A Constrained Minimal Flavour Violation model facing ...

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A Constrained Minimal Flavour Violation model facingflavour Physics data

Evan Machefer

To cite this version:Evan Machefer. A Constrained Minimal Flavour Violation model facing flavour Physics data. Physics[physics]. 2014. �dumas-01240491�

UFR Sciences et Technologies

MASTER SCIENCES DE LA MATIEREDEUXIÈME ANNÉE

SPÉCIALITÉ : Physique des Particules

RAPPORT DE STAGE

A Constrained Minimal Flavour Violationmodel facing flavour Physics data.

par

Evan MACHEFER

Responsable(s) de stage : Stéphane MONTEIL Jean ORLOFF

Juin 2014

Université Blaise Pascal - IN2P3 - CNRS 24 avenue des Landais BP 80026 63171 Aubière Cedex

E. Machefer

ii

Acknowledgement

I would like to express my gratitude to my supervisors, Stephane Monteil

and Jean Orloff, for their advices and their remarks that lead me to abetter understanding of the subject.

I would like to thank Marouen as well, for sharing his knowledge of sub-jects unknown to me, that helped me to work in a more efficient way.

iii

E. Machefer

iv

Contents

Acknowledgement iii

Introduction 1

1 CKM matrix and models of Minimal Flavour Violation 3

1.1 CKM matrix and Unitarity Triangles . . . . . . . . . . . . . . 31.1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . 31.1.2 Number of free parameters . . . . . . . . . . . . . . . . 41.1.3 Standard parametrisation . . . . . . . . . . . . . . . . 51.1.4 Wolfenstein parametrisation . . . . . . . . . . . . . . . 51.1.5 Unitarity triangles . . . . . . . . . . . . . . . . . . . . 5

1.2 Minimal Flavour Violation . . . . . . . . . . . . . . . . . . . . 7

2 Global fits of CKM matrix elements: the LHCb Bs inputs 9

2.1 Preliminary part . . . . . . . . . . . . . . . . . . . . . . . . . 92.2 Fit for the (ρs, ηs) plane . . . . . . . . . . . . . . . . . . . . . 10

3 CMFV in neutral meson mixing 13

3.1 No New Physics at tree-level assumption . . . . . . . . . . . . 133.2 Modification of tree-level processes . . . . . . . . . . . . . . . 15

Conclusion 19

v

E. Machefer

vi

Introduction

The Standard Model of particles physics is, so far, the best description thereis of the subatomic world, and the recent discovery of a new scalar particle,makes it now complete. It has, for more than sixty years, resisted to all thetests it was submitted to, and one of those tests is the consistency check of theCabibbo–Kobayashi–Maskawa (CKM) matrix, allowing flavour transitionswithin weak charged currents. This matrix can be described by four freeparameters, that need to be determined by the experiment.

Overconstraining the parameters describing the CKM matrix, one canprobe the consistency between the different experiments and the SM expec-tations. These tests of the CKM matrix are made by two main groups,one is the UTfit collaboration, and the other is the CKMfitter group, whoseframework is used hereafter.

CKMfitter is a group of a ten of people gathered to provide worldwiderenowned fits, using a minimiser written in fortran, and theories written inthe Mathematica language.

The first part of this work will focus on the basic knowledge needed tounderstand what are Constrained Minimal Flavour Violation (CMFV) mod-els, the second part will talk about how new measurements can constrain theCKM matrix elements, and the last part will deal with the CMFV imple-mentation within the CKMfitter package.

1

E. Machefer

2

Chapter 1

CKM matrix and models of

Minimal Flavour Violation

1.1 CKM matrix and Unitarity Triangles

1.1.1 Introduction

The Standard Model (SM) is described by a lagrangian density with a SU(3)c×SU(2)L × U(1)Y gauge invariance [1]. In this model, fermions are classifiedin left-handed doublets and in right-handed chirality singlets.

