Prepared for submission to JHEP

Fermiophobic Z′ model for simultaneouslyexplaining the muon anomalies RK(∗) and (g − 2)µ

Mario Fernández Navarro1 and Stephen F. King1

1School of Physics & Astronomy, University of Southampton, Southampton SO17 1BJ, UK

E-mail: M.F.Navarro@soton.ac.uk, S.F.King@soton.ac.uk

Abstract: We discuss a simple renormalisable, gauge invariant model with a fermiophobicZ ′ boson: it has no couplings to the three Standard Model (SM) chiral families, but doescouple to a fourth vector-like (VL) family. The SM Higgs couples to the fourth VL lepton,leading to an enhanced contribution to the muon anomalous magnetic moment (g − 2)µ.The latter contribution requires a non-vanishing coupling of Z ′ to right-handed muons,which arises within this model due to mixing effects between the SM and VL fermions,along with Z ′ couplings to the second generation SM lepton doublet and third generationSM quark doublet. This model can simultaneously account for the measured B-decayratios RK(∗) and (g − 2)µ. We identify the parameter space where this explanation isconsistent with existing experimental constraints coming from Bs − Bs mixing, neutrinotrident production and collider searches. We also check that the SM Higgs coupling to thefourth VL lepton does not produce a dangerous contribution to the Higgs diphoton decay.

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Contents

1 Introduction 1

2 The model 3

3 Mixing 5

4 Phenomenology and flavour constraints 74.1 (g − 2)µ 74.2 RK(∗) 84.3 Bs − Bs mixing 114.4 Neutrino trident 124.5 Constraints from lepton flavour violation 134.6 Collider constraints 134.7 Higgs diphoton decay 14

5 Results and discussion 16

6 Conclusions 19

1 Introduction

Although the vast majority of particle-physics data is consistent with the predictions ofthe Standard Model (SM), in recent times a conspicuous series of discrepancies in flavourobservables has been established. One example is the discrepancy in rare flavour-changingprocesses mediated by quark-level b(b)→ s(s)`¯ transitions, explored in the past by BaBar[1] and Belle [2], along with LHC [3, 4]. In particular, the ratio of B-mesons decaying toK`+`−, which involves a b → s`¯ transition, has been recently measured by LHCb [5] inthe dilepton mass-squared range 1.1 < q2 < 6 GeV2 for the final states µ+µ− over e+e−,

R[1.1,6]K =

Br (B → Kµ+µ−)

Br (B → Ke+e−)= 0.846+0.044

−0.041 , (1.1)

along with the ratio of B-mesons decaying to K∗`+`−, measured in the past by LHCb [6],

R[1.1,6]K∗ =

Br (B → K∗µ+µ−)

Br (B → K∗e+e−)= 0.69+0.16

−0.12 . (1.2)

Within the SM, lepton universality predicts RK(∗) = 1, up to corrections of order 1%[7–11] due to the different mass of muons and electrons. Hence, the previous observationsof RK(∗) seem to indicate the breaking of SM lepton universality, up to the 3.1σ [5] of the

– 1 –

most updated measurement of RK , while RK∗ is compatible with the SM expectations at2.4− 2.5σ [6].

The apparent discrepancy of RK(∗) with the SM may be a hint of new physics. Fol-lowing these recent measurements of LHCb, a number of phenomenological analyses of thisdata, see e.g. Refs. [12–24], favour new physics operators of the form sLγµbLµLγ

µµL orsLγµbLµRγ

µµR. In particular, RK(∗) can be explained by only the purely left-handed (LH)operator with a coefficient Λ−2 where Λ ∼ 40 TeV, or also by a linear combination of both.Promising candidates for the arise of such effective operators are tree-level exchange of ahypothetical, electrically neutral and massive Z ′ boson (see e.g. [25–30]) with non-universalcouplings to SM fermions, or the contribution of a hypothetical leptoquark (LQ) couplingwith different strengths to the different types of charged leptons (see e.g. [31–34]).

Independent of the RK(∗) anomaly, there also exists a discrepancy with the SM predic-tions in the experimentally measured anomalous magnetic moments a = (g − 2)/2 of boththe muon and the electron. The long-lasting non-compliance of the muon aµ with the SMwas first observed by the Brookhaven E821 experiment at BNL [35]. This discrepancy hasbeen recently confirmed by the most recent measurement of the Fermilab experiment [36],

∆aµ = aµ − aSMµ = (2.51± 0.59) · 10−9 , (1.3)

a result 4.2σ greater than the SM prediction [37–57] and in excellent agreement with theprevious BNL E821 measurement. Such a discrepancy can also be addressed by Z ′ models[58–66] or leptoquarks [67–69], along with models involving extended scalar content and/orvector-like (VL) fermions [70–73]. In particular, the minimal Z ′ explanations [58] requireto introduce τ − µ mixing in order to obtain an enhanced contribution proportional tomτ , along with dangerous contributions to the flavour-violating process τ → 3µ, unlessfurther model building is considered. The latter is the case of Refs. [59–66], which considera fermiophobic Z ′ model where the Z ′ couplings with SM fermions are obtained throughmixing with a 4th VL family. An enhanced contribution to ∆aµ is obtained through acoupling between the SM Higgs and a 4th VL lepton, although it has to be checked thatsuch a coupling would not spoil the existing Higgs diphoton decay data. Moreover, thiscontribution requires a non-vanishing coupling of Z ′ to right-handed (RH) muons, in sucha way that a purely left-handed explanation of RK(∗) , as in previous studies [29, 30], cannotbe performed in this case. The fact that the latest phenomenological analyses [23, 24] leavethe possibility to include an effective operator sLγµbLµRγµµR in the explanation of RK(∗)

