Global Fits of Third Family Hypercharge Models to Neutral ...

Eur. Phys. J. C manuscript No.(will be inserted by the editor)

Global Fits of Third Family Hypercharge Models to NeutralCurrent B-Anomalies and Electroweak Precision Observables

B.C. Allanach1, J. Eliel Camargo-Molina2, and Joe Davighi1

1 DAMTP, University of Cambridge, Wilberforce Road, Cambridge, CB3 0WA, United Kingdom2 Department of Physics and Astronomy, Uppsala University, Box 516, SE-751 20 Uppsala, Sweden

Received: date / Accepted: date

Abstract. While it is known that third family hypercharge models can explain the neutral currentB−anomalies, it was hitherto unclear whether the Z − Z′ mixing predicted by such models could si-multaneously fit electroweak precision observables. Here, we perform global fits of several third familyhypercharge models to a combination of electroweak data and those data pertinent to the neutral currentB−anomalies. While the Standard Model is in tension with this combined data set with a p−value of.00068, simple versions of the models (fitting two additional parameters each) provide much improved fits.

The original Third Family Hypercharge Model, for example, has a p−value of .065, with√∆χ2 = 6.5σ.

1 Introduction

The neutral current B−anomalies (NCBAs) consist of var-ious measurements in hadronic particle decays which, col-lectively, are in tension with Standard Model (SM) pre-dictions. The particular observables displaying such ten-sion often involve an effective vertex with an anti-bottomquark, a strange quark, a muon and an anti-muon, i.e.(bs)(µ+µ−), plus the charge conjugated version. Observ-

ables such as the ratios of branching ratiosR(∗)K = BR(B →

K(∗)µ+µ−)/BR(B → K(∗)e+e−) are not displaying thelepton flavour universality (LFU) property expected ofthe SM [1–3]. Such observables are of particular interestbecause much of the theoretical uncertainty in the pre-diction cancels in the ratio, leaving the prediction ratherprecise. Other NCBA observables display some disparitywith SM predictions even when their larger theoretical un-certainties are taken into account, for example BR(Bs →µ+µ−) [4–8], BR(Bs → φµ+µ−) [9, 10] and angular dis-tributions of B → K(∗)µ+µ− decays [11–16]. Global fitsfind that new physics contributions to the (bs)(µ+µ−) ef-fective vertex can fit the NCBAs much better than theSM can [17–23].

A popular option for beyond the SM (BSM) expla-nations of the NCBAs is that of a Z ′ vector boson withfamily dependent interactions [24–28]. Such a particle ispredicted by models with a BSM spontaneously brokenU(1) gauged flavour symmetry. The additional quantumnumbers of the SM fermions are constrained by the needto cancel local anomalies [29–31], for example muon minustau lepton number [32–37], third family baryon numberminus second family lepton number [38–40], third familyhypercharge [41–43] or other assignments [44–62].

The current paper is about the third family hyper-charge option. The Third Family Hypercharge Model [41](henceforth abbreviated as the ‘Y3 model’) explains the hi-erarchical heaviness of the third family and the smallnessof quark mixing. It was shown to successfully fit NCBAs,along with constraints from Bs−Bs mixing and LFU con-straints on Z0 boson interactions. The ATLAS experimentat the LHC has searched1 for the production of BSM res-onances that yield a peak in the di-muon invariant mass(mµµ) spectrum, but have yet to find a significant one [64].This implies a lower bound upon the mass of the Z ′ in theY3 model, MX > 1.2 TeV [65], but plenty of viable param-eter space remains which successfully explains the NCBAs.A variant, the Deformed Third Family Hypercharge Model(DY3 model), was subsequently introduced [43] in orderto remedy a somewhat ugly feature (ugly from a natu-ralness point of view) in the construction of the originalY3 model: namely, that a Yukawa coupling allowed at therenormalisable level was assumed to be tiny in order toagree with strict lepton flavour violation constraints. TheDY3 model can simultaneously fit the NCBAs and be con-sistent with the ATLAS di-muon direct search constraintfor 1.2 TeV< MX <12 TeV. We will also present resultsfor a third variant, the DY ′3 model, which is identical tothe DY3 model but with charges for second and third fam-ily leptons interchanged. As we show later on, this resultsin a better fit to data due to the different helicity structureof the couplings of the Z ′ boson to muons (see section 4.3for details).

1 During the final stages of preparation of this manuscript,the CMS experiment released a di-muon resonance search [63]with similar exclusion regions.

arX

iv:2

103.

1205

6v3

[he

p-ph

] 1

2 Ju

l 202

1

2 B.C. Allanach1 et al.: Global Fits of Y3 Models

In either of these third family hypercharge models, thelocal gauge symmetry of the SM is2 extended to SU(3)×SU(2) × U(1)Y × U(1)X . This is spontaneously brokento the SM gauge group by the non-zero vacuum expecta-tion value (VEV) of a SM-singlet ‘flavon’ field θ that hasa non-zero U(1)X charge. In each model, the third fam-ily quarks’ U(1)X charges are equal to their hyperchargeswhereas the first two family quarks are chargeless underU(1)X . We must (since it is experimentally determined tobe O(1) and is therefore inconsistent with a suppressed,non-renormalisable coupling) ensure that a renormalisabletop Yukawa coupling is allowed by U(1)X ; this impliesthat the SM Higgs doublet field should have U(1)X chargeequal to its hypercharge. Consequently, when the Higgsdoublet acquires a VEV to break the electroweak symme-try, this gives rise to Z0 − Z ′ mixing [41]. Such mixing issubject to stringent constraints from electroweak precisionobservables (EWPOs), in particular from the ρ-parameter,which encodes the ratio of the masses of the Z0 boson andthe W boson [66].

Third family hypercharge models can fit the NCBAsfor a range of the ratio of the Z ′ gauge coupling to itsmass gX/MX which does not contain zero. This meansthat it is not possible to ‘tune the Z −Z ′ mixing away’ ifone wishes the model to fit the NCBAs. As a consequence,it is not clear whether the EWPOs will strongly precludethe (D)Y3 models from explaining the NCBAs or not.

The purpose of this paper is to perform a global fitto a combined set of electroweak and NCBA-type data,along with other relevant constraints on flavour changingneutral currents (FCNCs). It is clear that the SM pro-vides a poor fit to this combined set, as Table 1 shows.A p−value3 of .00068 corresponds to ‘tension at the 3.4σlevel’. The (D)Y3 models are of particular interest as plau-sible models of new physics if they fit the data significantlybetter than the SM, a question which can best be settledby performing appropriate global fits.

Our paper proceeds as follows: we introduce the mod-els and define their parameter spaces in §2. At renormal-isation scales at or below MX but above MW , we en-code the new physics effects in each model via the Stan-dard Model Effective Theory (SMEFT). We calculate theleading (dimension-6) SMEFT operators predicted by ourmodels at the scale MX in §3. These provide the inputto the calculation of observables by smelli-2.2.0 [67]4,which we describe at the beginning of §4. The results of

2 Possible quotients of the gauge group by discrete subgroupsplay no role in our argument and so we shall henceforwardignore them.

3 All χ2 and p−values which we present here are calculatedin smelli-2.2.0. We estimate that the numerical uncertaintyin the smelli-2.2.0 calculation of a global χ2 value is ±1 andthe resulting uncertainty in the second significant figure of anyglobal p−value quoted is ±3.

4 We use the development version of smelli-2.2.0 that wasreleased on github on 8th March 2021 which we have up-dated to take into account 2021 LHCb measurements of RK ,BR(Bs → µ+µ−) and BR(B → µ+µ−).

data set χ2 n p−valuequarks 221.2 167 .0032

LFU FCNCs 35.3 21 .026EWPOs 35.7 31 .26global 292.2 219 .00068

Table 1. SM goodness of fit for the different data sets weconsider, as calculated by smelli-2.2.0. We display the totalPearson’s chi-squared χ2 for each data set along with the num-ber of observables n and the data set’s p−value. The set named‘quarks’ contains BR(Bs → φµ+µ−), BR(Bs → µ+µ−), ∆ms

and various differential distributions in B → K(∗)µ+µ− de-cays among others, whereas ‘LFU FCNCs’ contains RK(∗) andsome B meson decay branching ratios into di-taus. Our setsare identical to those defined by smelli-2.2.0 and we referthe curious reader to its manual [67], where the observablesare enumerated. We have updated RK and BR(Bs,d → µ+µ−)with the latest LHCb measurements as detailed in the text.

the fits are presented in the remainder of §4, before a dis-cussion in §5.

2 Models

In this section we review the models of interest to thisstudy, in sufficient detail so as to proceed with the cal-culation of the SMEFT Wilson coefficients (WCs) in thefollowing section. Under SU(3)×SU(2)×U(1)Y , we definethe fermionic fields such that they transform in the follow-ing representations: QLi := (uLi dLi)

T ∼ (3, 2, +1/6),LLi := (νLi eLi)

T ∼ (1, 2, −1/2), eRi ∼ (1, 1, −1),dRi ∼ (3, 1, −1/3), uRi ∼ (3, 1, +2/3), where i ∈1, 2, 3 is a family index ordered by increasing mass. Im-plicit in the definition of these fields is that we have per-formed a flavour rotation so that dLi, eLi, eRi, dRi, uRi aremass basis fields. In what follows, we denote 3-componentcolumn vectors in family space with bold font, for exam-ple uL := (uL, cL, tL)T . The Higgs doublet is a com-plex scalar φ ∼ (1, 2, +1/2), and all three models whichwe consider (the Y3 model, the DY3 model and the DY ′3model) incorporate a complex scalar flavon with SM quan-tum numbers θ ∼ (1,1, 0), which has a U(1)X chargeXθ 6= 0 and is used to Higgs the U(1)X symmetry, suchthat its gauge boson acquires a mass at the TeV scale orhigher.