Gauge invariance forbids mass terms for fermions or gauge bosons. Theintroduction of a scalar doublet of SU(2)L, denoted φ, breaks spontaneouslythe gauge symmetry when it acquires a vacuum expected value. Conversely,three out of the four scalar fields are absorbed to provide a longitudinalpolarisation for the Z and W bosons. At the same time, the φ field couples tothe quarks by the following most general gauge-invariant and renormalisableYukawa terms in the lagrangian density:

LY = −λdijQ

I3LiφD

I3Rj − λu

ijQI3Liǫφ

∗U I3Rj + h.c, (1.1)

where λu,d are 3 × 3 complex mass matrices, φ is a scalar SU(2) doubletfield, i, j are generation labels, and ǫ is the 2 × 2 antisymmetric tensor.QI3

L are left-handed quark doublets, and UR, DR are right-handed singletsin the weak eigenstate basis. When φ acquires a vacuum expectation value,〈φ〉 = (0, v/

√2), mass terms appear

−MdijD

I3LiD

I3Rj −Mu

ijUI3LiU

I3Rj + h.c,

with,

Mu(d) =λu(d)v√

2.

3

E. Machefer

The mass matrix can be diagonalised,

Vu(d)L Mu(d)V

u(d)R =

mu(d) 0 00 mc(s) 00 0 mt(b)

,

with Vu(d)L and V

u(d)R unitary matrices. One can then define,

QI3Li =

(

U I3Li

DI3Li

)

= (V u†L )ij

(

U′

Lj

(V uL V

d†L )jkD

′

Lk

)

.

The Cabbibo-Kobayashi-Maskawa (CKM) ([2], [3]) matrix, defined asV uL V

d†L , is a (3×3) unitary matrix allowing the transition between the flavours

of quarks. It relates the weak eigenstates DI3 = (d, s, b) to the mass eigen-states D′ = (d′, s′, b′),

d′

s′

b′

= V uL V

d†L

dsb

= VCKM

dsb

=

Vud Vus Vub

Vcd Vcs Vcb

Vtd Vts Vtb

dsb

. (1.2)

The matrix elements Vij (i = u, c, t ; j = d, s, b) are complex numbers.

1.1.2 Number of free parameters

For an (n × n) complex matrix, where n describes the number of quarkgenerations, there are 2n2 free real parameters, one modulus and one phasefor each matrix element. However, for unitary matrices, the relations

(V V †)jk =∑

i

VjiV∗ki = δjk, (1.3)

where δjk is the Kronecker delta1 , reduce the number of independent param-eters. The compact writing of Eq. (1.3) embodies constraints on real param-eters only, such as |Vud|2+ |Vus|2+ |Vub|2 = 1 (j = k), and real and imaginaryparts, as in the scalar product between columns j 6= k. Eq. (1.3) leads ton + (n(n− 1)) /2 independent constraints on the moduli, and (n(n− 1)) /2constraints on phases.

Moreover, physics is invariant under

V →

eiφU

1 00

0 eiφUn

V

eiφD

1 00

0 eiφDn

,

1δjk =

∣

∣

∣

∣

1 if i = j

0 if i 6= j

4

1.1. CKM MATRIX AND UNITARITY TRIANGLES

where φU,Di are phases of the up (U) or down (D) mass eigenstate fields.

This rephasing freedom of the 2n quark fields can in general be used tofix only 2n− 1 phases of V , the last rephasing freedom φU,D

i = φglobal leavingV unchanged. Thus, the total number of free parameters will be (n − 1)2.The (3×3) CKM matrix depends hence on 4 parameters, one being a phase.The latter implies that CP violation can be accounted for in this framework.

1.1.3 Standard parametrisation

The standard parametrisation [4] for VCKM uses 3 Euler angles (θij) and oneglobal phase (δ),

VCKM =

c12c13 s12c13 s13e−iδ

−s12c23 − c12s23s13eiδ c12c23 − s12s23s13e

iδ s23c13s12s23 − c12c23s13e

iδ −s23c12 − s12c23s13eiδ c23c13

, (1.4)

where cij = cos(θij) and sij = sin(θij).