opens the possibility to explain simultaneously RK(∗) and (g − 2)µ within this Z ′ model.However, it has to be checked whether such simultaneous explanation of both anomalies

can also preserve all currently released high energy experimental data, such as the mea-surement of the mass difference ∆Ms of neutral Bs mesons, the observations of neutrinotrident production and the most recent collider signatures. Ideally, such a model shouldbe imminently testable with well designed future searches. We know from some other Z ′

models trying to address both anomalies by considering a 4th VL family, however in [60]the couplings to muons are loop-induced, [61] contains an extra Z(1)

1 × Z(2)2 discrete sym-

– 2 –

metry and [62, 63] are not fermiophobic. Refs. [64–66] consider our same model but withmore general mixing between VL and SM fermions, which also implies a higher numberof parameters. Instead, we consider in this article the minimal mixing framework requiredto explain RK(∗) and (g − 2)µ simultaneously, where the 4th VL family only mixes withthe third generation of the SM quark doublet and with the second generation of the SMlepton doublet and singlet, hence the number of mixing parameters is only three and theparameter space is easier to explore.

The remainder of this article is organised as follows: in Section 2 we outline the renor-malisable and gauge invariant fermiophobic model in which the Z ′ only couples to a vector-like fourth family. In Section 3, we show how it is possible to switch on the couplings of theZ ′ to the muon and bs-quarks through mixing with the VL fermions, thereby eliminatingall unnecessary couplings and allowing us to focus on the connection between the RK(∗)

and (g − 2)µ anomalies. The phenomenology and the constraints that affect this model arepresented in Section 4. In Section 5 we systematically explore the parameter space of themodel, and we also display and discuss the results from our analysis. Finally, Section 6concludes the article.

2 The model

The model [28] (Table 1) includes the three chiral families of LH SU(2)L doublets (QLi, LLi)

and RH SU(2)L singlets (uRi, dRi, eRi) of the SM, i = 1, 2, 3; along with one vector-likefamily of fermions (formed by LH and RH SU(2)L doublets QL4, LL4, and QR4, LR4, to-gether with LH and RH SU(2)L singlets uR4, dR4, eR4, νR4 and uL4, dL4, eL4, νL4). Thevector-like fermions are charged under a gauge symmetry U(1)′, while the three chiral fami-lies remain neutral under this symmetry, which is the reason behind the model being calledfermiophobic. The scalar sector is augmented by gauge singlet fields φf with non-trivialcharge assignments −qf4 under the new symmetry, which are responsible for spontaneouslybreaking U(1)′ developing vacuum expectation values (VEVs) 〈φf 〉. The Z ′ boson generatedafter the symmetry breaking has a mass at the same scale 〈φf 〉.

The full renormalisable Lagrangian is

Lren = yuijQLiHuRj + ydijQLiHdRj + yeijLLiHeRj

+ yu4QL4HuR4 + yd4QL4HdR4 + ye4LL4HeR4 + yν4LL4HνR4

+ xQi φQQLiQR4 + xLi φLLLiLR4 + xui φ∗uuL4uRi + xdi φ

∗ddL4dRi + xeiφ

∗e eL4eRi

+MQ4 QL4QR4 +ML

4 LL4LR4 +Mu4 uL4uR4 +Md

4 dL4dR4 +M e4 eL4eR4 +Mν

4 νL4νR4 + h.c.

(2.1)

where H = iσ2H∗, i = 1, 2, 3. The requirement of U(1)′ invariance of the Yukawa interac-

tions involving the fourth family yields the following constraints on the U(1)′ charges:

qQ4 = qu4 = qd4 , qL4 = qe4 = qν4 . (2.2)

– 3 –

Field SU(3)c SU(2)L U(1)Y U(1)′

QLi =

(uLi,

dLi

)3 2 1/6 0

uRi 3 1 2/3 0dRi 3 1 -1/3 0

LLi =

(νLieLi

)1 2 -1/2 0

eRi 1 1 -1 0

QL4, QR4 3 2 1/6 qQ4

uR4, uL4 3 1 2/3 qu4dR4, dL4 3 1 -1/3 qd4LL4, LR4 1 2 -1/2 qL4

eR4, eL4 1 1 -1 qe4νR4, νL4 1 1 0 qν4

φf 1 1 0 −qf4

H =

(h+(

v + h0)/√

2

)1 2 1/2 0

Table 1: Particle assigments under SU(3)c × SU(2)L × U(1)Y × U(1)′ gauge symme-try, i = 1, 2, 3. The singlet scalars φf (f = Q, u, d, L, e, ν) have U(1)′ charges −qf4 =

−qQ4,u4,d4,L4,e4,ν4 [28].