In the following we will present our results for threevariants of third family hypercharge models, which differin the charge assignment for the SM fields:

– The Y3 model, introduced in [41]. Only third genera-tion fermions have non-zero U(1)X charges. The chargeassignments can be read in Table 2.

– The DY3 model as introduced in [43]. It differs fromthe Y3 model in that charges have been assigned tothe second generation leptons as well, while still beinganomaly free.

– The DY ′3 model, which differs from the DY3 model inthat the charges for third and second generation left-handed leptons are interchanged. The charge assign-ments can be read in Table 3.

B.C. Allanach1 et al.: Global Fits of Y3 Models 3

All three of these gauge symmetries have identical cou-plings to quarks, coupling only to the third family viahypercharge quantum numbers. This choice means that,of the quark Yukawa couplings, only the top and bottomYukawa couplings are present at the renormalisable level.Of course, the light quarks are not massless in reality; theirmasses, as well as the small quark mixing angles, must beencoded in higher-dimensional operators that come froma further layer of heavy physics, such as a suite of heavyvector-like fermions at a mass scale Λ > MX/gX , wheregX is the U(1)X gauge coupling.

Whatever this heavy physics might be, the structure ofthe light quark Yukawa couplings will be governed by thesize of parameters that break the U(2)3global := U(2)q ×U(2)u × U(2)d accidental global symmetry [68–72] of therenormalisable third family hypercharge lagrangians. Forexample, a minimal set of spurions charged under bothU(2)3global and U(1)X was considered5 in Ref. [73], whichreproduces the observed hierarchies in quark masses andmixing angles when the scale Λ of new physics is a factorof 15 or so larger than MX/gX . Taking this hierarchy ofscales as a general guide, and observing that the global fitsto electroweak and flavour data that we perform in thispaper prefer MX/gX ≈ 10 TeV, we expect the new physicsscale to be around Λ ≈ 150 TeV. This scale is high enoughto suppress most contributions of the heavy physics, aboutwhich we remain agnostic, to low energy phenomenologyincluding precise flavour bounds6. For this reason, we feelsafe in neglecting the contributions of the Λ scale physicsto the SMEFT coefficients that we calculate in § 3, andshall not consider it in any further detail.

Continuing, we will first detail the scalar sector, whichis common to (and identical in) all of the third familyhypercharge models, before going on to discuss aspectsof each model that are different (most importantly, thecouplings to leptons).

2.1 The scalar sector

The coupling of the flavon to the U(1)X gauge field isencoded in the covariant derivative

Dµθ = (∂µ + iXθgXXµ)θ, (1)

where Xµ is the U(1)X gauge boson in the unbroken phaseand gX is its gauge coupling. The flavon θ is assumed to ac-quire a VEV 〈θ〉 at (or above) the TeV scale, which spon-

taneously breaks U(1)X . Expanding θ = (〈θ〉 + ϑ)/√

2,its kinetic terms (Dµθ)

†Dµθ in the Lagrangian densitygive the gauge boson a mass MX = XθgX〈θ〉 through the

5 In the present paper we take a more phenomenological ap-proach in specifying the fermion mixing matrices, but nonethe-less, the quark mixing matrices that we use are qualitativelysimilar to those expected from the U(2)3global × U(1)X break-ing spurion analysis. For example, there is no mixing in theright-handed fields.

6 Kaon mixing is one possible exception where such heavyphysics might play a non-trivial role, however.

XQi = 0 XuRi= 0 XdRi

= 0XQ3 = 1/6 XuR3

= 2/3 XdR3= −1/3

XLi = 0 XeRi= 0 Xφ = 1/2

XL3 = −1/2 XeR3= −1 Xθ

Table 2. U(1)X charges of the gauge eigenbasis fields in theY3 model, where i ∈ 1, 2. The flavon charge Xθ is left unde-termined.

Higgs mechanism. After electroweak symmetry breaking,the electrically-neutral gauge bosons X, W 3 and B mix,giving rise to γ, Z0 and Z ′ as the physical mass eigen-states [41]. To terms of order (M2

Z/M2X), the mass and

the couplings of the X boson are equivalent to those ofthe Z ′ boson. Because we take MX MZ , the matchingto the SMEFT (§3) should be done in the unbroken elec-troweak phase, where it is the X boson that is properlyintegrated out. In the rest of this section we therefore spec-ify the U(1)X sector via the X boson and its couplings.Throughout this paper, we entreat the reader to bear inmind that in terms of searches and several other aspectsof their phenomenology, to a decent approximation the Xboson and the Z ′ boson are synonymous.

The covariant derivative of the Higgs doublet is

Dµφ =

(∂µ + i

g

2σaW a

µ + ig′

2Bµ + i

gX2Xµ

)φ, (2)

where W aµ (a = 1, 2, 3) are unbroken SU(2) gauge bosons,

σa are the Pauli matrices, g is the SU(2) gauge cou-pling, Bµ is the hypercharge gauge boson and g′ is thehypercharge gauge coupling. The kinetic term for theHiggs field, (Dµφ)†(Dµφ), contains terms both linear andquadratic in Xµ. It is the linear terms

L ⊃ −igX2Xµφ

†(∂µ + i

g

2σaW a

µ + ig′

2Bµ

)φ+ h.c. (3)

that, upon integrating out the Xµ boson, will givethe leading contribution to the SMEFT in the form ofdimension-6 operators involving the Higgs, as we describein §3.

The charges of the fermion fields differ between the Y3model and the DY ′3 model, as follows.

2.2 Fermion couplings: the Y3 model

The Y3 model has fermion charges as listed in Table 2 (inthe gauge eigenbasis), leading to the following Lagrangiandensity describing the X boson-SM fermion couplings [41]:

LψY3= − gX

(1

6QLΛ

(dL)ξ

/XQL −1

2LLΛ

(eL)ξ

/XLL

+2

3uRΛ

(uR)ξ

/XuR −1

3dRΛ

(dR)ξ

/XdR

−eRΛ(eR)ξ

/XeR

), (4)

whereΛ(I)P := V †I PVI (5)


XQ1 = 0 XuR1= 0 XdR1

= 0XQ2 = 0 XuR2

= 0 XdR2= 0

XQ3 = 1/6 XuR3= 2/3 XdR3

= −1/3XL1 = 0 XeR1

= 0 Xφ = 1/2XL2 = −4/3 XeR2

= 2/3 XθXL3 = 5/6 XeR3

= −5/3

Table 3. U(1)X charges of the gauge eigenbasis fields in theDY ′3 model. The original DY3 model charges (as introduced inRef. [43]) can be obtained by interchanging XL3 and XL2 . Theflavon charge Xθ is left undetermined.

are Hermitian 3-by-3 matrices. The index I ∈uL, dL, eL, νL, uR, dR, eR and the matrix P ∈ ξ,Ω, Ψ,where

ξ =

0 0 00 0 00 0 1

, (6)

and Ω and Ψ are described in §2.3. The VI are 3-by-3unitary matrices describing the mixing between fermionicgauge eigenstates and their mass eigenstates. Note thatthe quark doublets have been family rotated so that thedLi (but not the uLi fields) correspond to their mass eigen-states. Similarly, we have rotated Li such that eLi alignwith the charged lepton mass eigenstates, but νLi are not.This will simplify the matching to the SMEFT operatorsthat we perform in §3. We now go on to cover the X bosoncouplings in the DY ′3 model before detailing the fermionmixing ansatz (which is common to all three models).

2.3 Fermion couplings: the DY ′3 model

For the DY ′3 model with the charge assignments listed inTable 3, the Lagrangian contains the following X boson-SM fermion couplings [43]:

LψDY ′3 = −gX(

1

6QLΛ

(dL)ξ

/XQL −4

3LLΛ

(eL)Ω

/XLL

+2

3uRΛ

(uR)ξ

/XuR −1

3dRΛ

(dR)ξ

/XdR

+2

3eRΛ

(eR)Ψ

/XeR

). (7)

The matrices Λ(I)Ω and Λ

(I)Ψ are defined in (5), where

Ω =

0 0 00 1 00 0 − 5

8

, Ψ =

0 0 00 1 00 0 − 5

2

. (8)

2.4 Fermion mixing ansatz

The CKM matrix and the PMNS matrix are predicted tobe

V = V †uLVdL , U = U†νLVeL , (9)

respectively. For all of the third family hypercharge modelsthat we address here, the matrix element (V(dL))23 must

be non-zero in order to obtain new physics contributionsof the sort required to explain the NCBAs. Moreover, inthe Y3 model we need (VeL)23 6= 0 in order to generate acoupling (here left-handed) to muons7. These will lead toa BSM contribution to a Lagrangian density term in theweak effective theory proportional to (bγµPLs)(µγµPLµ),where PL is the left-handed projection operator, whichprevious fits to the weak effective theory indicate is essen-tial in order to fit the NCBAs [17–23].

