a summation over the generation indicesi, j, k = 1, 2, 3, andsummation over gauge indices is understood. One has for exampleLiLjE

ck ≡ (ǫabL

aiL

bj)E

ck =

(NiEj − EiNj)Eck andU c

iDcjD

ck ≡ ǫαβγU c

iαDcjβD

ckγ, wherea, b = 1, 2 areSU(2)L indices,

α, β, γ = 1, 2, 3 areSU(3)C indices, andǫab andǫαβγ are totally antisymmetric tensors. Gaugeinvariance enforces antisymmetry of theλijk couplings with respect to their first two indices.As a matter of fact, one has

λijk LiLjEck = λijk ǫabL

aiL

bjE

ck = −λjik ǫbaL

ajL

biE

ck = −λjik LiLjE

ck , (2.3)

which leads toλijk = −λjik . (2.4)

Gauge invariance also enforces antisymmetry of theλ′′ijk couplings with respect to their last twoindices:

λ′′ijk = −λ′′ikj . (2.5)

At this point one should make a comment on the bilinear6Rp superpotential termsµiHuLi.These terms can be rotated away from the superpotential uponsuitably redefining the lepton

1Other versions of the Supersymmetric Standard Model may allow for additional6Rp superpotential terms.

~l lll6L 00~q q

q6B0~q 6Lql

Figure 2.1: Basic tree diagrams associated with the trilinear6Rp superpotential interactionsinvolving the Yukawa couplingsλ or λ′ (6L), or λ′′ (6B). q (q) and l (l) denote (s)quarks and(s)leptons. The arrows on the (s)quark and (s)lepton lines indicate the flow of the baryon (resp.lepton) number.

and Higgs superfields (withHd → H ′d ∝ µHd + µiLi) [24]. However, this rotation will

generate6Rp scalar mass terms (see subsection 2.1.3) from the ordinary,R-parity conservingsoft supersymmetry-breaking terms of dimension 2, so that bilinear R-parity-violating termswill then reappear in the scalar potential. The fact that onecan makeµi = 0 in Eq. (2.2) doesnot mean that the Higgs-lepton mixing associated with bilinearR-parity breaking is unphysical,but rather that there is no unique way of parametrizing it, aswill be discussed insubsection 2.1.4.

Altogether, Eq. (2.2) involves 48 (a priori complex) parameters: 3 dimensionful param-etersµi mixing the charged lepton and down-type Higgs superfields, and 45 dimensionlessYukawa-like couplings divided into 9λijk and 27λ′ijk couplings which break lepton-numberconservation, and 9λ′′ijk couplings which break baryon-number conservation.

2.1.2 Lagrangian Terms Associated with the SuperpotentialCouplings

We now derive the interaction terms in the Lagrangian density generated from theR-parity oddsuperpotential of Eq. (2.2).

i) 6Rp Yukawa couplings

Let us first consider the terms involving fermions. They consist of fermion bilinears as-sociated with the bilinear superpotential termsµiHuLi, and of trilinear, Yukawa-like interac-tions associated with the superpotential couplingsλ, λ′ andλ′′. In two-component notation forspinors, the fermion bilinears read (see Appendix A for the definition of fields, and Ref. [25]for the two-component notation):

LHuLi= µi

(h0

uνi − h+u l

−i

)+ h.c. . (2.6)

These terms, which mix lepton with higgsino fields, will be discussed in section 2.3. Expandedin standard four-component Dirac notation, the trilinear interaction terms associated with theλ,λ′ andλ′′ couplings read, respectively (see Appendix A for the definition of fields, and AppendixB for the derivation of this Lagrangian density):

LLiLjEck

= −1

2λijk

(νiLlkRljL + ljLlkRνiL + l⋆kRν

ciRljL − (i↔ j)

)+ h.c. , (2.7)

LLiQjDck

= −λ′ijk(νiLdkRdjL + djLdkRνiL + d⋆

kRνciRdjL

−liLdkRujL − ujLdkRliL − d⋆kRl

ciRujL

)+ h.c., (2.8)

and2

LUci Dc

jDck

= −1

2λ′′ijk

(u⋆

iRdjRdckL + d⋆

kRuiRdcjL + d⋆

jRuiRdckL

)+ h.c. . (2.9)

In these equations, the superscriptc denotes the charge conjugate of a spinor (for instanceνciR =

(νci )R is the adjoint of the charge conjugate ofνiL), the superscript⋆ the complex conjugate of a

scalar field, and theR andL chirality indices on scalar fields distinguish between independentfields corresponding to the superpartners of right- and left-handed fermion fields, respectively.Like in Eq. (2.2), an implicit sum runs freely over the quark and lepton generations labelled bythe indicesi, j, k, and summation over gauge indices is understood. The trilinear interactions ofEqs. (2.7), (2.8) and (2.9) couple a scalar field and two fermionic fields (see Fig. 2.1), and areindeed forbidden by theR-parity symmetry defined by Eq. (1.3).

ii) 6Rp scalar interactions

Let us now consider the6Rp scalar interactions associated with the6Rp superpotential cou-plings (2.2). The bilinear superpotential terms induce dimension-2 and -3 terms,

V µi

6Rp= µ⋆µi h

†dLi − µ⋆

iλujk(L

†i Qj)u

ck +

∑

l

µ⋆l λ

eli(h

†uhd)l

ci + h.c. , (2.10)

while the trilinear superpotential couplings induce dimension-4 terms,

V λ,λ′,λ′′

6Rp= −

∑

l

λe⋆lj λlik(h

†dLi)l

c⋆j l

ck +

1

2

∑

l

λe⋆il λjkl(hdLi)

⋆(LjLk)

−∑

l

λe⋆lj λ

′lik(h

†dQi)l

c⋆j d

ck +

∑

l

λd⋆lj λ

′ilk(h

†dLi)d

c⋆j d

ck

−∑

l

λu⋆lj λ

′ilk(h

†uLi)u

c⋆j d

ck +

∑

l

λd⋆il λ

′jkl(hdQi)

⋆(LjQk)

+∑

l

λd⋆il λ

′′jkl(hdQi)

⋆ucjd

ck − 1

2

∑

l

λu⋆il λ

′′ljk(huQi)

⋆dcjd

ck + h.c.. (2.11)

iii) Additional Rp conserving scalar couplings

The superpotential (2.2) also yields scalar couplings thatdepend quadratically on the6Rp

couplings. SinceR-parity acts as aZ2 symmetry, these terms correspond in fact toR-parityconserving interactions, even if they vanish in the limit ofexactR-parity,

V µi

Rp=∑

i

|µi|2(h†uhu + L†

i Li

), (2.12)

2Due to the antisymmetry property of theλ′′ijk couplings and toSU(3)C invariance, the second and third terms

in Eq. (2.9) are actually identical.

V λ,λ′,λ′′

Rp=

∑

m

λ⋆mikλmjl(L

†i Lj)l

c⋆k l

cl +

1

4

∑

m

λ⋆ijmλklm(LiLj)

⋆(LkLl)

+∑

m

(λ⋆

mikλ′mjl(L

†i Qj)l

c⋆k d

cl + h.c.

)+∑

m

λ′⋆mikλ′mjl(Q

†i Qj)d

c⋆k d

cl

+∑

m

λ′⋆imkλ′jml(L

†i Lj)d

c⋆k d

cl +

∑

m

λ′⋆ijmλ′klm(LiQj)

⋆(LkQl)

+∑

m

(λ′⋆ijmλ

′′klm(LiQj)

⋆uckd

cl + h.c.

)+

1

2

∑

m

λ′′⋆mikλ′′mjl(d

c⋆i d

cj)(d

c⋆k d

cl )

+∑

m

λ′′⋆ikmλ′′jlm

[(uc⋆

i ucj)(d

c⋆k d

cl ) − (uc⋆

i dcl )(d

c⋆k u

cj)], (2.13)

andV µi,λ,λ′

Rp= −

∑

l

µ⋆l λlij(h

†uLi)l

cj −

∑

l

µ⋆l λ

′lij(h

†uQi)d

cj + h.c. . (2.14)

2.1.3 Soft Supersymmetry-Breaking Terms

Since supersymmetry is necessarily broken, a proper treatment ofR-parity violation must alsoinclude 6Rp soft terms. In the Supersymmetric Standard Model, we parametrize our ignoranceabout the mechanism responsible for supersymmetry breaking by introducing the most generalterms that break supersymmetry in a soft way, i.e. without reintroducing quadratic divergences.The possible soft terms were classified by Girardello and Grisaru [19]. They consist of massterms for the gauginos (Mλλ), analytic3 couplings for the scalar fields known as “A-terms”(Aijk φiφjφk,), analytic scalar mass terms known as “B-terms” (Bij φiφj), and scalar mass terms(m2

ij φ†iφj). This leads to the following soft supersymmetry-breakingLagrangian density for the

Supersymmetric Standard Model (see Appendix A), given by:

−LsoftRp = (m2

Q)ij Q

†i Qj + (m2

uc)ij uc†i u

cj + (m2

dc)ij dc†i d

cj + (m2

L)ij L

†i Lj + (m2

lc)ij l

c†i l

cj

+(Ae

ij hdLi lcj + Ad

ij hdQidcj − Au

ij huQiucj + h.c.

)

+ m2d h

†dhd + m2

u h†uhu + (B huhd + h.c.)

+1

2M1

¯BB +1

2M2

¯W 3W 3 + M2¯W+W+ +

1

2M3

¯gaga . (2.15)

In the absence ofR-parity, one must in principle also consider the most general soft termsassociated withW/Rp

:

V soft/Rp

=1

2Aijk LiLj l

ck + A′

ijk LiQj dck +

1

2A′′

ijk uci d

cjd

ck

+ Bi huLi + m2di h

†d Li + h.c.. (2.16)

Again Eqs. (2.15) and (2.16) assume the particle content of the MSSM; more general versionsof the Supersymmetric Standard Model may allow for additional soft terms, bothR-parity evenandR-parity odd.

The soft potential (2.16) introduces 51 new (a priori complex) 6Rp parameters:9 + 27 +9 = 45 6Rp A-terms with the same antisymmetry properties as the corresponding trilinear

3“Analytic” means here that these couplings involve only products of the scalar components of the (left-handed)chiral superfields, and not their complex conjugates.

superpotential couplings, 3Bi associated with the bilinear superpotential terms, and 36Rp softmass parametersm2

di mixing the down-type Higgs boson with the slepton fields . In the presenceof the bilinear6Rp soft terms, the radiative electroweak symmetry breaking generally leads tonon-vanishing sneutrino vacuum expectation values< νi >≡ vi/

√2, together with the familiar

Higgs VEVs< h0d > ≡ vd/

√2 and< h0

u > ≡ vu/√

2. Indeed, the bilinear terms in Eq. (2.16)yield linear terms in theνi fields after translation of the Higgs fields,h0

d → h0d + vd/

√2 and

h0u → h0

u + vu/√

2, which destabilizes the scalar potential and leads to< νi > 6= 0 – unlessparticular conditions are satisfied by the bilinear soft terms.

Since the sneutrino fields correspond toR-parity odd particles, these VEVs in turn in-duce new6Rp terms in the Lagrangian. In particular new mixing terms between lepton andchargino/neutralino fields (resp. between slepton and Higgs boson fields) are generated fromthe gauge andR-parity conserving Yukawa couplings (resp. from theR-parity conservingA-terms). It is important to keep in mind, however, that the slepton VEVs are not independent6Rp parameters, since they are functions of the6Rp couplings; as we shall see below it is alwayspossible to find a weak eigenstate basis in whichvi = 0.

2.1.4 Choice of the Weak Interaction Basis

In the absence ofR-parity and lepton-number conservation, there is noa priori distinctionbetween theY = −1 Higgs (Hd) and the lepton (Li) superfields, which have the same gauge in-teractions. One can therefore freely rotate the weak eigenstate basis by a unitary transformation[24]: (

Hd

Li

)→

(H ′

d

L′i

)= U

(Hd

Li

), (2.17)

whereU is anSU(4) matrix with entriesUαβ , α ≡ (0, i) = (0, 1, 2, 3), β ≡ (0, j) = (0, 1, 2, 3).Under (2.17),6Rp couplings and slepton VEVs transform as follows (we use the notationm2

ij

instead of(m2L)ij):

µ′i = U⋆

i0 µ +∑

j

U⋆ij µj , (2.18)

m′2di = U00U

⋆i0 m

2d +

∑

l,m

U0lU⋆im m

2lm +

∑

l

(U00U

⋆il m

2dl + U0lU

⋆i0 (m2

dl)⋆), (2.19)

B′i = U⋆

i0B +∑

j

U⋆ij Bj , v′i = Ui0 vd +

∑

j

Uij vj , (2.20)

(λijk)′ =

∑

l

(U⋆

i0U⋆jl − U⋆

ilU⋆j0

)λe

lk +∑

l,m

U⋆ilU

⋆jm λlmk , (2.21)

(λ′ijk)′ = U⋆

i0 λdjk +

∑

l

U⋆il λ

′ljk , (λ′′ijk)

′ = λ′′ijk . (2.22)

Since theA-terms transform exactly in the same way as the trilinear superpotential couplings,we do not write explicitly the corresponding transformation forAijk, A′

ijk andA′′ijk.

From the above equations it is clear that the values of the lepton-number-violating couplingsare basis-dependent. There is also a redundancy between thebilinear and trilinear6Rp parame-ters, as can be seen from the fact that 3 among the 9 bilinear6Rp parametersµi, Bi andm2

di canbe rotated away upon a suitable superfield redefinition, as expressed by Eq. (2.17). This leavesus with48 + 51 − 3 = 96 physically meaningful (in general complex) parameters.

As already noticed, it is always possible to choose a basis inwhichµ′i = 0, eliminating all

bilinearR-parity violation from the superpotential; but then in general non-vanishing sleptonVEVs are induced by the presence of the bilinear6Rp soft terms. Alternatively, one can choose abasis in which all slepton VEVs vanish4, v′i = 0, but then in generalµ′

i 6= 0. It is therefore cru-cial, when discussing6Rp effects, to specify in which basis one is working. A particular choice ofbasis, in which the Higgs-lepton mixing induced by bilinearR-parity violationis parametrizedin an economical and physically sensible way, will bepresented in section 2.3. Another option(to be discussed in subsection 6.1.3) is to define a complete set of basis-independent quantitiesparametrizing6Rp effects [26, 27, 28].

Before closing this subsection, let us write down explicitly the infinitesimalSU(4) trans-formation that rotates away small bilinear6Rp terms from the superpotential (2.2),µi ≪ µ.Definingǫi ≡ µi

µ, we can write the corresponding unitary matrixU , up toO(ǫ2) terms, as5 [24]:

U =

(1 −ǫiǫ⋆i 13×3

). (2.23)

In the new basisµ′i = 0, and the trilinear6Rp superpotential couplings are modified to(λijk)

′ =λijk +(ǫiλ

ejk − ǫjλe

ik) and(λ′ijk)′ = λ′ijk + ǫiλ

djk, whereλe

jk andλdjk are the Yukawa couplings in

the initial basis. In the supersymmetry-breaking sector, the mass parameters transform (omittingO(ǫ2) terms) as:

B′ = B −∑i ǫ⋆iBi , B′

i = Bi + ǫiB , (2.24)

m′2d = m2

d − (∑

i ǫ⋆i m

2di + h.c.) , m

′2ij = m2

ij + (ǫ⋆i m2dj + h.c.) , (2.25)

m′2di = m2

di + ǫim2d −

∑j ǫjm

2ji . (2.26)

As stressed above, such a pattern of soft mass parameters generally induces non-vanishing slep-ton vacuum expectation values, unless some very particularconstraints, that shall be discussedin section 2.3, are satisfied.

2.1.5 Constraints on the Size of6Rp Couplings

Being renormalizable, the6Rp couplings of Eqs. (2.2) and (2.16) are not expected to be sup-pressed by any large mass scale, and may thus induce excessively large baryon and lepton-number-violating effects. In particular, the combinationof couplingsλ′imkλ

′′⋆11k (i = 1, 2, 3,

m = 1, 2) would lead to proton decay via tree-level down squark exchange at an unacceptablerate, unless| λ′imkλ

′′⋆11k | is smaller than about10−26 for a typical squark mass in the300 GeV

range, smaller by more than twelve orders of magnitude than atypical GUT scale (see Sec-tion 6.4.4 for a discussion of this bound). There are many other constraints, both theoreticaland phenomenological, on the superpotential and soft supersymmetry-breaking6Rp parameters.Bounds on the6Rp superpotential couplings coming from the non-observationof baryon andlepton-number-violating processes and from direct searches at colliders are reviewed in chap-ters 6 and 7 respectively. Astrophysical and cosmological bounds are presented in chapter 4,and constraints associated with the effects of the renormalization group evolution (perturbativeunitarity bounds and unification constraints), in chapter 3.

4This does not mean, however, that the scalar potential is free from any term mixing the slepton and Higgsfields in this basis. InsteadBi andm2

di satisfy a particular relation that prevents the slepton fields from acquiringa VEV (see the discussion at the end of subsection 2.3.1).

5Forµi ∼ µ, Eq. (2.23) is no longer a good approximation. The explicit form of a unitary matrixU that rotatesaway arbitrarily large bilinear6Rp parameters from the superpotential can be found in Ref. [29].

2.2 Patterns ofR-Parity Breaking

Several patterns ofR-parity breaking can be considered, depending on whether the breaking isexplicit or spontaneous – and, in the case of an explicit breaking, on which type of couplings,bilinear, trilinear, or both, are present. Before classifying the various patterns considered in theliterature, let us make some general comments on subtletiesassociated with the lepton-number-violating couplings.

First of all, if lepton-number conservation is not associated with a symmetry of the theory,there is in general no preferred (Hd, Li) basis, as was seen in the previous section, and thestatement that only a certain class of lepton-number-violating 6Rp couplings are present is basis-dependent.

Still it is not an empty statement, especially if one thinks in terms of the number of indepen-dent parameters. Indeed, if there is a basis in which allλijk andλ′ijk superpotential couplingsas well as their associated soft terms vanish, lepton-number violation can be parametrized by 9parameters only (3µi and 6 bilinear soft terms). In a different basis theλijk andλ′ijk couplingsdo not vanish, but their values are determined, through the rotation (2.17), from the originaldown quark and charged leptonRp-conserving Yukawa couplings.

Secondly, in a consistent quantum field theory, all operators breaking a symmetry up tosome dimensiond must normally be included in the Lagrangian density, since if some of themare absent at tree level they are in general expected to be generated by radiative corrections.A well-known manifestation of this effect is the generationof bilinear 6Rp terms through one-loop diagrams involving lepton-number-violating trilinear 6Rp interactions [30, 31] (see alsoRef. [32]). For example, thed = 4 couplings (2.7) induce thed = 3 higgsino-lepton mixingtermsµihuLi as well as thed = 2 Higgs-slepton mixing mass termsBihuLi and m2

dih†dLi

(whereLi andLi denote the fermionic and scalar components of the lepton doublet superfieldsLi, respectively). This provides at least three consistent patterns ofR-parity violation in thelepton sector at the quantum level6:

(a) R-parity violation through d = 2, d = 3 and d = 4 terms: this corresponds to themost general explicit breaking ofR-parity, with all superpotential and soft6Rp terms al-lowed by gauge invariance and renormalizability present inthe Lagrangian density. Inthis case one has to deal with 99 new (in general complex) parameters beyond the (Rp-conserving) MSSM: 3 bilinear (µi) and 45 trilinear (9λijk, 27λ′ijk, 9 λ′′ijk) couplings inthe superpotential, together with their associatedB-terms (3Bi) andA-terms (9Aijk, 27A′

ijk, 9A′′ijk), and 3 additional6Rp soft masses (m2

di) in the scalar potential. (The 3 sleptonVEVs vi are not independent6Rp parameters, since they can be expressed in terms of theMSSM parameters and6Rp couplings.) Due to the ambiguity in the choice of the (Hd, Li)basis, however, only 6 among the 9 bilinear6Rp parameters are physical, thus reducing thenumber of physically meaningful6Rp parameters to 96.

(b) R-parity violation through d = 2 and d = 3 terms: in this case all soft6Rp termsare present in the scalar potential (both the bilinear termsBihuLi andm2

dih†dLi and the

triscalar couplingsAijk,A′ijk andA′′

ijk), while the superpotential contains only the bilinear6Rp termsµiHuLi, which corresponds to bothd = 2 andd = 3 terms in the Lagrangian

6We do not assume the conservation of lepton or baryon numbera priori. Of course, if lepton-number con-servation (resp. baryon-number conservation) were assumed, 6L couplings (resp.6B couplings) would never begenerated by radiative corrections.

density. We are thus left with 546Rp parameters (3µi in the superpotential; 3Bi, 3 m2di, 9

Aijk, 27A′ijk and 9A′′

ijk in the scalar potential), all of them physically meaningful.

(c) R-parity violation through d = 2 terms: in this case,R-parity violation originatessolely from the soft termsBihuLi andm2

dih†dLi, and can therefore be parametrized by 6

parameters only.

A closer look at the renormalization group equations (see chapter 3) shows that there actu-ally exist other consistent patterns ofR-parity violation. In particular the popular “bilinearR-parity breaking” scenario, in whichR-parity is explicitly broken by bilinear (superpotential andsoft) terms only, is perfectly consistent at the quantum level since6Rp dimension-3A-terms arenot generated from the bilinear superpotentialµi terms, also of dimension 3. On the other handanother popular scenario in whichR-parity is broken solely by trilinear terms is not consistentsince, as already mentioned, bilinear6Rp terms are then generated from quantum corrections.The absence of bilinear6Rp terms can only be assumed, strictly speaking, at the level oftheclassical Lagrangian.

With the above remarks in mind, we can now comment on the scenarios ofR-parity violationthat have received most attention in the literature: explicit 6Rp by trilinear terms, explicit6Rp bybilinear terms (“bilinearR-parity breaking”) and spontaneousR-parity violation.

(i) explicit 6Rp by trilinear terms: in this case one assumes that all bilinear6Rp terms are ab-sent from the tree-level Lagrangian. One is then left with 45trilinear 6Rp couplings inthe superpotential, and their associatedA-terms. This is the situation considered in mostphenomenological studies ofR-parity violation, and in the major part of this review. Theabsence of bilinear couplings and sneutrino VEVs also has the practical advantage ofremoving the ambiguity associated with the choice of the weak interaction basis (cf. sub-section 2.1.4). One must keep in mind, however, that bilinear 6Rp cannot be completelyabsent. Indeed, if only trilinear couplings are present at some energy scale, they willgenerate bilinear couplings at some other energy scale through renormalisation groupevolution7 [30, 31] (see chapter 3 for details). Thus, in a consistent quantum field theory,one must include bilinear6Rp terms as soon as trilinear terms are present. Still, since theexperimental and cosmological bounds on neutrino masses put strong constraints on thetolerable amount of bilinearR-parity violation (see next section), it makes sense to con-sider a situation in which the phenomenology of trilinear6Rp interactions is not affected bythe presence of bilinear6Rp terms – except for some specific phenomena such as neutrinooscillations.

(ii) explicit 6Rp by bilinear terms: assuming that, in an appropriately chosen basis,R-parity isbroken solely by bilinear terms, the number of independent6Rp parameters reduces to 9 (3µi or vi, 3Bi and 3m2

di). Since bilinear6Rp terms mix leptons with Higgs fields, however,trilinear 6Rp interactions of theλ andλ′ type (both triscalar and Yukawa-like interactions)are generated upon rotating the initial weak eigenstate basis to the mass eigenstate basis.Still those couplings can be expressed in terms of the initial bilinear 6Rp couplings and ofthe down-type Yukawa couplings, and are not independent6Rp parameters. Note that, due

7This is in fact a consequence of supersymmetry breaking. In the limit of exact supersymmetry, theµiHuLi

terms generated radiatively can be removed from the superpotential by a rotation (2.17), eliminating any bilinear6Rp term from the Lagrangian. In the presence of soft supersymmetry-breaking terms, however, the bilinear6Rp

terms generated in the superpotential and in the scalar potential cannot be rotated away simultaneously.

to the fact thatR-parity breaking originates here fromL-number-violating interactions,no B-violating λ′′-type interactions are generated, thus avoiding the problem of a toofast proton decay. The advantage of such a scenario resides in its predictivity (albeitit is difficult to motivate the absence of trilinear couplings by other considerations thansimplicity or predictivity); the main difficulty lies in suppressing the neutrino massesinduced by bilinear6Rp terms to an acceptable level without fine-tuning (see chapter 5). Adetailed discussion of bilinearR-parity breaking is given in section 2.3.

(iii) spontaneous6Rp: a completely different option is the spontaneous breakingof R-parityinduced by the vacuum expectation value of anR-parity odd scalar (i.e. necessarily ascalar neutrino in the Minimal Supersymmetric Standard Model). Such a scenario can beimplemented in various ways. Common features of the models are a constrained patternof 6Rp couplings, showing some analogy with the bilinear6Rp case, and - with the exceptionof the models where lepton number is gauged - a variety of new interactions involving amassless Goldstone boson (or a massive pseudo-Goldstone boson in the presence of asmall amount of explicit lepton-number violation) associated with the breaking of thelepton number. SpontaneousR-parity breaking is addressed in section 2.4.

Even within a restricted pattern ofR-parity violation like (i) or (ii), one is generally left witha very large number of arbitrary6Rp parameters. For this reason, it is often necessary to makesome assumptions on their flavour structure; for example, the bounds coming from particularprocesses are generally derived under the assumption that asingle coupling or combination ofcouplings gives the dominant contribution (see chapter 6).From a more theoretical perspectiveit is interesting to study models that can constrain the flavour structure of the6Rp couplings. Inparticular, the case of an abelian flavour symmetry is addressed in section 2.5. Furthermore,extensions of the Supersymmetric Standard Model with an enlarged gauge structure may havea more restricted pattern of6Rp couplings than the Supersymmetric Standard Model itself.R-parity violation in the context of Grand Unified Theories is discussed in section 2.6.

Finally, it should be kept in mind that despite its simplicity, R-parity may be viewed ashaving no clear theoretical origin, at least at this level. Thus, in the absence ofR-parity, itis a valid option to consider other discrete or continuous symmetries sharing withR-paritythe capability of protecting proton decay from renormalizable operators, while allowing somebaryon or lepton-number-violating couplings. Well-knownexamples of such symmetries are“baryon parities”, which allow only for lepton-number violation, and “lepton parities”, whichallow only for baryon-number violation. Some of these symmetries are even more efficientthanR-parity in suppressing proton decay from non-renormalizable operators which may begenerated from some fundamental theory beyond the Supersymmetric Standard Model. Theseother symmetries (”alternatives” toR-parity) are discussed in section 2.7.

2.3 Effects of BilinearR-Parity violation

As already argued in the previous section, bilinear6Rp terms are present in any consistent patternof R-parity violation, although they have often been neglectedin the literature; most of thestudies assumeR-parity breaking by trilinear terms only. The purpose of this section is todescribe the effects associated with the presence of bilinear 6Rp terms. The effects of trilinear6Rp

terms will be discussed in chapters 6 and 7. Let us stress thatthe following discussion does notonly apply to the bilinear and spontaneousRp-breaking scenarios, but also to the most general

scenario in which both bilinear and trilinear6Rp couplings are present. The phenomenology ofbilinearR-parity violation has been first investigated in Refs. [24, 33, 34].

2.3.1 Distinguishing Between Higgs and Lepton Doublet Superfields

As we have already said, in the limit of unbroken supersymmetry, the bilinear6Rp superpotentialtermsHuLi can always be rotated away by a suitable redefinition, as expressed by Eq. (2.17) ofthe four superfields(Hd, Li=1,2,3). This redefinition leaves a single bilinear term in the superpo-tential, theµ-termµHuHd, but generates new contributions to theλijkLiLjE

ck andλ′ijkLiQjD

ck

terms from the charged lepton and down quark Yukawa couplings. So, as long as supersymme-try is unbroken,R-parity violation can be parametrized by trilinear couplings only. In the pres-ence of soft terms, however, the scalar potential and the superpotential providetwo independentsources of bilinearR-parity violation thatcannot be simultaneously rotated away[24], unlesssome very particular conditions to be discussed below are satisfied. Furthermore these condi-tions are not renormalization group invariant. Therefore bilinear 6Rp terms, if absent at someenergy scale, are always regenerated at other energy scales. One is then left with a physicallyrelevant mixing between the Higgs and lepton superfields, which leads to specific signatures, asdiscussed in the next subsections.

The Higgs-lepton mixing associated with bilinearR-parity violationresults in an ambiguityin the definition of theHd andLi superfields,which carry the same gauge charges. The values ofthe lepton-number-violating couplings, in particular, depend on the choice of the (Hd, Li) basis(see subsection 2.1.4). It is therefore crucial, in phenomenological studies ofR-parity violation,to specify in which basis one is working. Of course any physical quantity one may computewill not depend on the choice of basis made; but the formula that expresses this quantity as afunction of the lepton-number-violating couplings will. In this subsection, we shall introducetwo basis-independent quantitiessin ξ andsin ζ that control the size of the effects of bilinearR-parity violation in the fermion and in the sfermion sectors, respectively.

Before doing so, let us rewrite the most general superpotential and soft scalar potential, Eqs.(2.1) – (2.2) and (2.15) – (2.16), in a form that makes apparent the freedom of rotating theHd

andLi superfields. For this purpose we group them into a 4-vectorLα, α = 0, 1, 2, 3 [35]. Therenormalizable superpotential, including all possibleR-parity-preserving and -violating terms,then reads:

W = µαHuLα +1

2λe

αβk LαLβEck + λd

αjk LαQjDck

− λujkHuQjU

ck +

1

2λ′′ijk U

ciD

cjD

ck , (2.27)

and the soft supersymmetry-breaking terms in the scalar sector read:

Vsoft =(Bα hu

ˆLα + h.c.)

+ m2αβ

ˆL†αˆLβ + m2

u h†uhu + . . .

+

(1

2Ae

αβk LαLβ lck + Ad

αjk LαQj dck +

1

2A′′

ijk uci d

cjd

ck + h.c.

), (2.28)

where the dots stand for the other soft (squark and ”right-handed” slepton) scalar mass terms.In the presence of the bilinear soft terms in Eq. (2.28), the radiative breaking of the electroweaksymmetry leads to a vacuum expectation value for each of the neutral scalar components of theLα doublet superfields,< ˆνα > ≡ vα/

√2. The covariance of Eqs. (2.27) and (2.28) under

SU(4) rotations of theLα superfields,Lα → UαβLα, dictates the transformation laws for theparametersµα, λe

αβk, λdαjk, Bα, m2

αβ, Aeαβk andAd

αjk.

Up to now, we have not chosen a specific basis for theLα superfields. Several bases havebeen considered in the literature. A first possibility [24] is to define the down-type Higgs su-perfieldHd as the combination of theLα superfields that couples toHu in the superpotential.This choice impliesµi = 0 for the orthogonal combinationsLi, i.e. all bilinear6Rp terms arecontained in the soft supersymmetry-breaking contribution to the scalar potential, but the leptonscalars have in general non-vanishing VEVs,vi 6= 0. A second, more physical possibility is todefineHd as the combination of theLα superfields whose vacuum expectation value breaks theweak hypercharge [35], implyingvi = 0 for all three sneutrino fields. In the following, wechoose the latter option and define

Hd =1

vd

∑

α

vαLα , (2.29)

wherevd ≡ (∑

α v2α)

1/2 (here and in the following, we neglect phases for simplicity, since takingthem into account is straightforward). The orthogonal combinationsLi, i = 1, 2, 3 correspondto the usual slepton fields with vanishing VEVs:

Lα =vα

vdHd +

∑

i

eαi Li , (2.30)

where the 4-vectors~ei ≡ eαiα=0...3 satisfy~v.~ei = 0 and~ei.~ej = δij. At this point onecan still rotate freely theLi superfields. This freedom can be removed by diagonalizing thecharged lepton Yukawa couplings [36]. The advantage of thischoice is that, in the physicallyrelevant limitsin ξ ≪ 1, to be discussed below, the charged leptons almost coincidewith theirmass eigenstates, and the lepton flavour composition of the massive neutrino (see below) canbe parametrized in terms of theµi. Alternatively, one may require that a single lepton doubletsuperfield, sayL3, couples toHu in the superpotential, i.e.~µ.~e1 = ~µ.~e2 = 0. [37]. The latterchoice allows one to rewrite the superpotential as (leavingaside the last two terms in Eq. (2.27),which are not modified):

W = λeik HdLiE

ck + λd

ik HdQiDck +

1

2λijk LiLjE

ck + λ′ijk LiQjD

ck

+ µ cos ξ HuHd + µ sin ξ HuL3 , (2.31)

whereµ ≡ (∑

α µ2α)

1/2, ξ is the angle between the 4-vectors~µ and~v, given by [35]

cos ξ ≡ 1

µvd

∑

α

µαvα , (2.32)

and the physical Yukawa and trilinear6Rp couplings are given by:

λeik =

∑

α, β

vα

vdeβi λ

eαβk , λd

ik =∑

α

vα

vdλd

αik , (2.33)

λijk =∑

α, β

eαi eβj λeαβk , λ′ijk =

∑

α

eαi λdαjk (2.34)

(similar relations hold for the associatedA-termsAeij , A

dij , Aijk andA′

ijk). The residual termHuL3 in Eq. (2.31) corresponds to a physical higgsino-lepton mixing that cannot be removed

by a field redefinition, unless the vacuum expectation valuesvα are proportional to theµα, sothat sin ξ = 0. Indeed, contrary to theµα andvα which are basis-dependent quantities and donot have any intrinsic physical meaning, their relative angle ξ does not depend on the choiceof basis for theLα superfields.This angle controls the size of the effects of bilinearR-parityviolation in the fermion sector(see section 2.3).

Similarly, the amount of bilinearR-parity violation in the scalar sector is measured by theangleζ formed by the 4-vectors~B and~v [38],

cos ζ ≡ 1

Bvd

∑

α

Bαvα , (2.35)

whereB ≡ (∑

αB2α)

1/2. Indeed, in thevi = 0 basis, the6Rp scalar potential reduces to:

V(2)/Rp

= µ⋆µi h†dLi + Bi huLi + m2

di h†dLi + h.c.

= Bi

(hu − tanβ h†d

)Li + h.c. , (2.36)

where tan β ≡ vu/vd, Bi =∑

αBαeαi, and we have used the minimization conditionm2

di + µ⋆µi + Bi tanβ = 0 to derive the last equality. This condition ensures that theLi

fields only couple to the combination ofhu andhd that does not acquire a VEV, thus allow-ing for vanishingvi’s. Since

∑iB

2i = B2 sin2 ζ , Eq. (2.36) tells us thatthe overall amount

of physical Higgs-slepton mixing is controlled by the angleζ ; in particular,it vanishes if andonly if the vacuum expectation valuesvα are proportional to the soft parametersBα, so thatsin ζ = 0.In this case all bilinear soft terms areR-parity conserving in thevi = 0 basis, i.e.V

(2)soft = (B huhd + h.c.) + m2

d h†dhd + m2

u h†uhu + m2

ij L†i Lj + . . ..Some phenomenological con-

sequences of the Higgs-slepton mixing are discussedat the end of subsection 2.3.4 and later insection 5.5.

In general the vacuum expectation valuesvα are neither aligned with the superpotentialmassesµα nor with the soft parametersBα. This results in a physical Higgs/lepton mixingboth in the fermion and in the scalar sectors, characterizedby the two misalignment anglesξandζ , respectively. The casesin ξ = 0 corresponds to the absence of mixing in the fermionsector: µi = vi = 0 in the same basis, and all effects of bilinearR-parity violation comefrom the Higgs-slepton mixing, Eq. (2.36). Conversely, thecasesin ζ = 0 corresponds tothe absence of mixing in the scalar sector:Bi = vi = 0 in the same basis, and all effects ofbilinearR-parity violation come from the higgsino-lepton mixing, Eq. (2.31). Finally, in thecase wheresin ξ = sin ζ = 0, all bilinear 6Rp terms can be simultaneously rotated away from theLagrangian, andR-parity violation is purely of thetrilinear type. Again, the statement thatsin ξor sin ζ vanishes is scale-dependent [30, 31].

As we shall discuss in subsection 2.3.3, there are particularly strong constraints onsin ξcoming from neutrino masses. Therefore, it is legitimate toask under which circumstancessin ξvanishes. To achieve this situation, it is sufficient (but not necessary) to impose the followingtwo conditions on the6Rp soft terms [35]: (i) theB terms are proportional to theµ-terms,Bα ∝ µα [24]; (ii) ~µ is an eigenvector of the scalar mass-squared matrix,m2

αβ µβ = m2d µα.

Although these relations are not likely to hold exactly at the weak scale, a strong correlationbetween the soft parameters and theµα may result from a flavour symmetry [35] or from someuniversality assumption at the GUT scale [39, 40], leading to an approximate alignment betweenthe 4-vectors~µ and~v, so thatsin ξ ≪ 1 (see chapter 3 for more details). Actually conditions (i)and (ii) do not only implysin ξ = 0, but alsosin ζ = 0; therefore they are sufficient to ensure

the absence of bilinearR-parity violation at the scale at which they are satisfied. Finally, whenonly condition (i) holds, one hassin ξ = sin ζ , hence bilinearR-parity violation is parametrizedby a single physical parameterξ (with Bi = B sin ξ δi3 in the basis whereµi = µ sin ξ δi3).

2.3.2 Trilinear Couplings Induced by Bilinear 6Rp Terms

Let us now comment on an interesting consequence of Eqs. (2.33)–(2.34) in scenarios where,in some particular basisLα, the trilinear 6Rp couplings vanish (i.e.λe

ijk = λdijk = 0, while

λe0jk = −λe

j0k andλd0jk are non-vanishing). This is the case, in particular, in the “bilinear”

and spontaneousRp-breaking scenarios. Then, in the (Hd, Li) basis defined by Eq. (2.30), thetrilinear 6Rp couplings are related to the ordinary Yukawa couplingsλe

ij andλdij by

λijk =vd

v0

(e0i λ

ejk − e0j λ

eik

), λ′ijk =

vd

v0e0i λ

djk , (2.37)

wherev0 ande0i are theα = 0 components of the 4-vectorsvα andeαi that define the (Hd, Li)basis. The flavour dependence of theλijk andλ′ijk is then determined by the charged lepton anddown quark Yukawa couplings, respectively.

Eq. (2.37) has two obvious consequences. First, unless there is a hierarchy of VEVsv0 ≪ vi

in the initial basis, the trilinear6Rp couplings are at most of the same order of magnitude as thedown-type Yukawa couplings. This suppression, which is stronger for smaller values oftan β,allows them to evade most of the direct and indirect bounds that shall be discussed in chapters 6and 7 (some particularly severe bounds require an additional suppression of the rotation pa-rameterse0i). Second, the contributions of the trilinear6Rp couplings (2.37) to flavour-changingneutral current (FCNC) processes either vanish or are strongly suppressed [37]. Indeed, theλ′ijk LiQjD

ck terms read, in the mass eigenstate basis of the quarks (we puta bar on the trilinear

6Rp couplings when they are defined in the mass eigenstate basis for fermions):

λ′ijk

(NiDj − Ei V

†jp Up

)Dc

k with λ′ijk = e0i

√2mdj

v0

δjk , (2.38)

whereV is the CKM matrix. It follows that theλ′ couplings induced by bilinear6Rp vanishfor quark-flavour-changing transitionsdj → dk, while uj → dk transitions are suppressed bythe CKM angles. Thus most of the constraints onλ′ couplings due to quark-flavour-violatingprocesses (see chapter 6) are trivially satisfied.

The case of lepton-flavour violation is more subtle, both because the relation between theλijk andλe

ij couplings is not an exact proportionality relation, and because the charged leptonmasses are not given by the eigenvalues of the Yukawa matrix,due to the lepton-higgsino mixing(see discussion below). However, for small mixing (sin ξ ≪ 1), one can write, in the masseigenstate basis of the charged leptons:

λijk NiEj Eck with λijk ≃

√2

v0

(Re⋆

Lile0l mejδjk − Re⋆

Ljle0l meiδik), (2.39)

where the matrixReL rotates the left-handed charged leptons fields to their corresponding mass

eigenstates8. These couplings violate lepton flavour, but there is a restricted number of them,8Strictly speaking, the couplingsλijk are defined in the basis in which the charged lepton Yukawa couplings are

diagonal, i.e.ReL is defined byRe

LλeRe†R = Diag(λe1

, λe2, λe3

). However, forsin ξ ≪ 1 this basis approximatelycoincides with the mass eigenstate basis, and thereforeλei

vd ≃√

2 mei. In the limit sin ξ = 0, both bases coincide

and Eq. (2.39) is exact.

sinceλijk 6= 0 only if i = k or j = k. Furthermore, the non-vanishing couplings are suppressedby the small lepton Yukawa couplings. This significantly reduces their contributions to lepton-flavour-violating processes, especially in the smalltanβ case.

The above conclusions hold when the trilinear6Rp couplings vanish in the originalLα basis,i.e. λe

ijk = λdijk = 0. If this is not the case, the physical trilinear6Rp couplingsλijk andλ′ijk

(defined in the (Hd, Li) basis) receive an additional contribution from the initial λeijk andλd

ijk

couplings, which modifies Eqs. (2.37), (2.38) and (2.39) as well as the resulting conclusions forFCNC processes.

2.3.3 Higgsino-Lepton Mixing

As already mentioned, the main effect of bilinearR-parity violation is a physical mixing be-tween sleptons and Higgs bosons in the scalar sector, and between leptons and neutralinos/char-ginos in the fermion sector. We do not discuss the details of the mixing in the scalar sector here,and refer the interested reader to the literature (see e.g. Refs. [41, 42, 43, 44, 45]). In the fermionsector, in an arbitrary basisLα, the superpotential mass parametersµα mix the fermionic com-ponents of theLα andHu superfields (i.e. the neutrino fieldsνα with the neutral higgsino fieldh0

u, and the charged lepton fieldsl−α with the conjugate of the charged higgsino fieldh+u ), and

the VEVsvα mix the neutrino fieldsνα with the gaugino associated with theZ gauge boson.

As a result the4 × 4 neutralino mass matrix (resp. the2 × 2 chargino mass matrix) of theMSSM becomes a7×7 neutralino-neutrino mass matrix (resp. a5×5 chargino- charged leptonmass matrix). With the notationLmass = −1

2ψ0TMNψ

0 + h.c., the7 × 7 neutralino-neutrinomass matrixMN reads, in theψ0 = (−iλγ ,−iλZ , h

0u, να)α=0···3 basis9 [35]:

MN =

M1c2W +M2s

2W (M2 −M1)sW cW 0 01×4

(M2 −M1)sW cW M1s2W +M2c

2W

g2cW

vu − g2cW

vα

0 g2cW

vu 0 −µα

04×1 − g2cW

vα −µα 04×4

, (2.40)

whereM1 andM2 are theU(1)Y andSU(2)L gaugino masses,g is theSU(2)L gauge coupling,cW = cos θW andsW = sin θW .

The5 × 5 chargino- charged lepton mass matrix, defined byLmass = −ψ−TMCψ+ + h.c.,

reads, in theψ− = (−iλ−, l−α )α=0···3 andψ+ = (−iλ+, h+u , l

ck) k=1···3 basis [35]:

MC =

(M2 gvu/

√2 01×3

gvα/√

2 µα λαβkvβ

), (2.41)

or, in a more explicit form:

MC =

M2 gvu/√

2 01×3

gv0/√

2 µ0 λ0βkvβ

gvi/√

2 µi λiβkvβ

. (2.42)

9We use here a two-component notation for spinors. With the conventions of Ref. [25], the two-componentspinorsλγ andλZ are related to the four-component Majorana spinorsγ andZ by γ = (−iλγ , iλγ)T andZ =

(−iλZ , iλZ)T . Similarly, the two-component spinorsλ+ andλ− are related to the Dirac spinorW+ by W+ =(−iλ+, iλ−)T .

In theR-parity conserving case, it is possible to find a (Hd, Li) basis in whichµα ≡ (µ, 0, 0, 0)andvα ≡ (vd, 0, 0, 0). Then Eqs. (2.40) and (2.41) reduce in the “ino” sector to thewell-known(4× 4) neutralino and (2× 2) chargino mass matrices of the MSSM withR-parity conservation(cf. Eq. (1.10)).

In order to discuss the physical implications of the neutrino-neutralino and charged lepton-chargino mixings caused by bilinearR-parity violation, we now turn to the basis defined by Eqs.(2.29) and (2.31), in whichµα ≡ (µ cos ξ, 0, 0, µ sin ξ) andvα ≡ (vd, 0, 0, 0). This choice ofbasis leaves only a physical mixing inMN andMC , proportional to the misalignment parametersin ξ. The structure of both matrices is schematically:

MSSMneutralino (chargino)

mass matrix6Rp terms

6Rp termsneutrino

(charged lepton)mass matrix

. (2.43)

More precisely,MN reads:

MN =

M1c2W + M2s

2W (M2 − M1)sW cW 0 0

(M2 − M1)sW cW M1s2W + M2c

2W MZ sinβ −MZ cosβ

0 MZ sin β 0 −µ cos ξ0 −MZ cosβ −µ cos ξ 0

00

−µ sin ξ0

0 0 −µ sin ξ 0 0

, (2.44)

wheretanβ ≡ vu/vd. As can also be seen from Eq. (2.31),ν1 and ν2 decouple from thetree-level neutrino-neutralino mass matrix, and only the mixing betweenν3 and the neutralinosremains (this is no longer the case at the one-loop level, where all three neutrinos mix with theneutralinos, see chapter 5). As for the chargino-charged lepton mass matrix, it reads:

MC =

M2 gvu/√

2

gvd/√

2 µ cos ξ

01×3

01×3

03×1 µδi3 sin ξ λeikvd/

√2

. (2.45)

As already mentioned above, the mixing inMN andMC , proportional tosin ξ, is suppressedwhen the VEVsvα are approximately aligned with theµα; we shall see below that experimentalconstraints on neutrino masses actually require a strong alignment, i.e. a very small value ofsin ξ. The matrixMC has five mass eigenstates; the three lightest ones are identified with thecharged leptonsl−i (which, due to the higgsino-lepton mixing, do not exactly coincide with theeigenstates of the Yukawa matrixλe

ik, although this mismatch can be neglected whensin ξ ≪1), and the two heaviest ones with the charginosχ−

1 and χ−2 . In addition to its two massless

eigenstatesν1 andν2, the matrixMN has five massive eigenstates; the four heaviest ones arethe neutralinosχ0

i , i = 1 · · · 4 ; the lightest one, which is mainlyν3 in thesin ξ ≪ 1 case, canbe identified with a Majorana neutrino with a mass [35]

mν3 = m0 tan2 ξ , (2.46)

wherem0 is given, in the casesin ξ ≪ 1, by the following expression [39]:

m0 ≃ M2Z cos2 β(M1c

2W +M2s

2W )

M1M2µ cos ξ −M2Z sin 2β(M1c2W +M2s2

W )µ cos ξ . (2.47)

For a rough estimate, we can take:

m0 ∼ (100 GeV) cos2 β

(100 GeVM2

). (2.48)

The exact value ofm0 depends on the gaugino masses,µ andtanβ, but Eq. (2.48) becomes agood approximation for a heavy supersymmetric spectrum or for large values oftan β. Thus wesee that the neutrino mass is proportional to the square of the 6Rp angleξ (it could not have beenlinear intan ξ, since the neutrino mass term isR-parity even), and therefore roughly measuresthe overall amount of bilinearR-parity violation in the fermion sector. The other two neutrinosremain massless at tree-level, but acquire masses at the one-loop level (see chapter 5).

Sincemν3 is proportional totan2 ξ, with a natural scale in the(1 − 100) GeV range de-pending on the value oftanβ, the known experimental and cosmological upper bounds on theheaviest neutrino mass provide strong constraints on bilinearR-parity violation. Indeed, thecosmological bound on neutrino masses inferred from CMB [46] and large scale structure data[47] (

∑imνi

. 1 eV, where the sum runs over all neutrino species) imposes a strong alignmentof thevα with theµα, typically10 sin ξ . 3 × 10−6

√1 + tan2 β.

We see that the presence of bilinear6Rp terms can only be tolerated at the expense of somesignificant amount of tuning in the parameters so as to keep neutrinos sufficiently light. It isimportant to keep in mind that the values ofµi andvi by themselves are not a good measurementof this tuning, since in an arbitrary weak eigenstate basis,large values of theµi andvi can becompatible with a strong alignment. However in thevi = 0 basis (resp.µi = 0 basis), theµi

(resp.vi) are constrained to be small, according to the estimateµi ∼ µ sin ξ (resp.vi ∼ vd sin ξ).

Finally, let us mention the fact that the neutrino sector also constrains the tolerable amountof bilinearR-parity violation in the scalar sector, yieldingsin ζ . (10−4 − 10−3) for the cos-mological bound (see section 5.1).

2.3.4 Experimental Signatures of BilinearR-Parity Violation

The mixing of ordinary leptons with charginos and neutralinos also leads to interactions that arecharacteristic of bilinearR-parity violation. These are:(i) 6Rp gauge interactions,(ii) lepton-flavour-violatingZ couplings, and(iii) trilinear 6Rp interactions distinct from those generatedfrom the superpotentialλ andλ′ terms. These interactions are suppressed by at least one powerof sin ξ, and are therefore very difficult to observe experimentally, with however a few excep-tions. We give a brief description of them below:

(i) Non-diagonal couplings of theZ andW bosons to a lepton and a supersymmetric fermion(chargino or neutralino) appear when the currents are written in terms of the mass eigen-states:Z χ±

i l∓j , Z χ0

i ν3, W−χ+i νj , W−χ0

i l+j . These6Rp gauge couplings are proportional

to sin ξ, and therefore correlated to the heaviest neutrino mass,mν3. They give rise to6Rp processes such as single production of charginos and neutralinos (e.g. through thedecaysZ → χ±

i l∓j andZ → χ0

i ν3 at LEP) and decays of the lightest neutralino into threestandard fermions (χ0

1 → ν3f f or χ01 → lif f

′) or, if it is heavier than the gauge bosons,10Some scenarios in which the heaviest neutrino has rapid enough decay modes could in principle evade this

bound, but this possibility looks rather unnatural in view of the experimental evidence for neutrino oscillations (seee.g. the discussion at the end of section 2.4).

into liW or ν3Z. Since the cross-section goes assin2 ξ, single production is unobservablein practice. The two-body decays are characteristic of bilinearR-parity violation, whilethe three-body decays are also induced by the trilinear6Rp couplings (except however forχ0

1 → ν3νν). Studies of the corresponding signals at LEP and at hadron colliders canbe found in Refs. [48] and [49, 50], respectively. More recently, the 6Rp decays of a neu-tralino LSP have been discussed in Refs. [50, 51]. See also Ref. [52] for the 6Rp decays ofa chargino LSP.

(ii) Together with the previous6Rp gauge interactions, bilinear6Rp also gives rise to flavour-violating couplings of theZ boson to the leptons,Z l−i l

+j . These couplings, which con-

tribute to FCNC processes such asµ → 3 e [36], are proportional tosin2 ξ, and theireffects are therefore extremely difficult to observe experimentally. In particular, theflavour-violatingZ-boson decaysZ → l+i l

−j are suppressed bysin4 ξ, well below the

experimental upper limits, BR(Z → l+i l−j ) < (10−6 − 10−5) [20].

(iii) Finally, the couplings of neutralinos and charginos to matter fields give rise, when writtenin terms of mass eigenstates, to6Rp trilinear interactions proportional tosin ξ [53]. Theones that originate from down-type higgsino couplings are similar to interactions gener-ated from the superpotentialλ andλ′ couplings; they are suppressed both by the smallnessof the Yukawa couplings and bysin ξ. The ones that originate from up-type higgsino orgaugino couplings, on the other hand, cannot arise from superpotentialλ or λ′ couplings.Examples of such interactions arelRdLu

⋆R andνc

RuLu⋆R.

Since all the above couplings are suppressed by at least one power ofsin ξ, which is constrainedto be very small by the cosmological bound on neutrino masses, the corresponding experimentalsignatures are very difficult to observe in practice, with a few exceptions like the decaysχ0

1 →lif f

′ when the lightest neutralinoχ01 is the LSP.

Finally, bilinearR-parity violation introduces a mixing in the scalar sector between theHiggs bosons and the sleptons, which leads to6Rp decay modes of these scalars, such ash,H →χ0 ν, χ+ l− for neutralCP -even Higgs bosons [26, 41, 42], orτ1 → l−ν, qq′ for the lighteststau [54].

2.4 Spontaneous Breaking ofR-Parity

Spontaneous breaking ofR-parity has been considered as an interesting alternative to explicitR-parity breaking, because of its predictivity and potentially rich phenomenology. Also, ifspontaneousR-parity breaking occurs below a few TeV, the strong cosmological bounds ontrilinear 6Rp couplings associated with the requirement that6Rp interactions do not erase anyprimordial baryon asymmetry (see section 4.2.2) can be evaded.

The simplest possibility to breakR-parity spontaneously is to give a vacuum expectationvalue to a sneutrino field [55, 56, 57], e.g.< ντ > 6= 0 (i.e. v3 6= 0 with our previous notations).This may occur since the squared masses of the sneutrinos receive negative contributions bothat tree-level from theD-terms and from radiative corrections. Due to the larger third familyYukawa couplings, radiative corrections are expected to generate a VEV for the tau sneutrinoonly. However, since the conservation of lepton number is associated with a global symmetry, itsspontaneous breaking would give rise to a massless Goldstone bosonJ , called the Majoron [58,

59], together with a scalarρ having a mass of the order of< ντ >. The decay mode11 Z → J ρwould then contribute to the invisible decay width of theZ boson like half a neutrino flavour,which is excluded by experimental data. There are several ways out: (i) introduce some amountof explicit lepton number breaking into the MSSM, so as to give to the pseudo-Majoron a massmJ > MZ [61]; (ii) enlarge the gauge group to include lepton number,so that the would-beMajoron becomes the longitudinal degree of freedom of a new gauge boson [62]; (iii) breakR-parity through the VEV of a right-handed sneutrino,< νc

τ > 6= 0, so that the Majoron is mainlyan electroweak singlet, and does not contribute sizeably totheZ decay width [63, 64, 65].

Let us illustrate approach (iii) by giving the main featuresof the model of Ref. [63]. Themodel contains, beyond the MSSM superfields, the followingSU(2) × U(1) singlets: threeright-handed neutrinosN c

i , three singletsSi with lepton numberL = 1, and an additional singletΦ whose role is to generate theµ-term – as in the “NMSSM” (Next to Minimal SupersymmetricStandard Model). The superpotential contains, beyond the quark and charged lepton Yukawacouplings, the following cubic terms [66]:

W = h0HuHdΦ + λΦ Φ3 + hνij HuLiNcj + hij SiN

cj Φ (2.49)

This superpotential preserves bothR-parity and total lepton number.R-parity is spontaneouslybroken provided that the lepton singlets acquire vacuum expectation values (in the following,we restrict ourselves to the one-family case):

< νRτ > ≡ vR√2, < Sτ > ≡ vS√

2. (2.50)

The most general minimum also involves, together with the Higgs VEVsvu/√

2 andvd/√

2required to break the electroweak symmetry, a sneutrino VEV< νLτ >≡ vL/

√2, assumed to

be small. The MajoronJ is given by the imaginary part of the linear combination:

1√v2

R + v2S

[v2

L

v2( vuhu − vdhd ) + vL ντ + vR ν

cτ + vS Sτ

]. (2.51)

The most stringent constraint onJ comes from astrophysics: in order to avoid a too largestellar energy loss via Majoron emission, one must impose [63] v2

L/vRmW . 10−7, i.e. vL .

100 MeV for vR ∼ 1 TeV. This small value ofvL ensures thatJ couples only very weaklyto theZ boson, and therefore does not affect its invisible decay width. Typical values for aviableRp-breaking minimum of the scalar potential are10 GeV . vR, vS, vΦ . 1 TeV andvL . (10 − 100) MeV. The hierarchyvL ≪ vR can be understood in terms of small Yukawacouplings, sincevL is found to be proportional to the neutrino Yukawa coupling constantshνij .

As can be seen from Eq. (2.49), a nonzerovR generates effective superpotential bilinearterms,µi = hνi3vR/

√2. As a result, upon a redefinition of the Higgs superfieldHd, small trilin-

ear couplingsλ andλ′ are generated with the same flavour structure as in the previous section,and 6Rp effects are induced in gauge and matter interactions of neutralinos and charginos, aswell as in slepton and Higgs decays. The magnitude of these effects is related to the value ofthe heaviest neutrino mass. However, there is a noticeable difference with the explicit bilinearR-parity breaking discussed in the previous section, due to the presence of the Majoron. This

11Several constraints on< ντ >, e.g. the LEP limit on the tau neutrino mass which requires< ντ > . 2 GeV(as can be seen by adapting formulae (2.46) and (2.47) to the caseµi = 0, v3 6= 0), or the even stronger bound< ντ > . 100 keV coming from stellar energy loss through Majoron emission in Compton scattering processeseγ → eJ [60], ensure thatmρ < MZ , so that this decay mode is indeed kinematically allowed.

results in new interactions, such as chargino and (invisible) neutralino decaysχ± → τ±J andχ0 → νiJ , invisible decay of the lightest Higgs bosonh → JJ (which may be sizeable [67]),flavour-violating decays of charged leptonsei → ejJ , and tau neutrino annihilationντντ → JJand decayντ → νµJ . It has been argued that the latter processes could be large enough to relaxthe energy density and nucleosynthesis constraints on the tau neutrino mass, and allow it to beas large as the LEP limit of18.2 MeV [68]. However such a possibility, quite popular at a timewhere atmospheric neutrino oscillations were not established on a very solid basis, does notlook very appealing today since it fails to accommodate bothsolar and atmospheric neutrinooscillations.

The feasibility of spontaneous6Rp along the lines of Ref. [63] has been investigated byseveral authors; it has in particular been shown that spontaneousR-parity breaking could beinduced radiatively together with electroweak symmetry breaking [66]. Numerous studies ofthe experimental signatures of spontaneous6Rp can be found in the literature, see e.g. Refs. [67,69, 70].

2.5 Constraining 6Rp Couplings from Flavour Symmetries

The trilinear terms in the6Rp superpotential of Eq. (2.2) are very similar to those associatedwith the quark and charged lepton Yukawa couplings in Eq. (2.1), known from the patternof fermion masses and mixing angles to have a rather hierarchical structure. It is conceivablethat the mechanism at the origin of this hierarchy also provides a hierarchical structure for6Rp

couplings.

A possible simple explanation for the fermion mass hierarchy has been provided long agoby Froggatt and Nielsen, who postulated the existence of a spontaneously broken, flavour-dependent abelian symmetry. In this section, we show that such a symmetry may also naturallygenerate a flavour hierarchy between6Rp couplings [29, 35, 37, 71, 72, 73, 74]. For the case ofa non-abelian flavour symmetry, see e.g. Ref. [75].

Let us first explain how a family-dependent symmetryU(1)X constrains the Yukawa sec-tor [73, 76]. Consider a Yukawa couplingHuQiU

cj and let us denote generically theX-charge

of a superfieldΦi by the corresponding small letterφi. Invariance underU(1)X implies thatHuQiU

cj appears in the superpotential only if itsX-charge vanishes, i.e.qi + uj + hu = 0.

To account for the large top quark mass, one assumes that thishappens only for the YukawacouplingHuQ3U

c3 ; thus all fermions but the top quark are massless before the breaking of this

symmetry. One further assumes that the flavour symmetry is broken by the VEV of a StandardModel singletθ with X-charge−1, and that the other Yukawa couplings are generated from(gauge-invariant) interactions of the form

yuij HuQiU

cj

(θ

M

)qi+uj+hu

, (2.52)

whereM is a mass scale,yuij is an unconstrained coupling of order one, andqi + uj + hu > 0.

Such non-renormalizable terms typically appear in the low-energy effective field theory of afundamental theory with heavy fermions of massM (one may also think of a string theory, inwhich caseM ∼MP ). If U(1)X is indeed broken below the scaleM , ǫ =< θ > /M is a smallparameter, and (2.52) generates an effective Yukawa coupling

λuij = yu

ij ǫqi+uj+hu , (2.53)

whose order of magnitude is fixed by the values of theX-charges. Similarly one has, for downquarks and charged leptons:

λdij ∼ ǫ qi+dj+hd , λe

ij ∼ ǫ li+ej+hd . (2.54)

Such a family-dependent symmetry thus naturally yields a hierarchy between Yukawa cou-plings, and therefore fermion masses.

For example, the charge assignmentq1 − q3 = 3, q2 − q3 = 2, u1 − u3 = 5, u2 − u3 = 2,d1 − d3 = 1, d2 − d3 = 0 yields quark Yukawa matrices of the form:

λu ∼

ǫ8 ǫ5 ǫ3

ǫ7 ǫ4 ǫ2

ǫ5 ǫ2 1

, λd ∼ ǫq3+d3+hd

ǫ4 ǫ3 ǫ3

ǫ3 ǫ2 ǫ2

ǫ 1 1

, (2.55)

where the symbol∼ indicates that the entries are known up to factors of order one only. Eq.(2.55) holds at the scale at which the abelian symmetry is spontaneously broken, usually takento be close to the Planck scale. With renormalisation group effects down to the weak scaletaken into account, these Yukawa matrices can accommodate the observed quark masses andmixings if the small numberǫ is of the order of the Cabibbo angle, i.e.ǫ ≈ Vus ≃ 0.22. Moregenerally, assuming that theX-charge associated with each Yukawa coupling is positive, onlya few structures forλu andλd, which differ from Eq. (2.55) by a±1 change in the powers ofǫ,are allowed by the data. In the lepton sector there is more freedom, as long as a mechanism forgenerating neutrino masses is not specified. The combination ofX-chargesq3 +d3 +hd, relatedto the value oftan β by mt/mb ∼ tanβ ǫ−(q3+d3+hd), is actually constrained if one imposesgauge anomaly cancellation conditions.

6Rp couplings are then constrained byU(1)X exactly as for Yukawa couplings. They aregenerated from the following non-renormalizable superpotential terms:

LiLjEck

(θ

M

)li+lj+ek

, LiQjDck

(θ

M

)li+qj+dk

. (2.56)

To avoid unnaturally large values of the quarkX-charges, we have assumed a baryon paritythat forbids the baryon-number-violating superpotentialtermsU cDcDc as well as the dangerousdimension-5 operators discussed in Section 2.7, thus preventing proton decay. One can see fromEq. (2.56) that abelian flavour symmetries yield a hierarchybetween6Rp couplings that mimics(in order of magnitude) the down quark and charged lepton mass hierarchies. Indeed, one has:

λijk ∼ ǫ li−hd λejk , λ′ijk ∼ ǫ li−hd λd

jk . (2.57)

Provided that the Yukawa matricesλd andλe are known, experimental limits onλ andλ′ canbe translated into a constraint onli − hd. We shall assume here that theX-charge carried byeach operator is positive, and take forλd the structure of Eq. (2.55). In the lepton sector, theλe

ij are less constrained; however, it is possible to derive upper bounds on theλijk couplingsfrom the three charged lepton masses. Assuming a small valueof tan β (corresponding toq3 + d3 + hd = 3), one finds that the experimental bounds on coupling products (including thelimit coming fromǫK , | ℑ (λ′i12 λ

′⋆i21) | ≤ 8 × 10−12) are satisfied as soon as [77]:

li − hd ≥ 2 − 3 . (2.58)

For moderate or large values oftanβ, larger values of theX-charges would be required.

The condition (2.58) can now be used, together with Eq. (2.57), to derive, in the frameworkconsidered,tan β-independent upper bounds on the individual couplingsλ andλ′. All of themare well below the experimental limits. Thus, if abelian flavour symmetries are responsiblefor the observed fermion mass spectrum along the lines discussed above, one expects the firstsignals for brokenR-parity to come from FCNC processes [77]. These conclusions, however,are not completely generic for abelian flavour symmetries: they would be modified ifU(1)X

were broken by a vector-like pair of singlets [74], or if we gave up the assumption that theX-charge associated with each operator is positive.

Let us now consider the most general scenario ofR-parity violation, with both bilinear andtrilinear 6Rp couplings (see section 2.3 for the notations used in the following, where we closelyfollow the discussion of Ref. [37]). Assuming that the bilinear terms are generated throughsupersymmetry breaking [78] (which ensures that theµα are of the order of the weak scale, asrequired by electroweak symmetry breaking), one finds:

µα ∼ m ǫ lα , vα ∼ vd ǫlα−l0 , (2.59)

wherem is the typical mass scale associated with the soft supersymmetry-breaking terms,lα ≡|lα + hu|, and the above estimates are valid for0 ≤ l0 < li, i = 1, 2, 3. Thus thevα areapproximatively aligned along theµα by the flavour symmetry [35], which implies (assumingwith no loss of generalityl3 ≤ l1,2):

sin2 ξ ∼ ǫ 2 (l3−l0) . (2.60)

Furthermore, the redefinition (2.30) is completely fixed by requiringL1 ≃ L1 andL2 ≃ L2,with

vα

vd∼ ǫ lα−l0 , eαi ∼ ǫ |lα−li| . (2.61)

Note thatHd ≃ L0, which allows us to definehd ≡ l0.

The low-energy6Rp couplings depend on the signs of the chargeslα +hu. In all phenomeno-logically viable cases, the order of magnitude relations (2.57) are modified to:

λijk ∼ ǫ li−l0 λejk , λ′ijk ∼ ǫ li−l0 λd

jk . (2.62)

By combining Eqs. (2.46), (2.60) and (2.62), we can write down a relation between the mass ofthe tau neutrino, the6Rp couplingsλ′3jk and the down quark Yukawa couplings (m0 is defined inEq. (2.48)),

mν3 ∼ m0

(λ′3jk

λdjk

)2

, (2.63)

which is a generic prediction of this class of models. Let us stress however that, in Eqs. (2.60)and (2.63), we have assumed that the suppression of the misalignment angleξ is only dueto the abelian flavour symmetry. In this case the cosmological bound on neutrino masses,mν ≤ O(1 eV), requires very large values of thelα, so thatR-parity violation should be verysuppressed and in practice unobservable. This conclusion can be evaded only if some othermechanism provides the required alignment between thevα and theµα, in which case Eqs.(2.60) and (2.63) are no longer valid.

Let us now concentrate on the caseli + hu ≥ 0 > l0 + hu, which leads to an enhancementof flavour-diagonal couplings relative to off-diagonal couplings. Indeed, the dominant terms in

Eq. (2.34) correspond toα = 0 or β = 0, which provides an alignment of the6Rp couplingsalong the Yukawa couplings:

λijk ≃(e0i λ

ejk − e0j λ

eik

), λ′ijk ≃ e0i λ

djk . (2.64)

As a consequence,6Rp couplings are almost diagonal in the basis of fermion mass eigenstates.Furthermore, they undergo an enhancement relative to the naive power counting, since e.g.

λ′ijk ∼ ǫ li−l0 λdjk ∼ ǫ−2 l0 ǫ li+qj+dk . (2.65)

This opens the phenomenologically interesting possibility thatR-parity violation be sizeablewhile its contribution to FCNC processes is suppressed, as required by experimental data –provided however that the misalignment angleξ is reduced to phenomenologically acceptablevalues by some other mechanism than the suppression by largeleptonX-charges.

Abelian flavour symmetries can also play a useful role in controlling the proton decay ratein supersymmetric theories with unbrokenR-parity. Indeed, they can suppress the coefficientsof the dangerous dimension-5 operators12 QQQL andU cU cDcEc [72, 79], which preserveR-parity but violate baryon number and lepton number, and are expected to be generated fromunknown Planck-scale physics (see Section 2.7). Flavour symmetry models that explain thefermion mass hierarchy and are compatible with the experimental lower limit on the protonlifetime generally predict proton decay close to the present experimental sensitivity [29, 79, 80].

2.6 R-Parity Violation in Grand Unified Theories

Up to now we discussedR-Parity Violation in the framework of the Supersymmetric StandardModel (possibly augmented by some flavour symmetries). Moresophisticated supersymmetricextensions of the Standard Model may lead to a different structure of6Rp couplings. In particular,nontrivial constraints on the allowed6Rp couplings generally result from an enlarged gaugestructure, as in Grand Unified Theories (see e.g. Refs. [24, 39, 81, 23, 82, 83, 84, 85]).

In Grand Unified Theories based on theSU(5) gauge group, all trilinear6Rp superpotentialcouplings originate, at the renormalizable level, from thesame operator [23]

1

2Λijk 5i5j10k , (2.66)

antisymmetric under the exchange of5i and 5j , where the antifundamental representation5i

contains theLi andDci superfields, while the antisymmetric representation10i contains theQi,

U ci andEc

i superfields13. As a result, all three types of trilinear6Rp couplings are simultaneoulypresent or absent, and related by

λijk =1

2λ′ikj = λ′′kij , (2.67)

12Of course, this statement also holds in supersymmetric theories withoutR-parity [72], but in this case one mayfind more appealing to invoke a discrete symmetry forbiddingbaryon number violation from both dimension-4 anddimension-5 operators (see Section 2.7), as we did in the above discussion.

13Ordinary quark and charged lepton Yukawa couplings are generated, at the renormalizable level, from thesuperpotential termsΛd

ij10i5j5d + Λuij10i10j5u, where the representations5d and5u contain the doublet Higgs

superfieldsHd and Hu, respectively. The first term leads to the relationλdij = λe

ji, which even after takinginto account the renormalization effects is in gross contradiction with the measured fermion masses, and must becorrected by terms (renormalizable or not) involving higher-dimensional Higgs representations.

with the resulting antisymmetry of theλ′ couplings,λ′ijk = −λ′kji. We are then left with 9superpotential couplings, as well as the 9 correspondingA-terms satisfying the same relation.Since bothλ′ and λ′′ couplings are simultaneously present, the experimental bound on theproton lifetime severely constrains these couplings, with|Λij1|, |Λ123| . 2 × 10−13 atMGUT

[81], where the constraint on theΛij1 comes from the familiar (B − L)-conserving operators,while the constraint onΛ123 comes from (B + L)-conserving operators generated by diagramsinvolving a left-right mixing mass insertion of third generation squarks [33]. The otherΛijk

couplings do not induce proton decay at tree level, but can contribute at the one-loop level, andmust therefore satisfy|Λijk(MGUT )| . 3 × 10−9 [81].

However Eq. (2.67) only applies when the5i5j10k operator arises at the renormalizablelevel, i.e. when theΛijk are field independent. On the contrary when these couplings areinduced by VEVs responsible for GUT symmetry breaking, a different pattern for trilinearR-parity breaking can be obtained (see e.g. Ref. [83] for a model leading toλ′′ijk couplings only,and Refs. [84, 85] for models leading toλ′ijk couplings only).SU(5) Grand Unified Theoriesalso potentially contain a bilinear6Rp superpotential operator

Mi 5u5i , (2.68)

where5u contains both the usual doublet Higgs superfieldHu and a (superheavy) Higgs colourtriplet Tu. This yields, together with the usualHuLi terms, a baryon-number-violating termTuD

ci . While the former is a source forλ andλ′ couplings, the latter is a source forλ′′ cou-

plings, which are the only surviving6B couplings at low energy. However these are small ifthe “doublet-triplet splitting problem” is solved (withmHu ∼ Mweak ≪ mTu ∼ MGUT ) andMi . Mweak [24]. Then the effective low energy6B couplings are suppressed relative to6L cou-plings by a typical factor ofMweak/MGUT , and one ends up with an approximate conservationof baryon number.

It is natural to ask whether one can avoid the generation of6Rp couplings – without im-posingR-parity or any other global symmetry – by considering a unification group larger thanSU(5). It is well known that one of the generators of theSO(10) group acts asB − L on theSupersymmetric Standard Model fields, with the matter superfields embedded into the spino-rial representation16i (which decomposes underSU(5) as16i = 10i ⊕ 5i ⊕ 1i, where1i

corresponds to a right-handed neutrino), and the Higgs doublet superfields embedded into therepresentation10 = 5 ⊕ 5. SinceR-parity can be rewritten as(−)2S+3(B−L), it follows that6Rp operators are forbidden by theSO(10) gauge symmetry (at least as long as it is unbroken).For instance, with the above field content, the only cubic operator compatible with theSO(10)gauge symmetry takes the form10 16 16 and preservesR-parity.

Still the process of gauge symmetry breaking down toSU(3)C × SU(2)L × U(1)Y maylead to a theory that does not conserveR-parity. The question of whetherR-parity remainsexact or not and of what types of6Rp couplings are generated crucially depends on the breakingscheme (for a discussion, see e.g. [86]). If theB−L symmetry is spontaneously broken by theVEV of (the right-handed neutrino-like component of) a Higgs boson in the spinorial represen-tation16H , thenR-parity gets broken in the low-energy effective theory. More precisely, therenormalizable operator

10u16H16i , (2.69)

gives rise, through<16H > 6= 0, to both baryon number and lepton-number-violating bilin-ear terms, contained in theSU(5) 6Rp operator5u5i. These are especially dangerous since

<16H> is generally much larger thanMweak. Trilinear 6Rp couplings are generated from higher-dimensional operators like

16i16j16k16H . (2.70)

If present, these operators generate trilinear6Rp couplingsλ, λ′ andλ′′ satisfying theSU(5)relations (2.67). Alternatively, theB − L symmetry can be broken by the VEV of a Higgsboson in the126 representation. In this case the selection rule∆(B − L) = 2 holds, andR-parity is automatically conserved.