It is clear from Eq. (2.1) that fields in the 4th, vector-like family obtain masses from twosources. Firstly, from Yukawa terms involving the SM Higgs field, such as ye4LL4HeR4, whichget promoted to chirality-flipping fourth family mass termsMC

4 once the SM Higgs acquiresa VEV. Secondly, from vector-like mass terms, like ML

4 LL4LR4. For the purpose of clarity,we shall treat MC

4 and ML4 as independent masses in the analysis of the physical quantities

of interest, rather than constructing the full fourth family mass matrix and diagonalisingit, since such quantities rely on a chirality flip and are sensitive to MC

4 rather than thevector-like masses ML

4 . Spontaneous breaking of U(1)′ by the scalars φf spontaneouslyacquiring VEVs gives rise to a massive Z ′ boson featuring couplings with the vector-likefermion fields. In the interaction basis such terms will be diagonal and of the followingform:

LgaugeZ′ = g′Z ′µ

(QLDQγ

µQL + uRDuγµuR + dRDdγ

µdR + LLDLγµLL + eRDeγ

µeR + νRDνγµνR

),

(2.3)

DQ = diag (0, 0, 0, qQ4) , Du = diag (0, 0, 0, qQ4) , Dd = diag (0, 0, 0, qQ4) ,

DL = diag (0, 0, 0, qL4) , De = diag (0, 0, 0, qL4) , Dν = diag (0, 0, 0, qL4) .

(2.4)

– 4 –

At this stage, the SM quarks and leptons do not couple to the Z ′. However, theYukawa couplings detailed in Eq. (2.1) have no requirement to be diagonal. Before we candetermine the full masses of the propagating vector-like states and SM fermions, we need totransform the field content of the model such that the Yukawa couplings become diagonal.Therefore, fermions in the mass basis (denoted by primed fields) are related to particles inthe interaction basis by the following unitary transformations

Q′L = VQLQL , u′R = VuRuR , d′R = VdRdR ,

L′L = VLLLL , e′R = VeReR , ν ′R = VνRνR .

(2.5)

This mixing induces couplings of SM mass eigenstate fermions to the massive Z ′, which canbe expressed as follows

D′Q = VQLDQV†QL

, D′u = VuRDuV†uR , D′d = VdRDdV

†dR,

D′L = VLLDLV†LL, D′e = VeRDeV

†eR , D′ν = VνRDνV

†νR .

(2.6)

3 Mixing

In this article, we assume that the 4th VL fermion family only mixes with the third gen-eration of the SM quark doublet and with the second generation of the SM lepton doubletand singlet,

VQL = V QL34 , VLL = V LL

24 , VeR = V eR24 , (3.1)

where

V QL34 =

1 0 0 0

0 1 0 0

0 0 cos θQ34 sin θQ34

0 0 − sin θQ34 cos θQ34

, (3.2)

V LL,eR24 =

1 0 0 0

0 cos θLL,eR24 0 sin θLL,eR24

0 0 1 0

0 − sin θLL,eR24 0 cos θLL,eR24

, (3.3)

so for the matrices in Eq. (2.6) we obtain

D′Q = V QL34 DQ

(V QL

34

)†= qQ4

0 0 0 0

0 0 0 0

0 0(

sin θQ34

)2cos θQ34 sin θQ34

0 0 cos θQ34 sin θQ34

(cos θQ34

)2

, (3.4)

– 5 –

D′L = V LL24 DL

(V LL

24

)†= qL4

0 0 0 0

0(

sin θLL24

)20 cos θLL24 sin θLL24

0 0 0 0

0 cos θLL24 sin θLL24 0(

cos θLL24

)2

, (3.5)

D′e = V eR24 De (V eR

24 )† = qL4

0 0 0 0

0 (sin θeR24 )2 0 cos θeR24 sin θeR24

0 0 0 0

0 cos θeR24 sin θeR24 0 (cos θeR24 )2

, (3.6)

hence in this basis the relevant Z ′ couplings read

LZ′ ⊃ Z ′µ(gbbbLγ

µbL + gLµµµLγµµL + gRµµµRγ

µµR), (3.7)

where

gbb = g′qQ4

(sin θQ34

)2, (3.8)

gLµµ = g′qL4

(sin θLL24

)2, (3.9)

gRµµ = g′qL4 (sin θeR24 )2 . (3.10)

When the mass matrix for quarks is diagonalised, an effective bs coupling is obtainedthrough CKM mixing,

Z ′gbssLγµbL , (3.11)

gbs = gbbVts = g′qQ4

(sin θQ34

)2Vts , (3.12)

where Vts ≈ −0.04. The parameters here are the same as in the model of Ref. [30] but withthe addition of gRµµ, which will be crucial for explaining (g − 2)µ, as we shall see.

– 6 –

4 Phenomenology and flavour constraints

4.1 (g − 2)µ

µR,L µL,RµL,R

Z ′

µL,R

(a)

µR,L µL,RµL,R

Z ′

E4L,R

(b)

µR,Lmµ

µL,R

Z ′

µR,L µL,R

(c)

µR,LMC

4µL,R

Z ′

E4R,L E4L,R

(d)

Figure 1: Feynman diagrams in the model contributing to (g − 2)µ, photon lines areimplicit.