In order to investigate the model further phenomeno-logically, we must assume a particular ansatz for the uni-tary fermion mixing matrices VI . Here, for VdL , we choosethe ‘standard parameterisation’ often used for the CKMmatrix [74]. This is a parameterisation of a family of uni-tary 3 by 3 matrices that depends only upon one complexphase and three mixing angles (a more general parame-terisation would also depend upon five additional complexphases):

c12c13 s12c13 s13e−iδ

−s12c23 − c12s23s13eiδ c12c23 − s12s23s13eiδ s23c13s12s23 − c12c23s13eiδ −c12s23 − s12c23s13eiδ c23c13

,

(10)where sij := sin θij and cij := cos θij , for angles θij , δ ∈R/(2πZ). To define our particular ansatz, we choose anglessuch that (VdL)ij = Vij for all ij 6= 23, i.e. we insert thecurrent world-averaged measured central values of θij andδ [74], except for the crucial mixing angle θ23, upon whichthe NCBAs sensitively depend. Thus, we fix the anglesand phase such that s12 = 0.22650, s13 = 0.00361 andδ = 1.196 but allow θ23 to float as a free parameter inour global fits. Following Refs. [41, 43], we choose simpleforms for the other mixing matrices which are likely toevade strict FCNC bounds. Specifically, we choose VdR =1, VuR

= 1 and

VeL =

1 0 00 0 10 1 0

(11)

in the Y3 model8, and VeL = 1 in the DY ′3 model. Fi-nally, VuL

and VνL are then fixed by (9) and the measuredCKM/PMNS matrix entries. For the remainder of this pa-per, when referring to the Y3 model or the DY ′3 model, weshall implicitly refer to the versions given by this mixingansatz (which, we emphasise, is taken to be the same forall third family hypercharge models aside from the assign-ment of VeL).

Next, we turn to calculating the complete set ofdimension-6 WCs in the SMEFT that result from inte-grating the X boson out of the theory.

7 The DY3 model (DY ′3 model) does not require (VeL)23 6= 0to fit the NCBAs, since it already possesses a coupling to µL.

8 Note that in the Y3 model, this is equivalent to insteadsetting VeL = 1 and switching the U(1)X charges of L2 and L3

in Table 2.


Xµ

ψ1

ψ2

ψ3

ψ4

Fig. 1. X-boson mediated process responsible for the effectivevertex between 4 fermionic fields ψi.

3 SMEFT Coefficients

So far, particle physicists have found scant direct evidenceof new physics below the TeV scale. This motivates thestudy of BSM models whose new degrees of freedom re-side at the TeV scale or higher. In such scenarios, it makessense to consider the Standard Model as an effective fieldtheory realisation of the underlying high energy model.If one wishes to remain agnostic about further details ofthe high energy theory, this amounts to including all pos-sible operators consistent with the SM gauge symmetriesand performing an expansion in powers of the ratio of theelectroweak and new physics scales.

The Standard Model Effective Field Theory [75–77] issuch a parameterisation of the effects of heavy fields be-yond the SM (such as a heavy X boson field of interestto us here) through d > 4 operators built out of the SMfields. In this paper we will work with operators up todimension 6 (i.e. we will go to second order in the powerexpansion). Expanding the SMEFT to this order gives us avery good approximation to all of the observables we con-sider; the relevant expansion parameter for the EWPOsis (MZ/MX)2 1, and in the case of observables derivedfrom the decay of a meson of mass m, the relevant ex-pansion parameter is (m/MX)2 1. By restricting to theMX > 1 TeV region, we ensure that both of these massratios are small enough to yield a good approximation.

The SMEFT Lagrangian can be expanded as

LSMEFT = LSM + C5O5 +∑dim 6

CiOi + . . . , (12)

where O5 schematically indicates Weinberg operators withvarious flavour indices, which result in neutrino massesand may be obtained by adding heavy gauge singlet chi-ral fermions to play the role of right-handed neutrinos.The sum that is explicitly notated then runs over all massdimension-6 SMEFT operators, and the ellipsis representsterms which are of mass dimension (in the fields) greaterthan 6. The WCs Ci have units of [mass]−2. In the fol-lowing we shall work in the Warsaw basis, which definesa basis in terms of a set of independent baryon-number-conserving operators [78]. By performing the matching be-tween our models and the SMEFT, we shall obtain the setof WCs Ci at the scale MX , which can then be used tocalculate predictions for observables.

To see where these dimension-6 operators come from,let us first consider the origin of four-fermion operators.We may write the fermionic couplings of the underlyingtheory of the X boson, given in (4) and (7) for the Y3

model and DY ′3 model respectively, as

Lψ = −JµψXµ, (13)

whereJµψ =

∑ψi,j

κijψiγµψj (14)

is the fermionic current that theX boson couples to, wherethe sum runs over all pairs of SM Weyl fermions ψi. Thecouplings κij are identified from (4) or (7), dependingupon the model. After integrating out the X boson in pro-cesses such as the one in Fig. 1, one obtains the followingterms in the effective Lagrangian:

LSMEFT ⊃ −JψµJψ

µ

2M2X

. (15)

We match the terms thus obtained with the four-fermionoperators in the Warsaw basis [78] in order to identify the4-fermion SMEFT WCs in that basis.

These 4-fermion operators are not the only SMEFToperators that are produced at dimension-6 by integrat-ing out the X bosons of our models. Due to the tree-levelU(1)X charge of the SM Higgs, there are also various op-erators in the Higgs sector of the SMEFT, as follows. The(linear) couplings of the X boson to the Higgs, as recordedin (3), can again be written as the coupling of Xµ to a cur-rent, viz.

Lφ = −JφµXµ, (16)

where this time

Jφµ = igX2φ†DSM

µ φ+ h.c. (17)

is the bosonic current to which theX boson couples, where

DSMµ = ∂µ + i g2σ

aW aµ + i g

′

2 Bµ. Due to the presence of Xboson couplings to operators which are bi-linear in boththe fermion fields (Jµψ) and the Higgs field (Jµφ ), integrat-ing out the X bosons gives rise to cross-terms

LSMEFT ⊃ −JφµJψ

µ

M2X

, (18)

which encode dimension-6 operators involving two Higgsfields, one SM gauge boson, and a fermion bi-linear cur-rent. Diagrammatically, these operators are generated byintegrating out the X boson from Feynman diagrams suchas that depicted in Fig. 2.

Finally, there are terms that are quadratic in thebosonic current Jµφ ,

LSMEFT ⊃ −JφµJφ

µ

2M2X

, (19)

which encode dimension-6 operators involving four Higgsfields and two SM covariant derivatives. The correspond-ing Feynman diagram is given in Fig. 3.

This accounts, schematically, for the complete set ofdimension-6 WCs generated by either the Y3 model or the


DY ′3 model9. We tabulate all the non-zero WCs generatedin this way in Table 4 for the Y3 model and in Table 5 forthe DY ′3 model.

φ

φ

DSMµ

Xµ

ψ1

ψ2

Fig. 2. X-boson mediated process responsible for the effectivevertex between two Higgs fields, one SM gauge boson, and afermion bi-linear operator.

φ

φ

DSMµ

Xµ

φ

φ

DSMµ

Fig. 3. X-boson mediated process responsible for dimension-6operators involving four Higgs insertions and either two deriva-tives or two SM gauge boson insertions.

4 Global Fits

Given the complete sets of dimension-6 WCs (Tables 4and 5) as inputs at the renormalisation scale MX ,10

we use the smelli-2.2.0 program to calculate hun-dreds of observables and the resulting likelihoods. Thesmelli-2.2.0 program is based upon the observable cal-culator flavio-2.2.0 [79], using Wilson-2.1 [80] for run-ning and matching WCs using the WCxf exchange for-mat [81].

In a particular third family hypercharge model, forgiven values of our three input parameters θ23, gX , MX ,the WCs in the tables are converted to the non-redundant

9 Note that in deriving the WCs we have assumed that thekinetic mixing between the X boson field strength and the hy-percharge field strength is negligible at the scale of its deriva-tion, i.e. at MX .10 Strictly speaking, the parameters gX and θ23 that we quoteare also implicitly evaluated at a renormalisation scale of MX

throughout this paper.

WC value WC value

C2222ll − 1

8(C

(1)lq )

22ij 112Λ

(dL)ξ ij

(C(1)qq )ijkl Λ

(dL)ξ ij Λ

(dL)ξ kl

δikδjl−2

72C3333ee − 1

2

C3333uu − 2

9C3333dd − 1

18

C3333eu

23

C3333ed − 1

3

(C(1)ud )3333 2

9C2233le − 1

2

C2233lu

13

C2233ld − 1

6

Cij33qe16Λ

(dL)ξ ij (C

(1)qu )ij33 − 1

9Λ

(dL)ξ ij

(C(1)qd )ij33 1

18Λ

(dL)ξ ij (C

(1)φl )22 1

4

(C(1)φq )ij − 1

12Λ

(dL)ξ ij C33

φe12

C33φu − 1

3C33φd

16

CφD − 12

Cφ − 18

Table 4. Non-zero dimension-6 SMEFT WCs predicted bythe Y3 model, in units of g2X/M

2X , in the Warsaw basis [78].