The breaking ofSO(10) into an intermediateSU(5) requires a16H , while if SO(10) firstbreaks into a left-right symmetric gauge groupSU(4)C × SU(2)L × SU(2)R or SU(3)C ×SU(2)L × SU(2)R × U(1)B−L, the breaking of theB − L gauge symmetry involves either a16H or a126H . (Actually supersymmetry demands a vector-like pair of Higgs representations,e.g. 16H ⊕ 16H .) We conclude that theSO(10) breaking chain with an intermediateSU(5)gauge symmetry, or with an intermediate left-right symmetry spontaneously broken by a16H ,leads to unacceptably large6Rp couplings, unless the dangerous operators (2.69) and (2.70) areforbidden by some symmetry or strongly suppressed by some mechanism.

2.7 Restrictions on R-Parity Violations from GeneralizedMatter, Baryon or Lepton Parities

As already alluded to before in this chapter, there exist discrete symmetries that can protectproton decay from renormalizable operators as efficiently asR-parity, while allowing for some6Rp couplings. Such symmetries therefore provide viable patterns ofR-parity violations, and itis useful to classify them, as we do in this section, following the discussion of Ref. [87].

Let us first note thatR-parity actually does not forbid all dangerous baryon-number- andlepton-number-violating couplings. Indeed, it is highly probable that the Supersymmetric Stan-dard Model is just an effective theory, to be embedded in a more fundamental theory includ-ing quantum gravity at some high energy scaleΛ. Gauge-invariant, higher-dimensional non-renormalizable operators are then generated in the low-energy theory by integrating out massiveparticles. In particular the effective superpotential is expected to contain the following quarticterms:

Wn.r. ∋ (κ1)ijkl

Λ(QiQj)(QkLl) +

(κ2)ijkl

Λ(U c

i UcjD

ck)E

cl +

(κ3)ijk

Λ(QiQj)(QkHd)

+(κ4)ijk

Λ(QiHd)(U

cjE

ck) +

(κ5)ij

Λ(LiHu)(LjHu) +

(κ6)i

Λ(LiHu)(HdHu) , (2.71)

whereΛ can be viewed as parametrizing the scale of new physics (e.g.the string scale, thePlanck scale or the GUT scale) beyond the Supersymmetric Standard Model. Other non-renormalizable6B and 6L operators can be present in the Kahler potentialK, the function thatdefines the matter kinetic terms in a non-renormalizable supersymmetric theory (for example,the kinetic terms for complex scalar fieldsφi readLkin =

(∂2K/∂φi∂φj

)∂µφi∂

µφj) [87, 88]:

Kn.r. ∋ (κ7)ijk

Λ(QiQj)D

c†k +

(κ8)i

Λ(H†

uHd)Eci

+(κ9)ijk

Λ(QiL

†j)U

ck +

(κ10)ijk

Λ(U c

iDc†j )Ec

k , (2.72)

where we have kept only the trilinear terms, which correspond to dimension-5 operators in theLagrangian density.

In Eqs. (2.71) and (2.72), we assumed the particle content ofthe MSSM; other versionsof the Supersymmetric Standard Model may allow for additional terms. It is easy to see thatthe operators parametrized byκ1, κ2 andκ5, while compatible withR-parity, still violate thebaryon number and lepton number symmetries. The operatorLHuLHu generates Majoranamasses for the neutrinos; as long asΛ & 5 × 1013 GeV, its contribution is compatible with theexperimental constraints on neutrino masses, and the(κ5)ij are not required to be small. TheoperatorsQQQL andU cU cDcEc, on the other hand, induce proton decay and their coefficientsκ1 andκ2 are therefore constrained to be small, even if the cutoff scale Λ is taken to be aslarge as the Planck mass (i.e.Λ ∼ 1019 GeV). In particular,(κ1)112l ≤ O(10−7) for typicalsuperpartner masses in the TeV range. Other operators in Eqs. (2.71) and (2.72) induce protondecay in conjunction with the dimension-46Rp operators of Eq. (2.2), implying severe boundson the productsκ4λ

′′

, κ9λ′′

andκ10λ′′

. This provides an additional motivation for searching fordiscrete symmetries protecting proton decay from renormalizable operators, while allowing forsome of the6Rp terms in Eq. (2.2). Some of these symmetries will eventuallyturn out to bemore efficient thanR-parity in forbidding the dangerous6B and 6L nonrenormalizable operatorsin Eqs. (2.71) and (2.72).

We are interested in discrete symmetries of theR-parity conserving superpotential of Eq.(2.1) (i.e. symmetries compatible with the presence of theµ-term and of quark and leptonYukawa couplings) protecting proton decay from renormalizable operators. These symmetries,which are either commuting with supersymmetry orR-symmetries also acting on the supersym-metry generator, can be divided into three general classes [87]:

i) Generalized matter (R-)parities(GMP) : discrete (R-)symmetries protecting both baryonand lepton number from dimension-4 operators, i.e. enforcingλ = λ′ = λ′′ = 0;

ii) Generalized baryon (R-)parities (GBP) : discrete (R-)symmetries protecting baryonnumber from dimension-4 operators and allowing for lepton-number violations, i.e. enforcingλ′′ = 0;

iii) Generalized lepton (R-)parities(GLP) : discrete (R-) symmetries protecting lepton num-ber from dimension-4 operators and allowing for baryon-number violations, i.e. enforcingλ = λ′ = 0.

For simplicity, we restrict our discussion to flavour-blindsymmetries (the case of flavour-dependent symmetries has been illustrated in section 2.5),and we assume the particle contentof the MSSM. To start with, let us consider discreteZN symmetries commuting with supersym-metry, which act on the chiral superfields as (k = 0 · · ·N − 1):

Q → e2kπi q/N Q , U c → e2kπi u/N U c , Dc → e2kπi d/N Dc ,

L → e2kπi l/N L , Ec → e2kπi e/N Ec , Hu,d → e2kπi hu,d/N Hu,d , (2.73)

whereq, · · · , hd, hu denote here theZN charges of the MSSM superfields, defined moduloN . Restricting the scan toZ2 andZ3 symmetries, one finds the generalized parities listed inTable 2.1 [87].

TheZ2 GMP (ZM2 ) is actually equivalent, after a weak hypercharge rotation, to a symmetry

X acting on MSSM superfields asX(Qi, Uci , D

ci , Li, E

ci ) = −(Qi, U

ci , D

ci , Li, E

ci ),X(Hd, Hu) =

+(Hd, Hu). This symmetry is identical to the matter parity symmetry already considered in

Generalized parities q u d l e hd hu

ZM2 , ZM

3 0 -1 1 0 1 -1 1ZM

3 0 -1 1 1 0 -1 1

ZL2 , ZL

3 0 0 0 -1 1 0 0ZB

2 0 -1 1 -1 0 -1 1

ZB3 0 -1 1 -1 -1 -1 1

Table 2.1: Matter, lepton and baryon generalized parities in the MSSM.Superscript indicesdistinguish between matter (M), baryon (B) and lepton (L) generalized parities. The lettersq,u, d, · · · denote theZN charges referring to the action of the generalized paritiesas expressedin Eq. (2.73).

chapter 1, and is therefore equivalent toR-parity. Contrary to the other generalized paritieslisted in Table 2.1, theZ2 andZ3 GBP’s do not forbid the bilinear lepton number violating op-eratorsLiHu. Some of the symmetries listed in Table 2.1 are actually not better thanR-parity inforbidding the dangerous non-renormalizable dimension-5operators in Eqs. (2.71) and (2.72).Indeed, the first two GMP’s (including the usual matter parity) allow for such terms.

A similar analysis can be done for discreteR-symmetries, i.e. discrete versions of thecontinuousR-symmetries discussed in chapter 1. SinceR-symmetries do not commute withsupersymmetry, the transformations (2.73) must be supplemented with a corresponding actionon Grassmann variablesθ, which defines also the total charge of the superpotential (see Eq.(1.12)),

W (x, θ) → e−2kπi/N W (x, ekπi/Nθ) . (2.74)

Scanning over all flavour-blindZ2 andZ3 discreteR-symmetries, one finds the generalizedR-parities [87] listed in Table 2.2. Like the generalized parities of Table 2.1, the generalizedR-parities do not necessarily forbid all dangerous dimension-5 operators. Actually only onegeneralized matterR-parity satisfies the requirement of forbidding all operators in Eqs. (2.71)and (2.72) butLHuLHu, the secondZM

3 symmetry listed in Table 2.2. The baryon and leptonR-parities listed in Table 2.2 are at this stage all safe, since they forbid at least one dangerouscoupling appearing in experimentally constrained products of couplings.

One may wonder whether the discrete symmetries considered above are likely to be exact atlow energy. Indeed, any global symmetry, continuous or discrete, might be broken by quantumgravity effects. Even if this happens only at the Planck scale, as noticed above, the strong con-straints coming from proton decay rule out the corresponding symmetry [89]. It is however wellknown that a discrete symmetry originating from the spontaneous breaking of some continuousgauge symmetry is protected against quantum-gravity violations. The original gauge quantumnumbers are of course very constrained by cancellation of triangle gauge anomalies, as well asby mixed gauge-gravitational anomalies. As noticed in [87,90], there are remnants of theseconditions, called “discrete gauge anomaly cancellation conditions” in the low-energy theoryafter spontaneous symmetry breaking. Discrete symmetriesthat respect these conditions aretherefore safe with respect to anomalies. Among the generalized parities of Table 2.1, only twoare discrete anomaly free in the MSSM, namely the standardZ2 matter parity (ZM

2 ), actuallyequivalent toR-parity, and theZ3 generalized baryon parity (ZB

3 ). If in addition we requirethe absence of the dimension-5 operators of Eqs. (2.71) and (2.72), we are left with theZ3

generalized baryon parity only, a remarkable degree of uniqueness.

Let us finally note that the standard matter parityX could also originate from a spontaneoulybroken anomalousU(1)X gauge symmetry under which all matter superfields have charge one

GeneralizedR-parities q u d l e hd hu

ZM2 0 0 1 1 0 0 1

ZM3 0 0 -1 -1 0 0 -1

ZM3 0 1 1 0 1 1 1

ZM3 0 1 1 -1 -1 1 1

ZL2 0 -1 0 0 0 -1 0

ZL3 0 -1 0 1 -1 -1 0

ZL3 0 -1 0 0 0 -1 0

ZB2 0 0 1 0 1 0 1

ZB3 0 1 1 1 0 1 1

ZB3 0 0 -1 0 -1 0 -1

Table 2.2: Matter, lepton and baryon generalizedR-parities in the MSSM. Superscript indicesdistinguish between matter (M), baryon (B) and lepton (L) generalizedR-parities. The lettersq, u, d, · · · denote theZN charges referring to the action of the generalizedR-parities asexpressed in Eq. (2.73).

and the two Higgs superfields have charge zero (however the Yukawa couplings are not invariantunder this symmetry, and should therefore be generated by a Froggatt-Nielsen mechanism, asin section 2.5). Indeed, compactifications of the heteroticstring often contain a seeminglyanomalous abelian gauge symmetry, whose mixed gauge anomalies are compensated for bythe Green-Schwarz mechanism [91]. For this mechanism to work, the following conditionmust be satisfied:A′g′ 2 = Ag2 = A3 g

23, whereA′, A, A3 are the coefficients of the mixed

gauge anomalies[U(1)Y ]2U(1)X , [SU(2)L]2U(1)X and[SU(3)C ]2U(1)X , andg′, g, g3 are thecoupling constants of the gauge groupsU(1)Y , SU(2)L andSU(3)C . This condition can beunderstood as a relation determining the relative normalization of the generators associated withdifferent gauge groups in terms of the anomaly coefficients.Now the above charge assigmentyieldsA3 = A = −12 andA′ = −20, implying g′ 2/g2 = A/A′ = 3/5 (or simplyg1 = g = g3

with g1 =√

5/3 g′) and therefore [92]

sin2 θW =g′ 2

g′ 2 + g2=

3

8, (2.75)

a relation known to be successful at the grand unification scale. We conclude that a gaugecontinuous version of the standard matter parity discussedin chapter 1, which is equivalent totheZ2 matter parity of Table 2.1 and has the same effect asR-parity, may be a good stringsymmetry. This provides a further motivation forR-parity conservation. Still the argumentsdeveloped earlier in this section as well as in section 2.5 motivate possible violations of theR-parity symmetry, with a hierarchy of6Rp couplings that could make them well compatible withexperimental data.

In this chapter, we studied from a theoretical point of view the possible violations of theR-parity symmetry introduced in chapter 1 in connection withbaryon and lepton-number con-servation. We first gave the most general form of the6Rp terms that may be present in thesuperpotential and in the soft supersymmetry-breaking scalar potential, and addressed issuesassociated with the choice of a basis for the Higgs and leptonsuperfieldsHd andLi, in thepresence of bilinearR-parity violation.

Then we classified the patterns ofR-parity breaking that are consistent at the quantum level,according to which types of6Rp terms are present in the Lagrangian density. We also discussedthree scenarios ofR-parity violation often considered in the literature, namely explicit 6Rp bytrilinear terms, explicit6Rp by bilinear terms and spontaneousR-parity breaking. We then dis-cussed the effects of the Higgs-lepton mixing associated with the presence of bilinear6Rp terms,which leads to a potentially very large (orderMZ) neutrino mass and to specific6Rp signaturesat colliders. To suppress the neutrino mass to aphenomenologically acceptable value, either astrong fine-tuning of bilinear6Rp parameters or an “alignment” mechanism is required. We alsopresented ascenario of spontaneousR-parity breaking involving a singletMajoron, thus avoid-ing conflict with the measured invisible decay width of theZ boson.

Finally, we discussed possible microscopic origins for terms violating lepton and baryonnumbers. We focussed essentially on abelian flavour symmetries, Grand Unified gauge symme-tries and discrete symmetries generalizingR-parity. The latter appear for example in string the-ories. The phenomenological constraints on the lepton- andbaryon-number violations, whichare crucial tests of any extension of the Standard Model, become therefore a window into thestructure of the underlying high energy theory.

Chapter 3

RENORMALIZATION GROUP SCALEEVOLUTION OF 6Rp COUPLINGS

One of the most interesting indications for supersymmetry is the unification of gauge couplingsat a high scale, obtained by renormalization group evolution of the measurements done by LEPat the “low” scale ofMZ [93]. The renormalization group allows one to evolve couplingsand mass parameters between two energy scales, thus providing a way to test at the availableenergies physical assumptions postulated at a higher scale, or vice-versa to translate availableexperimental data into quantities at high energy.

In this chapter, we shall focus on the renormalization groupevolution of constraints forthe 6Rp interactions and cover in particular those associated withthe perturbative unitarity orthe so-called “triviality bounds”, the infrared fixed points and the tests of grand unificationschemes. The role of supersymmetry-breaking effects willalso be discussed. In the limit ofunbroken supersymmetry the bilinear6Rp interactions can be recast only in terms of theµ-term by a suitable redefinition of the four superfieldsLα ≡(Hd, L1, L2, L3), thereforeR-parity violation can be parametrized by trilinear couplings only. In the presence of soft terms,as explained in chapter 2, the superpotential and the scalarpotential contain two independentsources of bilinearR-parity violation which, in general, cannot be simultaneously rotated awayby field redefinitions. In turn, the Higgs-lepton mixing due to bilinearR-parity violation leavesan arbitrariness in the choice of theLα basis. It is therefore crucial, when dealing with numericalresults, to specify the choice of basis. A detailed discussion is given in section 2.3.1. In thefollowing we shall keep the discussion general, and, when referring to numerical results, thereader will be guided to cited papers for the choice of basis and detailed assumptions.

Our discussion of renormalization group studies in presence of 6Rp will mostly concern theso-called supergravity framework. In this framework, the soft supersymmetry-breaking termsare supposed to be generated through gravity in the limit in which the gravitational couplingconstant (κ =

√8πGN ) is taken to be small. Since this is closely tied with a grand unified

theory approach, one is led to consider the scale evolution of parameters up to large energyscales of the order of the grand unification scaleMX (or the compactified string theory scaleMC , or the Planck scaleMP ). Conversely to this bottom-up type scale evolution, one may alsoenvisage a top-down type scale evolution, by assigning boundary condition values for the (fewerin number) unified parameters at the large scale and using therenormalization group equations(RGEs) to evolve the entire set of Minimal Supersymmetric Standard Model parameters valuesdown to the electroweak symmetry breaking mass scale.

3.1 Renormalization Group Equations

As remarked in chapter 2, in the absence ofR-parity and lepton-number conservation laws,there is noa priori distinction between theHd Higgs andLi lepton superfields, as they have thesame gauge quantum numbers. Only after electroweak symmetry breaking, the mass eigenstatebasis defines which combinations of theHd andLi component fields correspond to the physicalleptons and sleptons. The magnitude of6Rp couplings depends in particular on which directionin the space of weak doublet superfields with weak hypercharge−1 ultimately corresponds tothe Higgs. One possible strategy consists in constructing combinations of coupling constantsthat are invariant under these basis redefinitions [26, 27, 28], and parametrize the6Rp content ofthe Lagrangian density in a way similar to the Jarlskog invariants forCP violations [94].

In order to express the superpotential and the renormalization group equations (RGEs) in acompact way, we rewrite Eqs. (2.1) and (2.2) of section 2.1.1in the form of Eq. (2.27):

WRp +WR/p = µαHuLα +1

2λe

αβk LαLβEck + λd

αjk LαQjDck + λ′′ and λu

ij terms, (3.1)

whereα ≡ (0, i) = (0, 1, 2, 3), β ≡ (0, j) = (0, 1, 2, 3). Here we use the following notation:

Hd ≡ L0 , µ ≡ µ0 , (3.2)

λe0jk ≡ λe

jk , λd0jk ≡ λd

jk . (3.3)

This allows us to write theSU(4) transformation of Eq. (2.17) and following equations as:

Lα → Uαβ Lβ , (3.4)

µα → U⋆αβ µβ , (3.5)

λeαβk → U⋆

αγ U⋆βδ λ

eγδk , (3.6)

λdαjk → U⋆

αβ λdβjk , (3.7)

whereU is theSU(4) matrix with entriesUαβ associated with the basis rotation. It is clearfrom the above equations that the lepton-number-violatingcouplings are basis-dependent. Fora detailed discussion see chapter 2.

In the following, explicit expressions for the6Rp RGEs up to the two loop order will bewritten using the above notations. To facilitate a comparison with the existing literature we givein Table 3.1 the correspondence between our notations and the ones of [95].

3.1.1 Evolution of the Bilinearµ Terms

Following the general equations given in [96] the RGEs for the bilinearµ terms including all6Rp effects can be written as:

d

dtµα = µα Γuu + µβ Γαβ, (3.8)

wheret = log q2 andΓ are the anomalous dimensions. The indexα is defined as in the previoussection (α ≡ (0, i)). The notationΓuu is a shorthand for the anomalous dimensionΓHu Hu for

Our Notation Notation of [95]

Hd H1

Hu H2

λeijk (ΛEk)ij

λejk ≡ λe

0jk −(YE)jk

λdijk (ΛDk)ij

λdjk ≡ λd

0jk −(YD)jk

λ′′ijk (ΛUk)ij

λujk (YU)jk

Table 3.1: Correspondence among our notation and the one of [95].

theHu superfield. In a similar wayΓ00 stands forΓHd HdandΓij meansΓLi Lj

. ExpandingEq. (3.8) into its components one obtains

d

dtµ0 = µ0 Γuu + µ0 Γ00 + µi Γ0i , (3.9)

d

dtµi = µi Γuu + µ0 Γi0 + µj Γij . (3.10)

This set of equations implies that even if we start with allµi = 0, non-zeroµi will be generatedthrough the RGEs via a non-zeroµ0 and vice-versa.

The bilinear terms do not appear in the equations for the evolution of the Yukawa couplingsor the gauge couplings. Thus, they do not directly affect theunification of the latter. Theanomalous dimensionsΓ transform as follows under theSU(4) rotation of the fields:

Γuu → Γuu , (3.11)

Γαβ → UβγU⋆ατΓτγ . (3.12)

The anomalous dimensions are given by

Γij =1

16π2γ

(1)ij +

1

(16π2)2γ

(2)ij + . . . (3.13)

whereγ(1), γ(2), . . . are 1–loop, 2–loop,. . . contributions:

γ(1)ij =

1

2YimnY

⋆jmn − 2 δij

∑

a

g2aCa(i) , (3.14)

γ(2)ij = −1

2YimnY

⋆npqYpqrY

⋆mrj + YimnY

⋆jmn

∑

a

g2a [2Ca(p) − Ca(i)]

+ 2 δij∑

a

g2a

[g2

aCa(i)Sa(R) + 2∑

b

g2bCa(i)Cb(i) − 3g2

aCa(i)C(Ga)

]. (3.15)

Yijk is a generic Yukawa coupling (it stands forλe, λd or λ′′), Ca(f) is the quadratic Casimir ofthe representationf of the gauge groupGa. C(G) is an invariant of the adjoint representationof the gauge groupG andSa(R) is the second invariant of the representationR in the gaugegroupGa. Explicitly if tA are the representation matrices of a groupG one has:

(tAtA)ij = C(R)δij , (3.16)

for SU(3) tripletsQ andSU(2) doubletsL:

CSU(3)(Q) =4

3, CSU(2)(L) =

3

4, (3.17)

and for theU(1) weak hypercharge embedded inSU(5):

C(φ) =3

5y2(φ), (3.18)

wherey(φ) is the weak hypercharge of the fieldφ. The factor3/5 comes from theSU(5) grandunified normalisation of the weak hypercharge generator. For the adjoint representation:

C(G) δAB = fACDfBCD, (3.19)

wherefABC are the structure constants. For the groups under study:

C(SU(3)C) = 3, C(SU(2)L) = 2, C(U(1)Y ) = 0, C(SU(N)) = N . (3.20)

The Dynkin index is defined by

TrR(tAtB) = S(R) δAB. (3.21)

For the fundamental representationsf we have

SU(3), SU(2) → S(f) =1

2, (3.22)

U(1)Y → S(f) =3

5y2(f) . (3.23)

3.1.2 Evolution of the Trilinear 6Rp Yukawa Couplings

As already stated in chapter 2, scenarios in whichR-parity is broken only by trilinear interactionterms are in general not consistent since bilinear6Rp terms are generated by quantum correctionsand cannot be rotated away through aSU(4) redefinition of the superfields, due to the presenceof soft supersymmetry-breaking terms. One has therefore toconsider all the couplings whenstudying the renormalization group evolution. The generalRGEs for the Yukawa couplingsYijk

(whereY stands forλeαβk, orλd

αjk, λuik or λ′′ijk of Eq. (3.1)) are given by [96]:

d

dtYijk = Yijl Γlk + (k ↔ j) + (k ↔ i) . (3.24)

For the Yukawa couplingsλeαβk andλd

αjk this gives

d

dtλe

αβk = λeαβl ΓEl Ek

+ λeαδk Γδβ + λe

γβk Γγα , (3.25)

d

dtλd

αjk = λdαjl ΓDl Dk

+ λdαlk ΓQl Qj

+ λdγjk Γγα , (3.26)

while for the Yukawa couplingsλ′′ijk andλujk we have

d

dtλu

ij = λuik ΓUj Uk

+ λuij Γuu + λu

kj ΓQi Qk, (3.27)

d

dtλ′′ijk = λ′′ilk ΓDj Dl

+ λ′′ljk ΓUi Ul+ λ′′ijl ΓDk Dl

. (3.28)

Of course the two-loop RGEs for the Yukawa couplings preserve λ′′ijk = 0 (for all i, j, k)at all scales if they are zero at some scale (i.e. baryon parity is conserved). The same is trueif lepton parity is imposed at some scale forλe

ijk andλdijk. If however one imposes only one

coupling to be non-zero at some scale, this remains in general not true at all scales. Take forexample onlyλd

111 6= 0 at some scale. Then through the CKM mixing the other termsλd1ij will

be generated by the RGEs.

3.1.3 Evolution of the Gauge Couplings

The RGEs for the Standard Model gauge couplingsg1 [for U(1)Y , with g1 =√

5/3 g′ using forexample theSU(5) GUT normalisation],g2 [for SU(2)L] andg3 [for SU(3)c can be written:

d

dtga =

g3a

16π2B(1)

a +g3

a

(16π2)2

[ 3∑

b=1

B(2)ab g

2b − Ce

a Tr (λe†jkλ

ejk) − Cd

a Tr (λd†jkλ

djk) (3.29)

−Cua Tr (λu†

jkλujk) + Ae

a

3∑

k=1

Tr (λe†ijkλ

eijk) + Ad

a

3∑

k=1

Tr (λd†ijkλ

dijk) + Au

a

3∑

k=1

Tr (λ′′†ijkλ′′ijk)].

In the evolution equation for the gauge couplings, the equations are coupled only starting withthe two-loop term, while up to one loop each coupling has an independent evolution.

The coefficientsBa, Bab, andCxa are calculated in [97]:

B(1)a = (

33

5, 1,−3), (3.30)

B(2)ab =

199/25 27/5 88/59/5 25 2411/5 9 14

, (3.31)

Cu,d,ea =

26/5 14/5 18/56 6 24 4 0

, (3.32)

where the indexu, d, e refers to the lines of the matrix. The6Rp contributions to the running ofthe gauge couplings appear only at two-loops. They are givenin [95]:

Au,d,ea =

12/5 14/5 9/50 6 13 4 0

. (3.33)

In the Minimal Supersymmetric Standard Model there is a relation between the running of thetop-quark mass and the ratio of VEVs of the two Higgs doublets, tan β. Given the measureof the top-quark mass there is a restriction for the allowedtanβ range. In the presence of6Rp,however, there is no such a restrictive prediction. Furthermore, allowing the bilinear lepton-number-violating terms [106, 95, 40, 98, 42], bottom-tau Yukawa unification can occur for anyvalue oftanβ.

3.2 Perturbative Unitarity Constraints

It is possible to derive upper bounds on the6Rp coupling constants, without the need of specifyingfurther input boundary conditions, simply by imposing the requirement that the ultraviolet scaleevolution remains perturbative up to the large unification scale,

Y 2ijk(MX)

(4π)2< 1 , (3.34)

whereYijk is a generic trilinear6Rp coupling constant. In more general terms, the unitaritylimits concern the upper bound constraints on the coupling constants imposed by the conditionof a scale evolution between the electroweak and the unification scales, free of divergencesor Landau poles for the entire set of coupling constants. Theprincipal inputs here are theStandard Model gauge coupling constants, the superpartnerspectrum together with the ratioof Higgs bosons VEVs parameter,tanβ = vu/vd, and the quark and lepton mass spectra, asdescribed by the Yukawa coupling constants,λu,d,e

ij . Since the third generation Yukawa couplingconstants,λt = λu

33, λb = λd33, λτ = λe

33, are predominant, the influential6Rp coupling constantsare expected to be those containing the maximal number of third generation indices, namelyλe

233, λd333, λ

′′313, λ

′′323.

The first study developing perturbative unitarity bounds isdue to Brahmachari and Roy [99].The derived bounds for the baryon-number-violating interactions,[λ′′313, λ

′′323] < 1.12, turn out

to be very weakly dependent on the input value fortan β. These bounds increase smoothly withthe input value ofmt, diverging atmt ≈ 185 GeV. Bounds for the other configurations of thegeneration indices ofλ′′ijk have been obtained on the same basis by Goity and Sher [100],λ′′mjk <1.25, [m = 1, 2]. Allanach et al. [95] carry out a systematic analysis of the renormalization flowequations, up to the two-loop order, for the lepton and baryon-number-violating interactionsλe

ijk, λdijk andλ′′ijk. The resulting coupling constant bounds readλe

323(mt) < 0.93, λd333(mt) <

1.06, λ′′323(mt) < 1.07, at tanβ = 5. Choosing a highertan β lowers these bounds slightly.

3.3 Quasi-fixed points analysis for6Rp couplings

The RGEs describing the evolution of the Yukawa couplings down from a large scaleMX

may have fixed points which give information on the couplings. The existence of infraredfixed points (IRFP) for the third generation Yukawa couplingconstants and the relevant6Rp

coupling constants, is signalled by vanishing solutions for the beta-functions describing thescale evolution for the ratios of the above Yukawa coupling constants to the gauge interactioncoupling constants. In principle, one seeks fixed point solutions [101] characterized by the exactabsence in the infrared regime of a renormalization group flow for ratios such as, for instance,λ2

t/g23 or λe

3232/g2

3. In practice however, this fixed point regime may be inaccessible since itwould set in at a scale much lower than the electroweak scale,making it irrelevant. In that casethe values of the Yukawa couplings are determined by quasi-fixed points (QFP) [102] describingthe actual asymptotic behaviour of the couplings. In such a case, the values at the weak scaleare essentially independent of their values at the large scale, provided the initial values are large.For an analytical study see [103].

As an example let us consider a simplified renormalization group equation for the top-quarkYukawa couplingλt at one loop in the Standard Model:

16π2dλt

dt=λt

2

(9λ2

t − 16g23

), (3.35)

where t = log q2 and we have neglected the contributions from the lighter quarks and theelectroweak contributions. By forming the difference of the previous equation with the one forthe evolution of the QCD strong interaction coupling:

16π2dg3

dt= g3

3

(2

3Nf − 11

)(3.36)

whereNf is the number of flavours, one obtains:

16π2 d

dtlog(λt/g3) =

9

2λ2

t + g23

(3 − 2

3Nf

). (3.37)

When the value

λ2t =

2

9g23

(2

3Nf − 3

)(3.38)

is reached, Eq. (3.37) has zero on the right-hand side which implies a constant ratio of the twocouplings for subsequent decreasing values of the scalet. This is the Pendleton-Ross fixedpoint. However this behaviour would only set in at a very low scale, of the order of 1 GeV,while the region of interest is the one in which the scale is aroundmt. The reason why the fixedpoint is important only at a very low scale is that in Eq. (3.35) the strong coupling constantg3

becomes large only below 1 GeV. In the intermediate regionλt evolves with the lowering of themass scale until theg3 coupling becomes of the same order, i.e. :

9

2λ2

t ≃ 8g23 . (3.39)

In this intermediate region the right-hand side of equation(3.35) is close to zero, which inturn implies thatλt must remain relatively constant. The previous argument canbe made moreprecise by a detailed calculation or a graph ofλt(q) versusλt(MX) whereMX is the high scaleandq is in the intermediate range. In both cases the asymptotic behaviour is ascribed to thecondition (3.39) which is termed a quasi-fixed point.

Before considering the effect of6Rp, let us consider the quasi-fixed point regime for theMinimal Supersymmetric Standard Model. In this case the renormalization group flow pointstowards the value of the top quark coupling constantλt(mt) ≃ 1.1, which establishes a corre-lation between the top mass andtan β, described by the relation

mt(pole) =v sin β√

2λt(pole) . (3.40)

Substitution of the physical top massmt as an input value singles out a discrete range fortan β.

When the third generation6Rp interactions are switched on, individually or collectively, so-lutions of the quasi-fixed point type continue to exist. These fixed point values of the couplingconstants provide theoretical bounds under the assumptionthat the theory remains perturbative.By requiring a lower bound on the top mass, say,mt > 150 GeV, they would lead to excludeddomains in the parameter space ofλt and the6Rp Yukawa coupling constants [99]. Looking for asimultaneous quasi-fixed point inλt and/orλb and in the6Rp coupling constants one at a time, oneobtains [104, 105]:λt ≃ 0.94, λ′′323 ≃ 1.18, λd

333 ≃ 1.07, andλt ≃ 1.16, λe233 ≃ 0.64, at small

tanβ andλt ≃ 0.92, λb ≃ 0.92, λ′′323 ≃ 1.08, at largetan β. As the6Rp couplingsλe, λd, λ′′ aresuccessively switched on, the regular top Yukawa coupling varies as,λt ≃ 1.06 → 1.06 → 0.99,respectively in the smalltanβ regime. In the largetan β ≃ mt/mb ≈ 35 regime, the solution

for the quasi-fixed point predictions are modified as,λt ≃ 1.00 → 1.01 → 0.87, respectivelyandλb ≃ 0.92 → 0.78 → 0.85, respectively, the corresponding fixed point values for the6Rp

coupling constants beingλd333 ≃ 0.71, λ′′323 ≃ 0.92 [104, 105]. Further discussions of the fixed

point physics in connection with6Rp can be found in [106].

The stability condition with respect to small variations ofthe parameters for a renormaliza-tion group fixed point requires that the matrix of derivatives of the beta-functions with respectto the coupling constants has all its eigenvalues of the samefixed (positive in our conventions)sign. Discussion of the stability issue motivated by the supersymmetric models can be foundin [107, 108, 109]. The above stability condition is actually never satisfied in the MinimalSupersymmetric Standard Model even for the trivial fixed point at which the6Rp coupling con-stants tend to zero. Once one includes the6Rp interactions, a stable infrared quasi-fixed pointdoes exist, but only if one considers simultaneously the third generation regular Yukawa cou-pling constants,λt, λb, along withλ′′332 [110]. In particular there is no simultaneousB- andL-violating infrared fixed point. Note that the validity of these results is based on the extent ofwhat variation of the parameters is considered “small”.

The quasi-fixed points are reached for large initial values of the couplings at the GUT scale,therefore they reflect the assumption of perturbative unitarity of the corresponding couplings.Under this assumption, the quasi-fixed points provide upperbounds on the relevant Yukawacouplings, especially theB-violating couplingλ′′332 [111].

In the Minimal Supersymmetric Standard Model the coupling constantsg1, g2 andg3 unifyat a certain scaleMGUT thanks toR-parity conservation. The scale evolution of the gaugecouplings leads to a successful unification with the values of the unified coupling constantand the unification scale given by,αX(MX) ≃ 1/24.5 = 0.041, [gX(MX) ≃ 0.72], MX ≃2.3 × 1016 GeV. Besides gauge coupling unification, GUT theories reduce the number of freeparameters in the Yukawa sector.6Rp affects this picture: the feed-back effects of the6Rp trilinearinteractions on the regular Yukawa interactions may have significant implications for the con-straints set by grand unification on the Minimal Supersymmetric Standard Model parameters.

In the context of Grand Unified Theories one could consider a unification of the6Rp parame-ters. However, if the6Rp interactions arise from aSU(5) invariant term there would be a relationbetweenB- andL-violating terms and this in turn would imply non-zero contributions to theproton decay, either directly or at one-loop level through flavour mixing, therefore limiting the6Rp couplings to very small values as already discussed in section 2. The situation for the widelyused hypothesis of Yukawa coupling unificationλb = λτ is analogous, even if there is no di-rect link between the two sorts of coupling unification. Analyses avoiding the assumption ofλb = λτ unification have been performed and the quasi-fixed point values for the6Rp couplingconstants are found as [104, 105],λe

233 = 0.90, λd333 = 1.01, λ′′323 = 1.02, for tanβ < 30.

As a simplifying assumption one can take a hierarchy similarto the one between StandardModel Yukawa couplings, and therefore consider only one coupling at a time. Solving the two-loop RGEs, Allanach et al. [95] find that by turning on any one of the three relevant6Rp thirdgeneration related coupling constants, from zero to their maximally allowed values, the unifi-cation coupling constant,αX , is insignificantly affected by less than5%, while the unificationscale,MX , can be reduced by up to20%. Note also that for large values of theR-parity vi-olating coupling, the value ofαs(MZ) predicted from unification can be reduced by 5% withrespect to theR-parity conserving case.

3.4 Supersymmetry Breaking

The renormalization group studies in the presence of the soft supersymmetry-breaking becomefar less tractable. A proper treatment ofR-parity violation must also include6Rp soft terms,therefore a large number of additional parameters arise which all have a mutual influence onone another. Within the MSSM, these additional terms are given in Eq. (2.16) and introduce 51new 6Rp parameters: 3Bi associated with the bilinear superpotential terms,45 6Rp A-terms withthe same antisymmetry properties as the corresponding trilinear superpotential couplings, and3 6Rp soft mass termsm2

di mixing the down-type Higgs boson and slepton fields.

The inclusion ofR-parity violation in the superpotential allows the generation of lepton-Higgs mixing which leads to sneutrino VEVs and hence neutrino masses as discussed in chapter2. The indirect generation of sneutrino VEVs through the running of the RGEs for the soft termscan lead to large effects. They induce finite sneutrino VEVsvi, via the renormalization groupevolution of the6Rp trilinear interactions from the grand unification scale to the electroweakscale, as discussed in [30]. A renormalization group analysis, including the soft supersymmetry-breaking parameters, is developed within a supergravity framework [88], where the6Rp trilinearinteractions are specified at the grand unification scale,λe

ijk(MX), and one performs at eachenergy scale the requisite field transformation aimed at removing away the bilinear interactionsin the superpotential,µi(q) = 0. Since finitevi contribute, via the mixing with neutralinos,to the neutrino Majorana masses (see chapter 5), the condition that the experimental limitson these masses are satisfied leads to the following qualitative bounds at the unification scale,λe

i33 < (10−2 − 10−3) andλdi33 < (10−2 − 10−3). The 6Rp interactions also initiate, through the

renormalization group evolution, indirect contributionsto flavour changing soft mass and6Rp pa-rameters. An application to the prototype processµ→ e+γ indicates that these indirect effectsturn out to dominate over the direct effects associated withthe explicit contributions from theone-loop diagrams discussed in chapter 6. However, the situation cannot be described in termsof quantitative predictions, owing to the large number of free parameters and the occurrence ofstrong cancellations amongst contributions from different sources.

In another study [112] the role of the6Rp interactions in driving certain superpartner masssquared to negative values is examined. The sneutrinos are most sensitive to this vacuum stabil-ity constraint because of the weaker experimental bounds ontheir masses. The attractive contri-bution from the6Rp interactions readsδm2

ν ≃ −|λdijk(MX)|2(13m2

0 +49M212

−1.5M 12A−12A2),

wherem0, M 12

andA stand for the unification values of the soft scalar masses, the gauginomasses and theA-terms respectively, assumed to be universal. Invoking theexperimental con-straint onmν from LEP, one may derive bounds on the6Rp coupling constants at the unificationscale, valid for all flavour configurations, such asλd

ijk(MX) < 0.15, which translate into boundsat the electroweak scale, of the formλd

ijk ≈ λdijk(MZ) < 0.3. The indirect effects of the6Rp inter-

actions on the flavour changing parameters are also examinedfor the processb→ s+ γ. Thesecontributions appear to dominate over the direct perturbative contributions from the one-loopdiagrams. However, because of the large number of relevant parameters and the complicateddependence on the observables, one can again only infer conclusions of a qualitative nature.

In conclusion, the renormalization group evolution is a powerful tool to link theoreticalhypotheses and experimental data, by allowing the comparison of quantities such as couplingconstants at different scales. It is however difficult to draw general conclusions on the boundsthat one can obtain as they may strongly depend on assumptions, in a range of energy that isstill to be explored. Nonetheless if one allows to include the general picture of grand unificationand supersymmetry a number of interesting results can be obtained.

Chapter 4

COSMOLOGY AND ASTROPHYSICS

One of the first merit of a conservedR-parity is to provide, naturally, a stable lightest super-symmetric particle (LSP). IfR-parity is absolutely conserved, the LSP is absolutely stable –none of its possible decay channels being kinematically allowed – and therefore it constitutes apossible dark matter candidate.

A brokenR-parity supersymmetry could have important implications on this issue of thedark matter of the universe. The LSP can then decay through6Rp interactions into StandardModel particles only. Such an unstable LSP can still remain,however, a possible dark mattercandidate, provided its lifetime is sufficiently long; the corresponding6Rp couplings are thenrequired to be extremely small. On the other hand a short-lived LSP, irrelevant to the dark matterproblem, is required to decay sufficiently quickly so as not to affect the successful predictionsof Big Bang nucleosynthesis.

A second important issue concerns the cosmological baryon asymmetry, i.e. the fact thatthere is no significant amount of antibaryons in the universe. Understanding the origin of theobserved baryon-to-photon rationB/nγ = (6.1+0.3

−0.2) × 10−10 [46], or in other terms of thecosmological baryon asymmetryηB ≡ (nB − nB)/nγ , raises the question of how and whenthis baryon-antibaryon asymmetry was generated, and what the protection of this asymme-try against subsequent dilution by baryon-number-violating interactions over the history of theuniverse requires. Different solutions for the creation ofthe cosmological baryon asymmetryhave been proposed in the literature, either directly throughB-violating interactions, or indi-rectly throughL-violating interactions (the resulting lepton number asymmetry being turnedinto a baryon asymmetry throughB − L conserving butB + L violating processes known assphalerons). This also requires that the corresponding interactions should violate, in addition tobaryon and/or lepton number, theC andCP symmetries between particles and antiparticles.

Supersymmetric theories with brokenR-parity have the interesting feature of providing thebaryon and/or lepton-number non-conservation needed for baryogenesis. However, while these6Rp interactions may generate a baryon or lepton asymmetry all by themselves, in reverse, theymight also dilute a pre-existing baryon asymmetry.

4.1 Constraints from the lifetime of the Lightest Supersym-metric Particle

4.1.1 Decays of the Lightest Supersymmetric Particle

In supersymmetric extensions of the Standard Model with unbrokenR-parity, the LSP plays afundamental role as the sole supersymmetric relic from theBig Bang, and may then provide thenon-baryonic component of the dark matter of the universe.

In principle the LSP could be any supersymmetric particle, such as the lightest neutralinoor chargino, a sneutrino, a charged slepton, a squark or a gluino. There are however strongarguments in favour of an electrically neutral and uncoloured (stable) LSP [113]. Stable, elec-trically charged and uncoloured particles would combine with electrons (if they have charge+1) or with protons or nuclei (if they have charge−1) to form superheavy isotopes of thehydrogen or of other elements. Stable coloured particles would first bind into new hadrons(such as(tud)+ or (tdd)0 in the case of a stop LSP), which would then combine with elec-trons (in the case of a stable, charge+1 heavy hadron) or with nuclei to form superheavyisotopes of the hydrogen or of other elements. The relic number densities of such massivestable particles have been evaluated to benX/nB ≃ 10−6 (mX/1 GeV) for an electricallycharged, uncoloured particle [114] such as a charged slepton LSP, andnX/nB ≃ 10−10, inde-pendently of the hadron mass, for a coloured particle [114, 115], such as a squark or a gluinoLSP. However, terrestrial experiments searching for anomalously heavy protons or superheavyisotopes have placed stringent upper limits on the relic abundances of electrically charged orcoloured stable particles (for a review see Ref. [116]). Forexample “heavy proton” experimen-tal searches yield the limitnX/nB < 10−21 formX < 350 GeV [117]; heavy isotopes searches,nX/nB < (2 × 10−16 − 7 × 10−9), depending on the element, for102 GeV< mX < 104 GeV[118]; and searches for superheavy isotopes of hydrogen in water, nX/nB < 10−28 [119],3× 10−20 [118] and6× 10−15 [120] in the mass ranges(10− 103) GeV,(102 − 104) GeV and(104 − 108) GeV, respectively. The comparison of these negative experimental results with theabove predicted relic abundances almost certainly rules out charged or coloured superparticlesas suitable (stable) LSP candidates.

Among the possible electrically neutral and uncoloured LSPs, the lightest neutralinoχ01 ap-

pears to be the best candidate for the non-baryonic dark matter of the universe. The gravitinoremains a possible dark matter candidate, but it generally suffers from an abundance excessproblem, while the possibility of a sneutrino LSP has been excluded, in the Minimal Supersym-metric Standard Model, by direct dark matter searches in underground experiments [121, 122].The requirement that the relic density of the lightest neutralino falls within the range allowedby observations,ΩCDM = 0.23±0.04 [46], whereΩCDM ≡ ρCDM/ρc is the ratio of the presentcold dark matter (CDM) energy density to the critical energydensity, puts strong constraints onthe parameters of the Supersymmetric Standard Model. But the fact that satisfactory values ofthe relic abundance can be obtained constitutes one of the important motivations forR-parityconservation in supersymmetric extensions of the StandardModel1.

The above state of affairs gets drastically modified in the case of a brokenR-parity. Themost important effect of6Rp interactions is the resulting instability of the LSP. An unstable LSP

1If the strongCP problem is solved by the Peccei-Quinn mechanism [123], the supersymmetric partner of theaxion, the axino, could also be a viable dark matter candidate (see e.g. Ref. [124], assuming primordial axinos tohave been diluted by inflation).

can still be a dark matter component of the universe providedit is sufficiently long-lived, so asto retain most of its primordial abundance until the presenttime – but this requires extremelysmall values of the6Rp couplings. A LSP with lifetime shorter than a fraction of theage of theuniverse, on the other hand, would now have disappeared almost completely and can no longerplay a role as a dark matter component of the universe. In this case however, the constraintsassociated with experimental searches for anormalously heavy protons or superheavy isotopesno longer apply, and the LSP can be any superpartner – not necessarily an electrically neutraland uncoloured particle. We shall though restrict ourselves to the case of a neutralino LSP inthe following.

Depending on the lifetimeτ 0χ of the LSP (i.e. depending on the strength of the6Rp couplings

responsible for its decay), different types of cosmological constraints apply. The decays ofa long-lived LSP, with a lifetime comparable to, or slightlylarger than, the present aget0 ofthe universe,τ 0

χ & t0, can produce an excess of particles such as antiprotons or positrons inour galaxy at a level incompatible with observations. To avoid this problem, one must requireτ 0χ ≫ t0, i.e. extremely small values of the trilinear6Rp couplings, at the level ofO(10−20)

or below. These very strong constraints do not apply, of course, when the LSP lifetime isshorter than the age of the universe. In this case, the LSP must decay sufficiently quicklyso that its late decays do not modify the light element abundances successfully predicted byBig-Bang nucleosynthesis. This constraint results in an upper bound onτ 0

χ, or equivalentlyon a lower bound on trilinear6Rp couplings of the order ofO(10−12) [126]. For comparison,these couplings are required to be larger thanO(10−8) for the LSP to decay inside a laboratorydetector.

We now present a more detailed discussion of the constraintsoriginating from nucleosynthe-sis. The decay of an unstable relic particle after the nucleosynthesis epoch would have producedelectromagnetic and/or hadronic showers that could have either dissociated or created light nu-clei [125]. Hence, in order not to destroy the predictions ofBig-Bang nucleosynthesis, the LSPlifetime, if not greater than the age of the universe, must not exceed some upper limit. Focusingon the constraints arising from deuterium photo-dissociation, Kim et al. [126] estimate the max-imal allowed lifetime to be(τχ0)max ≃ 2.24 × 107s/ [ 4.92 + ln (mχ0/1 GeV) − ln (nB/nγ) ].Imposing that the neutralino LSP lifetime associated to itsdecays via trilinear6Rp couplings isshorter than the above value leads to a lower bound of the order of 10−12 on a weighted sum ofsquared couplings. As an example, for a60 GeV photino-like neutralino, assuming a universalsfermion mass of1 TeV, the constraint reads:

0.12∑

i,j,k |λijk|2 + 0.31∑

i,j 6=3,k |λ′ijk|2 + 0.04∑

i,k |λ′i3k|2+ 0.23

∑i<j,k 6=3 |λ′′ijk|2 > 7.7 × 10−24 . (4.1)

Let us now discuss the constraints applying to a neutralino LSP with a lifetime greater thanthe age of the universe (τ 0

χ > t0). A first set of constraints comes from the production ofantiprotons through LSP decays mediated by theλ′ijk andλ′′ijk couplings [127]. The observedflux of cosmic rays antiprotons places a strong bound on such decays, resulting in stringentupper limits on the corresponding6Rp couplings:

λ′ijk, λ′′ijk <

(10−24 − 10−19

), (4.2)

for all generation indices, exclusive ofλ′′3jk in the case where the LSP is lighter than the topquark. The upper bound on a given coupling strongly depends on the model parameters (espe-cially on the neutralino and squark masses), but is always smaller by some 3 orders of magnitude

than the upper bound corresponding to the condition that theLSP lifetime is greater than theage of the universe,τ 0

χ > t0 (see Ref. [127] for details).

A very long-lived LSP neutralino can also produce positronsthrough the three-body decaysχ0 → e+ + 2 fermions, which can be induced both by trilinear and bilinear 6Rp interactions.The experimentally measured positron flux in our galaxy imposes the following bound on thecorresponding partial lifetime of the neutralino [128]:τ(χ0 → e+ + 2 fermions)/t0 > 6 ×1010 h (mχ0/100 GeV)

12 (m/100 GeV)

12 , where all sfermion masses have been set tom, and

h is the reduced Hubble parameter defined byH0 = 100 h km/s/Mpc. This leads to stringentupper bounds on all trilinear and bilinear6Rp superpotential couplings [129]:

λijk, λ′ijk, λ

′′ijk < 4 × 10−23 N−1

1l

( mf

100 GeV

)2 ( mχ0

100 GeV

)−9/8(

1 GeVmf

)1/2

,

µi < 6 × 10−23 N−11l

( mχ0

100 GeV

)( m

100 GeV

)−7/4

GeV , (4.3)

wheremf is the emitted fermion mass and theN1l [l = 3, 4] parametrize the amount of thehiggsino components in the neutralino.

4.1.2 Gravitino Relics

It is well known that supergravity theories are plagued witha cosmological gravitino prob-lem. Indeed, since the gravitino interacts only gravitationally, it has a very small annihilationcross-section and tends to overclose the universe; or, if itis unstable, to destroy the successfulpredictions of Big-Bang nucleosynthesis through its late decays. In the case of a stable gravitino(or a quasi-stable one with a lifetime longer than the age of the universe), the annihilation rateis too weak to prevent the relic energy density of heavy gravitinos from exceeding the criticalenergy density [130, 131]. Then gravitinos must be very light, m3/2 . 1 keV [132], in orderfor their relic abundance not to overclose the universe2. In the case of an unstable gravitino, itsdecay must occur sufficiently early so as not to affect nucleosynthesis. Indeed, if the gravitinodecays after nucleosynthesis, its decay products will either dissociate or create light nuclei andmodify their relative abundances, thus destroying the agreement between Big-Bang nucleosyn-thesis predictions and observations. Furthermore the entropy release subsequent to gravitinodecays will wash out the baryon asymmetry and spoil the concordance between the observedbaryon-to-photon ratio and the light nuclei abundances. The second problem can be evaded ifthe gravitino is heavier than about104 GeV [130]. This lower bound assumes that the gravitinois not the LSP, so that it can decay to lighter supersymmetricpartners; if the gravitino is theLSP and decays via6Rp channels, it should be even heavier – but then all superpartners shouldbe extremely heavy. To summarize, there is no cosmological gravitino problem ifm3/2 . 1keV, or if the gravitino is unstable and heavy (m3/2 & 10 TeV if it is not the LSP).

The above constraints, however, were derived within standard cosmology (without infla-tion), and can be relaxed if there is an inflationary phase which dilutes the gravitino abundance

2Because the effective strength of its couplings are fixed by the ratioGN/m23/2

, a gravitino heavier than afew eV would have extremely small interaction cross-sections and decouple very early, allowing for its residualabundance to be higher than that of a neutrino with the same mass [133, 134]. The upper limit on the mass of such alight gravitino, obtained by demanding that its relic energy density be less that the critical density, is then increased(as compared to the corresponding limit for neutrinos), up to ∼ 1 keV [132] – its precise value depending on thenumber of particle species in thermal equilibrium at the gravitino decoupling time.

[135, 136, 137]. In this case one still has to face a cosmological gravitino problem associatedwith the gravitinos produced during the reheating phase after inflation, whose abundance is es-sentially proportional to the reheating temperatureTR. If the gravitino is stable, requiring thatits relic energy density is less than the critical density therefore results in an upper bound onTR. If it is unstable, its decays should not affect the light nuclei abundances successfully pre-dicted by Big-Bang nucleosynthesis, which requires valuesof TR lower than in the case of astable gravitino. One typically findsTR . 107 GeV form3/2 ∼ 100 GeV if the gravitino is notthe LSP [138]. Such a stringent upper bound is problematic for standard inflationary models[139, 140], which generally predict much higher values of the reheating temperature.

Let us consider in greater detail the case of an unstable gravitino. All decay rates of thegravitino are proportional to Newton’s constantGN = 1/M2

P , and may be expressed asΓG ≈αG m3

3/2/M2P , whereαG is a dimensionless coefficient. The fastest possible decay modes, for

which the coefficientαG is of order one, are theR-parity conserving two-body decay modes,such asG → χ∓ + W±, G → χ0 + γ(Z) andG → l∓ + l±. These are allowed only if thegravitino is not the LSP. In the presence of6Rp interactions, the gravitino can also decay intochannels solely comprising the ordinary (R-even) particles [141], but with much smaller ratesthan theR-parity conserving modes (αG ≪ 1) due to the smallness of the6Rp couplings. As aresults, the6Rp decay channels are relevant only for the case of a gravitino LSP, on which weshall concentrate now.

The case of bilinearR-parity violation has been discussed in Ref. [142]. Assuming thatthe lightest neutralino is essentially bino-like, the dominant decay mode of the gravitino LSP isG→ νγ, for whichαG ≃ 1

32πcos2 θWmν/mχ0

1, wheremν is the neutrino mass generated at tree

level by the bilinear6Rp terms (see chapter 5). The experimental and cosmological constraintson neutrino masses then imply that the gravitino lifetime ismuch longer than the age of theuniverse, even for a gravitino mass as high as100 GeV. The gravitino relic abundance and massare further constrained by the requirement that the photon flux produced in gravitino decaysdoes not exceed the observed diffuse photon background; fora relic abundance in the relevantrange for dark matter andmν ∼ 0.07 eV, this impliesm3/2 . 1 GeV. Thus, in the presenceof bilinearR-parity violation (at the level required to explain atmospheric neutrino data), agravitino LSP can constitute the dark matter of the universeonly if it is lighter than about1GeV, assuming in addition that the reheating temperature islow enough for the gravitino relicdensity to fall in the range relevant for dark matter. The case of trilinearR-parity violation hasbeen discussed in Ref. [143]. Assuming standard cosmology (without an inflationary phase),a gravitino LSP which decays via trilinear6Rp couplingsλijk, λ′ijk or λ′′ijk can evade the relicabundance problem, but it is excluded by nucleosynthesis constraints, unless the gravitino massis unnaturally large. This abundance problem, however, canbe solved by inflation. To conclude,R-parity violation does not seem to provide a natural solution to the cosmological gravitinoproblem.

Still a gravitino with6Rp decay can have interesting implications in astrophysics and cosmol-ogy. As mentioned above, nucleosynthesis severely constrains the possibility of a late decayingmassive particle, and the constraint is particularly strong for an unstable gravitino. Howeverone can consider an alternative scenario to Big-Bang nucleosynthesis which relies on such aparticle [144]. In this scenario, light element productiontakes place when the hadronic decayproducts interact with the ambient protons and4He. In order to reproduce the observed abun-dances, very specific properties of the decaying particle are required; it must in particular decayafter nucleosynthesis and have a small baryonic branching ratio, rB ∼ 10−2. The candidateproposed in Ref. [145] is a massive, not LSP gravitino decaying to hadrons predominantly via

theL-violating trilinear 6Rp couplings. One can arrange for the required small baryonic branch-ing ratio in gravitino decays by considering a sneutrino LSPand assuming non-vanishing6Rp

couplingsλ131, λ232 andλ′3jk with λ′3jk/λ131 ∼ 0.1 andλ′3jk/λ232 ∼ 0.1. The gravitino then

undergoes the following cascade decays:G → ν ¯ν, ν → (ee, µµ, · · · ) + (qq). The predictedabundances ofD, 4He and7Li can be made to match the observations even for a universe closedby baryons [144, 146] (i.e. withΩB ≃ 1), but the scenario overproduces6Li and is thereforedisfavoured [147].

4.2 Cosmological Baryon Asymmetry

4.2.1 Baryogenesis fromR-Parity-Violating Interactions

Generating the observed baryon asymmetry of the universe isone of the challenges of particlephysics. In order for a baryon-antibaryon asymmetry to be dynamically generated in an expand-ing universe, three necessary conditions, known as Sakharov’s conditions [148], must be met:(i) baryon-number violation; (ii)C andCP violation; (iii) departure from thermal equilibrium.In principle, all three ingredients are already present in the Standard Model, where baryon num-ber is violated by nonperturbative processes known as sphalerons [149, 150, 151] (which violateB + L but preserveB − L and are in thermal equilibrium above the electroweak scale), andthe departure from thermal equilibrium could be due to the electoweak phase transition. Thisleads to the standard electroweak baryogenesis scenario, which however has been excluded asa viable mechanism in the Standard Model [152], and works only in a restricted portion of theMinimal Supersymmetric Standard Model parameter space [153]. Other mechanisms, such asleptogenesis [154], in which a lepton asymmetry is generated by out-of-equilibrium decays ofheavy Majorana neutrinos and then partially converted intoa baryon asymmetry by sphalerontransitions, or Affleck-Dine baryogenesis [155], offer possible alternatives to the standard sce-nario. In this section, we review several attempts to generate the observed baryon asymmetryfromRp-violating interactions.

A first class of scenarios uses the trilinear6Rp couplingsλ′′ijk and their associatedA-termsA′′

ijk as the source of baryon-number violation. Theλ′′ijk couplings induce decays of a squark(resp. an antisquark) into two antiquarks (resp. two quarks), which violate baryon number byone unit. The differences between the various scenarios that rely on this process reside in theway departure from equilibrium is realized, and in the mechanism that produces squarks.

In the scenario proposed by Dimopoulos and Hall [156], squarks are produced far fromthermal equilibrium at the end of inflation as decay productsof the inflaton field. Their subse-quent decays into quarks and antiquarks induced by the6Rp couplingsλ′′ijk generate a baryonasymmetry directly proportional to theCP asymmetry in these decays,∆Γq = (Γ(qR →qRqR)−Γ(qc

L → qRqR))/(ΓqR+Γqc

L). The dominant contribution to thisCP asymmetry comes

from the interference between tree-level and two-loop diagrams involving theCP -violatingphases present in theA-terms (in the convention in which gaugino mass parameters are real). Inorder for this scenario to work, the reheating temperatureTR must be extremely low (typicallyTR . 1 GeV) so that scattering processes induced by theλ′′ijk couplings, which could dilute thebaryon asymmetry created in squark decays, are suppressed.

In the scenario considered by Cline and Raby [157], the departure from thermal equilibriumis provided by the late decays of the gravitino, and the baryon asymmetry is produced in two

steps. First an asymmetry in the number densities of squarksand antisquarks is produced byCP -violating decays of the neutral gauginos produced in the out-of equilibrium decays of thegravitino, or byCP -violating decays of the gravitino itself, but the total baryon number remainsconserved due to an opposite asymmetry in the number densities of quarks and antiquarks. Asin the previous model, the source ofCP violation is the relative phase between the gauginomass parameters and theA-terms, but now theCP asymmetry arises from the interferencebetween tree-level and one-loop diagrams. In the case of gluons, the asymmetry induced bynon-vanishingλ′′323 andA′′

323 couplings reads:

∆Γg ≡ Γ(g → ttc) − Γ(g → tt)

Γg≈ λ′′323

16π

ℑ(A′′⋆323mg)

|mg|2, (4.4)

whereℑ denotes the imaginary part. In the second step, this asymmetry gets partially convertedinto a baryon asymmetry by theB-violating decays of the (anti)squarks induced by theλ′′ijkcouplings – it is assumed here that the squarks are lighter than the gauginos, so that their onlyrelevant tree-level decay mode is into two quarks. At the time where the squark decays occur,the scattering processes that could erase theCP asymmetry are highly suppressed by low parti-cle densities. For this scenario to work, the reheating temperature must be high enough for therequired gravitino abundance to be regenerated after inflation, i.e. typicallyTR & 1015 GeV. Inaddition, the gravitino should be heavy enough (m3/2 & 50 TeV) so that its decay products donot affect nucleosynthesis.

To generate the observed baryon asymmetry, the previous twoscenarios require theCPasymmetry to be close to its maximal allowed value, i.e. theCP -violating phase should be closeto the upper bound associated with the electric dipole moment of the neutron, and the dominantB-violating coupling should be of order one. The source of baryon-number violation mustthen be a coupling that is not constrained by∆B = 2 processes such as neutron-antineutronoscillations and heavy nuclei decays (see chapter 6), e.g.λ′′323.

A variant of the scenario considered by Cline and Raby, whichalso works for smaller valuesof theCP asymmetry and for lower reheating temperatures, has been proposed by Mollerachand Roulet [158]. In this scenario, a large, out-of-equilibrium population of gluinos is createdin the decays of heavy axinos (a) and/or saxinos (s), the fermionic superpartners and scalarpartners of the pseudoscalar axions, respectively. This requiresma > mg (ms > 2mg), so thatthe decay channela → gg (s → gg) be kinematically allowed. The axinos (saxinos) decayat a temperature around1 GeV and thus do not interfere with nucleosynthesis. The baryonasymmetry is then generated in two steps from gluino decays,like in the scenario of Cline andRaby, but the present scenario is much more efficient and theCP asymmetry is not required tobe close to its maximal value. TheCP -violating phase can thus be small, and theB-violatingcoupling can be smaller than one. In the presence of an inflationary phase, the observed amountof baryon asymmetry can be obtained for reheating temperatures as low as104 GeV (in the caseof a largeCP asymmetry), due to the fact that (s)axinos are regenerated much more efficientlythan gravitinos.

In another scenario studied by Adhikari and Sarkar [159], the baryon asymmetry is gen-erated in out-of-equilibrium decays of the lightest neutralino induced by theλ′′ijk couplings,χ0

1 → uiR djR dkR, rather than in squark decays. TheCP asymmetry in these decays arises atthe one-loop level, and can be large even for small values of theB-violating couplings, whichare required by the out-of-equilibrium condition. Unlike in the previous scenarios,CP vio-lation is due to the complexity of theλ′′ijk couplings. The sfermions are assumed to be muchheavier than the lightest neutralino, so that the former have already decayed at the time where

the latter decays, and their6Rp decay modes do not erase the generated baryon asymmetry. Theother processes that could dilute the baryon asymmetry, such as the∆B = 1 scattering pro-cessesuiR djR → dkR χ

01, must be out of equilibrium. This requires rather small values of the

λ′′ijk couplings; still Adhikari and Sarkar estimate that it is possible to generate the observedbaryon asymmetry of the universe for values of theλ′′ijk couplings in the10−4−10−3 range (seehowever the footnote below).

In a second class of scenarios, the6Rp couplingsλijk andλ′ijk, which violate lepton num-ber, are used to create a lepton asymmetry at the electroweakscale. This one is then partiallyconverted into a baryon asymmetry by the sphaleron processes3.

In the scenario considered by Masiero and Riotto [161], the lepton asymmetry is generatedthrough the6Rp (andCP -violating) decays of the LSP, which is produced out of equilibrium inbubble collisions during the electroweak phase transition. The origin ofCP violation is simplythe presence of phases in the6Rp couplingsλijk andλ′ijk. Assuming that the LSP is the lightestneutralino, the dominant decay channel is expected to beχ0

1 → l−i tdk (l+i tdk), and theCPasymmetry is given by:

ǫ =Γ(χ0

1 → l−td) − Γ(χ01 → l+td)

Γ(χ01 → l−td) + Γ(χ0

1 → l+td)≃ 1

16π

∑i,k,l,m,n ℑ (λ′⋆inlλ

′mnlλ

′⋆m3kλ

′i3k)∑

i,k |λ′i3k|2. (4.5)

For this mechanism to work, the phase transition must be firstorder, i.e. it must proceeds bynucleation of bubbles of true vacuum in the unbroken phase. Furthermore, contrary to whatis required in the standard electroweak baryogenesis scenario, the sphaleron processes mustremain in equilibrium until the temperature drops below thecritical temperature. This canindeed be the case if the Higgs sector of the Supersymmetric Standard Model is extended bythe addition of one or more singlet superfields. Finally, the6Rp interactions responsible forthe generation of the lepton asymmetry must be in equilibrium at the electroweak scale, whilethe ∆L = 1 scattering processesliLdkR χ0

1 t, which could wash it out, must be out ofequilibrium. The first condition turns into a lower bound on theλ′i3k couplings:

|λ′i3k| & 5.3 × 10−5

(500 GeVmχ0

1

)(T0

150 GeV

)( mtL

1 TeV

)2

, (4.6)

and the second one into an upper bound on the same couplings:

|λ′i3k| . 2.4 × 10−4

(500 GeVmχ0

1

)(150 GeV

T0

)1/2 ( mtL

1 TeV

)2

. (4.7)

whereT0 is the critical temperature of the electroweak phase transition. For values of the6Rp

couplings in the range delimited by Eqs. (4.6) and (4.7), this scenario can generate the observedbaryon asymmetry, provided that the top squark is heavy. Indeed, relatively high values ofλ′i3k

are needed to provide a large enoughCP asymmetry (typically|λ′i3k| ∼ 10−2, which requiresmtL & 6 (3) TeV if mχ0

1≃ 500 (100) GeV).

Adhikari and Sarkar [162] noticed that, in the presence of flavour-violating couplings ofthe neutralinos, the generation of the lepton asymmetry canbe much more efficient. This is

3It has been noted in Ref. [160] that the lepton or baryon asymmetry generated at the electroweak scale from thedecay of gauge non-singlet particles can be strongly suppressed due to the efficiency of the annihilation of theseparticles into two gauge bosons. This effect had been overlooked or underestimated in the scenarios discussedbelow, as well as in the scenario of Ref. [159].

especially so if some sfermions are not much heavier than thelightest neutralino, so that theycan be produced in bubble collisions and contribute to the lepton asymmetry through their6Rp

decaysνiL → dkR djL, djL → dkR νiL, dkR → djL νiL, . . . .

Hambye, Ma and Sarkar [163] consider another scenario in which a lepton asymmetry iscreated in out-of-equilibrium decays of the lightest neutralino into a charged Higgs boson and alepton singlet,χ0

1 → l±R h∓. TheCP asymmetry is proportional to the square of the parameter

ξ that accounts for the mixing between the slepton singlets and the charged Higgs boson, andtheCP -violating phase comes from the neutralino mass matrix. Thesame parameterξ controlsthe out-of-equilibrium condition for∆L = 2 processes that could erase the generated leptonasymmetry. For this mechanism to produce enough baryon asymmetry, ξ must be close tothe upper bound associated with this condition. This can be achieved by introducing non-holomorphic6Rp soft terms of the formH+

u Hd lc, which contrary to the standard6Rp soft terms

are not constrained by experimental data.

4.2.2 Survival of a Baryon Asymmetry in the Presence of6Rp Interactions

We have seen in the previous subsection thatR-parity violation may be at the origin of thebaryon asymmetry of the universe. In general, however,6Rp interactions are considered as adanger since they can erase a baryon asymmetry that would be present before the electroweakphase transition. In order to avoid this, one requires6Rp interactions to be out of equilibriumabove the critical temperature of the electroweak phase transition, which results in strong upperbounds on the6Rp couplings.

Let us first review the standard conditions for a baryon asymmetry generated during thethermal history of the universe to be preserved until today,in the absence ofB- andL-violatinginteractions. Above the critical temperatureTc ∼ 100 GeV and up to temperatures of theorder of1012 GeV, nonperturbative processes which violateB + L but preserveB − L are inthermal equilibrium [150, 151]. These processes, known as sphalerons, tend to erase anyB+Lasymmetry present in the high temperature phase, in such a way that a baryon asymmetry canpersist only if it corresponds to aB −L asymmetry. More precisely, the existence of sphaleronprocesses in thermal equilibrium with the interactions of the Standard Model (or of the MinimalSupersymmetric Standard Model) leads to the following proportionality relation between theBandB − L asymmetries [164, 165]:

YB =24 + 4NH

66 + 13NHYB−L , (4.8)

whereYB ≡ (nB − nB)/s (YL ≡ (nL − nL)/s), with nB (nL) the baryon (lepton) numberdensity ands the entropy density of the universe, andNH is the number of Higgs doublets. Forthe Standard Model with one Higgs doublet, one hasYB/YB−L = 28/79, while for the MinimalSupersymmetric Standard Model one hasYB/YB−L = 8/23. As a result, there are only twoviable possibilities for generating the baryon asymmetry of the universe: one can generate it ator after the electroweak phase transition (when sphaleron transitions are suppressed), or abovethe electroweak phase transition in the form of aB − L asymmetry.

The above discussion must be modified in the presence of interactions that violateB − L,such as6Rp interactions [166, 167, 168, 169, 170, 171]. Assuming that aB − L asymmetry isgenerated by some mechanism above the electroweak phase transition, the only possibility forit to be preserved is that theB − L-violating interactions be out of equilibrium, i.e. that their

characteristic timescale be longer than the age of the universe. This condition can be written asΓB−L < H, whereΓB−L is the rate of a typicalB − L-violating process, andH is the Hubbleparameter. In the case ofR-parity violation, this yields strong upper bounds on6Rp parameters.For the trilinear couplings, the processes that yield the best bounds are the decays of squarks andsleptons into two fermions or sfermions, and the corresponding rates are given by [167, 169]:

Γλ ≃ 1.4 × 10−2 |λ|2 m2Φ

T, ΓA ≃ 1.4 × 10−2 |A|2

T, (4.9)

wheremΦ is the mass of the decaying sfermion,λ stands for any of the couplingsλijk, λ′ijk or

λ′′ijk, andA for Aijk, A′ijk or A′′

ijk. Since the temperature dependence of the Hubble parameter

is given byH ≃ 1.66 g1/2∗ T 2/MP (whereg∗ is the number of effectively massless degrees of

freedom at the temperatureT ), the out-of equilibrium conditionΓ < H is more easily satisfiedat high temperature, and the best bounds are obtained forT ∼ Tc. AssumingmΦ ∼ Tc ∼ 100GeV, one obtains [169]:

|λijk|, |λ′ijk|, |λ′′ijk| . 10−7 , (4.10)

|Aijk|, |A′ijk|, |A′′

ijk| . 10−5 GeV . (4.11)

For heavier sfermions, these constraints are slightly weakened, e.g. forT ∼ mΦ ∼ 1 TeVthe upper bounds (4.10) are increased by a factor of3. Also, the bounds (4.10) and (4.11) werederived under the assumption that the decays are kinematically allowed, which is not necessarilythe case for the decays induced byA-terms. The bounds on the6Rp A-terms associated withscattering processes can be estimated to be one order of magnitude weaker [172].

For bilinear6Rp couplings, the relevant interaction rates are [169]:

Γµi≃ 1.4 × 10−2 g2 |µi|

2

m2 T , ΓBi≃ 1.4 × 10−2 g2 |Bi|

2

m4 T ,

Γm2di

≃ 1.4 × 10−2 g2 |m2di|2

m4 T , (4.12)

wherem stands for the relevant scalar or supersymmetric fermion mass. As usual, the bilinear6Rp parametersµi, Bi andm2

di are expressed in the (Hd, Li) basis in which the sneutrino VEVs< νi > vanish and the charged lepton Yukawa couplings are diagonal(see subsection 2.3.1).Assumingm ∼ Tc ∼ 100 GeV, one obtains the bounds:

|µi| . 2 × 10−5 GeV, |Bi|, |m2di| . 2 × 10−3 GeV2 . (4.13)

There is however a little subtlety in deriving these bounds,due to the fact that the thermal masseigenstate basis for theHd andLi fields above the electroweak phase transition is not the sameas the zero temperature mass eigenstate basis [26, 27]. For adiscussion of the cosmologicalbounds on6Rp couplings in terms of basis-independent quantities, see Ref. [27].

As discussed in section 2.7, baryon- and lepton-number violation may also proceed throughnon-renormalizable operators, which are expected to be generated from some more fundamentaltheory than the Supersymmetric Standard Model. Writing a generic non-renormalizable opera-tor of dimension4+n asO4+n/M

n4+n, where all couplings have been absorbed in the definition

of the mass scaleM4+n, one can estimate the rate of theB − L-violating processes inducedby this operator to beΓ ∼ 10−3T 2n+1/M2n

4+n. The requirement that these processes are out ofthermal equilibrium at the temperatureT yields a lower bound on the mass scaleM4+n:

M4+n & 102+ 6n GeV

(T

100 GeV

)1− 12n

. (4.14)

The interaction rates for non-renormalizable operators increase faster with temperature than theHubble parameter; therefore, the strongest bounds onM4+n are obtained at high temperature. Inprinciple4, the bound (4.14) should be applied at the temperature at which the baryon asymmetryhas been created, but it is no longer valid aboveT ∼ 1012 GeV, where sphaleron processes areout of equilibrium. It should be noted that the bound (4.14) applies not only to6Rp operators,but also to theR-parity conserving operators which violateB − L, such as the superpotentialterm 1

M5LHuLHu, which induces Majorana masses for the neutrinos. The boundon M5 is

M5 & 108 GeV(T/100 GeV)1/2, which is compatible with the cosmological bound on neutrinomasses

∑imνi

. 1 eV [46], even if the bound onM5 applies up toT ∼ 1012 GeV.

In the above, we did not make any distinction between couplings which violate differentlepton flavours. However, since the sphaleron processes preserve each one of the three com-binationsB/3 − Li, i = 1, 2, 3, aB − L asymmetry generated before the electroweak phasetransition will survive as soon as the processes violating one of theB/3 − Li are out of equi-librium, even if the other two are violated by processes in thermal equilibrium. This means thatthe bounds (4.10) to (4.14) must be satisfied by all baryon-number-violating couplings and bythe couplings that violate, sayLe, while the6Rp couplings that violateLµ or Lτ can be arbitralylarge – provided however that the sources of lepton-flavour violation are out of equilibrium[170]. Explicitly, the conditions for preserving aB/3 − Le asymmetry generated before theelectroweak phase transition read (for superpartner masses of the order ofTc ∼ 100 GeV):

|λ1jk|, |λij1|, |λ′1pq|, |λ′′npq| . 10−7 , (4.15)

|A1jk|, |Aij1|, |A′1pq|, |A′′

npq| . 10−5 GeV , (4.16)

|µ1| . 2 × 10−5 GeV, |B1|, |m2d1| . 2 × 10−3 GeV2 , (4.17)

where i, j, k 6= 1. The 6Rp couplings that violateLµ or Lτ can be much larger if the off-diagonal slepton soft terms are small enough, so that they donot induce lepton-flavour-violatingprocesses at thermal equilibrium [170, 172]:

∣∣∣∣∣(m2

L)1j

(m2L)11

∣∣∣∣∣ ,∣∣∣∣∣(m2

lc)1j

(m2lc)11

∣∣∣∣∣ . 5 × 10−2 , |Ae1j | . 10−5 GeV , (4.18)

wherej = 2 or 3. The constraints (4.15) to (4.17) can be summarized by saying that baryon-number violation, as well as lepton-number violation in at least one generation, must be stronglysuppressed.