The diagrams displayed in Fig. 1 lead to Z ′-mediated contributions to the muon anomalousmagnetic moment, namely [59]

∆aµ = −m2µ

8π2M2Z′

[ (∣∣gLµµ∣∣2 +∣∣gRµµ∣∣2)F (m2

µ/M2Z′) +

(∣∣gLµE∣∣2 +∣∣gRµE∣∣2)F (m2

E/M2Z′)

+Re[gLµµ

(gRµµ)∗]

G(m2µ/M

2Z′) + Re

[gLµE

(gRµE

)∗]MC4

mµG(m2

E/M2Z′)

],

(4.1)where G(x) and F (x) are O(1) loop functions, and mE is the propagating mass of the4th lepton. In our case, mE ' ML

4 since we consider that the dominant source of massfor the 4th lepton is vector-like, i.e. ML

4 � MC4 . For the upcoming sections we shall fix

ML4 = 5 TeV, in order to preserve ML

4 � MC4 for a chirality-flipping mass MC

4 of orderGeV. The couplings between muons and VL leptons read

gLµE = g′qL4 cos θLL24 sin θLL24 = g′qL4

√1− gLµµ/(g′qL4)

√gLµµ/(g

′qL4) , (4.2)

gRµE = g′qL4 cos θeR24 sin θeR24 = g′qL4

√1− gRµµ/(g′qL4)

√gRµµ/(g

′qL4) , (4.3)

where from now on we will assume g′qL4 = 1 for simplicity.

Since the loop functions satisfy G(x) < 0 and F (x) > 0, the contributions proportionalto G(x) and F (x) in Eq. (4.1) interfere negatively. However, for a chirality-flipping massMC

4 of order v/√

2 (where v = 246 GeV is the SM Higgs VEV), the term proportional to

– 7 –

MC4 in Eq. (4.1) is dominant and positive due to G(x) < 0, matching the required sign to

explain the experimental measurement of ∆aµ by Fermilab [36] (see Eq. (1.3)). Hence, anon-vanishing coupling of Z ′ to RH muons is crucial to explain (g − 2)µ here: otherwise, ifwe assume gRµµ = 0, then gRµE vanishes and we lose the dominant contribution proportionalto MC

4 .

4.2 RK(∗)

bL

sL

µL,R

µL,R

Z ′

(a) Z′ exchange diagrams contributingto RK(∗) .

νµL

νµL

µL,R

µL,R

Z ′

(b) Z′ exchange diagrams contributingto neutrino trident production.

sL

bL µL,R

µL,R

Z ′

(c) Z′ exchange diagrams contributingto Bs → µµ.

sL

bL sL

bL

Z ′

(d) Z′ exchange diagrams contribut-ing to Bs − Bs mixing.

Figure 2

One possible explanation of the RK(∗) measurements in LHCb is that the low-energy La-grangian below the EW scale contains additional contributions to the effective 4-fermionoperator with left/right-handed muon, left-handed b-quark, and left-handed s-quark fields,

∆Leff ⊃ GLbsµ (sLγµbL) (µLγµµL) +GRbsµ (sLγµbL) (µRγ

µµR) + h.c. , (4.4)

arising in our model from integrating out the Z ′ boson at tree-level (Fig. 2a). The aboveoperators contribute to the flavour changing transitions bL → sLµLµL and bL → sLµRµR,respectively. A Z ′-mediated contribution to Bs → µµ (Fig. 2c) also arises.

We can express the coefficients GLbsµ and GRbsµ as a function of the couplings gbb, gLµµand gRµµ,

– 8 –

GLbsµ = −Vtsgbbg

Lµµ

M2Z′

=−Vts (g′)2 qQ4qL4

(sin θQ34

)2 (sin θLL24

)2

M2Z′

, (4.5)

GRbsµ = −Vtsgbbg

Rµµ

M2Z′

=−Vts (g′)2 qQ4qL4

(sin θQ34

)2(sin θeR24 )2

M2Z′

, (4.6)

where it can be seen that both GLbsµ and GRbsµ have the same sign in our model.In Ref. [24], the vector and axial effective operators

Heff ⊃ N [δC9 (sLγµbL) (µγµµ) + δC10 (sLγµbL) (µγµγ5µ)] + h.c. , (4.7)

N = −4GF√2VtbVts

e2

16π2, (4.8)

had been fitted to explain RK(∗) up to the 1σ level, as shown in Tables 2 and 3. From theresults for δC9 and δC10 we have computed the numerical values of GLbsµ and GRbsµ that fitRK(∗) up to the 1σ level,

δC9 = −GLbsµ +GRbsµ

2N⇒ GLbsµ = N (δC10 − δC9) , (4.9)

δC10 =GLbsµ −GRbsµ

2N⇒ GRbsµ = −N (δC9 + δC10) . (4.10)

The results displayed in Table 2 consider the so-called “theoretically clean fit” which,as explained in Ref. [24], displays the values of GLbsµ and GRbsµ that simultaneously fit RK(∗)

and the Bs → µµ data. This fit is denoted as theoretically clean since all the observablesincluded are free from theoretical uncertainties. On the other hand, the global fit inTable 3 also includes the fit of angular observables in B → K∗µµ data reported by LHCb,ATLAS and CMS, which are afflicted by larger theoretical uncertainties than the ratios oflepton universality violation and the Bs → µµ data [24].