We have highlighted the coefficient (for i = 2, j = 3) that isprimarily responsible for the NCBAs in bold font.

WC value WC value

C2222ll − 25

72C2233ll

109

C3333ll − 8

9(C

(1)lq )

22ij− 5

36Λ

(dL)ξ ij

(C(1)lq )33ij 2

9Λ

(dL)ξ ij (C

(1)qq )ijkl Λ

(dL)ξ ij Λ

(dL)ξ kl

δikδjl−2

72

C2222ee − 2

9C2233ee

109

C3333ee − 25

18C3333uu − 2

9

C3333dd − 1

18C2233eu − 4

9

C3333eu

109

C2233ed

29

C3333ed − 5

9(C

(1)ud )3333 2

9

C2222le − 5

9C2233le

2518

C3333le − 20

9C3322le

89

C2233lu − 5

9C3333lu

89

C2233ld

518

C3333ld − 4

9

Cij22qe − 19Λ

(dL)ξ ij Cij33qe

518Λ

(dL)ξ ij

(C(1)qu )ij33 − 1

9Λ

(dL)ξ ij (C

(1)qd )ij33 1

18Λ

(dL)ξ ij

(C(1)φl )22 − 5

12(C

(1)φl )33 2

3

(C(1)φq )ij − 1

12Λ

(dL)ξ ij C22

φe − 13

C33φe

56

C33φu − 1

3

C33φd

16

CφD − 12

Cφ − 18

Table 5. Non-zero dimension-6 SMEFT WCs predicted by theDY ′3 model, in units of g2X/M

2X , in the Warsaw basis [78]. We

have highlighted the coefficient that is primarily responsible(for i = 2, j = 3) for the NCBAs in bold font. WCs for theoriginal DY3 model may be obtained by switching the l indices2↔ 3 everywhere.


Xµ

b

s s

b

Fig. 4. An X-boson mediated contribution to Bs−Bs mixing.

basis [67] assumed by smelli-2.2.011 (a subset of theWarsaw basis). The renormalisation group equations arethen solved in order to run the WCs down to the weakscale, at which the EWPOs are calculated. Most of theEWPOs and correlations are taken from Ref. [82] bysmelli-2.2.0, which neglects the relatively small theo-retical uncertainties in EWPOs. The EWPOs have notbeen averaged over lepton flavour, since lepton flavournon-universality is a key feature of any model of theNCBAs, including those built on third family hyperchargewhich we consider.

We have updated the data used by flavio-2.2.0 with2021 LHCb measurements of BR(Bd,s → µ+µ−) taken on9 fb−1 of LHC Run II data [8] by using the two dimen-sional Gaussian fit to current CMS, ATLAS and LHCbmeasurements presented in Ref. [83]:

BR(Bs → µ+µ−) = (2.93± 0.35)× 10−9,

BR(B0 → µ+µ−) = (0.56± 0.70)× 10−10, (20)

with an error correlation of ρ = −0.27. The most recentmeasurement by LHCb in the di-lepton invariant masssquared bin 1.1 < Q2/GeV2 < 6.0 is

RK = 0.846+0.042−0.039

+0.013−0.012, (21)

where the first error is statistical and the second system-atic [3] (this measurement alone has a 3.1σ tension withthe SM prediction of 1.00). We incorporate this new mea-surement by fitting the log likelihood function presentedin Ref. [3] with a quartic polynomial.

The SMEFT weak scale WCs are then matched to theweak effective theory and renormalised down to the scaleof bottom mesons using QCD×QED. Observables relevantto the NCBAs are calculated at this scale. smelli-2.2.0then organises the calculation of the χ2 statistic to quan-tify a distance (squared) between the theoretical predic-tion and experimental observables in units of the uncer-tainty. In calculating the χ2 value, experimental correla-tions between different observables are parameterised andtaken into account. Theoretical uncertainties are modelledas being multi-variate Gaussians; they include the effectsof varying nuisance parameters and are approximated tobe independent of new physics. Theory uncertainties andexperimental uncertainties are then combined in quadra-ture.

We note that an important constraint on Z ′ modelsthat explain the NCBAs is that from ∆ms (included by

11 Jupyter notebooks (from which all data files and plots maybe generated) have been stored in the anc/ subdirectory of thearXiv version of this paper.



Table 6. Goodness of fit for the different data sets we con-sider for the Y3 model as calculated by smelli-2.2.0 forMX = 3 TeV. We display the total χ2 for each data set alongwith the number of observables n and the data set’s p−value.The data set named ‘quarks’ contains BR(Bs → φµ+µ−),BR(Bs → µ+µ−), ∆ms and various differential distributionsin B → K(∗)µ+µ− decays among others, whereas ‘LFU FC-NCs’ contains RK(∗) and some B meson decay branching ra-tios into di-taus. Our data sets are identical to those definedby smelli-2.2.0 and we refer the curious reader to its man-ual [67], where the observables are enumerated. We have up-dated to RK and BR(Bs,d → µ+µ−) with the latest LHCbmeasurements as detailed in §4. Two free parameters of themodel were fitted: θ23 = −0.145 and gX = 0.426.

smelli-2.2.0 in the category of ‘quarks’ observables),deriving from measurements of Bs − Bs mixing, becauseof the tree-level BSM contribution to the process depictedin Fig. 4. The impact of this constraint has significantlyvaried over the last decade, to a large degree because ofnumerically rather different lattice or theory inputs usedto extract the measurement [84–86]. Here, we are weddedto the calculation and inputs used by smelli-2.2.0, al-lowing some tension in ∆ms to be traded against tensionpresent in the NCBAs.

As we shall see, in all the models that we consider theglobal fit is fairly insensitive to MX , provided we specifyMX > 2 TeV or so in order to be sure to not contraveneATLAS di-muon searches [43, 64, 65]. We will demonstratethis insensitivity to MX below (see Figs. 9 and 15), butfor now we shall pick MX = 3 TeV and scan over the pair(gX×3 TeV/MX) and θ23. Since the WCs at MX all scalelike gX/MX , the results will approximately hold at differ-ent values of MX provided that gX is scaled linearly withMX . The running between MX and the weak scale breaksthis scaling, but such effects derive from loop corrections∝ (1/16π2) ln(MX/MZ) and are thus negligible to a goodapproximation.

4.1 Y3 model fit results

The result of fitting θ23 and gX for MX = 3 TeV is shownin Table 6 for the Y3 model. The ‘global’ p−value is cal-culated by assuming a χ2 distribution with n− 2 degreesof freedom, since two parameters were optimised. The fitis encouragingly of a much better quality than the one ofthe SM. We see that the fits to the EWPOs and NCBAsare simultaneously reasonable.

The EWPOs are shown in more detail in Fig. 5, inwhich we compare some pulls in the SM fit versus theY3 model best-fit point. We see that there is some im-provement in the prediction of the W -boson mass, whichthe SM fit predicts is almost 2σ too low (as manifest in


2 1 0 1 2 3pull

AbAcAeAA

AFBb

AFBc

AFBe

AFB

AFB

BR(W e )BR(W )BR(W )

WZ

RbRcReRR

MWhad0 Al

lana

ch, C

amar

go-M

olin

a an

d Da

vigh

i, 20

21

SMY3

Fig. 5. Pulls in the EWPOs for the Y3 model MX = 3 TeVbest-fit point: θ23 = −0.145, gX = 0.426. The pull is defined tobe the theory prediction minus the central value of the obser-vation, divided by the combined theoretical and experimentaluncertainty, neglecting any correlations with other observables.

the ρ-parameter being measured to be slightly larger thanone [74], for MZ taken to be fixed to its SM value). Theeasing of this tension in MW is due precisely to the Z−Z ′mixing in the (D)Y3 models. The non-zero value of theSMEFT coefficient CφD breaks custodial symmetry, re-sulting in a shift of the ρ-parameter away from its tree-level SM value of one, to [66]

(ρ0)Y3 = 1− CφDv2/2 = 1 + v2g2X/(4m2X). (22)

where v is the SM Higgs VEV. Rather than being danger-ous, as might reasonably have been guessed, it turns outthat this BSM contribution to ρ0 is in large part respon-sible for the Y3 model fitting the EWPOs approximatelyas well as the SM does.

We also see that σhad0 , the e+e− scattering cross-section to hadrons at a centre-of-mass energy of MZ , isbetter fit by the Y3 model than the SM. Although theother EWPOs have some small deviations from their SMfits, the overall picture is that the Y3 model best-fit pointhas an electroweak fit similar to that of the SM.

0.0 0.5 1.0 1.5 2.0gX(3 TeV/MX)

0.4

0.2

0.0

23

MX = 3 TeV

Allanach, Camargo-Molina and Davighi, 2021

quarksLFU FCNCsEWPOsglobal

Fig. 6. Two parameter fit to the Y3 model for MX = 3 TeV.Shaded regions are those preferred by the data set in the leg-end at the 95% confidence level (CL). The global fit is shownby the solid curves, where the inner (outer) curves show the70%(95%) CL regions, respectively. The set named ‘quarks’contains BR(Bs → φµ+µ−), BR(Bs → µ+µ−), ∆ms and var-ious differential distributions in B → K(∗)µ+µ− decays amongothers, whereas ‘LFU FCNCs’ contains RK(∗) and some B me-son decay branching ratios into di-taus. Our sets are identicalto those defined by smelli-2.2.0 and we refer the curiousreader to its manual [67], where the observables are enumer-ated. We have updated RK and BR(Bs,d → µ+µ−) with thelatest LHCb measurements as detailed in §4. The black dotmarks the locus of the best-fit point.