The constraints presented in this subsection should be regarded as sufficient conditions forthe baryon asymmetry of the universe not to be erased by6Rp interactions, rather than strictbounds. First of all, they do not apply if the baryon asymmetry of the universe is generated at orafter the electroweak phase transition, like in the standard electroweak baryogenesis scenario,or in all baryogenesis scenarios discussed in the previous subsection, where the6Rp interac-tions act as the source of the baryon asymmetry. Indeed, the sphaleron processes come out ofequilibrium just below the critical temperature, so that they can no longer erase aB + L asym-metry. Furthermore, there are several loopholes in the cosmological arguments used to derive

4For operators that do not involve the right-handed electronfield, it may be enough to require that the corre-sponding interactions are out of equilibrium up toTeR

∼ (104 − 105) GeV [173, 174]. Indeed, aboveTeR, the

electron Yukawa coupling is out of equilibrium, so that an asymmetry stored ineR cannot be transferred to otherparticle species. In baryogenesis scenarios that generateaneR asymmetry, this can protect the baryon asymmetrydown to the temperatureTeR

.

the bounds on6Rp couplings displayed above, and these (or some of them) can beevaded in sev-eral baryogenesis scenarios, even if the baryon asymmetry is generated above the electroweakphase transition (see e.g. Refs. [170, 175]).

In this chapter, we studied the implications of a brokenR-parity in cosmology and astro-physics. The most dramatic change with respect to theR-parity conserving SupersymmetricStandard Model is the unstability of the LSP, which rules it out as a natural candidate for thenon-baryonic dark matter of the universe, unless6Rp couplings are unrealistically small. An evenmore damaging effect ofR-parity violation is the potential erasure of the baryon asymmetry ofthe universe by6Rp interactions, if it has been generated before the electroweak phase transition.These could be two good reasons to stick to a conservedR-parity. On the other hand,R-parityviolation can help solving the cosmological gravitino problem, although with some difficulty,and provide new mechanisms for generating the baryon asymmetry of the universe at or afterthe electroweak phase transition.

Chapter 5

NEUTRINO MASSES AND MIXINGS

R-parity forbids lepton-number (L) violation from renormalizable interactions. Allowing forviolation ofL-conservation law has several important effects in the neutrino sector. The mostdramatic implication of non-vanishing couplingsλijk, λ′ijk and/or bilinear6Rp parameters is theautomatic generation of neutrino masses and mixings. As a consequence, the possibility thatthe atmospheric and solar neutrino data, now interpreted interms of neutrino oscillations, beexplained by6Rp interactions has motivated a large number of studies and models. Besidesneutrino masses,R-parity violation in the lepton sector leads to neutrino transition magneticmoments, new contributions to neutrinoless double beta decays, neutrino-flavour transitionsin matter and sneutrino-antisneutrino oscillations. In this chapter, we shall concentrate on thequestion of neutrino masses and mixings in supersymmetric models withoutR-parity, and on therelated issues of neutrino transition magnetic moments,6Rp-induced neutrino-flavour transitionsin matter and sneutrino-antisneutrino mixing . Neutrinoless double beta decay will be discussedin Chapter 6.

5.1 6Rp Contributions to Neutrino Masses and Mixings

5.1.1 R-Parity Violation as a Source of Neutrino Masses

In order to account for nonzero neutrino masses and mixings,the Standard Model (with two-component, left-handed neutrino fields) has to be supplemented with additional particles. Thesimplest possibility is to add right-handed neutrinos, which either leads to Dirac neutrinos or,if the right-handed neutrinos have heavy Majorana masses, to Majorana neutrinos through thewell-known seesaw mechanism [176]. Other mechanisms, which directly generate a Majoranamass term for the standard two-component neutrino fields, donot involve right-handed neutri-nos but require an enlarged Higgs sector, involving additionalSU(2)L triplet [59] or singlet anddoublet [177, 178] Higgs fields.

A new possibility arises in the Supersymmetric Standard Model. Indeed, in the absence ofR-parity, lepton-number-violating couplings induce Majorana masses for neutrinos without theneed of right-handed neutrinos or exotic Higgs fields. In other words,supersymmetry withoutR-parity automatically incorporates massive neutrinos. This may be regarded both as an appealingfeature of6Rp models and as a potential problem, since the contribution of6Rp couplings mayexceed by several orders of magnitude the experimental bounds on neutrino masses.

One can distinguish between two types of contributions to neutrino masses and mixings insupersymmetric models withoutR-parity1 [24]: (i) a tree-level contribution arising from theneutrino-neutralino mixing due to bilinearR-parity violation; (ii) loop contributions inducedby the trilinear6Rp couplingsλijk andλ′ijk and/or bilinear6Rp parameters. These contributions,if present, are generally expected to be large and potentially in conflict with the experimentallimits on neutrino masses [20],

mνe < 3 eV , mνµ < 190 keV , mντ < 18.2 MeV , (5.1)

and with the cosmological bound on stable neutrinos,∑

imνi. 1 eV [46]. The effective

Majorana masses generated through mechanisms (i) and (ii) can be written as (with the neutralgauginos and higgsinos integrated out)

− 1

2Mν

ij νLi νcRj + h.c. , (5.2)

wherei, j = 1, 2, 3 are generation indices, and the3×3 matrixMν is symmetric by virtue of theproperties of the charge conjugation matrix. The relative rotation between charged lepton andneutrino mass eigenstates defines a lepton mixing matrixU , 3 × 3 and unitary, the Pontecorvo-Maki-Nakagawa-Sakata (PMNS) matrix [179, 180], responsible for neutrino oscillations. Withthe conventionsRe

LMeRe+

R = Diag(me1 , me2 , me3) andMν = Rν Diag(mν1 , mν2, mν3)RνT ,

the lepton mixing matrix readsU = ReLR

ν , and the weak eigenstate neutrinosνα=e,µ,τ – i.e.theSU(2)L partners of the mass eigenstate charged leptons – are related to the mass eigen-state neutrinosνi=1,2,3 by να =

∑i Uαi νi. HereMe is an effective mass matrix obtained after

integrating out the charged gauginos and higgsinos, which mix with charged leptons throughbilinearR-parity violation (see subsection 2.3.3). In general it is not simply proportional to thecharged lepton Yukawa matrix, althoughMe = λevd/

√2 remains a good approximation in the

phenomenologically relevant limit of small bilinearR-parity violation.

In the following, we shall neglectCP violation in the lepton sector, and therefore assumethe charged lepton Yukawa couplingsλe

ij and the6Rp couplingsµi, λijk andλ′ijk to be real. TheMNS matrix is then real and can be parametrized by three angles θ12, θ13 andθ23, responsiblefor solar, reactor and atmospheric neutrino oscillations respectively:

U =

c13c12 c13s12 s13

−c23s12 − s13s23c12 c23c12 − s13s23s12 c13s23

s23s12 − s13c23c12 −s23c12 − s13c23s12 c13c23

. (5.3)

Let us finally notice that since we are assuming non-vanishing L-violating couplings, weshould make sure that theB-violating couplingsλ′′ijk are absent from the theory – otherwisethe proton would decay at a much too rapid rate. This can be ensured by imposing, instead ofR-parity, theZ3 baryon parity discussed in section 2.7. The advantage of this symmetry, whichis discrete anomaly free in the MSSM, is that it forbids not only theλ′′ couplings, but also thedangerous dimension-5 operators leading to proton decay.

5.1.2 Tree-Level Contribution Generated by Neutrino-Neutralino Mixing

Let us first consider the tree-level contributions. As discussed in section 2.3.3, bilinearR-parityviolation induces a mixing between neutrinos and neutralinos, which yields a single massive

1For early discussions on neutrino masses in supersymmetricmodels with explicitly or spontaneously brokenR-parity, see Refs. [24, 33, 34, 55, 56].

neutrino state at tree level. This can be understood as a kindof seesaw mechanism, in whichthe neutral gauginos and higgsinos play the role of the right-handed neutrinos. Indeed, in thelimit of a small neutrino-neutralino mixing, the7×7 neutrino-neutralino mass matrixMN has a“seesaw” structure, with a strong hierarchy2 between the gaugino-higgsino diagonal4×4 blockMχ and the off-diagonal3 × 4 blockm:

MN |tree =

(Mχ mT

m 03×3

). (5.4)

The effective mass matrix obtained by integrating out the neutralinos, which yields the neutrinomass and mixing angles, is given byMν

tree ≃ −mM−1χ mT . The fact that only one neutrino

becomes massive is most easily seen in a basis in which the superpotential6Rp mass parametersµ1 andµ2 (together with the sneutrino VEVsvi) vanish; then bothL0

1 andL02 decouple from

MN |tree, as can be seen from Eq. (2.44). Of course, when quantum corrections are included, allthree neutrinos acquire a mass, as explained in the next sections.

In order to determine the flavour composition of the single neutrino that acquires a mass atthis level, one has in principle to diagonalize the charginomass matrix. Indeed, since bilinearR-parity violation mixes charged leptons with charginos, the physical charged leptons are thethree lightest eigenstates of the5 × 5 charged lepton-chargino mass matrixMC , Eq. (2.45). Ingeneral these do not coincide with the eigenstates of the Yukawa matrixλe

ij. However, in thelimit of a small charged lepton-chargino mixing we are interested in, one can identify the twobases to a good approximation. We therefore choose to writeMN andMC in the basis in whichthe sneutrino VEVsvi vanish and the charged lepton Yukawa couplingsλe

ij are diagonal. In thisbasis, the effective neutrino mass matrix reads [181]:

Mνtree ≃ −mM−1

χ mT = − mνtree∑i µ

2i

µ21 µ1µ2 µ1µ3

µ1µ2 µ22 µ2µ3

µ1µ3 µ2µ3 µ23

, (5.5)

wheremνtree is given by Eqs. (2.46) and (2.47),

mνtree ≃ M2Z cos2β (M1c

2W +M2s

2W )µ cos ξ

M1 M2 µ cos ξ −M2Z sin 2β (M1c

2W +M2s

2W )

tan2 ξ . (5.6)

The misalignment angleξ, defined in subsection 2.3.1, is a basis-independent quantity thatcontrols the size of the bilinear6Rp effects in the fermion sector (in particular, assuming smallneutrino-neutralino and charged lepton-chargino mixingsamounts to requiresin ξ ≪ 1). Inthe basis in which we are working, it is given bysin ξ =

√∑i µ

2i /µ. As already noticed in

section 2.3.3, phenomenologically relevant values ofmνtree require a strong alignment betweenthe 4-vectorsµα ≡ (µ0, µi) andvα ≡ (v0, vi), sin ξ ∼ 3 × 10−6

√1 + tan2 β

√mνtree/1 eV.

Several authors [39, 40] have studied the possibility of obtaining the desired amount of align-ment from GUT-scale universality in the soft terms. With this assumption the LEP bound onthe tauneutrino mass can be satisfied in a relatively large region of the parameterspace, but asignificant amount of fine-tuning in the bilinear6Rp parameters is necessary in order to reach theeV region (one would typically needµi/µ ∼ 10−4 at the GUT scale). Another possibility is toinvoke an abelian flavour symmetry [35]; however rather large values of the associated chargesare necessary to yield the desired alignment (see section 2.5).

2As in section 2.3, we are working in a (Hd, Li) basis in which the sneutrino VEVs vanish,vi = 0. The seesawstructure may be less obvious in an arbitrary basis where both µi 6= 0 andvi 6= 0, since large values ofµi andvi

are in principle compatible with a small physical neutrino-neutralino mixing (see subsection 2.3.3).

ekLeelR jLiL ekR

eemLikl jmk iL jLedlR edmL0jmk0ikl dkRdkL(a) (b)Figure 5.1: One-loop contributions to neutrino masses and mixings induced by the trilinear6Rp couplingsλijk (a) andλ′ijk (b). The cross on the sfermion line indicates the insertion of aleft-right mixing mass term. The arrows on external legs follow the flow of the lepton number.

The massive neutrino is mainly a superposition of the electroweak neutrino eigenstates, andits flavour composition is given, in the basis we are considering, by the superpotential6Rp massparametersµi [181]:

ν3 ≃ 1√∑i µ

2i

(µ1νe + µ2νµ + µ3ντ ) . (5.7)

In terms of mixing angles, this gives the relations

sin θ13 =µ1√∑

i µ2i

, sin θ23 =µ2√µ2

2 + µ23

, (5.8)

while sin θ12 is undetermined.

5.1.3 One-Loop Contributions Generated by Trilinear 6Rp Couplings

At the one-loop level, a variety of diagrams involving the trilinear 6Rp couplingsλ andλ′ and/orinsertions of bilinear6Rp masses contribute to the neutralino-neutrino mass matrix,thus correct-ing Eq. (5.5). In this subsection, we concentrate on the diagrams involving trilinear6Rp cou-plings only. These diagrams represent the dominant one-loop contribution to neutrino massesand mixings when bilinearR-parity violation is strongly suppressed (i.e. whensin ξ ≃ 0 andsin ζ ≃ 0 in the language of subsection 2.3.1, where the angleζ formed by the 4-vectorsBα ≡ (B0, Bi) andvα ≡ (v0, vi) controls the Higgs-slepton mixing ). The one-loop diagramsinvolving bilinear6Rp masses will be discussed in the next subsection.

The trilinear6Rp couplingsλijk andλ′ijk contribute to each entry of the neutrino mass matrixthrough the lepton-slepton and quark-squark loops of Fig. 5.1, yielding [24, 182]

Mνij |λ =

1

16π2

∑

k,l,m

λiklλjmk mek

(me 2LR

)ml

m2eRl

−m2eLm

ln

(m2

eRl

m2eLm

)+ (i↔ j) , (5.9)

Mνij |λ′ =

3

16π2

∑

k,l,m

λ′iklλ′jmk mdk

(md 2LR

)ml

m2dRl

−m2dLm

ln

(m2

dRl

m2dLm

)+ (i↔ j) , (5.10)

Here the couplingsλijk (resp. λ′ijk) and the left-right slepton mixing matrixme 2LR

= (Aeij −

µ tanβ λeij) vd/

√2 (resp. the left-right squark mixing matrixmd 2

LR= (Ad

ij −µ tan β λdij) vd/

√2)

are expressed in the basis in which the charged lepton masses(resp. the down quark masses) as

well as the mass matrices for the associated doublet and singlet scalars are diagonal. The aboveexpressions simplify if, as is customary, one assumes that the sfermion masses are approxi-mately degenerate, and that theA-terms are proportional to the Yukawa couplings,Ae

ij = Aeλeij

andAdij = Adλd

ij. Then Eqs. (5.9) and (5.10) reduce to:

Mνij |λ ≃ 1

8π2

Ae − µ tanβ

m2e

∑

k,l

λiklλjlk mekmel

, (5.11)

Mνij |λ′ ≃ 3

8π2

Ad − µ tanβ

m2d

∑

k,l

λ′iklλ′jlkmdk

mdl, (5.12)

whereme (resp.md) is an averaged scalar mass parameter, and the couplingsλijk (resp.λ′ijk)are now expressed in the superfield basis corresponding to the charged lepton (resp. downquark) mass eigenstate basis. Even after those approximations, the neutrino mass matrix stilldepends on a large number of trilinear6Rp couplings (9λijk and 27λ′ijk). To obtain a morepredictive scheme, one has to make assumptions on the generational structure of the trilinear6Rp

couplings.

One may for instance assume that, for a given generation index i, there is no strong hierarchyamong the couplingsλijk andλ′ijk, or that their flavour structure in the indicesj andk is linkedto the fermion mass hierarchy [35, 71, 37]. The second assumption is natural in models wherethe fermion mass hierarchy is explained by flavour symmetries (see section 2.5). In both casesthe contributions withk, l = 2 or 3 dominate in Eqs. (5.11) and (5.12), and one obtains

Mνij|λ ≃ 1

8π2

λi33λj33

m2τ

m+ (λi23λj32 + λi32λj23)

mµmτ

m+ λi22λj22

m2µ

m

, (5.13)

Mνij |λ′ ≃ 3

8π2

λ′i33λ

′j33

m2b

m+ (λ′i23λ

′j32 + λ′i32λ

′j23)

msmb

m+ λ′i22λ

′j22

m2s

m

, (5.14)

where we have set all sfermion mass parameters equal tom. The term proportional tom2τ in Eq.

(5.13) comes from the tau-stau loop and givesMνij|λ ∼ λi33λj33 (4 × 105 eV) (100 GeV/m);

similarly, the term proportional tom2b in Eq. (5.14) comes from the bottom-sbottom loop and

givesMνij |λ′ ∼ λ′i33λ

′j33 (7.7 × 106 eV) (mb/4.5 GeV)2 (100 GeV/m).

This shows that trilinear6Rp couplings of order 1 would lead to large entries in the neutrinomass matrix, grossly conflicting with experimental data. This in turn puts strong constraints onthe trilinear6Rp couplings. The most stringent upper bound comes from the non-observation ofneutrinoless double beta decay, whose rate is directly related to the(i, j) = (11) element ofMν

[183]:

|λ133| ≤ 9.4 × 10−4(<mν>

0.35 eV

) 12

(m

100 GeV

) 12

, (5.15)

|λ′133| ≤ 2.1 × 10−4(<mν>

0.35 eV

) 12

(4.5 GeVmb

) (m

100 GeV

) 12

, (5.16)

where<mν> is the effective neutrino mass, bounded by neutrinoless double beta decay exper-iments. From the other terms in Eqs. (5.13) and (5.14) one canalso extract (weaker) boundson the couplingsλ1kl andλ′1kl, (k, l) 6= (3, 3). From a different perspective it is quite remark-able that a small amount ofR-parity violation through trilinear couplings, withλijk andλ′ijkcomparable in strength with the charged lepton and down quark Yukawa couplings, can induceneutrino masses in the phenomenologically interesting range, namely10−3 eV . mν . 1 eV.

Let us now discuss the flavour structure of the neutrino mass matrix. Assuming furtherthat theλ-type couplings are not greater than theλ′-type couplings, the leading contribution toMν

loop ≡ Mν |λ +Mν |λ′ comes from the bottom-sbottom loop. Then

Mνloop =

mνloop∑i λ

′2i33

λ′2133 λ′133λ′233 λ′133λ

′333

λ′133λ′233 λ′2233 λ′233λ

′333

λ′133λ′333 λ′233λ

′333 λ′2333

+ · · · , (5.17)

where

mνloop=

3m2b

8π2m

∑

i

λ′2i33 (5.18)

and the dots stand for corrections of order

m2τ

8π2mλi33λj33 ,

3mbms

8π2m(λ′i23λ

′j32 + λ′i32λ

′j23) ,

mµmτ

8π2m(λi23λj32 + λi32λj23) · · ·

(with i, j 6= 3 for them2τ corrections, and(i, j) 6= (2, 2), (3, 3) for themµmτ corrections).

The structure (5.17) generally leads to a hierarchical massspectrum. Indeed, only one neutrinobecomes massive at leading order, with a massmν3 = mνloop

and a flavour composition given

by the mixing anglessin θ13 = λ′133/√∑

i λ′2i33 andsin θ23 = λ′233/

√λ′2233 + λ′2333 (θ12 remains

undetermined at dominant order). Once sub-dominant contributions toMν are included, allthree neutrinos become massive. Let us stress again that theabove discussion does not take intoaccount the contributions of bilinear6Rp parameters to the neutrino mass matrix.

5.1.4 One-Loop Contributions Generated by both Bilinear and Trilinear6Rp Couplings

In the above, we discussed one-loop contributions to neutrino masses in the limit where bilin-earR-parity violation can be neglected. However this is generally not a valid approximation,since bilinear6Rp terms are always generated radiatively from trilinear6Rp interactions, and thepresence of bilinear6Rp terms drastically changes the discussion of one-loop neutrino masses.First of all, the neutrino mass matrix receives contributions already at tree level, as discussedin section 5.1.2. Secondly, in addition to the lepton-slepton and quark-squark loops already en-countered, one-loop diagrams involving insertions of bilinear6Rp masses or slepton VEVs mustbe considered [24, 39, 184, 38, 185, 186, 187, 188, 189, 190, 191, 192]. Here we shall onlypresent briefly the main classes of loop contributions, and comment on the level of suppressionof bilinear 6Rp parameters required by neutrino mass constraints. We referthe interested readerto Ref. [191] for a detailed classification and evaluation ofthe various diagrams.

Let us first notice that there are two ways of computing the one-loop neutrino masses andmixing angles. The first one is to compute one-loop corrections to the full7 × 7 neutralino-neutrino mass matrix, Eq. (5.4) [39, 186, 189]. The second one is to compute one-loop correc-tions to the tree-level effective3 × 3 neutrino mass matrix, Eq. (5.5) [190, 191]; in this casethe Feynman rules are written in terms of tree-level MSSM mass eigenstates (i.e. the tree-levelmass matrices are diagonalized neglectingR-parity violation) and the bilinear6Rp masses areincluded in the diagrams as mass insertions, both on internal and external lines. The secondapproach is more suitable for a discussion of the various contributions to neutrino masses andmixings. Leaving aside gauge boson loops and diagrams with two 6Rp mass insertions on the

RL L Rg

RL (e) L R0(d)

e; ~

(ia) (ib)

(ii) (iii)f0

e; f+ee; hd(e) 0(d)

(e)g gh;H;A; e

ded

~e; h+u

Figure 5.2: Schematic description of the one-loop diagrams contributing to neutrino massesand mixings, divided into three classes as described in the text. 6Rp mass insertions on internaland/or external lines are not shown. The arrows on external legs follow the flow of the leptonnumber.

external legs, which renormalize the tree-level neutrino mass, one can divide the one-loop con-tributions to the neutrino mass matrix into three classes (see Fig. 5.2), depending on whichcouplings appear at the two vertices (a diagram with two couplings λ1 andλ2 at the verticeswill be denoted by (λ1, λ2)) [191]:

(i) diagrams involving trilinear6Rp couplings and/or Yukawa couplings at the vertices, withcharged fermions and scalars in the loop; in addition to the (λ, λ) and (λ′, λ′) diagramsdiscussed above, there are (λ, λe) and (λ′, λd) diagrams with one6Rp mass insertion, and(λe, λe) and (λd, λd) diagrams with two6Rp mass insertions;

(ii) diagrams involving two gauge couplings, with a neutralino and neutral scalars in the loop[184, 38]; these diagrams have two6Rp mass insertions;

(iii) diagrams involving a trilinear6Rp coupling or a Yukawa coupling at one vertex and a gaugecoupling at the other vertex, with a chargino and charged fermions and scalars in the loop;the (g, λ) and (g, λ′) diagrams have one6Rp mass insertion, while the (g, λe) and (g, λd)diagrams have two6Rp mass insertions.

Each of these diagrams contains two6Rp interactions, which can be trilinear (λ andλ′ couplings),mass insertions on lepton or higgsino lines (µi mixing parameters or slepton VEVsvi), LRmixing mass insertions on scalar lines (slepton VEVsvi) or soft 6Rp mass insertions on scalarlines (Bi andm2

di parameters). Note that the mass insertion approximation isvalid only in a

jRBi h;H;A BjeiL ~jLiL g gf0

Figure 5.3: Neutral loop with gauge couplings at the vertices and two6Rp mass insertions on thescalar line. The cross on the neutralino line indicates a Majorana mass insertion. The arrowson external legs follow the flow of the lepton number.

basis in which6Rp masses are indeed small, e.g. in thevi = 0 basis. We adopt this basis for therest of the section.

Depending on the relative size of the various6Rp parameters, some diagrams can be ne-glected. For example, assuming that the trilinear couplings always give a significant contribu-tion, one can consider three representative cases [191] (itmay also happen that the contributionof the bilinear terms in the loops dominates over the contribution of the trilinear couplings, seeRef. [193]):(a) the contribution of bilinear terms in the loops is negligible. This is the situation discussed inthe previous subsection, where only the (λ, λ) and (λ′, λ′) diagrams were considered;(b) the bilinear soft terms, but not the bilinear superpotential masses, induce sizeable loop con-tributions. In addition to the previous diagrams, the neutral loop from class (ii) with two6Rp softmass insertions on the scalar line (see Fig. 5.3) must be taken into account;(c) both the bilinear soft terms and the bilinear superpotential masses induce sizeable loop con-tributions. In this case, all diagrams listed above must be considered a priori.

The contribution of the diagrams of Fig. 5.3 can be estimatedto be [191],

∆Mνij ∼ g2

64π2

BiBj

m3

ǫHcos2 β

, (5.19)

wherem is the typical mass of the particles in the loop, andǫH takes into account the cancel-lation between theh, A andH loops. As noticed in Ref. [193], the different Higgs loops tendto cancel partially, and the cancellation becomes strongerwhen the pseudoscalar Higgs bosonbecomes heavier (ǫH → 0 in the decoupling limitMA → ∞). Furthermore,ǫH decreases whentanβ increases, which softens the dependence of Eq. (5.19) ontan β. Assumingm ∼ 100GeV andǫH/ cos2 β ∼ 0.1, one can see from Eq. (5.19) that soft6Rp masses

√Bi ∼ 1 GeV

are enough to generate neutrino mass matrix entries of the order of the cosmological bound(∆Mν

ij ∼ 1 eV). Thus, neutrino masses do not only constrain the misalignment angle in the

fermion sector (sin ξ . 3×10−6√

1 + tan2 β for mν ≤ 1 eV), but also the misalignment anglein the scalar sector. Namely, one hassin ζ . (10−4 − 10−3) (m/100 GeV)3/2 (100 GeV/

√B)2

for mν ≤ 1 eV, assuming a partial cancellation between the different Higgs loops in the rangeǫH/ cos2 β = 0.01 − 1.

5.2 Explicit Models of Neutrino Masses

We have seen in the previous section how the violation ofR-parity improves our understandingof the generation of neutrino masses. The questions that immediately arise is how well6Rp

models can account for the observed neutrino oscillation parameters, and whether these modelslead to specific experimental signatures that could allow totest them. Numerous studies havebeen devoted to these questions, and we shall not attempt to give an exhaustive account of theexisting literature on the subject. Rather we would like to stress the main characteristics of6Rp

models of neutrino masses through a detailed discussion of some representative examples.

Before doing so, let us summarize the experimental status ofneutrino masses and mixings.

5.2.1 Experimental Constraints on Neutrino Masses and Mixings

Atmospheric neutrino data [194, 195, 196] strongly suggestoscillations of atmosphericνµ’sinto ντ ’s, with a squared mass difference∆m2

atm ≡ m2ν3

− m2ν2

≃ (1.5 − 3.9) × 10−3 eV2,and a large-to-maximal mixing angle,tan2 θ23 = (0.45 − 2.3), both at the3σ level [198]. Theresults of the K2K long-baseline neutrino oscillation experiment [197] are consistent with theseoscillation parameters.

Solar neutrino data [199, 200, 201, 202, 203] combined with the results of the KamLANDexperiment [204] provide evidence for oscillations of solar νe’s into νµ’s and ντ ’s. Beforethe SNO and KamLAND results, four different solutions to thesolar neutrino deficit were al-lowed, out of which three are now excluded. We nevertheless list them for future reference(the following pre-SNO allowed ranges of parameters are taken from Ref. [205]): (i) a smallmixing angle solution (SMA) in the MSW regime, in which neutrino oscillations inside the sunare enhanced by matter effects (known as the Mikheev-Smirnov-Wolfenstein, or MSW effect[206, 207]), with a squared mass difference∆m2

⊙ ≡ m2ν2

− m2ν1

≃ (4 × 10−6 − 10−5) eV2

and a mixing angletan2 θ12 ≃ (10−4 − 2 × 10−3); (ii) a large mixing angle MSW solution(LMA) with ∆m2

⊙ = (10−5 − 5 × 10−4) eV2 and tan2 θ12 ≃ (0.2 − 1.); (iii) a solutionwith low squared mass difference (LOW), which extends to quasi-vacuum oscillations, with∆m2

⊙ = (4 × 10−10 − 4 × 10−7) eV2 and tan2 θ12 ≃ (0.2 − 4.); (iv) a tower of regions inthe vacuum oscillation regime (VO), in which the oscillations occur during the propagation ofthe neutrinos from the sun to the Earth, with∆m2

⊙ ∼ (10−11 − 5 × 10−10) eV2 and a largemixing angle. After the results from the SNO and KamLAND collaborations, LMA is the onlyallowed solution (see e.g. Refs. [208, 209, 210, 211]), withsmaller allowed regions in the os-cillation parameter space:∆m2

⊙ ≃ (5.4−10.)×10−5 eV2 and∆m2⊙ ≃ (14.−19.)×10−5 eV2

(high-LMA region),tan2 θ12 ≃ (0.29 − 0.82) at the3σ level [198].

The CHOOZ reactor experiment [212] provides an upper limit on the mixing angleθ13 thatconnects the solar and atmospheric neutrino sectors,|Ue3| = | sin θ13| < 0.2 (90% CL).

One should also mention the LSND experiment [213], which claims evidence forνµ ↔ νe

oscillations with parameters∆m2 = (0.2 − 2) eV2 and sin2 2θ = (3 × 10−3 − 3 × 10−2).However this result, which cannot be accounted for togetherwith the other neutrino oscillationdata within the standard scheme of three-neutrino oscillation, is still controversial.

Finally, upper bounds on the absolute neutrino mass come from direct mass measurements,mνe < 3 eV [20]; CMB [46] and large scale structure data [47],

∑imνi

. 1 eV; and fromneutrinoless double beta decay experiments which are sensitive to the effective neutrino mass<mν>≡∑imνi

U2ei, found to verify<mν>≤ (0.35 − 1.05) eV (90% CL) [214].

5.2.2 Classification of Models

One can classify the6Rp neutrino mass models according to the pattern ofR-parity violationthat they assume. We shall distinguish between models with trilinear couplings only, modelswith both bilinear and trilinear couplings, and models withbilinear couplings only. In thefirst class of models, bilinearR-parity violation is neglected, and neutrino masses and mixingsarise at the one-loop level. A more realistic variant assumes the absence of bilinearR-parityviolation at the GUT scale, and takes into account the tree-level contribution of the bilinear6Rp

terms generated from the renormalization group evolution.In the second class of models, bothbilinear and trilinear6Rp terms are present, and the neutrino mass matrix receives both tree-leveland loop contributions. In the third class of models,R-parity violation can be parametrized interms of bilinear6Rp couplings only. The neutrino mass matrix receives both tree-level and loopcontributions, as in the second class of models.

5.2.2 a) Models with Trilinear Couplings only

In the limit where bilinearR-parity violation can be neglected, the only contributionstothe neutrino mass matrix come from the (λ, λ), or lepton-slepton, and (λ′, λ′), or quark-squarkloop diagrams, and are given by Eqs. (5.9) and (5.10) (cf. subsections 5.1.3 and 5.1.4). Theseexpressions simplify to Eqs. (5.11) and (5.12) under the (very common) assumptions of propor-tionality of theA-terms to the Yukawa couplings and flavour-independence of sfermion masses.Despite the large number of arbitrary parameters involved in those formulae, trilinearR-parityviolation leads to an interesting structure for the neutrino mass matrix, given by Eq. (5.17),when the hierarchy among trilinear couplings is mild or linked to the fermion mass hierarchy,and theλ-type couplings are not greater than theλ′-type couplings. Indeed the structure (5.17)can account for both the large atmospheric mixing angle (withλ′233 ≈ λ′333) and the hierarchy ofoscillation frequencies∆m2

⊙ ≪ ∆m2atm (with ∆m2

⊙ determined by the subdominant contribu-tions toMν

loop, governed by the charged lepton and down quark mass hierarchies). The correctscale of atmospheric neutrino oscillations is obtained forvalues of the6Rp couplings that couldgive rise to FCNC processes (see chapter 6) and to observablesignals at colliders (see chapter7).

As a prototype of model trying to explain atmospheric and solar neutrino oscillations interms of trilinear6Rp couplings, let us discuss a model by Drees et al. [215]. Besides thestandard assumptions on soft terms, the authors require that all 6Rp trilinear couplings that are notforbidden by a symmetry be of comparable magnitude. While theoretically not well motivated,this hypothesis gives them some control on the subdominant contributions to the neutrino massmatrix. Note that the only way to account for the smallness of|Ue3| consistently with theirhypothesis is to setλ′133 = 0. Then, leaving aside the contribution ofλ couplings, one obtainsat leading order:

Mνloop ∼ 3m2

b

8π2mλ′2333

ms

mb

ms

mb

ms

mbms

mb1 1

ms

mb1 1

, (5.20)

where each entry in Eq. (5.20) should be multiplied by an arbitrary factor of order one, andthe determinant of the lower right2 × 2 sub-matrix is of orderms/mb. This structure is notaltered when the contribution ofλ-type couplings, which involves the charged lepton masses,is included. One typically obtainsmν2/mν3 ∼ ms/mb ∼ 0.04 (takingms = 200 MeV andmb = 5 GeV), yielding a∆m2

⊙ roughly in the MSW range, and a moderate to large solar mixingangle. This points towards the large mixing angle solution,which is precisely the only allowed

solution to the solar neutrino problem after the KamLAND results. Such a scenario wouldrequireλ′ couplings in the(5×10−5−10−4) (m/100 GeV)1/2 range, in order for∆m2

atm ≃ m2ν3

to fall in the allowed interval. Due to the hierarchymν2 ≪ mν3 (a frequent feature of6Rp

neutrino mass models), no measurable signal is expected forneutrinoless double beta decay.The required value of6Rp trilinear couplings, on the other hand, may give rise to sizeable FCNCdecays such asµ → eγ andK0 → µe, depending on the model. Signals at colliders are alsoexpected, due to the decay of the LSP inside the detector – probably without an observabledisplaced vertex (see also Ref. [216]). In order to motivatethe structure of Eq. (5.20), theauthors of Ref. [215] try to find a discrete flavour symmetry allowing for the desired couplings,while forbiddingλ′133 as well as theB-violating couplingsλ′′ijk and the bilinear6Rp parametersµi. They are able to identify such aZ3 symmetry, which however must be explicitly broken bythe strange quark Yukawa coupling. This problem is generic for abelian discrete symmetriesand makes their model less natural.

As discussed in sections 2.2 and 3.4, neglecting bilinearR-parity violation in the presenceof trilinear 6Rp couplings is not always a valid assumption, since the latterinduce all types ofbilinear 6Rp parameters at the one-loop level [30, 31]. In particular, even if only trilinear 6Rp

couplings are present at the GUT scale, the renormalizationgroup induced bilinear6Rp termsgenerally give the dominant contribution to the heaviest neutrino mass. This is the situationstudied by Joshipura and Vempati in Ref. [217]. For simplicity only theλ′ijk couplings andtheir associatedA-terms are considered, and universality among soft terms isassumed at theGUT scale. At the weak scale, bilinear6Rp terms are generated and give a tree-level contri-bution to the neutrino mass matrix, which upon neglecting the scale-dependence of the softparameters in the renormalization group evolution takes the form (Mν

tree)ij = m0 aiaj, whereai ≡

∑k λ

′ikkmdk

/vd in theµi = 0 basis, andm0 is determined by solving the renormalizationgroup equations. At the one-loop level, the trilinear couplings give an additional contribu-tion (∆Mν

loop)ij = m1

∑k,l λ

′iklλ

′jlkmdk

mdl/v2

d, see Eq. (5.12) (other loop contributions areneglected). Depending on the MSSM parameters, the ratiom1/m0 varies between typical val-ues of10−3 and10−1; in some regions of the parameter space cancellations inm0 can lead tom1/m0 > 1. Form1/m0 ∼ (10−2 − 10−1) andλ′ijk ∼ 10−4 (for small values oftanβ), onenaturally obtains a neutrino mass spectrum compatible withboth atmospheric neutrino data andthe – now excluded – vacuum oscillation solution of the solarneutrino problem. MSW solutionscan also be obtained providedm1/m0 ∼ 1, which happens in a particular region of the MSSMparameter space.

Other examples of neutrino mass models based on trilinearR-parity breaking can be foundin Refs. [186, 218, 219, 220]. The last reference also discusses two-loop contributions inducedby both superpotential and soft trilinear6Rp couplings.

5.2.2 b) Models with both Bilinear and Trilinear Couplings

In the presence of all types of6Rp couplings, the neutrino mass matrix receives a tree-levelcontribution from bilinear6Rp terms, Eq. (5.5), and one-loop corrections involving both bilinearand trilinear 6Rp couplings, as explained in section 5.1.3. In practive however most studieshave omitted the loop diagrams containing bilinear6Rp mass insertions (such as the neutral loopdiagrams of Fig. 5.3), thus keeping only the(λ, λ) and(λ′, λ′) loop contributions, Eqs. (5.9)and (5.10) (see however Ref. [187]).

Since flavour symmetries can constrain6Rp couplings, it is interesting to study their pre-dictions for 6Rp models of neutrino masses. The case of aU(1)X flavour symmetry has been

considered by Borzumati et al. [71] and by several other authors (see e.g. Refs. [37, 193, 221]).In this framework the order of magnitude of each6Rp coupling is determined by theX-chargeof the corresponding operator (see section 2.5 for details and notations):λijk ∼ ǫli−l0 λe

jk,

λ′ijk ∼ ǫli−l0 λdjk, m2

αβ ∼ m2ǫ|lα−lβ | andBα/m ∼ µα ∼ m ǫlα , wherem is the typical scale

associated with the soft supersymmetry-breaking terms,ǫ ≃ Vus = 0.22, lα ≡ |lα + hu|, andthe generations of leptons are labelled in such a way thatl0 < l3 ≤ l1,2. The flavour symmetryensures an approximate alignment of the doublet VEVsvα along the superpotential mass pa-rametersµα, resulting in a misalignment anglesin2 ξ ∼ ǫ2(l3−l0); still large leptonX-chargesare necessary in order for neutrino masses to reach the phenomenologically interesting range.Keeping only the (λ, λ) and (λ′, λ′) diagrams at the one-loop level, one finds the followingstructure for the neutrino mass matrix,

Mνij ∼ (m0δi3δj3 +m1) ǫ

li+lj−2l0 , (5.21)

wherem0 ∼ (100 GeV) (100 GeV/m) ǫ2l0 is associated with the tree-level contribution, andm1 ∼ (5 keV) (mb/4.5 GeV)4 (100 GeV/m) ǫ−2l0 with the one-loop contribution. The neu-trino masses and mixing angles are then given bymν3 ∼ m0 ǫ

2(l3−l0), mν2 ∼ m1 ǫ2(l2−l0),

mν1 ∼ m1 ǫ2(l1−l0) andsin θij ∼ ǫ|li−lj | (i 6= j). The mass spectrum is characterized by a large

hierarchymν2 ≪ mν3 . For this model to account for atmospheric and solar neutrino oscilla-tions, one would need both large values of the lepton charges, with l3 = 9 or 10, and l0 ≥ 2,which corresponds totan β & 20 (indeedtan β ∼ ǫ−l0). Since the large atmospheric mixingangle favoursl2 = l3, the – now excluded – low∆m2

⊙ solution is selected forl0 = 2. Then,in order to account for the large solar mixing angle,|Ue3| should be close to its present limit.The large mixing angle solution can be accommodated forl2 = l3 + 1, which is marginallycompatible with the large atmospheric mixing angle, andl0 = 3. This discussion neglects thecontribution of the diagrams with6Rp mass insertions, however. As noticed in Ref. [193], thediagrams of Fig. 5.3 dominate over the (λ′, λ′) diagrams in a large portion of the parameterspace. If one assumes a moderate cancellation between the different Higgs loops, it becomespossible to accommodate the large mixing angle solution to the solar neutrino problem togetherwith the large atmospheric mixing angle.

Other examples of neutrino mass models based on both trilinear and bilinearR-parity break-ing can be found in Refs. [187, 222, 223, 224, 225].

5.2.2 c) Models with Bilinear Couplings only

The above discussion clearly shows that models of neutrino masses based on the mostgeneral6Rp mass terms and couplings suffer from a lack of predictivity.This led several au-thors [39, 40, 185, 186, 189, 226, 227, 228, 229] to consider the so-called “bilinearR-paritybreaking” scenario, in which one assumes that the only seed of R-parity violation resides inthe bilinear superpotential and soft terms (see section 2.2). This scenario yields only onemassive neutrino at tree level with the flavour composition of Eq. (5.7). Radiative correc-tions to the neutralino-neutrino mass matrix then generatethe other two masses and the so-lar mixing angle, while slightly modifying the heaviest neutrino state. In order to accom-modate the atmospheric neutrino mass scale, these studies generally assume universality ofthe soft terms at the GUT scale [39, 40, 189] (or at the messenger scale in the context ofgauge-mediated supersymmetry breaking [185, 186]) together with the smallness of the6Rp

parameters, with typicallyµi/µ (MGUT ) ≤ 10−3. The second assumption is crucial for ob-tainingmνtree =

√∆m2

atm ∼ 0.05 eV. Indeed, while the universality conditions lead to an

exact alignment between the superpotential mass parameters µα and the doublet VEVsvα atthe GUT scale, radiative corrections induce some amount of non-universality among the softterms at the weak scale, which spoils this alignment and induces a nonzero neutrino mass.For µi (MGUT ) ∼ µ (MGUT ), however, the resulting neutrino mass would lie in the range100 eV ≤ mνtree ≤ 100 MeV [39, 40], well above the atmospheric neutrino scale. This isthe reason whyµi ≪ µ is required atMGUT . Note that the universality assumption reducesthe number of independent6Rp parameters to only 3, which one can choose to be the three6Rp

supersymmetric masses at the GUT scale,ǫi ≡ µi (MGUT ).

A detailed study of this model has been presented in Ref. [189]. The authors define analignment vectorΛi ≡ (µvi − vdµi)|Mweak

, which parametrizes the misalignment induced atthe weak scale by the renormalization group evolution of thesoft terms (in theǫi ≪ µ limitthat we are considering, this alignment vector is related tothe misalignement angleξ definedin subsection 2.3.1 by

∑i Λ

2i ≃ µ2v2

d sin2 ξ). The tree-level neutrino mass matrix is givenby Eq. (5.5), with the replacementµi → Λi. In the regime where

∑i ǫ

2i ≪

√∑i Λ

2i and

ǫ2Λ2/ǫ3Λ3 < 0, the one-loop corrections are small and do not spoil the structure of the tree-levelneutrino mass matrix . Thus the atmospheric neutrino parameters are essentially determined bythe Λi, with mν3 ∼ ∑

i Λ2i /µ

2M2, tan θ23 ≈ Λ2/Λ3 and |Ue3| = sin θ13 ≈ Λ1/√∑

i Λ2i .

Consistency with experimental data requiresΛ1 ≪ Λ2 ≈ Λ3 and√∑

i Λ2i ∼ 0.1 GeV2 (the

second constraint depends on the values of the supersymmetry parameters). As for the solarneutrino parameters, they are determined by the one-loop corrections to the neutralino-neutrinomass matrix, controlled by the ratiosǫi/µ : ∆m2

21 is a function of√∑

i ǫ2i /µ, whileθ12 depends

onǫ1/√ǫ22 + ǫ23. In the case of universal boundary conditions atMGUT , theΛi/Λj are correlated

with theǫi/ǫj, so that the CHOOZ limit on|Ue3| constrains the solar mixing angle to be small.A departure from the universality hypothesis is therefore necessary to accommodate the largemixing angle solution (see Ref. [230] for a more recent analysis of this model).

Interestingly, this scenario can be checked at colliders such as the LHC or a future linearcollider [50, 51, 189]. Indeed, for the required values of the ǫi, the lightest neutralino (assumedto be the LSP) should decay within the detector. Furthermore, the ratios of branching ratiosfor semi-leptonic decays into different charged leptons show some correlation with the leptonmixing angles. In particular, BR(χ0

1 → µqq′)/BR(χ01 → τqq′) is strongly correlated with

tan θ23, irrespective of the lightest neutralino mass [50, 189]. The experimentally allowed rangefor the atmospheric mixing angle indicate that this ratio should be of order one. Similarly,BR(χ0

1 → eqq′)/BR(χ01 → τqq′) and to a smaller extent BR(χ0

1 → eτνi)/BR(χ01 → µτνi)

are correlated with|Ue3| andtan θ12, respectively [51]. Since the solar mixing angle is large,BR(χ0

1 → eτνi)/BR (χ01 → µτνi) should be of order one. All above branching ratios, except

BR(χ01 → eqq′), are larger than(10−4−10−3) and it should be possible to measure them. Other

collider signatures of this scenario are discussed in Refs.[52, 54, 231].

The weakness of this scenario is that there is no a priori reason for the absence of trilinear6Rp couplings, nor for the smallness of theǫi. To cure this problem, one may invoke spontaneousR-parity breaking, or an abelian flavour symmetry [232].

A model with soft bilinear couplings only

In Ref. [233], another option has been explored, namely the possibility thatR-parity is bro-ken by soft bilinear terms only. This scenario has 6 parameters (3Bi and 3m2

di, or equivalently3 Bi and 3vi) but does not assume anything about the structure of the softterms, contrary tothe previous “bilinearR-parity breaking” scenario, whose predictivity relies on the universal-ity assumption at the GUT scale. In addition to the tree-level contribution, the neutrino mass

matrix receives contributions from the neutral loop diagrams from class (ii), in the classifica-tion of subsection 5.1.4. The other loop contributions, class (i) and class (iii), are negligiblefor low values oftanβ. The tree-level and loop contributions are governed by the quantitiesδiµ ≡ vi/vd andδi

B ≡ (Bvi − Bivd)/vd

√B2 +

∑iB

2i , respectively. To make the connection

with the misalignment anglesξ andζ defined in subsection 2.3.3, note that∑

i(δiµ)2 = sin2 ξ

and∑

i(δiB)2 = sin2 ζ .

In the limit where the subleading loop diagrams are neglected, only two neutrinos are mas-sive. Depending on the values of the various parameters, theheaviest neutrino mass is deter-mined either by the tree-level or the loop contribution. Both atmospheric and solar neutrino datacan be accommodated, but the required values of the soft bilinear6Rp parameters are very small,with |δi

µ| ≤ 8 × 10−7 and|δiB| ≤ 3 × 10−5. In the absence of a specific mechanism that would

explain the weakness ofR-parity violation in the soft terms, it is difficult to motivate such smallvalues.

5.3 Neutrino Transition Magnetic Moments

Massive neutrinos can have magnetic dipole moments (and also electric dipole moments ifCPis violated). Since the magnetic moment of a neutrino is induced by loop diagrams involving achirality flip, it is generally proportional to its mass and therefore very tiny. For instance, a Diracneutrino with no interaction beyond the Standard Model hasµν = 3eGFmν/8

√2π2 ≃ 3.2 ×

10−19 (mν/ eV)µB [234, 235], whereµB ≡ e/2me is the Bohr magneton, to be compared witha laboratory limit ofµν < 1.0×10−10 µB at90% C.L. [20]. In the case of Majorana neutrinos –relevant for supersymmetry withoutR-parity – only transition dipole momentsµνiνj

, i 6= j areallowed (the coefficientsµνiνj

of the effective operatorsνiσµννjFµν are antisymmetric in(i, j)

due to the Majorana nature of the neutrino). These correspond to transitions between a left-handed neutrino and a right-handed (anti)neutrino with different flavours, and mediate radiativeneutrino decaysνi → νjγ. Neutrino transition magnetic moments have several implications inastrophysics; in particular, they can induce spin-flavour transitions such asνeL → νc

µR or νµL →νc

eR in the solar magnetic field [236, 237]. However this possibility is strongly constrained bysolar neutrino and KamLAND data, from which the boundµν . (10−12 − 10−11)µB, similarto other astrophysical bounds, has been derived [238]. Still obtaining such a large value forµν

while keeping small neutrino masses in explicit models is a theoretical challenge.

In 6Rp models, transition magnetic moments are generated from thetrilinear couplingsλijk

andλ′ijk via lepton-slepton and quark-squark loops [182, 239]. The corresponding diagramsonly differ from the ones responsible for neutrino masses byan additional photon vertex at-tached to an internal line. As a consequence, an upper bound on µνiνj

depending on the neu-trino masses can be derived. Barring accidental cancellations between different contributionsto the neutrino mass matrix, and assuming a conservative upper bound of10 eV for eachMν

ij ,one finds [239]|µνeνµ| ≤ O (10−13µB) for light sleptons (squarks), much above the StandardModel value. This bound can be saturated only if the mass scales involved in atmospheric andsolar neutrino oscillations result from accurate degeneracies among neutrino masses.

An interesting mechanism for avoiding the constraining proportionality between the massand magnetic moment of a Dirac neutrino is to postulate an approximateSU(2)ν symmetryunder whichνL andνc

L (theCP conjugate ofνR) form a doublet [240]. The Lorentz structureof interaction terms is then such that the electromagnetic dipole moment operatorνT

LCσµννc

L

is antisymmetric underνL ↔ νcL and transforms as a singlet ofSU(2)ν , while the Dirac mass

operatorνTLCν

cL is symmetric and transforms as a triplet. Thus, in theSU(2)ν symmetric limit,

µν can be nonzero whilemν vanishes.

Babu and Mohapatra [241] have generalized this mechanism toMajorana neutrinos by re-placingSU(2)ν with an horizontalSU(2)H flavour symmetry acting on the first two generationsof leptons. In a6Rp model [182], they consider a discrete version of this symmetry, namely theZ2 flavour group acting on the electroweak doublet and singlet lepton fields of the first two gen-erations as(Le, Lµ) → (Lµ,−Le), (ec

L, µcL) → (µc

L,−ecL), with all other fields left invariant.

In combination with the assumption of conservation of the lepton number differenceLe − Lµ,this leads to a large transition magnetic momentµνeνµ together with vanishing masses for thefirst two generation neutrinos. At the same time, however, theZ2 symmetry yieldsme = mµ.This can be cured by assuming a soft breaking ofZ2 in the slepton sector. Then a contributionto the transition moment as large asµνeνµ ≈ (10−11 − 10−10)µB, consistent with light neutrinomasses,mνe , mνµ < 10 eV, and with the observed value of the splitting between the electronand muon masses,mµ −me, can be achieved at the price of a fine-tuning in the slepton massmatrices.

Motivated by the desire of avoiding an unnatural fine-tuning, Barbieri et al. [239] considerthe continuousSU(2)H flavour symmetry to be broken solely by the Yukawa couplings of theelectron and the muon. The mismatchλe

11 6= λe22 results in the splitting(me2

LR)11 − (me2

LR)22 6= 0

necessary to generate masses for the first two generation neutrinos, while a large transitionmagnetic momentµνeνµ can be obtained by requiring(me2

LR)11 ≃ (me2

LR)22 ≫ |(me2

LR)11 −

(me2LR

)22|. However(me2LR

)11 is bounded by its contribution to the one-loop corrections to theelectron mass, resulting in the upper boundµνeνµ ≤ O (10−12µB). Therefore, even with the helpof suitable symmetries, the neutrino transition magnetic moment generated by6Rp couplingshappens to be at least 2 orders of magnitude smaller than the present laboratory upper bound.

5.4 Neutrino Flavour Transitions in Matter Induced by 6Rp

Interactions

The oscillations of neutrinos in matter are affected by their interactions with the medium. Themost familiar illustration of this phenomenon is the Mikheev-Smirnov-Wolfenstein (MSW)mechanism [206, 207], i.e. the enhancement of neutrino oscillations inside the sun due totheir coherent forward scatterings on electrons and nucleons. Since the charged current interac-tions only contribute to scatterings of electron neutrinos, the electron neutrinos on one side, andthe muon and tau neutrinos on the other side have different scattering amplitudes on electrons,which results in oscillation parameters in matter different from the oscillation parameters invacuum.

Similarly, any non-standard interaction of neutrinos withthe charged leptons and downquarks, such as6Rp interactions [242, 243], modifies neutrino oscillations inmatter. For instancethe couplingsλ′131 andλ′331 contribute to the scattering processesνed → νed andντd → ντd,respectively, and the corresponding amplitudes are different as soon asλ′131 6= λ′331. Moreoverthese couplings, if both present, induce the flavour-changing scatteringsνed → ντd. It followsthat 6Rp interactions can induce flavour transitions of neutrinos inside the sun or the Earth, evenif neutrinos are massless and do not oscillate in vacuum.

Several authors have studied the possibility of accountingfor the solar and atmosphericneutrino data with6Rp-induced flavour transitions, or more generally with non-standard neu-trino interactions. While flavour-changing non-standard neutrino interactions can only play a

subleading role with respect to oscillations in the atmospheric neutrino sector [244], several au-thors have found that they could be responsible for the solarneutrino deficit (for recent studies,see Refs. [245, 246, 247, 248, 249]), although the case of pure 6Rp-induced flavour transitionsis strongly disfavoured by the SNO data [249]. However, the KamLAND results have showedthat neutrinos in the solar neutrino energy range oscillatein vacuum with parameters consistentwith the large mixing angle MSW solution, leaving only the possibility that 6Rp-induced flavourtransitions contribute as a subdominant effect.

5.5 ∆L = 2 Sneutrino Masses and Sneutrino-AntisneutrinoMixing

Supersymmetry breaking∆L = 2 sneutrino mass terms, parametrized by the Lagrangian terms−1

2(m2

∆L=2)ij νiνj + h.c. [184, 250], are expected in any supersymmetric model with nonzeroneutrino Majorana masses. These terms induce a mass splitting and a mixing between the sneu-trino and the antisneutrino of a same generation, which gives rise to characteristic experimentalsignatures such as sneutrino-antisneutrino oscillations. In the one-generation case, the sneutrinomass splitting is given by∆m2

ν ≡ m2ν2−m2

ν1= 2 m2

∆L=2, where the mass eigenstatesν1 andν2 are linear combinations ofν andνc.

In supersymmetric models with bilinearR-parity violation, the sneutrinos mix with theneutral Higgs bosons, which leads to6L sneutrino-antisneutrino mixing at the tree level. Underthe assumption ofCP conservation, it is convenient to deal with the sneutrinoCP eigenstatesrather than with the lepton number eigenstatesν andνc. TheCP -even sneutrinosν+i ≡ (νi +νc

i )/√

2 mix with the CP -even Higgs bosonsh andH, and theCP -odd sneutrinosν−i ≡−i(νi − νc

i )/√

2 with theCP -odd Higgs bosonA and with the Goldstone boson that in theabsence ofR-parity violation is absorbed by theZ boson. As a result the mass degeneracybetween theCP -even andCP -odd sneutrinos of each generation is broken. The sneutrinomass splitting within each generation reads, if one neglects flavour mixing [38]:

∆m2νi

≡ m2ν+i

−m2ν−i

=4B2

i M2Z m

2νi

sin2 β

(m2νi−m2

H)(m2νi−m2

h)(m2νi−m2

A), (5.22)

wherem2νi

= (M2L)ii + µ2

i − 18(g2 + g′2)(v2

u − v2d), and as usual we are working in a (Hd, Li)

basis in which the sneutrino VEVsvi vanish. In this basis∆m2νi

is proportional to the square ofthe 6Rp soft termsBi; therefore the contribution of bilinearR-parity violation to the sneutrino-antisneutrino mixing is controlled by the misalignment between the 4-vectorsBα ≡ (B0, Bi)andvα ≡ (v0, vi) (see subsection 2.3.1). At the one-loop level,∆m2

νireceives additional con-

tributions from the trilinear6Rp couplingsλ andλ′ and from their associated soft parametersAandA′. In practice∆m2

νi≪ m2

νiand one can write∆mνi

≡ mν+i−mν−i

≃ ∆m2νi/ 2mνi

.

Sneutrino-antisneutrino mixing and neutrino masses are closely linked at the one-loop order.A Majorana neutrino mass term induces radiatively a sneutrino mass splitting term and vice-versa. Taking these effects into account, one finds [38] thatfor generic model parameters thesneutrino mass splitting to neutrino mass ratio∆mν

mνfalls in the interval

10−3 .∆mν

mν. 103 . (5.23)

Cancellations between the tree-level and one-loop contributions tomν may enhance this ratio,thus allowing for larger sneutrino mass splittings at the price of a fine-tuning. Furthermore therearise strong pairwise correlations, of nearly linear character, between the contributions to theMajorana neutrino masses, the∆L = 2 sneutrino masses and the neutrinoless double beta decayamplitudes. Hirsch et al. [251] find in the framework of theR-parity conserving Supersymmet-ric Standard Model that the induced effect ofm∆L=2 on neutrinoless double beta decay imposesthe boundm∆L=2 < 2 GeV (m/100 GeV)3/2, resp.m∆L=2 < 11 GeV (m/100 GeV)7/2, in theextreme case where the lightest neutralino is a pure bino, resp. a pure higgsino (all superpartnermasses assumed equal tom). Another bound,m∆L=2 < (60 − 125) MeV (mν/1 eV)

12 , is as-

sociated with the one-loop contribution to the neutrino mass induced bym∆L=2. These bounds,which can be converted into bounds on∆mν via ∆mν ≃ m∆L=2/mν , also apply in the6Rp case.

The phenomenological implications of the sneutrino-antisneutrino mass splittings and mix-ings have been examined in recent works [184, 38, 250]. For large mass splittings,∆mν >1 GeV, the sneutrino pair production at colliders could be tagged through the leptonic decaysof the sneutrinos resulting in characteristic charged dilepton final states [184] through the decaymodesν → e±χ∓. Interesting signals could also arise from the resonant sneutrino or antis-neutrino production [252] ine+e− or qq collisions. The corresponding off-shell sneutrino orantisneutrino exchange processes could also be observed via the fermion-antifermion pair pro-duction reactions,e+e− → ν, νc → f f , at high energy lepton colliders and similarly at hadroncolliders [38, 253].

For small mass splittings,∆mν << 1 GeV, sneutrino-antisneutrino oscillations could riseto a measurable level provided that the oscillation time is shorter than the sneutrino lifetime,corresponding toxν ≡ ∆mν

Γν> 1. By analogy with theB − B system, the production of a

sneutrino-antisneutrino pair would be signaled by characteristic like-sign dileptons, providedthat the branching ratios of the decay modesν → e±χ∓ are appreciable [184].

Bar-Shalom et al. [253] have developed an interesting test for the 6Rp-induced resonant pro-duction of sneutrinos at leptonic and hadronic colliders. Based on the current sensitivity reaches,a mere observation of deviations with respect to the Standard Model predictions for the tau-antitau lepton pair production reactionse+e− → ν → τ+τ−, resp.pp→ ν → τ+τ−+X, wouldlead to bounds on coupling products of the formλ232λ

′311 < 0.003, resp.λ232λ

′322 < 0.011. In

the presence ofCP violation among6Rp couplings, theν-νc mixing could contribute to theCP -odd double spin correlation observables associated with the spin polarisation of theτ+τ− pair.Nonzero and large contributions arise already at the tree level, thanks to the complex phase de-pendence provided by the spatial azimuthal angle. AnalogousCP -even double spin correlationobservables may also be induced through the same type of6Rp interactions.

Altogether, we have seen how the violation ofR-parity in supersymmetric theories naturallyleads to massive neutrinos. The fact that6Rp models automatically incorporate massive neutri-nos, while allowing for observable signals at colliders forthe values of6Rp couplings suggestedby neutrino data, is certainly an appealing feature ofR-parity violation. However, the smallnessof neutrino masses dictates strong constraints on6Rp couplings, especially on bilinear6Rp cou-plings, while 6Rp supersymmetric models also suffer from a lack of predictivity in the neutrinosector. This motivates the study of restricted scenarios such as the “bilinearR-parity break-ing” scenario, which are more predictive. We have also seen that, given the acceptable neutrinomasses, the neutrino transition magnetic moments generated from 6Rp couplings generally liewell below the current laboratory upper limit. Finally,6Rp couplings induce∆L = 2 sneutrinomass terms, which leads to mass splittings and mixings between sneutrinos and antisneutrinosand gives rise to sneutrino-antisneutrino oscillations.

Chapter 6

INDIRECT BOUNDS ON R-parity ODDINTERACTIONS

The assumption of a brokenR-parity introduces in the Supersymmetric Standard Model newinteractions between ordinary particles and supersymmetric particles, which can contribute toa variety of low, intermediate and high energy processes that do not involve the production ofsuperpartners in the final state. Requiring that the6Rp contribution to a given observable doesnot exceed the limit imposed by the precision of the experimental measurement and by thetheoretical uncertainties on the Standard Model prediction (or, for a process that has not beenobserved, that it does not exceed the experimental upper limit), yields upper bounds on the6Rp

couplings involved. In addition to the bounds associated with renormalization group effectsand with astrophysics and cosmology, discussed in chapters3 and 4, and with the direct limitsextracted from supersymmetry searches at colliders, to be discussed in chapter 7, these indirectbounds provide useful information on the possible patternsof R-parity violation.

In this chapter, we give a comprehensive review of the indirect bounds on6Rp interactionscoming from low, intermediate and high energy particle phenomenology, as well as from nuclearand atomic physics observables. This complements and updates other existing reviews, e.g.Refs. [254, 255, 256, 257, 258, 259, 260].

The chapter is organised in five main sections. In section 6.1, the assumptions under whichthe bounds on6Rp couplings have been extracted are presented, and issues concerning quark andlepton superfield bases are addressed, in particular in connection with the single coupling dom-inance hypothesis. A basis-independent parametrization of R-parity violation is also presented,as an alternative to the choice made in this review. Section 6.2 deals with the constraints onbilinear 6Rp parameters, both for an explicit breaking and in the case of aspontaneousR-paritybreaking. Section 6.3 reviews the indirect bounds on trilinear6Rp couplings associated with fun-damental tests of the Standard Model in charged current and neutral current interactions, withCP violation and with high precision measurements of electroweak observables. In section 6.4the constraints on trilinear6Rp interactions coming from a variety of hadron flavour or leptonflavour violating processes, andB or L violating processes are presented. Finally, section 6.5provides a list of the indirect bounds on trilinear6Rp couplings presented in chapters 3 and 5 andin this chapter. The robustness and significance of these bounds are discussed.

6.1 Assumptions and Framework

Our main focus in this chapter will be on theR-parity odd renormalizable superpotential cou-plingsµi, λijk, λ′ijk andλ′′ijk, defined in Eq. (2.2). The simultaneous presence of bilinearandtrilinear 6Rp terms results in an ambiguity in the choice of a basis for the down-type Higgs andlepton doublet superfields (Hd, Li), an issue already addressed in subsections 2.1.4 and 2.3.1.In this chapter we adopt a superfield basis in which the VEVs ofthe sneutrino fields vanish and,unless otherwise stated, the charged lepton Yukawa couplings are diagonal (see the discussionat the beginning of subsection 6.2.1).

6.1.1 The Single Coupling Dominance Hypothesis

Most of established indirect bounds on the trilinear couplingsλijk, λ′ijk andλ′′ijk have beenderived under the so-called single coupling dominance hypothesis, where a single6Rp couplingis assumed to dominate over all the others. This useful working hypothesis can be rephrasedby saying that each of the6Rp couplings contributes one at a time [254, 261]. The perturbativeconstraints arising from contributions at loop orderl then involve upper bounds on combinationsof 6Rp couplings and superpartner masses of the formλp

mq ( e2

(4π)2)l, whereλ, e, m are generic

symbols for 6Rp couplings, gauge couplings and superpartner masses, and the power indicesp ≥ 2, q ≥ 2 depend on the underlying mechanism. Apart from a few isolated cases, the boundsderived under the single coupling dominance hypothesis arein general moderately strong. Thetypical orders of magnitude areλ, λ′, λ′′ < (10−2−10−1) m

100 GeV, involving generically a lineardependence on the superpartner mass.

The constraints based on the single coupling dominance hypothesis deal with a somewhatrestricted set of applications, such as charged current or neutral current gauge interactions,neutrinoless double beta decay and neutron-antineutron oscillation. By contrast, a much largerfraction of the current constraints on6Rp interactions are derived from extended hypotheseswhere the dominance is postulated for quadratic or quartic products of couplings. The literatureabounds with bounds involving a large variety of flavour configurations for quadratic productsof the coupling constants. The processes that yield constraints on products of6Rp couplings canbe divided into four main classes: (i) hadron flavour changing processes, such as oscillations ofneutral flavoured mesons, and leptonic or semileptonic decays ofK orB mesons likeK → eiej

andK → πνν; (ii) lepton flavour changing processes, such asµ− → e− conversion in nuclei, orradiative decays of charged leptons; (iii)L-violating processes, such as neutrinoless double betadecay, neutrino Majorana masses and mixings (cf. chapter 5), or three-body decays of chargedleptons,l±l → l±l−n l

+p ; (iv) B-violating processes, such as nucleon decay, neutron-antineutron

oscillations, double nucleon decay or some rare decays of heavy mesons.

6.1.2 Choice of the Lepton and Quark Superfield Bases

When discussing specific bounds on coupling constants, it isnecessary to choose a definite basisfor quark and lepton superfields, especially if the single coupling dominance hypothesis is used.Two obvious basis choices can be made for quark superfields: the current (or weak eigenstate)basis, in which left-handed quarks have flavour diagonal couplings to theW gauge boson,and the “super-CKM” basis, in which quark mass matrices are diagonal. A similar distinctionbetween weak eigenstate basis and mass eigenstate (or “super-MNS”) basis exists for leptons

when neutrino masses are taken into account. In most studies, it is tacitly understood that thesingle coupling dominance hypothesis applies in the mass eigenstate basis. It may appear morenatural, however, to apply this hypothesis in the weak eigenstate basis when dealing with modelsin which the hierarchy among (weak-eigenstate-basis) couplings originates from some flavourtheory. In this case, a single process may allow to constrainseveral couplings, provided one hassome knowledge of the rotations linking the weak eigenstateand mass eigenstate bases [262].

It is useful to write down the trilinear6Rp superpotential terms in the two superfield bases. Letus first consider theLQDc terms. We denote byλ′ijk the corresponding couplings expressed inthe weak eigenstate basis, and byλ

′Aijk andλ

′Bijk the same couplings in two useful representations

of the mass eigenstate basis:

W (λ′) = λ′ijk(NiDj − EiUj)Dck = λ

′Aijk(NiD

′j − EiUj)D

ck = λ

′Bijk(NiDj − EiU

′j)D

ck ,

λ′Aijk = λ′imn(V u†

L )mj(VdTR )nk , D′

j = VjlDl ,

λ′Bijk = λ′imn(V d†

L )mj(VdTR )nk , U ′

j = V †jlUl . (6.1)

In (6.1), hatted superfields are in the weak eigenstate basisand unhatted superfields in the masseigenstate basis;V u

L , V dL andV d

R are the matrices that rotate, respectively, the left-handed upquarks, the left-handed down quarks and the right-handed down quarks to their mass eigenstatebasis (one has e.g.Ui =

∑j(V

uL )ijUj); andV = VCKM = V u

L Vd†L is the Cabibbo-Kobayashi-

Maskawa (CKM) matrix. Since the three sets of couplingsλ′A, λ

′B andλ′ are related by unitarymatrices, the following sum rules hold:

∑jk |λ

′Aijk|2 =

∑jk |λ

′Bijk|2 =

∑jk |λ′ijk|2. The hypoth-

esis of a single dominant6Rp coupling, when applied to the coupling setsλ′Aijk or λ′B

ijk, mayallow for flavour changing transitions in the down quark or upquark sectors, respectively, butnot in both simultaneously. The flavour mixing may be formally obtained by replacements ofthe formb → b′ = V33b+ V32s + V31d in the former case, andt → t′ = V †

33t + V †32c + V †

31u inthe latter case (with similar relations for the first two generations).

Analogous expressions can be written for theLLEc interaction terms:

W (λ) =1

2λijk(NiEj − EiNj)E

ck =

1

2λA

ijk(NiE′j − E ′

iNj)Eck =

1

2λB

ijk(N′iEj − EiN

′j)E

ck ,

λAijk = λlmn(V ν†

L )li(Vν†L )mj(V

eTR )nk , E

′j = V ′

jmEm ,

λBijk = λlmn(V e†

L )li(Ve†L )mj(V

eTR )nk , N

′i = V

′†il Nl , (6.2)

whereV νL , V e

L andV eR are the rotation matrices for left-handed neutrinos, left-handed charged

leptons and right-handed charged leptons, respectively, and the Maki-Nakagawa-Sakata matrixis UMNS = V

′† = V eLV

ν†L . For theU cDcDc interactions, the mass eigenstate basis couplings

λ′′ijk are related to the current basis couplingsλ′′ijk by λ′′ijk = λ′′lmn(V uTR )li(V

dTR )mj(V

dTR )nk.

As mentioned above, if the single coupling dominance hypothesis applies in the weak eigen-state basis, several bounds may be derived from a single process. Indeed, let us assume that, inthe weak eigenstate basis, the single6Rp couplingλ′IJK is dominant. Then, in the mass eigen-state basis, this coupling generates the operatorλ

′AIJKEIUJD

cK and, due to the flavour mixing,

subdominant operatorsλ′AijkEiUjD

ck, (i, j, k) 6= (I, J,K), with couplingsλ

′Aijk suppressed rela-

tive toλ′AIJK by fermion mixing angles. Explicitly, we haveλ′IJKEI UJD

cK ≃ λ′IJKEIUJD

cK +∑

i6=I(Ve⋆L )iI λ

′IJK EiUJD

cK +

∑j 6=J(V u⋆

L )jJ λ′IJK EIUjD

cK +

∑k 6=K(V d

R)kKλ′IJK EIUJD

ck +

· · · , where we have assumed that the rotation matricesV(e,u,d)L,R are close to the unit matrix.

Sinceλ′AIJK ≃ λ′IJK, the following relations among couplings in the mass eigenstate basis

hold (from now on, we drop the upper indexA in the mass eigenstate basis couplingsλ′Aijk.):

λ′iJK ≃ (V e⋆L )iIλ

′IJK , λ′IjK ≃ (V u⋆

L )jJλ′IJK , λ′IJk ≃ (V d

R)kKλ′IJK , etc. It may then hap-

pen that the most severe bound on the dominant couplingλ′IJK comes from a process in-volving a subdominant operatorEiUjD

ck. For instance an upper bound on the couplingλ′IjK,

|λ′IjK| < |λ′IjK|upper, yields|λ′IJK | < |λ′IjK|upper / |(V u⋆L )jJ |. This constraint may be stronger

than bounds extracted from processes involving the dominant operatorEIUJDcK .

More generally, one can derive a sequence of bounds on6Rp couplings from a single pro-cess without any reference to the single coupling dominancehypothesis, provided that one hassome knowledge of the rotation matricesV (e,u,d)

L,R . For instance, in the model of Ref. [74],in which the quark rotation matrices are controlled by an abelian flavour symmetry, the con-straintsλ′111 < 0.002 andλ′133 < 0.001 (associated with the non-observation of neutrinolessdouble beta decay and with the upper limit on the electron neutrino mass, respectively, and de-rived assuming superpartner masses of200 GeV) are used to derive upper bounds on allλ′1jk

couplings. Combining the latter with the bounds extracted from other observables, one obtainsλ′1jk < (10−3−2×10−2), depending on the coupling. Such bounds obtained from flavour mix-ing arguments, however, are model-dependent. In Ref. [88],upper bounds on theλ′ijk couplingsare derived from the cosmological neutrino mass bound undervarious assumptions regardingthe quark rotation matrices and(V e

L,R)ij = δij .

The same arguments allow to transform a bound on a product of6Rp couplings into a boundon a single coupling. For instance, in the model of Ref. [74],the contraint|λ′⋆

i13λ′i31| < 3.2 ×

10−7, associated withB − B mixing, yields|λ′i13| < 6 × 10−4 (again for sfermion masses of200 GeV).

6.1.3 A Basis-Independent Parametrization ofR-Parity Violation

As explained in subsection 2.1.4, there is no a priori distinction between the (Y = −1) Higgssuperfield and the lepton superfields in the absence ofR-parity and lepton number. The choiceof a basis forHd and theLi becomes even a delicate task if bilinear6Rp terms and/or sneutrinoVEVs are present. As a result, the definition of the6Rp parameters is ambiguous (except forthe baryon number violating couplingsλ′′ijk, which are not discussed here). This ambiguity isremoved either by choosing a definite (Hd, Li) basis, as explained in subsection 2.3.1, or byparametrizingR-parity violation by a complete set of basis-invariant quantities instead of theoriginal 6Rp Lagrangian parameters.

In this subsection, we briefly present the second approach, as an alternative to the choicemade in this review1. We use the notations of subsection 2.3.1, with the 4 doubletsuperfieldsHd andLi grouped into a 4-vectorLα, α = 0, 1, 2, 3. The renormalizable, baryon number con-serving superpotential then readsW = µαHuLα+ 1

2λe

αβkLαLβEck+λ

dαpqLαQpD

cq+λ

upqQpU

cqHu.

The couplingsλeαβk = (λk

e)αβ , λdαpq = (λpq

d )α andµα define 3 matrices and 10 vectors in the

Lα field space (each value ofk defines a4 × 4 matrix, and each value of (p, q) a 4-vector).In addition, the6Rp soft termsBα, m2

αβ, Aeαβk = (Ak

e)αβ andAdαpq = (Apq

e )α define 4 matricesand 10 vectors. At this stage the “ordinary” Yukawa couplings and the trilinear6Rp interactionscannot be disentangled; the distinction between lepton number conserving and lepton numberviolating interactions arises once a direction in theLα field space has been chosen for the Higgssuperfield.