Table 2: Fit of RK(∗) and the Bs → µµ data (Theoretically Clean Fit) [24]

Best fit 1σ range

(δC9, δC10) (−0.11, 0.59) δC9∈ [−0.41, 0.17] , δC10∈ [0.38, 0.81](GLbsµ/N , GRbsµ/N

)(0.7,−0.48) GLbsµ/N∈ [0.64, 0.79] , GRbsµ/N∈ [−0.98, 0.03]

(GLbsµ, G

Rbsµ

) (1

(42.5 TeV)2 , −

1

(51.3 TeV)2

) GLbsµ∈

[1

(44.44 TeV)2 ,

1

(40 TeV)2

],

GRbsµ∈

[− 1

(35.9 TeV)2 ,

1

(205 TeV)2

]

– 9 –

Table 3: Fit of RK(∗) , Bs → µµ data and angular observables of B → K∗µµ data (GlobalFit) [24]

Best fit 1σ range

(δC9, δC10) (−0.56, 0.30) δC9∈ [−0.79, −0.31] , δC10∈ [0.15, 0.49](GLbsµ/N , GRbsµ/N

)(0.86, 0.26) GLbsµ/N∈ [0.8, 0.94] , GRbsµ/N∈ [−0.18, 0.64]

(GLbsµ, G

Rbsµ

) (1

(38.34 TeV)2 ,

1

(69.73 TeV)2

) GLbsµ∈

[1

(39.75 TeV)2 ,

1

(36.67 TeV)2

],

GRbsµ∈

[− 1

(83.8 TeV)2 ,

1

(44.44 TeV)2

]

On one hand, GLbsµ shows similar best fit values of order (40 TeV)−2 in both fits, al-though the 1σ region is slightly tighter in the global fit (Table 3) than in the theoreticallyclean fit (Table 2). On the other hand, GRbsµ shows the largest differences between bothfits. For the theoretically clean fit, GRbsµ < 0 is favoured, although GRbsµ > 0 is still allowed.For the global fit, the situation is the opposite: GRbsµ > 0 is favoured, although GRbsµ < 0 isalso allowed. As a consequence, in both fits GRbsµ is compatible with zero and hence RK(∗)

can also be explained with only the purely left-handed operator sLγµbLµLγµµL, as in pre-vious Z ′ models [28–30]. However, we have shown that we need a non-vanishing couplingof right-handed muons to Z ′ in order to explain (g − 2)µ, hence within this model we havea non-zero right-handed contribution to RK(∗) . Therefore, we need to be aware of keepingsuch contribution, i.e. GRbsµ, within the 1σ region of the considered fit.

Moreover, the best fit value ofGRbsµ is negative within the theoretically clean fit, but pos-itive within the global fit. This indicates that the extra angular observables of B → K∗µµ

data are relevant and drastically change the picture for explaining RK(∗) with effective op-erators sLγµbLµRγµµR. However, the fact that these angular observables are affected byimportant theoretical uncertainties lead to some tension in the community about whetherangular observables of B → K∗µµ data should be considered or not in the global fits. Be-cause of this, during the remainder of this work we will consider both fits for computing ourresults. On the other hand, in our model GLbsµ and GRbsµ must have the same relative sign.Therefore, we shall keep the product qQ4qL4 positive and then fit GRbsµ in the positive regionallowed within the 1σ. We shall study whether this can be challenging in the theoreticallyclean fit, where the positive region allowed by the 1σ range of GRbsµ is tiny. In other words,the requirement of keeping GRbsµ within the 1σ range of the theoretically clean fit constitutesan extra effective constraint over this model.

– 10 –

10-4 0.001 0.010 0.100 110-4

0.001

0.010

0.100

1

gμμL

gbb

bL→sLμLμL (Theoretically Clean Fit)

Bsmixing Avg.'19

Bsmixing FLAG'19

Trident

ATLAS

MZ' = 10 GeV

MZ' = 100 GeV

MZ' = 1 TeV

gμμR = 0

(a)

10-4 0.001 0.010 0.100 110-4

0.001

0.010

0.100

1

gμμL

gbb

bL→sLμLμL (Theoretically Clean Fit)

Bsmixing Avg.'19

Bsmixing FLAG'19

Trident

ATLASbL→sLμ RμR(Clean Fit)

1σ excluded

MZ' = 10 GeV

MZ' = 100 GeV

MZ' = 1 TeV

gμμR = 0.0001

(b)

10-4 0.001 0.010 0.100 110-4

0.001

0.010

0.100

1

gμμL

gbb

bL→sLμLμL (Theoretically Clean Fit)

Bsmixing Avg.'19

Bsmixing FLAG'19

Trident

bL→sLμ RμR (Clean Fit)

1σ excluded

ATLAS

MZ' = 10 GeV

MZ' = 100 GeV

MZ' = 1 TeV

gμμR = 0.01

(c)

10-4 0.001 0.010 0.100 110-4

0.001

0.010

0.100

1

gμμL

gbb

bL→sLμLμL (Global Fit)

Bsmixing Avg.'19

Bsmixing FLAG'19

Trident

bL→sLμ RμR (Global Fit)

1σ excluded

ATLAS

MZ' = 10 GeV

MZ' = 100 GeV

MZ' = 1 TeV

gμμR = 0.01

(d)

Figure 3: The parameter space in the (gLµµ, gbb) plane compatible with RK(∗) anomaliesand flavour constraints (white). The Z ′ mass varies over the plane, with an unique Z ′ massfor each point in the plane as required to match the best fit value for GLbsµ(Eq. (4.5)) of thetheoretically clean fit in Table 2 (Fig. 3a, 3b, 3c) and the global fit in Table 3 (Fig. 3d).We show the recent Bs − Bs mixing constraints (blue and light blue) [74], the neutrinotrident bounds (orange) [75], and the region excluded by LHC dimuon resonance searches(purple) [76]. When a non-vanishing gRµµ is considered, the red-shaded region is excludedof the 1σ range of GRbsµ(Eq. (4.6)) for the considered fit. The dashed lines correspond toconstant values of MZ′ as specified in the plots.