0.0 0.2 0.4 0.6 0.8 1.0gX

20

0

20

40

2 SM2

MX=3.0 TeV23=-0.147



Fig. 7. Y3 model ∆χ2 contributions in the vicinity of the best-fit point as a function of gX .

In order to see which areas of parameter space arefavoured by the different sets of constraints, we provideFig. 6. The figure shows that the EWPOs and differentsets of NCBA data all overlap at the 95% CL. The best-fit point has a total χ2 of 43 less than that of the SM andis marked by a black dot. The separate data set contribu-tions to χ2 at this point are listed in Table 6. In order tocalculate 70% (95%) CL bounds in the 2-dimensional pa-rameter plane, we draw contours of χ2 equal to the best-fitvalue plus 2.41 (5.99) respectively, using the combined χ2

incorporating all the datasets.We further study the different χ2 contributions for the

Y3 model in the vicinity of the best-fit point in Figs. 7-


0.3 0.2 0.1 0.023

2010

01020304050

2 SM2

MX=3.0 TeVgX=0.418



Fig. 8. Y3 model ∆χ2 contributions in the vicinity of the best-fit point as a function of θ23.

2 4 6 8 10MX/TeV

0

20

40

2 SM2

23=-0.147gX=0.418MX/3 TeV



Fig. 9. Y3 model ∆χ2 contributions in the vicinity of the best-fit point as a function ofMX , where gX has been scaled linearly.

9. From Fig. 7, we see that large couplings gX > 0.6 aredisfavoured by EWPOs as well as the NCBAs. From Fig. 8we see that the EWPOs are insensitive to the value of θ23in the vicinity of the best-fit point but the NCBAs are not.At large −θ23 the Y3 model suffers due to a bad fit to theBs−Bs mixing observable ∆ms. In Fig. 9, we demonstratethe approximate insensitivity of χ2 near the best-fit pointto MX , provided that gX is scaled linearly with it.

Finally, we display some individual observables of in-terest in Fig. 10 at the Y3 model best-fit point. While someof the prominent NCBA measurements (for example RKin the bin of m2

µµ between 1.1 GeV2 and 6 GeV2) fit con-siderably better than the SM, we see that this is partlycompensated by a worse fit in ∆ms, as is the case formany Z ′ models for the NCBAs. The P ′5 observable (de-rived in terms of angular distributions of B0 → K∗µ+µ−

decays [87, 88]) shows no significant change from the SMprediction in the bin that deviates the most significantlyfrom experiment: m2

µµ ∈ (4, 6) GeV2, as measured by

LHCb [12] and ATLAS [13]. The fit to BR(Λb → Λµ+µ−)is slightly worse than that of the SM in one particu-lar bin, as shown in the figure. Some other flavour ob-servables in the flavour sector, notably various bins ofBR(B → K(∗)µ+µ−), show some small differences in pullsbetween the SM and the Y3 model. Whilst there are manyof these and in aggregate they make a difference to theoverall χ2, there is no small set of observables that pro-

3 2 1 0 1 2 3 4pull

RK(1, 6)RK * (0.045, 1.1)

RK * (1.1, 6)P ′5(4, 6)

BR(Bs-> )ms

BR(Bs-> )(1,6)BR( b-> )(15,20)

Alla

nach

, Cam

argo

-Mol

ina

and

Davi

ghi,

2021

SMY3

Fig. 10. Pulls of interest for the Y3 model MX = 3 TeV best-fit point: θ23 = −0.145, gX = 0.426. The pull is defined tobe the theory prediction minus the central value of the obser-vation, divided by the combined theoretical and experimentaluncertainty, neglecting any correlations with other observables.Numbers in brackets after the observable name refer to min-imum and maximum values of m2

µµ of the bin in GeV2, re-spectively (many other bins and observables are also used tocompute the global likelihood).



Table 7. Goodness of fit for the different data sets we con-sider for the DY ′3 model, as calculated by smelli-2.2.0 forMX = 3 TeV. We display the total χ2 for each data setalong with the number of observables n and the data set’sp−value. The set named ‘quarks’ contains BR(Bs → φµ+µ−),BR(Bs → µ+µ−), ∆ms and various differential distributionsin B → K(∗)µ+µ− decays among others, whereas ‘LFU FC-NCs’ contains RK(∗) and some B meson decay branching ra-tios into di-taus. Our sets are identical to those defined bysmelli-2.2.0 and we refer the curious reader to its man-ual [67], where the observables are enumerated. We have up-dated RK and BR(Bs,d → µ+µ−) with the latest LHCb mea-surements as detailed in §4. Two free parameters of the modelwere fitted: θ23 = −0.181 and gX = 0.253.

vide the driving force and so we neglect to show them12.We shall now turn to the DY ′3 model fit results, wherethese comments about flavour observables also apply.

4.2 DY ′3 model fit results

We summarise the quality of the fit for the DY ′3 modelat the best-fit point, for MX = 3 TeV, in Table 7. We

12 The interested reader can find values for all observablesand pulls in the Jupyter notebook in the ancillary informationassociated with the arXiv version of this paper.


0.0 0.5 1.0 1.5 2.0gX(3 TeV/MX)

0.4

0.2

0.0

23

MX = 3 TeV



Fig. 11. Two parameter fit to the DY ′3 model for MX = 3TeV. Shaded regions are those preferred by the data set in thelegend at the 95% confidence level (CL). The global fit is shownby the solid curves, where the inner (outer) curves show the70% (95%) CL regions, respectively. The set named ‘quarks’contains BR(Bs → φµ+µ−), BR(Bs → µ+µ−), ∆ms and var-ious differential distributions in B → K(∗)µ+µ− decays amongothers, whereas ‘LFU FCNCs’ contains RK(∗) and some B me-son decay branching ratios into di-taus. Our sets are identicalto those defined by smelli-2.2.0 and we refer the curiousreader to its manual [67], where the observables are enumer-ated. We have updated RK and BR(Bs,d → µ+µ−) with thelatest LHCb measurements as detailed in §4. The black dotmarks the locus of the best-fit point.

see a much improved fit as compared to the SM (by a∆χ2 = 39) and a similar (but slightly worse) quality of fitcompared to the Y3 model, as a comparison with Table 6shows.

The constraints upon the parameters θ23 and gX areshown in Fig. 11. Although the figure is for MX = 3TeV, the picture remains approximately the same for2 < MX/TeV < 10. We see that, as is the case for theY3 model, there is a region of overlap of the 95% CL re-gions of all of the constraints.

The pulls in the EWPOs for the best-fit point of theDY ′3 model are shown in Fig. 12. Like the Y3 model above,we see a fit comparable in quality to that of the SM. Again,the DY ′3 model predicts MW to be a little higher than inthe SM, agreeing slightly better with the experimentalmeasurement.

The behaviour of the fit in various directions in param-eter space around the best-fit point is shown in Figs. 13-15.Qualitatively, this behaviour is similar to that of the Y3model: the EWPOs and NCBAs imply that gX should notbecome too large. The mixing observable ∆ms prevents−θ23 from becoming too large, and the fits are insensitiveto MX varied between 2 TeV and 10 TeV so long as gX isscaled linearly with MX .

Fig. 16 shows various pulls of interest at the best-fitpoint of the DY ′3 model. Better fits (than the SM) to sev-

2 1 0 1 2pull

AbAcAeAA

AFBb

AFBc

AFBe

AFB

AFB

BR(W e )BR(W )BR(W )

WZ

RbRcReRR

MWhad0 Al

lana

ch, C

amar

go-M

olin

a an

d Da

vigh

i, 20

21

SMDY ′3

Fig. 12. Pulls in the EWPOs for the MX = 3 TeV DY ′3 modelbest-fit point: gX = 0.253, θ23 = −0.181. The pull is defined tobe the theory prediction minus the central value of the obser-vation, divided by the combined theoretical and experimentaluncertainty, neglecting any correlations with other observables.

0.0 0.2 0.4 0.6 0.8gX

20

0

20

40

2 SM2

MX=3.0 TeV23=-0.181



Fig. 13. ∆χ2 contributions in the vicinity of the DY ′3 modelbest-fit point as a function of gX .


0.3 0.2 0.1 0.023

100

10203040

2 SM2

MX=3.0 TeVgX=0.253



Fig. 14. ∆χ2 contributions in the vicinity of the DY ′3 modelbest-fit point as a function of θ23.

2 4 6 8 10MX/TeV

0

20

40

2 SM2

23=-0.181gX=0.253MX/3 TeV



Fig. 15. ∆χ2 contributions in the vicinity of the DY ′3 modelbest-fit point as a function of MX , where gX has been scaledlinearly.

3 2 1 0 1 2 3 4pull

RK(1, 6)RK * (0.045, 1.1)

RK * (1.1, 6)P ′5(4, 6)

BR(Bs-> )ms

BR(Bs-> )(1,6)BR( b-> )(15,20)

Alla

nach

, Cam

argo

-Mol

ina

and

Davi

ghi,

2021

SMDY ′3

Fig. 16. Pulls of interest for the MX = 3 TeV DY ′3 modelbest-fit point: gX = 0.253, θ23 = −0.181. The pull is defined tobe the theory prediction minus the central value of the obser-vation, divided by the combined theoretical and experimentaluncertainty, neglecting any correlations with other observables.Numbers in brackets after the observable name refer to mini-mum and maximum values of m2

µµ in GeV2, respectively (manyother bins and observables are also used to compute the globallikelihood).