1.1.4 Wolfenstein parametrisation

In practice, s13 ≪ s23 ≪ s12 ≪ 1, and a more convenient way to expressVCKM is due to Wolfenstein[5]. The following definitions [6] are adopted:

s212 = λ2 =|Vus|2

|Vud|2 + |Vus|2, s223 = A2λ4 =

|Vcb|2|Vud|2 + |Vus|2

,

s13eiδ = V ∗

ub = Aλ3(ρ+ iη), ρ+ iη = −VudV∗ub

VcdV ∗cb

,

which ensures a parametrisation that is both phase-independent and unitaryto all. It can be noted that λ ≃ Vus. For the sake of illustration, Eq. (1.5)shows a truncated expression of VCKM up to O(λ4),

VCKM =

1− λ2/2 λ Aλ3(ρ− iη)−λ 1− λ2 Aλ2

Aλ3(1− ρ− iη) −Aλ2 1

+O(λ4). (1.5)

1.1.5 Unitarity triangles

Two relations coming from Eq. (1.3) will be used in the following,

VudV∗ub + VcdV

∗cb + VtdV

∗tb = 0, (1.6)

VusV∗ub + VcsV

∗cb + VtsV

∗tb = 0. (1.7)

5

E. Machefer

Notice that the magnitude of each component of Eq. (1.6) is of the sameorder O(λ3).

Dividing each term of Eq. (1.6) by VcdV∗cb, one obtains,

VudV∗ub

VcdV ∗cb

+ 1 +VtdV

∗tb

VcdV ∗cb

= 0. (1.8)

Defining then

ρ+ iη = −VudV∗ub

VcdV ∗cb

,

this sum of three complex numbers can be seen as the closing of an unitarytriangle, i.e. a triangle with unit basis, in the complex plane (ρ, η) as shownin Fig. 1.1. The different parameters displayed are

Ru =

∣

∣

∣

∣

VudV∗ub

VcdV ∗cb

∣

∣

∣

∣

, Rt =

∣

∣

∣

∣

VtdV∗tb

VcdV ∗cb

∣

∣

∣

∣

,

α = Φ2 = arg

(

− VtdV∗tb

VudV ∗ub

)

, β = Φ1 = arg

(

−VcdV∗cb

VtdV ∗tb

)

, γ = Φ3 = arg

(

−VudV∗ub

VcdV ∗cb

)

.

In the following, only the notation (α, β, γ) will be used for the angles.

C = (0, 0)✲γ

Ru

A = (ρ, η)

❩❩

❩❩

❩❩

❩❩

❩❩

❩❩⑥α

Rt

B = (1, 0)

✂✂✂✂✂✂✂✂✂✌ β

Figure 1.1: Unitary Triangle

We will scrutinise in the section 2.1 the Eq. (1.7) which sides and anglescan be obtained from observables related to the Bs meson. With an adequatenormalisation, one obtains

VusV∗ub

VcsV ∗cb

+ 1 +VtsV

∗tb

VcsV ∗cb

= 0, (1.9)

with ρs + iηs = −VusV∗ub

VcsV ∗cb

. Because the terms in this equation have different

powers of λ, the corresponding triangle will be squashed, contrarily to theone of Eq. (1.8) related to the Bd meson.

6

1.2. MINIMAL FLAVOUR VIOLATION

1.2 Minimal Flavour Violation

The Standard Model (SM) describes three out of the four elementary inter-actions in the framework of gauge theories, and it has not been proven wrongtill now. It indeed explains most of the phenomena that have been observed,but it fails at naturally explaining the baryonic asymmetry in the Universeand does not provide a dark matter candidate. The remarkable descriptionof all flavour physics and CP violation observables within the KM frame-work suggests that any New Physics (NP) model should not bring additionalflavour violation beyond that present in the SM. This data-driven assumptiondefines classes of NP models denoted as Minimal Flavour Violation (MFV).