4.3 Bs − Bs mixing

The Z ′ coupling to bs-quarks in Eq. (3.11) leads to an additional tree-level contribution(Fig. 2d) to Bs − Bs mixing,

∆Leff ⊃ −Gbs2

(sLγµbL)2 + h.c. (4.11)

where

– 11 –

Gbs =g2bs

M2Z′

=g2bbV

2ts

M2Z′

. (4.12)

Such a new contribution is constrained by the results of the mass difference ∆Ms ofneutral Bs mesons. The theoretical determination of the mass difference is limited by ourunderstanding of non-perturbative matrix elements of dimension six operators, which canbe computed with lattice simulations or sum rules. Here we follow the recent analysis ofRef. [74], which displays two different results for ∆Ms,

∆MFLAG′19s =

(1.13+0.07

−0.07

)∆M exp

s , (4.13)

∆MAverage′19s =

(1.04+0.04

−0.09

)∆M exp

s . (4.14)

∆MFLAG′19s is obtained using lattice results, and is about two standard deviations above

the experimental numbers. This result for the mass difference sets the strong bound

Gbs .1

(330 TeV)2 . (4.15)

On the other hand, MAverage′19s , obtained as a weighted average from both lattice simula-

tions and sum rule results, shows better agreement with the experiment, and a reductionof the total errors by about 40%. This result for the mass difference sets a less constrainingbound

Gbs .1

(220 TeV)2 . (4.16)

The resulting constraints will be shown as blue regions over the parameter space.

4.4 Neutrino trident

The Z ′ couplings to the second generation of the SM lepton doublet and singlet lead to anew tree-level contribution (Fig. 2b) to the effective 4-lepton interaction

∆Leff ⊃ −(gLµµ)2

2M2Z′

(µLγµµL) (νµLγµνµL)−

gRµµgLµµ

M2Z′

(µRγµµR) (νµLγµνµL) . (4.17)

This operator is constrained by the trident production νµγ∗ → νµµ+µ− [77–79] . Using the

results of the global fit in Ref. [75], the bound over gLµµ and gRµµ is given by

− 1

(390 GeV)2 >

(gLµµ)2

+ gLµµgRµµ

M2Z′

>1

(370 GeV)2 , (4.18)

whereas in our case only the right side of (4.18) applies, since according to Eqs. (3.9) and(3.10) gLµµ and gRµµ have the same relative sign in our model and hence the product gLµµgRµµis positive. The resulting constraints will be shown as orange regions over the parameterspace.

– 12 –

4.5 Constraints from lepton flavour violation

Within this model we are assuming zero mixing in the charged-lepton sector of the SM.Therefore, in our Z ′ model there are no contributions to flavour-violating processes such asµ→ eγ or τ → 3µ.

4.6 Collider constraints

Our model is not constrained by electron collider searches since our Z ′ does not couple toelectrons. However, further constraints on our model come from LHC searches. For light Z ′

masses, the LHC measurements of the Z decays to four muons, with the second muon pairproduced in the SM via a virtual photon [80, 81], pp → Z → 4µ, set relevant constraintsin the low mass region of Z ′ models, 5 . MZ′ > 70 GeV [30, 79, 82, 83]. We avoid such aconstraint by keeping MZ′ > 75 GeV in our analysis.

For heavier Z ′ masses, the strongest constraints come from LHC dimuon resonancesearches, pp→ Z ′ → µ+µ−, see also [84, 85]. In our model, the Z ′ is dominantly producedat the LHC through its coupling to bottom quarks, bb→ Z ′. The cross section σ

(bb→ Z ′

)from bb collisions is given for gbb = 1 in Fig. 3 of Ref. [86], we multiply it by g2

bb in order toobtain the cross section for any gbb. We neglect a further contribution coming from bs→ Z ′

since it is CKM suppressed by V 2ts. Therefore, we assume that σ (pp→ Z ′) is dominated by

the subprocess bb→ Z ′. The Z ′ boson can subsequently decay into muons, muon neutrinos,bottom quarks, bottom-strange quark pair, and also into top quarks when kinematicallyallowed. The partial decay widths are given by

ΓZ′→µµ =1

24π

[(gLµµ)2

+(gRµµ)2]

MZ′ ,

ΓZ′→νµνµ =1

24π

(gLµµ)2MZ′ ,

ΓZ′→bb =1

8πg2bbMZ′ , ΓZ′→bs =

1

8πg2bbV

2tsMZ′ ,

ΓZ′→tt =1

8πg2bbMZ′

(1− m2

t

M2Z′

)√1− 4m2

t

M2Z′,

(4.19)

from which we compute Br (Z ′ → µµ) analytically,

Br(Z ′ → µµ

)=

ΓZ′→µµΓZ′→µµ + ΓZ′→νµνµ + ΓZ′→bb + ΓZ′→bs + ΓZ′→tt

. (4.20)

Then σ (pp→ Z ′ → µ+µ−) is estimated using the narrow-width approximation,

σ(pp→ Z ′ → µ+µ−

)≈ σ

(pp→ Z ′

)Br(Z ′ → µµ

), (4.21)

and compared with the limits obtained from the dimuon resonance search by ATLAS [76],which allows us to constrain Z ′ masses between 150 GeV and 5 TeV. Previous studies[30] verified that the analogous Tevatron analyses give weaker constraints than LHC. Allthings considered, the resulting ATLAS constraints will be shown as purple regions overthe parameter space.