1Although we frequently use the basis-independent quantitiessin ξ andsin ζ to measure the total amount ofbilinearR-parity violation in the fermion and scalar sector, respectively (cf. subsection 2.3.1), we do not make asystematic use of invariants to parametrizeR-parity violation.

However, an intrinsic definition ofR-parity violation is possible with the help of basis-invariant products of the6Rp parameters [26, 27]. These invariants, being independent of thebasis choice in theLα field space, can be defined in a geometrical way. Let us give an exampleof an invariant involving only superpotential parameters,together with its geometrical interpre-tation. We first notice that each of the 10 vectors(λpq

d )α andµα defines a (would-be) Higgsdirection in theLα field space (if(Hd)α ∝ (λpq

d )α, all λ′ijk vanish, and if(Hd)α ∝ µα, all µi

vanish). If the directions defined byµα and one of the(λpqd )α differ, R-parity is violated. An-

other, slightly different criterion forR-parity violation is when one of the(λpqd )α has a nonzero

component along one of the vectors∑

β µ⋆β(λk

e)βα (to be interpreted as the three lepton direc-tions, labeled by the flavour indexk, the direction for the Higgs superfield being defined byµα).The projection of the vector(λpq

d )α on the direction∑

β µ⋆β(λk

e)βα is then a measure ofR-parityviolation in thekth lepton number. The corresponding invariant can be defined as[26]:

δkpq1 =

|µ†λkeλ

pq⋆d |2

|µ|2|λke |2|λpq

d |2 , (6.3)

whereµ†λkeλ

pq⋆d ≡∑α,β µ

⋆α(λk

e)αβ(λpqd )⋆

β, |µ|2 ≡∑α |µα|2, |λke |2 ≡

∑α,β |(λk

e)αβ |2 and|λpqd |2 ≡∑

α |(λpqd )α|2. If δkpq

1 6= 0 for some (p, q),R-parity and thekth lepton number are violated (sum-mingδkpq

1 over the right-handed lepton indexk provides a measure of the breaking of total leptonnumber). However,δkpq

1 = 0 for all (k, p, q) does not imply the absence ofR-parity violationfrom the superpotential, since other invariants involvingthe 6Rp parameters(λk

e)αβ, (λpqd )α and

µα can be constructed.

The invariants that may be constructed out of the6Rp Lagrangian parameters are much morenumerous than the6Rp parameters themselves, but not all of them are independent.After remov-ing the redundancies, one finds a set of 36 independent invariants parametrizingR-parity viola-tion from the superpotential. Once soft supersymmetry breaking terms are included, one obtainsa total number of 78 independent invariants, in agreement with the counting of independent6Rp

parameters that do not break baryon number (3µi, 9 λijk and 27λ′ijk in the superpotential; 3Bi, 3 m2

di, 9Aijk and 27A′ijk in the scalar potential; but due to the freedom of redefining the

(Hd, Li) basis, only 6 among the 9 bilinear6Rp parameters are physical).

The basis-independent approach has been used for the derivation of cosmological boundsonR-parity violation [26, 27] and for the computation of neutrino masses and mixings in6Rp

models [190, 191, 233].

6.1.4 Specific Conventions Used in this Chapter

In presenting numerical results for coupling constants, weneed at times to distinguish betweenthe first two families and the third. For this purpose, when quoting numerical bounds only,we assume the following conventions for the alphabetical indices:l,m, n ∈ [1, 2] andi, j, k ∈[1, 2, 3]. The mass of superpartners are fixed at the reference value ofm = 100 GeV unlessotherwise stated. A notation likedp

kR in a numerical relationship, such asλ′ijk < 0.21 dpkR,

stands for the expression,λ′ijk < 0.21 (m

dkR

100GeV)p.

The following auxiliary parameters for the6Rp coupling constants arise in the main body ofthe text [254]:

rijk(l) =M2

W

g22m

2l

|λijk|2 , r′ijk(q) =M2

W

g22m

2q

|λ′ijk|2 . (6.4)

Finally, unless otherwise stated, we also rely on the Reviewof Particle Physics from theParticle Data Group [20] as a source for the experimental data information as well as for shortreviews on the main particle physics subjects. Our notations and conventions are given inappendix A. We also assume familiarity with the standard textbooks and reviews, such as [263]for general theory and [25, 264, 265] for phenomenology.

6.2 Constraints on Bilinear 6Rp Terms and on SpontaneouslyBroken R-Parity

In this section, we summarize the main constraints on bilinear 6Rp parameters, both for an ex-plicit breaking (cf. section 2.3) and for a spontaneous breaking of R-parity (cf. section 2.4).The spontaneous breaking ofR-parity is characterised by anR-parity invariant Lagrangianleading to non-vanishing VEVs for someR-parity odd scalar field, which in turn generates6Rp

terms. Since such a spontaneous breakdown ofR-parity generally also entails the breaking ofthe global continuous symmetry associated lepton number conservation, this scenario is distin-guished by a non-trivial scalar sector including a masslessGoldstone boson, the Majoron, anda light scalar field, partner of the pseudo-scalar Majoron. Some scenarios of spontaneousR-parity breaking also involve a small amount of explicit lepton number breaking, in which casethe Majoron becomes a massive pseudo-Goldstone boson. By contrast, the explicitR-paritybreaking case may lead to finite sneutrino VEVs< νi >≡ vi/

√2, but the Lagrangian density

always includes terms that violateR-parity intrinsically.

6.2.1 Models with Explicit R-Parity Breaking

The bilinear6Rp parameters consist of 36Rp superpotential massesµi and 6 soft supersymmetrybreaking parameters mixing the Higgs bosons with the sleptons (3 6Rp B-termsBi associatedwith theµi, and 36Rp soft mass parametersm2

di). In the presence of these parameters, the sneu-trinos generally acquire VEVsvi, which in turn induce new bilinear6Rp interactions. Howeverthevi are not independent parameters, since they can be expressedin terms of the Lagrangianparameters – or, alternatively, they can be chosen as input parameters, while 3 among the 9 bi-linear 6Rp parametersµi,Bi andm2

di are functions of thevi and of the remaining6Rp parameters.

As explained in subsection 2.3.1, in the presence of bilinear R-parity violation, the (Hd, Li)superfield basis in which the Lagrangian parameters are defined must be carefully specified.Indeed, due to the higgsino-lepton and Higgs-slepton mixings induced by the bilinear6Rp terms,there is no preferred basis for theHd andLi superfields, and a change in basis modifies thevalues of all lepton number violating parameters, including the trilinear couplingsλijk andλ′ijk(cf. subsection 2.1.4). In practice the most convenient choice is the basis in which the VEVs ofthe sneutrino fields vanish and the Yukawa couplings of the charged leptons are diagonal. In the(phenomenologically relevant) limit of small bilinearR-parity violation, this superfield basis isvery close to the fermion mass eigenstate basis, and therefore allows for comparison with theindirect bounds on trilinear6Rp couplings derived later in section 6.3. ¿From now on we shallassume that this choice of basis has been made. Therefore:

vi = 0 and λe = Diag(me, mµ, mτ ) , (6.5)

and the bilinear6Rp parametersµi, Bi andm2di, as well as the trilinear6Rp couplingsλijk, λ′ijk

and their associatedA-terms, are unambiguously defined.

Let us now turn to the bounds that can be put on bilinear6Rp parameters, or equivalentlyon the induced mixings between leptons and neutralinos/charginos, and between sleptons andHiggs bosons.

In the fermion sector, the neutralino-neutrino and chargino-charged lepton mixings lead toa variety of characteristic signatures (cf. subsection 2.3.4) which in principle can be used toconstrain the superpotential6Rp massesµi. In practice however, the strongest bounds on theseparameters come from the neutrino sector. Indeed, the neutralino-neutrino mixing induces atree-level neutrino mass, given by Eqs. (2.46) and (2.47). In the absence of a fast decay modeof the corresponding neutrino, this mass is subject to the cosmological boundmν . 1 eV [46],which in turn requires a strong suppression of bilinearR-parity violation in the fermion sector:

sin ξ . 3 × 10−6√

1 + tan2 β , (6.6)

whereξ is the misalignment angle introduced in subsection 2.3.1 toquantify in a basis-indepen-dent way the size of the neutralino-neutrino and chargino-charged lepton mixings. Since we areworking in a basis wherevi = 0, sin ξ is related to the6Rp superpotential mass parametersµi bysin2 ξ =

∑i µ

2i /µ

2. The bound (6.6) is strong enough to suppress the experimental signaturesof bilinear 6Rp violation in the fermion sector below observational limits2.

In the scalar sector, the strongest contraints on the Higgs-slepton mixing comes again fromthe neutrino sector. Indeed the6Rp soft masses contribute to the neutrino mass matrix at theone-loop level, through the diagrams discussed in subsection 5.1.4. According to Eq. (5.19),the cosmological bound on neutrino masses yields the following upper limit on bilinear6Rp inthe scalar sector:

sin ζ . 10−4 , (6.7)

whereζ is the misalignment angle in the scalar sector introduced inEq. (2.35) of subsection2.3.1. Since we are working in a basis wherevi = 0, sin ζ is related to the6Rp B-termsBi bysin2 ζ =

∑iB

2i /B

2.

Finally, one should mention that some physical quantities receive contributions involvingsimultaneously bilinear and trilinear6Rp parameters, especially in the neutrino sector (cf. sub-sections 5.1.4 and 6.4.3). This results in upper bounds on products of6Rp parameters likeλijk µi

or λ′ijk µi.

6.2.2 Models with SpontaneousR-Parity Breaking

The main constraints on models of spontaneously brokenR-parity are essentially due to theexistence of a Goldstone boson (or, in the presence of interactions breaking explicitly leptonnumber, of a pseudo-Goldstone boson) associated with the breakdown of lepton number, theMajoronJ . The first constraint comes from the invisible decay width oftheZ boson, which

2This conclusion would have been different if the heaviest neutrino mass could have been as large as theντ

LEP limit of 18.2 MeV [20], as was often assumed in the early literature on bilinear6Rp. Indeed, in most models ofspontaneous6Rp the tau neutrino was unstable enough to evade the cosmological energy density bound. Since thenmany of these models have been excluded by the invisible decay width of theZ boson, and the scenario of a heavydecaying neutrino has become less attractive after the discovery of atmospheric and solar neutrino oscillations (seealso subsection 6.2.2).

excludes a massless doublet or triplet Majoron. In viable models, the Majoron must be eithermainly an electroweak singlet (i.e. it contains only a very small doublet or triplet component)like in the model of Ref. [63], or a massive pseudo-Goldstoneboson like in the model of Ref.[61]. In the second case,mJ should be large enough for the decayZ → Jρ, whereρ is thescalar partner of the Majoron, to be kinematically forbidden; in practicemJ & MZ is required.A third option, not discussed here, is to gauge the lepton number, in which case the Majorondisappears from the mass spectrum by virtue of the Higgs mechanism.

Due to its electroweak non-singlet components, the Majoronpossesses interactions withquarks and leptons of the formLffJ = gffJ fγ5fJ + h.c., where the couplinggffJ is model-dependent but related to the electroweak non-singlet VEVvL involved in the breaking of leptonnumber – generally the VEV of a sneutrino field. In the case of alight Majoron, these couplingsinduce physical processes that can be used to put upper bounds onvL, such as exotic semilep-tonic decay modes ofK andπ mesons, likeK+ → l+νJ [266]; neutrino-hadron deep inelasticscattering with Majoron emission initiated by the subprocessνµu → l+dJ [266]; or leptonflavour violating decays of charged leptons, likee → µJ . In practice however, the strongestconstraints onvL come from astrophysical considerations. Indeed light Majorons can be pro-duced inside the stars via processes such as the Compton scatteringe+ γ → e+ J [60]. Beingweakly coupled, Majorons, once produced, escape from the star, carrying some energy out. Therequirement that the corresponding energy loss rate shouldnot modify stellar dynamics beyondobservational limits puts a severe bound on the couplingsgffJ , therefore onvL. The strongestbounds come from red giant stars:

geeJ . 5 × 10−13 , (6.8)

if the Majoron mass does not exceed a few times the characteristic temperature of the process,mJ . 10 keV [267]. In the model of Ref. [63], whereJ is mainly an electroweak singlet, thisbound translates into

v2L

vRMW. 10−7 , (6.9)

where< ντ >≡ vL/√

2, and< νRτ >≡ vR/√

2 is the VEV of the right-handed sneu-trino field involved in the spontaneous breaking ofR-parity (cf. section 2.4 for details). ForvR ∼ 1 TeV, this is satisfied as soon asvL . 100 MeV. Models involving a doublet or triplet(pseudo-)Majoron are not subject to the constraint (6.8), since such Majorons are too heavy tobe produced in stars.

Finally, since spontaneousR-parity breaking involves the VEV of a left-handed sneutrinofield and/or generates bilinear6Rp terms through the VEV of a right-handed sneutrino field, theconstraints on models with explicit bilinearR-parity breaking also apply here. In particular, asingle neutrino becomes massive at tree level. This neutrino, if cosmologically stable, is subjectto the boundmν . 1 eV [46], which in turn requires a strong suppression of the misalignmentangleξ as expressed by Eq. (6.6). The misalignment angleξ, defined by Eq. (2.32), can beexpressed in terms of the parameters of the model. In the model of Ref. [61], sin ξ = vL/vd andthe constraint (6.6) translates into

< ντ > ≡ vL√2

. 500 keV . (6.10)

This bound is independent3 of tan β, due to the fact thatvd = v cosβ, which turns out to be astrong constraint on such models.

3In principle models with a massless or light Majoron can evade such a cosmological bound, hence the con-

6.3 Constraints on the Trilinear 6Rp Interactions

In this section we discuss the subset of the charged and neutral electroweak current phenomenawhich forms the basis for the high precision measurements. We also consider applications atthe interface ofCP violation andR-parity violation and review some miscellaneous topicsassociated with high precision observables (anomalous magnetic moments or electric dipolemoments). Unless otherwise stated, the various numerical results quoted in this section employStandard Model predictions which include either tree and/or one-loop level contributions. Thelimits on the6Rp coupling constants quoted in this section are2σ bounds unless otherwise stated.

6.3.1 Charged Current Interactions

Two important issues associated with the Standard Model charged current interactions are: (1)the universality with respect to theW± gauge boson couplings to quarks and leptons, andbetween the couplings of different lepton families; (2) therelations linking the independentrenormalised physical parameters of the Standard Model at the quantum level.

Charged Current Universality in Lepton Decays

The presence of aL1L2Eck operator leads to the additional contribution to the muon decay

shown in Fig. 6.1b. The effective tree-level Fermi couplingGF which determines theµ lifetime

eL(b)

e12k 12k~ekR eL W e(a)

Figure 6.1: Contributions toGF from (a) the standard model and (b) an6Rp operatorL1L2Eck.

becomes:

GF√2

=g2

8M2W

(1 + r12k(ekR)), (6.11)

the auxiliary parameterrijk being defined in section 6.1.4. The direct measurement ofGF to-gether with the tree-level relation (6.11) cannot be used, however, to set conservative constraints

straint (6.6), since the heavy neutrino can decay into a lighter one plus a Majoron, as originally suggested inRef. [63]. However such a scenario, quite popular at a time where oscillations of atmospheric neutrinos were notestablished on a firm basis, no longer appears to be very appealing, since it cannot be reconciled with both solarand atmospheric neutrino oscillations.

onλ12k, due to the large effects induced by radiative corrections.A study of the one-loop quan-tum relations linking the basic set of renormalized input parametersα, GF , MZ , with the weakangle and/or theW boson mass parametermW has to be performed. For an estimate see [261].We shall examine here two different versions associated with the off-shellMS and on-shellregularization schemes, respectively [268]:

off-shell (MS) : M2W =

πα(1 + r12k(ekR))√2GF sin2 θW (MZ)|MS(1 − ∆r(MZ)|MS)

,

on-shell: sin2 θW ≡ 1 − (MW

MZ)2 =

πα(1 + r12k(ekR))√2GFM2

W (1 − ∆r). (6.12)

The quantities labeled byMS refer to the modified minimal subtraction scheme and those with-out a label refer to the on-shell renormalization scheme. The off-shell scheme relation can beinterpreted as a prediction for theW boson massmW depending on the weak interaction param-etersin2 θW (MZ)|MS, and the on-shell scheme relation as a prediction for this weak interactionparameter, linked to theW mass to all orders of perturbation theory bysin2 θW = 1−M2

W /M2Z.

The auxiliary parameters in these two schemes,∆r, ∆r(MZ)|MS are calculable renormaliza-tion scheme dependent functions which depend on the basic input parameters and the StandardModel mass spectrum.

We evaluate both relations by using the experimental valuesfor the input parameters [269].The parameters common to both relations are set as:

α = 1/137.035, GF = 1.16639 × 10−5 GeV−2,MZ = 91.1867 ± 0.0020 GeV, MW = 80.405 ± 0.089 GeV (6.13)

. The weak angle in the off-shellMS relation is set assin2 θW (MZ)|MS = 0.23124 ± 0.00017,while in the on-shell it is in principle determined in terms of the W mass bysin2 θW = 1 −M2

W/M2Z . For the auxiliary parameters, we use the values:∆r = 0.0349 ± 0.0019 ± 0.0007,

and∆r(MZ)|MS = 0.0706 ± 0.0011.

Let us now quote the results of the calculations. We find that the off-shell scheme relationtends to rule out the existence ofλ12k. However, taking into account the uncertainties on theinput parameters leaves still the possibility of inferringbounds on the6Rp coupling constants.The uncertainty inMW dominates by far all the other uncertainties. A calculationat the1σlevel leads to the coupling constant boundλ12k < 0.038 ekR.

For the on-shell scheme relation, we still find that this tends to rule outλ12k, but yields the1σlevel boundλ12k < 0.046 ekR. To illustrate the importance of the uncertainties in theW bosonmass in this context, we consider the alternative calculation in the on-shell scheme where weuse the experimental value for the on-shell renormalized weak anglesin2 θW = 0.2260±0.0039and evaluate theW mass from the relationM2

W = M2Z(1 − sin2 θW ). This prescription is now

found to yield definite values for the6Rp coupling constants given byλ12k = 0.081 ekR .

The main conclusion here is that the constraints for the coupling constantsλ12k extractedfrom the 6Rp correction toGF depend sensitively on the input value of theW boson mass. Thecomparison of results obtained with the off-shell and on-shell regularization schemes serves,however, as a useful consistency check.

New contributions to theµ decay can be probed by comparing the measurement of the ratioof rates:

Rτµ = Γ(τ → µνν)/Γ(µ→ eνν)

to its SM expectationRSMτµ . This was first considered in [254] where it was shown that the

small experimental value forRτµ reported in [270] could be accounted for by6Rp muon decaysfor coupling valuesλ12k ∼ 0.15. An updated analysis [271] using more precise measurementsof Rτµ [272] andO(α) values forRSM

τµ now yields the bound

λ12k < 0.07 ekR. (6.14)

More generally, the ratioRτµ can also be affected byL2L3Eck operators modifying theτ

leptonic decays viaekR exchanges, similarly to the process shown in Fig. 6.1b. The expressionof Rτµ reads as:

Rτµ ≃ RSMτµ [1 + 2(r23k(ekR) − r12k(ekR))] , (6.15)

while the ratio of both leptonicτ decay widths is:

Rτ =Γ(τ → eνeντ )

Γ(τ → µνµντ )≃ RSM

τ [1 + 2(r13k(ekR) − r23k(ekR))] . (6.16)

The comparison of the experimental measurements with the SMvalues yields the followingbounds [271] on the coupling constantsλi3k:

λ13k < 0.07 ekR [Rτ ]; λ23k < 0.07 ekR [Rτ ]; λ23k < 0.07 ekR [Rτµ]. (6.17)

Aside from the muon lifetime, the energy and angular distributions of the charged leptonemitted in muon decay offer useful observables in order to test the Lorentz covariant structureof the charged current interactions, through the presence of either nonV −A couplings or tenso-rial couplings. The information is encoded in terms of the Michel parameterρ and the analogousparameters,η, ξ, δ, functions of the independent FermiS, V, T invariant couplings, which enterthe differential (energy and angle) muon decay distributions [273]. TheekR exchange depictedin Fig. 6.1b and induced by theλ12k coupling alone initiates6Rp contributions to the Lorentz vec-tor and axial vector couplings. However the corresponding corrections in this case are maskedby a predominant Standard Model contribution of the same structure, so that no useful con-straints can be inferred. In contrast, the tree level exchange of a stauτL initiates corrections tothe scalar Lorentz coupling, yielding the bound:|λ⋆

232λ131| < 0.022 τ 2L [274]. While the above

quadratic bound actually turns out to be weaker than those deduced by combining tentativelythe individual bounds on the coupling constantsλ13k andλ23k given in equation (6.17), it hasthe advantage of providing a more robust bound, not exposed to invalidating cancellations.

Charged Current Universality in π and τ Decays

Leptonic decays of theπ as well asτ− → π−ντ can be mediated at the tree level by6Rp in-teractions, as shown by the diagrams of Fig. 6.2. At low energies, these6Rp contributions canbe represented by four fermion interactions between pairs of quarks and leptons,(lΓl)(qΓq).Writing the effective interactions in this form allows a systematic calculation of the6Rp inducedmeson leptonic decays andτ semi-hadronic decays. For a general and complete study of boundsfrom meson decays we refer to [275]. In comparing with the experimental data for theπ-mesondecay widthΓ(π− → µ−νµ), it is advantageous to eliminate the dependence on the pion decaycoupling constant,Fπ, by considering the ratio [254]:

Rπ =Γ(π− → e−νe)

Γ(π− → µ−νµ)= RSM

π [1 +2

Vud(r′11k(dkR) − r′21k(dkR))] , (6.18)

(a) (b)

031k01ik

01ikd

u

j

ei~dkR 031k~dkR u d

Figure 6.2: Contributions of6Rp interactions to (a) the leptonicπ decays and (b) theτ → πντ

decay.

where the auxiliary parametersr′ijk are defined in section 6.1.4. The inferred coupling constantbounds are [275]:

λ′21k < 0.059 dkR, λ′11k < 0.026 dkR . (6.19)

The closely related two-body decayτ− → π−ντ also offers an additional useful test of thelepton universality [276]. A model-independent analysis based on a comparison with the exper-imental results for the ratio ofτ lepton andπ meson decay widths,

Rτπ =Γ(τ− → π−ντ )

Γ(π− → µ−νµ)= RSM

τπ

|Vud + r′31k(dkR)|2|Vud + r′21k(dkR)|2

(6.20)

yields the coupling constant bound:

λ′31k < 0.12 dkR , λ′21k < 0.08 dkR . (6.21)

Charged Current Universality in the Quark Sector

In the quark sector, the presence of aLQD operator leads to additional contributions to quarksemileptonic decays, via processes similar to that shown inFig. 6.2 where the incoming anti-quark line is reversed. The experimental value of the CKM matrix elementVud, determined bycomparing the nuclearβ decay to the muon decay, is then modified according to:

|Vud|2 =|V SM

ud + r′11k(dkR)|2|1 + r12k(ekR)|2 . (6.22)

Similarly, the rates fors → ulνl and b → ulνl, measured inK and charmlessB decaysrespectively, are affected by6Rp corrections induced byλ′ couplings. The values ofVus andVub extracted from these rates depend again onr12k via the effect ofλ12k couplings onGF .Summing over the down quark generations yields [254]:

3∑

j=1

|Vudj|2 =

1

|1 + r12k(eR)|2 [ |V SMud + r′11k(dkR)|2

+ |V SMus + [r′11k(dkR)r′12k(dkR)]

12 |2

+ |V SMub + [r′11k(dkR)r′13k(dkR)]

12 |2 ] (6.23)

= 1 − 2r12k(ekR)V SMud + 2r′11k(dkR)V SM

ud

+ 2[r′11k(dkR)r′12k(dkR)]12V SM

us + 2[r′11k(dkR)r′13k(dkR)]12V SM

ub , (6.24)

the last equality resulting from the unitarity of the CKM matrix 4. At the lowest order in the6Rp corrections into which we specialize, it is consistent to identify the flavour mixing matrixelements appearing in the right-hand side with the measuredCKM matrix elements,V SM

udj≃

Vudj. Setting the various CKM matrix elementsVudj

and in the sum∑

j |Vudj|2 at their measured

values [272], the following bounds are obtained in the single and quadratic coupling constantdominance hypothesis, respectively:

λ12k < 0.05 ekR, λ′11k < 0.02 dkR,

(λ′⋆11kλ

′12k)

1/2 < 0.04 dkR, (λ′⋆11kλ

′13k)

1/2 < 0.37 dkR.(6.25)

For more bounds on leptonic meson decays see [360]. A consideration of the unitarity con-straints on the other sums of CKM matrix elements,∑

j=1,2,3 |Vcdj|2 and

∑j=1,2,3 |Vcdj

|2, could also be used with the same prescriptions as in theabove comparison, to derive bounds on the single coupling constantsλi2k andλi3k, respectively.

Semileptonic and Leptonic Decays of Heavy Quark Hadrons

An experimental information on the three-body decay channels of charmed mesons is availablefor the following three classes of semileptonic processes,differing in the final state by the leptongeneration or the strange meson type:D+ → K0l+i νi, D

+ → K0⋆l+i νi, D0 → K−l+i νi, [li =

e, µ; νi = νe, νµ]. These decays could be enhanced by6Rp contributions involving aλ′i2k cou-pling as shown in Fig. 6.3. Denoting the branching fraction ratiosB(D → µνµK

(∗)) / B(D →

0i2k 0i2k~dk;R s l+i iD+f K0dd g

Figure 6.3: 6Rp contributions to the semileptonic decay of a charmed meson.

eνeK(∗)) byR(∗)

D+ , RD0 respectively, one can write the6Rp corrections as [277]:

RD+

(RD+)SM=

R⋆D+

(R⋆D+)SM

=RD0

(RD0)SM=

|1 + r′22k(dkR)|2|1 + r′12k(dkR)|2

. (6.26)

Following [277], we use(R(∗)D )SM = 1/1.03 to account for the phase space suppression in the

muon channel. ¿From the experimental values ofR(∗)D given in [272], one deduces the following

4We have corrected the formula for the unitarity constraint used in the work by Ledroit and Sajot [271] bynoting that the6Rp corrections to the flavour mixing matrix elementsVus andVub are given by quadratic productsof the coupling constants.

2σ coupling constant bounds:

|λ′12k| < 0.44 dkR, |λ′22k| < 0.61 dkR [RD+ ];

|λ′12k| < 0.23 dkR, |λ′22k| < 0.38 dkR [R⋆D+ ];

|λ′12k| < 0.27 dkR, |λ′22k| < 0.21 dkR [RD0 ].

(6.27)

By invoking the existence of a flavour mixing in the up-quark sector, within the current basissingle coupling constant dominance hypothesis, the6Rp contribution to the inclusive semilep-tonicB meson inclusive decay process,B− → Xqτ

−ν, may be expressed solely in terms of thesingle coupling constantλ′333. The comparison with experiment yields the following estimatefor the bound [278]:

|λ′333| < 0.12 bR. (6.28)

The same process as considered in [279] leading toλ′333 < 0.32 bR. The different predictionsfurnish an indication of the dependence on the input hadronic parameters.

The two-body leptonic decay channels of the charmed quark mesons,D−s → l−ν, also serve

a good use in testing the lepton universality. A comparison with the experimental results for theratios ofτ to µ emission,

RDs(τµ) =B(Ds → τντ )

B(Ds → µνµ)=

|Vcs + r′32k(dkR)|2|Vcs + r′22k(dkR)|2

, (6.29)

yields the coupling constant bounds [271]:

|λ′22k| < 0.65 dkR, |λ′32k| < 0.52 dkR . (6.30)

The∆S = 1 decays of strange baryons, e.g.Λ → pl−νe, · · · [l = e, µ] , provide bounds onquadratic products of theλ′ interactions. We quote the2σ bounds obtained by Tahir et al. [280]:

|λ′⋆11kλ

′12k| < 1.3 × 10−1 (5.3 × 10−3) d2

kR [Λ → pl−νl] ;

|λ′⋆11kλ

′12k| < 8.5 × 10−2 (1.6 × 10−2) d2

kR [Σ− → nl−νl] ;

|λ′⋆11kλ

′12k| < 1.2 × 10−1 (5.0 × 10−2) d2

kR [Ξ− → Λl−νl] ,

(6.31)

from the upper limits on the branching ratios of the indicated decays withl = e (l = µ).

6.3.2 Neutral Current Interactions

Neutrino-Lepton Elastic Scattering and Neutrino-NucleonDeep Inelastic Scattering

Most of the experimental information on neutrino interactions with hadron targets or with lep-tons is accessed via experiments usingνµ and νµ beams. One may consult [281, 282] forreviews. The elastic scatteringνµe → νµe, νµe → νµe has been studied by the CHARM IIexperiment, which provides measurements for the ratioR = σ(νµ)/σ(νµ), whereσ(νµ(νµ)) de-notes the cross-sectionσ(νµ(νµ)e → νµ(νµ)e). Neutral current (NC) and charged current (CC)deep inelastic scattering on nucleons or nuclei,νµN(A) → νµX andνµN(A) → µX, has been

studied by the CDHS and CHARM experiments at CERN, and by the CCFR Collaboration atFermilab. These experiments measure the following rates:

Rν =< σ(νN)NC >

< σ(νN)CC >, Rν =

< σ(νN)NC >

< σ(νN)CC >, (6.32)

where the brackets stand for an average over the neutrino beam energy flux distribution. Usefulinformation is also collected through the elastic scattering of neutrinos on a proton target [283].Each of these observables presents its own specific advantages by providing highly sensitivemeasurements of the Standard Model parameters.

The neutrinoνµ scattering on quarks and charged leptons is described by thes-channelZboson exchange diagram. At energies well belowMZ , the relevant neutral current couplings areencoded in the parametersgνf

L,R for charged leptons, andǫL,R(f) for quarks, as defined in termsof the effective Lagrangian density,

L = −4GF√2

(νLγµνL)[∑

l=e,µ

gνfL (lLγ

µlL) + gνlR (lRγ

µlR)

+∑

q=u,d

ǫL(f)(qLγµqL) + ǫR(f)(qRγ

µqR)]. (6.33)

The 6Rp contributions to neutrino elastic scattering arise at the tree level order. Examples areshown in Fig. 6.4 in the case ofνµe and νµe scattering. Similar contributions induced by aλ′21k (λ′2j1) coupling affect theνµd (νµ)d scattering via the exchange of adkR (djL) squark. The

~ejL2j1eR

eR2j1

(b)~ekR12k eL

eL

12k(a)

Figure 6.4: Examples of6Rp contributions to (a)ν − e and (b) νµ − e scattering. Othercontributions, as coming from thet-channel exchange of a selectron, are not represented.

results for the combined Standard Model and6Rp contributions read [254]:

gνeL = (−1

2+ xW )(1 − r12k(ekR)) − r12k(ekR),

gνeR = xW (1 − r12k(ekR)) + r211(eL) + r231(τL),

ǫL(d) = (−1

2+

1

3xW )(1 − r12k(ekR)) − r′21k(dkR),

ǫR(d) =xW

3(1 − r12k(ekR)) + r′2j1(djL), (6.34)

with xW = sin2 θW . Note that although aλ12k coupling does not lead to sfermion exchange con-tributing toνq → νq scattering, it affectsǫR andǫL via its effects onGF (see equation (6.11)).¿From these relations and using the experimental values forgνe

L andgνeR given in [272], which

rely on theσ(νµ(νµ)) measurements from the CHARM II experiment [284], one obtains thebounds:

λ12k < 0.14 ekR, λ231 < 0.11 τL, λ121 < 0.13 eL . (6.35)

The fits to the current data from the CDHS and CCFR Collaborations [285] determine thenumerical values for the parametersǫL,R [272]. Comparing with the Standard Model values,suitably including the radiative corrections, yields the following limits on the6Rp coupling con-stants [271]:

λ12k < 0.13 ekR, λ′21k < 0.15 dkR, λ′2j1 < 0.18 djL . (6.36)

The elastic scattering ofνµ and νµ on a proton target is known to furnish an independentsensitive means to measure the weak angle,sin2 θW [283]. Although this case has been includedin the global studies of the electron quark four fermion contact interactions [286], the presentexperimental accuracy and the theoretical uncertainties on the nucleon weak form factors donot warrant a detailed study of the constraints on the6Rp interactions.

Fermion-Antifermion Pair Production and Z-Boson Pole Observables

The fermion pair production reactions,e+e− → f f , [f = l, q] have been studied over a widerange of incident energies at the existing leptonic colliders, PEP, PETRA, TRISTAN, LEP andSLC. These reactions provide sensitive tests of the Standard Model [281]. For the high energyregime, the data for theZ boson pole resonant production,e+e− → Z0 → f f , as collected atthe CERN LEP and the SLC colliders, have provided a wealth of experimental measurementsof the Standard Model parameters.

At low energies, the basic parameters are conventionally defined in terms of the followingparametrisation for the effective Lagrangian density,

L = −4GF√2

∑

f=l,q

eγµ(geLPL + ge

RPR)e fγµ(gfLPL + gf

RPR)f. (6.37)

The high energy scattering regime,√s ≥ MZ , is described by analogous transition amplitudes,

differing in form only by the insertion of an energy dependent Z boson resonance propaga-tor factor. TheZ-pole measurements provide information on a large set of observables. Ofparticular interest here are the forward-backward asymmetries

AfFB =

(σ)> − (σ)<

(σ)> + (σ)<

∣∣∣∣ee→ff

which, in the Standard Model, are related to the vector and axial couplings of fermions to theZboson via :

AfFB =

3

4AeAf where Af = 2gf

V gfA/(g

f2V + f f2

A )tree level

= −T f3L .

The tree levelt-channel exchange of a sneutrino (squark) induced by aλijk (λ′ijk) couplingaffects the forward-backward asymmetriesAl

FB (AqFB) as shown in Fig. 6.5. Note that the

ijkijk

~je+i

ei eke+k(+ei $ ek)

01jk01jk

~dkRe+e uj

uj01jk01jk

~ujLe+e dk

dkFigure 6.5: 6Rp contributions to the forward-backward asymmetries.

s-channel exchange of a sneutrino also affects the Bhabha cross-section, but leavesAeFB un-

changed since the sneutrino decays isotropically in its rest frame. The SM values ofAfFB are

modified according to:

(AlkFB)SM/A

lkFB = |1 + r1jk(νj)|2 lk = e : ijk = 121, 131

lk = µ : ijk = 121, 122, 132, 231lk = τ : ijk = 123, 133, 131, 231

(Auj

FB)SM/Auj

FB = |1 + r′1jk(dkR)|2(Adk

FB)SM/Adk

FB = |1 + r′1jk(ujL)|2 .

(6.38)

Taking from [272] the experimental values forAfFB, as well as the SM predictions which include

radiative corrections, the following2σ bounds are obtained:

λijk < 0.37 νjL ; (ijk) = (121), (131) [AeFB]

λijk < 0.25 νjL ; (ijk) = (122), (132), (211), (231) [AµFB]

λijk < 0.11 νjL ; (ijk) = (123), (133), (311), (321) [AτFB]

λ′12k < 0.21 dkR [AcFB]

λ′1j2 < 0.28 ujL [AsFB]

λ′1j3 < 0.18 ujL [AbFB]

(6.39)

At the one-loop order,6Rp interactions lead toZff [f = q, l] vertex corrections. The dia-grams of the dominant6Rp processes contributing at the one-loop order to the leptonicZ decaywidth Γl are shown in Fig. 6.6. Theλ′ couplings which lead to these contributions also affecttheZ → bb decay widthΓb, via loop processes propagating internal top and slepton lines. Thesubsequent change of the hadronic decay width of theZ has thus to be taken into account whencalculating the6Rp induced corrections toRl = Γh /Γl. Those corrections read [287]:

δRl ≡Rl

RSMl

− 1 = −RSMl ∆l +RSM

l RSMb ∆b

where

Rb ≡ Γb /Γh, ∆f ≡ Γ(Z → f f)

ΓSM(Z → f f)− 1,

The comparison with the CERN LEP-I measurements [272] ofRZl [l = e, µ, τ ] leads to the

following 2σ bounds [271], valid form(dkR) = 100 GeV:

λ′13k < 0.47, λ′23k < 0.45, λ′33k < 0.58. (6.40)

01jk0i3k 0i3k

0i3k~dkR ~dkR

~dkRZZ tt t ei

eieiei

Figure 6.6: 6Rp contributions to the leptonicZ decay width. The (subleading) self-energydiagrams are not represented.

The dependence of these bounds on thedkR mass is not trivial and can be found in [287].

The 6Rp λ′′ interactions affectΓh via diagrams similar to those shown in Fig. 6.6, and thus

affectRl andRb as well [288]. From the measurement of the average leptonic decay widthRl =20.795± 0.040, the following bounds are obtained assuming squark masses of 100 GeV [288]:

[λ′′313, λ′′323] < 1.46(0.97) , λ′′312 < 1.45(0.96), (6.41)

the first (second) number corresponding to the2σ (1σ) upper bound. Less stringent upper limitson the sameλ′′ couplings were also obtained in [288] from an old measurement of Rb. Theseare out-of-date since the measuredRb is now in good agreement with its SM value.

Lebedev et al. [289] have performed a global statistical fit of the one-loop levelR-parity vi-olating contributions to the experimental data for the asymmetry parametersAb, Ab

FB gatheredat the CERN LEP and Stanford SLC colliders. The pattern of signs from the6Rp corrections issuch that certain contributions enter with a sign opposite to the one that would be required byobservations. This circumstance lead these authors to conclude that out the whole set of cou-pling constantsλ′i31, λ

′i32, λ

′′321 andλ′i33, λ

′′331 was ruled out at the1σ and2σ level respectively.

The pair-production of leptons with different flavours at the Z pole provides quadraticbounds on a large subset of the6Rp coupling constants,λijk, λ

′ijk. Lepton-flavour violating

(LFV) decays of theZ, with branching ratios defined by:

Bii′ ≡ Γ(Z → eiei′) + Γ(Z → ei′ ei)

Γ(Z → all), [i 6= i′] (6.42)

may occur through6Rp induced one-loop processes similar to those shown in Fig. 6.6, whoseamplitude is proportional to quadratic products of the relevant 6Rp coupling constants. Thecontribution of such processes induced byLQD operators to the LFV decays of theZ wasstudied by Anwar Mughal et al. in [290], focusing again on thedominant diagrams involving atop quark in the internal loop. A comparison with the currentexperimental limits onBeµ, Beτ

andBµτ [272] yields the bounds:

∑

k

|λ′⋆13kλ

′23k| < 6.2 × 10−2 [Z → eµ];

∑

k

|λ′⋆33kλ

′13k| < 1.5 × 10−1 [Z → eτ ];

∑

k

|λ′⋆23kλ

′33k| < 1.7 × 10−1 [Z → µτ ] (6.43)

established assumingm(dkR) = 200 GeV. The non trivial dependence of these bounds onm(dkR) can be found in [290].

A more general study of the LFVZ decays induced by6Rp process can be found in [291].This analysis is not restricted to diagrams involving a top quark in the loop, andλ couplingsare also considered. When LFV decays are mediated byλ couplings, it is shown thatBJJ ′ ≈(|λijJλ

⋆ijJ ′|/0.01)2 4. × 10−9. Under the hypothesis of a pair of dominant coupling constants,

one deduces from the current experimental limits:|λijJλ⋆ijJ ′| < [0.46, 1.1, 1.4] for [(J J ′) =

(12), (23), (13)] andm = 100 GeV, m being the common mass of the sfermions involvedin the contributing loops. LFV decays mediated byλ′ interactions are enhanced by an extracolour factor and by the possibility of accommodating an internal top quark in the loops. Anapproximate estimate reads:BJJ ′ ≈ (|λ′⋆

Jjkλ′J ′jk|/0.01)21.17 × 10−7. The comparison with

the experimental limits yields the coupling constant bounds: |λ′⋆Jjkλ

′J ′jk| < [3.8 × 10−2, 9.1 ×

10−2, 1.2 × 10−1], for [(J J ′) = (12), (23), (13)] andm = 100 GeV.

Thesee+e− collider bounds are not yet competitive with bounds obtained from fixed targetexperiments but they are expected to be improved in the context of futuree+e− linear colliders.The 6Rp contributions to Flavour Changing Neutral Currents ate+e− colliders for centre of massenergies above theZ boson pole will be discussed in chapter 7.

Atomic Parity Violation and Polarisation Asymmetries

Atomic Parity Violation (APV) has been observed via the6S → 7S transitions of13355 Cs [292].In the SM, parity violating transitions between particularatomic levels occur viaZ-exchangebetween the nucleus and the atomic electrons. The underlying four fermion contact interactions,flavour diagonal with respect to the leptons and quarks, are conventionally represented by theeffective Lagrangian:

L =GF√

2

∑

i=u,d

[C1(i)(eγµγ5e)(qiγµqi) + C2(i)(eγµe)(qiγ

µγ5qi)] (6.44)

with

C1(u) = 2geAg

uV = −1

2+ 4

3xW , C1(d) = 2ge

AgdV = 1

2− 2

3xW

C2(u) = 2geV g

uA = −1

2+ 2xW , C2(d) = 2ge

V gdA = 1

2− 2xW ,

(6.45)

andxW = sin2 θW . In the presence of aλ′11k (λ′1j1) coupling, thes-channel exchange of adkR

(ujL) between an electron and au (d) quark in the nucleus, as shown by the crossed processesdepicted in Fig. 6.5b(c), leads to additional parity violating interactions. Moreover, the coeffi-cientsC1 andC2 are affected by a non vanishingλ12k coupling via its effect onGF given byequation (6.11). The expression ofC1 andC2 in the presence of6Rp interactions reads as [254]:

C1(u) = (−1

2+

4

3xW )(1 − r12k(ekR)) − r′11k(dkR),

C2(u) = (−1

2+ 2xW )(1 − r12k(ekR)) − r′11k(dkR),

C1(d) = (1

2− 2

3xW )(1 − r12k(ekR)) + r′1j1(ujL),

C2(d) = (1

2− 2xW )(1 − r12k(ekR)) − r′1j1(ujL). (6.46)

Instead of using theC coefficients directly, one can use the measurement of the weak chargeQW , defined asQW = −2[(A + Z)C1(u) + (2A − Z)C1(d)] whereZ (A) is the number of

protons (nucleons) in the considered atom, or its differenceδQW = QW −QSMW to its SM value.

In Cs atoms [293]:

δ(QW (Cs)) = −2[72.07 r12k(ekR) + 376 r′11k(dkR) − 422 r′1j1(ujL)] .

Comparison with the experimental measurement :

δ(QW (Cs)) = 0.45 ± 0.48

in the single coupling dominance hypothesis leads to the2 − σ bounds:

|λ12k| < 0.05 ekR, |λ′11k| < 0.02 dkR, |λ′1j1| < 0.03 ujL . (6.47)

In Tl atoms,δ(QW (Tl)) = −2[116.89 r12k(ekR)+ 570 r′11k(dkR)− 654 r′1j1(qjL)]. Howeverthe resulting bounds on the6Rp couplings are less stringent than the ones given above, due to thelarge error on the measurement ofδ(QW (Tl)).

Closely related to APV measurements in atoms, polarisationasymmetries in elastic andinelastic scattering of longitudinally polarised electrons on proton or nuclear targets can alsobe used to constrain6Rp interactions [294]. Below, we shall use the summary of experimentalresults for the parametersC1,2(q) as quoted in [286].

A relevant observable is the asymmetry with respect to the initial lepton longitudinal polar-isation for the elastic scattering on scalarJP = 0+ nuclear targets,

Apol =dσR − dσL

dσR + dσL; [Apol =

GF q2

√2πα

3

2(C1(u) + C1(d))(1 +R(q2))]. (6.48)

The elastic electron scatteringe− +12 C → e− +12 C is studied at the BATES accelerator. Thedifference between the experimental measurement and the Standard Model prediction [295,296] is given byδ(C1(u) + C1(d)) = (0.137 ± 0.033) − (0.152 ± 0.0004) = −0.015 ± 0.033.Fitting this to the6Rp contribution yields the coupling constant boundsλ12k < 0.255 ekR, λ

′11k <

0.10 dkR, and the1σ level boundλ′1j1 < 0.11qjL.

The polarisation asymmetry of inelastic electron scattering on deuterone+ d→ e′ +X, asmeasured by the SLAC experiment [297], is described by,

Apol =3GFQ

2

5√

2πα[(C1(u)−

1

2C1(d))+(C2(u)−

1

2C2(d))

1 − (1 − y)2

1 + (1 − y)2] [y =

Ee −E ′e

Ee]. (6.49)

The differences between the experimental values and the Standard Model predictions are,δ(2C1(u) − C1(d)) = (−0.22 ± 0.26), δ(2C2(u) − C2(d)) = (−0.77 ± 1.23). Compar-ing with the 6Rp contribution for the first quantity yields the coupling constant boundsλ′11k <0.29dkR, λ

′1j1 < 0.38qjL, and the1σ level bound,λ12k < 0.20ekR. Comparing for the second

quantity yieldsλ12k < 2.0 × ekR, λ′1j1 < 0.71qjL and at the1σ level,λ′11k < 0.39dkR.

The electron polarisation asymmetry, as measured for theeL,R +9 Be → p + X quasi-elastic scattering Mainz accelerator experiment [298], exhibits a discrepancy with respect to theStandard Model prediction,AMainz = 12.7 C1(u) − 0.65 C1(d) + 2.19 C2(u) − 2.03 C2(d) =(−0.065 ± 0.19). Comparing with the6Rp contribution yields the coupling constant boundsλ′11k < 0.93 ×10−1dkR, and at the1σ level,λ12k < 3.0 ×10−1ekR, λ

′1j1 < 2.4 ×10−1qjL. The

above results clearly show that the strongest bounds fromγ − Z interference effects are thoseemanating from the APV experiments.

6.3.3 Anomalous Magnetic Dipole Moments

The anomalous magnetic dipole (M1) moments of the quarks and leptons represent valuableobservables that may be accessed in both low and energy experiments. For the light leptons,these observables are determined with high precision thanks to the high sensitivity currently at-tained by the experimental measurements. Other moments such as theZ boson current anoma-lous magnetic dipole moments for theτ -leptonaτ (m

2Z) or theb-quarkab(m

2Z) are currently

accessed in experiments at the leptonic and hadronic colliders.

The discrepancies with respect to the Standard Model expectations for the leptons or quarks,defined asδal = aEXP

l − aSMl , reflect the corrections arising from the perturbative higher-loop

orders electroweak contributions, the virtual hadronic corrections and possibly the MinimalSupersymmetric Standard Model loop corrections. The comparison between theory and exper-iment is expected to provide a sensitive test for new physics[299]. The electron momentae isa basic data for the purpose of extracting the experimental value of the hyperfine constantα. Asmall finite discrepancy is present for the electron in the difference between the experimentaland Standard Model value given byδae ≡ aEXP

e − aSMe = 1. × 10−11.

The measurement of the muon anomalous magnetic moment [300]exhibits a finite deviationfrom the Standard Model prediction,δaµ = aEXP

µ − aSMµ = 33.7 (11.2) × 10−10, based on

e+e− data, orδaµ = aEXPµ −aSM

µ = 9.4 (10.5) × 10−10, based onτ data. The theoretical value(aµ)SM includes the contribution from the hadronic radiative effects [301]. The above quoteddeviation with respect to the Standard Model is deduced on the basis of calculations of multi-loop diagrams which have not been verified in totality. Recent works by Knecht et al. [302] haveresolved the long-standing problem associated with the sign in the muon anomalous magneticmoment [303] for the non-perturbative contribution from the pion-pole term in the light-by-lightscattering amplitude. We refer to the review cited in [302] for more details. Although the exactsize of the hadronic contributions toaSM

µ remains an unsettled problem, the comparison of thevarious existing calculations [304] indicates that the corresponding uncertainties do not affectsignificantly the Standard Model prediction.

An early study by Frank and Hamidian [305] of the6Rp contribution to the leptons anoma-lousM1 moments indicated that the resulting constraints on the6Rp coupling constants wererelatively insignificant. The recently reported measurement of the muon anomalous magneticmoment [300] has stimulated two detailed studies of the6Rp effects [306, 307] focused on themuon anomalousM1 moment.

The study by Kim et al., [306] is performed within the so-called effective supersymmetryframework. One retains the contributions from the third generation sfermions only, based onthe assumption that the first and second generation sfermions decouple as having large massesof orderml = O(20) TeV, [l = 1, 2]. Such a radical hypothesis would, of course, relax sig-nificantly the various bounds on the6Rp coupling constants with superpartner indices associatedto the first two generations. The one-loop diagram contributions to the anomalousM1 momentafJ

of a fermionfJ enter in two types depending on whether the required chirality flip betweenthe external fermions takes place on the external lines themselves (giving an external fermionmass overall factormJ ) or on the internal fermion and sfermion lines (giving an overall factormfJ

m2LR/m2

f). For the muon case, it turns out that the contributions of the first type, with a

chirality flip on the external line, are the predominant ones. The bounds inferred by assumingthe single coupling constant dominance hypothesis are:

[λ32k, λ3j2, λi23, λ2j3] < 0.52mf

100GeV. (6.50)

Alternatively, if one focuses solely on the coupling constant λ322, based on the observation thatthis is the least constrained of all the coupling constants involved, then the predicted value ofthe anomalous moment associated with the6Rp effects,

(aµ)RPV ≃ 34.9 × 10−10(100 GeV

m)2|λ322|2, (6.51)

is seen by comparison with the experimental result to be compatible with the perturbativitybound on the corresponding coupling constant.

A comparison of the6Rp effects on the muon magnetic moment and the neutrino masses isof interest, despite the fact that the one-loop contributions to the neutrino masses are of the typeinvolving chirality flip mass insertion terms for the fermion and sfermion internal lines. Indeed,some correlation still exists between the one-loop contributions in these two cases owing to theidentity of the 6Rp coupling constant factors. Adhikari and Rajasekaran [307]observe that inorder to get6Rp contributions to the muon anomalous magnetic moment and neutrino mass ofthe size required by the current experiments,aµ = O(10−9), mνµ = O(1) eV, one needs tosuppress in some way the one-loop contribution to the neutrino mass. This can be achievedby postulating for the chirality flip slepton mass parameters (me2

LR)ij either reduced values or a

degeneracy with respect to the first two generations,(me2LR

)11 ≃ (me2LR

)22. A natural resolutionof this issue can be achieved in a model using a discrete symmetry acting on the lepton sector.

6.3.4 CP Violation

The existence of a possible connection betweenCP violation andR-parity violation has re-ceived increased attention in the literature [253, 308, 309, 310, 311, 312, 313, 314, 315, 316,317]. In this section, we discuss a non-exhaustive set of physical applications which lie at theinterface betweenR-parity violation andCP or T violation. The topics to be adressed includediscussions on observables in the neutralKK system, the electric dipole moment (EDM) ofleptons and quarks, theCP -odd asymmetries inB meson hadronic decay rates and theCP -odd asymmetries inZ boson decay rates into fermion-antifermion pairs. A study of theCP -violation effects in association with sneutrino flavour oscillations was carried out in chapter 5.

General Considerations onCP Violation

As is known, the violation ofCP symmetry is revealed in the context of field theories bycomplex phases present in VEVs of scalar fields, in particlesmass parameters or in Yukawainteraction coupling constants which cannot be removed by field redefinitions.

It is conventional to distinguish between soft and hardCP violation, depending on whetherthe dimensionality of theCP -odd operators in the effective lagrangian is≤ 3 and≥ 4 re-spectively. The spontaneousCP violation case, as characterized by the presence ofCP -oddcomplex phases in scalar fields VEVs resulting from aCP conserving lagrangian, falls natu-rally within the soft violation category. The distinction between soft and hardCP violationsis motivated by the different impact that the quantum and thermal fluctuations effects have inthe two cases. The softCP violation interactions cannot renormalize the hard interactions,unlike the hardCP violation interactions which indeed can renormalize the operators of lowerdimensions. The softCP violation parameters may also be suppressed by thermal fluctua-tions, eventually leading to a restoration ofCP symmetry at high temperatures. By contrast,

the thermal effects do not affect significantly the couplingconstant parameters of operators ofdimension≥ 4, a fact which makes the hardCP violation mechanisms more robust candidatesfor generating the baryon or lepton asymmetry in the early Universe.

The Standard Model includes two sources of hardCP violation One is the complex phasein the CKM matrix with three quark generations. The other is the QCD theta-vacuum angle.For supersymmetric models, new sources of softCP violation appear with theµ term and softsupersymmetry breaking interactions. With the known structure of the constrained MSSM clas-sical action, assuming fully universal soft supersymmetrybreaking, the unremovableCP -oddphases are restricted to a pair of phases given by the relative complex phasesφA = arg(AM⋆

12

)

andφB = arg(Bµ⋆). The experimental constraints give strong individual bounds on thesephases i.e.φA,B < O(10−3). However several other additional phases arise once one relaxesthe universality hypothesis for the supersymmetry breaking parameters for the quark and leptongenerations or the different gaugino mass parameters. It has been found [318] that in caseswhere some built-in correlations between the various phases are included, the evaluation ofphysical observables includes such strong cancellation effects that the experimental bounds re-lax to valuesO(1).

With brokenR-parity, new sources ofCP violation can contribute through complex phasesin the parametersµ, µi andB, Bi for the bilinear interactions and in the parametersλijk, λ

′ijk, λ

′′ijk

andAijk, A′

ijk, A′′

ijk for the trilinear interactions. Each of these coupling constants can carrya complex phase although only the subset of these phases invariant under the fields rephasingis physical. For products of the trilinear6Rp coupling constants only, aCP -odd phase invariantunder complex phase redefinitions of the fields can be defined starting from the quartic or-der. Examples encountered in the calculations of scattering amplitudes for processes involvingfour fermion fields such ase+ + e− → fJ + fJ ′ are given by:arg(λi1Jλ

⋆i1J ′λ⋆

i′jJλi′jJ ′), andarg(λijJλ

⋆ijJ ′λ⋆

i′j′Jλi′j′J ′).

Although basis-independent studies have been performed for specific cases [26, 319, 320]a full systematic discussion of a basis independent parametrisation ofCP violation for the6Rp

interactions would be useful in characterizing the naturalsize of the relevant parameters as canbe emphasized from the example described in [321].

Concerning the context of bilinearR-parity violation, a partial study of the parametrisationof CP violation is performed in [322]. For that part of the scalar potential which determinesthe Higgs bosons VEVs, theCP -odd phase is included through the coupling constant prod-uct µ⋆µiBiB. Including the trilinear terms, one characteristicCP violation condition can beidentified in terms of the phase in the coupling constant product of λ interactions and regularYukawa interactions given by:ℑ(λ⋆

nmkλijkλenlλ

e⋆il λ

empλ

e⋆jp) 6= 0. Another interesting conclusion

concerning a spontaneous violation ofCP in the presence of6Rp interactions is that complexvalued sneutrino VEVs can occur in a natural way without fine-tuning of the parameters [322].

Neutral KK System

The possibility of embedding aCP -odd complex phase in the6Rp coupling constants has beenenvisaged at an early stage in a work by Barbieri and Masiero [323] and then further discussedin [359, 371, 112, 324, 325] and references therein. A complex relative phase present in aquadratic product ofλ′′ijk coupling constants can contribute to theKS −KL mass difference.

The most general∆S = 2 effective lagrangianL∆S=2eff including contributions of charginos

and charged Higgs boson neglected in earlier works [323, 371, 112] has been considered in [324]

~bsR dR

dLt~b

~bs

dt

d

s W (t)sL

(t)(a) (b)

Figure 6.7: Box diagrams leading toKK mixing induced byλ′′ couplings.

(a) (b)u

s ~bq

gu

d

qt~bs dt

Figure 6.8: Tree level diagram (a) and gluonic penguin one-loop diagram(b) contributing tothe direct∆S = 1 CP violation involvingλ′′ couplings.

(see also [325]). Its contribution to theKS − KL mass difference is related to the matrix el-ement< Ko|L∆S=2

eff | ¯Ko > and involves the products ofλ′′313λ′′323 couplings (see for example

Fig. 6.7) as well as CKM matrix elements5. This 6Rp coupling’s contribution to theKS −KL

mass difference has been calculated in [324] using NLO QCD evolution of Wilson coefficientalso included inL∆S=2

eff [326] as well as lattice calculations for long-distance hadronic processeswhich cannot be evaluated pertubatively and also contribute to the above matrix element. Re-quiring that this contribution to theKS −KL mass difference is not larger than the experimentalvalue [272]6 allows one to set an upper limit [324]:

λ′′313λ′′⋆323 < O(0.033) (6.52)

by performing a general scan over the parameter space on the minimimal supersymmetric ex-tension of the standard model at the weak scale and taking into account the contraints fromdirect searches for supersymmetric particles.

Theλ′′ interactions contribute also at the tree level to the direct∆S = 1 CP violation (seeFig. 6.8(a)), as described by the observable parameterǫ′. The 6Rp contribution toǫ′ is described

5In [323] the charm contribution and in consequence theλ′′232λ

′′213 products have also been considered where

the t-quark in the loop is replaced by a c-quark.6Actually the upper bound derived in [324] comes from the experimental value published in [327] on the

KS − KL mass difference. However the difference with the publishedvalue in [272] being marginal for thepresent purpose, the conclusion of the analysis presented in [324] onλ′′

232λ′′⋆213 is unchanged.

by the relation [323]:

ℑ(λ′′123λ′′⋆113) ≈ |ǫ′ | 101 q2. (6.53)

The gluonic penguin one-loop diagram, see Fig. 6.8(b), provides a competitive contributionto that of the box diagram due to the existence of a logarithmic enhancement factor in theamplitude. The resulting bound reads [323]:

ℑ(λ′′313λ′′⋆323) ≈ |ǫ′| 10−2 q2. (6.54)

In order to match the currently observed value forǫ′ = O(10−6) one should require valuesfor the quadratic coupling constantsℑ(λ′′123λ

′′⋆113) (respectively ℑ(λ′′313λ

′′⋆323)) of O(10−5) q2

(respectivelyO(10−8) q2).

Since different generational configurations of the coupling constants contribute to theCPviolation parametersǫ andǫ′, one concludes that the6Rp interactionsλ′′ might be relevant can-didates for milliweak typeCP violation contributing solely to the indirectCP violation.

dLsLdL

dLsL

(a) (b)

(d)( )

sL dRsRdRsL~

~g~ ~ dR

sRdRdLdR

~~g

sL~sR

~dR~sL

~dL

Figure 6.9: Contributions to∆S = 2 (a, b and c) and∆S = 1 (d) observables involvingλ′

couplings.

An interesting proposal [310] is to incorporate theCP -odd phase in the scalar superpart-ner interactions corresponding to the soft supersymmetry breaking6Rp interactions of the formVsoft = A

′

ijkLiQjDck + h.c. . The contribution to the imaginary part of the∆S = 2 mass

shift is estimated qualitatively asǫ ≃ 10−2 ℑ(A′

i21 − A′

i12)/mg. The comparison with the ex-perimental value indicates that some cancellation betweenthe above two flavour non-diagonalconfigurations i.e.A

′

i21and A′

i12, may be required to take place. A contribution to the fourquark interactionsRdLdRsL arises from a sneutrino exchange penguin type diagram involvinga one-loop correction to theνds vertex as shown in Fig. 6.9(b) and 6.9(c). The predicted effecton the directCP violation∆S = 1 observableǫ′, see Fig. 6.9(d), reads [310]:

|ǫ′

ǫ| ≃ 10−7 λ

′i11

λ′i12> 102 λ′i11λ

′i21, (6.55)

where the inequality obtained at the second step uses the bounds on the coupling constant prod-ucts,λ

′⋆i12λ

′i21 < 10−9ν2

i .

Asymmetries in Hadron Decay Rates and Polarisation Observables

The polarisation of the muon emitted in theK-meson three-body semileptonic decayK+ →π0 + ν + µ+ (Kµ3) or in the radiative decayK+ → µ+ + ν + γ (Kµ2γ) constitute usefulobservables for testingT and/orCP violation.

The transverse muon polarizationPT (Kµ2γ) can be related7 to |ℑ(λ⋆2i2λ

′i12)|/m2

eiLas shown

in [314].

Under various simplifying assumptions [314], rough estimates on upper bounds onPT (Kµ2γ)may be derived from bounds on the branching ratio ofµ → eγ and from the measured value8

of BR(K+ → π+νν). TheKµ2γ decay has been measured in [329] (see also [330]) and shouldprovide useful handles in constraining these6Rp couplings.

The transverse muon polarizationPT (Kµ3 is O(10−10) in the standard model [331]. It canbe related to(λ⋆

232λ′312)/m

2τiL

and (λ⋆122λ

′112)/m

2eiL

as shown in [332] (see also [333, 334]).These contributions from6Rp couplings can be as large as the present experimental limits9. onPT (Kµ3 [335, 336] (see also [337, 338]).

As discussed in [339] future projects may reach uncertainties approaching the O(10−4 ) levelfor the measurement ofPT (Kµ3) thus allowing also to test6Rp couplings further.

Z boson partial decay

TheZ boson partial decay channels into fermion-antifermion (up-quark, down-quark, chargedlepton) pairs of different flavours,Z → fJ + fJ ′ [J 6= J ′; f = u, d, l], may exhibit potentiallyobservableCP violating decay asymmetries. These are defined by the normalised differencesof flavour non-diagonal, spin-independent rates,

AJJ ′ =BJJ ′ − BJ ′J

BJJ ′ +BJ ′J

, (6.56)

where the branching ratiosBJJ ′ are defined in equation (6.42). A finite contribution to theflavour decay asymmetry is rendered possible by the existence of a finiteCP -odd complexphase,ψ, embedded in the6Rp coupling constants. The asymmetries [291] are proportional toratios of the coupling constants of the form:

ℑ(λ

′⋆iJkλ

′iJ ′k

λ′⋆1Jk′λ′1J ′k′

) ∝ sinψ. (6.57)

Should the6Rp coupling constants exhibit generational hierarchies, onecould then expect largeenhancement or suppression of the asymmetries, depending on the flavour of the emitted fermions.Assuming that the above ratios of6Rp coupling constants products take values of order unity,the resulting asymmetries for the emission of charged leptons, down-quarks and up-quarks arefound to be of the order ofAJJ ′ ≈ (10−1 − 10−3) sinψ.

7Various expressions forPT (Kµ3) can be found in [328] in the context of leptoquark models.8At the time of [314] only upper bounds onBR(K+ → π+νν) were known.9The limits used in [332] come from [335] and should be replaced by those published in [336]. However the

general conclusion drawn in [332] should not be significantly affected.

Neutron Electric Dipole Moment

The 6Rp interactionsλ′′ may contribute to the neutron electric dipole momentdγn [323] via a

quark electric dipole moment described by a two-loop vertexFeynman diagram involving thecrossed exchange ofW and d internal particle lines as seen in Fig. 6.10. Note that no contri-butions from the6Rp interactions to the neutron dipole moment can arise at the one-loop orderlevel [340, 341].

s(b)

~b (t) WdL dRs(b) (t)

Figure 6.10:Contributions to the neutron electric dipole moment involving λ′′ couplings.

The needed suppression to account for a sufficiently small electric dipole moment is pro-vided in part by the light quark mass factors reflecting the chirality flip selection rules of the6Rp couplings. A relativeCP -odd complex phase embedded in a pair product ofλ′′ couplingconstants is required in order to obtain a finite contribution to the electric dipole moment. Thecontribution is maximised by choosing third generationb− b quarks configuration for the inter-nal fermion and sfermion particles. The results derived by Barbieri and Masiero [323] on thebasis of the double coupling constant dominance hypothesisread:

ℑ(λ′′213λ′′⋆232) ≃ 10−2(

dγn

10−25 e× cm)q2 (6.58)

ℑ(λ′′312λ′′⋆332) ≃ 10−1(

dγn

10−25 e× cm)q2 (6.59)

for the internal charm quark and top quark cases, respectively. Using the current experimentalbound on the neutron electric dipole moment,dγ

n < 1.2 × 10−25 e cm from [342] one obtainsa bound for the top quark box diagram contribution only whichis |ℑ(λ′′312λ

′′⋆332)| < 0.12q2.

The discussion on systematic uncertainties concerning themore recent bounddγn < 6.3 ×

10−26 e · cm recently published in [343] has been criticized in [344].

Contributions from products ofλ′ couplings such asℑ(λ′⋆i33λ′i11) are discussed in [345].

New contributions involving both bilinear and trilinear couplings can lead to a neutron elec-tric dipole moment as discussed in [346, 347].

Electron Electric Dipole Moment

Even if one assumes purely realCP conserving6Rp coupling constants, a non-vanishingCPviolating contribution could possibly be induced by invoking the existence of other possiblesources of complex phases present in the (minimal) Supersymmetric Standard Model. Thus theλijk andλ′ijk 6Rp couplings may induce one-loop contributions to the electric dipole momentof leptons (and quarks) through an interference with a complex valuedCP -odd soft super-symmetry breaking parametersAu

ijand Adij associated with the regular Yukawa interactions.

A non-vanishing amplitude associated with the Feynman diagram with a pair of sfermion andfermion internal lines requires the presence of aL − R chirality flip mass-mixing insertionm2

LRfor the internal sfermion. Stated equivalently, it requires a mass splitting between the

opposite chiralities sfermion eigenstates. The bounds arestrongest for the electron electricdipole momentdγ

e and can lead to several strong individual coupling constantbounds as shownin [305]. A representative subset is tentatively summarized by λ′111 < 5.5 × 10−5, λ′121 <8.7 × 10−6, λ′213 < 9.5 × 10−2, λ′233 < 1.5 × 10−2 usingdγ

e = (3 ± 8) × 10−27e cm [378]which sould be superseded by the recently [348] measured valuedγ

e = (6.9± 7.4)× 10−28e cmi.e. |dγ

e | = 1.6 × 10−27e cm

Focusing on the contribution from a complex6Rp coupling constantλ′133 = |λ′133|eiβ , inter-fering with a complex soft coupling constantAq = |Aq|eiαA , the current bound on the exper-imental electron electric dipole moment can admit solutions with large values for both of theaboveCP -odd phasesβ andαA as shown in [349].

However the conclusions drawn from the above two studies [305, 349] have been recentlychallenged by the observation that no contributions from the 6Rp interactions to the electricdipole moment can arise at the one-loop order level [340, 341]. Due to the chirality selectionrules, aneR particle line can never be emitted nor absorbed. For similarreasons, adR − dL

mass insertion on the squark line must be accompanied by a neutrino Majorana mass insertion,resulting in a strongly suppressed contribution to the electric dipole moment which exactlyvanishes in the zero neutrino mass limit. A one-loop contribution could only be possible througha sneutrino Majorana mass term,mij νiLνjL as can be seen in Fig. 6.11.

~ekL(b)

j j eReL ;Z

~j(a)

ekR ekL eR ;ZeL ~ j ~ekR

Figure 6.11: One-loop diagrams contribution to the electron dipole moment involvingλ cou-plings.

Similar chirality selection rules would also apply for the analogous chirality flip observables,such as involved in the contributions to the neutrino Majorana mass, the neutrinoM1 or E1diagonal or off-diagonal moments, or the charged fermionsM1 transition moments. On the

other hand,E1 transition moments for Majorana neutrinos may possibly be initiated by the6Rp

interactions at the one-loop order. The above observationsfigure in [340, 341].

~tR(a)

tR dkR dR ;ZHdL ~bR

(b)dRdL

;ZujLujR dkR umL

HH ~eiL

Figure 6.12:Examples of two-loops diagrams contribution to the electric dipole moment of theelectron with Higgs and sfermions exchangesλ′ couplings (a) andλ andλ′′ couplings (b).

(a)

ejL ;ZeL

ejLejL ;ZeR~ ejR eR

(b)

djL ;ZdjLdjL ;ZeR~ djReL eR

Figure 6.13:Examples of two-loops diagrams contribution to the electric dipole moment of theelectron involvingλ couplings (a) andλ andλ′ couplings (b).

As discussed in [340, 341], at the two-loop order many possible mechanisms can con-tribute to the electric dipole moment. We have already discussed above the contributions fromtheλ′′ interactions. The two-loop diagrams with overlapping or crossed exchanges of Higgsbosons and sfermions (see Fig. 6.12), or of gauge boson and sfermions (see Fig. 6.13 andFig. 6.14), yield contributions proportional to quadraticproducts of the6Rp coupling constantstimes quadratic products of the CKM matrix elements, entering in appropriately rephased in-variant flavour configurations.

New contribution involving both bilinear and trilinear couplings can also lead to electronelectric dipole moment as discussed in [346, 347].

dL dRujL ujR dlR dlL umL umRW ;Z

~dkR

Figure 6.14: Example of two-loop diagram contribution to the electron dipole moment withgauge boson and sfermion exchange involvingλ′′ couplings.

Atomic Electric Dipole Moment

The electric dipole moment of atoms also present a strong potential interest [350, 351] owing tothe high experimental sensitivity that can be attained in the experimental measurements of theelectric dipole moment of atoms such as133 Cs or205 Tl. The 6Rp contributions to the electronelectric dipole moment from the other mechanism described by the two-loop diagram with onefermion closed loop leads to the coupling constant bounds [340, 341, 352] (for J=1,2,3):

|ℑ(λ⋆1J1λ

′J33)| < 6. × 10−7 (6.60)

These bounds should be revisited in view of new experimentalresults [348].

The 6Rp contributions from the two-loop diagram with two crossed sfermionic loops attachedto the external line, yield bounds on quartic coupling constant products of the form [341]:

ml

mτ

|ℑ(λ1mnλ⋆jlnλ

⋆imlλij1)| < 10−6 (6.61)

ml

mt|ℑ(λ1mnλ

⋆jlnλ

⋆imlλij1)| < 3.× 10−6 (6.62)

using experimental bounds on the electric dipole moment of both the electron and the neutron.

These bounds on quartic coupling constant products should also be revisited in view of thenew experimental result [348] on the electric dipole momentof the electron and the discussionpublished in [344] on the experimental value of the electricdipole moment of the neutron.

Finally, useful information on theP andT violatinge−N interactions as parametrised bythe effective Lagrangian:

L = −GF√2(CSpeiγ5epp+ iCTpeσαβepσ

αβp) + (p↔ n), (6.63)

with:

σαβ =1

2ǫαβγδσγδ, (6.64)

can be obtained from the experimental limits for the electric dipole moment of atoms such ascurrently available for the133Cs or 205Tl atoms [350]. The comparison can result for examplein the bound [352]|ℑ(λ⋆

1I1λ′I11)| < 1.7 × 10−8ν2

I (I=2,3).

Hadronic B Meson Decay Asymmetries

The formalism forB meson physics as well asCP violation in theB meson system can befound in [272].

The∆b = 1 non-leptonic decay transition amplitudes arise from tree level diagrams and,when these are forbidden, from one-loop box type and penguintype diagrams associated withthe quark subprocessb → diq

′q′′, [q′, q′′ = (u, c, d, s)]. The relevant effective Lagrangianconsists of10 independent operators quartic in the quark fields.

The tree level contribution from the6Rp interactions having the specific form:

L =∑

i

λ′ijkλ′⋆ij′k′

m2eiL

(dkRdjL)(dj′Ldk′R). (6.65)

Assuming the mixing decay amplitudes to consist of the two additive contributions from theStandard Model and6Rp interactions, one can parametrise the off-diagonal elements of the massmatrixM12 of the neutralB mesons and the decay amplitudeA in terms of real parameters andcomplex phasesrX andθX whereX stands either forM (mixing) orD (decay) as:

M12 = MSM12 (1 + rMe

iθM ) (6.66)

A = ASM(1 + rDeiθD) (6.67)

Under the simplifying approximation where the final stateCP -even strong interactionsphase is the same for all the additive terms in the decay amplitudes, the ratioA/A = e−2iφD ,whereA is theCP mirror conjugate decay amplitude becomes a pure complex phase, so thatone expresses the basic asymmetry parameter as:

rf(CP ) = e−2i(φM +φD) (6.68)

The 6Rp corrections may be represented in terms of shifts in the mixing and decay complexphasesφX = φSM

X + δφX (with X= M, D) such that:

δφX = tan−1 rX sin θX

1 + rX cos θX. (6.69)

For illustration, note that in the Standard Model the mixingphase for theBd system is describedby φM = −1

2arg(V ⋆

tbVtd), and the decay mode to the final statef(CP ) = J/Ψπ0 by:

rf(CP ) = exp i[arg(V ⋆tbVtd) + arg(V ⋆

csVcb) + arg(V ⋆cdVcs)] = e−2iβ (6.70)

The decay channels such asB → K0K0, φπ0, φKS are of special interest since theirassociated quark subprocesses,b → ddd, dds, sss, respectively, are tree level forbidden in theStandard Model10.

The 6Rp contribution to the mixing parameter readsrM ≃ 108 λ′⋆i13λ

′i31ν

−2 [311]. Predictionsfor the 6Rp contributions to theCP -odd asymmetry parameterrD in the various decay channelsare provided in refs. [309, 311, 312].

10See for example [272, 353] for experimental results.

Using bounds on quadratic products of theλ′ andλ′′ coupling constants from experimentalconstraints in [378], one can observe [309] that the predicted bounds onrD follow different pat-terns for the heavy meson decay channels such asB → J/ΨKS, B → D+D−, in comparisonto the light meson decay channels such asB → φKS, φπ

0, KSKS, the latter generally yieldingmore favourable signals with(1 + rD) ≃ |A 6Rp/RSM | >> 1. One can also incorporate system-atically the mixing effects and updated values for the Wilson coefficients of the operators [312].The contributions to the asymmetry parametersrD in the various decays from theλ′ interactionsare typically of order,10−3 − 10−4. By contrast, those from theλ′′ interactions turn out to arisewith a more interesting order of magnitude i.e.O(1 − 10−1).