In MFV models [7], all flavour changing transitions and Charge Parity(CP) violation come from the CKM matrix. The decay amplitudes in MFVmodels can be written as:

A(M → F ) = Pc(M → F ) +∑

r

Pr(M → F )Fr(v), (1.10)

where Fr(v) are real process-independent master functions, that are reducedto the Inami-Lim functions [8] in the SM, and Pc and Pr are process-dependent,but model-independent, coefficients. Pc summarises the light quarks contri-butions, in particular the charm quark, and Pr in the sum, incorporates theremaining contributions.

The MFV master functions and Inami-Lim functions come from penguinand box diagram computations as the ones shown on Fig. 1.2.

(a) (b)

Figure 1.2: Box and penguin diagrams.

Their knowledge is required for the determination of the predictions of thebranching fractions. The master functions associated with various processesare given in Table 1.1.

As an example, the branching fraction of Bs → µ+µ− is given by

B(Bs → µ+µ−) = C|VtsV∗tb|2ηY Y (v) (1.11)

where ηY are the QCD corrections, and

C =τBs

G2FmBs

m2µα

2EM(MZ)

16π3 sin2 θW~

√

1−4m2

µ

mB2s

f 2Bs. (1.12)

7

E. Machefer

K0 − K0-mixing (ǫK) S(v)B0

d,s − B0d,s-mixing (∆md,s) S(v)

K → πνν, B → Xd,sνν X(v)KL → µ+µ−, B → l+l− Y (v)KL → π0e+e− Y (v), Z(v), E(v)ǫ′, Non-leptonic ∆B = 1, ∆S = 1 X(v), Y (v), Z(v), E(v)B → XSγ D′(v), E ′(v)B → XSgluon E ′(v)B → XSl

+l− Y (v), Z(v), E(v), D′(v), E ′(v)

Table 1.1: The observables related to the master functions.

GF is the Fermi constant, αEM(MZ) is the electromagnetic structure constanttaken at the mass of the Z boson and fBs

is a form factor, the decay constant,given by lattice QCD computations.

The Y (v) master function absorbs any CMFV constrained NP contribu-

tions. Conversely, the v parameter generalises the parameter xt =m2

t

M2W

. A

specific CMFV model will add up to v the mass dependencies on the hypo-thetical new particles.

In this paper, the first implementation of CMFV model will focus on themixing processes, since S(v) is only related to them, as seen on Table 1.1.

8

Chapter 2

Global fits of CKM matrix

elements: the LHCb Bs inputs

2.1 Preliminary part

Before any implementation of CMFV models in the theory packages of CKM-fitter, one should understand how the fit is done. The most simple case, isthe fit for the (ρ, η) plane using all the standard observables. We have triedto perform a similar fit but using the (ρs, ηs) instead.

The result given by CMS and LHCb on the branching ratio of Bs → µ+µ−

[9] gives us information on the |Vts| matrix element. Because it does not giveany direct constraint in the (ρ, η) plane, it is instructive to implement itseffect on a fit in the (ρs, ηs) plane instead.

This fit is usually done on (ρ, η) plane then transposed to the (ρs, ηs)plane, but since Bs → µ+µ− does not contribute directly in the first, con-straint displayed on the latter might suffer from numerical convergency prob-lems. In order to see the constraint of this decay, it is more convenient (butfully equivalent) to re-express all the parameters in terms of ρs and ηs insteadof ρ and η.

From the definition

ρs + iηs = −VusV∗ub

VcsV ∗cb

,

V ∗ub can be extracted in terms of ρs and ηs:

V ∗ub = −Aλ

√1− λ2

√1− A2λ4(ρs + iηs)

1− A2λ4(ρs + iηs). (2.1)

Since (ρ + iη) = −VudV∗ub

VcdV ∗cb

, ρ and η can in turn be expressed in terms of ρs

9

E. Machefer

and ηs1,

ρ =λ2 − 1

λ2

ρs − A2λ2(ρ2s + η2s)

A4λ4η2s + (1− A2λ2ρs)2, (2.2)

and,

η =λ2 − 1

λ2

ηsA4λ4η2s + (1− A2λ2ρs)2

. (2.3)

Including this change of variables, the fit will be done using ρs and ηsinstead of ρ and η.