– 13 –

4.7 Higgs diphoton decay

h0

γ

γ

fSM

(a)

h0

γ

γ

E4

(b)

h0

γ

γ

W+

(c)

h0

γ

γ

W+

(d)

Figure 4: Diagrams contributing to the Higgs diphoton decay (h0 → γγ) where fSM =

ui, di, ei, i = 1, 2, 3 and E4 is the 4th family VL lepton.

After Spontaneous Symmetry Breaking (SSB), the Yukawa term in Eq. (2.1) involvingthe SM Higgs field and the 4th VL lepton gives rise to the chirality-flipping mass MC

4 ,which gives a very important contribution in Eq. (4.1) for accommodating ∆aµ with theexperimental measurements. On the other hand, MC

4 is also expected to give an extracontribution to the decay of the Standard Model Higgs to two photons, a process thathas been explored in colliders. Firstly, within the SM, fermions (Fig. 4a) and W± bosons(Figs. 4c, 4d) contribute to the decay channel h0 → γγ [87]

Γ(h0 → γγ)SM =α2m3

h

256π3v2

∣∣∣∣∣∣F1(τW ) +∑f ε SM

NcfQ2fF1/2(τf )

∣∣∣∣∣∣2

, (4.22)

where α = 1/137, mh = 126 GeV, v = 246 GeV, Ncf = 1 (leptons), 3 (quarks), Qf is theelectromagnetic charge of the fermion f in units of e, and the loop functions are defined as

F1 = 2 + 3τ + 3τ (2− τ) f(τ) , (4.23)

F1/2 = −2τ [1 + (1− τ) f(τ)] , (4.24)

with

τi = 4m2i /m

2h (4.25)

– 14 –

and

f(τ) =

[arcsin

(1/√τ)]2

, if τ ≥ 1 ,

−1

4

[ln

(1 +√

1− τ1−√

1− τ

)− iπ

]2

, if τ < 1 .

(4.26)

Note here that for large τ , F1/2 → −4/3. The dominant contribution to Γ(h0 → γγ)SM isthe contribution of the W bosons,

F1(τW ) ' 8.33 , (4.27)

and it interferes destructively with the top-quark loop

NctQ2tF1/2(τf ) = 3 (2/3)2 (−1.37644) = −1.83526 , (4.28)

therefore

Γ(h0 → γγ)SM =α2m3

h

256π3v2|8.33− 1.83526|2 ' 9.15636 · 10−6 GeV . (4.29)

The exact result by taking into account the contribution of all SM fermions is

Γ(h0 → γγ)SM = 9.34862 · 10−6 GeV , (4.30)

and if we take Γ(h0 → all)PDG 2021 = 3.2+2.8−2.2 MeV, then

BR(h0 → γγ

)SM

=Γ(h0 → γγ)SM

Γ(h0 → all)PDG 2021× 100 ' 0.29 % . (4.31)

Now we add the contribution of a fourth VL lepton (Fig. 4b) with VL mass ML4 that

couples to the Higgs via the chirality-flipping mass MC4 , where ML

4 �MC4 (in such a way

that the propagating mass of the fourth lepton can be approximated by the VL mass) [88],

Γ(h0 → γγ) =α2m3

h

256π3v2

∣∣∣∣∣∣F1(τW ) +∑f ε SM

NcfQ2fF1/2(τf ) +

MC4

ML4

F1/2 (τE4)

∣∣∣∣∣∣2

. (4.32)

We can see that the new contribution proportional to the chirality-flipping mass is sup-pressed by the heavier VL mass. Moreover, this new contribution decreases Γ(h0 → γγ),since it interferes destructively with the most sizable contribution of the W bosons. Letus now compare with the experimental results for the h0 signal strength in the h0 → γγ

channel [89],

Rγγ =Γ(h0 → γγ)

Γ(h0 → γγ)SM, (4.33)

RPDG, 2020γγ = 1.11+0.1

−0.09 . (4.34)

– 15 –

In the case of MC4 = 200 GeV,

RVLγγ =

|8.33− 1.83526− 0.0533333|2

|8.33− 1.83526|2= 0.983813 . (4.35)

In the case of MC4 = 600 GeV,

RVLγγ =

|8.33− 1.83526− 0.16|2

|8.33− 1.83526|2= 0.951336 . (4.36)

Therefore, even for a value ofMC4 close to the perturbation theory limitMC

4 .√

4πv/√

2 '616.8 GeV, the chirality-flipping mass contribution to h0 → γγ is within the 2σ range ofRPDG, 2020γγ .