Table 8. Goodness of fit for the different data sets we considerfor the original DY3 model, as calculated by smelli-2.2.0

for MX = 3 TeV. We display the total χ2 for each data setalong with the number of observables n and the data set’sp−value. The set named ‘quarks’ contains BR(Bs → φµ+µ−),BR(Bs → µ+µ−), ∆ms and various differential distributionsin B → K(∗)µ+µ− decays among others, whereas ‘LFU FC-NCs’ contains RK(∗) and some B meson decay branching ra-tios into di-taus. Our sets are identical to those defined bysmelli-2.2.0 and we refer the curious reader to its man-ual [67], where the observables are enumerated. We have up-dated RK and BR(Bs,d → µ+µ−) with the latest LHCb mea-surements as detailed in §4. Two free parameters of the modelwere fitted: θ23 = 0.122 and gX = 0.428.

eral NBCA observables are partially counteracted by aworse fit to the ∆ms observable.

4.3 Original DY3 model fit results

We display the overall fit quality of the original DY3model in Table 8. By comparison with Tables 1 and 6we see that although its predictions still fit the data sig-nificantly better than the SM (∆χ2 is 32), the originalDY3 model does not achieve as good fits as the othermodels. For the sake of brevity, we have refrained fromincluding plots for it13. Instead, it is more enlighten-ing to understand the reason behind this slightly worsefit, which is roughly as follows. The coupling of the Xboson to muons in the original DY3 model is close tovector-like, viz. L = gX/6(µ /X(5PL + 4PR))µ+ . . . (wherePL, PR are left-handed and right-handed projection op-erators, respectively), which is slightly less preferred bythe smelli-2.2.0 fits than an X boson coupled morestrongly to left-handed muons [67]. This preference is inlarge part due to the experimentally measured value ofBR(Bs → µ+µ−), which is somewhat lower than the SMprediction [4–7], and is sensitive only to the axial com-ponent of the coupling to muons. Compared to the Y3model and the DY ′3 model, the fit to BR(Bs → µ+µ−)is worse when the DY3 model fits other observables well.The p−value is significantly lower than the canonical lowerbound of 0.05, indicating a somewhat poor fit.

4.4 The θ23 = 0 limit of our models

The NCBAs can receive sizeable contributions even whenthe tree-level coupling of the X boson to bs vanishes. Forexample, non-zero and O(1) TeV2 values for Clu

2233 (as

13 These may be found within a Jupyter notebook in the anc

sub-directory of the arXiv submission of this paper.


µ+

µ−

bL

sL

t

t

W

Fig. 17. Example of the W−loop process dominating theSMEFT contribution to the NCBAs in the θ23 = 0 case. Thefilled disc marks the location of the BSM operator.

well as non-zero C2233eu in the case of the DY3 model and

the DY ′3 model) can give a reasonable fit to the NCBAdata [89] via a W -boson loop as in Fig. 1714 Such a sce-nario would require that Vts originates from mixing en-tirely within the up-quark sector. This qualitatively dif-ferent quark mixing ansatz therefore provides a motivationto consider the θ23 = 0 scenario separately. In the θ23 = 0

limit, we have that Λ(dL)ξ 23 is proportional to s13 1 mean-

ing that we also predict a negligible (C(1)lq )2223 at MX .

Meanwhile, C2233lu in the Y3 model (as well as C2233

eu for theDY ′3 model and the DY3 model) remains the same as in theθ23 6= 0 case shown in Tables 4 and 5. We note that con-tributions to the NCBAs arising from W loops such as theone in Fig. 17 are nevertheless always included (throughrenormalisation group running) by smelli-2.2.0, evenfor θ23 6= 0.

While it is true that much of the tension with theNCBAs can be ameliorated by such W loop contributions,we find from our global fits that the corresponding valuesfor g2X/M

2X are far too large to simultaneously give a good

fit to the EWPOs in this θ23 = 0 limit. As our resultswith a floating θ23 suggest (see e.g. Fig. 6 along θ23 = 0),as far as the EWPOs go, SM-like scenarios are stronglypreferred, since EWPOs quickly exclude any region thatmight resolve the tension with NCBAs. This is even moreso than for the simplified model studied in [89], wherealready a significant tension with Z → µµ was pointedout. In our case, besides several stringent bounds fromother observables measured at the Z0-pole for gX ≈ 1 andMX ≈ O(3) TeV (which predicts C2223

lq just large enough

to give a slightly better fit to RK and RK∗ than the SM)the predicted W−boson mass is more than 5σ away fromits measured value. This stands in stark contrast with theoverall better-than-SM fit we find for MW when θ23 6= 0.

14 The connection between the EWPOs and minimal flavour-violating one-loop induced NCBAs has recently been addressedin Ref. [90], encoding the inputs in the SMEFT framework.Third family hypercharge models are in a sense orthogonal tothis analysis, since they favour particular directions in non-minimal flavour violating SMEFT operator space generatedalready at the tree-level. The possibility of accommodating theNCBAs through such loop contributions was first pointed outin Ref. [91] with a more general study for b → s`` transitionspresented in Ref. [92].

5 Discussion

Previous explorations of the parameter spaces of thirdfamily hypercharge models [41, 43] capable of explainingthe neutral current B−anomalies showed the 95% confi-dence level exclusion regions from various important con-straints, but these analyses did not include electroweakprecision observables. Collectively, the electroweak pre-cision observables were potentially a model-killing con-straint because, through the Z0 − Z ′ mixing predictedin the models, the prediction of MW in terms of MZ issignificantly altered from the SM prediction. This was no-ticed in Ref. [66], where rough estimates of the absolutesizes of deviations were made. However, the severity of thisconstraint on the original Third Family Hypercharge (Y3)Model parameter space was found to depend greatly uponwhich estimate15 of the constraint was used. In the presentpaper, we use the smelli-2.2.0 [67] computer programto robustly and accurately predict the electroweak preci-sion observables and provide a comparison with empiricalmeasurements. We have thence carried out global fits ofthird family hypercharge models to data pertinent to theneutral current B−anomalies as well as the electroweakprecision observables. This is more sophisticated than theprevious efforts because it allows tensions between 217 dif-ferent observables to be traded off against one another ina statistically sound way. In fact, at the best-fit points ofthe third family hypercharge models, MW fits better thanin the SM, whose prediction is some 2σ too low.

One ingredient of our fits was the assumption of afermion mixing ansatz. The precise details of fermion mix-ing are expected to be fixed in third family hyperchargemodels by a more complete ultra-violet model. This couldlead to suppressed non-renormalisable operators in thethird family hypercharge model effective field theory, forexample which, when the flavon acquires its VEV, lead tosmall mixing effects. Such detailed model building seemspremature in the absence of additional information com-ing from the direct observation of a flavour-violating Z ′, orindeed independent precise confirmation of NCBAs fromthe Belle II [93] experiment. Reining in any urge to delveinto the underlying model building, we prefer simply toassume a non-trivial structure in the fermion mixing ma-trix which changes the observables we consider most: thoseinvolving the left-handed down quarks. Since the neutralcurrent B−anomalies are most sensitive to the mixing an-gle between left-handed bottom and strange quarks, wehave allowed this angle to float. But the other mixing an-gles and complex phase in the matrix have been set tosome (roughly mandated but ultimately arbitrary) val-ues equal to those in the CKM matrix. We have checkedthat changing these arbitrary values somewhat (e.g. set-ting them to zero) does not change the fit qualitatively: achange in χ2 of up to 2 units was observed. It is clear thata more thorough investigation of such variations may be-

15 In more detail, the strength of the constraint was stronglydependent on how many of the oblique parameters S, T , andU were allowed to simultaneously float.


model χ2 p−valueSM 292.2 .00068Y3 249.3 .065DY3 260.5 .023DY ′3 253.9 .044

Table 9. Comparison of p−values resulting from our globalfits of the SM and various third family hypercharge models(with MZ′ = 3 TeV) to a combination of 219 neutral currentB−anomaly and electroweak data.

come interesting in the future, particularly if the NCBAsstrengthen because of new measurements.

We summarise the punch line of the global fits in Ta-ble 9. We see that, while the SM suffers from a poorfit to the combined data set, the various third familyhypercharge models fare considerably better. The modelwith the best fit is the original Third Family HyperchargeModel (Y3). We have presented the constraints upon theparameter spaces of the Y3 model and the DY ′3 model indetail in §4. The qualitative behaviour of the Y3 modeland the DY ′3 model in the global fit is similar, althoughthe regions of preferred parameters are different.