The important decay modes with the final statesf = φKS and J/ΨKS have been con-sidered in [311]. The Standard Model predicts equal decay phasesφD along with controlleduncertainties for the difference of phases [354]:

∆φD = |φD(Bd → φKS) − φD(Bd → J/ΨKS)| < O(10−1). (6.71)

Anticipating the possibility that the experimental errorsmay reach a sensitivity at this level ofaccuracy or higher, an important question is the expected size for the6Rp contributions. Theseare found to be [311]:

rD(Bd → φKS) ≃ 8. × 102|λ′⋆i23λ

′i22 + λ

′⋆i22λ

′i32|(

mW

mνi

)2

rD(Bd → J/ΨKS) ≃ 2. × 102|λ′⋆i23λ

′i22|(

mW

meiL

)2 (6.72)

Using the current bounds on the6Rp coupling constants, especially those coming fromBR(b→Xsνν) [355], yields an encouraging prediction for the above difference of phases,∆φD ≃O(1) [311].

Let us finally note that analogous methods have been developed to extract experimentalinformation for neutral or chargedB meson decays into non-pureCP channels. TheCP de-cay rate asymmetries are obtained by forming differences between the decay rates for theCPconjugate transitions,B0 → f, B0 → f . Interesting signals from the6Rp contributions are alsoexpected for the chargedB meson decaysCP asymmetries in the transitionsB+ → f andB− → f [309] such as, for example,Bd → J/Ψρ0, D±π∓, K+π− orB+

d → J/ΨK+, π+π0.The leptonic or semileptonic decay modes of theB mesons also deserve a due consideration.

For the decay modeB± → π±K, the 6Rp induced amplitudeA 6Rp ∝ [λ′i23λ′⋆i12/m

2](bs)(dd)could yield a nearly 100 % contribution to theCP -odd asymmetry much larger than the Stan-dard Model contributions which are expected not to exceed a 40 % effect [317].

TheCP violating asymmetries in the decay and polarization observables of hyperon non-leptonic weak decay modesΛb → p+ π− are examined in [313].

6.4 Trilinear 6Rp Interactions in Flavour Violating Processesand in 6B and 6L Processes

A very large number of bounds for the trilinear6Rp couplings have been deduced from studies oflow and intermediate energy processes. In particular, raredecays involving either hadron flavour

violation or lepton flavour violation (LFV), or both combined, constitute a nearly inexhaustiblesource of constraints on the trilinear6Rp couplings. Processes that violate lepton number orbaryon number also provide strong constraints on trilinearR-parity violation. To review theresults obtained in the current literature, we shall organise the discussion into four subsections,where we discuss in succession hadron flavour violating processes, lepton flavour violatingprocesses, lepton number violating processes and baryon number violating processes.

6.4.1 Hadron Flavour Violating Processes

Mixing of Neutral Mesons

The contribution from6Rp coupling to the mass difference and mixing observables for the neutralBB meson system (i.e.[∆b = 2]), has been considered in [112, 356] and further updatedin [357, 325].

Sneutrino exchange can contribute to theBB mixing as well asKK mixing through twoλ′

couplings via the tree level diagrams shown in Fig. 6.15a,b.

(b) db

db 0i310i13

(a) ds

ds 0i210i12 ~ i ~ i

Figure 6.15: 6Rp contributions to (a)KK and (b)BB mixing involving sneutrino in thes-channel. Ws

d sdVtdVts

t t0131 0132~ei

Figure 6.16: Box diagram leading toKK mixing induced byλ′ couplings.

Individual coupling constant bounds involving theλ′ interactions alone, based on the sin-gle coupling constant dominance hypothesis in the current basis for quark fields can be ob-tained [262]. Applying the transformation from current to the mass basis, one may express thetransition amplitudes so that only a single6Rp coupling constant appears. The bounds deducedin [262] deserve an update in view of the experimental results published in [272] including theresults from the BABAR and BELLE collaborations. These bounds would however depend on

the absolute mixing in the quark sector and would be valid if the relative mixing of the up anddown quark sectors was entirely due to the absolute mixing inthe down sector. However inthis case, noDD mixing can be induced by a single6Rp coupling. Alternatively, if the CKMmixing comes only from the mixing in the up quark sector thenDD mixing can provide a verystringent bound onλ′ijk.

Rare Leptonic Decays of Mesons

The study of rare leptonic decay modes of theK andB mesons offers distinctive probes fornew physics beyond the Standard Model [358]. We shall consider in this subsection the leptonictwo-body decay channels corresponding to final states with acharged lepton-antilepton pair,M0 → l−i l

+j (with M0 = K0

L, K0S, B0

d or B0s ), as well as chargedB meson decays into a

charged lepton and a (anti)neutrino,B− → l−ν.

The decaysK0, B0 → l−i l+j arise via the underlying quark flavour violating subprocess

dk + dl → ei + ej (k 6= l). In the Standard Model, the transitions that preserve lepton flavour(i = j), such asKL → µ+µ− or B0 → µ+µ−, arise through loop diagrams and are stronglysuppressed, while the transitions that violate lepton flavour (i 6= j), such asKL → µ+e−, areunobservable due to the smallness of neutrino masses. On theexperimental side, the decaysKL → e+e− andKL → µ+µ− have been measured, while only upper bounds are available onthe correspondingKS decays. In theB meson sector, the experimental upper bounds on thedecaysB0

d,s → µ+µ− andB0d,s → e+e− are still several orders of magnitude above the Standard

Model predictions, whileB0d,s → τ+τ− is yet unconstrained.

6Rp interactions contribute to the subprocessdk + dl → li + lj via tree-level sneutrino and upsquark exchange, as shown in Fig. 6.17. This allows to extract significant bounds on quadraticproducts of6Rp couplings from rare leptonic meson decays. Specifically, the decayM0 → l−i l

+j ,

whereM0 = dkdl, constrains the following quantities:

Aklij ≡

∑

n,p,q

VnpV†qn

λ′⋆ipkλ′jql

m2unL

, Bklij ≡

∑

n,p,q

U †npUqn

λ⋆pijλ

′qkl

m2νnL

, (6.73)

whereAklij andBkl

ij are associated with up squark and sneutrino exchange, respectively. InEq. (6.73), the couplingsλijk andλ′ijk are expressed in the mass eigenstate bases of downquarks and charged leptons, which explains the presence of the CKM and MNS mixing angles(see subsection 6.1.2), and the sfermion mass matrices are assumed to be diagonal in the masseigenstate basis of their fermion partners. In the following, we shall further assume that themasses of the exchanged sfermions are degenerate, i.e.munL

≡ muLandmνnL

≡ mνL; then

Eq. (6.73) reduces toAklij = 1

m2uL

∑p λ

′⋆ipkλ

′jpl andBkl

ij = 1m2

νL

∑p λ

⋆pijλ

′pkl.

For mesons that have wave functions of the formMkl = (dkdl ±dldk)/√

2, likeKL andKS

in the limit whereCP violation is neglected,Aklij andBkl

ij must be replaced by(Aklij ±Alk

ij )/√

2

and(Bklij ± Blk

ij )/√

2, respectively. One then defines, for the kaon system [359]:

ALij ≡ 1

m2uL

∑

p

(λ′⋆ip1λ′jp2 − λ′⋆ip2λ

′jp1) , BL

ij ≡ 1

m2νL

∑

p

λ⋆pij(λ

′p12 − λ′p21) , (6.74)

ASij ≡ 1

m2uL

∑

p

(λ′⋆ip1λ′jp2 + λ′⋆ip2λ

′jp1) , BS

ij ≡ 1

m2νL

∑

p

λ⋆pij(λ

′p12 + λ′p21) , (6.75)

~n0nkl ejeidk

dlnij 0jkl

0ink~unLeidk

dl ejFigure 6.17: 6Rp contributions to the processdk + dl → ei + ej .

whereALij,B

Lij are relevant forKL decays, andAS

ij ,BSij are relevant forKS decays. SinceAL

ji =−AL⋆

ij , AL11 andAL

22 vanish for real6Rp couplings. As a result, the lepton flavour conservingdecaysKL → e+e− andKL → µ+µ− only constrain the imaginary part of the productsλ′⋆ip1λ

′ip2

(i = 1, 2)11. By requiring that the6Rp contribution itself does not exceed the2σ upper boundon the branching ratios ofKL → e+e− andKL → µ+µ−, measured to be(9 + 6

− 4) × 10−12

and(7.25 ± 0.16) × 10−9 [272], respectively, one obtains the following bounds, which updatethe ones given in Ref. [359]:|BL

11| < 1.0 × 10−8 ν2L, |BL

22| < 2.2 × 10−7 ν2L, |ℑ(AL

11)| <8.1 × 10−5 u2

L, |ℑ(AL22)| < 7.8 × 10−6 u2

L. Under the double coupling dominance hypothesis,these bounds yield:

|λ⋆121λ

′212|, |λ⋆

121λ′221| < 1.0 × 10−8 ν2

L [KL → e+e−] ,|λ⋆

131λ′312|, |λ⋆

131λ′321| < 1.0 × 10−8 ν2

L [KL → e+e−] ,|λ⋆

122λ′112|, |λ⋆

122λ′121| < 2.2 × 10−7 ν2

L [KL → µ+µ−] ,|λ⋆

232λ′312|, |λ⋆

232λ′321| < 2.2 × 10−7 ν2

L [KL → µ+µ−] ,

(6.76)

|ℑ(λ′⋆1j1λ′1j2)| < 8.1 × 10−5 u2

L [KL → e+e−] ,|ℑ(λ′⋆2j1λ

′2j2)| < 7.8 × 10−6 u2

L [KL → µ+µ−] .(6.77)

The bounds associated with the lepton flavour violating decay KL → e±µ∓ have been derivedin Ref. [359] and updated in Ref. [360] with the90% CL experimental limitB(KL → e±µ∓) <4.7 × 10−12 given in Ref. [272]:

|λ⋆122λ

′212|, |λ⋆

122λ′221| < 6 × 10−9 ν2

L ,|λ⋆

132λ′312|, |λ⋆

132λ′321| < 6 × 10−9 ν2

L ,|λ⋆

121λ′112|, |λ⋆

121λ′121| < 6 × 10−9 ν2

L ,|λ⋆

231λ′312|, |λ⋆

231λ′321| < 6 × 10−9 ν2

L ,|λ′⋆1j1λ

′2j2|, |λ′⋆1j2λ

′2j1| < 3 × 10−7 u2

L .

(6.78)

Significantly better bounds are obtained forλλ′-type products of couplings; the reason for thatis that the contribution ofBL

ij to the decay amplitude is enhanced with respect to the contributionof AL

ij by a factor2m2K0/ml(md +ms), whereml = mµ orme. In updating the bounds (6.76),

we have used the central values of the estimated ranges forms andms/md given in Ref. [272],ms = (80 − 155) MeV andms/md = (17 − 22).

11This conclusion remains true when the exchanged sfermions are not degenerate in mass, which is the caseconsidered in Ref. [359]. In this case however, the productsλ′⋆

ip1λ′iq2 andλ′⋆

ip2λ′iq1 (p 6= q) also contribute to the

decaysKL → l+i l−i , but their contribution is suppressed by CKM mixing angles.For the imaginary part of theseproducts, the order of magnitude of the suppression is givenby |Vpq| (and is therefore rather mild forℑ(λ′⋆

i11λ′i22)

andℑ(λ′⋆i21λ

′i12), but the latter is much more constrained byCP violation in theKK system, see subsection 6.3.4),

while it can be much stronger for the real part. This results in weaker bounds onℑ(λ′⋆ip1λ

′iq2), and especially on

ℜ(λ′⋆ip1λ

′iq2), than onℑ(λ′⋆

ip1λ′ip2).

The bounds derived fromKS leptonic decays are less stringent than the ones derived fromKL leptonic decays, due to the weaker experimental sensitivity toKS decays, and we do not listthem here. We just mention that, sinceAS

ji = AS⋆ij , the decaysKS → e+e− andKS → µ+µ−

provide bounds on the real part of the productsλ′⋆ip1λ′ip2 (i = 1, 2), while the decaysKL → l+i l

−i

only constrain their imaginary part. Stronger bounds onℜ(λ′⋆ip1λ′ip2) can however be derived

from KK mixing by considering the contribution of box diagrams withan internalW boson,charged Higgs or charged Goldstone boson [356].

The decaysB0 → l−i l+j provide bounds on the coupling productsλ⋆

pijλ′pkl, λ

⋆pjiλ

′plk and

λ′⋆ipkλ′jpl with (k, l) = (1, 3), (3, 1) (B0

d decays) and(k, l) = (2, 3), (3, 2) (B0s decays). Since

leptonicB meson decays are less constrained experimentally than leptonic kaon decays, thebounds on coupling products associated with the former are less stringent than those associatedwith the latter, Eqs. (6.76) – (6.78). Nevertheless leptonic B meson decays provide the bestbounds (with some exceptions) on coupling products of the form λ⋆

pijλ′pkl, with k = 3 or l = 3.

These bounds were derived in Ref. [361] and updated in Refs [360, 362] with the90% CLexperimental limits given in Ref.[272]. We list below the bounds on theλλ′-type couplingproducts given in Ref. [362]:

|λ⋆i11λ

′i13|, |λ⋆

i11λ′i31| < 1.7 × 10−5 ν2

L [B0d → e+e−] ,

|λ⋆i22λ

′i13|, |λ⋆

i22λ′i31| < 1.5 × 10−5 ν2

L [B0d → µ+µ−] ,

|λ⋆i12λ

′i13|, |λ⋆

i12λ′i31|, |λ⋆

i21λ′i13|, |λ⋆

i21λ′i31| < 2.3 × 10−5 ν2

L [B0d → e±µ∓] ,

|λ⋆i13λ

′i13|, |λ⋆

i13λ′i31|, |λ⋆

i31λ′i13|, |λ⋆

i31λ′i31| < 4.9 × 10−4 ν2

L [B0d → e±τ∓] ,

|λ⋆i23λ

′i13|, |λ⋆

i23λ′i31|, |λ⋆

i32λ′i13|, |λ⋆

i32λ′i31| < 6.2 × 10−4 ν2

L [B0d → µ±τ∓] ,

|λ⋆i11λ

′i23|, |λ⋆

i11λ′i32| < 1.4 × 10−4 ν2

L [B0s → e+e−] ,

|λ⋆i22λ

′i23|, |λ⋆

i22λ′i32| < 2.7 × 10−5 ν2

L [B0s → µ+µ−] ,

|λ⋆i12λ

′i23|, |λ⋆

i12λ′i32|, |λ⋆

i21λ′i23|, |λ⋆

i21λ′i32| < 4.7 × 10−5 ν2

L [B0s → e±µ∓] .

(6.79)

By contrast, the bounds onλ′λ′-type coupling products associated with rare leptonic decaysof B mesons are generally weaker than the products of bounds on individual couplings. Wenevertheless list the bounds on the coupling productsλ′⋆ipkλ

′jpl given in Ref. [362] (there is no

significant bound associated with the decay modesB0d,s → e+e−):

|λ′⋆2j1λ′2j3| < 2.1 × 10−3 u2

L [B0d → µ+µ−] ,

|λ′⋆1j1λ′2j3|, |λ′⋆1j3λ

′2j1| < 4.7 × 10−3 u2

L [B0d → e±µ∓] ,

|λ′⋆1j1λ′3j3|, |λ′⋆1j3λ

′3j1| < 5.9 × 10−3 u2

L [B0d → e±τ∓] ,

|λ′⋆2j1λ′3j3|, |λ′⋆2j3λ

′3j1| < 7.3 × 10−3 u2

L [B0d → µ±τ∓] ,

|λ′⋆2j2λ′2j3| < 3.9 × 10−3 u2

L [B0s → µ+µ−] ,

|λ′⋆1j2λ′2j3|, |λ′⋆1j3λ

′2j2| < 9.6 × 10−3 u2

L [B0s → e±µ∓] .

(6.80)

The bounds (6.79) and (6.80) have been derived usingfBd= fBs = 200 MeV; they scale as

(200 MeV/fBd,s). Alsomb +md ≈ MB0

dandmb +ms ≈ MB0

shave been used for the bounds

(6.79).

No dedicated search for the decaysB0 → τ+τ− has yet been carried out explicitely at theexisting colliders. However, such decays would manifest themselves at the LEP experiments asbb events associated with large missing energy, due to the neutrinos emerging from theτ decays.Such events were studied at LEP to set constraints on the branching ratio forB− → τ ν, andfrom an analysis of the same data Grossman et al. infer the upper boundsB(B0

d → τ+τ−) <0.015 andB(B0

s → τ+τ−) < 0.05 [363], four orders of magnitude above the Standard Modelpredictions. These results in the following bounds on the coupling productsλ⋆

i33λ′ikl, with k = 3

or l = 3 [363]:

|λ⋆i33λ

′i13|, |λ⋆

i33λ′i31| < 6.4 × 10−3 ν2

L [Bd → τ+τ−] , (6.81)

|λ⋆i33λ

′i23|, |λ⋆

i33λ′i32| < 1.2 × 10−2 ν2

L [Bs → τ+τ−] . (6.82)

For completeness, we also mention the bounds that have been derived from the non-observa-tion of the lepton flavour violating neutral pion decayπ0 → µ+e− in Ref. [360], using the90%CL experimental upper limitB(π0 → µ+e−) < 3.8 × 10−10 [272]. The following bounds arebetter than previous bounds:

|λ⋆312λ

′311|, |λ⋆

321λ′311| < 3 × 10−3 ν2

L . (6.83)

We now consider leptonic decays of chargedB mesons,B− → l−ν. In the Standard Model,these decays are suppressed by the CKM angleVub and by charged lepton masses, and the exper-imental upper bounds on their branching ratios are still well above the theoretical predictions,except for the decay modeB− → τ−ν. 6Rp interactions contribute to these decays via similartree-level diagrams to those of Fig. 6.17, with the exchanged sneutrino (resp. up squark) re-placed by a charged slepton (resp. down squark). Specifically, the decayB− → l−i ν constrainsthe following quantities [364]:

Cij ≡∑

n,p

V1p

λ′⋆ipnλ′j3n

m2dnR

, Dij ≡∑

n,p

V1p

λ′⋆np3λnji

m2enL

, (6.84)

where, as in Eq. (6.73), the couplingsλijk andλ′ijk are expressed in the mass eigenstate bases ofdown quarks and charged leptons, and the sfermion mass matrices are assumed to be diagonal.Due to the impossibility of distinguishing experimentallythe flavour of the neutrino produced, asingle decay modeB− → l−i ν constrains the six quantitiesCij andDij , j = 1, 2, 3. Then, froma single constraint|Dij| < B, one can derive, under the double coupling dominance hypothesis,the following set of bounds (n = 1, 2, 3): λ′⋆n13λnji < B d2

nR, λ′⋆n23λnji < (B/Vus) d2nR and

λ′⋆n33λnji < (B/Vub) d2nR. A similar statement holds for the bounds on the coupling products

λ′⋆i1nλ′j3n, λ′⋆i2nλ

′j3n, andλ′⋆i3nλ

′j3n derived from|Cij| < B′.

The bounds on quadratic products of6Rp couplings associated with the90% CL experimentalupper limit onB(B− → l−i ν) have been derived in Ref. [364]. Ref. [360] obtained weakerbounds from a more careful analysis based on a conservative treatment of the experimentalerrors. We list below the bounds that are not weaker than bounds associated with other processesor products of individual bounds [360]:

|λ′⋆i13λi31| < 6 × 10−4 l2iL [B− → e−ν] ,

|λ′⋆i13λi32| < 7 × 10−4 l2iL [B− → µ−ν] ,

|λ′⋆313λ233| < 2 × 10−3 l23L [B− → τ−ν] ,

−6 × 10−4 l22L < ℜ(λ′⋆213λ233) < 1 × 10−3 l22L [B− → τ−ν] .

(6.85)

The bounds onλ′λ′-type products associated with leptonic decays of chargedB mesons are notcompetitive, since the contribution ofCij to the decay amplitude is suppressed by a factor ofmli/mB± with respect to the contribution ofDij .

Rare Semileptonic Decays of Mesons

The rare semileptonic FCNC decayK+ → π+νν is often regarded as a hallmark for tests ofthe Standard Model and searches for new physics. Indeed, this process is theoretically veryclean, since the hadronic matrix element can be extracted from the well-measured decayK+ →π0e+ν, and the long-distance hadronic physics contributions areknown to be small [365, 366].The present experimental value,B(K+ → π+νν) = 1.6+1.8

−0.8 × 10−10 [20], is compatible withthe SM predictions, but has still large errors.

s ii00ij10i0j2 d~djL

us i0i

0i01k0i2k d~dkRu

Figure 6.18: 6Rp contributions toK+ → π+νν.

The 6Rp interactions can contribute to the processK+ → π+νν through the tree-level di-agrams shown in Fig. 6.18, which involve adkR or a dkL exchange. The dependance of thebranching ratio on the6Rp couplings is encapsulated in the auxiliary parametersEii′ [359]:

Eii′ =∑

k

λ′⋆i2kλ′i′1k d

−2kR −

∑

j

λ′⋆ij1λ′i′j2 d

−2jL . (6.86)

In Ref. [359], an experimental upper limit was used to put an upper bound on∑

ii′ |Eii′|2, ne-glecting the Standard Model contribution. A warning is in order concerning this method. Sincethe present experimental valueB(K+ → π+νν) = (1.47 +1.3

−0.8) 10−10 [367] is close to theexpected Standard Model value, it is no longer legitimate toneglect the Standard Model contri-bution when deriving constraints on6Rp couplings.

Since then a detailed analysis, including all relevant contributions, was performed [368],yielding the upper bound

∑ii′ |Eii′|2 < 4.45 × 10−10.

One can use the bound on∑

ii′ |Eii′|2 to infer bounds on products ofλ′-type couplings.Applying the double coupling dominance hypothesis in the mass eigenstate basis, one obtains[368]:

|λ′⋆i2kλ′i′1k| < 2.11 × 10−5 d2

kR ,

|λ′⋆ij1λ′i′j2| < 2.11 × 10−5 d2jL .

[K+ → π+νν] (6.87)

Bounds on individual coupligs may also be obtained if, instead of applying the double cou-pling dominance hypothesis in the mass eigenstate basis, itis assumed that a single cou-pling is nonzero in the weak eigenstate basis [262], i.e.λ′ipq 6= 0 in the notation of subsec-tion 6.1.2. Then, upon rotating the down quarks to their masseigenstate basis, several couplingsλ′ijk = (V d†

L )pj(VdTR )qkλ

′ipq are generated, and one has

∑k λ

′⋆i2kλ

′i′1k = δi′i(V

dL )2p(V

d⋆L )1p|λ′ipq|2.

In a similar manner, bounds on the coupling productsλ′⋆i3kλ′i′2k andλ′⋆ij2λ

′i′j3 can be extracted

from the non-observation of the rare semileptonicB meson decayB → Xsνν [278]. We have

updated the result of Ref. [278] with the90% CL experimental upper limitB(B → Xsνν) <7.7 × 10−4 [369], which lies an order of magnitude above the Standard Model prediction:

|λ′⋆i3kλ′i′2k| < 1.5 × 10−3 d2

kR ,

|λ′⋆ij2λ′i′j3| < 1.5 × 10−3 d2jL .

[B → Xsνν] (6.88)

Finally, the rare semileptonic decaysB → Xsl+i l

−j can also be used to set bounds onλ′λ′-

type andλλ′-type coupling products [370].

Rare Hadronic Decays of theB Mesons

The hadronicB meson decays that do not proceed through ab → c transition are suppressedin the Standard Model, and offer potentially promising constraints on the6Rp interactions. Un-like the rare leptonic and semileptonic decays discussed before, however, these processes areplagued with large hadronic uncertainties and the bounds onproducts of6Rp couplings presentedbelow should be considered as indicative.

In Ref. [371], the decaysB+ → K0K+ andB+ → K0π+ have been used to set boundson the products of baryon number violating couplingsλ′′i23λ

′′⋆i12 andλ′′i13λ

′′⋆i12, which contribute to

these processes via tree-level exchange of an up squark. We have updated the bound estimates ofRef. [371] by using the90% CL experimental upper limitB(B+ → K0K+) < 2.4×10−6 [272]and by requiring that the6Rp contribution toB+ → K0π+ does not exceed by more than2σ themeasured value of the branching ratio,B(B+ → K0π+) = (1.73+0.27

− 0.24) × 10−5 [272], andfound:

|λ′′i23λ′′⋆i12| < 1.7 × 10−3 u2iR [B+ → K0K+] ,

|λ′′i13λ′′⋆i12| < 6.4 × 10−3 u2iR [B+ → K0π+] .

(6.89)

Ref. [372] improves the results of Ref. [371] by consideringa large sample of hadronicdecay modes of theB mesons, which receive contributions of the baryon number violating 6Rp

interactions through tree-level exchange of either a down squark or an up squark. Assumingnaive factorization of the hadronic matrix elements, Ref. [372] obtains the following allowedranges at90% CL (we give only the stronger constraints):

−1.1 × 10−3 d21R < λ′′113λ

′′112 < 7.8 × 10−4 d2

1R [B0→π0K0⋆ B+→π0K+] ,

−1.2 × 10−3 d22R < λ′′123λ

′′212 < 1.4 × 10−3 d2

2R [B+→π+D0, ρ+D0;B0→D0π0] ,

−1.4 × 10−2 d21R < λ′′213λ

′′112 < 2.0 × 10−2 d2

1R [B+→D+s π0] ,

−7.9 × 10−4 u2iR < λ′′i13λ

′′i12 < 1.2 × 10−3 u2

iR [B+→π+K0, π0K+, π+K0⋆] ,

−1.9 × 10−3 u2iR < λ′′i23λ

′′i12 < 2.8 × 10−3 u2

iR [B0→K0K0] .

(6.90)

Ref. [373] considers the decay modeB− → φπ−, which, using QCD factorization, theyestimate to be suppressed at the level ofB(B− → φπ−) = (2.0+0.3

−0.1) × 10−8 in the StandardModel. From the90% CL upper limitB(B− → φπ−) < 1.6 × 10−6 [272], they derive thefollowing upper bounds:

|λ′′⋆i23λ′′i12| < 6 × 10−5 u2

iR ,|λ′i32λ′⋆i12| < 4 × 10−4 ν2

iL ,|λ′⋆i23λ′i21| < 4 × 10−4 ν2

iL .[B− → φπ−] (6.91)

FCNC Top Quark Decays

The flavour changing neutral current decays of the top quark,which will be best constrainedin future Tevatron experiments at Fermilab and at the CERN LHC, might provide bounds onproducts of6Rp couplings that are not constrained by other processes. Ref.[374] considered theFCNC top quark decayst → c + V (V = Z, γ, g), for which an experimental sensitivity of(10−5−10−3) is expected, depending on the decay mode. These decays, which are negligible inthe Standard Model, are induced at the one-loop level by the6Rp couplingsλ′ijk andλ′′ijk; how-ever, given the constraints from other processes on theλ′ijk couplings, only theλ′′ijk couplingsare likely to give an observable contribution. The corresponding branching ratios are estimatedto be, for squark massesmdkR

. 170 GeV [374]:

B(t→ c+ [Z, γ, g]) = [3.6 × 10−5, 9 × 10−7, 1.6 × 10−4] |∑

j<k

λ′′⋆3jkλ′′2jk|2 , (6.92)

and scale as1/m4dkR

for larger squark masses. Although modest, the bounds that might beinferred from the expected experimental sensitivities arecomplementary with the other boundson theλ′′ijk couplings discussed in this chapter.

The Rare Decayb → sγ

0i3k 0i2k~ i sLbR bL

dkR

Figure 6.19: Example of the6Rp contributions to the decayb→ sγ.

The measured inclusiveb → sγ rate,B(B → Xsγ) = (3.3 ± 0.4) × 10−4 [20], is ingood agreement with the Standard Model prediction [375, 376, 377]. This constrains newphysics contributions, and in particular implies restrictions on the Supersymmetric StandardModel spectrum and on some combinations of6Rp couplings. Ref. [112] considered both thedirect contribution of the6Rp interactions, which can mediateb→ sγ through one-loop diagramssuch as the one shown in Fig. 6.19, and their indirect contribution through the renormalizationgroup evolution of the soft supersymmetry breaking masses.Indeed, the6Rp couplings enterthe renormalization group equations for the supersymmetric parameters, and can generate largeflavour violating entries in the squark mass matrices which then induce the decayb → sγ.This indirect contribution can enhance the branching ratioby up to an order of magnitude withrespect to the direct contribution, but it is difficult to derive bounds on the6Rp couplings fromthis effect due to its complicated dependence on the supersymmetric mass spectrum. The directcontribution yields the following upper bounds [112]:

|λ′i3kλ′⋆i2k| < 0.09 (2ν−2

iL − d−2iR )−1 ,

|λ′⋆ij3λ′ij2| < 0.035 (l−2iL − d−2

jL )−1 ,|λ′′⋆i3kλ

′′i2k| < 0.16 q2

R .

(6.93)

Since the experimental value ofB(B → Xsγ) has significantly changed with respect to the onegiven in Ref. [378] that was used in Ref. [112], these bounds should be considered as indicative.Using more recent data, Ref. [379] derives a weaker bound on the coupling productsλ′′⋆33kλ

′′32k.

An update of their results, taking into account the reduction of the experimental error, yields the2σ upper bound:

|λ′′⋆33kλ′′32k| < 0.35 d2

iR . (6.94)

6.4.2 Lepton Flavour Violating Processes

In the Standard Model, lepton flavour violating (LFV) processes occur at a negligible rate dueto the smallness of neutrino masses. They are therefore verysensitive probes of new physics,and can be used to place bounds on6Rp couplings. In order to disentangle the effect of6Rp

interactions from the effect of possible flavour non-universalities in the slepton sector, we shallassume in this subsection that the slepton mass matrices arediagonal and proportional to theidentity matrix, i.e.m2

lRi

≡ m2lR

,m2lLi

≡ m2lL

,m2νLi

≡ m2νL

.

Lepton Flavour Violating Radiative Decays of Charged Leptons

The 6Rp interactions can induce LFV radiative decays of charged leptons,lj → li + γ (i 6= j),through one-loop diagrams analogous to the one shown in Fig.6.19. The most constrained ofthese decays,µ→ eγ, yields the following upper bounds onλλ andλ′λ′-type coupling products(the bounds given in Refs. [30] and [380] have been updated inRef. [381] using the90% CLexperimental upper limitB(µ→ eγ) < 1.2 × 10−11 [327]):

|λ⋆ij2λij1| < 8.2 × 10−5 (2ν−2

L − l−2L )−1 ,

|λ23kλ⋆13k| < 2.3 × 10−4 (2ν−2

L − l−2R )−1 ,

|λ′2jkλ′⋆1jk| < 7.6 × 10−5 d2

kR (j = 1, 2) .

(6.95)

Due to the large top quark mass, the bound on|λ′23kλ′⋆13k| does not scale asm2

dkR. Indeed,

updating the bounds of Ref. [380], one obtains:

|λ′23kλ′⋆13k| < 1.3 × 10−3 , 2.0 × 10−3 , 9.9 × 10−3 (k = 1, 2) ,

|λ′233λ′⋆133| < 1.7 × 10−3 , 2.0 × 10−3 , 9.9 × 10−3 ,(6.96)

for mdkR= mtL = 100 GeV,300 GeV and1 TeV, respectively. In Eqs. (6.95) and (6.96), left-

right mixing in the squark and charged slepton mass matriceshas been neglected. The boundsthat can be inferred fromτ → µγ and fromτ → eγ are much weaker.

Ref. [30] also investigates the indirect contribution of6Rp interactions toµ → eγ throughtheir effect on the renormalization group evolution of the slepton masses. The indirect contri-bution often dominates over the direct contribution discussed above; however, due to its com-plicated dependence on the supersymmetric parameters, it is not possible to derive bounds on6Rp couplings from this effect.

Lepton Flavour Violating Decays ofµ and τ into three Charged Leptons

The lepton flavour violating decayl−m → l−i + l−j + l+k , wherelm = µ or τ , can be mediated bytree-levelt- andu-channel sneutrino exchange when the involved leptons havenonzeroλ-typecouplings. The non-observation of these processes yield bounds on products of6Rp couplings ofthe formλnmiλ

⋆njk, λ⋆

nimλnkj, λnmjλ⋆nik andλ⋆

njmλnki [256, 359]. We have updated the boundsof Ref. [359] using the90% CL experimental upper limits onB(l−m → l−i + l−j + l+k ) given inRef. [272]:

|λ321λ⋆311| , |λ⋆

i12λi11| < 6.6 × 10−7 ν2L [µ → eee] ,

|λ231λ⋆211| , |λ⋆

i13λi11| < 2.7 × 10−3 ν2L [τ → eee] ,

|λ231λ⋆212| , |λ⋆

313λ321| < 2.0 × 10−3 ν2L [τ− → µ+e−e−] ,

|λ232λ⋆211| , |λ⋆

323λ311| , |λ131λ⋆121| , |λ⋆

i13λi12| < 2.1 × 10−3 ν2L [τ− → µ−e+e−] ,

|λ132λ⋆121| , |λ⋆

323λ312| < 2.0 × 10−3 ν2L [τ− → e+µ−µ−] ,

|λ131λ⋆122| , |λ⋆

313λ322| , |λ232λ⋆212| , |λ⋆

i23λi21| < 2.1 × 10−3 ν2L [τ− → e−µ+µ−] ,

|λ132λ⋆122| , |λ⋆

i23λi22| < 2.2 × 10−3 ν2L [τ → µµµ] .

(6.97)

The decaysl−m → l−i +l−j +l+j (k = j) can also be induced through photon penguin diagramsby the sameλλ- andλ′λ′-type coupling products as the radiative decayslm → liγ. In the case ofµ → eee, the associated bounds are stronger than the ones extractedfrom the non-observationof µ → eγ. We list below the bounds given in Ref. [381]:

|λ⋆232λ231| < 4.5 × 10−5 , |λ232λ

⋆132| < 7.1 × 10−5 , |λ233λ

⋆133| < 1.2 × 10−4 ,

|λ′211λ′⋆111| < 1.3 × 10−4 , |λ′212λ′⋆112| < 1.4 × 10−4 , |λ′213λ′⋆113| < 1.6 × 10−4 ,|λ′221λ′⋆121| < 2.0 × 10−4 , |λ′222λ′⋆122| < 2.3 × 10−4 , |λ′223λ′⋆123| < 2.9 × 10−4 .

(6.98)

In Eq. (6.98), the bounds onλλ-type (resp.λ′λ′-type) coupling products have been derivedassuming that all slepton (resp. squark) masses are degenerate and equal tom = 100 GeV(resp. m = 300 GeV) and neglecting left-right mixing in sfermion mass matrices. Thesebounds do not simply scale asm2.

Muon to Electron Conversion in Nuclei

µ− → e− conversion in a nucleus can be induced byλλ′- andλ′λ′-type coupling products viathe exchange of a sneutrino (resp. a squark) in thet-channel (resp.s- andu-channels) [382].Experimentally, stringent bounds are set on the rate ofµ− e conversion in a nucleusA relativeto the ordinary muon capture,Rµe ≡ Γ(µ− + A → e− + A)/Γ(µ−capture inA). Using the90% CL upper limitRµe < 6.1 × 10−13 obtained by the SINDRUM II experiment on a48Titarget [383], the following bounds are deduced, updating those given in Ref. [382]:

|λ⋆i12λ

′i11| , |λi21λ

′⋆i11| < 2.1 × 10−8 ν2

L ,|λ′2j1λ

′⋆1j1| < 4.3 × 10−8 u2

jL (j = 2, 3) ,

|λ′21kλ′⋆11k| < 4.5 × 10−8 d2

kR (k = 2, 3) .

(6.99)

The combinationλ′211λ′⋆111 can also induceµ − e conversion in48Ti, but cancellations may

occur between the up squark and the down squark contributions, resulting in a weaker bound,|λ′211λ′⋆111| < 4.3 × 10−8 (u−2

L − 7074d−2

R )−1.

6Rp-inducedµ− → e− conversion in a nucleus can also proceed through photon penguindiagrams [384], in the same way as the LFV decayµ → eee. The associated bounds are

stronger than the ones extracted from the non-observation of µ→ eγ andµ → eee, if the latterdoes not occur at tree level. We list below the bounds given inRef. [381]:

|λ⋆122λ121| < 6.1 × 10−6 , |λ⋆

132λ131| < 7.6 × 10−6 , |λ⋆232λ231| < 8.3 × 10−6 ,

|λ231λ⋆131| < 1.1 × 10−5 , |λ232λ

⋆132| < 1.3 × 10−5 , |λ233λ

⋆133| < 2.3 × 10−5 ,

|λ′222λ′⋆122| < 4.3 × 10−5 , |λ′223λ′⋆123| < 5.4 × 10−5 .(6.100)

In Eq. (6.100), the bounds onλλ-type (resp.λ′λ′-type) coupling products have been derivedassuming that all slepton (resp. squark) masses are degenerate and equal tom = 100 GeV (resp.m = 300 GeV) and neglecting left-right mixing in sfermion mass matrices. These bounds donot simply scale asm2.

Muonium to Antimuonium Conversion

The conversion reaction of a muonium atom into an antimuonium atom,M(µ+e−) → M(µ−e+),has been initially proposed as a test of a multiplicative lepton number symmetry [385] whichwould forbid ∆Lµ = ±1 transitions, but would allow for∆Lµ = ±2 transitions such asM → M . Experimental limits on this process are conventionally expressed in terms of aneffective couplingGMM defined by [386]:

Leff(M → M) =4GMM√

2(µLγ

µeL)(µLγµeL) + h.c. . (6.101)

The current90% CL experimental limit [387] isGMM < 3.0 × 10−3GF . In the presence ofR-parity violation, muonium to antimuonium conversion can bemediated by tree-level exchangeof a tau sneutrino in thes- or theu-channel as shown in Fig. 6.20. The associated effective

321 312(a)~

eL

ReR

L312

~R

eReL

L 321

(b)Figure 6.20: 6Rp contributions to muonium-antimuonium conversion. Similar diagrams involv-ing an incomingµR are not shown.

interaction is of the(V − A)(V + A) form (after a Fierz transformation) and is described byan effective couplingGMM distinct from the couplingGMM defined by Eq. (6.101). Explicitly,one has [382, 388, 389]:

GMM√2

=λ312λ

⋆321

8m2νL

. (6.102)

The90% CL experimental limit on muonium-antimuonium conversion [387], in the case of a(V −A)(V +A) interaction, translates intoGMM < 2.0 × 10−3GF . This yields the followingupper bound on the coupling productλ312λ

⋆321, which updates the bound given in Ref. [382]:

|λ312λ⋆321| < 1.9 × 10−3 ν2

L . (6.103)

Lepton Flavour Violating Semileptonic Decays of theτ

Theτ−lepton decay modes include a variety of lepton flavour violating processes which yieldconstraints on several products of6Rp couplings. Of special interest are the two-body decaymodes into pseudoscalar and vector mesons,τ → l + P 0 and τ → l + V 0, with l = e, µ,P = π0, η,K0 andV = ρ0, ω,K⋆0, φ. The 6Rp interactions contribute to these processes viatree-level sneutrino or squark exchange, induced byλλ′-type orλ′λ′-type coupling pairs, re-spectively [382]. For sneutrinos or up-type squarks, the corresponding diagrams are the time-reversed of the diagrams shown in Fig. 6.17; the exchange of adown-type squark, which is notshown, corresponds to the subprocessei + ej → uk + ul.

The sneutrino exchange mediates tau decays into pseudoscalar mesons only; hence theλλ′-type coupling products are only constrained by these decays. Using the90% CL experimentalupper limits onB(τ → l + P 0) given in Ref. [272], one obtains the following bounds, whichupdate the bounds of Ref. [382]:

|λi31λ′⋆i11| , |λ⋆

i13λ′i11| < 1.6 × 10−3 ν2

iL [τ− → e− + η0] ,|λi31λ

′⋆i22| , |λ⋆

i13λ′i22| < 1.6 × 10−2 ν2

iL [τ− → e− + η0] ,|λi31λ

′⋆i12| , |λ⋆

i13λ′i21| < 8.5 × 10−2 ν2

iL [τ− → e− +K0] ,|λi32λ

′⋆i11| , |λ⋆

i23λ′i11| < 1.7 × 10−3 ν2

iL [τ− → µ− + η0] ,|λi32λ

′⋆i22| , |λ⋆

i23λ′i22| < 1.7 × 10−2 ν2

iL [τ− → µ− + η0] ,|λi32λ

′⋆i12| , |λ⋆

i23λ′i21| < 7.6 × 10−2 ν2

iL [τ− → µ− +K0] .

(6.104)

λ′λ′-type coupling products induce both decaysτ → l+P 0 andτ → l+V 0, but the latter aremore constrained experimentally and therefore provide stronger bounds than the former. Usingthe90% CL experimental upper limits onB(τ → l + V 0) given in Ref. [272], one obtains thefollowing bounds, which update the bounds of Ref. [382]:

|λ′3j1λ′⋆1j1| < 2.4 × 10−3 u2

jL [τ− → e− + ρ0] ,|λ′3j1λ

′⋆1j2| < 2.7 × 10−3 u2

jL [τ− → e− +K⋆0] ,|λ′3j1λ

′⋆2j1| < 4.4 × 10−3 u2

jL [τ− → µ− + ρ0] ,|λ′3j1λ

′⋆2j2| < 3.4 × 10−3 u2

jL [τ− → µ− +K⋆0] ,

|λ′31kλ′⋆11k| < 2.4 × 10−3 d2

kR [τ− → e− + ρ0] ,

|λ′31kλ′⋆21k| < 4.4 × 10−3 d2

kR [τ− → µ− + ρ0] .

(6.105)

6.4.3 Lepton Number Non-Conserving Processes

Neutrinoless Double Beta Decay

Searches for neutrinoless double beta decay (ββ0ν) of nuclei ((Z,N) → (Z+2, N−2)+l−i +l−j )are performed using76Ge, 48Ca, 82Se, 100Mo. They are mainly carried out in underground lab-oratories and make use of various detection techniques. Thecurrent experimental informationand some of the promising future prospects are reviewed in Ref. [390].

The nucleon level transition,n+ n→ p+ p+ e− + e− is induced at the quark level by thesubprocessd+d→ u+u+e+e. The 6Rp operatorL1Q1D

c1 would allow such a transition to occur

at the tree level, via processes involving the sequentialt-channel exchange of two sfermions anda gaugino, where the sfermion may be a slepton or a squark,eL or uL, dR, and the gaugino, aneutralino or a gluino [391, 392]. Corresponding diagrams are shown in Fig. 6.21. In the limit

eLuL dR

dRdReLuL eL

uLeLuL~dR

~dR0; ~g

dR ~eL0~eL

Figure 6.21: Contributions to the neutrinoless double beta decay induced by aλ′111 coupling.

of large masses for the exchanged sparticles, these mechanisms can be described in terms ofpoint-like six fermion effective interactions. Relying onsuch an effective Lagrangian and usingan approximate evaluation of the nuclear operator matrix element, an early study by Mohapatraled to the bounds [391]:|λ′111| < 0.48 × 10−9/4f 2g

12 , |λ′111| < 2.8 × 10−9/4f 2χ

12 . Meanwhile,

detailed calculations of theββ0ν amplitudes have been performed including all contributingdiagrams, and the relevant nuclear matrix elements have been calculated in the proton-neutronQuasiparticle Random Phase Approximation (QRPA) [392]. ¿From the lower limit on the half-life of 76Ge measured by the Heidelberg-Moscow experiment [393]:

T ββ0ν

1/2 (76Ge) > 1.1 × 1025yr

the following bound is obtained in the minimal supergravityframework [394]:

|λ′111| < 3.3 × 10−4 q2g12 . (6.106)

Since the upper bound onλ′111 scales with(T ββ0ν

1/2 )−1/4lim , most recent bounds on the half-life of

76Ge do not significantly improve the above result. Slightly moreserere bounds onλ′111 havebeen obtained in Ref. [395] by including the pion-exchange contributions to the6Rp inducedββ0ν decay.

Babu and Mohapatra [396] identified another6Rp contribution toββ0ν , based on thet-channel scalar-vector type exchange of a sfermion and a chargedW boson linked togetherthrough an intermediate internal neutrino exchange. The corresponding diagram is shown inFig. 6.22. The amplitude for this process is closely relatedto that of the familiar SM neutrinoexchange, except for the important fact that no chirality flip is required for the intermediate inter-nal neutrino line propagation. The strong suppression factor arising within the Standard Modelcontribution from the neutrino propagator factor is replaced as,mν/q

2 → 1/g · q = γ · q/q2,whereq is the intermediate neutrino four momentum. The chirality flip penalty is transferredinstead to the exchanged down-squark, as seen in Fig. 6.22. The contribution shown in Fig. 6.22thus disappears in case of a vanishing mixing betweendkR anddkL. The bound on(T ββ0ν

1/2 ) leadsthen to upper limits on the productsλ′1k1λ

′11k which scale withm4

dkR/ (Ak − µ tanβ), the de-

nominator determining the left-right mixing in thedk sector. The resulting bounds for the third,second and first generations, as quoted in Ref. [397], read:

|λ′113λ′131| < 3.8 × 10−8 (m/100 GeV)3 ,

|λ′112λ′121| < 1.1 × 10−6 (m/100 GeV)3 ,(6.107)

d

~dkL ~dkR 011k ue

eW u

e01k1dR

Figure 6.22: Contributions to the neutrinoless double beta decay induced by squark andWexchanges.

assuming the input values for the down squark mass parametersmdkR≃ (Ak − µ tanβ) ≡ m.

A systematic discussion including both the bilinear and trilinear 6Rp interactions has beengiven by Faessler et al. in Ref. [398]. It includes a detailedstudy of the validity of the differ-ent approximation schemes in the determination of the relevant nuclear matrix elements. Thebilinear 6Rp terms give rise to several contributions toββ0ν , either alone or in combination withtrilinear 6Rp interactions. The dominant contribution turns out to be theneutrino exchange dia-gram controlled by the effective neutrino mass parameter induced by bilinearR-parity violation,< mν >≡

∑imνi

U2ei ∝ (v1µ− vdµ1)

2. The comparison of their predicted results with the ex-perimental limit forββ0ν yields the bounds (assuming a common superpartner mass parameterm = 100 GeV andtanβ = 1):

|µ1| < 470 keV , |µ1λ′111| < 100 eV , (6.108)

|v1| < 840 keV , |v1λ′111| < 55 eV . (6.109)

Strictly speaking, the bounds (6.108) (resp. the bounds (6.109)) apply in a (Hd, Li) basis inwhichvi ≡< νi >= 0 (resp.µi = 0).

In Ref. [399], Hirsch studied the contribution of bilinearR-parity violation toββ0ν both atthe tree-level and at the one-loop level. The considerationof the tree-level contribution led himto exclude values forµ1 or v1 in the intervalO(0.1) − O(1) MeV, in agreement with Faessleret al. [398]. He further observed that even in the case of a perfect alignment between theµ1

andv1 parameters (such thatv1µ− vdµ1 = 0, hence bilinear6Rp violation does not contribute to< mν > at the tree level), there can occur finite contributions to< mν > arising at the one-looplevel, which leads to the bound|µ1/µ| < 0.01.

A lepton number violating process which is closely related to theββ0ν reaction concernstheµ+ → e− conversion reaction taking place in atomic nuclei via the atomic orbit capturereaction of muons [264],µ+ + (Z,N) → e− + (Z + 2, N − 2). The numerical result for thepredicted branching fraction,(B(µ− → e+)/10−12) ≃ |λ′213λ′131|/2.3 × 10−2, as representedby scaling with respect to the currentO(10−12) experimental sensitivity, indicates the extentto which future improvement in the measurements ofµ+ → e− conversion could bring usefulinformation on the6Rp interactions.

6.4.4 Baryon Number Non-Conserving Processes

Single Nucleon Decay

Matter instability, as would be implied by a non-conservation of baryon number, is a well-documented subject thanks to the extensive research developed in connection with Grand Uni-fied Theories [8, 400, 401, 402, 403]. The combined contributions of theλ′ andλ′′ interactionslead to an effective interaction of the formL = (λ′′λ′⋆/m2

dR)[(ucdc)†(νd)−(ucdc)†(eu)]+ h. c.,

which we have written using a two-component Weyl spinor representation for the fermion fields.This effective interaction is obtained by contracting a pair of down-squark fields as(dc)†dc, andtherefore yields a(B − L)-conserving amplitude. This is illustrated by the the tree-level dkR

squarks-channel exchange diagrams shown in Fig. 6.23. The comparison with the experimental

~dkRuu; de+; eu

duFigure 6.23: 6Rp contributions to the proton decay.

limits on nucleon decays yields extremely severe bounds on the 6Rp coupling productsλ′imkλ′′⋆11k

(i, k = 1, 2, 3, m = 1, 2). Adapting to the6Rp case the computation of the proton decay ratefrom dimension-6 operators done in the context of Grand Unified Theories in Ref. [404], andusing the experimental lower bounds on partial nucleon lifetimes given in Ref. [20], we obtain:

|λ′l1kλ′′⋆11k| . (2 − 3) × 10−27 d2

kR (l = 1, 2) [p→ π0l+] ,

|λ′31kλ′′⋆11k| . 7 × 10−27 d2

kR [n→ π0ν] ,

|λ′i2kλ′′⋆11k| . 3 × 10−27 d2

kR [p→ K+ν] ,

(6.110)

which update the estimate given in Ref. [256]. ThedkR squark exchange can also occur in thet-channel, yielding a bound on the6Rp coupling productsλ′l1kλ

′′⋆12k (l = 1, 2, k = 1, 2, 3):

|λ′l1kλ′′⋆12k| . (6 − 7) × 10−27 d2

kR [p→ K0l+] . (6.111)

One could also alternatively contract the down-squark fields asd − dc† by including a left-right mixing mass insertion termmd 2

LR, which then yields a(B+L)-conserving amplitude [405].

The bounds derived from the experimental limits on the nucleon decay channelsp→ K+ν andn→ π0ν [20] read (i, j = 1, 2, 3):

|λ′ij1λ′′11j | . 7 × 10−27 d2jL

(m2

djR

(md 2LR)jj

)[n→ π0ν] ,

|λ′ij2λ′′11j | . 3 × 10−27 d2jL

(m2

djR

(md 2LR)jj

)[p→ K+ν] ,

|λ′i31λ′′123| . 3 × 10−27 b2L

(m2

bR

(md 2LR)33

)[p→ K+ν] .

(6.112)

These bounds are less stringent than the previous ones due tothe presence of the left-rightmass term(md 2

LR)jj = (Ad − µ tanβ)mdj

in the denominator; the best bounds are obtainedfor j = 3 (bL − bR exchange). The exchanged scalar field can also be an up-squark, with theinsertion of a left-right mass term(mu 2

LR)jj = (Au − µ cotβ)muj

[405]. The bounds derivedfrom the experimental limits on the neutron decay modesn → K+l− [20] read (l = 1, 2,j = 1, 2, 3):

|λ′lj1λ′′j12| . 10−26 u2jL

(m2

ujR

(mu 2LR)jj

)[n→ K+l] . (6.113)

Again the best bounds are obtained forj = 3 (tL − tR exchange).

The above very stringent bounds concernλ′λ′′(⋆) products involving dominantly the firsttwo light generation indices. It was observed by Smirnov andVissani [406] that an appropriateextension of the above analysis to the one-loop level could be used to set strong bounds oncoupling products for all possible configurations of the generation indices. The contributionscome from one-loop diagrams obtained from the above tree-level diagrams by adding a vertexdiagram dressing for the couplingνidjLd

⋆kR or for the couplingui′Rdj′Rdk′R, or a box diagram

dressing for both couplings, where the internal lines propagating in the loops are charged orneutral Higgs bosons, winos or sfermions. The loop and flavour mixing suppression factors inthe transition amplitudes result in much weaker bounds thanEqs. (6.110)–(6.113). Assumingsquark masses around1 TeV, one obtains the following conservative bound on any product ofλ- andλ′′-type couplings [406]:

|λ′ijkλ′′⋆i′j′k′| < O(10−9) . (6.114)

For squark masses around100 GeV, this bound would beO(10−12).

Single nucleon decays can also be induced by products ofλ-type andλ′′-type couplings,through tree-level diagrams involving the sequential exchange of a squark, a neutralino orchargino, and a slepton, or through one-loop or two-loop diagrams obtained from the dress-ing of the former tree-level diagrams [371, 407, 408]. Bhattacharyya and Pal [408] considerproton decay mediated by diagrams involving the exchange ofa neutralino. Assuming a com-mon superpartner massm = 1 TeV, they obtain the following bounds on theλλ′′⋆ productsinvolving a couplingλ′′112:

|λ231λ′′⋆112|, |λ132λ

′′⋆112| . 10−16 [p→ K+e±µ∓ν] ,

|λ123λ′′⋆112| . 10−14 [p→ K+ννν] ,

|λ121λ′′⋆112|, |λ131λ

′′⋆112| . 10−17 [p→ K+ν] ,

|λ122λ′′⋆112|, |λ232λ

′′⋆112| . 10−20 [p→ K+ν] ,

|λ133λ′′⋆112|, |λ233λ

′′⋆112| . 10−21 [p→ K+ν] .

(6.115)

The bounds obtained from four-body decay modes could actually be relaxed by about two ordersof magnitude, due to phase space factors. The constraints onproducts involving any otherλ′′ijkcoupling are much weaker, since the corresponding vertex must be dressed by a loop with acharged Higgs boson in order to induce proton decay. The resulting bounds read, assumingmH+ = m = 1 TeV ((i, j, k) 6= (1, 1, 2)) [408]:

|λ231λ′′⋆ijk|, |λ132λ

′′⋆ijk| . (10−7 − 10−5) [p→ π+(K+)e±µ∓ν] ,

|λ123λ′′⋆ijk| . (10−5 − 10−3) [p→ π+(K+)ννν] ,

|λ121λ′′⋆ijk|, |λ131λ

′′⋆ijk| . (10−8 − 10−6) [p→ π+(K+)ν] ,

|λ122λ′′⋆ijk|, |λ232λ

′′⋆ijk| . (10−11 − 10−9) [p→ π+(K+)ν] ,

|λ133λ′′⋆ijk|, |λ233λ

′′⋆ijk| . (10−12 − 10−10) [p→ π+(K+)ν] .

(6.116)

Considering nucleon decay mediated by tree-level diagramsinvolving a chargino exchange,Carlson et al. [371] find stronger bounds than Eqs (6.116) fortheλλ′′ products involving thecouplingsλ′′113, λ

′′123, λ

′′212 andλ′′312:

|λijkλ′′113| . 10−13, |λijkλ

′′123| . 10−12, |λijkλ

′′212| . 10−13, |λijkλ

′′312| . 10−12 . (6.117)

These bounds correspond to(B + L)-conserving decays such asp → l+k νiνj. The boundson λλ′′ products involving the otherλ′′ijk couplings are weaker than Eqs (6.116), since a loopdressing of the corresponding vertex is necessary in order to induce nucleon decay; we thereforedo not give them here.

Other independent bounds resulting from the combined effects of the trilinear interactionsλ′′ and bilinear interactionsµi are obtained by Bhattacharyya and Pal [409], based on a tree-level mechanism involving the intermediate action of the Yukawa interaction of quarks with theup-type Higgs boson. The relevant effective Lagrangian is(B +L)-conserving and contributesto the proton decay channelsp → K+ν andp → K+π+l−. The bound associated with thechannelp→ K+ν reads (i = 1, 2, 3) [409]:

|λ′′112µi

µ| . 10−23 u2

R [p→ K+ν] . (6.118)

At the one-loop level, the same type of Higgs or gaugino dressing as described above can beinvoked to deduce analogous bounds involving also the heavyquark generations. The associatedmechanism requires the intermediate action of the Yukawa interactions of quarks with Higgsbosons, which results in an extra suppression factor(λd)2. The associated upper bounds onλ′′ijnµi′/µ, n = 1, 2, (i, j, n) 6= (1, 1, 2), vary inside the following range [409]:

|λ′′ijnµi′

µ| . (10−16 − 10−12) d2

nR [p→ π+ν, p→ K+ν] . (6.119)

We quote below a representative subset of the derived bounds:

|λ′′321µi′

µ| . 10−16 d2

R , |λ′′331µi′

µ| . 10−15 d2

R , |λ′′332µi′

µ| . 10−16 s2

R . (6.120)

If the production of superpartner particles in single nucleon decays were energetically al-lowed, additional exotic decay modes could arise from the baryon number violating interac-tions alone. A familiar example is furnished, for the case ofa very light neutralino,mχ0 <<mp −mK+, by the exotic proton decay channelp→ χ0K+, which can proceed vias tree-levelexchange. This decay mode sets the bound|λ′′112| . 10−15 [410].

For the case of an ultralight gravitinoG or axinoa [411, 412], as characteristically arises inthe low-energy gauge-mediated supersymmetry breaking approach, additional single nucleondecay channels may appear where the6Rp interactions initiate processes involving the emissionof a light strange meson accompanied by anR-parity odd gravitino or axino. The tree-levels ex-change graph for the relevant subprocesses,ud→ sG andud→ sa, leads to the bounds [413],

|λ′′112| . 6 × 10−17 s2R

(m3/2

1 eV

)[p→ K+G] , (6.121)

|λ′′112| . 8 × 10−17 C−1q s2

R

(Fa

1010 GeV

)[p→ K+a] , (6.122)

applying to the gravitino and axino emission cases, respectively. For the axino case,Fa des-ignates the axionic symmetry breaking mass scale, and the parameterCq, which describes the

model dependence of the axino couplings to quark and lepton fields, is assigned an order onevalue or values in the rangeC−1

q = O(102 − 103), depending on the type of axino considered.

Pursuing along the same lines as above with the study of the two-body single nucleon decaymodes at the one-loop level, one can derive strong bounds on all couplingsλ′′ijk. Accountingapproximately for the loop and flavour suppression factors associated with the one-loop dressingof the previous tree-level diagrams, Choi et al. obtain bounds in the ranges [414]:

|λ′′ijk| .(10−11 m3 − 10−8 m2

) (m3/2

1 eV

), (6.123)

|λ′′ijk| .(10−11 m3 − 10−8 m2

)C−1

q

(Fa

1010 GeV

), (6.124)

wherem denotes a common superpartner mass, for the gravitino and axino emission cases,respectively. A representative subset of these bounds reads:[ |λ′′113|m3

,|λ′′212|m2

,|λ′′323|m2

].

[2 × 10−11, 3 × 10−9, 6 × 10−9

] (m3/2

1 eV

), (6.125)

[ |λ′′113|m3

,|λ′′212|m2

,|λ′′323|m2

].

[3 × 10−11, 4 × 10−9, 8 × 10−9

]C−1

q

(Fa

1010 GeV

), (6.126)

for the gravitino and axino emission cases, respectively.

Nucleon-Antinucleon Oscillations and Double Nucleon Decay

Then→ n transition is governed by the effective Lagrangian,

L = −(n nc)

(m δmδm⋆ m

)(nnc

), (6.127)

where the inputs needed to determine the mass shift parameter δm, involve the couplingsλ′′,the superpartner mass parameters, and the hadronic matrix elements of the relevantD = 9local operatorsdRdRdRuRqLqL anddRdRqLqLqLqL. While the neutron-antineutron oscillationtime, defined approximately byτosc ≃ 1/δm, is strongly hindered by the nuclear interactions,one hopefully anticipates to find observable manifestations of∆B = 2 baryon number violationunder the guise of nuclear two-nucleon disintegration processes,N+(A−1) → N+(A−1) →X + (A − 2), whereX denotes the possible decay channels for the nucleon-antinucleon pairannihilation reactionnn → X, np → X andpp → X, with X = π, 2π, 3π, 2K, · · · [415].The role of the hadronic and nuclear structure effects in the estimation of the dimension-9operators matrix elements is discussed in Refs. [416, 417, 418].

Two competitive tree-level mechanisms for the6Rp contributions were originally discussedin an initiating study by Zwirner [419]. The dominant process is shown in Fig. 6.24. Thebounds inferred by comparison with the experimental limit due to the non-observation ofn-n oscillations, using the tentative estimate|ψN (0)|4 ≈ 10−4 GeV6 for the wave function, arerather strong [156, 256, 419]:

|λ′′11k| . (10−8 − 10−7)108 sτosc

(m

100 GeV

)5/2

. (6.128)

uR

dR dL~dkR ~g ~dkR

dL dR

uR

Figure 6.24: 6Rpcontribution to nucleon-antinucleon oscillation.

An alternative estimate reads:|λ′′11k| . (0.3 − 1.7) × 10−10 g1/2d2kR. However, these bounds

should be taken as indicative only, since an unknown suppression factor from the flavour off-diagonal entries of the left-right mixing squark mass matrix was ignored. The second mecha-nism discussed by Zwirner [419] is described by an intermediate vertex at which three sfermions,which are emitted by quark lines viaλ′′ interactions, jointly annihilate via a soft supersymmetrybreaking interaction of the typeA′′

ijkuci d

cj d

ck [419]. This contribution faces the same problem re-

garding the unknown input for the interaction trilinear in the squark fields. Being of orderλ′′4,it should be subdominant compared to the above one.

Goity and Sher [100] have challenged the view thatn-n oscillations do actually constrain thecoupling constantλ′′121, in view of the uncertain information on the input supersymmetry break-ing mass parameters. They argue that one can identify a competitive mechanism, with a fullycalculable transition amplitude, which sets a bound onλ′′131. This alternative mechanism [100]is based on the sequence of reactionsuRdR+dL → b⋆R+dL → b⋆L+dL → dL+bL → dL+uRdR,where the intermediate transitionb⋆L +dL → dL + bL is due to aW boson and gaugino exchangebox diagram [100]. The choice of intermediate bottom squarks is the most favourable one inorder to maximise factors such asm2

d/MW , which arise from the electroweak interactions ofd-quarks in the box diagram amplitude. The resulting bound must be evaluated numericallyand lies in the wide interval|λ′′131| . (2 × 10−3 − 10−1), for squark masses varying in therangemq = (200 − 600) GeV. The bound onλ′′121 is a factorms/mb ≈ 4 × 10−2 weaker,|λ′′121| . (5 × 10−2 − 2.5), and is of marginal physical interest [100]. Chang and Keung[410]observe that the above mechanism actually includes three other analogous one-loop box dia-grams involving the exchange of gaugino-W boson and quark-squark pairs. A single one ofthese dominates and yields bounds for the associated couplings of the form [410],

|λ′′321| . [2.1 × 10−3, 1.5 × 10−2](

ms

200 MeV

)−2

,

|λ′′331| . [2.6 × 10−3, 2 × 10−2] ,(6.129)

where the two numbers inside brackets are in correspondencewith the two input values usedfor the squark mass,mq = [100, 200] GeV, andms is the strange quark mass.

The generational structure of theλ′′ijk couplings imposes non-diagonal flavour configura-tions for thedc quarks, which disfavours the strangeness conservingn → n transition. Basedon an observation by Dimopoulos and Hall [156], Barbieri andMasiero [323] propose instead toapply the same mechanism to the∆S = 2 processudd→ ucscsc, which contributes to the tran-sitionn → Ξ. One avoids in this way the penalty of two flavour off-diagonal left-right mixingsquark mass insertions, but the counterpart of this advantage is an energetically suppressed, off-shelln→ Ξ transition. Another hadronic physics aspect in the comparison between then→ Ξ

andn → n systems resides in their different short distance hadronicinteractions. Ann − Ξoscillation initiates∆B = ∆S = 2 double nucleon decay processes such asp+p→ K+ +K+

or n + n → K0 + K0, which could become the predominant channel for a double nucleoninduced nuclear decay. Application to the nuclear decay reaction 16O →14 C+K+ +K+ yieldsa relationship for the double nucleon decay lifetime,τNN ≃ (10−2 y ) λ

′′−4112 (q4 g)2, which

results in the bound [323]:

|λ′′121| . 10−8.5 g12 q2

(τNN

1032 yr

)−1/4(10−6 GeV6

. N |ududss|Ξ >

)1/2

. (6.130)

An alternative treatment of the nuclear decay process is proposed by Goity and Sher [100],where one bypasses the intermediate step of then → Ξ transition by dealing directly withthe transitionNN → KK. These authors identify a mechanism where the6Rp interactionscontribute through a Feynman diagram involving thes-channel production of a pair of squarksmediated by thet-channel exchange of a gluino, based on the reaction scheme(qiqj)(qlqm) →q⋆kq

⋆n → qkqn, where (qi, qj , qk) is a permutation of (uR, dR, sR), and similarly for (ql, qm, qn).

The decay amplitude for the nuclear reaction16O →14 C + K+ + K+ is evaluated within animpulse approximation nuclear Fermi gas model for the nuclei, where the nuclear momentumintegral contains the folded product of the elementary process cross-section with the nuclearmomentum distributions of nucleons. The resulting bound reads [100]:

|λ′′121| . 10−15 R−5/2 , R ≡ Λ

(mgm4q)

1/5, (6.131)

where the parameterΛ in the overall scale factor parameterR describes a hadronic scale repre-senting the dimensional analysis estimates of the hadronicand nuclear matrix elements. Vary-ing R inside the range(10−3 − 10−6), one finds a bound spanning a wide interval:|λ′′121| .

(10−7 − 100). In spite of the strong dependence on the hadronic and nuclear structure inputs,the preferred estimates are quoted in Ref. [100] as|λ′′121| . 10−6 and |λ′′131| . 10−3, for thechoice of a common superpartner massm = 300 GeV.

6.5 General Discussion of Indirect Trilinear Bounds

In this section we shall attempt to assess from a more global perspective the current situationregarding the indirect bounds on the trilinear6Rp coupling constants.

6.5.1 Summary of Main Experimental Bounds

We have collected together a sizeable subset of the strongest bounds available in the literature.Table 6.1 displays the results for single coupling constants and Table 6.2 to 6.5 for quadraticcoupling constant products. These lists recapitulate bounds already encountered in the pre-ceding sections. The bounds derived from the perturbative unitarity conditions following thediscussion of the renormalisation group constraints in chapter 3 have also been included, aswell as the bounds inferred from the neutrino mass discussedin chapter 5. Self-evident abbre-viations (described in the caption of Table 6.1) are used to identify the associated observableprocesses from which the bounds are inferred.

Charged Neutral Other ProcessesCurrent Current

λ12k 0.05 ekR [Vud] (6.25) 0.14 ekR [νµe] (6.35)0.13 ek=1L [νµe] (6.35)

0.07 ekR[Rµτ ](6.14) [0.37, 0.25, 0.11]νk = 1, 2, 3 [AFB] (6.39)

0.05ekR [QW (Cs)] (6.47)0.13 ekR [νµq] (6.36)

λ13k 0.07 ekR [Rτ ] (6.17) [0.37, 0.25, 0.11]νk = 1, 2, 3 [AFB] (6.39)

λ23k 0.07 ekR [Rτ ] (6.17) 0.11 τL [νµe] (6.35)0.07 ekR [Rτµ] (6.17) k = 1

λ233 0.90 [RG]

λi22 2.7 × 10−2 µm− 12 [mν < 1 eV]

(me 2LR22 = mmµ) (5.11)

λi33 1.6 × 10−3 τ m− 12 [mν < 1 eV]

(me 2LR33 = mmτ ) (5.11)

λ′11k 0.02 dkR [Vud] (6.25) [0.28, 0.18] uL

k = 2, 3 [AFB] (6.39)0.03 dkR [Rπ](6.19) 0.02dkR [QW (Cs)](6.47)

λ′111 3.3 × 10−4 q2g12 [ββ0ν] (6.106)

λ′12k 0.44 dkR[RD+ ] (6.27) 0.21 dkR [AFB] (6.39)0.27 dkR[RD0 ] (6.27) [0.28, 0.18] cL0.23 dkR[R⋆

D+ ] (6.27) k = 2, 3 [AFB] (6.39)λ′13k [0.28, 0.18] tL

k = 2, 3 [AFB] (6.39)0.47 [Re] (6.40)

(m(dkR) = 100 GeV)λ′1j1 0.03 ujL [QW (Cs)] (6.47)λ′2j1 0.18 djL [νµq] (6.36)λ′21k 0.06 dkR [Rπ] (6.19) 0.15 dkR [νµq] (6.36)

0.08 dkR [Rτπ] (6.21)λ′22k 0.61 dkR[RD+ ] (6.27)

0.38dkR [R⋆D+ ] (6.27)

0.21dkR [RD0 ] (6.27)0.65 dkR[RDs(τµ)] (6.30)

λ′23k 0.45 [Rµ] (6.40)(mdkR

= 100 GeV)

Charged Neutral Other ProcessesCurrent Current

λ′31k 0.12 dkR [Rτπ] (6.21)λ′32k 0.52 dkR [RDs(τµ)] (6.30)λ′33k 0.58 [Rτ ] (6.40)

(mdkR= 100 GeV)

λ′333 0.32 bR [B → τνX] (6.28) 1.06 [RG]

λ′i11 0.2 dm− 12 [mν < 1 eV]

(md 2LR11

= mmd) (5.12)λ′i22 10−2 sm− 1

2 [mν < 1 eV](md 2

LR22= mms) (5.12)

λ′i33 4 × 10−4 bm− 12 [mν < 1 eV]

(md 2LR33 = mmb) (5.12)

λ′′11k (10−8 − 10−7)(108s/τosc)m5/2

[nn] (6.128)λ′′112 10−6 [NN ] (m = 300 GeV) (6.131)

6 × 10−17 s2R (m3/2/1 eV)

[p→ K+G] (6.121)8 × 10−17C−1

q s2R (Fa/1010 GeV)

[p→ K+a] (6.122)λ′′113 10−3 [NN ] (m = 300 GeV) (6.131)λ′′123 1.25 [RG]λ′′212 1.25 [RG]λ′′213 1.25 [RG]λ′′223 1.25 [RG]λ′′312 1.45 [Rl] (6.41) 4.28 [RG]

(m = 100 GeV) 2.1 × 10−3 [nn] (6.129)λ′′313 1.46 [Rl] (6.41) 1.12 [RG]

(m = 100 GeV) 2.6 × 10−3 [nn] (6.129)λ′′323 1.46 [Rl] (6.41) 1.12 [RG]

(m = 100 GeV)λ′′ijk (10−11 m3 − 10−8 m2)

×(m3/2/1 eV) [p→ K+G] (6.123)×(Fa/1010 GeV) [p→ K+a] (6.124)

Table 6.1: Single bounds for the6Rp coupling constants at the 2σ level. We use the notationVij

for the CKM matrix,Rl, Rl l′ , RD, RZl for various branching fractions or ratios of branching

fractions as defined in the text,QW for the weak charge,νq, νl for the neutrino elastic scat-tering on quarks and leptons,mν for the neutrino Majorana mass,RG for the renormalisationgroup,AFB for forward-backward asymmetry,QW (Cs) for atomic physics parity violation,nnfor neutron-antineutron oscillation andNN for two nucleon nuclear decay,[KK], forK0−K0

mixing . The generation indices denotedi, j, k run over the three generations while those de-notedl,m, n run over the first two generations. The dependence on the superpartner massfollows the notational conventionmp = ( m

100 GeV)p. Aside from a few cases associated with

one-loop effects, we use the reference valuem = 100 GeV. The quoted equation labels refer toequations in the text.

Lepton Hadron L and/or BFlavour Flavour violation

|λ⋆ij2λij1| 8.2 × 10−5(ν2

L, l2L) [µ→ eγ] (6.95)

|λ23kλ⋆13k| 2.3 × 10−4(ν2

L, l2R) [µ → eγ] (6.95)

|λ312λ⋆321| 1.9 × 10−3ν2

L

[µ+e− → µ−e+] (6.103)|λ⋆

i12λi11| 6.6 × 10−7ν2L [µ → 3e] (6.97)

|λ321λ⋆311| 6.6 × 10−7ν2

L [µ → 3e] (6.97)|λ⋆

i23λi22| 2.2 × 10−3ν2L [τ → 3µ] (6.97)

|λ132λ⋆122| 2.2 × 10−3ν2

L [τ → 3µ] (6.97)|λi12λj21| 0.15 l2m−1 [mν < 1 eV]

|λi13λj31| 8.7 × 10−3 l2m−1 [mν < 1 eV]|λi22λj22| 7 × 10−4 µ2m−1 [mν < 1 eV]

|λi23λj32| 4.2 × 10−5 l2m−1 [mν < 1 eV]|λi33λj33| 2.5 × 10−6 τ 2m−1 [mν < 1 eV]

(me 2LR

= mMe) (5.11)

|λ⋆i12λ

′i11| 2.1 × 10−8ν2

L

[µ → e (Ti)] (6.99)|λi21λ

′⋆i11| 2.1 × 10−8ν2

L

[µ → e (Ti)] (6.99)|λ⋆

1j1λ′j33| I 6 × 10−7ν2

j [dγe ] (6.60)

|λi31λ′⋆i11| 1.6 × 10−3ν2

iL [τ → eη] (6.104)|λ⋆

i13λ′i11| 1.6 × 10−3ν2

iL [τ → eη] (6.104)|λi32λ

′⋆i11| 1.7 × 10−3ν2

iL [τ → µη] (6.104)|λ⋆

i23λ′i11| 1.7 × 10−3ν2

iL [τ → µη] (6.104)

Table 6.2: Quadratic coupling constant product bounds. We use the sameconventions as inthe preceding table for the single coupling constant bounds. The presence of a symbolI meansthat the bound applies to the imaginary part of the coupling constant products.