2.2 Fit for the (ρs, ηs) plane

With the definition given with Eq. (1.9), the |Vts| constraint given by B(Bs →µ+µ−) = (2.9 ± 0.7)10−9 will lead to a circle centred on (1, 0) as shown onFig. 2.1.

rhobars

-2 -1 0 1 2

eta

bars

-2

-1

0

1

2

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0p-value

excluded area has CL > 0.95

CKMf i t t e r

Figure 2.1: Fit of ρs, ηs using |Vcb|, |Vcs| and B(Bs → µµ) as inputs.

If |Vud| and |Vus| are included as inputs to get a constraint on A and λ,the authorised region will greatly be reduced, only allowing (ρs, ηs) to be

1This kind of change of variable can also be applied to ρ and η.

10

2.2. FIT FOR THE (ρS , ηS) PLANE

near the origin. The reason is that A and λ greatly constrain ρs and ηs dueto the relation between |Vts|, A and λ:

|Vts|2 =Aλ2

1− A2λ4

[

(ρs − 1)2 + η2s]

|Vcs|2. (2.4)

Including also ∆ms = 17.762±0.023 ps−1, the mass difference in B0s − B0

s

mixing, and φs = 0.01 ± 0.07 rad, the CP violating phase in B0s decays, in

the fit for (ρs, ηs) plane, Fig. 2.2a is obtained. With the expected precision

sρ

-0.15 -0.10 -0.05 0.00 0.05 0.10

sη

-0.15

-0.10

-0.05

-0.00

0.05

0.10

0.15

0.20

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0p-value

excluded area has CL > 0.95

bs plane

CKMf i t t e r

(a) Fit for ρs, ηs using φs, ∆ms andB(Bs → µµ) as inputs. The pink dotis the region allowed with the standardfit.

sρ

-0.15 -0.10 -0.05 0.00 0.05 0.10

sη

-0.15

-0.10

-0.05

-0.00

0.05

0.10

0.15

0.20

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0p-value

excluded area has CL > 0.95

bs plane

CKMf i t t e r

(b) Fit for ρs, ηs using the expected pre-cision of φs, ∆ms and B(Bs → µµ) asinputs. The pink dot is the allowed re-gion with the standard fit.

for the LHCb upgrade with the second run of the LHC, φs = 0.01±0.01 rad,∆ms = 17.762 ± 0.023 ps−1 and B(Bs → µ+µ−) = (3.0 ± 0.3)10−9. Adifference between the value of ρs and ηs obtained with the usual fit (inpink in Fig. 2.2a and Fig. 2.2b) and the allowed region with φs, ∆ms andB(Bs → µ+µ−), might occur as shown in Fig. 2.2b. It can be seen that theexpected precision of φs can radically change the allowed region, and thatthe precision attained with B(Bs → µ+µ−) will not have any effect on thefit.

It is interesting to remark that the precision attained with the fit withB0

s systems is not as good as the precision attained with Bd systems relatedfit.

11

E. Machefer

12

Chapter 3

CMFV in neutral meson mixing

3.1 No New Physics at tree-level assumption

B0d,s−B0

d,s oscillations is an effective way to measure the matrix elements |Vtd|and |Vts| precisely in the framework of the Standard Model. Experimentally,the mass difference between the two states will be measured, ∆md and ∆ms,for B0 and B0

s respectively, from time-dependent analysis of B oscillations.In the SM, ∆F = 2 transitions are described by the Inami-Lim function

S0(xq), with xq = m2q/M

2W . For example, the explicit mathematical expres-

sion of a ∆S = 2 transition, as a possible process is shown in Fig. 3.1, is:

Box(∆S = 2) = λ2i

G2F

16π2M2

WS0(xt, xc)(sd)V−A(sd)V−A, (3.1)

where λi = V ∗isVid, (i = u, c, t). In fact, that box is linked to the hamiltonian

governing the evolution of the ∆S = 2 system.