It is well known that ∆ms provides a strong constrainton models which fit the NCBAs and ours are no excep-tion: in fact, we see in Figs 10,16 that this variable has apull of 2.7σ (2.1σ) in the Y3 model (DY ′3 model), whereasthe SM pull is only 1.1σ, according to the smelli-2.2.0calculation. The dominant beyond the SM contributionto ∆ms from our models is proportional to the Z ′ cou-

pling to sb quarks squared, i.e. [gX(Λ(dL)ξ )23/6]2. The cou-

pling (Λ(dL)ξ )23 is adjustable because θ23 is allowed to

vary over the fit, and ∆ms provides an upper bound upon

|gX(Λ(dL)ξ )23|. On the other hand, in order to produce a

large enough effect in the lepton flavour universality vio-lating observables to fit data, the product of the Z ′ cou-plings to sb quarks and to µ+µ− must be at least a certainsize. Thus, models where the Z ′ couples more strongly tomuons because their U(1)X charges are larger fare bet-ter when fitting the combination of the LFU FCNCs and∆ms. The Z ′ coupling to muons is 1/2 for the Y3 modeland 2/3 for the DY ′3 model, favouring the DY ′3 model inthis regard.

Despite the somewhat worse fit to ∆ms for the Y3model as compared to the DY ′3 model, Table 9 shows that,overall, the Y3 model is a better fit. Looking at the flavourobservables in detail, it is hard to divine a single causefor this: it appears to be the accumulated effect of manyflavour observables in tandem. The difference in χ2 be-tween the Y3 model and the DY ′3 model of 4.4 is not largeand might merely be the result of statistical fluctuations inthe 219 data; indeed 1.2 of this comes from the differenceof quality of fit to the EWPOs.

All of the usual caveats levelled at interpretingp−values apply. In particular, p−values change depend-ing upon exactly which observables are included or ex-cluded. We have stuck to pre-defined sets of observablesin smelli-2.2.0 in an attempt to reduce bias. However,

we note that there are other data that are in tensionwith SM predictions which we have not included, namelythe anomalous magnetic moment of the muon (g − 2)µand charged current B−anomalies. If we were to includethese observables, the p−values of all models in Table 9would lower. Since third family hypercharge models givethe same prediction for these observables as the SM, eachmodel would receive the same χ2 increase as well as theincrease in the number of fitted data. However, since ourmodels have essentially nothing to add to these observ-ables compared with the SM, we feel justified in leavingthem out from the beginning. We could have excludedsome of the observables that smelli-2.2.0 includes inour data sets (obvious choices include those that do notinvolve bottom quarks, e.g. εK) further changing our cal-culation of the p−values of the various models.

As noted above, as far as the third family hyperchargemodels currently stand, the Z ′ contribution to (g− 2)µ issmall [28]. In order to explain an inferred beyond the SMcontribution to (g−2)µ compatible with current measure-ments ∆(g− 2)µ/2 ≈ 28± 8× 10−10, one may simply adda heavy (TeV-scale) vector-like lepton representation thatcouples to the muon and the Z ′ at one vertex. In that case,a one-loop diagram with the heavy leptons and Z ′ runningin the loop is sufficient and is simultaneously compatiblewith the neutral current B−anomalies and measurementsof (g − 2)µ [28].

Independent corroboration from other experimentsand future B−anomaly measurements are eagerly awaitedand, depending upon them, a re-visiting of global fits toflavour and electroweak data may well become desirable.We also note that, since electroweak precision observablesplay a key role in our fits, an increase in precision uponthem resulting from LHC or future e+e− collider data,could also prove to be of great utility in testing third fam-ily hypercharge models indirectly. Direct production of thepredicted Z ′ [43, 65] (and a measurement of its couplings)would, along with an observation of flavonstrahlung [40],ultimately provide a ‘smoking gun’ signal.

Acknowledgements

This work has been partially supported by STFC Con-solidated HEP grants ST/P000681/1 and ST/T000694/1.JECM is supported by the Carl Trygger foundation (grantno. CTS 17:139). We are indebted to M McCullough forraising the question of the electroweak precision observ-ables in the Y3 model. We thank other members of theCambridge Pheno Working Group for discussions and DStraub and P Stangl for helpful communications aboutthe nuisance parameters in smelli-2.2.0.

References

1. R. Aaij, et al., JHEP 08, 055 (2017). DOI 10.1007/JHEP08(2017)055

2. R. Aaij, et al., Phys. Rev. Lett. 122(19), 191801 (2019).DOI 10.1103/PhysRevLett.122.191801


3. R. Aaij, et al., (2021)4. M. Aaboud, et al., JHEP 04, 098 (2019). DOI 10.1007/

JHEP04(2019)0985. S. Chatrchyan, et al., Phys. Rev. Lett. 111, 101804 (2013).

DOI 10.1103/PhysRevLett.111.1018046. V. Khachatryan, et al., Nature 522, 68 (2015). DOI 10.

1038/nature144747. R. Aaij, et al., Phys. Rev. Lett. 118(19), 191801 (2017).

DOI 10.1103/PhysRevLett.118.1918018. K. Petridis, M. Santimaria. New results on theoreti-

cally clean observables in rare b-meson decays from lhcb(2021). URL https://indico.cern.ch/event/976688/.LHC seminar

9. R. Aaij, et al., JHEP 09, 179 (2015). DOI 10.1007/JHEP09(2015)179

10. CDF collaboration, CDF-NOTE-10894 (2012)11. R. Aaij, et al., Phys. Rev. Lett. 111, 191801 (2013). DOI

10.1103/PhysRevLett.111.19180112. R. Aaij, et al., JHEP 02, 104 (2016). DOI 10.1007/

JHEP02(2016)10413. M. Aaboud, et al., JHEP 10, 047 (2018). DOI 10.1007/

JHEP10(2018)04714. A.M. Sirunyan, et al., Phys. Lett. B 781, 517 (2018). DOI

10.1016/j.physletb.2018.04.03015. V. Khachatryan, et al., Phys. Lett. B 753, 424 (2016). DOI

10.1016/j.physletb.2015.12.02016. C. Bobeth, M. Chrzaszcz, D. van Dyk, J. Virto, Eur.

Phys. J. C 78(6), 451 (2018). DOI 10.1140/epjc/s10052-018-5918-6

17. M. Alguero, B. Capdevila, A. Crivellin, S. Descotes-Genon, P. Masjuan, J. Matias, M. Novoa Brunet, J. Virto,Eur. Phys. J. C 79(8), 714 (2019). DOI 10.1140/epjc/s10052-019-7216-3. [Addendum: Eur.Phys.J.C 80, 511(2020)]

18. A.K. Alok, A. Dighe, S. Gangal, D. Kumar, JHEP 06, 089(2019). DOI 10.1007/JHEP06(2019)089

19. M. Ciuchini, A.M. Coutinho, M. Fedele, E. Franco,A. Paul, L. Silvestrini, M. Valli, Eur. Phys. J. C 79(8),719 (2019). DOI 10.1140/epjc/s10052-019-7210-9

20. J. Aebischer, W. Altmannshofer, D. Guadagnoli, M. Re-boud, P. Stangl, D.M. Straub, Eur. Phys. J. C 80(3), 252(2020). DOI 10.1140/epjc/s10052-020-7817-x

21. A. Datta, J. Kumar, D. London, Phys. Lett. B 797, 134858(2019). DOI 10.1016/j.physletb.2019.134858

22. K. Kowalska, D. Kumar, E.M. Sessolo, Eur. Phys. J. C79(10), 840 (2019). DOI 10.1140/epjc/s10052-019-7330-2

23. A. Arbey, T. Hurth, F. Mahmoudi, D.M. Santos, S. Ne-shatpour, Phys. Rev. D 100(1), 015045 (2019). DOI10.1103/PhysRevD.100.015045

24. R. Gauld, F. Goertz, U. Haisch, Phys. Rev. D 89, 015005(2014). DOI 10.1103/PhysRevD.89.015005

25. A.J. Buras, F. De Fazio, J. Girrbach, JHEP 02, 112 (2014).DOI 10.1007/JHEP02(2014)112

26. A.J. Buras, J. Girrbach, JHEP 12, 009 (2013). DOI 10.1007/JHEP12(2013)009

27. A.J. Buras, F. De Fazio, J. Girrbach-Noe, JHEP 08, 039(2014). DOI 10.1007/JHEP08(2014)039

28. B. Allanach, F.S. Queiroz, A. Strumia, S. Sun, Phys.Rev. D 93(5), 055045 (2016). DOI 10.1103/PhysRevD.93.055045. [Erratum: Phys.Rev.D 95, 119902 (2017)]

29. J. Ellis, M. Fairbairn, P. Tunney, Eur. Phys. J. C 78(3),238 (2018). DOI 10.1140/epjc/s10052-018-5725-0

30. B.C. Allanach, J. Davighi, S. Melville, JHEP 02, 082(2019). DOI 10.1007/JHEP02(2019)082. [Erratum: JHEP08, 064 (2019)]

31. B.C. Allanach, B. Gripaios, J. Tooby-Smith, Phys. Rev.Lett. 125(16), 161601 (2020). DOI 10.1103/PhysRevLett.125.161601

32. W. Altmannshofer, S. Gori, M. Pospelov, I. Yavin, Phys.Rev. D 89, 095033 (2014). DOI 10.1103/PhysRevD.89.095033

33. A. Crivellin, G. D’Ambrosio, J. Heeck, Phys. Rev. Lett.114, 151801 (2015). DOI 10.1103/PhysRevLett.114.151801

34. A. Crivellin, G. D’Ambrosio, J. Heeck, Phys. Rev. D 91(7),075006 (2015). DOI 10.1103/PhysRevD.91.075006

35. A. Crivellin, L. Hofer, J. Matias, U. Nierste, S. Pokorski,J. Rosiek, Phys. Rev. D 92(5), 054013 (2015). DOI 10.1103/PhysRevD.92.054013