Lepton Hadron L and/or BFlavour Flavour violation

|λ⋆122λ

′112| 2.2 × 10−7ν2

L [KL → µ+µ−] (6.76)|λ⋆

122λ′121| 2.2 × 10−7ν2

L [KL → µ+µ−] (6.76)|λ⋆

121λ′212| 1.0 × 10−8ν2

L [KL → e+e−] (6.76)|λ⋆

121λ′221| 1.0 × 10−8ν2

L [KL → e+e−] (6.76)|λ⋆

i12λ′i12| 6 × 10−9ν2

L [KL → e±µ∓] (6.78)|λ⋆

i12λ′i21| 6 × 10−9ν2

L [KL → e±µ∓] (6.78)|λ⋆

i21λ′i12| 6 × 10−9ν2

L [KL → e±µ∓] (6.78)|λ⋆

i21λ′i21| 6 × 10−9ν2

L [KL → e±µ∓] (6.78)|λi31λ

′⋆i13| 6 × 10−4l2iL [B− → e−ν] (6.85)

|λi32λ′⋆i13| 7 × 10−4l2iL [B− → µ−ν] (6.85)

|λ233λ′⋆313| 2 × 10−3l23L [B− → τ−ν] (6.85)

|λ⋆i11λ

′i13| 1.7 × 10−5ν2

L [B0d → e+e−] (6.79)

|λ⋆i11λ

′i31| 1.7 × 10−5ν2

L [B0d → e+e−] (6.79)

|λ⋆i22λ

′i13| 1.5 × 10−5ν2

L [B0d → µ+µ−] (6.79)

|λ⋆i22λ

′i31| 1.5 × 10−5ν2

L [B0d → µ+µ−] (6.79)

|λ⋆i12λ

′i13| 2.3 × 10−5ν2

L [B0d → e±µ∓] (6.79)

|λ⋆i12λ

′i31| 2.3 × 10−5ν2

L [B0d → e±µ∓] (6.79)

|λ⋆i21λ

′i13| 2.3 × 10−5ν2

L [B0d → e±µ∓] (6.79)

|λ⋆i21λ

′i31| 2.3 × 10−5ν2

L [B0d → e±µ∓] (6.79)

|λ231λ′′⋆112| 10−16 [p→ K+e±µ∓ν]

|λ132λ′′⋆112| 10−16 [p→ K+e±µ∓ν]

|λ123λ′′⋆112| 10−14 [p→ K+ννν]

|λi11λ′′⋆112| 10−17 [p→ K+ν]

|λi22λ′′⋆112| 10−20 [p→ K+ν]

|λi33λ′′⋆112| 10−21 [p→ K+ν]

(m = 1 TeV) (6.115)|λijjλ

′′⋆i′j′k′| (10−12 − 10−6) [p→ π+(K+)ν]

(mh+ = m = 1 TeV) (6.116)|λijkλ

′′113| 10−13 [p→ l+νν] (6.117)

|λijkλ′′123| 10−12 [p→ l+νν] (6.117)

|λijkλ′′212| 10−13 [p→ l+νν] (6.117)

|λijkλ′′312| 10−12 [p→ l+νν] (6.117)

Table 6.3: Quadratic coupling constant product bounds. We use the sameconventions as in thepreceding table.

Lepton Hadron L and/or BFlavour Flavour violation

|λ′⋆i21λ

′i12| 4.5 × 10−9 ν2

iL [KK]|λ′i31λ

′⋆i22| 1.× 10−4 [KK] (m = 100 GeV)

|λ′⋆i31λ

′i32| 7.7 × 10−4 [KK] (m = 100 GeV)

|λ′⋆i2kλ′i′1k| 2.11 × 10−5 d2

kR [K+ → π+νν] (6.87)|λ′⋆ij1λ′i′j2| 2.11 × 10−5 d2

jL [K+ → π+νν] (6.87)|λ′⋆

i31λ′i13| 3.3 × 10−8ν2

iL [BB]|λ′i31λ

′⋆i33| 1.3 × 10−3 [BB]

|λ′⋆i3kλ′i′2k| 1.5 × 10−3d2

kR [B → Xsνν] (6.88)|λ′⋆ij2λ′i′j3| 1.5 × 10−3d2

jL [B → Xsνν] (6.88)|λ′2mkλ

′⋆1mk| 7.6 × 10−5d2

kR [µ→ eγ] (6.95)|λ′23kλ

′⋆13k| 2.0 × 10−3 [µ→ eγ] (6.96)

(mdkR= mtL = 300 GeV)

|λ′⋆1j1λ′1j2| I 8.1 × 10−5u2

L [KL → e+e−] (6.77)|λ′⋆2j1λ

′2j2| I 7.8 × 10−6u2

L [KL → µ+µ−] (6.77)|λ′⋆1j1λ

′2j2| 3 × 10−7u2

L [KL → e±µ∓] (6.78)|λ′⋆1j2λ

′2j1| 3 × 10−7u2

L [KL → e±µ∓] (6.78)|λ′⋆2j1λ

′2j3| 2.1 × 10−3u2

L [B0d → µ+µ−] (6.80)

|λ′⋆1j1λ′2j3| 4.7 × 10−3u2

L [B0d → e±µ∓] (6.80)

|λ′⋆1j3λ′2j1| 4.7 × 10−3u2

L [B0d → e±µ∓] (6.80)

|λ′2j1λ′⋆1j1| 4.3 × 10−8u2

jL [µ → e (Ti)] (6.99)|λ′3j1λ

′⋆1j1| 2.4 × 10−3u2

jL [τ → eρ] (6.105)|λ′⋆

11kλ′12k| 5.3 × 10−3d2

kR [Λ → pl−νl] (6.31)|λ′21kλ

′⋆11k| 4.5 × 10−8d2

kR [µ→ e (Ti)] (6.99)|λ′31kλ

′⋆11k| 2.4 × 10−3d2

kR [τ → eρ] (6.105)

Table 6.4: Quadratic coupling constant product bounds. We use the sameconventions as in thepreceding table.

Lepton Hadron L and/or BFlavour Flavour violation

|λ′113λ′131| 3.8 × 10−8 [ββ0ν] (6.107)|λ′112λ′121| 1.1 × 10−6 [ββ0ν] (6.107)|λ′i3kλ

′⋆i2k| 0.09 (ν2

iL, d2iR) [B → Kγ] (6.93)

|λ′⋆ij3λ′ij2| 0.035 (e2iL, d2jL) [B → Kγ] (6.93)

|λ′i11λ′j11| 5 × 10−2 d2m−1 [mν < 1 eV]|λ′i12λ′j21| 3 × 10−3 q2m−1 [mν < 1 eV]|λ′i13λ′j31| 8 × 10−5 q2m−1 [mν < 1 eV]|λ′i22λ′j22| 2 × 10−4 s2m−1 [mν < 1 eV]|λ′i23λ′j32| 5 × 10−6 q2m−1 [mν < 1 eV]

|λ′i33λ′j33| 10−7 b2m−1 [mν < 1 eV](md 2

LR= mMd) (5.12)

|λ′l1kλ′′⋆11k| (2 − 3) × 10−27d2

kR [p→ π0l+] (6.110)|λ′31kλ

′′⋆11k| 7 × 10−27d2

kR [n→ π0ν] (6.110)|λ′i2kλ

′′⋆11k| 3 × 10−27d2

kR [p→ K+ν] (6.110)|λ′l1kλ

′′⋆12k| (6 − 7) × 10−27d2

kR [p→ K0l+] (6.111)|λ′ij1λ′′11j | 7 × 10−26 d2

jL(m2djR/(md 2

LR)jj)

|λ′ij2λ′′11j | 3 × 10−27 d2jL(m2

djR/(md 2

LR)jj)

|λ′i31λ′′123| 3 × 10−27 b2L(m2bR/(md 2

LR)33)

[n→ π0ν, p→ K+ν] (6.112)|λ′lj1λ′′j12| 10−26 u2

jL(m2ujR/(mu 2

LR)jj)

[n→ K+l] (6.113)|λ′ijkλ′′⋆i′j′k′| 10−9 [p→ Xν(Xν)]

(m = 1 TeV) (6.114)

|λ′′⋆232λ

′′231| Min[6.× 10−4c, 2.× 10−4c2]

[KK]|λ′′⋆

332λ′′331| Min[6.× 10−4t, 3.× 10−4t2]

[KK]|λ′′i13λ′′⋆i12| 6.4 × 10−3u2

iR [B+ → K0π+] (6.89)|λ′′⋆i23λ

′′i12| 6 × 10−5u2

iR [B− → φπ−] (6.91)|λ′′213λ′′⋆232| I 10−2q2 [dγ

n] (6.58)|λ′′312λ′′⋆332| I 10−1q2 [dγ

n] (6.59)|λ′′⋆i3kλ

′′i2k| 0.16q2

R [B → Kγ] (6.93)

Table 6.5: Quadratic coupling constant product bounds. We use the sameconventions as in thepreceding table.

It appears clearly from the tables that the low energy phenomenology is a rich and valuablesource of information on the6Rp interactions. The most robust cases include the single nucleondecay channels, neutrinoless double beta decay, double nucleon decay, the neutralK, B mesonmixings and rare leptonic or semileptonic decays, the lepton number and/or flavour violatingdecays of leptons. The strongest single coupling constant bounds arise, in order of decreasingstrength, from the baryon number violating processes ofn − n oscillation andNN decay,from neutrinoless double beta decay, neutrino masses, semileptonic decays ofK mesons, andfrom neutral current (APV) and charged current lepton universality. For the quadratic couplingconstant bounds, a similar classification ordered with respect to decreasing strength, placesbaryon number violating processes in first position, followed byK andB meson mixing,µ → econversion, leptonic or semileptonic decays ofK andB mesons, neutrinoless double beta decay,three-body lepton decays, and neutrino masses.

The consideration of the loop level contributions is a very effective way to deduce comple-mentary bounds on coupling constants involving the heaviergenerations of quarks and leptons.For instance, the single nucleon stability bounds|λ′ijkλ′′⋆i′j′k′| < O(10−9), which are valid for allthe generation indices, have far-reaching implications. Should a single lepton number violatingcoupling constantλ′ijk be sizeable, then one would conclude a strong suppression ofthe full setof baryon number violatingλ′′i′j′k′ coupling constants. An analogous converse statement wouldhold for all theλ′ijk if a singleλ′′i′j′k′ were sizeable.

6.5.2 Observations on the Bound Robustness and Validity

Before discussing the impact on supersymmetric model building of the indirect bounds, we startwith some general preliminary observations aimed at appreciating their potential usefulness.

6.5.2 a) Natural Order of Magnitude for 6Rp Couplings

First, it is important to ask what might be considered as natural values for the dimensionlessR-parity violating trilinear interactions coupling constants. (For convenience, we shall denotethese collectively asλijk.) In the absence of any symmetry, the anticipated natural values areO(1) or O(g). If one assumed instead a hierarchical structure with respect to the quarks andleptons generations, analogous to that exhibited by the regular R-parity conserving Yukawainteractions, an educated guess could be, for instance,λijk = O((mimjmk/v

3)1/3), wheremj denote theq, l masses. A variety of alternative forms for the generation dependence aresuggested by considerations based on the physics of discrete symmetries or grand unified andstring theories. Just on the basis of the existing experimental constraints, one can check thatthe individual coupling constant bounds, as given in Table 6.1 fall in an interval of valuesO(10−1) − O(10−2) which interpolates between the above two extreme estimates.

6.5.2 b) Impact of the SUSY Masses on the Bounds

A second observation concerns the dependence of the indirect bounds with respect to thesuperpartner mass parameters. Our reference value for the supersymmetry breaking mass pa-rameter is set uniformly atm = 100 GeV, apart from a very few exceptions. The tree levelmechanisms have, of course, a transparent dependence on thesuperpartners masses, such thatthe single or quadratic coupling constant bounds scale linearly or quadratically withm. Severaltree level dominated observables involve a single superpartner species, which then allows usto identify the relevant sfermion by indicating explicitlyits particle name. The bounds associ-ated with one-loop effects have, in general, a weaker mass dependence over the mass interval,

m = 100 − 500 GeV. With increasing values of the superpartner masses, both the tree andloop level bounds gradually get weaker. In fact, if the supersymmetry breaking mass scale hap-pened to reach theO(1) TeV extrem limit for the Standard Model naturalness, a largenumberof the existing individual indirect bounds would become useless. For the so-called ”More Min-imal Supersymmetric Standard Model” [420] where the third generation squarks or sleptonsconstitute the lightest scalar superpartners and the first and second generation sfermion massesare raised up to the TeV scale, one would be led to strongly weakened bounds for the first twogenerations of sfermions. Several quadratic bounds would still remain of interest, especiallythose associated with a simultaneousB andL number violation, due to the extreme severity ofthe nucleon stability bounds.

There are two other important exceptions to the suppressioneffect of the bounds from largesfermions masses. The first concerns the process independent bounds derived from the renor-malisation group considerations, as was discussed in chapter 3. Because the perturbative unitar-ity or quasi-fixed points bounds originate from indirect effects associated with the resummationof large logarithms, they are practically insensitive to the value of the supersymmetry breakingscale as long as this does not extend beyond the TeV decade. The second exception concernsthe class of observables governed by dimension nine operators, such as the amplitudes for theββ0ν or n − n processes, where several contributions from different intermediate states, asso-ciated with sfermions and gauginos, compete with one another. The destructive interference ofthese contributions renders the inferred bounds sensitiveto the supersymmetry breaking massspectrum as a whole.

6.5.2 c) Validity of the Assumption of one or two Dominant Couplings

A third observation concerns the validity of the single or double coupling constant domi-nance hypotheses. When applied to the6Rp interactions, the dominance hypotheses rest on thepremise that some hierarchy exists either between theB andL number violations or between thedifferent quark and lepton generations. The conclusions from certain studies might be alteredif the dominance hypotheses were not justified or if certain unexpected finely tuned cancella-tions were at work. One could imagine, for example, that a subset of the coupling constantsexhibited generational degeneracies that would induce cancellations between the contributionsof different component interactions part of the predominant subset. An indirect evidence for apossible correlation between different coupling constants is furnished by the observation thatthe strongest constraints arise for quadratic products rather than the individual coupling constantbounds. This is clearly not surprising since the latter entail less demanding model-dependentassumptions.

An examination of Table 6.1 reveals that a few amongst the charge current and neutralcurrent single coupling constant bounds are immune to invalidating cancellation effects. Ex-amples of robust bounds comprise for the charged current interactions, those deduced from therenormalised observable parametersGµ andMW , and for the neutral current interactions, thosededuced from the forward-backward asymmetry parameterAFB, and the auxiliary parameter,C2(d). Partly responsible for this state of affairs is the use of ratios of rates or branching frac-tions. While the comparison with experimental data for suchratios removes the dependence onsome poorly known hadronic matrix elements parameters, this has the drawback of introducingcancelling contributions. As a result, these ratios obtaincorrections from different6Rp interac-tions which often combine together destructively, with opposite signs. The quadratic couplingconstant bounds are exposed to a much lesser extent to cancellations since they are often de-rived for observables where the contributions from different sets of coupling constants add upincoherently.

6.5.2 d) Bound Robustness in Regards to Model Dependence

As a final remark we should emphasise that not all coupling constant bounds are to betreated indiscriminately. One must exercise a critical eyeon the model-dependent assumptions.It is important to keep track of the superpartner generationindex in light of the possibility ofa large splitting between the sfermion generations. The generational structure of the sfermionchirality-flip mass matricesm2

LRis a crucial input for the one-loop contributions to the neutrinos

Majorana masses or then− n oscillation amplitude. Deviations from a generation universalityyield large off-diagonal contributions(m2

LR)ij which could modify the ensuing predictions.

6.5.3 Phenomenological Implications of Bounds

What implications on theoretical models beyond the Standard Model can be drawn from theexisting bounds? As discussed in chapitre 3, works have beendone on model building usingrenormalization group equations (RGE) to get bounds at theMGUT scale [95]. One might askwhether the results hint at any of the known alternatives, especially the gauged horizontal con-tinuous or discrete symmetries. The existence of some correlations with the flavour symmetriesis clearly suggested. Indeed, it is generally the case that the coupling constants in the first andsecond generations are more constrained, although this might just reflect the lack of direct ex-perimental data for the heavy flavoured hadrons or leptons. The fact that the strongest boundsare for productsλ′λ′′ andλλ′′ hints at an incompatibility between simultaneousB andL num-ber violations. SeparateB orL number violation, as described by interactions governed bythecoupling constant productsλλ, λλ′, λ′λ′ or λ′′λ′′, may also be disfavoured but in a way whichdepends on flavour, the first and second generations being those most strongly constrained.Ready solutions to prevent a coexistence ofB andL number non-conservation are offered bytheB andL parities or the corresponding generalised discrete symmetries versions.

The existence of a strong hierarchical structure in the6Rp coupling constants does not ex-clude the presence of certain unexpected degeneracies withrespect to the quarks and leptonsgenerations, as would be implied by the presence of unbrokendiscrete symmetries. To estab-lish this possibility one would need global studies of the6Rp interactions effects encompassinga large body of experimental data. One way to infer robust bounds in cases where one suspectscancellation effects to be at work, is by choosing suitable observables which depend selectivelyon fixed products of the coupling constants. Such an example was encountered with the6Rp in-duced contributions to the nonV −A charged current interactions (see section 6.3.1). Anotherattractive idea would be to fit a selected subset of the6Rp interactions coupling constants to acorrespondingly selected subset of experimental constraints. While global studies along theselines are routinely performed in the context of the contact interactions physics [286] or the mir-ror fermions physics [421] their application to the6Rp physics appears problematical in view ofthe proliferation of the coupling constants. Interesting partially global studies of the6Rp interac-tions have been recently reported in the literature regarding fits to the APV observables [422]or theZ boson partial decay width observables [289]. The recent accumulated experimentalinformation on the neutrino oscillations has also allowed to implement in part such a programby envisaging global fits to the data for the neutrino Majorana masses based on the6Rp con-tributions. Even if these studies must appeal to some assistance from theory, through certainspecialised assumptions on the generational structure of the sfermion mass parameters, theyhave yielded a wealth of useful, although model-dependent,information on the6Rp interactions.Of course, one should keep in mind the alternative options toexplain the neutrino physics ex-perimental data, which include in fact the6Rp mechanism of flavour changing neutral currentneutrino interactions with quarkonic and leptonic matter.

The model-independent studies devoted to the four fermionscontact interactions may beof some use in establishing the existence of possible cancellation effects amongst different6Rp interactions. Special attention, motivated by searches ofcompositeness [423, 424, 425],has been devoted in recent years to the flavour diagonal contact interactions. Along with thelow energy neutral and charged current interactions (neutrino or (polarised) electron elasticand inelastic scattering data) the current fits [286, 426] include the high energy data for theDrell-Yan dilepton production and largepT jet production at the Fermilab Tevatron collider, thedijet production (e.g.,e−e+ → ss) at the CERN LEP collider, and the deep inelastic electronand positron scattering at the HERA collider at DESY. Based on the initial studies [423], onegenerally restricts the consideration to the dominantD = 6 Lorentz vector interactions, usingan effective Lagrangian of the general form,

LNC =∑

[(i,j)=(L,R);q=(u,d)]

4πηqij

Λq2ij

eiγµeiqjγµqj , (6.132)

where a sum over the fermions flavour and chirality is understood andηq = ±1 stand for asign phase. One may express the relationship between the different scale parameters, denotedgenerically byΛ, and the6Rp interactions coupling constants by the order of magnitude relation,λ2/m2 ≈ 4π/Λ2. More quantitatively, an identification with the neutral current interactions,for instance, yields [427],

C1(q) =

√2π

GF(ηq

RL

Λq2RL

− ηqLL

Λq2LL

− ηqLR

Λq2LR

+ηq

RR

Λq2RR

).

An important observation here is that the high energy collider experimental data favour lowvalues of the energy scales. Some currently quoted experimental bounds are,Λ[−,+]d

[LR,RL] > [1.4,

1.6] TeV , from the dijet production data [428],Λ[+,+]u[LR,RL] > [2.5, 2.5] TeV from the Drell-

Yan production data [429] andΛ ≈ 1 TeV from the anomalous deep inelastic scattering eventsdata [277, 430, 431]. In contrast to these results, it appears that the low energy experimentaldata consistently favours larger values of the energy scales. This is most explicit in the Cesiumatom APV data, where assuming that no cancellations occur between the different terms inC1(q), leads to the strong bound,Λ > 10 TeV. More quantitatively, the simultaneous fits ofthe flavour diagonaleeqq contact interactions to both low and high energy experimental data,as completed by incorporation of the HERA highQ2 data [286], infer large values of the scaleparameters with a non-trivial trend of relative signs between the different interactions. Quotingfrom [286] one finds the fitted values:

Λ−euLL = 12.4

(+50.6

−34.8

)TeV, Λ+eu

LR = 3.82

(+0.93

−1.62

)TeV, Λ+eu

RL = 5.75

(+5.06

−6.88

)TeV.

These quantitative analyses indicate that cancellation effects are taking place at low energiesbetween the contributions from different interactions. Such cancellations would clearly passunnoticed within analyses based on a single coupling constant dominance hypothesis. Ten-tative explanations have been sought in terms of a short distance parity conserving interac-tion [427], implying the relations ηq

iL

(ΛqiL)2

= − ηqiR

(ΛqiR)2

, or an extended global flavour symmetrygroup [432]. Applied to the6Rp interactions, the implications would be in the existence ofde-generacies amongst the subset of relevant6Rp coupling constants.

The Lorentz vector component of the charged current (CC) electron-quark four fermionscontact interactions appears also to lead to similar conclusions. The fits to the leptonic or

hadronic colliders data based on the conventional parametrisation of the effective Lagrangian,

LCC =4πη

Λη2CC

eLγµνLuLγµd′L

yield the typical bounds,Λ−CC > 1.5 TeV. The recent deep inelastic scattering events observed

at HERA also favour low scales,ΛCC = (0.8 − 1) TeV [277, 431]. By contrast, the fits tothe low energy experimental data associated with the leptons universality or for meson decaysfavour larger scales [431],ΛCC ≈ (10. − 30.) TeV. A recent study of the nonV − A chargedcurrent interactions, based on the high precision measurements of the muon decay rate differ-ential distributions, also predicts strong bounds for the scales [433],Λ±ll

CC > [7.5, 10.2] TeV,for the four lepton interactions andΛ±lq

CC > [5.8, 10.1] TeV for the two lepton two quark inter-actions.

Chapter 7

PHENOMENOLOGY AND SEARCHESAT COLLIDERS

7.1 Introduction

The search for6Rp supersymmetry processes has been a major analysis activityat high energycolliders over the past decade, and is likely to remain so at existing and future colliders unlessthe idea of supersymmetry itself somehow becomes falsified.

We have seen in chapters 1 and 2 that on the theory side,6Rp is (and will) remain a centralissue since gauge invariance and renormalizability do not ensure lepton- and baryon-numberconservation in supersymmetric extensions of the StandardModel. A consequence is that ageneral superpotential allows for trilinear terms corresponding to6Rp fermion-fermion-sfermioninteractions involvingλ, λ′ or λ′′ Yukawa couplings. It moreover possibly allows for additionalexplicit (bilinear) orspontaneoussources of lepton-number violation.

The presence of6Rp interactions could have important consequences on the phenomenologyrelevant for supersymmetry searches at high energy colliders. This is because6Rp entails afundamental instability of supersymmetric matter, thus opening up new decay channels forsparticles. Especially crucial in this respect will be the fate of the lightest supersymmetricparticle (LSP). Even for relatively weak6Rp interaction strengths, the decay of the LSP will leadto event topologies departing considerably from the characteristic ”missing momentum” signalof Rp conserving theories. But6Rp could be more than a mere observational complication. Itcould also enlarge the discovery reach for supersymmetry itself as it allows for the creation orexchange of single sparticles.

In this chapter, essential ingredients of the phenomenology and search strategies for6Rp

physics at colliders are presented. Extensive references to related detailed studies for specific su-persymmetry models are provided. Existing experimental constraints established at LEPe+e−,HERA ep and Tevatronpp colliders are reviewed and discovery prospects in future collider ex-periments are discussed. The analyses and prospective studies in the literature have generallybeen carried in the context of a given existing of future collider project. The Table 7.1 gives alist of the machine parameters considered in the studies reviewed in this chapter.

Collider Beams√s

∫Ldt Years

LEP 1 e+e− MZ ∼ 160 pb−1 ⊗ 4 1989-95LEP 2 e+e− > 2 ×MW ∼ 620 pb−1 ⊗ 4 1996-00

HERA Ia e−p 300 GeV O(1 pb−1) ⊗ 2 1992-93HERA Ib e±p <∼ 320 GeV O(100 pb−1) ⊗ 2 1994-00

Tevatron Run Ia pp 1.8 TeV O(10 pb−1) 1987-89Tevatron Run Ib pp 1.8 TeV O(100 pb−1) ⊗ 2 1992-96

HERA II e±L,Rp ∼ 320 GeV ∼ 1 fb −1 ⊗ 2 ≥ 2002Tevatron Run II pp ∼ 2.0 TeV 1 − 10 fb −1 ⊗ 2 ≥ 2002

LHC pp 14.0 TeV 10 − 100 fb −1 ⊗ 2 >∼ 2007

Future LC e+e− ∼ 0.5 − 1.0 TeV 50 fb −1 . . . NLC [434]∼ 0.5 − 1.0 TeV 500 fb −1 . . . TESLA [435]

FutureµC µ+µ− ∼ 0.35 − 0.5 TeV 10 fb −1 . . . FMC [436]∼ 1.0 − 3.0 TeV 1000 fb −1 . . . NMC [436]

Table 7.1: Main contemporary and future collider facilities which areconsidered in the searchanalyses and prospective studies described in this chapter. The facilities are listed together withthe nature of the colliding beams, the available centre-of-mass energies

√s, and the integrated

luminosities∫Ldt accumulated (or the range of

∫Ldt expected) per experiment. The mul-

tiplicative factors after the⊗ sign denotes the number of multi-purpose collider experimentsoperating (or expected to be operating) simultaneously around each collider.

7.2 Interaction Strength and Search Strategies

The way supersymmetry could become manifest at colliders crucially depends both on the struc-ture and parameters of the model followed by Nature and on thea priori unknown magnitudes(individual and relative) of the new6Rp couplings. The weakest6Rp coupling values are likely tobe felt mostly through the decay of sparticles otherwise pair produced via gauge couplings. Thestrongest6Rp coupling values could contribute to direct or indirect single sparticle production.The best search strategy at a given collider will ultimatelydepend on the specific signal andbackground environment.

In the absence of definite theoretical predictions for the values of the 45 independent trilinearYukawa couplingsΛ (λijk, λ′ijk andλ′′ijk), and facing the formidable task of testing245 − 1possible non-vanishing coupling combinations, it is necessary in practice to assume a stronghierarchy among the couplings. For the ”hierarchy” betweendifferent types of couplings thisis an arbitrary choice sinceλ, λ′ andλ′′ appear fundamentally independent. Empirically, it canbe partially justified by the fact that indirect bounds are particularly stringent on non-vanishingcoupling products involving a6L and a6B coupling as was seen in section 6.4.4. For example,the lower limit on the proton lifetime translates [406] intovery stringent bounds on theλ′ ×λ′′ < O(10−9) applicable to all possible flavour combinations. Restrictions on combinationsof couplings of a given type can be legitimized by analogy with the strong hierarchy of theHiggs Yukawa couplings structure in the Standard Model [255, 261]. It may also be empiricallyjustified by the fact that indirect bounds (chapter 6) are generally more stringent on the productof two different couplings than on the square of individualλ, λ′ or λ′′ couplings.

Thus, a reasonable simplifying assumption for the search strategy at colliders is to postulatethe existence of a single (dominant)6Rp coupling. Most of the prospective studies on6Rp and

actual search analyses at colliders rely on this assumption, i.e. that only one6Rp coupling existswhich can connect sleptons or squarks to ordinary fermions.By doing so, it is in addition as-sumed implicitly or explicitly (through some mixing angles[255] connecting the squark currentand mass basis) that flavour mixing relating various couplings (see section 2.1.4) is suppressed.

Having chosen (somehow) a single dominant couplingΛ, the next question is that of therange of coupling values relevant for collider physics. As to what concerns lower bounds,cosmology considerations do not provide much help. As discussed in section 4.1.1, a lightestsupersymmetric particle (LSP) can no more be considered as acold dark matter candidate inpresence of a single non-vanishing6Rp Yukawa coupling with values even as small asO(10−20).Strengthened lower bounds of[λ, λ′, λ′′] > O(10−12) are obtained from the argument (sec-tion 4.1.1) that an unstable LSP ought to decay fast enough inorder not to disrupt nucleosyn-thesis. But even these still lie many orders of magnitude below the sensitivity reach of colliderexperiments. ForΛ coupling values belowO(10−8 − 10−6) (depending in detail on model pa-rameters), the lifetime of the LSP is so large that it is likely to completely escape detectionin a typicalO(10)m diameter collider experiment. An immediate consequence is that for awide range of coupling values, the phenomenology at colliders would appear indistinguishablefrom that ofRp conserving theories. Only a discovery that the LSP turns outto be coloured orcharged, a fact forbidden by cosmological constraints for astable LSP, could be an indirect hintof the existence of6Rp interactions beyond the collider realm. Otherwise, there exist no knowndirect observational tests for suchvery long-livedLSPs [259]. This inaccessible coupling rangewill not be discussed further in this chapter.

In case a non-vanishing6Rp coupling does exist with a magnitude leading to distinct phe-nomenology at colliders, the optimal search strategy will then depend on the absolute couplingvalue and the relative strength of the6Rp and gauge interactions, as well of course on the natureof the supersymmetric model considered (sparticle spectrum and parameter space depending onthe supersymmetry breaking mechanism, etc.). Sparticle direct and indirect6Rp decay topolo-gies will be discussed on general grounds in section 7.3. Anticipating this discussion, a directsensitivity to along-livedLSP might be provided by the observation of displaced vertices in anintermediate range of coupling values up toO(10−5 − 10−4).

For even largerΛ values, the presence of6Rp supersymmetry could become trivially manifestthrough the decay ofshort-livedsparticles pair produced via gauge couplings. A possible searchstrategy in such cases consists of neglecting6Rp contributions at production (in non-resonantprocesses). This is valid provided that the6Rp interaction strength remains sufficiently smallcompared to electromagnetic or weak interaction strengths, i.e. for Λ values typically belowO(10−2−10−1). Such a strategy has been thoroughly explored at existing colliders to study howthe experimental constraints on basic model parameters in specific supersymmetry models beaffected by the presence of6Rp interactions. This and the question of whether and how differenttypes of couplings could be distinguished at colliders in such a scenario will be discussed insection 7.4.

In a similar range of (or for larger) coupling values,6Rp could manifest itself most strikinglyat colliders via single resonant or non-resonant production of supersymmetric particles. Singlesparticle production involving6Rp couplings and how it allows the extension of the discoverymass reach for supersymmetric matter at a given collider is discussed in section 7.5.

Coupling values corresponding to interactions stronger than the electromagnetic interactionmight still be allowed for sufficiently large masses. For masses beyond the kinematic reach of agiven collider,6Rp could contribute to observable processes through virtual sparticle exchange.This is discussed in section 7.6.

Realistic search strategies at colliders must take into account the upper bounds on theΛcouplings derived from indirect processes. As was seen in chapter 6, these bounds all be-come weaker with increasing scalar masses but each possiblywith a specific functional mass-dependence and each depending on a specific type of scalar [259].

Ultimately, the question of whether or not a given6Rp process is truly allowed by existingconstraints must be answered at each point of the parameter space of a given supersymmetrymodel. A review of the huge number of publications dealing with specific aspects of6Rp inspecific supersymmetry models would clearly be beyond the scope of this chapter. In the fol-lowing, essential aspects of the phenomenology will be discussed and references to detailedstudies provided.

7.3 Decay of Sparticles Involving6Rp Couplings

In the scenario with6Rp due to the trilinear terms, the supersymmetric particles are allowedto decay into standard particles through one6Rp coupling. For sparticles other than the LSP,these6Rp-decays will in general compete with ”cascade decays” initiated by standard gaugecouplings. The review of possibly allowed direct and cascade 6Rp-decays for sfermions andgauginos-higgsinos is presented in this section. This willlater on allow us to easily characterizethe essential event topologies expected in6Rp-SUSY searches. Direct decays are discussed insubsections 7.3.1 and 7.3.2. Indirect cascade decays are discussed in subsection 7.3.3. Forcompleteness, decays involving bilinear interactions arediscussed in 7.3.4.

7.3.1 Direct 6Rp Decays of Sfermions

TheLLE, LQD, or UDD couplings allow for a6Rp direct decay into two standard fermions of,respectively, sleptons, sleptons and squarks, or squarks.The allowed decays become evidentwhen considering the Lagrangian for the trilinear Yukawa interactions written in expended no-tations in Eq. (2.7) to Eq. (2.9) and discussed in more details in Appendix B. For conveniencethe corresponding list of decay channels is given in Table 7.2. The 6Rp decay of a sfermionof a particular family will be possible only for specific indicesi, j, k of the relevant Yukawacoupling.

The partial widths ofνi’s when decaying viaλijkLLE or λ′ijkLQD are given (neglectinglepton and quark masses) by [437]:

Γ(νi → ℓ+j ℓ−k ) =

1

16πλijk

2mν i, (7.1)

Γ(νi → djdk) =3

16πλ′ijk

2mν i, (7.2)

where the factor 3 between relations 7.1 and 7.2 is a colour factor. For sneutrinos undergoingdirect decays, the mean decay lengthL in centimetres can then be numerically estimated from:

L(cm) = 10−12(βγ)

(1GeV

mν i

)1

λijk2, 3λ′ijk

2 . (7.3)

Supersymmetric Couplingsparticles λijk λ′ijk λ′′ijkνi,L ℓ+j,Lℓ

−k,R dj,Ldk,R

ℓ−

i,L νj,Lℓ−k,R uj,Ldk,R

νj,L ℓ+i,Lℓ−k,R

ℓ−

j,L νi,Lℓ−k,R

ℓ−

k,R νi,Lℓ−j,L , ℓ−i,Lνj,L

ui,R dj,Rdk,R

uj,L ℓ+i,Ldk,R

dj,L νi,Ldk,R

dj,R ui,Rdk,R

dk,R νi,Ldj,L, ℓ−i,Luj,L ui,Rdj,R

Table 7.2: Direct decays of sleptons and squarks via trilinear6Rp operatorsλijkLiLjEk,λ′ijkLiQjDk and λ′′ijkUiDjDk.

Similar formulae hold for charged sleptons in absence of mixing.

In the case of squarks or sleptons of the third generation, a possible mixing between su-persymmetric partners of left- and right-handed fermions has to be taken into account. Forinstance, the width of the lightest stopt1 writes [438]:

Γ(t1 → ℓ+i dk) =1

16πλ′ijk

2cos2(θt)mt1, (7.4)

whereθt is the mixing angle of top squarks. The case of the stop is somewhat special. Thetypical decay time of a100 GeV stop via a6Rp decay mode is roughly3×10−23 s for a couplingvalue of 10−1, and3 × 10−21 s for a coupling value of 10−2. So the stop6Rp decay time is ofthe same order or even greater than its hadronization time which from the strong interaction isO(10−23) s. Thus, the stop may hadronize before it decays.

7.3.2 Direct 6Rp Decays of Gauginos-Higgsinos

In a direct6Rp decay, the neutralino (chargino) decays into a fermion and avirtual sfermion withthis virtual sfermion subsequently decaying to standard fermions via a6Rp coupling. Thus, direct6Rp decays of gauginos-higgsinos are characterized by three fermions in the final state with thefermion type depending on the dominant coupling. The possible decays are listed in Table. 7.3.The corresponding diagrams are shown for theLiLjE

ck interactions in Figs. 7.1 and 7.2.

A collection of general expressions for three-body decays and matrix elements entering inthe calculation of partial widths can be found in appendix C.In the case of a pure photinoneutralino decaying withλijk , the expression for the partial width simplifies [34] to

Γ = λijk2 α

128π2

m5

χ01

m4f

(7.5)

with mf the mass of the virtual slepton in the decay. Further detailscan be found in Ref. [439].

Supersymmetric Couplingsparticles λijk λ′ijk λ′′ijkχ0 ℓ+i νjℓ

−k , ℓ

−i νjℓ

+k , ℓ+i ujdk , ℓ

−i ujdk, uidjdk , uidjdk

νiℓ+j ℓ

−k , νiℓ

−j ℓ

+k νidjdk , νidjdk

χ+ ℓ+i ℓ+j ℓ

−k , ℓ

+i νjνk ℓ+i djdk , ℓ

+i ujuk uidjuk , uiujdk

νiℓ+j νk , νiνjℓ

+k νidjuk , νiujdk didjdk

Table 7.3: Direct decays of neutralinos and charginos with trilinear6Rp operatorsλijkLiLjEk,λ′ijkLiQjDk and λ′′ijkUiDjDk.

~0l ~ii l+j

lk~0l

l+j

lk~lj i ~0l

lk

l+j~l+k i

Figure 7.1: Diagrams for the direct decays of the neutralinoχ0l via the couplingλijk of the

6Rp trilinear LiLjEck interaction. The indexl = 1 . . . 4 determines the mass eigenstate of the

neutralino. The indicesi, j, k = 1, 2, 3 correspond to the generation. Gauge invariance forbidsi = j. The indexα = 1, 2 gives the slepton mass eigenstate (i.e. the chirality of theSM leptonpartner in absence of mixing).

In practice, the LSP lifetime is a crucial observable when discussing the final state topologyto be expected for supersymmetric events. The experimentalsensitivity of collider experimentsis often optimal if the LSP has a negligible lifetime so that the production and decay verticescoincide. Otherwise the LSP decay vertex is displaced. If the lifetime is sufficiently large,the LSP decays may occur outside the detector, giving rise tofinal states characteristic ofRp

conserving models.

The mean decay lengthL in centimetres for the lightest neutralino can be numerically esti-mated [255] from:

L(cm) = 0.3(βγ)( mf

100 GeV

)4(

1 GeVmχ0

1

)5

[1

λijk2 ,

3

λ′ijk2 ,

3

λ′′ijk2 ] . (7.6)

Figure 7.3 illustrates the behaviour of the LSP lifetime as presented in M2 versusµ planes fordifferent values oftan β and m0, and considering a dominantλ133 coupling. A translation interms ofL as a function ofmχ0

1for a fixedmf is shown in Fig. 7.4. Measurements of6Rp

coupling values can be performed through displaced vertex associated to the6Rp decay of theLSP.

The sensitivities on the6Rp couplings obtained via a displaced vertex depend of course onthe specific detector geometry and performances. Let us estimate the largest values of the6Rp

coupling constants that can be measured via the displaced vertex analysis. The LSP is assumed

l+j~l+k i i

j~l+i l+k j

i~l+j l+kk~+l ~+l ~+l

lj~i l+k

l+i~+l l+j l+k~+l ~jli

Figure 7.2: Diagrams for the direct decays of the charginoχ+l via the couplingλijk of the

6Rp trilinear LiLjEck interaction. The indexl = 1 . . . 4 determines the mass eigenstate of the

neutralino. The indicesi, j, k = 1, 2, 3 correspond to the generation. Gauge invariance forbidsi = j. The indexα = 1, 2 gives the slepton mass eigenstate (i.e. the chirality of theSM leptonpartner in absence of mixing).

Figure 7.3: LSP lifetime for different values of the MSSM parameters, and with a dominantλ133 coupling; For this illustration, the coupling has been set to λ133 = 0.004.

to be the lightest neutralino (χ01). Since a displaced vertex analysis is an experimental chal-

lenge at hadron colliders, the performance typically achievable at a futuree+e− linear collideris considered here. Assuming that the minimum distance between two vertices necessary to

0 10 20 30 40 50 60 70 80 90M(χ) (GeV/c2)

λ

10-2

10-3

10-4

10-5

10-6

M(χ) (GeV/c2)

λ

Excluded

χ decays within 1cm

χ decays inside Detector(Displaced Vertices)

χ decays outside Detector

(a)

0 10 20 30 40 50 60 70 80 90M(l

~),M(ν) (GeV/c2)

λ

10-2

10-3

10-4

10-5

10-6

10-7

10-8

10-9

Excluded

l~ / ν decay within 1cm

l~ / ν decay inside Detector

(Displaced Vertices)

l~ / ν decay outside Detector

(b)

Figure 7.4: Regions in theλ versus sparticle mass plane where the sparticle has a mean decaylength ofL < 1 cm, 1 < L < 3m (displaced vertices), andL > 3m (decay outside a typicalHEP detector) for a)χ0

1 assumingmf = 100 GeV and b) sleptons and sneutrinos. The dashedlines show an indirect limit onλ133 [440].

distinguish them experimentally is of orderO(2 × 10−5)m, it can be seen from Eq. (7.6) thatthe 6Rp couplings can be measured up to the values,

Λ < 1.2 × 10−4γ1/2(mf

100 GeV)2(

100 GeVmχ0

1

)5/2. (7.7)

whereΛ = λ, λ′/√

3 or λ′′/√

3, andγ is the Lorentz boost factor.

There is a gap between these values and the sensitivity of low-energy experiments whichrequires typically6Rp coupling values in the rangeΛ ∼ O(10−1 − 10−2) for superpartnersmasses of100 GeV. However, the domain above the values of Eq. (7.7) can be tested throughthe study of the single production of supersymmetric particles as will be discussed in section 7.5.Indeed, the cross-sections of such reactions are directly proportional to a power of the relevant6Rp coupling constant(s), which allows the determination of the values of the6Rp couplings.Therefore, there exists a complementarity between the displaced vertex analysis and the studyof singly produced sparticles, since these two methods allow to investigate different ranges ofvalues of the6Rp coupling constants.

7.3.3 Cascade Decays Initiated by Gauge Couplings

In an indirect decay, the supersymmetric particle first decays through aRp conserving vertex(i.e. through gauge couplings) to an on-shell supersymmetric particle, thus initiating a cascadewhich continues till reaching the LSP. The LSP then decays asdescribed above via one6Rp

coupling.

The sfermions may for example decay indirectly (i.e. undergo first a gauge decay) into afermion plus aχ0

1 if the lightest neutralinoχ01 is the LSP, as shown for example in the case

of sleptons in Fig. 7.5a and b. Theχ01 will subsequently undergo a6Rp decay via one of the

trilinear couplings. In the squark sector, such decays havemainly been considered for the stopand sbottom in actual searches at colliders as they possess amass eigenstate which can be

(a)~~01

(b)~l l~01 ( ) Z~02 ~01 f

fW ~W

W(d) (e) f

f 0f 000f 00~01

e02~f01 f

f 0Figure 7.5: Slepton (a, b) and gauginos (c, d, e) indirect decay diagrams.

among the lightest for squarks. If the lightest stop mass eigenstatet1 is not the LSP but thelightest charged supersymmetric particle, the cascade will be initiated through a decay tot1 →cχ0

1 . If mt1>mχ±+mb then the decayt1 → bχ± is possible. In the case of the sbottom, the

indirect decayb1 → bχ01 is generally treated as the dominant one.

In the gaugino-higgsino sector, the heavy neutralino and chargino mass eigenstates can de-cay, depending on their mass difference with theχ0

1 , either directly into three standard fermions,or indirectly toχ0

1 via a virtualZ orW , as illustrated in Fig. 7.5c, d and e.

Assuming a small value for the6Rp coupling, the indirect decay mode will generally domi-nate as soon as there is enough phase space available between“mother” and “daughter” spar-ticles. For searches at existing colliders this happens when the mass difference between thesetwo sparticles is larger than about5 to 10 GeV. As an example, the Fig. 7.6 shows the6Rp decaybranching fraction of theτR via theλ133 coupling as a function of the stau mass, for differentvalues of the neutralino mass. If the slepton is lighter thanthe neutralino, only the6Rp mode isopened. As soon as the indirect decay mode is possible, it dominates.

Nevertheless, there exist regions of the SUSY parameter space where theRp conserving de-cay (initiating the cascade) suffers from a “dynamic” suppression. This is the case for exampleif the field component of the two lightest neutralinos is mainly the photino, in which case theindirect decayχ0

2 → χ01 Z∗ is suppressed. In these regions, even if the sparticle is notthe LSP

it will decay through a direct6Rp mode.

It should be emphasized here that the indirect decays lead tofinal state topologies whichdiffer strongly from direct6Rp decays. For example, the direct decay of a neutralino via aLLEcoupling leads to a purely leptonic final state, while the indirect decay adds (mainly) jets to suchfinal state. The allowed indirect decays and their branchingratios depend on the parametersvalues for the specific supersymmetry model (e.g. MSSM, mSUGRA, ...) as well as on thevalue of the6Rp coupling considered.

Detailed strategies for practical searches at colliders will be discussed starting in section 7.4.

τR mass (GeV/c2)

Br(

RpV

)

~

Figure 7.6: Branching ratio of the6Rp decayτR → ee,eτ, ττ+ 6E as a function of the mass ofthe τR and for different values of the neutralino mass.

7.3.4 Decays Through Mixing Involving Bilinear Interactions

For completeness, the effects of theµiLiHu bilinear terms in the superpotential which violatelepton-number conservation must be discussed in the context of collider physics. The theo-retical motivations for the appearance of such terms in the superpotential of supersymmetricmodels were discussed in detail in chapter 2. They appear forinstance as a consequence of non-zero right handed sneutrino vacuum expectation values in models with spontaneousR-parityviolation, or in models with explicitly brokenR-parity.

As discussed in section 2.3, there are strong constraints onthe bilinear6Rp parameters fromthe neutrino sector. For this reason, theµi are often considered to be suppressed [259] and thusneglected in collider analyses, with the notable exceptionof the LSP decays, where the LSP isnot necessarily a neutralino in this context. Indeed, values of theµi suggested by the solar andatmospheric neutrino experiments can lead to observable decays of the LSP at colliders, withcorrelations between the neutrino mixing angles and the branching ratios into different leptonflavours [50, 51, 52, 54, 231] (see section 5.2.2).

However, exotic scenarios with non-negligibleµi parameters have been considered, in par-ticular in the context of spontaneousR-parity violation [63, 64, 65]. Although these scenariosare disfavoured by neutrino oscillation data, we mention for completeness the expected effectof e.g. a non-negligibleµ3. Such a term introduces a tau component in the chargino mass eigen-state. As a consequence, the tree level decayZ → χ−τ+ becomes for instance possible. In thetop sector, bilinear terms could rise to additional decay modes for top-quarks and stop-squarkssuch ast → τ+

1 + b or t1 → τ+ + b. Most of these new decay modes result, through cascadedecays, in final states with twoτ ’s and two b-quarks plus the possibility of additional jets andleptons. B-tagging andτ identification are therefore important tools for the analysis. Note thatthe t→ τ + b decay could also occur via a trilinearλ′333 coupling.

SpontaneousR-parity violation has been studied in the context of a singletau-lepton-number-violating bilinear term at the Tevatronpp collider in Ref. [441] and thet → τ + b decay was

found to be competitive with the decay in cχ01 for a discovery of thet. For the LHCpp collider,

the possibility to observe spontaneousR-parity violation through multilepton and same signdilepton signatures in gluino pair-production has been considered in Ref. [442].

7.4 6Rp Phenomenology from Pair Produced Sparticles

In this section, we are interested in the way the collider phenomenology for Supersymmetricmodels is affected by the presence of an individual6Rp coupling with a valueΛ2 ≪ 4πα suchthat 6Rp contributions can be neglected at production.

In such a configuration, a first question of interest for searches at colliders is whether or notthe nature of a specific non-vanishing6Rp coupling can be identified (within a range of allowedvalues) starting from the characteristics of the observed final states. Assuming that the presenceof specific6Rp interactions is eventually established, a next important question is to understandwhether and how the sensitivity to the fundamental parameters of a given supersymmetric theoryis affected.

7.4.1 Gaugino-Higgsino Pair Production

7.4.1 a) Production and final states

Gaugino-higgsino pair production via standard gauge couplings at colliders has been thor-oughly studied in the literature and a detailed review is clearly outside the scope of this paper.Here, only the key ingredients shall be summarized. Otherwise we concentrate on the phe-nomenology associated with the presence of6Rp interactions.

At l+l− lepton colliders, the neutralinos are produced by pairs vias-channelZ exchange(provided they have a higgsino component), or viat-channell± exchange (provided they have agaugino component). The charginos are produced by pairs in thes-channel viaγ or Z exchange,or in thet-channel via sneutrino (νl) exchange if the charginos have a gaugino component. Ofcourse, thet-channel contributions are suppressed for high slepton masses.

In the case of neutralinos, thet-channel exchange contributes to an enhancement which canbe significant for slepton masses typically below

√sll/2 (i.e. me

<∼ 100 GeV in the case ofselectron exchange at LEP 2e+e− collider). In constrast, the chargino pair production cross-section can decrease due to destructive interference between thes- and t-channel amplitudes(i.e. betweene andνe exchange at ae+e− collider) if the l± andνl masses are comparable.

As an example, one can consider pair production in the framework of the MSSM, assumingin addition that scalars have a common mass m0 at the GUT scale. In such a case m0 deter-mines the slepton masses at EW scale and the relevant MSSM parameters are M2, µ, tanβ andm0. In such a framework the production cross-sections are generally found to be large. If thedominant component of neutralinos and charginos is the higgsino (|µ| ≪M2) the productioncross-sections are also insensitive to slepton masses. Over a wide range of MSSM parame-ter values, the pair production cross-sections at LEP 2 for

√see ≃ 200 GeV is found to vary

typically from 0.1 to10 pb. Investigations of gaugino pair production in a similar constrainedMSSM framework and in presence of6Rp have been performed for the case of a future 500 GeVe+e− collider in Ref. [443].

At pp and pp hadron colliders, the main production process which has been studied forneutralinos and charginos is the associated productionqq′ → χ±χ0. In Rp-conserving theorieslike the MSSM or mSUGRA, measurable rates are expected mainly in the case ofχ±

1 χ02 and

only for certain regions of the parameter space. In presenceof 6Rp interactions of course, theprocessqq′ → χ±

1 χ01 involving the lowestχ0 mass eigenstate could also become observable.

In addition, 6Rp could allow for pair production of neutralinos and charginos in qq → γ, Zannihilation processes to become observable. The production cross-sections would then dependon the gaugino and higgsino components as discussed above for l+l− annihilation.

The final states resulting from the decay of pair-produced neutralinos or charginos are listedin Table 7.4 for the three different couplingsLLE, LQD andUDD.

gauginos decay mode LLE LQD UDDχ0

1 χ01 direct 4ℓ + 6E 1ℓ + 4j + 6E 6j

2ℓ + 4j4j + 6E

χ+1 χ

−1 direct 2ℓ + 6E 1ℓ + 4j + 6E 6j

4ℓ + 6E 2ℓ + 4j6ℓ 4j + 6E

χ02 χ

01 indirect 4ℓ + 6E 1ℓ + 4j + 6E 8j

4ℓ + 2j + 6E 1ℓ + 6j + 6E 6j + 2ℓ6ℓ + 6E 2ℓ + 4j + 6E 6j + 6E

2ℓ + 6j3ℓ + 4j +6E

4ℓ + 4j6ℓ +6E

χ+1 χ

−1 indirect 4ℓ + 4j + 6E 1ℓ + 6j + 6E 10j

5ℓ + 2j + 6E 1ℓ + 8j + 6E 8j + 1ℓ + 6E6ℓ + 6E 2ℓ + 4j + 6E 6j + 2ℓ +6E

2ℓ + 6j + 6E2ℓ + 8j

3ℓ + 4j + 6E3ℓ + 6j + 6E4ℓ + 4j + 6E

8j + 6E

Table 7.4: Neutralino and chargino pair production final states in caseof 6Rp decays. Thenotationsl, 6 E and j correspond respectively to charged lepton, missing energyfrom at leastone neutrino and jet final states.

At first approximation, withλ 6= 0 the final states are characterized by multi-lepton (chargedleptons and escaping neutrinos) event topologies. In contrast, withλ′ 6= 0, the final states arelikely to contain multi-jets and several more or less isolated leptons. One exception concernshere slepton pair production which could lead to four jet final states. Finallyλ′′ 6= 0, leads tofinal states with very high jet multiplicities. Thus, the existence of either a non-vanishingλ (LLE ), λ′ ( LQD ), orλ′′ ( UDD ) can indeed be readily distinguished.

Of course such a simple picture applies essentially in the framework of MSSM of mSUGRAmodels where theχ0

1 is likely to be the lightest supersymmetric particle. It moreover has to bemodulated in the presence of indirect (cascade) decays. Forinstance indirect gaugino decayswhen involving an intermediate Z∗ or W∗ might lead to final states containing jets forLLEinteractions or, symmetrically, containing leptons and neutrinos forUDD interactions.

7.4.1 b) Searches for Gaugino Pair Production ate+e− Colliders

It is interesting to review what has been learned from studies by the experiments at the LEPcollider. The analyses have been performed assuming ”shortlived” sparticles such that the6Rp

decays occur close enough to the production vertex and are not observable. In practice thisimplies a LSP flight path of less thanO(1) cm. Considering the upper limits on theλijk derivedfrom low energy measurements (chapter 6), and according to Eq. 7.6, the analyses are thusinsensitive to a lightχ of massMχLSP

≤ 10 GeV (due first to the termmχ−5 and second to the

term(βγ) which becomes important). When studyingχ decays, for typical masses considered,the LEP analyses have a lower limit in sensitivity on theλ coupling of the order of10−4 to 10−5.

In most of the6Rp analyses, the main background contributions come from the four-fermionprocesses andf fγ events. A discussion of such SM background contributions can be found inRef. [444]. Thef fγ cross-section decreases with the increase of the centre-of-mass energy; onthe other hand, the cross-sections of the four-fermion processes increase; in particular beyondthe W+W− and theZZ thresholds, these processes contribute significantly to the background,and lead to final states very similar to several6Rp signal event topologies.

TheLLE searches at LEP were mainly multi-lepton analyses, with themissing energy inthe final states most often coming from neutrinos. However, the indirect decay topologies mayalso contain hadronic jets (indirect decay of gauginos, seeTable 7.4). The number of chargedleptons in the final state varied between 4 and 6 except for direct decays of charginos withtwo neutrinos which lead to 2 charged leptons in the final state. Therefore, the crucial step inthe selection of the signal events was the electron, muon or tau identification. Electrons andmuons are typically identified by well isolated charged tracks in combination with either dE/dxmeasurements and deposits in the electromagnetic calorimeters (e) or information from hadroncalorimeters and muon chambers (µ, τ ) of the experiments. Tau decays may also be identifiedthrough isolated thin hadronic jets. In case ofW or tau jets, signal selection has been performedwith the additional discrimination provided by topological variables like theycut jet resolutionvariable. The two-lepton final states are difficult to separate from the SM background mainlycoming fromf fγ andγγ events and have been considered in the analyses. The other multi-lepton final states on the other hand, provided almost background free analyses with efficienciestypically between 20 and 60 %. The decays producing taus in the final states were found tohave lower efficiencies and rejection power. The analyses designed for signals produced with adominantλ133 can be applied to signals produced with otherλijk, and the efficiencies are eitherof the same order or higher. Therefore, the weakest limits which have been derived are thoseresulting from analyses performed assuming a dominantλ133 coupling [445].

For theLQD searches at LEP, the analyses of gaugino decays all includedhadronic activityin the final state (at least 4 hadronic jets, see Table 7.4) such that topological variables likejet resolution and thrust were used to select events in combination with missing energy and/orone or two identified leptons. For the topologies with missing energy the polar angle of themissing momentum has also been used to select candidate events. The analysis of gauginodecays via the couplingsλ′i3k andλ′ij3 also benefits from the presence of b-quarks, and therebyfrom possible background reduction via b-tagging. The topologies of the indirect chargino

decays depend heavily on the mass difference between chargino and neutralino, which is a freeparameter in the model. Sensitivity to a large range of topologies is hence needed in orderto completely cover all possibleLQD scenarios. Several of the gaugino final state topologiesclosely resemble those of6Rp sfermion decays and the same event selection may therefore beused to cover these channels too, e.g. sneutrino and sleptondecays. The SM backgroundconsists mainly of four-fermion events decaying either to hadronic or semileptonic final states.Signal efficiencies typically range from over 50% down to a few %, depending on topologyand selection criteria. The worst efficiency is generally obtained for decays into taus and lightquarks. The couplings generating these final states (e.g.λ′311) are therefore used to evaluateconservative constraints on the production cross-sections. The excludedLQD gaugino cross-sections at a 95% confidence level are typically of the order of 0.5 pb [446].

For the LEP searches in the case of a single dominantUDD coupling, the gauginos de-cay into mainly hadronic final states, however, the indirectgaugino decays may also includeleptons and missing energy, depending on the decay mode of the W. As seen from Table 7.4,the number of jets expected in the final states varies betweensix and ten. The selection ofcandidate events typically depends on topological variables like jet resolution, thrust and jetangles, thereby rejecting the major part of the SMf fγ , W+W− andZZ background events.The absence of neutrinos and missing energy in the fully hadronic final states also enables di-rect reconstruction of the gaugino masses. The mass reconstruction consists of assigning eachreconstructed jet to its parent gaugino and thereafter applying a kinematic fitting algorithm.These algorithms are also used to reconstruct the mass of theW bosons produced at LEP 2and impose constraints on conserved energy and momenta in combination with equal masses ofthe pair produced gauginos. The indirect chargino decays, again, strongly depend on the massdifference between chargino and neutralino, thereby making it difficult to use the same eventselection to cover all possible scenarios. Decays into light quarks (λ′′112 andλ′′122 couplings) gen-erally have lower efficiencies and are therefore used to derive conservative cross-section limitsat a 95% confidence level (CL) [445].

Since none of the6Rp gaugino searches at LEP 2 show any excess of data above the StandardModel expectations, the results are interpreted in terms ofexclusions of the MSSM parameters.The gaugino pair production cross-sections are, as previously discussed, mainly determinedby the MSSM parametersµ, M2, m0 and tanβ. The excluded gaugino cross-section at a 95%CL for each experimental search channel is therefore compared to the production cross-sectionprovided by the MSSM for each set of these four parameters. Hereby an exclusion of experi-mentally disproved combinations of the parameters is obtained for each of the performed searchchannels. This exclusion is then typically presented as contours in theµ, M2 plane for differentfixed values of tanβ and m0 [446, 445, 447, 448]. The LEP 1 excluded region of the (µ, M2)contours is obtained from theZ line-shape measurement. Examples of such (µ, M2) exclusioncontours are shown in Figures 7.7, 7.8 and 7.9. The dominant contribution to the exclusioncontours comes from the chargino pair production analyses in any6Rp couplings. The neutralinopair production analysis becomes relevant in case of lowtanβ, low m0, small M2 and negativeµ values (Fig. 7.9) which means when the chargino pair production cross-section is suppressedby destructive interferences betweens- andt-channels.

From the exclusion plots in the (µ, M2) plane the extraction of the minimum gaugino masseswhich is not excluded for the investigated range of parameters within the MSSM is performed.These limits on the lightest chargino and neutralino are obtained for high m0 value which corre-sponds to the disappearance of neutralino pair production contribution. The mass of the lightestnon-excluded gaugino is naturally shifted when tanβ is changed. In Fig. 7.10, the lightest non-

0

100

200

300

400

-200 -100 0 100 200 µ (GeV/c 2 )

M 2

(GeV

/c 2 )

LEP2 LEP1

tan β =1.41

m 0 = 500 GeV/c 2

(a) LLE −

ALEPH

excluded at 95% C.L.

Figure 7.7: Regions in the (µ, M2) plane excluded at95% CL at tanβ= 1.41 and m0 = 500 GeVforLLE coupling [446]. The dotted line is the kinematic limit for pair production of the lightestchargino.

0

50

100

150

200

250

300

350

400

-200 -100 0 100 200 µ (GeV/c 2 )

M 2

(GeV

/c 2 )

LEP2 LEP1

excluded at 95% C.L.

tan β =1.41

m 0 = 500 GeV/c 2

(b) LQD −

ALEPH

Figure 7.8: Regions in the (µ, M2) plane excluded at95% CL for tanβ= 1.41 andm0 = 500 GeV in the case of the dominantLQD coupling [446]. The black line is the kinematiclimit for pair production of the lightest chargino.

excluded neutralino mass as a function of tanβ for theLLE searches in DELPHI is shown. Ithas been checked that this result is independent of m0 values.

In this context, one of the most important results in the searches for supersymmetry ob-tained with data taken up to a centre-of-mass energy of208 GeV at LEP 2 by the four LEPexperiments is that the lightest chargino mass is excluded at 95% CL up to 103 GeV and thelightest neutralino mass is excluded at95% CL up to39 GeV in the framework of the MSSMwith 6Rp assuming that theχ0

1 decays in the detector. These results are formally only valid in

DELPHI √ s = 189 GeV

tanβ=1.5

m0= 90.

LEPI

χ1+ χ1

-

χ10 χ1

0

µ (GeV)

M2

(GeV

)

tanβ=1.5

m0=300.

LEPI

χ1+ χ1

-

µ (GeV)

tanβ=30.

m0=90.

LEPIχ1

0 χ10

χ1+ χ1

-

µ (GeV)

M2

(GeV

)

tanβ=30.

m0=300.

LEPI

χ1+ χ1

-

µ (GeV)

50

100

150

200

250

300

350

400

-200 -100 0 100 200-200 -100 0 100 200

50

100

150

200

250

300

350

400

-200 -100 0 100 200-200 -100 0 100 200

50

100

150

200

250

300

350

400

-200 -100 0 100 200-200 -100 0 100 200

50

100

150

200

250

300

350

400

-200 -100 0 100 200-200 -100 0 100 200

Figure 7.9: Regions in the (µ, M2) plane excluded at95% CL for tanβ= 1.5, 30 andm0 = 90, 300 GeV in the case of a dominantUDD coupling [445].

the scanned MSSM parameter space, i.e. for1 ≤ tanβ ≤ 35, m0 ≤ 500 GeV, |µ| ≤ 200 GeV,M2 ≤ 500 GeV, and for any coupling value from 10−4 up to the existing limits.

We hinted above of situations where indirect (cascade) decays could play a significant role.This could be the case for instance at future lepton colliders where centre-of-mass energiesfar beyond the current lower mass limits on the LSP are being contemplated. In view of theconstraints established at LEP 2, the possibility of opening up large production of heavier neu-

tanβ

Mas

s of

χ 10 (

GeV

/c2 )

χ10 mass limit (LLE

– coupling)

√s– = 183 GeV

√s– = 189 GeV

DELPHI

Figure 7.10:The lightest non-excluded neutralino mass as a function of tanβ at 95% CL. Thislimit is independent of the choice of m0 in the explored range and of the generation indicesi, j, k of theλijk coupling [445] and it assumes that theχ0

1 in the detector.

tralinos and charginos at a future 500 GeVe+e− linear collider (LC) has been studied inRef. [443] assuming that the lightest neutralino is the LSP and in presence of6Rp. The studyshowed in this case (for a representative but finite number ofpoints in the constrained MSSMparameter space) that only the production modes involving the χ0

1 , χ02, χ

±1 need to be consid-

ered. Theχ03, χ

04, andχ±

2 being almost always beyond the reach of a500 GeV machine. Asa consequence, the analysis would remain relatively simple, with a limited amount of cascadedecays to take into account. Moreover, for a large part of theparameter space, theχ0

2 is nearlydegenerate with the lighter chargino, and then, the number of decay chains to be consideredis futher reduced. The signals produced byLLE, LQD, UDD operators have been studiedin Ref. [443] and retain the basic characteristics listed above. ForLLE the dominating signalremains an excess of multi-lepton final states, with possibly substantial missing energy. ForLQD the final states contain again leptons and jets. In this case,an algorithm to reconstructthe LSP and higher neutralino masses enables to identify thesignal as due to supersymmetrywith specific6Rp operators. The existence of like-sign dilepton final states, originating from theMajorana nature of neutralinos, appears to be a very promising signal practically free of trueStandard Model background sources. ForUDD the final states consist again of multiple jetswhich are more difficult to disentangle from a large number ofStandard Model backgroundsources. Mass reconstruction nevertheless appears promising here also to allow to identify thesignal as due to supersymmetry with specific6Rp operators.

7.4.1 c) Searches for Gaugino Pair Production at Hadron Colliders

Cascade decays involving trilinear6Rp couplings could also play a major role at future hadroncolliders.

Outstanding multi-lepton event signatures are expected inthe presence of aλ Yukawa cou-pling. The case of gaugino pair production with a trilepton signature has been investigated fortheD∅ experiment at the Tevatron collider in Ref. [449], in the framework of mSUGRA.

D∅ [450] also considered the dimuon and four-jets channel occuring after χ01 decay via

theλ′2jk coupling (j = 1, 2; k = 1, 2, 3) where theχ01 can be produced either directly in pair

or through cascade decays from squarks or gluinos. Gluinos masses below224 GeV (for allsquark masses and fortanβ = 2) are excluded. For equal masses of squarks and gluinos themass limit is265 GeV.

Theλ′′ coupling implies multi-jet final states for sparticle decays which severely challengesthe sensitivity atpp andpp colliders.

However the 1-lepton and various dileptons and multi-lepton event topologies that resultfrom simultaneous production of all sparticles at the Tevatron assuming the LSP decays viabaryon number violating operators have been studied in [451] giving reaches on the gluinomass from 150 GeV up to 360 GeV depending on the specific topology.

The study of the decay chainqL → χ02q → lRlq → χ0

1llq followed by the decay of theχ01

into three quarks assuming a non-zeroλ′′212 at the LHC has been performed in [452]. This studyshows that even in the choice of a non-zeroλ′′212, which is considered as the hardest choice, theχ0

2, χ02 and qL can be detected and their masses measured and that the mass ofthe lR can be

obtained in much of the parameter space.

7.4.2 Sfermion Pair Production

7.4.2 a) Production and final states

As for the gaugino-higgsino production discussed above, weshall concentrate in this sec-tion on the phenomenology associated with the presence of6Rp interactions. But we shall firstbriefly review the key ingredients for sfermion pair production via standard gauge couplings atcolliders.

The sfermion mass eigenstates,f1 and f2 (f: q or ℓ, f1 lighter thanf2), are obtained fromthe two supersymmetric scalar partnersfL and fR of the corresponding left and right-handedfermion [453, 454]:

f1 = fL cosθf + fR sinθff2 = –fL sinθf + fR cosθf

whereθf is the mixing angle with 0≤ θf ≤ π. According to the equations which give thesfermion masses (see for example in [455]), the left-handedsfermions are most often heavierthan their right-handed counterparts. ThefL–fR mixing is related to the off-diagonal terms ofthe scalar squared-mass matrix. It is proportional to the fermion mass, and is small compared tothe diagonal terms, with the possible exception of the thirdfamily sfermion [456]. The lightestsquark is then probably the lighter stopt1. This is due not only to the mixing effect but alsoto the effect of the large top Yukawa coupling; both tend to decrease the mass oft1 [457]. Thelightest charged slepton is probably theτ1. For small values of tanβ, τ1 is predominantly aτR,and it is not so much lighter thaneR andµR.

Sleptons and squarks can be pair produced ine+e− collisions via the ordinary gauge cou-plings of supersymmetry with conservedR-parity provided that their masses are kinematicallyaccessible. They can be produced vias-channelZ or γ exchange with a cross-section dependingon the sfermion mass. Theνe (e) can also be produced via the exchange of a chargino (neu-tralino) in thet-channel, provided that the gaugino component of the chargino (neutralino) is thedominant one. Thet-channel contributes if the chargino (neutralino) mass is sufficiently low,and then the cross-section depends also on theχ± (χ0) mass and field composition and therebyon the relevant parameters of the supersymmetric model. Thecoupling between the squarksand theZ boson depends on the mixing angleθq, and it is minimal for a particular angle value.For example in the case of the stop, the decoupling between the θt and theZ is maximal forθt = 0.98 rad such that the stop pair production cross-section is minimal.

In general, both direct and indirect decays of sfermions canbe studied in sparticle pairproduction at colliders. The direct decay of a sfermion via agiven 6Rp coupling involves specificstandard fermions and can be (e.g. when involving the top quark) kinematically closed. Thefinal states resulting from the decay of pair-produced sleptons or squarks are listed in Table 7.5and 7.6 for the three different couplingsλ ( LLE ), λ′ ( LQD ), orλ′′ ( UDD ).

When considering sfermion pair production and the decays ofTable 7.5 and 7.6 relevantfor e+e− colliders, it should be noticed that in general the indirectdecays into a chargino willnot be considered. This is because the chargino search itself provides a mass limit close to thekinematic limit. There is no phase space left for the production of (e.g.) two sleptons followedby decays involving charginos. This explains why for instance at LEP, the most general sfermionindirect decay studied has been the decay into the lightest neutralino considered as the LSP(ν→ ν χ0

1 , ℓ→ ℓ χ01 , q→ q′ χ0

1 ). The final states are then composed of two fermions plus thedecay products of the6Rp decay of the neutralino pair (see Table 7.6).

7.4.2 b) Slepton Searches at lepton colliders

sfermions LLE LQD UDDν ˜ν 4 ℓ 4 j not possibleℓ+Rℓ

−R ℓℓ′ +6E not possible not possible

ℓ+L ℓ−L 2 ℓ +6E 4 j not possible

uL¯uL, dR

¯dR not possible 2ℓ + 2 j 4 j

dR¯dR 1ℓ + 2 j + 6E

dL¯dL, dR

¯dR 2 j + 6E

Table 7.5: Sfermion pair production final states in case of direct6Rp decays. The notationsl, 6Eandj correspond respectively to charged lepton, missing energyfrom at least one neutrino andjet final states.

sfermions LLE LQD UDDν ˜ν 4 ℓ + 6E 2ℓ+ 4j + 6E 6j + 6E

2ℓ+ 4j + 6E4j + 6E

ℓ+ℓ− 6 ℓ+ 6E 4ℓ+ 4j 2ℓ+ 6j3ℓ+ 4j + 6E2ℓ+ 4j + 6E

q˜q 4 ℓ+ 2j + 6E 2ℓ+ 6j 8jℓ+ 6j + 6E

6j + 6E

Table 7.6: Sfermion pair production final states in case of indirect decays when the LSP is thelightest neutralino. The notationsl, 6E andj correspond respectively to charged lepton, missingenergy from at least one neutrino and jet final states.

Here again one can profit from what has been learned from actual studies by the experi-ments at LEP collider where both productions of sneutrino and charged slepton pairs have beensearched for.

Early discussions on severalR-parity-violating processes ate+e− colliders including chargedslepton pairs can be found in [458, 459].

Lets first consider the case of sneutrino pair production. Inthe presence ofLLE interac-tions, searches for four charged lepton final states are performed. The six possible configura-tions are listed in Table 7.7. Event topologies containing muons and electrons allow for a highselectivity by applying conditions on two lepton invariantmasses. The highest efficiencies areobtained when there are at least two muons in the final states,and the lowest when there aretaus. In the latter case, a certain amount of energy is missing in the final state, due to the neutri-nos produced in the tau decays. Then, most often, two extremecases in the coupling choice arestudied: the first one considering that theλ122 or λ232 is dominant, leading to the most efficientanalyses, the second considering that theλ133 or λ233 is dominant, leading to the least efficientanalyses. With these two analysis types, all the possible final states are probed.

In the presence ofLQD interactions, searches for four jets final states are performed. Herealso, the absence of missing energy in the final state offers the possibility to reconstruct thesneutrino mass. Depending on the generation indices, 0, 2, and 4 jets can contain a b-quark (Ta-ble 7.8). With the possibility to tag the jets generated by b-quarks, the analyses are very efficient.

final states processes and couplingsνe˜νe νµ˜νµ ντ ˜ντ

eeee λ121 λ131

eeµµ λ121 λ122 λ132, λ231

eeττ λ131 λ123, λ231 λ133

µµµµ λ122 λ232

µµττ λ123, λ132 λ232 λ233

ττττ λ133 λ233

Table 7.7: Four lepton final states produced by the direct decay via aLLE term of a sneutrinopair.

final states processes and couplingsνe˜νe νµ˜νµ ντ ˜ντ

4 q (no b-quarks) λ′1jk,j,k 6=3 λ′2jk,j,k 6=3 λ′3jk,j,k 6=3

2 q 2 b λ′1j3,j 6=3, λ′13k,k 6=3 λ′2j3,j 6=3, λ

′23k,k 6=3 λ′3j3,j 6=3, λ

′33k,k 6=3

4 b λ′133 λ′233 λ′333

Table 7.8: Four jet final states produced by the direct decay via aLQD term of a sneutrinopair.

The LEP experiments have presented results on the lower limit on the sneutrino electronmass derived by searching for the direct decay of sneutrinosinthe data collected in 1999 and2000 up to a centre-of-mass energy of208 GeV. With aLLE (LQD) coupling, the mostconservative lower limits are98 (91) GeV [446, 445, 448]. For the derivation of limits, ef-ficiencies are determined as function of the sneutrino mass.In case ofνe, due to the possiblet-channel contribution, they are considered for a specific set of MSSM parameters, generally forµ = −200 GeV, M2 = 100 GeV. When the contribution of thet-channel becomes negligible,the νe˜νe production cross-section is similar to theνµ˜νµ or ντ ˜ντ ones. Taking into account thenumber of expected events from the Standard Model processes, the number of observed events,and the analysis efficiencies, upper limits at 95% of confidence level on the sneutrino cross-section are obtained as function of the sneutrino mass. Comparing these upper limits to theexpected MSSM cross-section, limits on the sneutrino mass have been derived, as illustrated inFig 7.11. The limits obtained are much stronger that those existing in the hypothesis ofR-parityconservation, in which the sneutrino pair production is invisible.