✛

✲s t, c, u d

d t, c, u s

W W

Figure 3.1: Box diagram of a ∆F = 2 transition.

Including CMFV models, one free parameter must be added to describethese mixing. This free parameter is the master function S(v), that will beexpressed as r × S0(xt) in the following.

13

E. Machefer

The fit for the r parameter gives us an allowed region for NP that doesnot exclude the SM, where r is exactly equal to one by definition. Theordinates are the probability value expressed in terms of Confidence Level(CL) of exclusion:

p− value = 1− CL

SM value

r

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0

1-C

L

0.0

0.2

0.4

0.6

0.8

1.0

CMFV

CKMf i t t e r

with no NP at tree levelr

in the SMr

(a) Comparison between CMFV imple-mentation in a LO package and the SMpoint.

SM value

r

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0

1-C

L

0.0

0.2

0.4

0.6

0.8

1.0

CMFV

CKMf i t t e r

using CMFV implementationr

using NP assumptionsr

(b) Comparison between the NP packageat NLO and the CMFV implementationat LO.

Figure 3.2: Fit for r using |Vud|, |Vus|, |Vub|, |Vcb|, sin 2β, α, γ, ∆md and ∆ms

as inputs.

To test the validity of our implementation, we have compared our resultswith those of a pre-existing so-called ”NP package” [10], where differentassumption were made. It must be noticed that, for the sake of the devel-opment, the CMFV implementation has been made at Leading Order (LO)(and will be extended to the Next to Leading Order (NLO) at the end of theinternship), whereas the ”NP package” is at NLO.

In order to be able to compare the two packages, a change of variablesmust be made for the NP part. The ”NP package” is described by six realparameters,

Re(∆d), Re(∆s), Re(∆ttK), Im(∆d), Im(∆s), Im(∆tt

K),

where:

∆q =〈Mq|Heff

q |Mq〉〈Mq|HSM

q |Mq〉, (3.2)

with Heffq the effective hamiltonian of the Mq − Mq system (Mq = B0

d,s, K0),

and HSM the SM hamiltonian. The ∆q are free parameters added in B0

14

3.2. MODIFICATION OF TREE-LEVEL PROCESSES

mixing, B0s mixing, and K0 mixing. B0

d,s mixing will be multiplied by |∆q|,whereas in ǫK , only the top loop will be affected with the term Im[(VtsV

∗td)

2∆ttK ].

In order to have an equivalent of the r parameter from CMFV, the followingassumptions are made:

Re(∆d) = Re(∆s) = Re(∆ttK) = r, (3.3)

and,

Im(∆d) = Im(∆s) = Im(∆ttK) = 0, (3.4)

so that no new phase is added, and only one parameter describes NP contri-butions.

Fig. 3.2b shows that our implementation is compatible with the ”NP

package”, for the same assumptions as in CMFV models.With the assumption that no NP appears in tree level processes, CMFV

deviations of order twenty percent are allowed, as seen in Fig. 3.2a andFig. 3.2b.

3.2 Modification of tree-level processes

Since there is a tension between inclusive and exclusive determinations ofboth |Vub| and |Vcb|, it is worth exploring the possibility that tree level pro-cesses are affected by NP as well. Numerically,

|Vub|incl = (4.36± 0.18± 0.44)10−3, |Vub|excl = (3.23± 0.20± 0.26)10−3,

|Vcb|incl = (41.88± 0.44± 0.59)10−3, |Vcb|excl = (39.55± 0.51± 1.42)10−3.

These values should then be excluded from the fit in order to see how theallowed region for r is modified with this assumption.Fig. 3.3 shows the effectof relaxing the constraints from |Vub| and |Vcb| on the r parameter.