36. W. Altmannshofer, I. Yavin, Phys. Rev. D 92(7), 075022(2015). DOI 10.1103/PhysRevD.92.075022

37. J. Davighi, M. Kirk, M. Nardecchia, JHEP 12, 111 (2020).DOI 10.1007/JHEP12(2020)111

38. R. Alonso, P. Cox, C. Han, T.T. Yanagida, Phys. Lett. B774, 643 (2017). DOI 10.1016/j.physletb.2017.10.027

39. C. Bonilla, T. Modak, R. Srivastava, J.W.F. Valle, Phys.Rev. D 98(9), 095002 (2018). DOI 10.1103/PhysRevD.98.095002

40. B.C. Allanach, Eur. Phys. J. C 81(1), 56 (2021). DOI10.1140/epjc/s10052-021-08855-w

41. B.C. Allanach, J. Davighi, JHEP 12, 075 (2018). DOI10.1007/JHEP12(2018)075

42. J. Davighi, in 54th Rencontres de Moriond on QCD andHigh Energy Interactions (ARISF, 2019)

43. B.C. Allanach, J. Davighi, Eur. Phys. J. C 79(11), 908(2019). DOI 10.1140/epjc/s10052-019-7414-z

44. D. Aristizabal Sierra, F. Staub, A. Vicente, Phys. Rev. D92(1), 015001 (2015). DOI 10.1103/PhysRevD.92.015001

45. A. Celis, J. Fuentes-Martin, M. Jung, H. Serodio, Phys.Rev. D 92(1), 015007 (2015). DOI 10.1103/PhysRevD.92.015007

46. A. Greljo, G. Isidori, D. Marzocca, JHEP 07, 142 (2015).DOI 10.1007/JHEP07(2015)142

47. A. Falkowski, M. Nardecchia, R. Ziegler, JHEP 11, 173(2015). DOI 10.1007/JHEP11(2015)173

48. C.W. Chiang, X.G. He, G. Valencia, Phys. Rev. D 93(7),074003 (2016). DOI 10.1103/PhysRevD.93.074003

49. S.M. Boucenna, A. Celis, J. Fuentes-Martin, A. Vicente,J. Virto, Phys. Lett. B 760, 214 (2016). DOI 10.1016/j.physletb.2016.06.067

50. S.M. Boucenna, A. Celis, J. Fuentes-Martin, A. Vi-cente, J. Virto, JHEP 12, 059 (2016). DOI 10.1007/JHEP12(2016)059

51. P. Ko, Y. Omura, Y. Shigekami, C. Yu, Phys. Rev. D95(11), 115040 (2017). DOI 10.1103/PhysRevD.95.115040

52. R. Alonso, P. Cox, C. Han, T.T. Yanagida, Phys. Rev. D96(7), 071701 (2017). DOI 10.1103/PhysRevD.96.071701

53. Y. Tang, Y.L. Wu, Chin. Phys. C 42(3), 033104(2018). DOI 10.1088/1674-1137/42/3/033104. [Erratum:Chin.Phys.C 44, 069101 (2020)]

54. D. Bhatia, S. Chakraborty, A. Dighe, JHEP 03, 117(2017). DOI 10.1007/JHEP03(2017)117

55. K. Fuyuto, H.L. Li, J.H. Yu, Phys. Rev. D 97(11), 115003(2018). DOI 10.1103/PhysRevD.97.115003

https://indico.cern.ch/event/976688/


56. L. Bian, H.M. Lee, C.B. Park, Eur. Phys. J. C 78(4), 306(2018). DOI 10.1140/epjc/s10052-018-5777-1

57. S.F. King, JHEP 09, 069 (2018). DOI 10.1007/JHEP09(2018)069

58. G.H. Duan, X. Fan, M. Frank, C. Han, J.M. Yang, Phys.Lett. B 789, 54 (2019). DOI 10.1016/j.physletb.2018.12.005

59. Z. Kang, Y. Shigekami, JHEP 11, 049 (2019). DOI 10.1007/JHEP11(2019)049

60. L. Calibbi, A. Crivellin, F. Kirk, C.A. Manzari, L. Ver-nazza, Phys. Rev. D 101(9), 095003 (2020). DOI 10.1103/PhysRevD.101.095003

61. W. Altmannshofer, J. Davighi, M. Nardecchia, Phys. Rev.D 101(1), 015004 (2020). DOI 10.1103/PhysRevD.101.015004

62. B. Capdevila, A. Crivellin, C.A. Manzari, M. Montull,Phys. Rev. D 103(1), 015032 (2021). DOI 10.1103/PhysRevD.103.015032

63. A.M. Sirunyan, et al., (2021)64. G. Aad, et al., Phys. Lett. B 796, 68 (2019). DOI 10.1016/

j.physletb.2019.07.01665. B.C. Allanach, J.M. Butterworth, T. Corbett, JHEP 08,

106 (2019). DOI 10.1007/JHEP08(2019)10666. J. Davighi, Topological effects in particle physics phe-

nomenology. Ph.D. thesis, Cambridge U., DAMTP (2020).DOI 10.17863/CAM.47560

67. J. Aebischer, J. Kumar, P. Stangl, D.M. Straub, Eur.Phys. J. C 79(6), 509 (2019). DOI 10.1140/epjc/s10052-019-6977-z

68. A. Pomarol, D. Tommasini, Nucl. Phys. B 466, 3 (1996).DOI 10.1016/0550-3213(96)00074-0

69. R. Barbieri, G.R. Dvali, L.J. Hall, Phys. Lett. B 377, 76(1996). DOI 10.1016/0370-2693(96)00318-8

70. R. Barbieri, G. Isidori, J. Jones-Perez, P. Lodone, D.M.Straub, Eur. Phys. J. C 71, 1725 (2011). DOI 10.1140/epjc/s10052-011-1725-z

71. G. Blankenburg, G. Isidori, J. Jones-Perez, Eur. Phys. J.C 72, 2126 (2012). DOI 10.1140/epjc/s10052-012-2126-7

72. R. Barbieri, D. Buttazzo, F. Sala, D.M. Straub, JHEP 07,181 (2012). DOI 10.1007/JHEP07(2012)181

73. J. Davighi, arXiv:2105.06918 (2021)74. P. Zyla, et al., PTEP 2020(8), 083C01 (2020). DOI 10.

1093/ptep/ptaa10475. W. Buchmuller, D. Wyler, Nucl. Phys. B268, 621 (1986).

DOI 10.1016/0550-3213(86)90262-276. I. Brivio, M. Trott, Phys. Rept. 793, 1 (2019). DOI 10.

1016/j.physrep.2018.11.00277. A. Dedes, W. Materkowska, M. Paraskevas, J. Rosiek,

K. Suxho, JHEP 06, 143 (2017). DOI 10.1007/JHEP06(2017)143

78. B. Grzadkowski, M. Iskrzynski, M. Misiak, J. Rosiek,JHEP 10, 085 (2010). DOI 10.1007/JHEP10(2010)085

79. D.M. Straub, (2018)80. J. Aebischer, J. Kumar, D.M. Straub, Eur. Phys. J. C

78(12), 1026 (2018). DOI 10.1140/epjc/s10052-018-6492-781. J. Aebischer, et al., Comput. Phys. Commun. 232, 71

(2018). DOI 10.1016/j.cpc.2018.05.02282. S. Schael, et al., Phys. Rept. 427, 257 (2006). DOI 10.

1016/j.physrep.2005.12.00683. W. Altmannshofer, P. Stangl, (2021)84. A. Bazavov, et al., Phys. Rev. D 93(11), 113016 (2016).

DOI 10.1103/PhysRevD.93.113016

85. L. Di Luzio, M. Kirk, A. Lenz, Phys. Rev. D 97(9), 095035(2018). DOI 10.1103/PhysRevD.97.095035

86. D. King, A. Lenz, T. Rauh, JHEP 05, 034 (2019). DOI10.1007/JHEP05(2019)034

87. J. Matias, F. Mescia, M. Ramon, J. Virto, JHEP 04, 104(2012). DOI 10.1007/JHEP04(2012)104

88. S. Descotes-Genon, L. Hofer, J. Matias, J. Virto, JHEP06, 092 (2016). DOI 10.1007/JHEP06(2016)092

89. J.E. Camargo-Molina, A. Celis, D.A. Faroughy, Phys. Lett.B 784, 284 (2018). DOI 10.1016/j.physletb.2018.07.051

90. L. Alasfar, A. Azatov, J. de Blas, A. Paul, M. Valli, JHEP12, 016 (2020). DOI 10.1007/JHEP12(2020)016

91. D. Becirevic, O. Sumensari, JHEP 08, 104 (2017). DOI10.1007/JHEP08(2017)104

92. R. Coy, M. Frigerio, F. Mescia, O. Sumensari, Eur. Phys. J.C 80(1), 52 (2020). DOI 10.1140/epjc/s10052-019-7581-y

93. W. Altmannshofer, et al., PTEP 2019(12), 123C01 (2019).DOI 10.1093/ptep/ptz106. [Erratum: PTEP 2020, 029201(2020)]

Global Fits of Third Family Hypercharge Models to Neutral ...

Documents