Right-handed charged sleptons are mainly studied, becausetheir production cross-section,for a given slepton mass, is lower than for the left-handed ones, therefore leading to more con-servative results. The direct decay of a pair of charged sleptons lead to two charged leptonsand some missing energy. This low multiplicity final state isdifficult to analyse due to thehigh background of low multiplicity processes. With a dominantλijk coupling constant, onlythe pair producedℓkR and ℓiL or ℓjL are allowed to directly decay. The decay ofℓ+kRℓ

−kR gives

ℓiℓi, ℓiℓj, ℓjℓj + 6E , and sincei 6= j two lepton flavours are mixed in the final state (see Ta-ble 7.9). It is not the case in the direct decay of the supersymmetric partners of the left-handedcharged leptons, for which there is only one lepton flavour inthe2ℓ+ 6E final state (Table 7.10).In case of selectrons, theeLeR production is possible in thet-channel; direct decay of bothselectrons is possible only viaλ121 (ee, eµ +6E final state) or viaλ131 (ee, eτ +6E final state).

Similarly to the sneutrino decay, the search for final statescontaining mainly taus is the

ν mass (GeV/c2)

95%

C.L

. σ18

9 (pb

)

DELPHI√s = 189 GeV

σMSSM(e+e- → ντντ, νµνµ)

σMSSM(e+e- → νeνe)µ = -200 GeV/c2

M2= 100 GeV/c2

ττττeeττ

Figure 7.11: Sneutrino direct decay searches withLLE coupling: limit on theν ˜ν productioncross-section as a function of the mass for two different final states. The MSSM cross-sectionsare reported in order to derive a limit on the sneutrino mass in the case of direct6Rp decay. Thedashed lower curve corresponds to bothνµ˜νµ and ντ ˜ντ cross-sections, which depend only onthe sneutrino mass. The dashed upper curve on the plot is theνe˜νe cross-section obtained forµ = −200 GeV and M2 = 100 GeV, the corresponding chargino mass lies between90 and120 GeV [445].

final states processes and couplingseReR µRµR τRτR

ee , eµ, µµ + 6E λ121 λ122 λ123

ee , eτ, ττ + 6E λ131 λ132 λ133

µµ, µτ, ττ + 6E λ231 λ232 λ233

Table7.9: Final states produced by the direct decay via aLLE term of a pair of supersymmetricpartners of the right-handed leptons.

least efficient one. An upper limit on the cross-section is obtained as a function of the sleptonmass. Comparing this upper limit to the expected MSSM cross-section, limits on the sleptonmass is deduced. In case of selectron production, the limit depends also on the chosen MSSMparameters, since the neutralino exchange in thet-channel may also contribute to the cross-section. From the data collected at LEP 2, the ALEPH experiment derived a lower limit of96 GeV on theµR [446], and OPAL obtained a limit of74 GeV for the same slepton mass [448],

final states processes and couplingseLeL µLµL τLτL

ee+6E λ121,131 λ121,231 λ131,231

µµ+6E λ122,132 λ122,232 λ132,232

ττ+6E λ123,133 λ123,233 λ133,233

Table 7.10:Two lepton with missing energy final states produced by the direct decay via aLLEterm of a pair of supersymmetric partners of left-handed charged leptons.

0

0.1

0.2

0.3

0.4

0.5

60 80 100

ee+eµ+µµ limitee+eτ+ττ limitµµ+µτ+ττ limit

µ~

R+µ

~

R-

e~

R+e

~

R-

ALEPH

M(l~

R) (GeV/c2)

σ 202 (

pb)

(a)

Figure 7.12: Charged slepton direct decay searches withLLE couplings: the95% CL exclu-sion cross-sections for sleptons. The MSSM cross-section for pair production of right-handedselectrons and smuons are superimposed [446].

when aLLE coupling is considered to be dominant.

0

50

100

150

200

0 100 200 300

m0 (GeV)

M2

(GeV

)

λ

λ″

λ′

L3

Excluded

tanβ = 2b)

758085

100

M =

200 GeV

eR

∼

Figure 7.13:Exclusion contours in the m0–M2 plane, fortanβ =2, at95% CL. The lines rep-resent the isomasses of the supersymmetric partner of the right-handed electron (labelled withthe corresponding value inGeV). The solid and dotted curves show the95% CL lower limitson M2 as a function of m0 from which the limits on the scalar electron mass were derived [447].

From exclusion contours in theµ–M2 plane, determined with the MSSM interpretation ofthe gaugino pair production results, after the analysis of the data collected up to189 GeV, the

L3 experiment sets indirect lower limit on the scalar leptonmasses [447]. Figure 7.13 showshow the lower limits on the mass of the supersymmetric partner of the right-handed electron arederived, taking into account the limits on M2 as a function of m0.

Contrary to the direct decays, the slepton indirect decays can be studied in any choice ofthe dominant coupling. As previously said, mainly the indirect decay into a neutralino (LSP) issearched for:

• indirect decayν→ νχ01 , χ0

1 6Rp decay via any coupling,

• indirect decayℓ→ ℓχ01 , χ0

1 6Rp decay via any coupling.

Then the final state depends on the slepton type (and flavour incase of charged sleptons), andmainly on theχ0

1 LSP decay. The efficiencies are determined in a mχ versus mν (mℓ ) plane aswell as the upper limit on the cross-section, which is compared to the MSSM cross-section, inorder to exclude domains in the same mχ versus mν (ml ) plane (see Fig. 7.14). The limit onthe neutralino mass is used to set the limit on the sneutrino (slepton) mass in case of indirectdecay. The results obtained on data collected in 1998, 1999 and 2000 [446, 445, 447, 448] aresummarized in Table 7.11.

experiments ALEPH DELPHI L3 OPALDATA 1998-2000 1998-2000 1998-2000 1998-2000

LLE 89 85 78 81νµ,τ LQD 78

UDD 65 70LLE 96 95 79 99

eR LQD 93 92UDD 94 92 96LLE 96 90 87 94

µR LQD 90 87UDD 85 87 86LLE 95 86 92

τR LQD 76UDD 70 75

Table 7.11: 95% CL lower limits (in GeV) on the slepton mass, considering the sleptonindirect decay in lepton and lightest neutralino only.

The pair production of right selectrons followed by their decay in the presence ofR-parity-violating couplings has been investigated for a 500 GeVe−e− linear collider (with possiblyhighly polarized beams) in Ref. [460].

At such a collider, a very strong suppression of the StandardModel background is expectedand this could be further reduced by exploiting specific beampolarizations. The conservationof electric charge and lepton number actually forbid the pair production of most of supersym-metric particles at ae−e− collider: only selectrons can be produced via thet–channel exchangeof a neutralino. The pair production of selectrons has been studied in the hypothesis ofR-parityviolation. In case ofLLE operator, the decay of the pair produced right selectrons will lead tofinal state consisting ine−e− + 4ℓ± + Emiss, where the flavour ofℓ± depend on the particular

ν mass (GeV/c2)

χ 10 m

ass

(GeV

/c2 )

DELPHIe+e- → νν

Direct decayonlyν → l+l′-

Dominance ofindirect decayν → ν χ1

0

excludedat √s=183 GeV

excludedat √s=189 GeV

20

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) (G

eV/c

2 )

ALEPH

(c)

OPAL

0102030405060708090

50 60 70 80 90m(eR) (GeV)

m(χ

0 ) (G

eV)

Indirect

(a)tg β = 1.5µ = -200 GeV

λ

~

~ 1 ~

DirectDirect

0102030405060708090

50 60 70 80 90

λ´3jk

λ´2jk

tg β = 1.5µ = -200 GeV

λ´1jk

m(e) (GeV)

m(χ

0 ) (G

eV)

Indirect

(b)

~

1

Figure 7.14: Top: Search for sneutrino decaying via a dominantLLE coupling in DELPHI;excluded region at95% CL in mν , mχ0 plane byν pair production for direct and indirect decays.The dark grey area shows the part excluded by the searches at

√s = 183 GeV, the light grey

area the one excluded by the searches at√s = 189 GeV. Middle: Search for smuon decaying

via a dominantUDD coupling in ALEPH; excluded region at95% CL in mχ0 , mµRplane.

Bottom: Search for selectron decaying indirectly via a dominantLLE (left) andLQD (right)coupling in OPAL; MSSM exclusion region for selectron pair production in the (mχ0 , mℓ) planeat 95% CL.

type of coupling. This kind of final state is free from Standard Model background and permitan easy detection at a500 GeV e−e− collider. In case of a dominantLQD operator, the finalstate consists of charged leptons, multiple jets and/or missing energy, and in addition, the Ma-jorana nature of the LSP gives rise to like-sign dilepton signal, with almost no background fromStandard Model. In case of a dominantUDD operator, the final state consists of multiple jetsassociated to like-sign dielectrons. In bothLQD andUDD cases, it might be possible to givean estimate for the LSP mass from invariant mass reconstruction.

7.4.2 c) Squark Searches at lepton colliders

6Rp decays of pair-produced squarks have been searched for ine+e− collisions at LEP 2.Special emphasis has been given to thet andb as they could possibly be the lightest squarks.

For squarks of the third generation, thefL–fR mixing cannot be neglected. Hence a mixingangle must be taken into account for the pair production cross-section. This parameter willconsequently enter as a free parameter when deriving experimental squark mass limits.

The results obtained at LEP 2 for the searches of both squark direct and indirect decays arereviewed in the following.

For small couplings (< O(10−1)) the t hadronises into colourless bound states before de-caying (see section 7.3.1), producing additional hadronicactivity in the decay. Another conse-quence of the small width of thet indirect decay (t → cχ0

1 ) is that, unusually, the direct decaydominates over the indirect decay for a large range of coupling values (> O(10−5)). On thecontrary the indirect decay of theb (b → bχ0

1 ) dominates whenever kinematically possible.

As no quark superfield enters in theLLE term of the6Rp superpotential, there is no directtwo-body decay of squark via aλ coupling. TheLQD andUDD terms can be responsible forsquark direct decays. In the first case, the final states consist of two jets and charged lepton(s)and/or missing energy: the three possibilities are listed in Table 7.6. In the stop pair productionsearches, only the channels with two charged leptons were considered. Highest efficienciesare obtained for final states containing electrons (λ′13k) or muons (λ′23k); final states with twotaus (λ′33k) are more problematic and consequently have lower efficiency and lead to weakestlimits. Using the efficiencies determined for different stop masses, an upper limit on the stoppair production cross-section can be set at95% CL as a function of the stop mass. Then,considering the cross-section for the stop pair production(e.g. in the framework of the MSSM)and in case of no mixing and maximal decoupling to theZ boson, a lower limit on the stopmass can be derived. In the tau channel, a stop mass lower than96 GeV has been excluded byOPAL [448] for any mixing angle using the data recorded in 1999 and 2000. At a centre-of massenergy from 189 to209 GeV, considering also the tau channel, but in a no mixing scenario, stopmasses lower than97 GeV are excluded by ALEPH [446].

Via a UDD term, the stop decays directly into two down quarks and the sbottom into an upand a down quark. The signature for the pair production of squarks is therefore four hadronicjets. Selections for these types of topologies rely mainly on reconstructing the mass of thedecaying squark by forcing the event to four jets and formingthe invariant masses betweenpairs of jets. For couplings involving a b-quark in the final state “b-tagging” algorithms basedon requiring large impact parameter tracks are also helpfulto separate signal from the largebackground coming from two-quark and four-quark Standard Model processes. In “flavourblind” searches, the cross-section limits degraded in the range of W mass. For direct decayvia a UDD interaction (λ′′ coupling), stop masses lower than77 GeV have been excluded byOPAL [448] for any mixing angle using the data recorded in 1999 and 2000.

Assuming that the lightest neutralinoχ01 is the LSP, the squark indirect decay into a quark

and a neutralino with the subsequent6Rp decay of the neutralino, has been studied. As anysquark field can couple to theχ0

1 , there are no restrictions related to the squark “chirality”.Final states for each coupling consist of the correspondingfermions from the neutralino pairdecay plus two jets.

In case ofLLE coupling, the relevant final states are two hadronic jets + 4 leptons + missingenergy. Six jets are expected with aLQD coupling, together with two charged leptons and nomissing energy or 0-1 lepton + missing energy. Pure hadronicfinal states consisting of eightjets are expected in case ofUDD couplings.

Using the efficiencies determined in the (mt, mχ01) plane, upper limit on the stop pair pro-

duction cross-section can be derived as a function of the stop and neutralino masses (takingµ = −200 GeV andtanβ = 1.5). Considering the MSSM cross-section for the stop pair pro-duction in case of no mixing and in case of a maximal decoupling to theZ boson exclusioncontours were derived in the mt, mχ0

1plane. By combining the exclusion contours with the

result on the neutralino mass limit, a lower bound on stop mass can be derived.

From the analysis of the events collected at a centre-of-mass energy from189 GeV to209 GeV, in the hypothesis of a dominantLLE coupling, a left-handed stop lighter than91 GeVat95% CL has been excluded by ALEPH [446]. ALEPH, DELPHI and L3 haveperformed thesearch for stop and sbottom indirect decays in the hypothesis of a dominantUDD coupling.Using the limit on the neutralino mass (32 GeV) from the gaugino searches, DELPHI set lowerbounds on the squark masses with mq− mχ0

1> 5 GeV (Fig. 7.15). The lower mass limit on

the stop (sbottom) is87 GeV (78 GeV) in case of no mixing, and77 GeV in case of maximalZ-decoupling. The study of indirect decay of left-handed stop and sbottom by ALEPH lead toexclude stop and sbottom lighter than71 GeV at 95% CL (Fig. 7.15).

In view of the limitations posed on centre-of-mass energiesand luminosities bye+e− col-lider technologies, the case of a high energyµ+µ− collider using storage rings has been consid-ered. The phenomenology of supersymmetry with6Rp at aµ+µ− collider resembles very muchto the one at ae+e− collider.R-parity violation can manifest itself via either a) pair-productionof supersymmetric particles followed by6Rp decays or b) resonant and non-resonant productionof a single supersymmetric particle or finally c) virtual effects in four fermions processes. Thecase (a) is discussed below. The cases (b) and (c) will be discussed in sections 7.5 to 7.6.

The discussion of pair-production of supersymmetric particles followed by6Rp decays forµ+µ− colliders is analogous in most of the aspects to the one fore+e− colliders and can befound in 7.3. However, unlikee+e− colliders, a particular feature ofµ+µ− colliders stems fromthe possibles-channel production of theCP -even (h0 andH0) or CP -odd (A0) Higgs bosonsof e.g. the MSSM. The Higgs boson produced would then decay into a pair of supersymmetricparticles in processes likeh0(H0, A0) → χ+χ−, χoχo, ucu, dcd, lcl, etc, whereu andd denotegenerically up-type down-type squarks respectively. The supersymmetric particles themselveswould then undergo6Rp decays according to Table 7.2 and 7.3. The analysis of these specific pairproduction of supersymmetric particles is governed by the analysis of the Higgs bosons decaywidths with respect to the parameters of the minimal supersymmetric extension of the StandardModel into consideration [461]. The results of this analysis have then to be merged with themore familiar analysis of pair production of supersymmetric particles from ordinary processesarising fromµ+µ− annihilation (eitherγ andZ in the s-channel or sfermions or gauginos in thet-channel) as in the case ofe+e− annihilation.

Additionnal Higgs bosons decays may also come into consideration such asH± → W±h0,A0 → Zh0, H0 → h0h0 andH0 → A0A0 which may lead to the production of a pair of

DELPHI, √s = 189 GeV

M(t∼1) (GeV)

M(χ∼

0 1) (

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)

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ALEPH

(b)

Figure 7.15:Top: Search for stop decaying indirectly via a dominantUDD operator in DEL-PHI; excluded region at 95% CL in mχ0 versus mθ plane; the largest excluded area correspondsto the case of no mixing and the smallest one to the case with the mixing angle which gives aminimum cross-section. Bottom: Search for left-handed sbottom decaying indirectly via a dom-inant UDD operator in ALEPH; the 95% CL exclusion cross-sections is shown in the mχ0

versus mbLplane.

supersymmetric particles in association with a gauge bosonor to the production of four super-symmetric particles and thus to more complicated signaturewhen 6Rp decays are switched on.We refer the reader to [461] for the calculation of the cross-sectionµ+µ− → higgs as well asthe Higgs bosons total width.

7.4.2 d) Sfermion and Gluino Pair Production at Hadron Colliders

Following the observation of an excess of highQ2 events at HERA experiments [462, 463],the CDF collaboration examined two scenarii withλ′121 6= 0 using107 pb−1 of data [464]:

pp→ gg → (ccL)(ccL) → c(e±d)c(e±d) (7.8)

200

300

400

500

600

700

800

200 210 220 230 240 250 260 270 280 290 300

Br(c∼

L → ed)≥0.5

Br(c∼

L → ed)=1.0

M(g∼) (GeV/c2)

M(q∼ )

(Ge

V/c2 )

Excluded at95% C.L.

∫ L dt = 107 pb-1

M(c∼

L) = 200 GeV/c2

tan β = 2

Figure 7.16: Exclusion region in the planemg–mq for the charm squark analysis of the CDFcollaboration.

andpp→ qq → (qχ0

1)(qχ01) → q(dce±)q(dce±) . (7.9)

For process (7.8) assumptions made weremq > mg > mcL= 200 GeV, where degenerate mass

for all up-type squarks (exceptcL) and all right-handed down-type squarks is denoted bymq.The masses of all left-handed down-type squarks were obtained by using the HERA motivatedrelations given in [465]. For analysing process (7.9) assumptions made weremχ±

1> mq > mχ0

andmχ±

1≃ 2mχ0 . Five degenerate squarks and the stop were treated separately. In the case

of the process involving stop to ensure 100 % branching ratiofor the decayt1 → cχ01 when

mt1 < mt, an additional condition thatmχ±

1> mt1 −mb was imposed.

The Majorana nature of gluino and neutralino implies processes (7.8) and (7.9) each yieldlike sign dielectron and opposite sign dielectron with equal probability. Since the like-signdilepton signature has very little SM background, for both processes (7.8) and (7.9) CDFsearched for events with like sign dielectrons and at least two jets.

Analysis of107 pb−1 data yields no event with an expected contribution of0.3±0.3 eventsfrom backgrounds. Exclusion contours obtained for process(7.8) are shown in Fig. 7.16 in themg–mq plane for different assumptions for theBr(cL → ed). The region belowmq ≤ 260 GeVis not excluded because in this regionbL becomes lighter thancL, hence suppressing the decayg → ccL. Figure 7.17 (bottom) shows the 95% CL upper limit on the cross-section timesbranching ratios (obtained for process (7.9) ) along with the NLO prediction [466] for the cross-section, as a function of squark masses. 95% CL lower limits are given for two different massesof the lightest neutralino.

A lower limit on the degenerate squark mass was found to be in the range of200–260 GeVdepending on the mass of the lightest neutralino and gluino (range of gluino mass consideredwas200–1000 GeV). Figure 7.17 (top) shows also a similar plot in the case of the stop. Themass of the stop was excluded below135(120) GeV for a heavy (light) neutralino. The analysisfor process (7.9) has been performed for the Tevatron Run II in [467]: it shows that squarkmasses up to380 GeV should be tested. Finally, one point to note here is that,although the

10-1

1

100 110 120 130 140 150M(t

∼1) (GeV/c2)

σ⋅B

r(t∼ 1

t∼_ 1→ e

± e± + ≥

2j) (

pb

)

∫ L dt = 107 pb-1

M(χ∼ 0

1)=M(t∼1)/2

M(χ∼ 0

1)=M(t∼1)-M(c)

10-1

1

100 150 200 250 300 350M(q

∼) (GeV/c2)

σ⋅B

r(q∼ q∼_ →

e± e± +

≥ 2

j) (

pb

)

∫ L dt = 107 pb-1

M(χ∼ 01 )=M(q

∼)/2

M(χ∼ 01 )=M(q

∼)-M(q)

Figure 7.17:Bottom) upper limits on the cross-section times branching ratio for the productionof 5 degenerate squark flavours decaying to electrons and jets via neutralinos (solid lines). Alsoshown is the theoretical prediction forσ ·Br for three values of the gluino mass:200 GeV (dot-ted line),500 GeV (dot-dashed line), and1 TeV (dashed line). Top) upper limits on the cross-section times branching ratio for stop pair production decaying to electrons and jets via neu-tralinos (solid lines). The dashed curve is the theoreticalprediction.

analysis for process (7.9) assumed only one6Rp couplingλ′121 to be non-zero, as the analysisdoes not depend on the quark flavours, the results are equallyvalid for anyλ′1jk (j=1, 2 andk =1, 2, 3) couplings.

D∅ [468] considered squark pair production leading in6Rp-SUGRA to like-sign dielectronevents accompanied by jets, and has ruled outMq < 243 GeV (95 % CL) when assuming fivedegenerate squark flavours. TheD∅ analysis covers allλ′1jk couplings. From a similar analysisby CDF [469] but restricted toλ′121 6= 0, one can infer that a cross-section five times smallerwould lead to aMq limit of ≃ 150 GeV depending on the gluino andχ0 masses.

CDF also considered separately [469, 464] the pair production of a light stopt1 assuminga decay intocχ0

1 and excludedMt < 135 GeV. To translate this constraint in one relevantfor λ′13k 6= 0, it should be noted that in this latter case,6Rp-decays of thet would dominateover loop decays intocχ0

1 . Moreover, 6Rp-decays would themselves be negligible comparedto t → bχ+

1 decays as soon as this becomes allowed, i.e. ifM(t1) > M(χ+1 ) and if the t1

eigenstate possesses a sizeable admixture oftL. The subsequent decays of theχ+1 would then

lead to final states similar to those studied by CDF fort1 → cχ01. Thus, 130 − 150 GeV

appears to be reasonable rough estimate of the Tevatron sensitivity to a light t for λ′13k 6= 0.In summary, Tevatron and HERA sensitivities are competitive in 6Rp-SUSY models with fivedegenerate squarks, but models predicting a lightt are better constrained at HERA providedthatλ′13j is not too small.

In the above mentioned search by theD∅ experiment [450] in the dimuon plus four-jetschannel (occuring afterχ0

1 decay via theλ′ coupling, see section 7.4.1), it was seen that squarkmasses below240 GeV (for all gluino masses and fortanβ = 2) are excluded. For equal massesof squarks and gluinos the mass limit is265 GeV

In contrast with the above studies of theλ′ coupling constants which were based on theanalysis of a given superpartner pair production, theD∅ Collaboration has performed a study oftheλ′1jk andλ′2jk (j =1, 2 andk = 1, 2, 3) couplings based on the Monte Carlo simulation of allthe superpartner pair productions [467, 470]. In this work,it was assumed that the LSP is thelightest neutralino. The exclusion plot obtained in [467, 470] in the case of a single dominant6Rp

coupling of typeλ′1jk is presented in Fig.7.18. In this case, the studied final state is composed of2 e± and at least 4 jets. We observe in Fig.7.18 that theD∅ Collaboration is expected to searchfor squarks of mass up to575 GeV and gluinos of mass up to520 GeV. In the case of a singledominant6Rp coupling of typeλ′2jk, it was shown in [467, 470] that the analysis of the dimuonplus four jets signature leads to the expectation that squarks of mass up to640 GeV and gluinosof mass up to560 GeV will be tested by theD∅ Collaboration during the Tevatron Run II.

7.4.3 Effects of Bilinear 6Rp Interactions

Search for spontaneous6Rp violation has been performed at LEP 2 by the DELPHI experi-ment [471].

This search is based on the model described in Refs. [63, 64],in which 6Rp breaking isparametrized by effective bilinear termsµiLiHu (see section 2.4 for a theoretical review of thismodel). The most important phenomenological implication of the model is the existence of amassless MajoronJ which is the LSP. Therefore the Majoron enters in the chargino and neu-tralino decays and the branching ratios of these new processes depend on an effective bilinearterm parameter denoted byǫ in Refs. [63, 64, 471]. In the case where theǫ parameter is suf-ficiently high, roughly forǫ > 10 GeV the chargino decay is fully dominated by the Majoron

Figure 7.18:Estimated exclusion contour for Tevatron Run I and II, within the mSUGRA frame-work, in the (m0,m1/2) plane fortanβ =2, A0 = 0 andµ < 0, from the dielectron and four jetschannel. Scenario I corresponds to a background of36±4±6 events (direct scaling from Run I)while scenario II uses the background of15± 1.5± 1.5 events (scaling, but with improvementsin the detector taken into account).

chanel:χ± → τ±J . Therefore, the experimental signature of the chargino pair production inthis scenario is two acoplanar taus and missing momentum from the undetected Majorons. Oneshould note, however, that such values of theǫ parameter are incompatible with the cosmologi-cal bound on stable neutrino masses, and can arise only in thecontext of exotic scenarios witha heavy decaying neutrino (mν ∼ 1 MeV).

The search for spontaneous6Rp violation in DELPHI was performed in the MSSM frame-work [471]. An upper limit at95% CL on the chargino production cross-section of 0.3 pb andon the chargino mass of94 GeV (close to the kinematic limit for 1998 data) has been obtained.

Before closing the discussion on the possible effects of bilinear 6Rp interactions at leptoncolliders, it is interesting to come back to the case of resonant higgs production atµ+µ− col-liders discussed above in the presence of trilinear6Rp interactions. The effects of bilinear termsfrom spontaneously brokenR-parity would lead to invisible signature whenh0 → JJ whereJ stands for the Majoron. Furthermore, possible signatures with missing energies, when forexampleµ+µ− → H0 → h0h0, followed byh0 → JJ for one of theh0 and byh0 → χ+

1 χ−1 for

the otherh0, with the subsequent decayχ+1 → τ+J , deserve further studies.

Effects of bilinear6Rp interactions could be observed already at existing hadron colliders.Current data from Tevatron Run I are already sensitive to these decays and effectively limits thetotal branching ratio of top decays in different channels thant → W+b to approximately 25%.The Tevatron Run II data will enhance the sensitivity for alternative top decays to branchingratios of10−3 − 10−2 depending on the decay mode. If the stop is lighter than the lightestchargino it may decay dominantly throught1 → τ+ + b. By interpreting the stop as a thirdgeneration leptoquark, the exclusion obtained from leptoquark searches can in this case beapplied [472, 473]. This leads to an exclusion of scalar-stop masses below80 − 100 GeV fromthe Run I data. Here too, the Run II data will improve the sensitivity to a wider region of theSUSY parameter space.

7.5 Single Sparticle Production

New 6Rp trilinear interactions enter directly in sparticle production and decays via basic treediagrams as was illustrated in Fig. 2.1. The corresponding complete interaction Lagrangian isdiscussed in detail in section 2.1 and appendix B. The most striking feature of6Rp is to allow forsingle production of supersymmetric particles. For given centre-of-mass collider energies, thisextends the discovery mass reach for supersymmetric matterbeyond that of superpartner pairproduction inR-parity conserving models.

A list of the s-channel processes allowed at lowest order ine+e−, ep andpp collisions isgiven in Table 7.12. TheL-violating (6L) termLLE couples sleptons and leptons (Fig. 2.1a).It allows for resonant production ofν at l+l− colliders and for the direct6Rp-decay of sleptonsl± → l′±ν andν → l+l′−. The 6L termLQD couples squarks to lepton-quark pairs (Fig. 2.1b)and sleptons to quark pairs. It allows for resonant production of q at ep colliders andν or l±

at pp colliders. Direct6Rp-decay of squarks viaq → lq′; νq′′ or sleptons vial± or ν → qq′ aremade possible. TheB-violating (6B) term UDD couples squarks to quark pairs (Fig. 2.1c). Itallows for resonant production ofq atpp colliders and direct6Rp-decay of squarks viaq → q′q′′.

Moreover, as seen in sections 7.3.1 and 7.3.2, there is a gap between the constraints ob-tained via the detection of the displaced vertex and the low-energy experimental constraints on

Resonant Production of Sfermions at Colliders

(lowest-order processes)

Collider Coupling Sfermion Elementary Process

Type

e+e− λ1j1 νµ, ντ l+i l−k → νj i = k = 1 , j = 2, 3

pp, pp λ′ijk νe, νµ, ντ dkdj → νi i, j, k = 1, . . . , 3

e, µ, τ ujdk → liL i, k = 1, . . . , 3 , j = 1, . . . , 2

λ′′ijk d, s, b uidj → dk i, j, k = 1, . . . , 3 , j 6= k

u, c, t dj dk → ui i, j, k = 1, . . . , 3 , j 6= k

ep λ′1jk dR, sR, bR l−1 uj → dkR j = 1, 2

λ′1jk uL, cL, tL l+1 dk → ujL i, j, k = 1, . . . , 3

Table 7.12:Sfermionss-channel resonant production at colliders. Charge conjugate processes(not listed here) are also possible. Realν production at ane+e− collider can only proceed viaλ121 or λ131. In eγ collisions allλijk couplings wherei, j or k is equal to one can be probed.Theeγ collision mode opens new possibilities, such as the single production of theνe via λi11

or the single slepton production. Single squark productionis also possible via aλ′1jk coupling.Realq production at anep collider is possible via any of the nineλ′1jk couplings.

the 6Rp couplings. This domain can be tested through the study of thesingle production of su-persymmetric particles. Indeed, the cross-sections of such reactions are directly proportional toa power of the relevant6Rp coupling constant(s), which allows to determine the valuesof the 6Rp

couplings. Therefore, there exists a complementarity between the displaced vertex analysis andthe study of singly produced sparticles, since these two methods allow to investigate differentranges of values of the6Rp coupling constants.

For values beyondO(10−4), single sparticle production will allow in favorable casestodetermine or constrain specificλ, λ′ or λ′′ couplings as will be discussed in more details insections 7.5.1 and 7.5.2. Otherwise, the presence of such a large 6Rp coupling will becometrivially manifest through the decay of pair produced sparticles.

7.5.1 Single Sparticle Production at Leptonic Colliders

At leptonic colliders resonant (as well as non-resonant) production of single supersymmetricparticles involve theλijk couplings.

Early discussions on severalRp-violating processes ate+e− colliders including single su-persymmetric particles production can be found in [458, 459].

At a e+e− leptonic colliders, the sneutrinosνµ andντ can be produced at resonance throughthe couplingsλ211 andλ311, respectively. The sneutrino may then decay either via an6Rp inter-action [261], for example throughλijk as,νi → ljlk, or via gauge interaction as,νi

L → χ+a l

i, or,νi

L → χ0aν

iL. The sneutrino partial width is given in Eq. (7.1) for the leptonic decay channel and

is given in the following equation for the gauge decay channel [254]:

Γ(νiL → χ+

a li, χ0

aνiL) =

Cg2

16πmνi

L(1 −

m2χ+

a

m2νi

L

)2, (7.10)

whereC = |Va1|2 for the decay into chargino andC = |Na2|2, for the neutralino case, withVa1

andNa2 the mixing matrix elements written in the notations of [25].For reasonable values ofλijk (≤ 0.1) and most of the region of the supersymmetric parameter space, the decay modes ofEq. (7.10) are dominant, if kinematically accessible [474,254]. In the SUGRA parameter space,if mνi

L> 80 GeV, withM2 = 80 GeV,µ = 150 GeV andtan β = 2, the total sneutrino width is

higher than100 MeV which is comparable to or greater than the typical expected experimentalresolutions. The cross-section formula, for the sneutrinoproduction in thes-channel, is thefollowing [254],

σ(e+e− → νiL → X) =

4πs

m2νi

L

Γ(νiL → e+e−)Γ(νi

L → X)

(s−m2νi

L

)2 +m2νi

L

Γ2νi

L

, (7.11)

whereΓ(νiL → X) generally denotes the partial width for the sneutrino decayinto the final state

X. At sneutrino resonance, Eq. (7.11) takes the form,

σ(e+e− → νiL → X) =

4π

m2νi

L

B(νiL → e+e−)B(νi

L → X), (7.12)

whereB(νiL → X) generally denotes the branching ratio for sneutrino decay into a final state

X.

Diagrams for the single sparticle production at leptonic colliders are shown in Fig. 7.19.

The case of resonant production of single supersymmetric particles atµ+µ− colliders re-sembles very much the one ofe+e− colliders described above. The relevant decay widths andcross-section formulae are obtained respectively from Eq.7.10, 7.11 and 7.12 by replacinge+

ande− by µ+ andµ−.

The gauge decays of a resonantly produced sneutrino at leptonic colliders lead to singlechargino or neutralino production, or to the production of alighter slepton in association withan electroweak gauge boson when this is kinematically allowed. Away from the sneutrino reso-nance, other diagrams contribute to the single production of sparticles. Thet-channel exchangeof a slepton can lead to single chargino or neutralino production, and thet- or u-channel ex-change of a lepton allows for single slepton production.

Single chargino and neutralino productions both receive contributions from the resonantsneutrino production ate+e− colliders (see Fig. 7.19a and b). The single production of achargino,e+e− → χ±

a l∓j (via λ1j1), receives a contribution from thes-channel exchange of

a νjL sneutrino and another one from the exchange of aνeL sneutrino in thet-channel (seeFig. 7.19a). The single neutralino production,e+e− → χ0

aνj (via λ1j1), occurs through thes-channelνjL sneutrino exchange and also via the exchange of aeL slepton in thet-channel ora eR slepton in theu-channel (see Fig. 7.19b). The singleχ±

1 (χ01) production rate is reduced in

the higgsino dominated region|µ| ≪ M1,M2 where theχ±1 (χ0

1) is dominated by its higgsinocomponent, compared to the wino dominated domain|µ| ≫ M1,M2 in which theχ±

1 (χ01)

is mainly composed by the higgsino [475]. In addition, the single χ±1 (χ0

1) production cross-section depends weakly on the sign of theµ parameter at large values oftanβ. However, as

(a)(b)

(d)(e)

( )

~m

j

l+jlj ~j ~

l+m l+jlj l+m

~l+jlj ~0

m~lj ~mljl+j

m~0 l+j

lj ~lj

~lmW+~ml+j

lj~lmW+l+j

ljlj lj

l+j ~m Z~m ~m

Zljl+jlj

ljl+jlj

~m

l+jlj

m~0

~mZ

l+jlj ~m

ljFigure 7.19: Diagrams for the single production processes at leptonic colliders, namely,l+j l

−j → χ−l+m (a), l+j l

−j → χ0νm (b), l+j l

−j → l−mLW

+ (c), l+j l−j → νmLZ (d) andl+j l

−j → νmL γ

(e). The circled vertex corresponds to the6Rp interaction, with the coupling constantλmjj , andthe arrows indicate the flow of the lepton number.

tanβ decreases the rates increase (decrease) forsign(µ) > 0 (< 0). This evolution of the rateswith thetan β andsign(µ) parameters is explained by the evolution of theχ±

1 andχ01 masses

in the supersymmetric parameter space [475].

Forλ1j1 = 0.05, 50 GeV < m0 < 150 GeV and50 GeV < M2 < 200 GeV in a SUGRAparameter space, the off-pole values of the cross-sectionsare typically [475] of the order of100 fb (10 fb ) for the single chargino production and of10 fb (1 fb ) for the single neutralinoproduction at

√s = 200 GeV (500 GeV) (see Fig. 7.20).

(c)(b)(a)

(g) (h)(f)(e)

(d)

Figure 7.20:The integrated cross-sections [475] for the processes (a,b,c and d)e+e− → χ−1 l

+j

and (e,f,g and h)e+e− → χ01νj , at a centre-of-mass energy of500 GeV, are shown as a function

of µ for discrete choices of the remaining parameters: (a,e)tan β = 2, m0 = 50 GeV, (b,f)tanβ = 50, m0 = 50 GeV, (c,g) tanβ = 2, m0 = 150 GeV, and (d,h)tanβ = 50, m0 =50 GeV, with λ1j1 = 0.05. The windows conventions are such thattanβ = 2, 50 horizontallyandm0 = 50, 150 GeV vertically. The different curves refer to the value ofM2 of 50 GeV(continuous line),100 GeV (dot-dashed line),150 GeV (dotted line), as indicated at the bottomof the figure.

At the sneutrino resonance, the cross-sections of the single gaugino production are impor-tant: using Eq. (7.12), the rate for the neutralino production in association with a neutrinois of the order of3 103 in units of the QED point cross-section,R = σpt = 4πα2/3s, forM2 = 200 GeV,µ = 80 GeV, tanβ = 2 andλ1j1 = 0.1 at

√s = mνj

L= 120 GeV [254]. The

cross-section for the single chargino production reaches2 10−1 pb at√s = mνj

L= 500 GeV,

for λ1j1 = 0.01 andmχ± = 490 GeV [458, 476]. The Initial State Radiation (ISR) lowersthe single gaugino production cross-section at theν pole but increases greatly the single gaug-ino production rate in the domainmgaugino < mν <

√s. This ISR effect can be observed

in Fig. 7.21 which shows the single charginos and neutralinos productions cross-sections as afunction of the centre-of-mass energy for a given MSSM point[474].

The experimental searches of the single chargino and neutralino productions have been per-formed at the LEP collider at various centre-of-mass energies [477, 478, 479]. The off-pole ef-fects of the single gaugino productions rates are at the limit of observability at the LEP collider

Cross sections for Rpv Sneutrino S-channel Resonance

10-2

10-1

1

10

10 2

10 3

200 250 300 350 400 450 500

√s (GeV)

σ (p

b) e

+e- to

Cha

rgin

o Le

pton

10-2

10-1

1

10

10 2

10 3

200 250 300 350 400 450 500

√s (GeV)

σ (p

b) e

+e- to

Neu

tral

ino

Neu

trin

o

Figure 7.21:Cross sections of the single charginos and neutralinos productions as a function ofthe centre-of-mass energy formν = 300 GeV (full curves) andmν = 450 GeV (dashed curves)with me = 1 TeV, M2 = 250 GeV, µ = −200 GeV, tanβ = 2 andλ1j1 = 0.1. The ratesvalues are calculated by including the ISR effect and by summing over the productions of thedifferentχ±

i andχ0j eigenstates which can all be produced for this MSSM point.

even with the integrated luminosity of LEP 2. Therefore, theexperimental analyses of the sin-gle gaugino productions have excluded values of theλ1j1 couplings smaller than the low-energybounds only at the sneutrino resonance point

√s = mν and, due to the ISR effect, in a range

of typically ∆mν ∼ 50 GeV around theν pole. Finally, for the various sneutrino resonances,the sensitivities on theλ1j1 couplings which have been derived from the LEP data reach valuesof order10−3. The experimental analyses of the single chargino and neutralino productionswill be continued at the future linear collider. Using its polarisation capability as well as thespecific kinematics of the single chargino production allows to reduce the expected backgroundfrom pair productions of supersymmetric particles. As an example, this background reductionallows to improve the sensitivity to theλ121 coupling obtained from theχ±

1 µ∓ production study

at linear colliders [435] for√s = 500 GeV andL = 500 fb−1 which then amounts to values

of the order of10−4 at the sneutrino resonance and can improve the low-energy constraint overa range of∆mν ≈∼ 500 GeV around theν pole [480]. Due to the high luminosities reachedat linear colliders, the off-resonance contributions to the cross-section play an important role inthe singleχ±

1 production analysis.

The slepton and the sneutrino can also be singly produced viathe couplingλ1j1 in the (non-resonant) reactionse+e− → l∓jLW

±, e+e− → νjLZ

0 ande+e− → νjLγ. Those reactions receive

contributions from the exchange of a charged or neutral lepton of the first generation in thet-or u-channel (see Fig. 7.19). The single productions of a sneutrino accompanied by aZ or aW boson also occur through the exchange in thes-channel of aνj

L sneutrino which can notbe produced on-shell. When kinematically allowed, these processes have some rates of order

100 fb at√s = 200 GeV and10 fb at

√s = 500 GeV, forλ1j1 = 0.05 and various masses of

the scalar supersymmetric particles [475].

The production of single gaugino and the non-resonant production of single slepton (eithercharged or neutral) similar to those in Fig 7.21 are relevantatµ+µ− colliders. The only differ-ence stems from the initial states particles which, being muons instead of electrons, allows totest differentλ couplings as given in Table 7.13.

e+e− colliders µ+µ− colliders

coupling final state exchange channel coupling final state exchange channelλ121 χ±

a µ∓ νe t λ212 χ±

a e∓ νµ t

χ±a µ

∓ νµ s χ±a e

∓ νǫ sχ0

b νµ e t+u χ0b νe µ t+u

χ0b νµ e s χ0

b νe µ sµ±W∓ νe t e±W∓ νµ tµ±W∓ νµ s e±W∓ νe sνµZ e t+u νeZ µ t+uνµZ νµ s νeZ νe sνµγ e t+u νeγ µ t+u

λ131 χ±a τ

∓ νe t λ232 χ±a τ

∓ νµ tχ±

a τ∓ ντ s χ±

a τ∓ ντ s

χ0b ντ e t+u χ0

b ντ µ t+uχ0

b ντ τ s χ0b ντ τ s

τ±W∓ νe t τ±W∓ νµ tτ±W∓ ντ s τ±W∓ ντ sντZ e t+u ντZ µ t+uντZ ντ s ντZ ντ sντγ e t+u ντγ µ t+u

Table 7.13: Resonant and non-resonant single production of gauginos and sleptons ate+e−

colliders andµ+µ− colliders. The indicesa and b run as followa = 1, 2 and b = 1, 4. Theχ±

a and theχ0b can further cascade decay through ordinary gauge decays till either theχ±

1 orthe χ0

1 is reached. Theχ±1 can either decay intoW±χ±

1 or via virtual sfermion exchange andthen with theλ coupling involved in the single production. Theχ0

1 can also further decay withtheλ coupling involved in the single production. The sleptons can also decay either directly viatheλ coupling involved in their single production or into leptons and gauginos followed by thegauginos decay via the sameλ coupling. The multilepton final state can then be deduced usingTable 7.2 and Table 7.3.

Leptonic colliders allows also for lepton-photon collisions in which sleptons and squarkscan also be singly produced thus opening additional perspectives for 6Rp coupling studies. Forexample [481] the processese±γ → l±ν, l±ν, involving an on-shell photon radiated from oneof the colliding leptons, allow to probe theλ122, λ123, λ132, λ133 andλ231 6Rp couplings whichare otherwise not involved in the single sparticle productions frome+e− reactions.

The slepton or sneutrino production occurs either via the exchange of a charged lepton in thes-channel or the exchange of a charged slepton or lepton in thet-channel. Since thet-channel

Figure 7.22: Example diagram forsingle squark production in electron-photon collisions. q

e+e+

~q0e q

is dominant andml >> ml, the slepton production is about two order of magnitude lessthanthe sneutrino production which isσ(e+e− → νjeτ) = 300 fb at

√s = 500 GeV .

In lepton-photon collisions, single squark production occurs viaλ′ couplings as shown forexample in Fig. 7.22 foreγ collisions. When the produced squark directly decays viaλ′ into alepton and a quark the final state consists of one hard mono-jet with one well isolated energeticelectron, and eventually a soft jet in the forward region of the detector in the case where theinitial electron which scatters the quasi real photon escapes detection.

7.5.2 Single Sparticle Production at Lepton-Hadron Colliders

An lp collider provides both leptonic and baryonic quantum numbers in the initial state and isthus ideally suited for6Rp SUSY searches involvingλ′ijk. Among the twenty-seven possibleλ′ijkcouplings, each of the nine couplings withi = 1 can lead to direct squark resonant productionthroughe-q fusion at anep collider such as HERA. The phenomenology of such processeswas first investigated theoretically in Refs. [482, 483, 484, 485]. Search strategies taking intoaccount in general the indirect6Rp squark decay modes were discussed in Refs. [485, 486, 487].

The production processes are listed in Table 7.14 in the caseof an incidente+ beam.

Table 7.14: Direct resonant produc-tion of squarks at anep collider viaa 6Rp λ′1jk coupling. The processesare listed for an incidente+ beam.Charge conjugate processes are ac-cessible for an incidente− beam.

λ′1jk production process

111 e+ + u → ¯dR e+ + d→ uL

112 e+ + u→ ¯sR e+ + s→ uL

113 e+ + u→ ¯bR e+ + b→ uL

121 e+ + c→ ¯dR e+ + d→ cL

122 e+ + c→ ¯sR e+ + s→ cL

123 e+ + c→ ¯bR e+ + b→ cL

131 e+ + t→ ¯dR e+ + d→ tL

132 e+ + t→ ¯sR e+ + s→ tL

133 e+ + t→ ¯bR e+ + b→ tL

In e+p collisions, the production ofujL squarks of thejth generation viaλ′1j1 is especially

interesting as it involves a valenced quark of the incident proton. In contrast, fore−p collisionswhere charge conjugate processes are accessible, theλ′11k couplings become of special interest

Figure 7.23: Squark productioncross-sections ine±p collisions calcu-lated [487] for a couplingλ′ = 0.1and a collider centre-of-mass energyof

√sep = 300 GeV .

~ ~

~

~–

~

~–

as they allow for the production, involving a valenceu quark, ofdkR squarks of thekth genera-

tion. As an illustration, the Fig. 7.23 shows the productioncross-sections ine±p collisions forthe “up”-like squarksuj

L via λ′1j1 (j = 1 . . . 3) compared to that for the “down”-like squarks¯dkR

via λ′11k (k = 1 . . . 3)). The cross-sections are calculated [487] for coupling values ofλ′ = 0.1and for an available centre-of-mass energy of

√sep of 300 GeV characteristic of the HERA

collider. By gauge symmetry, onlyuL-like or dR-like squarks (or their charge conjugates) canbe produced inep collisions. Since superpartners of left- and right-handedfermions may havedifferent allowed or dominant decay modes, the dichotomy between the resonant production ofuL-like squarks ine+p collisions and that ofdR-like squarks ine−p collisions implies that thedetailed analysis will differ betweene− ande+ incident beams.

In the case of a direct6Rp decay through aλ′ coupling, the squarks which have been reso-nantly produced inep collisions behave as leptoquarks [488]. TheuL may couple to ane+ + dpair via a Yukawa couplingλ′111 in a way similar to the coupling of the first generationS1/2,L

leptoquark of charge|Qem| = 2/3. Via the same coupling, thedR couples toe− + u or νe + dpairs, thus behaving like the first-generationS0,L leptoquark of charge|Qem| = 1/3. As a gen-eral consequence, it is possible to translate constraints on theλ couplings of leptoquarks intoconstraints on theλ′1jk couplings of squarks inRp-violating supersymmetry. For real squarkproduction, this translation is limited to coupling valuesλ′>∼

√4πα. For smaller values, the

branching ratio into leptoquark-like final states rapidly drops as squarks will prefer indirect6Rp

decays. Such a re-interpretation of leptoquark constraints from early HERA data has been per-formed in Refs. [489, 490, 491]. For virtual squark exchangein the case whereMq ≫ √

sep,constraints can be established via four-fermion leptoquark-like contact interaction analysis aswill be discussed in section 7.6.1.

In the case of indirect6Rp decays, the squarks in a first stage decay through gauge couplingsinto a quark and a gaugino-higgsino (χ0, χ+) or, if Mg ≪ Mq, into a quark and a gluino.Such squark decays involving6Rp couplings were discussed in detail in section 7.3. Here again,at anep collider, the dichotomy between the production ofuL and dR will have importantphenomenological consequences. While theuL might decay viauL → uχ0

l or χ+m, the dR

mainly decays viadR → dχ0l , thebR decay into a chargino being also possible via the higgsino

component of the latter.

Figure 7.24: Cross-sections forχ0

production in e±p collisions fromt-channel selectron exchange calcu-lated [492] for a couplingλ′1j1 = 0.5and for collider centre-of-mass energyof

√sep = 300 GeV .

M(selectron) (GeV)σ

(pb

)

e- u → d χ0, M ( χ0 ) = 55 GeV

e+ d → u χ0, M ( χ0 ) = 55 GeV

e- u → d χ0, M ( χ0 ) = 105 GeV

e+ d → u χ0, M ( χ0 ) = 105 GeV10

-3

10-2

10-1

1

10

60 80 100 120 140 160 180 200 220 240

Depending on the mixing parameters in the neutralino and chargino sectors, the dominatingevent topologies to be expected might depend on whether the collider is running ine+p or ine−p mode. Decays ofχ0 andχ+ mass eigenstates were discussed in detail in section 7.3.

Detailed dicussion on the event topologies expected at anep collider for single squark pro-duction in the presence of6Rp couplings can be found in Refs. [485, 486, 487].

Real or virtual squark exchange in thes-channel contributes to the single production ofneutralinos or charginos. These can also be singly producedvia 6Rp interactions in lowest or-der processes involving sleptons or sneutrinos. Theχ0 can be produced viat-channel sleptonexchange or viau-channel squark exchange. Theχ+ can be produced viat-channel sneutrinoexchange.

The Fig. 7.24 shows the neutralino production cross-sections ine±p collisions expected for acoupling valueλ′11k = 0.5 and for an available centre-of-mass energy of

√sep of 300 GeV char-

acteristic of the HERA collider. The cross-sections are calculated [492] for selectron exchangeonly in the framework of the MSSM augmented by a single non-vanishingλ′ coupling, for twovalues ofMχo and fortanβ = 1. Such cross-sections could be expected in caseMe ≪ Mq.When boths-channelq exchange andt-channele contribute, the interference between thesecannot be neglected. For example at HERA II, constructive interference between squark andselectron exchange processes could contribute [492] up to20% of the totalχo production

Searches for single squark production have been performed at HERAI under the hypothesisof a single dominantλ′1jk coupling. The constraints obtained [493, 494, 495] by the H1exper-iment are shown in Fig. 7.25. Similar results were obtained [496] by the ZEUS experiment.All possible event topologies (multijets and lepton and/ormissing energy) resulting from thedirect or indirect sparticle decays involving such coupling have been considered in the analysis.The HERAI results are compared to the best existing indirect bounds [259] from low-energyexperiments. Theλ′111 coupling is seen to be very severely constrained by the non-observationof neutrinoless double-beta decay. The most stringent low-energy constraints onλ′121 andλ′131come from atomic-parity violation measurements. From these HERA I results, it can be inferedthat HERA II could offer a sensitivity reach beyond the domain excluded by indirect constraintsfor 2nd and 3rd generation squarks.

10-2

10-1

1

100 125 150 175 200 225 250 275

Unconstrained MSSM, j=1,2

tan β = 6-300 < µ < 300 GeV70 < M2 < 350 GeV

MLSP > 30 GeV imposed

(a)

EXCLUDED

EXCLUDED IN P

ART OF

PARAMETER SPACE

λ, 111

(ββ0ν)

λ, 121

(APV)

M squark (GeV)

λ, 1j1

H1

10-2

10-1

1

100 125 150 175 200 225 250 275

Unconstrained MSSM, j=3

tan β = 6-300 < µ < 300 GeV70 < M2 < 350 GeV

MLSP > 30 GeV imposed

(b)

EXCLUDED

EXCLUDED IN P

ART OF

PARAMETER SPACE

λ, 131

(APV)

H1

M squark (GeV)λ, 13

1(a) (b)

Figure 7.25: Upper Limits (95% CL) on a) the couplingλ′1j1 with j = 1, 2 and b)λ′131 as a func-tion of the squark mass fortan β = 6 in the unconstrained MSSM. The limits are obtained froma scan of theµ andM2 parameters within−300 < µ < 300 GeV and70 < M2 < 350 GeV andimposing that the lightest sparticle (LSP) has a massMLSP above30 GeV. The dark shadedarea is excluded for any parameter values. The light shaded area is excluded for some param-eters values. The dashed-dotted curve is the indirect upperbound [259] onλ′111 derived fromconstraints on neutrinoless double-beta decays [497, 498]. The dashed curves are the indirectupper bounds [259] onλ′1j1 derived from constraints on atomic-parity violation [499].

The HERA results analysed in the framework of6Rp mSUGRA are shown in Fig. 7.26 andcompared to complementary6Rp SUSY searches made at LEP 2 and Tevatron Run I colliders.The searches were performed here also under the hypothesis of a single dominantλ′1jk cou-pling. The results are presented as excluded domains in the parameter space of the model. Theconstraints from the D∅ [468] experiment at the Tevatron were obtained from a searchfor q pairproduction through gauge couplings. The D∅ analysis profits in this framework from an ap-proximate mass degeneracy implicitly extended to fiveq flavours (d,u,s,c,b) and both (partners)chiralities (qL,qR). The 6Rp couplings are assumed to be significantly smaller than the gaugecouplings, so that direct6Rp decays are suppressed and each squark rather decays back into aquark and the LSP through gauge couplings. The only effect ofthe 6Rp couplings is then tomake the LSP unstable. The D∅ analysis is further restricted to6Rp coupling values>∼10−3 toguarantee a negligible decay length of the LSP. In the domains considered, the LSP is almostalways the lightest neutralinoχ0

1 . The χ01 decays viaλ′1jk into a first-generation lepton (e or

νe) and two quarks. The analysis is restricted toj = 1, 2 andk = 1, 2, 3 and, in practice, theD∅ selection of event candidates requires like-sign di-electrons accompanied by multiple jets.The constraints from the L3 experiment at LEP were obtained from a search for pair productionthrough gauge couplings of neutralinos (e+e− → χ0

mχ0n with m = 1, 2 andn = 1, . . . , 4),

charginos (e+e− → χ+1 χ

−1 ) and scalar leptons (e+e− → l+R l

−R, νν). The 6Rp couplings contribute

here again in opening new decay modes for the sparticles. A negligible decay length of thesparticles through these decay modes is ensured by restricting the analysis to coupling values>∼10−5. All possible event topologies (multijets and lepton and/or missing energy) resultingfrom the direct or indirect sparticle decays involving theλ′ijk couplings have been considered in

0

20

40

60

80

100

120

140

160

180

0 50 100 150 200 250 300 350

not allowed

m(t∼1 )=200 GeV

m(t∼1)=265 GeV

m(u∼)=260 GeV

m(u ∼

)=275 GeV

m0 (GeV)

m1/

2 (G

eV)

D0 (j=1,2)

L3 limit

(a)µ < 0, A 0 = 0

mSUGRA, λ, 1j1 = 0.3, tanβ=2

MLSP < Mb

H1

excluded for j=3

excluded for j=1,2

m0 (GeV)

m1/

2 (G

eV)

0

20

40

60

80

100

120

140

160

180

0 50 100 150 200 250 300 350

m(u∼)=260 GeV

m(u ∼)=275 GeV

D0 (j=1,2)

not allowed

m(t∼

1 )=240 GeV

m(t∼

1 )=270 GeV

H1

(b)µ < 0, A 0 = 0

mSUGRA, λ, 1j1 = 0.3, tanβ=6

MLSP < Mb

excluded for j=3

excluded for j=1,2

m0 (GeV)

m1/

2 (G

eV)

Figure 7.26:Constraints on squark production viaλ′1j1 in Rp-violating SUSY in the parameterspace of Minimal Supergravity. Excluded domains obtained by the H1 [495] (shaded area) andD∅ (dotted curves) experiments are shown for a)tanβ = 2 and b)tanβ = 6. In a) the limitobtained by the L3 experiment at LEP 2 is also shown as the upper dotted curve. Contours ofconstant values for the light stop mass are drawn as dashed curves. The shaded region marked“not allowed” corresponds to points in the parameter space where the radiative electroweaksymmetry breaking does not occur (or which lead to unphysical Higgs or sfermion masses). Alsomarked as “not allowed” in this particular analysis are cases where the LSP is the sneutrino.

the L3 analysis.

For the set of mSUGRA parameters withtan β = 2, the Tevatron experiment excludessquarks with massesMq < 243 GeV (95 % CL) for any value ofMg and a finite value (>∼10−3)of λ′1jk with j = 1, 2 andk = 1, 2, 3. The sensitivity decreases for the parameter set with alarger value oftanβ due in part to a decrease of the photino component of the LSP, which im-plies a decrease of the branching fraction of the LSP into electrons, and in part to a softening ofthe final-state particles for lighter charginos and neutralinos. The best sensitivity attan β = 2is offered by LEP for any of theλ′ijk couplings. HERA offers a best complementary sensitiv-ity to the couplingλ′131 which allows for resonant stop production via positron-quark fusione+d → t1. The HERA constraints (shown here for a coupling of electromagnetic strength, i.e.λ′131 = 0.3) extend beyond LEP and Tevatron constraints towards largertan β.

7.5.3 Single Sparticle Production at Hadron-Hadron Colliders

The SUSY particles can be produced as resonances at hadron colliders through the6Rp inter-actions. This is particularly attractive as hadron colliders allow to probe for resonances over awide mass range given the continuous energy distribution ofthe colliding partons. If a single6Rp coupling is dominant, the resonant SUSY particle may decay through the same couplinginvolved in its production, giving a two quark final state at the partonic level. However, it isalso possible that the decay of the resonant SUSY particle ismainly due to gauge interactions,giving rise to a cascade decay. A review focusing on TevatronRun-II can be found in [467].

• Single sparticle production viaλ′

First, a resonant sneutrino can be produced indd annihilations through the constantλ′ijk.The associated formula can be written as follows [500]:

σ(dkdj → νi → X1X2) =4

9

s

m2νi

πΓdkdjΓf

(s−m2νi)2 +m2

νiΓ2νi

, (7.13)

whereΓdk dj, andΓf are the partial width of the channels,νi → dkdj, and, νi → X1X2,

respectively,Γνi is the total width of the sneutrino,mνi is the sneutrino mass ands is the squareof the parton centre-of-mass energy. The factor1/9 in front is from matching the initial colours,andΓdk dj

is given by,

Γdk dj=

3

4αλ′

ijkmνi , (7.14)

whereαλ′ijk

= λ′2ijk/4π. To compute the rate at app collider, the usual formalism of the partonmodel of hadrons can be used [501]:

σ(pp→ νi → X1X2) =∑

j,k

∫ 1

τ0

dτ

τ(1

s

dLjk

dτ) s σ(dkdj → νi → X1X2), (7.15)

wheres is the centre-of-mass energy squared,τ0 is given byτ0 = (MX1 + MX2)2/s andτ is

defined byτ = s/s = x1x2, x1, x2 denoting the longitudinal momentum fractions of the initialpartonsj andk, respectively. The quantitydLjk/dτ is the parton luminosity defined by,

dLjk

dτ=

∫ 1

τ

dx1

x1

[f pj (x1)f

pk (τ/x1) + f p

j (x1)fpk (τ/x1)], (7.16)

where the parton distributionfhj (x1) denotes the probability of finding a partonj with mo-

mentum fractionx1 inside a hadron h, and generally depends on the Bjorken variable,Q2, thesquare of the characteristic energy scale of the process under consideration. In order to seethe effects of the parton distributions on the resonant sneutrino production, some values ofthe rates are given in the following [502]: For instance, with an initial statedd for the hardprocess, the cross-section value isσ(pp → νi) = 8.5 nb for a sneutrino mass of100 GeVand a coupling,λ′i11 = 1 at

√s = 2 TeV. For identical values of the parameters and of the

centre-of-mass energy, the cross-section isσ(pp → νi) = 4 nb with an initial state,ds, andσ(pp → νi) = 0.8 nb with an initial state,db. The charged slepton can also be producedas a resonance at hadron colliders from an initial stateujdk and via the constantλ′ijk. The

cross-section value isσ(pp → liL) = 2 nb formliL= 100 GeV,

√s = 2 TeV andλ′i11 = 1

([502, 503]).

The single production of SUSY particles viaλ′ occurring throughtwo-to-two-body pro-cesses, offers the opportunity to study the parameter spaceof the 6Rp models with a quite highsensitivity at hadron colliders.

In Fig. 7.27, all the single superpartner productions whichoccur viaλ′ijk throughtwo-to-two-body processes at hadron colliders and receive a contribution from a resonant SUSY particleproduction are presented [504]. The spin summed amplitudesof those reactions including thehiggsino contributions have been calculated in [504]. In a SUGRA model, the rates of thereactions presented in Fig. 7.27 depend mainly on them0 andM2 parameters.

In Fig. 7.28, the variations of theσ(pp → χ+1,2µ

−) cross-sections withm0 for fixed valuesof M2, µ andtanβ and various6Rp couplings of the typeλ′2jk at Tevatron Run II in a SUGRA

(a)(b)

( )

(d)

~dk

~dk

~dj

~ujdk

~l+i

ukuj

i

l+i

dkdj

~l+i

djdk ~uj

~l+i~+i

~dji~+

ujdk~

idjuk

dkdj

~dk~0i dj

dk ~0i~0l+i

~0dkdjdkuj

~0

dkuj l+i

~0~dk uj

dk uj

~ i

~l+i

~ i

~l+iFigure 7.27: Diagrams for the four single superpartner production reactions involvingλ′ijk athadron colliders which receive a contribution from a resonant supersymmetric particle produc-tion. Theλ′ijk coupling constant is symbolised by a small circle and the arrows indicate the flowof the lepton or baryon number.

model are shown [504]. The6Rp couplings giving the highest cross-sections have been consid-ered. Theσ(pp→ χ+

1,2µ−) rates decrease whenm0 increases since then the sneutrino becomes

heavier and more energetic initial partons are required in order to produce the resonant sneu-trino. A decrease of the cross-sections also occurs at smallvalues ofm0, the reason being thatwhenm0 approachesM2 the ν mass is getting closer to theχ± masses so that the phase spacefactors associated to the decaysνµ → χ±

1,2µ∓ decrease. The differences between theχ+

1 µ−

production rates occurring via the variousλ′2jk couplings are explained by the different partonicluminosities. Indeed, as shown in Fig. 7.27 the hard processassociated to theχ+

1 µ− production

occurring through theλ′2jk coupling constant has a partonic initial stateqjqk. The χ+1 µ

− pro-duction via theλ′211 coupling has first generation quarks in the initial state which provide themaximum partonic luminosity.

σ(p p– → χ

~+ µ-)

σ(χ~

1+ µ-)

σ(χ~

2+ µ-)

m 0 (GeV)

σ (p

b)

λ’ 211=0.09

λ’ 231=0.22

λ’ 211=0.09

λ’ 221=0.18

λ’ 212=0.09

M 2 = 200 GeV

µ = -200 GeV

tan β = 1.5

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

100 200 300 400 500 600

Figure 7.28: Cross-sections (inpb) of the single chargino productionspp → χ+1,2µ

− as afunction of them0 parameter (in GeV). The centre-of-mass energy is taken at

√s = 2 TeV

and the considered set of parameters is:λ′211 = 0.09, M2 = 200 GeV, tanβ = 1.5 andµ = −200 GeV. The rates for theχ+

1 production via the6Rp couplingsλ′212 = 0.09, λ′221 = 0.18and λ′231 = 0.22 are also given. The chosen values of the6Rp couplings correspond to thelow-energy limits for a squark mass of100 GeV [258].

In Fig. 7.29, the variations of the rates of the reactionspp → χ−1 ν, pp → χ0

1,2µ− and

pp → χ01ν with them0 parameter in a SUGRA model are shown [504]. From this figure one

can see that the single neutralino productions do not decrease at smallm0 values in contrastwith the single chargino productions (see also Fig. 7.28). This is due to the fact that in SUGRAscenarios theχ0

1 and lL (lL = l±L , νL) masses are never close enough to induce a significantdecrease of the phase space factor associated to the decaylL → χ0

1l (l = l±, ν). By analysingFig. 7.28 and Fig. 7.29, one can also see that theχ−ν (χ0µ−) production rate is larger thanthe χ+µ− (χ0ν) production rate. The explanation is that inpp collisions the initial states ofthe resonant charged slepton productionujdk, ujdk have higher partonic luminosities than theinitial states of the resonant sneutrino productiondjdk, djdk.

The neutralino production in association with a charged lepton viaλ′ (see Fig. 7.27d) is aninteresting case at Tevatron [502]. The topology of the events consists of an isolated leptonin one hemisphere balanced by a lepton plus two jets in the other hemisphere, coming fromthe neutralino decay viaλ′. The Standard Model background arising from the productionoftwo jets plus aZ0, decaying into two leptons, has a cross-section of order10−3 nb [501], andcan be greatly reduced by excluding lepton pairs with an invariant mass equal to theZ0 mass.The other source of Standard Model background, which is the Drell-Yan mechanism into2leptons accompanied by 2 jets, is suppressed by a factor,10−6/αλ. Moreover, the signal can beenhanced by looking at the invariant mass of the 2 jets and thelepton in the same hemisphere,

σ(χ~

10 µ-)

σ(χ~

20 µ-)

σ(χ~

10 νµ)

σ(χ~

1- νµ)

m 0 (GeV)

σ (p

b)

λ’ 211=0.09

M 2 = 200 GeV

µ = -200 GeV

tan β = 1.5

0

5

10

15

20

25

100 150 200 250 300 350 400 450 500

Figure 7.29:Cross-sections (inpb) of theχ−1 ν, χ0

1,2µ− andχ0

1ν productions at Tevatron Run IIas a function of them0 parameter (inGeV). The centre-of-mass energy is taken at

√s = 2 TeV

and the considered set of parameters is:λ′211 = 0.09, M2 = 200 GeV, tanβ = 1.5 andµ = −200 GeV.

which should peak around the neutralino mass.

The single production viaλ′ of the neutralino together with a charged lepton can also gen-erate clean signatures free from large Standard Model background, containing two like-signcharged leptons [255, 467, 505, 506, 507, 504, 508, 509]. As amatter of fact, the neutralinohas a decay channel into a lepton and two jets through the coupling λ′ijk and due to its Ma-jorana nature, the neutralino decays to the charge conjugate final states with equal probability:Γ(χ0 → liujdk) = Γ(χ0 → liujdk). Therefore, the lepton coming from the production can havethe same sign than the one coming from the neutralino decay. Sinceλ′111 has a strong indirectbound, it is interesting to consider the coupling constantλ′211, which corresponds to the dimuonsproduction with an initial stateud or ud (see Fig. 7.27d) composed of first generation quarks.The analysis of the like sign di-taus signature generated bythe χ0τ± production through theλ′311 coupling (see Fig. 7.27d) suffers from a reduction of the selection efficiency due to the tau-lepton decay. Besides, the study of theχ0

1µ± production viaλ′211 in a scenario where theχ0

1 isthe LSP is particularly attractive since then theχ0

1 can only undergo6Rp decays. It was found thatin a SUGRA model, such a study can probe values of theλ′211 coupling at the5σ discovery leveldown to2 10−3 (10−2) for a muon-slepton mass ofmµL

= 100 GeV (mµL= 300 GeV) with

M2 = 100 GeV,2 < tan β < 10 and|µ| < 103 GeV at Tevatron Run II assuming a luminosityof L = 2fb−1 [467, 505], and down to2 10−3 (10−2) for mµL

= 223 GeV (mµL= 540 GeV)

with m1/2 = 300 GeV,A = 300 GeV, tan β = 2 andsign(µ) > 0 at the LHC assuming aluminosity ofL = 10fb−1 [506, 507]. It was also shown in [504], by using a detector responsesimulation, that the study of the single LSP production at Tevatron Run IIpp→ χ0

1µ± would al-

λ′211 λ′212 λ′213 λ′221 λ′222 λ′223 λ′231 λ′232 λ′2330.01 0.02 0.02 0.02 0.03 0.05 0.03 0.06 0.09

Table 7.15:Sensitivities on theλ′2jk coupling constants fortan β=1.5,M1 = 100 GeV, M2 =200 GeV, µ = −500 GeV, mq = ml = 300 GeV andmν = 400 GeV, assuming an integratedluminosity ofL = 30fb−1.

low to probem1/2 values up to∼ 850 GeV andm0 values up to∼ 550 GeV at the5σ discoverylevel, in a SUGRA scenario wheresign(µ) < 0,A = 0, tanβ = 1.5 λ′211 = 0.05 and assuminga luminosity ofL = 2fb−1. In the case where one considers the Standard Model backgroundcombined with the background generated by the superpartnerpair production [509], the singleχ0

1 production study based on the like sign dilepton signature analysis still allows to test largeranges of the SUGRA parameter space at Tevatron Run II or LHC,for λ′211 values of the sameorder of its present limit.

Besides, the like sign dilepton signature analysis based ontheχ01µ

± production (see Fig. 7.27d)allows theχ0

1 andµ±L mass reconstructions since the decay chainµ±

L → χ01µ

±, χ01 → µ±ud can

be fully reconstructed [504, 509]. Based on the like sign dilepton signature analysis, theχ01 (µ±

L )mass can be measured with a statistical error of∼ 11(20) GeV at the Tevatron Run II [504].

The singleχ±1 production in association with a charged lepton (see Fig. 7.27a) is another in-

teresting reaction at hadron colliders. In a scenario whereχ01 is the LSP andmν , ml, mq > mχ±

1,

this single production receives a contribution from the resonant sneutrino production and thesingly produced chargino decays into quarks and leptons with branching ratios respectivelyof B(χ±

1 → χ01dpup′) ≈ 70% (p = 1, 2, 3; p′ = 1, 2) andB(χ±

1 → χ01l

±p νp) ≈ 30% due to the

colour factor. The neutralino decays viaλ′ijk either into a lepton as,χ01 → liujdk, liujdk, or into

a neutrino as,χ01 → νidjdk, νidjdk. Hence, if both theχ±

1 andχ01 decay into charged leptons,

theχ±1 l

∓i production can lead to the three charged leptons signature which has a small Standard

Model background at hadron colliders [510, 507, 504, 508, 511]. The study of the three lep-tons signature generated by theχ±

1 µ∓ production via theλ′211 coupling constant is particularly

interesting for the same reasons as above. The sensitivity to theλ′211 coupling obtained fromthis study at Tevatron Run II would reach a maximum value of∼ 0.04 for m0 ≈ 200 GeV ina SUGRA model withM2 = 200 GeV, sign(µ) < 0, A = 0 and tanβ = 1.5, assuming aluminosity ofL = 2fb−1 [504]. The sensitivities on theλ′2jk couplings that can be obtainedfrom the trilepton analysis based on theχ±

1 µ∓ production at the LHC for a given set of MSSM

parameters are shown in Table 7.15 [510]. For each of theλ′2jk couplings the sensitivity hasbeen obtained assuming that the considered coupling was thesingle dominant one. The differ-ence between the various results presented in this table is due to the fact that eachλ′2jk couplinginvolves a specific initial state (see Fig. 7.27a) with its own parton density. Besides, all the sen-sitivities shown in Table 7.15 improve greatly the present low-energy constraints. The trileptonanalysis based on theχ±

1 e∓ (χ±

1 τ∓) production would allow to test theλ′1jk (λ′3jk) couplings

constants. While the sensitivities obtained on theλ′1jk couplings are expected to be of the sameorder of those presented in Table 7.15, the sensitivities ontheλ′3jk couplings should be weakerdue to the tau-lepton decay. The results presented in Table 7.15 illustrate the fact that even ifsome studies on the single superpartner production viaλ′ at hadron colliders (see Fig. 7.27)only concern theλ′211 coupling constant, the analysis of a given single superpartner productionat Tevatron or LHC allows to probe manyλ′ijk coupling constants down to values smaller thanthe corresponding limits from low-energy data.

Besides, the three leptons final state study based on theχ±1 µ

∓ production (see Fig. 7.27a)

allows to reconstruct theχ01, χ

±1 and ν masses [510, 507, 504, 508, 511]. Indeed, the decay

chainνi → χ±1 l

∓i , χ±

1 → χ01l

±p νp, χ0

1 → l±i ujdk can be fully reconstructed since the producedcharged leptons can be identified thanks to their flavours andsigns. Based on the trilepton sig-nature analysis, theχ0

1 mass can be measured with a statistical error of∼ 9 GeV at the TevatronRun II [508, 504] and of∼ 100 MeV at the LHC [511, 507, 510]. Furthermore, the width of theGaussian shape of the invariant mass distribution associated to theχ±

1 (ν) mass is of∼ 6 GeV(∼ 10 GeV) at the LHC for the MSSM point defined byM1 = 75 GeV, M2 = 150 GeV,µ = −200 GeV,mf = 300 GeV andA = 0 [511, 507, 510].Let us make a general remark concerning the superpartner mass reconstructions based on thesingle superpartner production studies at hadron colliders. The combinatorial background asso-ciated to these mass reconstructions is smaller than in the mass reconstructions analyses basedon the supersymmetric particle pair production since in thesingle superpartner production stud-ies only one cascade decay must be reconstructed.

At hadron colliders, some supersymmetric particles can also be singly produced throughtwo-to-two-body processes which generally do not receive contribution from resonant super-partner production [504]. Some single productions of squark (slepton) in association with agauge boson can occur through the exchange of a quark in thet-channel or a squark (slepton)in the s-channel viaλ′′ (λ′). From an initial stateg q, a squark (slepton) can also be singlyproduced together with a quark (lepton) with a coupling constantλ′′ (λ′) via the exchange of aquark or a squark in thet-channel, and of a quark in thes-channel. Finally, a gluino can be pro-duced in association with a lepton (quark) through a coupling constantλ′ (λ′′) via the exchangeof a squark in thet-channel (and in thes-channel).

Let us enumerate the single scalar particle and gluino productions occurring via thetwo-to-two-body processes which involve theλ′ijk coupling constants [504] (one must also add thecharge conjugate processes):

• The gluino productionujdk → gli via the exchange of aujL (dkR) squark in thet-(u-)channel.

• The squark productiondjg → d∗kRνi via the exchange of adkR squark (dj quark) in thet-( s- ) channel.

• The squark productionujg → d∗kRli via the exchange of adkR squark (uj quark) in thet-(s-) channel.

• The squark productiondkg → djLνi via the exchange of adjL squark (dk quark) in thet-(s-) channel.

• The squark productiondkg → ujLli via the exchange of aujL squark (dk quark) in thet-(s-) channel.

• The sneutrino productiondjdk → ZνiL via the exchange of adk or dj quark (νiL sneu-trino) in thet-(s-) channel.