In this fit, the observables |Vud|, |Vus|, the angles α, β, γ, and the mesonmixing ∆md, ∆ms and ǫK constrain the free parameters. |Vud| and |Vus| areused in λ =

|Vus|2|Vud|2 + |Vus|2

, hence λmight not be affected by NP contribution.

The parameter A, usually constrained by |Vcb|, is constrained here by thecombination of ∆md, ∆ms, and ǫK , processes which also constrain the rparameter.

It can be seen on Fig. 3.3 that it is |Vcb|, and hence the parameter A,that brings the most important information on the fit for r. CMFV devia-tions from the SM of the order of forty percent become possible when thisinformation is lost.

15

E. Machefer

SM value

r

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0

1-C

L

0.0

0.2

0.4

0.6

0.8

1.0

CMFV

CKMf i t t e r

cbV and

ubV withr

cbV withoutr

cbV and

ubV withoutr

Figure 3.3: Fit for r using |Vud|, |Vus|, sin 2β, α, γ, ∆md and ∆ms, andwithout |Vub| and |Vcb| as inputs.

The fit for |Vub| in CMFV models, is shown in Fig. 3.4: the expected valueis |Vub|CMFV = (3.70±0.52)10−3, to be compared with |Vub|SM = (3.70±0.12±0.26)10−3.

Since the (small) slope of the highest Confidence Level (CL) plateausin Fig. 3.3 and Fig. 3.4 are inverted, it worth investigating the correlationbetween r and |Vub|. This is shown in Fig. 3.5. The small correlation be-tween r and |Vub| is indeed negative, and the allowed region includes the SMexpectation.

A question arises at this stage: is it possible to cook a NP model workingat tree level and satisfying the CMFV first assumptions (i.e. where flavourchanging transition and CP violation only originate in the CKM matrix)?

A first attempt would be to add a new SU(2)′L broken gauge symmetryinto the description, but a problem would appear with the Yukawa couplings.

Another attempt would be to implement a new SU(2)N acting only onthe scalars, and whose mixing with SU(2)L would alter tree level amplitudes.However, fixing the effective GF to its value from muon decay would fix alltree-level low energy processes to their SM value. To overcome this difficulty,one would be forced to venture into leptophobic models, so that NP wouldonly affect quarks.

The implementation of the master function solely related to the mixingprocesses was the first step in implementing CMFV models into the frame-work of CKMfitter. To complete this work, one should include observables

16

3.2. MODIFICATION OF TREE-LEVEL PROCESSES

InclusiveExclusive

ubV

0.0020 0.0025 0.0030 0.0035 0.0040 0.0045 0.0050 0.0055

1-C

L

0.0

0.2

0.4

0.6

0.8

1.0

CMFV

CKMf i t t e r

in CMFV ub

V

from inclusive decays ub

V

from exclusive decays ub

V

in SM ub

V

Figure 3.4: Prediction fit for |Vub| in CMFV models

depending on other CMFV master functions, adding new free parametersinto the fit, as well as using new observables.

17

E. Machefer

SM

ubV

0.000 0.001 0.002 0.003 0.004 0.005 0.006

r

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0p-value

excluded area has CL > 0.95

CMFV

CKMf i t t e r

Figure 3.5: Correlation between r and |Vub|.

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Conclusion

To conclude, we have seen in this paper a way to introduce a ConstrainedMinimal Flavour Violation model into the CKMfitter package, as well as theeffect of the new LHCb measurement of B(Bs → µ+µ−) and φs on the (ρs, ηs)plane. It is observed that the most important measurement as far as the nulltests of Standard Model hypothesis are concerned is the φs measurement.

Secondly, it has been seen that slightly modifying the theory by addinga multiplicative free parameter in the B0

d,s − B0d,s and K − K mixing, order

twenty percent CMFV contributions are allowed, going up to forty percentif New Physics appears at tree level. For the other processes, the implemen-tation will be of the same type, adding in total 7 new free parameters to theStandard Model (more if New Physics appears at tree level).