• The charged slepton productionujdk → ZliL via the exchange of adk or uj quark (liLslepton) in thet-(s-) channel.

• The sneutrino productionujdk → W−νiL via the exchange of adj quark (liL slepton) inthet-(s-) channel.

• The charged slepton productiondjdk → W+liL via the exchange of auj quark (νiL

sneutrino) in thet-(s-) channel.

One must also add to this list thegdk → tli reaction which occurs via theλ′i3k coupling throughthe exchange of adk quark in thes-channel and a top quark in thet-channel [512].

Among these single productions only theujdk → W−νiL and djdk → W+liL reactionscan receive a contribution from a resonant sparticle production. However, in most of the SUSYmodels, as for example the supergravity or the gauge mediated models, the mass differencebetween the so called left-handed charged slepton and the left-handed sneutrino is due to the D-terms so that it is fixed by the relationm2

l±L−m2

νL= cos 2βM2

W [513] and thus it does not exceed

theW boson mass. In scenarios with largetanβ values, a scalar particle of the third generationproduced as a resonance can generally decay into theW boson due to the large mixing in thethird family sfermions sector. For instance, in the SUGRA model with a largetanβ a tau-sneutrino produced as a resonance indkdj → ντ throughλ′3jk can decay asντ → W±τ∓1 , τ∓1being the lightest stau.

Similarly, the single scalar particle and gluino productions occurring via thetwo-to-two-body processes which involve theλ′′ijk coupling constants cannot receive a contribution from aresonant scalar particle production for lowtan β. Indeed, the only reactions among thesetwo-to-two-body processes which can receive such a contribution are ofthe typeqq → q → qW . Inthis type of reaction, the squark produced in thes-channel, is produced viaλ′′ijk and is thus either

a Right squarkqR, which does not couple to theW boson, or the squarkst1,2, b1,2. However, thesingle gluino productions occuring via thetwo-to-two-body processes which involve theλ′′ijkcoupling constants can receive a contribution from a resonant scalar particle production.

Therefore, the single scalar particle and gluino productions occurring via thetwo-to-two-body processes are generally non resonant single superpartner productions, as already men-tioned at the beginning of this section. These non resonant single superpartner productionshave typically smaller cross-sections than the reactions receiving a contribution from a resonantsuperpartner production. For instance, withmq = 250 GeV, the cross-sectionσ(pp → uLµ)is of order∼ 10−3pb at a centre-of-mass energy of

√s = 2 TeV, assuming an6Rp coupling of

λ′211 = 0.09 [504]. However, the non resonant single productions can lead to interesting signa-tures. For instance, the production,qq → fW leads to the final state2l + 2j + W for a nonvanishing6Rp coupling constantλ′ and to the signature4j +W for aλ′′ [255]. Furthermore, thenon resonant single productions are interesting as their cross-section involves only few SUSYparameters, namely one or two scalar superpartner(s) mass(es) and one6Rp coupling constant.

TheD∅ collaboration searched for single slepton production through theλ′211 coupling inthe two muons and two hadron jets channel [514]. In the absence of any evidence for an excessof events with respect to expectation from Standard Model processes, bounds on supergravityparametersm1/2, mo have been set. Sneutrinos and smuons masses up to280 GeV have beenexcluded.

• Single sparticle production viaλ′′

TheB-violating couplingsλ′′ijk allows for resonant production of squarks at hadron collid-

ers. Either a squarkui or dk can be produced at the resonance from an initial state,djdk or uidj,respectively. Formdk

R= 100 GeV,

√s = 2 TeV andλ′′11k = 1, the rate of the down squark

production at the Tevatron isσ(pp→ dkR) = 25 nb [502].

dj(d)( )~dj ~dj

dkuidj

ui; dk

ui

~0; ~g

(b)(a)~ui

dk dkdjui

dj~0; ~g ~0; ~g

~0; ~g

dk; ui

~ui

Figure 7.30:Diagrams for the single neutralino production reactions involvingλ′′ijk at hadroncolliders. Theλ′′ijk coupling constant is symbolised by a small circle and the arrows indicatethe flow of the baryon number.

Formt1 = 600 GeV,√s = 2 TeV andλ′′323 = 0.1, the rate of the resonant stop production is

σ(pp → t1) = 10−3 picobarns [515]. Note that this rate is higher than the stop pair productionrate at the same centre-of-mass energy and for the same stop mass, which is of orderσ(pp →t1t1) = 10−6 picobarns.

The single superpartner production can also occur as atwo-to-two-body process, through an6Rp couplingλ′′ and an ordinary gauge interaction vertex: In baryon-number-violating models,any gaugino (including gluino) can be produced in association with a quark, in quark-quarkscattering, by the exchange of a squark in thes-, t- or u-channel.

For example, let us consider the photino and gluino production [502]: The rate values inthe t- andu-channel are,σ(pp → γq) = 2 10−2 nb , and,σ(pp → gq) = 3 10−1 nb , for,mq = mg = mγ = 100 GeV,

√s = 2 TeV andλ′′111 = 1. The photino or gluino which is

produced will then decay into three jets via theλ′′ coupling, resulting in a four jets final state.The corresponding QCD background is strong: It is estimatedto be about10 nb for

√s =

2 TeV [516]. Of course, the ratio signal over background can beenhanced considerably bylooking at the mass distribution of the jets: the QCD 4 jets are produced relatively uncorrelated,while the trijet mass distribution of the signal should peakaround the gaugino mass. However,one of the three jets may be too soft to be measured or jet coalescence may occur, especiallyfor small values of the gaugino mass. The study of this example bring us to the conclusion that,

~uj

~+

(d)( )~dj ~dj

dkuiuj

ui; dk

~+di

~

(b)(a)~ui ~ui

dk dkdjdi

dj

dk; uiFigure 7.31: Diagrams for the single chargino production reactions involving λ′′ijk at hadroncolliders. Theλ′′ijk coupling constant is symbolised by a small circle and the arrows indicatethe flow of the baryon number.

due to high QCD background, the analysis of the single production viaλ′′ remains difficult.

Nevertheless, there are some specific cases where the final state can be clear and free froma large background. For instance, aχ+

1 chargino can be produced viaλ′′3jk through the resonantproduction of a top squark asdjdk → t1 → bχ+

1 , t1 being the lightest top squark, and thendecay into the lightest neutralino plus leptons asχ+

1 → liνiχ01 [515, 467]. Due to the stop

resonance, this reaction can reach high rate values. The cascade decay demands the mass hier-archy,mt1 > mχ+

1> mχ0

1, to be respected, and by consequence is not allowed in all regions

of the supergravity parameter space. Assumingλ′′3jk to be the single dominant6Rp couplingconstant and theχ0

1 = LSP to be lighter than the top quark, theχ01 should then be treated as

a stable particle. Then, the signal for our process would be very clear since it would consistof a tagged b-quark jet, a lepton and missing transverse energy. The Standard Model back-ground for such a signature comes from the single top quark production, viaW g fusion, andthe production of aW gauge boson in association withbb, cc or a jet faking a b-quark jet. Ex-perimental studies lead to the conclusion that values ofλ′′ > 0.03 − 0.2 andλ′′ > 0.01 − 0.03can be excluded at the95% confidence level for,180 GeV < mt1 < 285 GeV, at the Teva-tron Run I (

√s = 1.8 TeV and

∫Ldt = 110 pb−1) and for,180 GeV < mt1 < 325 GeV,

at the Run II of the Tevatron (√s = 2 TeV and

∫Ldt = 2 fb −1), respectively. This result

is based on the leading-order CTEQ-4L parton distribution functions [517] and holds for thenormalisation,λ′′ = λ′′312 = λ′′313 = λ′′323, and for the point of a minimal supergravity model,

~uidjg

dk ( )dk~dj

dkgui (a)

ui ~uidkg

dj (b)dj

Figure 7.32:Diagrams for the resonant production of squarks involvingλ′′ijk at hadron collid-ers. Theλ′′ijk coupling constant is symbolised by a small circle and the arrows indicate the flowof the baryon number.

m1/2 = 150 GeV, A0 = −300 GeV andtanβ = 4. The constraints obtained onλ′′ are strongerthan the present low energy bounds.

Another particularly interesting reaction has been studied in [518]: the single gluino pro-ductiondjdk → tg which can receive a contribution from the resonant stop production via theλ′′3jk coupling. In certain regions of the mSUGRA parameter space,this single gluino produc-tion can reach rates at LHC of order102fb (for λ′′3jk = 10−1) thanks to the contribution comingfrom the resonantt2 production,t2 being the heavier top squark. The interesting point is that inthese mSUGRA domains the branching ratios of the decaysg → tbχ±

1 andg → ttχ01 reach also

significant values thanks to the exchange of the virtualt1 (the lighter top squark) which is thelighter squark and has a mixing angle nearπ/2. By consequence, the processpp→ tg (tg) cansimultaneously have large cross section values at LHC and produce in a significant way a clearsignature containing 3b quarks, at least 2 charged leptons and some missing energy (due to thetop quark decayt → blν). Since the background associated to this final state can be greatlyreduced thanks to the largeb-tagging efficiency available at the LHC (∼ 50%), the study of thereactionpp→ tg (tg) should provide an effective test of theλ′′3jk coupling constant.

7.6 Virtual Effects involving 6Rp Couplings

In a scenario where none of the supersymmetric particles canbe directly produced at colliderswith a significant cross-section, because of very high masses or unfavorable couplings with theStandard Model particles, the effects induced by6Rp could turn out to be felt only in indirectprocesses involving virtual sparticle exchange.

In contrast to single sparticle production for which a6Rp coupling only enter at one vertexwhen calculating total production rates,6Rp contributions (via additional sparticle exchange) to

(d)ui

g

dj

ui

~dk( )dk

g

dj

dk

~ui

(a)ui

g

dk

ui

~dj (b)dj

g

dk

dj

~ui

Figure 7.33: Diagrams for the non-resonant production of squarks involving λ′′ijk at hadroncolliders. Theλ′′ijk coupling constant is symbolised by a small circle and the arrows ndicate theflow of the baryon number.

Standard Model processes are suppressed in proportion to the square of the Yukawa coupling.These processes generally imply high statistics inclusivemeasurements as in the case of fermionpair production and effective four-fermion contact interactions discussed in section 7.6.1.

7.6.1 Fermion Pair Production

For sparticle masses far above the kinematical reach of a given collider,6Rp interactions couldmanifest themselves through effective four-fermion contact interactions interfering with Stan-dard Model fermion pair production processes.

At leptonic colliders dilepton production can occur in the presence of a unique (or largelydominant)6Rp coupling. The resonant sneutrinoνµ or ντ production viaλ121 orλ131 respectivelyfollowed by a decay through the same coupling constant (i.e.νi → ljlk viaλijk) would lead to aspectacular signature such as an excess of events Bhabha scattering events [261]. For examplethe cross-section of Bhabha scattering including theν(µ,τ) sneutrinos-channel exchange andthe interference terms reaches3 pb at

√s = mν(µ,τ)

= 200 GeV [519, 520, 521] forΓν(µ,τ)=

1 GeV andλ1(2,3)1 = 0.1.

Table 7.16 shows the accessibleλ couplings ate+e− andµ+µ− colliders and fermion pairproduction to which a single dominantλ coupling can contribute. Except few exceptions,e+e−

( )ui

W

~dj (d)ui

W

~dk

(b)dk

W

~ui(a)dj

W

~uidi

dj

uj

djdk

uj

dk

di

Figure 7.34:Diagrams for the associatedq−W production involvingλ′′ijk at hadron colliders.Theλ′′ijk coupling constant is symbolised by a small circle and the arrows indicate the flow ofthe baryon number.

andµ+µ− colliders allow to access the sameλ couplings. The difference in center-of massenergies and luminosities between these two types of leptonic colliders will determine the ex-plorable domain of these couplings.

The observation of an excess of highQ2 events at HERA experiments [462, 463] and its in-terpretation in terms of6Rp interactions has been followed by numerous discussions on dileptonproduction at LEP [519, 520, 521, 522, 523, 524] which are beyond the scope of this review.

Di-jets production can also occur at leptonic colliders in the presence of a uniqueλ′ cou-pling through the exchange of a squark in thet-channel. Thebb andcc production viaλ′1k3 andλ′12k respectively are of particular interest due to the possibility of tagging bottom, charm orlight quarks (u,d,s) at the experiment level [525]. Table 7.17 shows the accessibleλ′ couplingsat e+e− andµ+µ− colliders and fermion pair production to which a single dominantλ′ cou-pling can contribute. In contrast to the case ofλ couplings,e+e− andµ+µ− colliders accesscompletely different sets ofλ′ couplings. More specifically, in the case ofλ′ couplings, fermionpair production allows to explore onlyλ′1jk at e+e− colliders andλ′2jk atµ+µ− colliders.

Preliminary studies have been performed in [252] focusing on the study ofµ+µ− → µ+µ−

via ντ involving theλ232 coupling andµ+µ− → bb via ντ involving the productλ232λ′333.

( )dk

~ui (d)ui

~dk

(b)dj

~ui(a)ui

;Z; h; g

~djui

dj

dj

dkdk

ui

dj

dk ;Z; h; g ;Z; h; g

;Z; h; g

Figure 7.35: Diagrams for the associatedq − γ, (−Z, −h and−g) production involvingλ′′ijkat hadron colliders. Theλ′′ijk coupling constant is symbolised by a small circle and the arrowsindicate the flow of the baryon number.

In this case it has been found that once the mass of theντ is known from earlier stage of ae+e− collider or theµ+µ− collider and once fixing the center-of-mass energy atντ resonanceor around the resonance with theµ+µ− collider, one can exploreλ232 down to10−4 with anintegrated luminosity of3 fb−1 and a beam energy resolution of0.1 %.

Further preliminary studies have been performed in [526].

At hadron colliders6Rp reactions can induce contributions to Standard Model di-jets or di-leptons production processes. First, the jets pair production receives contributions from reac-tions involving eitherλ′ or λ′′ coupling constants. As a matter of fact, a pair of quarks can beproduced through theλ′′ couplings with an initial stateud or ud (dd or dd) by the exchange of ad (u) squark in thes-channel, and also with an initial stateuu or dd (ud or ud) by the exchangeof a u or d (d) squark in thet-channel. If thes-channel exchanged particle is produced on shellthe resonant diagram is of course dominant with respect to the t-channel diagram. The dijetchannel can also be generated via theλ′ couplings from an initial stateud, ud or dd through theexchange of al or ν slepton (respectively) in thes-channel.

If the dominant mechanism for either the slepton or the squark decay leads to two jets, theresonant production of such a scalar particle would result in a bump in the two-jet invariant massdistribution [261, 502] which would be a very clean signature. However the dijet production

e+e− colliders µ+µ− colliders

coupling final state exchange channel final state exchange channelλ121 e+e− νµ s e+e− νe t

µ+µ− νe t - - -λ122 µ+µ− νµ t e+e− νµ t

- - - µ+µ− νe s+tλ123 τ+τ− νµ t τ+τ− νe tλ131 τ+τ− νe t - - -

e+e− ντ s - - -λ132 µ+µ− ντ t e+e− ντ t

- - - τ+τ− νe tλ133 τ+τ− ντ t - - -λ231 τ+τ− νµ t e+e− ντ t

µ+µ− ντ t - - -λ232 - - - µ+µ− ντ s+t

- - - τ+τ− νµ tλ233 - - - τ+τ− ντ t

Table 7.16: Accessibleλ couplings ate+e− andµ+µ− colliders and fermion pair productionto which a single dominant coupling can contribute.

through 6Rp coupling constants will be hard to study at LHC unless the narrow resonances arecopiously produced given the severe expected QCD background [255, 527]. This was discussedin more details above in section 7.5.

Top quark pair production appears to be a particular case of fermion pair production athadron colliders because if kinematically allowed new decay channels such ast → ddR andt → dlL can open up. The amplitudes for top quark pair production involve diagrams withan initial statedkdk with either aliL slepton exchange (viaλ′i3k) or a di

R squark exchange Thesupersymmetric parameter space region allowed at a95% confidence level by the D0 and CDFdata [528] ontt production cross-section have also been obtained in [529] and are shown inFig. 7.36 in the planeλ′i31/mliL

and in Fig. 7.37 in the planeλ′′31i/mdiR. Furthermore,6Rp interac-

tions being chiral, one expects the two top quarks to be polarized thus providing an additionalhandle to probe the details of6Rp couplings [530] since the polarization of the top quark pairs isvery small in the Standard Model.

More complicated decay chains of the top quark such as the double cascade decayst →l+i dk, l

+i → χ0+ei, χ

0 → νibdk + νibdk where the top quark and neutralino6Rp decay processesare both controlled by the coupling constantsλ′i3k can lead to two potentially observable effectsin the leptonic events namely a deviation from lepton universality and (fork = 3) an excessof b quark hadron events A study based on the comparison of the ratio of branching fractionsfor singlee to singleµ eventsB(tt → (e + jets)/B(tt → µ + jets) to the experimentalratio of eventsN(e + jets)/N(µ + jets) = 1

(+a−b

)from the one charged lepton and two b-

quark jets final state of the CDF top quark sample of Tevatron Run I gives the bound [262]λ′13n < 0.41, [n = 1, 2].

Another method of analysis based on an identification of thisratio with the ratio of theexperimental to theoretical total production cross sections yieldsλ′13n < 0.48. An analysis ofthe hadronb quarks events yields [262]λ′133 < 0.41.

0

0.5

1

1.5

2

2.5

100 200 300 400 500

λ/

m~eL (GeV)

D0(2σ)CDF(2σ)

ALLOWED

DISALLOWED

DIS

ALL

OW

ED

Figure 7.36: Allowed regions in the plane ofλ′i3k and the mass of the left slepton in a lepton-number-violating scenario. Solid (dashed) lines correspond to the 2-σ bounds from the CDF(D0) collaborations.

0

0.5

1

1.5

2

2.5

100 200 300 400 500

λ//

m~dR

(GeV)

D0(2σ)CDF(2σ)

ALLOWED

DISALLOWED

DIS

ALL

OW

ED

Figure 7.37: Allowed regions in the plane ofλ′′3ki and the mass of the rightd-squark in abaryon-number-violating scenario. Solid (dashed) lines correspond to the 2-σ bounds from theCDF (D0) collaborations.

e+e− colliders µ+µ− colliders

coupling final state exchange channel final state exchange channelλ′111 dd uL t - - -λ′111 uu dR t - - -λ′112 ss uL t - - -λ′112 uu sR t - - -λ′113 bb uL t - - -λ′113 uy bR t - - -λ′121 dd cL t - - -λ′121 cc dR t - - -λ′122 ss cL t - - -λ′122 cc sR t - - -λ′123 bb cL t - - -λ′123 cc bR t - - -λ′131 dd tL t - - -λ′131 tt dR t - - -λ′132 ss tL t - - -λ′132 tt sR t - - -λ′133 bb tL t - - -λ′133 tt bR t - - -

λ′211 - - - dd uL tλ′211 - - - uu dR tλ′212 - - - ss uL tλ′212 - - - uu sR tλ′213 - - - bb uL tλ′213 - - - uy bR tλ′221 - - - dd cL tλ′221 - - - cc dR tλ′222 - - - ss cL tλ′222 - - - cc sR tλ′223 - - - bb cL tλ′223 - - - cc bR tλ′231 - - - dd tL tλ′231 - - - tt dR tλ′232 - - - ss tL tλ′232 - - - tt sR tλ′233 - - - bb tL tλ′233 - - - tt bR t

λ′3jk - - - - - -

Table 7.17: Accessibleλ′ couplings ate+e− andµ+µ− colliders and fermion pair productionprocesses to which a single dominant coupling can contribute.

Alternatively [104] the top quark6Rp decay channelt → bτ+ initiated by theλ′333 couplingleads to signature which can not be confused with the Standard Model decay channel and cancompete with it. This induces a reduction of the observed Standard Modeltt event rates. The

correction factor reads:

RB ≃ 1.12 λ′2333(1 − m2

τL

m2t

)−2 (7.17)

Similarly the hadron two-body decay channelst→ dj +¯dkR with λ′′ couplings have an impact

on thett events through a modification in the fraction of hadron top quark decays. Performinga similar analysis to the one above for the6Rp decay modest→ b˜s initiated by theλ′′323 couplingwhere thett pairs cascade down to a 5 jets final state leads to an induced reduction factor onthe multiple jet signal of(1 + 0.16λ

′′2323). Aside from ruling out the associated6Rp coupling

constants, one can evade a conflict with the experimental observations by closing the relevantdecay channels by assuming stau or squark masses larger than150 GeV.

Before closing this subsection on fermion pair production one has to keep in mind thatallowing for more than one dominant6Rp coupling leads to further possibilities for fermion pairproduction at both leptonic and hadron colliders.

At leptonic colliders, dilepton production involving two dominantλijk couplings such as forexamplee+e− → µ+µ− involving λ131 andλ232 with s-channelντ exchange ore+e− → τ+τ−

involving λ131 andλ232 with a s-channelνµ exchange have been considered [519, 523]. Di-jet production can also occur in processes involvingλijk andλ′ijk couplings withs-channelν exchange. For examplee+e− → bb involving λ131 andλ′333 or e+e− → dd involving λ131

andλ′311 both withs-channelντ exchange have been discussed in [477, 503]. Since the angulardistribution of thed andd jets is nearly isotropic on the sneutrino resonance, the strong forward-backward asymmetry in the Standard Model continuum,AFB(b) ≈ 0.65 at

√s = 200 GeV, is

reduced to≈ 0.03 on top of the sneutrino resonance [503].

Studies involving products ofλ coupling, products ofλ′ and products ofλwith λ′ couplingsatµ+µ− colliders have been performed in [526].

At hadron colliders the third generation slepton resonant production i.e. ντ tau-sneutrino(neutral current) andτ stau (charged current) involving weakly constrainedλ′311λ3jk couplingconstants thus leading to lepton pair production, have beenconsidered in [531] for both theTevatron and LHC colliders. The reach in terms of the sleptonmass ranges from 800 GeV atthe Tevatron Run II to 4 TeV at the LHC for sizeable values ofX = λ′311λ3jkBl, Bl being theleptonic branching ratio, fromX ≈ 10−3 down toX = 10−(5−8) the latter for small sleptonmasses of the order of hundred GeV . In the particular case ofe+e− production, existingTevatron data [532] from the CDF detector on thee+e− production have been exploited in [503]to derive the following bounds on the productλ′311λ311 (with some theoretical uncertaintiescoming from the knowledge of K factor for slepton production):

(λ′311λ311)1/2 < 0.08 Γ

1/4ντ

(7.18)

for sneutrino masses in the range120−250 GeV whereΓντ denotes the sneutrino width in unitsof GeV. The particular cases ofµ+µ− andτ+τ− productions have been considered in [315]based on the total cross-section studies above a given threshold on the dilepton invariant massin order to get rid of the background from thes-channelZ resonance contribution.

Futhermore, the distinction between a scalar or a new gauge boson resonance can be per-formed [531] by testing the lepton universality and by measuring the forward-backward asym-metry which is expected to be zero in the case of a resonant scalar production and non zero inthe case of a new gauge boson resonance as well as the leptoniccharge asymmetry defined as:

A(η) =

dN+

dη− dN−

dη

dN+

dη+ dN−

dη

(7.19)

whereN± are the number of positively/negatively charged leptons ofa given rapidityη. Thepresence of the slepton tends to drive the leptonic charge asymmetry to smaller absolute valueswhile a newW ′ gauge boson substantially increases the magnitude of this asymmetry. Atthe Tevatron Run II, the minimum value of the productλλ′ for which the asymmetry differssignificantly from the Standard Model expectation is0.1, for a luminosity of2 fb −1, assumingml = 750 GeV andΓl/ml = 0.004.

7.6.2 6Rp Contributions to FCNC

In the Standard Model flavour changing neutral current effects arise through loop diagrams.They are strongly suppressed [533, 534, 535] because of the CKM matrix unitarity and thequark mass degeneracy (except the top quark) relative to theZ boson mass. In several super-symmetric extensions of the Standard Model like the MSSM thelarge flavour changing neutralcurrent effects are expected to be reduced by assuming either a degeneracy of the soft super-symmetry breaking scalars masses or an alignment of the fermion and scalar superpartners massmatrices [536]. In addition flavour changing decay rates such asZ → qJ qJ ′ through trianglediagrams involving squarks and gluinos have been found to besmall with respect the StandardModel predictions [537, 538].

The 6Rp interaction, because of its non-trivial flavour structure,opens up the possibility ofobservable flavour changing effects at the tree level.

The 6Rp interactions contributions to theZ boson flavour off-diagonal decays branchingratios were discussed in Section 6.3.2.

At colliders, these flavour changing6Rp processes occur through the exchange of a super-symmetric scalar particle in thes- or t-channel and lead to fermion pair productionsfJfJ ′ withJ 6= J ′.

Furthermore, in minimal supersymmetric extension of the Standard Model without degen-eracies for sleptons masses, flavour changing effects can beinduced in the supersymmetricparticle pair production involving6Rp interactions.

At leptonic colliders, with centre-of-mass energies abovetheZ boson pole, single top quarkproduction such asl+i l

−i → tc, tc occuring via the exchange of adkR squark in thet-channel

through the6Rp couplingsλ′i2k andλ′i3k offers a clean opportunity to observe one of these treelevel flavour changing neutral current effects [539, 540, 541, 542, 543]. Indeed single top quarkproduction occuring at the one loop level in the Standard Model [533, 534, 535] is suppressedwith respect tobs production since it does not receive large contributions from heavy fermionsin the loop. Moreover the MSSM contribution has been shown tobe small compared to theStandard Model one [537, 538]. The cross-section ofe+e− → tc + tc is shown in Fig. 7.38from [540] for λ′12kλ

′13k = 0.01 which is the order of magnitude of the low-energy constraint

on this product of6Rp couplings formf = 100 GeV.

The reactione+e− → tc+ tc receives also contributions at one loop level from theλ′′ inter-actions [542, 540] in which adR squark is involved with theλ′′2jk andλ′′3jk coupling constants. Inparticular the combinationλ′′223 λ

′′323 with a low energy constraint ofλ′′223 λ

′′323 < 0.625 which is

less stringent than the constraints of the otherλ′′2jk λ′′3jk combinations can lead to cross-sections

as big as 1 fb formdkR= 100 GeV.

Thetc/tc production can also occur at one loop level via photon-photon reactionse+e− →γγ → tc + tc which involve the products of6Rp couplingsλ′i2kλ

′i3k when liL sleptons ordkR

Figure 7.38: Cross section of the reactione+e− → tc + tc as a function of the centre-of-massenergy forλ′12kλ

′13k = 0.01. The solid line corresponds tomdkR

= 100 GeV and the dashedline tomdkR

= 150 GeV.

squarks are exchanged in the loop and the productsλ′′2jkλ′′3jk whendR squarks run in the loop.

Again the combinationsλ′323λ′333 (λ′′223λ

′′323) which have less stringent low-energy constraints

than the otherλ′i2kλ′i3k (λ′′2jkλ

′′3jk) combinations lead to cross-sections which are about an order

of magnitude below the cross-sections of Fig. 7.38 from treelevel diagrams involvingλ′12k λ′13k.

A combination of the results from thee+e− andγγ collisions would allow to distinguish be-tween theλ′ andλ′′ effects on thetc/tc production.

On the experimental side thetc or tc production can lead tobclν final state. The backgroundfrom Standard Model processes such ase+e− → W+W− → bclν can then be significantlyreduced by observing that the c-quark has a fixed energy givenby [541]:

E(c) = (s+m2t −m2

c)/2√s. (7.20)

Searches fortc or tc production have been performed at LEPII along these lines. However theyhave not yet allowed to put a more stringent constraint onλ′12kλ

′13k than those coming from

low energy. Searches fortc or tc production will be performed at the futur linear collider. Thestudy of the final statebclν would allow to probe values of the productλ′12kλ

′13k down to∼ 0.1

for mdkR= 1 TeV at a linear collider with a centre-of-mass energy of

√s = 500 GeV and a

luminosity ofL = 100fb−1 [541].

The tc or tc production can occur atµ+µ− colliders as well. The cross-section for sucha production is shown in Fig. 7.39 from [540] forλ′223λ

′233 = 0.065 which is equal to its low-

energy limit formf = 100 GeV. An additionnal motivation for this choice of6Rp couplings isprovided by the observation that among the possibleλ′22kλ

′23k combinations theλ′223λ

′233 one

has the less stringent low-energy constraint.

Finally flavour changing effects in sfermion pair production can be investigated in highprecision measurements planned to be performed for exampleat future leptonic linear collid-ers [544]. The6Rp interactions can generate such effects through the exchange of a neutrino

Figure 7.39:Cross section of the reactionµ+µ− → tc + tc as a function of the centre-of-massenergy forλ′223λ

′233 = 0.065. The solid line corresponds tomdkR

= 100 GeV and the dashedline tomdkR

= 150 GeV.

in the t-channel in slepton pair productione+e− → lJ l∗J ′ (J 6= J ′). The flavour non-diagonal

rates vary in the rangeσJJ ′ ≈ ( Λ0.1

)4(2 − 20) fb [545] with Λ = λ, λ′ for sleptons massesml < 400 GeV as one covers centre-of-mass energy regions from theZ boson pole up tothe TeVregion. Due to the strong dependence on the6Rp couplings, the flavour non-diagonalrates reach smaller values than the rates obtained in the flavour oscillations approach [546]which range between250(100) and0.1(0.01) fb for

√s = 190(500) GeV.

At hadron colliders flavour changing lepton pair productions ljlj′ as well as quark pair pro-ductionsqjqj′ (j 6= j′) are both expected to be challenging to search for since the environmentin terms of background is not as clean as the environment at leptonic colliders.

For example flavour changing lepton pair productions occur from an initial statedjdk (dkdk)through the exchange of aνi

L sneutrino (ujL squark) in thes-channel (t-channel) via the cou-

plings productλ′ijkλiJJ ′ (λ′Jjkλ′J ′jk), or, from an initial stateujuj through the exchange of adk

R

squark in thet-channel via the couplings productλ′Jjkλ′J ′jk. More specific studies on flavour

changing lepton pair productions remain to be done.

More striking signatures of6Rpinduced flavour changing neutral current effects could beobserved in rare decays of the top quark as discussed in section 6.4.1.

Finally the possibility of single top quark production via squark and slepton exchanges toprobe several combinations of6Rp couplings at hadron colliders has been studied in [547, 548,549, 550]. Initial state partons such asud are particularly relevant forpp colliders such as theTevatron while theud initial state system is more relevant forpp colliders such as the LHC.

The single top quark productionuidj → tb can occur via the exchange of adkR squark in the

t-channel, through the product of couplingsλ′′i3kλ′′3jk. The choice of the initial state of the reac-

tion uidj → tb fixes the flavour indices of the coupling constants productλ′′i3kλ′′3jk because of

msRin GeV 100 200 300 400 500 600 700 800

λ′′132 λ′′312 0.01 0.02 0.03 0.04 0.06 0.08 0.1 0.13

Table 7.18: Sensitivities on the productλ′′132 λ′′312 for variousmsR

at the upgraded Tevatronfrom the processu1d1 → sR → tb from [548].

the antisymmetry of the generation indices of the coupling constantsλ′′. Furthermore, becauseof the low energy constraints and the low parton luminosities, the only product of interest isλ′′132λ

′′312. Assuming the observability criteria∆σ/σ0 > 20% where∆σ is the 6Rp cross-section

andσ0 is the Standard Model cross-section, Table 7.18 from [548] shows the sensitivities onλ′′132 λ

′′312 at the upgraded Tevatron for variousmsR

.

The single top quark productionujdk → tb can also occur through the exchange of aliL slep-ton in thes-channel via the couplings productλ′ijkλ

′i33. The dominant processud→ liL → tb

which involves the sum of couplingsλ′111λ′133 + λ′211λ

′233 + λ′311λ

′333 has been considered in

[549]. According to [549] values ofλ′ couplings below the low energy bounds can be probed ifthe slepton mass lies in the range200 GeV < mliL

< 340 GeV for the upgraded Tevatron andin the range200 GeV < mliL

< 400 GeV for the LHC. Although larger parton momenta areallowed at the LHC the result is not really improved at LHC because of the relative suppressionof the d quark structure function compared to thed quark one.

Turning to the case ofuidj initial state partons, the single top quark production can alsooccur through the exchange of adk

R squark in thes-channel via the couplings productλ′′ijkλ′′33k.

Table 7.19 gives an example of the cross-section obtained from different initial parton statesat the LHC. Sensitivities on the coupling productλ′′212λ

′′332 at the upgraded Tevatron and at the

Initial partons cd cs ub cb

Exchanged particle s d s d sCouplings λ′′212λ

′′332 λ′′212λ

′′331 λ′′132λ

′′332 λ′′231λ

′′331 λ′′232λ

′′332

Cross-section in pb 3.98 1.45 5.01 0.659

Table 7.19: Cross section inpb of the reactionuidj → dkR → tb at LHC for a squark of mass

of 600 GeV assuming andλ′′ijk = 0.1 andΓRp(q) = 0.5 GeV whereΓRp(q) is the width of theexchanged squark due toR-parity conserving decay.

LHC have been obtained in [549]. A more detailed simulation has been performed in [550]and the sensitivities on the coupling productλ′′212λ

′′332 are shown in Fig. 7.40. The reaction

ujdk → tb receives also a contribution from the exchange of al±iL slepton in theu-channel viatheλ′ij3 andλ′i3k couplings [550].

Supersymmetric particle masses reconstruction have been also performed within the frame-work of single top production in [550].

To summarize, the studies of single top quark production at hadron colliders [547, 548, 549,550] tend to indicate that the LHC is better at probing theB-violating couplingsλ′′ whereas theTevatron and the LHC have a similar sensitivity toλ′ couplings. Furthermore, this is the onlyframework in which the constraints onλ′′ from physics at colliders are comparable or betterthan the low energy bounds on theλ′′ coupling constants.

0.01

0.02

0.03

0.04

0.05

0.06

10-1

1

- Oakes et al.

- Oakes cuts

- Chiappetta et al.

Γ ⁄msquark (GeV)

λ”21

2λ” 33

2

Figure 7.40: Sensitivity limits on theλ′′212λ′′332 Yukawa couplings obtained from the analysis of

the reactioncd → s∗ → tb at the LHC after 1 year with low luminosity forms = 300 GeV,found in [550] (circles) and in [549] (triangles). The squares indicate the results obtained in[550] by applying the simplified cuts used in [549].

7.6.3 6Rp Contributions to CP Violation

As already discussed in section 6.3.4 the6Rp coupling constants can have a complex phase andhence be by themselves an independent source ofCP violation motivating many studies on lowenergy6Rp physics. These can still lead to new tests ofCP violation in combination with theother possible source of complex phase in supersymmetric extensions of the Standard Modelsuch as the MSSM even if one assumes that the6Rp interactions areCP conserving. For instancethe 6Rp couplings can bring a dependence on the CKM matrix elements due to the fermion massmatrix transformation from current basis to mass basis.

A study ofCP violation effects in association with sneutrino flavour oscillations has beencarried out in section 5.5. TheCP violation effects from6Rp couplings in theKoKo system andin hadrons decays asymmetries has been discussed in section6.3.4. TheCP asymmetries at theZ boson pole has been discussed in section 6.3.4.

At colliders, CP violation effects from6Rp couplings can also be further studied fromfermion pair productions either flavour changing or non flavour changing. These effects areeither controlled by interference terms between tree and loop amplitudes in the case ofCPasymmetries or directly considered from tree level processes.

Furthermore, if both non-degeneracies and mixing angles between all slepton flavours andif theCP odd phase do not vanish,CP violation asymmetries can also be observable in super-symmetric particles pair production. TheR-parity odd interactions can provide an alternative

mechanism for explainingCP violation asymmetries in such productions through possibleψCP odd phase incorporated in the relevant dimensionless coupling constant.

At leptonic colliders the effects of6Rp interactions on theCP asymmetries in the processesl+l− → fJ fJ ′ , with J 6= J ′, were calculated in [542]. The6Rp contributions to theseCPasymmetries are controlled by interference terms between tree and loop level amplitudes. Thediscussion of loop amplitudes was restricted to the photon andZ boson vertex corrections. Theoff Z boson pole asymmetries is given by:

AJJ ′ =σJJ ′ − σJ ′J

σJJ ′ + σJ ′J, (7.21)

whereσJJ ′ = σ(l+l− → fJ fJ ′). Definingψ as theCP odd phase, these asymmetries lie atAJJ ′ ≈ (10−2 − 10−3) sinψ for leptons and quarks irrespective of whether one deals with lightor heavy flavours.

TheCP asymmetriesAJJ ′ depend on a ratio of different6Rp coupling constants and aretherefore less sensitive to the absolute magnitude of thesecouplings than the flavour chang-ing ratesσJJ ′ which involve higher power of the6Rp constants. The particular dependenceof theCP asymmetries on the couplings is of the formIm(λλ∗λλ∗)/λ4 and may thus lead tostrong enhancement or suppression factors depending on thelargely unknown flavour hierarchi-cal structure of the involved Yukawa couplings. For examplethe study of single top productionl+l− → tc with t → bW → blν allows to learn aboutCP violation in the quark sector. In thisreaction theCP violation can be probed through the asymmetry defined in Eq. (7.21) or via thefollowing flavour off-diagonalCP asymmetry [541]:

A+− =dσ+

dEl− dσ−

dEl

dσ+

dEl+ dσ−

dEl

, (7.22)

whereσ+ = σ(l+l− → tc→ bclν), σ− = σ(l+l− → tc→ bclν) andEl is the energy of theproduced charged lepton. The values of thisCP asymmetryA+− range typically inA+− ≈(10−2 − 10−3) sinψ for El < 300 GeV [541]. TheseAJJ ′ andA+− CP asymmetries can beenhanced up to∼ 10−1 sinψ if the 6Rp coupling constants exhibit large hierarchies with respectto the generations.

Turning now briefly toCP violation asymmetries in supersymmetric particles pair pro-duction, as in the case of flavour changing fermion pair production, the 6Rp contributions totheseCP asymmetries in scalar particles pair production are controlled by interference termsbetween tree and loop level amplitudes. For example the flavour non-diagonalCP asym-metriesAJJ ′ for the slepton pair production,e+e− → lJ l

∗J ′ (J 6= J ′) are predicted to be

AJJ ′ ≈ (10−2 − 10−3) sinψ [545].

Finally, the6Rp interactions can give rise toCP violation effects at tree level in the non flavourchanging reactione+e− → τ+τ− via the observation of the double spin correlations of the pro-duced tau-leptons pair.

This possibility studied in [253] stands out as an very interesting issue by itself since previ-ous studies ofCP -violating effects in the processe+e− → τ+τ− which can happen for instancein models with multi-Higgs doublet or in leptoquark, Majoranaν or supersymmetry models, alloccur at one loop level.

Here theCP asymmetries are generated from the exchange of a resonantνµ sneutrino in thes-channel via the real couplingλ121 and the complex couplingλ323 if a νµ − ˜νµ mixing exists.

This sneutrino mixing can generate bothCP -even andCP -odd spin asymmetries which areforbidden in the Standard Model and that can be measured forτ leptons at leptonic colliders.The observation of such asymmetries would provide explicitinformation about three differentaspects of new physics:νµ − ˜νµ mixing, CP violation and6Rp . The sneutrino-antisneutrinomixing phenomena which have been gaining some interest recently [250, 184, 551] is interestingsince it is closely related to the generation of neutrino masses [250, 184]. The polarisationasymmetries from double spin correlations of the produced tau-leptons pair provide a feasiblealternative for establishing the mass splitting between theCP evenνµ

+ andCP odd νµ− muon-

sneutrino mass eigenstates [253]. These polarisation asymmetries depend on the relative valuesof the real part,a, and the imaginary part,b, of the complex coupling constantλ323. At thenext linear collider with

√s = 500 GeV [253], with the simultaneous measurement of theCP

conserving andCP -violating asymmetries, the whole range0 ≤ ba+b

≤ 1 can be probed to atleast3σ in themνµ mass range of20 GeV around resonance i.e.

√s− 10 GeV< mνµ <

√s+

10 GeV even for a small mass splitting of 1 GeV.

At hadron colliders, in analogy to the case of the leptonic colliders, the resonant productionof a sneutrino gives also rise to the possibility of havingCP violation effects at tree level [315].

If the τ spins can be measured,CP violation effects in the polarisation asymmetries ofthe hard processdjdk → νµ → τ+τ− can be observed at the Tevatron. The6Rp couplingconstantλ′2jk which enters this subprocess is chosen real, whileλ233 is taken complex in orderto generateCP asymmetries. However, at hadron colliders, spin asymmetries deserves a carefuldiscussion. The spin asymmetries change sign around

√s ≈ mνµ

±so that one has to integrate

over√s of the initial parton system. In consequence the spin asymmetries seem too small to

be measurable. Nevertheless a two-step measurement helps in overcoming this problem. In afirst step one has to determine the mass of the resonant sneutrino by measuring theττ invariantmass distribution and in a second step one has to integrate the absolute values of the polarizationasymmetriesl [315].

At the Tevatron Run IIA (IIB) withL = 2 fb −1 (30 fb −1), taking their low energy boundsas the values ofλ′2jk and |λ233| and including allj, k combinations indjdk fusion, theCPconserving andCP -violating asymmetries may be detected with a sensitivity above3σ overthe mass range155 GeV < mνµ

±< 400 GeV (155 GeV < mνµ

±< 300 GeV) if ∆mνµ = Γνµ

(∆mνµ = Γνµ/10) whereΓνµ is the sneutrino width.

Moreover the entire range0 ≤ ba+b

≤ 1 can be practically covered formνµ−

= 200 GeV atthe Tevatron with at least3σ standard deviations for∆mνµ = Γνµ (∆mνµ = Γνµ/4).

These results show that in contrast to the case of leptonic colliders [253], theCP odd andCP even spin asymmetries can be observed over a wideνµ sneutrino mass range of about300 GeV.

Chapter 8

CONCLUSIONS AND PROSPECTS

After the great successes of spontaneously broken gauge theories and of the Standard Model,supersymmetric theories of particles and interactions constitute one of the best motivated frame-works for the discussion of new physics beyond the Standard Model. The reasons are profoundand fundamental – although none is definitely conclusive, especially in view of the fact that thenewR-odd superpartners have escaped, for a long time now, all experimental efforts to disclosetheir existence.

Among the reasons to consider supersymmetry is our desire tosee bosons and fermionsplay similar roles, although this is against all immediateevidence, known bosons and fermionshaving very different properties ! Indeed the supersymmetry algebra did not allow us to relatedirectly known bosons with known fermions, and we had to invent, instead, a whole new zoo of“supersymmetric particles”, squarks and sleptons, gluinos, charginos and neutralinos, etc., so asto allow us to view the world as possibly supersymmetric. These objects are, precisely, the newR-odd particles. Not only do we have to “double everything” – which was once considered asevidence against supersymmetry – but in the usual frameworkof spontaneously broken gaugetheories additional Higgses should also be introduced, with their associated higgsinos ! Andthe whole construction assumes the existence of new self-conjugate Majorana fermions, oftenconsidered as ugly beasts, only Dirac fermions being known in Nature ! Is all this too high aprice to pay ? Only the future – and experiments – will tell.

But what can such supersymmetric theories do for us ? Plenty of things, many of themwell-known, according to different arguments all based on the nice and attractive features of su-persymmetric theories. There are also, unfortunately, less nice features, as the reader who wentthrough detailed discussions of the many possible supersymmetry-breaking terms inR-parityconserving andR-parity-violating theories will certainly have noticed.

Among the attractive features is the fact that, in supersymmetric theories – which are closelyrelated with gravitation – the Higgs potential is largely determined by the supersymmetry. Thequartic Higgs boson self-coupling (λ in the Standard Model), or rather self-couplings (for twoHiggs doublets), instead of being arbitrary, are now fixed byg2 and g2 + g′ 2, a fact at theorigin of many relations involving massive gauge bosons andHiggs bosons. The new particlesintroduced also allow for an appropriate high-energy convergence of the threeSU(3), SU(2)andU(1) gauge couplings, whose values get unified, as it would be the case in a grand-unifiedtheory. Supersymmetric theories also have improved convergence properties at the quantumlevel, leading to hopes of solving or alleviating the hierarchy problems associated with theextreme smallness of the cosmological constantΛ, or the smallness ofmW andmZ compared

to the GUT or Planck scales (although these hints towards solutions would still have to survivesupersymmetry-breaking). Supersymmetry usually also appears as a necessary ingredient in theconstruction of consistent string (and brane) theories – and, even without having to considerstrings and branes at all, shows us the way towards new spacetime dimensions...

The fundamental motivations for supersymmetric theories are and remain strong, even if westill don’t know which particular model, within the generalclass of supersymmetric theories ofweak, electromagnetic and strong interactions, should effectively be chosen. While the allowedparameter space of the popular Minimal version of the Supersymmetric Standard Model hasnow very seriously shrinked, we have known from the beginning that other ingredients (suchas an extra singlet superfield coupled to the two Higgs doublets Hd and Hu) could naturallybe present, with no special reason to stick to the “MSSM”. In addition, we still have verylittle insight on how supersymmetry should be broken. In theabsence of a really satisfactory,consistent and predicting mechanism, one generally chooses the option of parametrizing thevarious possible supersymmetry-breaking terms. Even softterms are numerous, and this led tothe introduction of a large number of arbitrary parameters in supersymmetric theories, as theprice to pay for our ignorance.

Yes, but what aboutR-parity, the subject of this review ? As discussed in chapter1, oneof the initial difficulty with supersymmetry was the absenceof Majorana fermions in Nature,all known fermions appearing as Dirac particles carrying additive quantum numbers, baryonnumberB and lepton numberL, both very well conserved. When trying to implement su-persymmetry we had to cope with the fact that these conservedB andL appear as carried byfundamental fermions only – quarks and leptons – not by bosons ! Still an additiveR-quantumnumber might tentatively have been interpreted as a lepton number, if we could have used su-persymmetry to relate the photon with a neutrino. However, once supersymmetry transformsthe photon into a “photino”, the gluons with gluinos, quarkswith squarks, etc., thisR-number,if it survives at all (under the form of a discreteR-parity character), must be given a differentinterpretation. While the ordinary particles of the Standard Model areR-even, their superpart-ners, including the various squarks and sleptons, areR-odd – withRp = (−1)R. But we mayhave introduced the wolf inside the sheephold sinceB andL get now carried, not only by fun-damental fermions (which would make it easy to understand their conservation), but also byfundamental bosons, the new (R-odd) squarks and sleptons ! If these are not well behaved weshall certainly face severe problems withB andL non-conservation.

Good behavior is, as we saw, closely connected withR-parity, even ifR-parity may ulti-mately turn out not to be exactly conserved. IndeedR-parity conservation, or possibly non-conservation, is related withB andL conservation laws, as easily seen by reexpressingR-parity as (−1)2 S (−1)3B+L. A conservedR-parity would prevent unwanted direct exchangesof spin-0 squarks and sleptons between ordinary particles.It would, also, prevent neutrinosfrom mixing with the photino or, more generally, the variousneutralinos, etc..

With noR-parity at all, i.e. ifR-parity is not even an approximate symmetry of the super-potential and of the supersymmetry-breaking terms (or in the absence of analogous symmetriesthat would play a similar role in excluding unwanted interactions), supersymmetric theories arenot phenomenologically viable, since they would lead, in general, to much too largeB- and/orL-violating processes – e.g. a much too fast proton decay, or too large neutrino masses.

R-parity, on the other hand, naturally excludes unwantedB- and/orL-violating terms fromthe superpotential, and from the supersymmetry-breaking terms. It leads to the famous “missing-energy” signature of supersymmetric theories at colliders, and to the stability of the Lightest Su-persymmetric Particle, the LSP, generally thought to be thelightest neutralino. It then provides

us, for free, with a stable weakly-interacting non-baryonic Dark Matter candidate. Quite re-markably, such a candidate is naturally present for structural reasons, without being introduced“by hand” for the sole purpose of obtaining Dark Matter.

R-parity may well be viewed as having a very fundamental origin, in relation with thereflection symmetryθ → − θ in superspace, or with the existence of extra dimensions whichmay be responsible for supersymmetry breaking by dimensional reduction. It is, on the otherhand, often criticized by tenants of an opposite attitude, explaining that they don’t see anythingfundamental in this symmetry. And that all possible terms compatible with gauge symmetriesshould therefore be included in the superpotential; and also added in the Lagrangian density, assupersymmetry-breaking terms.

This certainly leads, in general, to a complete disaster, which necessitates the reintroduc-tion of R-parity orR-parity-like symmetries, at least for parts of the Lagrangian density or asapproximate symmetries. Actually some other symmetries (coming e.g. from higher energy...) could mimic the effects of anR-symmetry orR-parity in excluding a certain number ofterms from the superpotential and the supersymmetry-breaking terms, while still allowing oth-ers, possibly with small or even extremely small coefficients. This makes it worthwhile to studypossible violations ofR-parity, within supersymmetric theories. And to discuss ina systematicway the constraints existing on the possible6Rp terms, taking into account all data, originatingfrom astrophysics and cosmology as well as from acceleratorexperiments.

It is clear thatR-parity violations are certainly allowed, but only provided they are suffi-ciently well hidden and therefore not too large ! From the cosmological point of view the mostdrastic – and in general regretted – consequence ofR-parity violation is that the LSP shouldnormally be unstable, and must then in general be abandoned as a favorite Dark Matter candi-date (unless of course its lifetime were extremely long, at least of the order of the age of theuniverse). If the LSP is really unstable, however, one has tomake sure that the6Rp interactionsresponsible for its decay are sizeable enough for this decayto occur before nucleosynthesis. Inthis case, the LSP is no longer constrained to be electrically neutral and uncolored.6Rp inter-actions, which would also violate theB and/orL symmetries, may allow for new baryogenesisscenarios. Conversely, one has to make sure that these new6Rp interactions are sufficientlysmall so as not to erase the baryon asymmetry needed to understand the origin of matter in ouruniverse.

The most flagrant penalties for too much6Rp are, as we have discussed, too largeB- and/orL-violating processes, leading for example to a too short lifetime for the proton, or too largemasses for the neutrinos. But neutrinos are now known to havesmall masses anyway, and itis tempting to speculate that these very small neutrino masses may have something to do witha very small mixing between the neutrino and neutralino sectors, that would be induced in6Rp

theories, in the presence ofL-violating interactions.

Small neutrino masses, as well as neutrino oscillations from one flavour to another, couldthen be viewed as originating from the effect of large neutralino masses, transmitted to the neu-trino sector through (sufficiently small)6Rp interactions. This certainly constitutes an appealingalternative to the familiar see-saw mechanism, as a framework in which to discuss the propertiesof neutrinos, masses and oscillations, as well as possible magnetic moments. It may be in factclosely related to the general question of the origin of the mixing between the three quark andlepton families. The question ofR-parity conservation, or non-conservation (or of how it mightturn out to be slightly violated), may then simply appear as one of the aspects of a much moregeneral “flavour problem”. This is indeed quite crucial, butalso not easy to solve!

Ongoing experiments such as MiniBoone at FERMILAB, possible future experimentationclose to a nuclear reactor a la CHOOZ and future long baseline projects as the US NUMI, CERNto Gran Sasso in Europe and T2K in Japan are expected to give highly valuable informations onneutrino oscillations. Exploiting theβ-decay of tritium as in the futur KATRIN spectrometerin Germany and using the search for0νββ decay as planned by the NEMO3 and the GENIUSexperiments will also bring fundamental informations for the understanding of neutrino massspectrum. On the longer term, projects involving very high intensity neutrino beams like beta-beams and megaton Cerenkov detectors are expected to further help determine the parametersof the neutrino sector and hopefully bring some informationon CP violation in this sector.These data will be exploited in order to solve at least a part of this “flavour problem”.

Supersymmetric particles have been searched for intensively in a large variety of acceleratorexperiments, most notably ate+e− , e+p and p p colliders – both under the assumptions of aconserved, or violated,R-parity. No direct experimental sign of supersymmetry has been foundyet, and it is known, from LEP, HERA and Tevatron experiments, that superpartners should beheavier than about 100 GeV at least, excepted may be for some of the neutral ones, both inRp-conserving andRp-violating theories.

Furthermore, sets of bounds for the parameters of6Rp interactions have been discussed, bothfrom the indirect searches for such interactions, and from the direct production of the new spar-ticles – either isolated or in pairs – in a situation ofR-parity violation. The most characteristicsignature of supersymmetry is then no longer the missing energy-momentum carried away bythe two unobserved LSP’s. These bounds have been obtained and discussed, either as boundson every single6Rp coupling constant considered isolately, or as bounds on products of twosuch 6Rp coupling constants.

A large number of the many6Rp coupling constants and parameters still remain uncon-strained. Imaginative efforts to find new processes that might fill in the remaining gaps in theinformation will require a concerted effort between theoryand experiment. One needs to iden-tify processes, allowing for significant contributions from the 6Rp interactions, where a highexperimental sensitivity, also taking into account the uncertainties in the Standard Model pre-dictions, is attainable. Several measurements aiming at detecting rare processes are expected tobe performed soon and should further extend the search for6Rp effects. Just to cite a few, thesearches forµ→ e conversion either with theµ→ eγ decay as chased by the MEG experimentat PSI or withµN → eN conversion processes as considered by the MECO project at BNLare expected to gain 2 to 3 orders of magnitude in sensitivitywith respect to their predecessors(the MEGA and SINDRUM2 experiments at PSI). Other promisingexamples are offered byBmeson andτ lepton rare decay processes. If some coupling constants happen to be of the orderof 10−1, this could be enough to lead to observable effects at high energy colliders.

The prospects on the long term are encouraging. Thanks to theongoing experiments such asBABAR at SLAC and BELLE at KEK, both aiming at very highB meson production statisticscorresponding to several hundreds fb−1 of integrated luminosity, experimental measurementsof rare (“forbidden”) decay processes are expected to gain several orders of magnitude in sen-sitivity. This kind of gain is also expected for planned projects such as CKM at FERMILAB,CLEO upgrades at Cornell, KOPIO at BNL and JHF at Tokai for high intensity kaon beams, aswell as detectors such as LHCB at CERN and BTEV at FERMILAB, dedicated toB physics.In parallel the CDF andD∅ experiments at Tevatron Run II at FERMILAB are expected to gain2 to 3 orders of magnitude in sensitivity for B physics with respect to Run I thus providingfurther tests for6Rp interactions. As for more direct searches both CDF andD∅ are expectedto extend their searches for supersymmetry with6Rp effects in both the single supersymmetric

particle production mode and the more conventional pair production mode followed by6Rp de-cays. Factors of10 − 100 improvements in accuracy are also anticipated for high precisionmeasurements of magnetic or electric dipole moments as, forexample, the10−28e.cm region forthe electric dipole moment of the neutron (to be explored with the spallation ultra-cold neutronsource (SUNS) at PSI). Some progress is expected thanks to the high energy leptonic collid-ers for high precisionZ boson physics observables, especially with the possibility of the highluminosity option of a future linear electron collider running at theZ boson resonance. Our the-oretical understanding of supersymmetry and of physics beyond the Standard Model is likely todeepen in the meantime. On a different front it is likely, also, that we shall learn more about theproperties of the Dark Matter and possibly its nature.

Ultimately if supersymmetry is indeed a symmetry of Nature along the lines presented here,there is no substitute for a direct observation of the superpartners. The best hope is that super-partners could show up directly, in a few years from now, at the LHC p p collider at CERN,revealing directly the presence of supersymmetry as a fundamental symmetry of the world ofparticles and interactions. One would then expect a wealth of new results, on the mass spectrumof the new particles as well as on their production and decay properties, which should all bemore precisely measured at a future linear electron collider. These data should be crucial to helpus understand the actual mechanism which breaks supersymmetry, and to discover whetherR-parity is conserved or not. And, in the last case, how and how much it turns out to be violated.In particular the unstability of the LSP associated with6Rp could be observable, especially if6Rp interactions were effectively responsible for neutrino masses. Beyond the possible discov-ery of supersymmetry, the knowledge about the conservationor possible violations ofR-parityis expected to be essential for the understanding of severalfundamental problems in particlephysics, and cosmology.

Acknowledgements

This review emerged from common efforts initiated in the framework of theR-Parity WorkingGroup of the FrenchGroupement de Recherche en Supersymetrie (GDR). We wish to thankP. Binetruy who headed the GDR for his continuous support. We are grateful to all members oftheR-Parity Working Group for useful discussions. We wish to acknowledge in particular thecontributions at early stages of this review work of F. Brochu, P. Coyle, D. Fouchez, P. Jonsson,F. Ledroit-Guillon, A. Mirea, E. Nagy, R. Nicolaidou, N. Parua and G. Sajot.

We also whish to thank theInstitut National de Physique Nucleaire et de Physique des Par-ticules(IN2P3), theCentre National de la Recherche Scientifique(CNRS) and theCommissariata l’Energie Atomique(CEA) for their support.

Appendix A

Notations and Conventions

In the following, the notations and conventions used throughout this review are presented.

The threeSU(3)C × SU(2)L × U(1)Y gauge couplings of the Standard Model are de-noted byg3, g andg′ respectively and the electroweak mixing angle byθW . We use the metric(+,−,−,−).

The superpartners of matter, Higgs and gauge fields in the Supersymmetric Standard Modelare denoted as follows:

• Scalar partners of left-handed quark fields(

uiL

diL

)(squarks) byQi =

(uiL

diL

), and scalar

partners of right-handed quark fieldsuiR, diR by uiR, diR (i = 1, 2, 3 is a family index). Simi-

larly, the superpartners of the left-handed lepton fields(

νiL

liL

)(sleptons) are denoted byLi =

(νiL

liL

), and those of the right-handed leptonsliR by liR. The corresponding superfields are

denoted with capital lettersQi =(

Ui

Di

), Li =

(Ni

Ei

), U c

i , Dci , E

ci . Since we use left-handed

chiral superfields only, right-handed fermion fields and their scalar partners are described by thecorrespondingCP conjugate fields (for example the scalar and fermion components ofU c

i areuc

i ≡ (uiR)⋆ anduci ≡ C(uiR)T , respectively).

• The two Higgs doublets of the Supersymmetric Standard Modelare denoted byhd =(h0

d

h−d

)andhu =

(h+

u

h0u

), the corresponding Weyl fermions (higgsinos) byhd and hu, and

the corresponding superfields byHd andHu. The Higgs VEVs are< h0d >= vd/

√2 and

< h0u >= vu/

√2 (we adopt the normalizationφ = (a+ib)/

√2 for complex scalar fields), and the

ratio of VEVs istanβ = vu/vd. The five physical Higgs states of the Minimal SupersymmetricStandard Model, in which no other superfield than the ones mentioned here are introduced,include two neutral scalars (CP -even) denoted byh (for the lightest one) andH, a chargedHiggs bosonH± and a pseudoscalar (CP -odd) Higgs bosonA.

• The Majorana fermion partners of the gluons (gluinos) are denoted byga (a = 1 · · ·8);similarly, the superpartners of theSU(2)L × U(1)Y gauge bosons are gaugino fields denotedby W i (i = 1, 2, 3) andB. Alternatively, one can define the fermionic partners of thephoton,ZandW± gauge fields: two Majorana spinorsγ ≡ sin θW W

3 + cos θW B andZ ≡ cos θW W3 −

sin θW B, and a Dirac spinorW± ≡ (W 1 ∓ iW 2)/√

2. The mass eigenstates of the higgsino-gaugino system, the neutralinos and the charginos, are denoted by χ0

l (l = 1 · · ·4) and χ±l′

(l′ = 1, 2), respectively.

The chiral superfields are normalized so that the lowest termin the θ, θ expansion of theleft-handed chiral superfieldΦ associated with the complex scalar fieldφ = (a + ib)/

√2 is

Φ|θ=θ=0 = (a + ib)/√

2. We adopt the following convention for the contraction of two SU(2)L

doubletsΦ andΨ: ΦΨ ≡ ǫabΦaΨb = Φ1Ψ2 − Φ2Ψ1, wherea, b = 1, 2 areSU(2)L indices,

ǫab = −ǫba is the totally antisymmetric tensor (withǫ12 = +1), andΦ1 (resp.Φ2) denotes theT3 = +1

2(resp.T3 = −1

2) component ofΦ.

The discussion ofR-parity violation does not depend, in general, of the particular version ofthe Supersymmetric Standard Model considered. In the following, we nevertheless specializefor clarity on the minimal supersymmetric extension of the Standard Model (MSSM). With theabove notations, the renormalizable superpotential of theMSSM reads

WMSSM = µHuHd + λeij HdLiE

cj + λd

ij HdQiDcj − λu

ij HuQiUcj , (A.1)

whereµ is the supersymmetric Higgs mass parameter, andλu,d,eij denote the quark and charged

lepton Yukawa coupling matrices. In Eq. (A.1), like in most equations of this review, a sum-mation over the generation indicesi, j = 1, 2, 3, and over gauge indices is understood. In theabsence ofR-parity, the following6Rp terms may also be added to the superpotential (A.1) :

W6Rp= µiHuLi +

1

2λijk LiLjE

ck + λ′ijk LiQjD

ck +

1

2λ′′ijk U

ciD

cjD

ck . (A.2)

The supersymmetric mass parametersµi as well as the trilinear couplingsλijk andλ′ijk violatelepton-number conservation law, while the couplingsλ′′ijk violate baryon-number conservationlaw. Gauge invariance enforces antisymmetry of theλijk (λ′′ijk) couplings in their first (last) twoindices:λijk = −λjik (λ′′ijk = −λ′′ikj). To avoid unwanted factors of 2 in scattering amplitudes,a factor1/2 has been introduced in the definition of theλijk andλ′′ijk couplings in Eq. (A.2). Itshould be noted that some authors omit these factors but restrict the sum over generation indicesto i < j (resp.j < k) in theλijk LiLjE

ck (resp.λ′′ijk U

ciD

cjD

ck) terms; this alternative writing is

equivalent to our definition (A.2).

The Supersymmetric Standard Model makes use of a large number of parameters describingour ignorance about the mechanism which breaks supersymmetry. As is customary, we considerthe most general terms that break supersymmetry in a soft way, i.e. without reintroducingquadratic divergences. In the MSSM, these “soft supersymmetry-breaking parameters” consistof the following:

• The mass parametersM1,M2 andM3 for theU(1)Y , SU(2)L andSU(3)C gauginos.

• 3 × 3 hermitian mass matrices for each type of squarks and sleptons, both left- and right-handed:m2

Q, m2

uc, m2dc , m

2L, m2

lc. We shall sometimes use the alternative notationmij for

(m2L)ij. WhenR-parity is broken, there may also be Higgs-slepton mixing soft massesm2

di.

• The ”analytic” (i.e. involving only the scalar components of chiral superfields, and nottheir complex conjugates) trilinear scalar couplingsA, with the same structure as the Yukawacouplingsλ. For example, the up-quark-type Yukawa couplingsλu

ijHuQiUcj have associated

trilinear soft termsAuij huQiu

cj. WhenR-parity is explicitly broken, there are also trilinear

couplingsAijk, A′ijk andA

′′

ijk corresponding to the6Rp superpotential couplingsλijk, λ′ijk andλ

′′

ijk, with the same symmetry properties. TheA parameters have mass dimension 1.

• The soft mass parametersm2d andm2

u for the two Higgs doubletshd andhu, and a bilinearanalytic mass termBhuhd, corresponding to the supersymmetric Higgs mass termµHuHd in

the superpotential. There are also6Rp bilinear soft termsBihuLi corresponding to the6Rp masstermsµiHuLi in the superpotential. TheB parameters have mass dimension 2.

The soft supersymmetry-breaking terms in the Lagrangian density of the MSSM are thengiven by

−LsoftRp

= (m2Q)ij Q

†i Qj + (m2

uc)ij uc†i u

cj + (m2

dc)ij dc†i d

cj + (m2

L)ij L

†i Lj + (m2

lc)ij l

c†i l

cj

+(Ae

ij hdLi lcj + Ad

ij hdQidcj − Au

ij huQiucj + h.c.

)

+ m2d h

†dhd + m2

u h†uhu + (Bhuhd + h.c.)

+1

2M1

¯BB +1

2M2

¯W 3W 3 + M2¯W+W+ +

1

2M3

¯gaga , (A.3)

where we have written the gaugino soft mass terms in a four-component notation, with MajoranaspinorsB, W 3, ga and a charged Dirac spinorW+. In the absence ofR-parity, additional softsupersymmetry-breaking terms may also be introduced in theLagrangian density, as given by:

−LsoftR/p

= V softR/p

=1

2Aijk LiLj l

ck + A′

ijk LiQjdck +

1

2A′′

ijk uci d

cjd

ck

+ Bi huLi + m2di h

†d Li + h.c. . (A.4)

Appendix B

Yukawa-like 6Rp Interactions Associatedwith the Trilinear 6Rp Superpotential

In the following, the Yukawa-like (fermion-fermion-scalar) 6Rp interactions associated with thetrilinear 6Rp superpotential couplings of Eq. (A.2) are derived.. The latter also give rise toRp

conserving scalar interactions that are quartic in the squark and slepton fields. However thesehave no significant phenomenological effects on the low-energy physics for heavy superpart-ners, so we do not discuss them here (see section 2.1.2).

Let us first derive explicitly the couplings trilinear in thefields generated by the part of thesuperpotential (A.2) given by

WLiLjEck

=1

2λijkLiLjE

ck. (B.1)

The trilinear couplings coming fromWLiLjEck

are obtained by differentiatingWLiLjEck, ex-

pressed in term of the scalar componentsz of the superfields, over all the scalar fields:

LLiLjEck

= −1

2

∑

α,β

∂2WLiLjEck(z)

∂zα ∂zβψαψβ − 1

2

∑

α,β

∂2W ∗LiLjEc

k(z)

∂z∗α ∂z∗β

ψαψβ , (B.2)

where the two-component spinorsψ are the superpartners of the scalar fieldsz. The two-component spinorsψ andψ belong respectively to the(1/2, 0) and(0, 1/2) representations ofthe Lorentz group. Eq.(B.1) and Eq.(B.2) lead together to,

LLiLjEck

= − 1

2

∑

α,β

∂2[

12λijk

(νiL ljL − liLνjL

)lckR

]

∂zα ∂zβψαψβ

− 1

2

∑

α,β

∂2[

12λ∗ijk

(ν∗iL l

∗jL − l∗iLν

∗jL

)lc∗kR

]

∂z∗α ∂z∗β

ψαψβ , (B.3)

whereν andl denote the sneutrinos and charged sleptons, respectively,the superscriptsc denotethe charge conjugate fields and the superscripts∗ the complex conjugate fields. The ‘R’ and ‘L’chirality indices for the scalar fields distinguish independent fields corresponding to superpart-ners of right- and left-handed fermions, respectively. TheLagrangian density (B.3) is equivalentto

LLiLjEck

= − 1

2λijk

(χνiχlj l

ckR + χνi

ηlk ljL + χljηlk νiL − (i↔ j)

)

− 1

2λ∗ijk

(χνiχlj l

c∗kR + χνi

ηlk l∗jL + χlj ηlk ν

∗iL − (i↔ j)

). (B.4)

In our notations, the two-component spinorsχl (χν) andηl (ην) associated with the chargedlepton (neutrino) are related to the four-component Dirac spinors describing the charged leptonsl (neutrinosν) and antileptonslc (antineutrinosνc) by

l =

(χl

ηl

), lc =

(ηl

χl

), ν =

(χν

ην

), νc =

(ην

χν

). (B.5)

The products of two-component spinorsψ and ψ and the products of four-component DiracspinorsΨ andΨ = Ψ†γ0 are related through the equations,

Ψ1PLΨ2 = η1χ2, Ψ2PRΨ1 = η1χ2, (B.6)

wherePL andPR are respectively the left and right chirality projectors. By applying the rela-tions (B.6), one can express the Lagrangian density (B.4) interms of the four-component Diracspinors:

LLiLjEck

= −1

2λijk

(νiLlkRljL + ljLlkRνiL + l⋆kRν

ciRljL − (i↔ j)

)+ h.c., (B.7)

where for instanceνciR = (νc

i )R.

Similarly, the couplings trilinear in the fields generated by the superpotential termsWLiQjDck

=

λ′ijkLiQjDck andWUc

i DcjDc

k= 1

2λ′′ijkU

ci D

cjD

ck are found to be

LLiQjDck

= −λ′ijk(νiLdkRdjL + djLdkRνiL + d⋆

kRνciRdjL

−liLdkRujL − ujLdkRliL − d⋆kRl

ciRujL

)+ h.c., (B.8)

and

LUci Dc

jDck

= −1

2λ′′ijk

(u⋆

iRdjRdckL + d⋆

jRuiRdckL + d⋆

kRuiRdcjL

)+ h.c. , (B.9)

respectively.

Appendix C

Production and Decay Formulae

In the following, some useful formulas relevant forR-parity violation searches at colliders arelisted. This section is based to a large extent on appendix B of [552]. The formulas for thedecays are organized here by particle families.

Mixing

The mixing for the first two generations of sleptons and squarks is expected to be small to a goodaccuracy due to the small fermion masses in the off-diagonalelements of the mass matrices. Onthe other hand, a large mixing between the left and right handed stops is expected because ofthe large top-quark mass.

For the current eigenstatesqiL,R and the mass eigenstatesqi

1,2 the mixing matrix is

(qiL

qiR

)=

(cos θi

q sin θiq

− sin θiq sin θi

q

)(qi1

qi2

). (C.1)

It will be denoted asQijk wherei = u, d, s, c, b, t is the quark flavour index. The slepton

mixing matrix is similar and will be denoted asLijk, wherei = e−, νe, µ

−, νµ, τ−, ντ is the

lepton flavour index. Sfermion mixing between generations will be neglected.

Two body decays

The two-body decay rate corresponding to an averaged matrixelementM(A → B + C) withno angular dependence is:

Γ(A→ B + C) =|M(A→ B + C)|2

16πm3A

√[m2

A − (mB +mC)2] [m2A − (mB −mC)2] . (C.2)

Three body decays

The partial width is given by

Γ(A→ 1 + 2 + 3) =1

(2π)3

1

32m3A

∫ (m212)max

(m212)min

d m212

∫ (m223)max

(m223)min

d m223|M |2, (C.3)

wherem212 ≡ (p1 + p2)

2 = m2A +m2

3 − 2mAE3 andp1, p2 are the 4-momenta of particles 1 and2 respectively, whileE3 is the energy of particle 3 in the rest frame of particleA. m23 is definedin a similar way. Therefore we obtain(m2

12)max = (mA −m3)2, (m2

12)min = (m1 +m2)2,

(m2

23

)max

= (E2 + E3)2 −

(√E2

2 −m22 −

√E3

2 −m23

)2

,

(m2

23

)min

= (E2 + E3)2 −

(√E2

2 −m22 +

√E3

2 −m23

)2

.

E2 = (m212 −m2

1 +m22) /2m12 andE3 = (m2

A −m212 −m2

3) /2m12 are now the energies ofparticles 2 and 3 in the rest frame of the reduced variablem12.

C.1 Sfermions

Two-bodyR-parity-violating decays of sfermions are given by (C.2) with the following matrixelements (averaged over spin and colour).α is the mass eigenstate of the sfermion if there ismixing. i, j, k are the generation indices.

For sneutrinos we have:

|M(νj → ℓ+i ℓ−k )|2 = |λijk|2(m2

ν −m2ℓi−m2

ℓk) ,

|M(νi → djdk)|2 = Nc|λijk′|2(m2

ν −m2dj−m2

dk) , (C.4)

for sleptons:

|M(e−jα → νiℓ−k )|2 = |λijk|2|L2j−1

1α |2(m2e −m2

ℓk) ,

|M(e−kα → νiℓ−j )|2 = |λijk|2|L2k−1

2α |2(m2e −m2

ℓj) ,

|M(e−iα → ujdk)|2 = Nc|λijk′|2|L2i−1

1α |2(m2e −m2

uj−m2

dk) , (C.5)

for squarks:

|M(ujα → e+i dk)|2 = |λijk′|2|Q2j

1α|2(m2u −m2

ei−m2

dk) ,

|M(uiα → djdk)|2 = (Nc − 1)!|λijk′′|2|Q2i

2α|2(m2u −m2

dj−m2

dk) ,

|M(djα → νidk)|2 = |λijk′|2|Q2j−1

1α |2(m2d−m2

dk) ,

|M(dkα → νidj)|2 = |λijk′|2|Q2k−1

2α |2(m2d−m2

dj) ,

|M(dkα → e−i uj)|2 = |λijk′|2|Q2k−1

2α |2(m2d−m2

ei−m2

uj) ,

|M(dkα → uidj)|2 = (Nc − 1)!|λijk′′|2|Q2k−1

2α |2(m2d−m2

ui−m2

dj) , (C.6)

whereNc is the number of colours.

Stops

A large mixing between the left and right handed stops is expected because of the large top–quark mass. We show here explicitly the effect of the mixing given in general terms above. Themass eigenstates are [438]

(t1t2

)=

(tL cos θt − tR sin θt

tL sin θt + tR cos θt

), (C.7)

whereθt denotes the mixing angle of the stops:

sin 2θt =2atmt√

(m2

tL−m2

tR)2 + 4a2

t m2t

, (C.8)

cos 2θt =m2

tL−m2

tR√(m2

tL−m2

tR)2 + 4a2

t m2t

. (C.9)

ThemfL,R

andaf are the SUSY mass parameters andmt is the top-quark mass. The mass

eigenvalues are given by:

m2

t1=

1

2

m2

tL+m2

tR−[(m2

tL−m2

tR)2 + (2atmt)

2

]1/2,

m2

t2=

1

2

m2

tL+m2

tR+

[(m2

tL−m2

tR)2 + (2atmt)