The next step is to implement CMFV in the Next to Leading Orderpackage, as well as including an additional parameter governing tree levelamplitudes modifications, so that the parameters |Vub| and |Vcb| could beincluded in the fit once again.

Eventually a complete CMFV analysis does require new observables tobe introduced in order to constrain the other master function. This is aperspective made possible by the exploration work made in this document.

19

E. Machefer

20

List of Figures

1.1 Unitary Triangle . . . . . . . . . . . . . . . . . . . . . . . . . 61.2 Box and penguin diagrams. . . . . . . . . . . . . . . . . . . . 7

2.1 Fit of ρs, ηs using |Vcb|, |Vcs| and B(Bs → µµ) as inputs. . . . . 10

3.1 Box diagram of a ∆F = 2 transition. . . . . . . . . . . . . . . 133.2 Fit for r using |Vud|, |Vus|, |Vub|, |Vcb|, sin 2β, α, γ, ∆md and

∆ms as inputs. . . . . . . . . . . . . . . . . . . . . . . . . . . 143.3 Fit for r using |Vud|, |Vus|, sin 2β, α, γ, ∆md and ∆ms, and

without |Vub| and |Vcb| as inputs. . . . . . . . . . . . . . . . . . 163.4 Prediction fit for |Vub| in CMFV models . . . . . . . . . . . . . 173.5 Correlation between r and |Vub|. . . . . . . . . . . . . . . . . . 18

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22

Bibliography

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[2] N. Cabibbo, “Unitary symmetry and leptonic decays,” Phys. Rev. Lett.,vol. 10, pp. 531–533, Jun 1963.

[3] M. Kobayashi and T. Maskawa, “CP Violation in the RenormalizableTheory of Weak Interaction,” Prog.Theor.Phys., vol. 49, pp. 652–657,1973.

[4] J. Beringer et al., “Review of Particle Physics (RPP),” Phys.Rev.,vol. D86, p. 010001, 2012.

[5] L. Wolfenstein, “Parametrization of the kobayashi-maskawa matrix,”Phys. Rev. Lett., vol. 51, pp. 1945–1947, Nov 1983.

[6] J. Charles, O. Deschamps, S. Descotes-Genon, R. Itoh, H. Lacker, et al.,“Predictions of selected flavour observables within the Standard Model,”Phys.Rev., vol. D84, p. 033005, 2011.

[7] A. J. Buras, “Minimal flavor violation,” Acta Phys.Polon., vol. B34,pp. 5615–5668, 2003.

[8] T. Inami and C. Lim, “Effects of Superheavy Quarks and Leptons inLow-Energy Weak Processes KL → µµ, K+ → π+νν, and K0 ↔ K0,”Prog.Theor.Phys., vol. 65, p. 297, 1981.

[9] CMS and L. Collaborations, “Combination of results on the rare decaysB0

(s) → µ+µ− from the CMS and LHCb experiments,” 2013.

[10] A. Lenz, U. Nierste, J. Charles, S. Descotes-Genon, A. Jantsch, et al.,“Anatomy of New Physics in B − B mixing,” Phys.Rev., vol. D83,p. 036004, 2011.

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E. Machefer

Abstract

The aim of the present work is to implement a Constrained Minimal FlavourViolation (CMFV) model into the B0

d,s and K0 mixing processes. To do so,the use of an additional free parameter is needed to describe our theory.Adding this parameter, CMFV deviations of order between twenty or fortypercent are allowed, depending whether New Physics appears or not at treelevel.

Resume

L’objectif du travail ici present est d’implementer un modele de violation desaveur minimale contraint (CMFV) dans les processus de melange B0

d,s etK0.

Pour ce faire, un parametre libre est utilise afin de decrire notre theorie. Enajoutant ce parametre, des deviations d’ordre vingt a quarante pourcent sontautorisees dans le cadre de CMFV, selon si la Nouvelle Physique apparait al’arbre ou non.

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