5 Results and discussion

In Fig. 3 we have displayed the parameter space in the (gLµµ, gbb) plane for gRµµ = 0, 0.0001,

0.01, considering the theoretically clean fit (Figs. 3a, 3b, 3c) and the global fit (Fig. 3d).In every case, there is parameter space free from all the constraints that is able to explainRK(∗) . If we set gRµµ = 0 (Fig. 3a), we are making a purely left-handed explanation of RK(∗) ,hence recovering the same results as in Ref. [30]. As gRµµ is increased, the condition of keep-ing the contribution to bL → sLµRµR (namely the Wilson coefficient GRbsµ) within the 1σ

range becomes constraining over the parameter space, especially when the theoreticallyclean fit is considered (Figs. 3b, 3c). On the other hand, if the global fit is considered(Fig. 3d), then larger values of gRµµ are accessible.

In Figs. 5 and 6, for light and heavy Z ′ masses respectively, it can be seen that boththe contribution to bL → sLµLµL that explains RK(∗) (namely the Wilson Coefficient GLbsµ)and ∆aµ can be produced simultaneously within their 1σ region by a Z ′ with a mass in therange of 75 GeV to 2 TeV, for both the theoretically clean fit and the global fit. Within thisrange of masses, GLbsµ, G

Rbsµ and ∆aµ can be simultaneously fitted with the same parameters

up to the 1σ ranges of both the theoretically clean fit and the global fit, since in all theconsidered cases the parameter space where GLbsµ and ∆aµ are simultaneously explained isalso within the 1σ range of GRbsµ. The upper bound for the latter in the theoretically cleanfit is displayed in Figs. 5 and 6 as a red horizontal line, there is no lower bound displayedsince GRbsµ is compatible with zero in both fits. In all the explored cases, the condition offitting GRbsµ is less constraining than Bs − Bs mixing.

For light Z ′ masses around 75− 200 GeV (Fig. 5), both anomalies RK(∗) and ∆aµ canbe explained simultaneously with the condition of small gRµµ and MC

4 . On the other hand,as displayed in Fig. 6, for heavy Z ′ masses of 1−2 TeV both anomalies can also be explainedsimultaneously but it is required to increase either gRµµ (two lower panels of Fig. 6) or MC

4

(two upper panels of Fig. 6). In every case, MC4 is kept below the perturbation theory limit

of MC4 >

√4πv/

√2 ≈ 618 GeV, and its contribution to Higgs diphoton decay has been

previously proven to be within the 2σ range of the experimental signals even for values of

– 16 –

Δaμ 1σ

bL→sLμLμL (Clean fit) 1σ

bL→sLμLμL (Global fit) 1σ

10-4 0.001 0.010 0.100 110-4

0.001

0.010

0.100

1

gμμL

gbb Bs mix. Avg.'19

Bs mix. FLAG'19

Trident

bL→sLμ RμR (Clean Fit) 1σ limit

MZ'= 75 GeV

gμμR = 0.0001

M4C= 15 GeV

(a)

Δaμ 1σ

bL→sLμLμL (Clean fit) 1σ

bL→sLμLμL (Global fit) 1σ

10-4 0.001 0.010 0.100 110-4

0.001

0.010

0.100

1

gμμL

gbb

Bs mix. Avg.'19Bs mix. FLAG'19

Trident

bL→sLμ RμR(Clean Fit) 1σ limit

MZ'= 100 GeV

gμμR = 0.0001

M4C= 20 GeV

(b)

Δaμ 1σ

bL→sLμLμL (Clean fit) 1σ

bL→sLμLμL (Global fit) 1σ

10-4 0.001 0.010 0.100 110-4

0.001

0.010

0.100

1

gμμL

gbb

Bs mix. Avg.'19Bs mix. FLAG'19

Trident

bL→sLμ RμR (Clean Fit) 1σ limit

ATLAS

MZ'= 200 GeV

gμμR = 0.0001

M4C= 45 GeV

(c)

Δaμ 1σ

bL→sLμLμL (Clean fit) 1σ

bL→sLμLμL (Global fit) 1σ

10-4 0.001 0.010 0.100 110-4

0.001

0.010

0.100

1

gμμL

gbb

Bs mix. Avg.'19Bs mix. FLAG'19

ATLAS

MZ'= 500 GeV

gμμR = 0.0001

M4C= 200 GeV

(d)

Figure 5: Bounds on the parameter space in the (gLµµ, gbb) plane for fixed Z ′ masses: 75,100, 200 and 500 GeV, as indicated on each panel. Each panel also displays the con-sidered MC

4 and gRµµ, while the propagating mass of the VL lepton is always kept asmE 'ML

4 = 5 TeV. The green region explains ∆aµ up to 1σ. The yellow and pink re-gions fit the Wilson coefficient GLbsµ (4.5) up to 1σ for the theoretically clean fit and theglobal fit [24], respectively. The red horizontal line shows the limit of the 1σ region for theWilson coefficient GRbsµ (4.6) in the more restrictive theoretically clean fit, in such a waythat the parameter space above the red line is excluded. The blue and orange areas showthe Bs − Bs mixing [74] and neutrino trident [75] exclusions, respectively, while the purpleregion is excluded by LHC dimuon resonance searches [76].

MC4 close to the perturbation theory limit. Moreover, in every case gRµµ is set to values for

which GRbsµ can be simultaneously fitted, hence RK(∗) is explained. Fitting both anomaliessimultaneously for MZ′ > 2 TeV could in principle be possible but would require chirality-flipping masses too close to the perturbation theory limit, and/or values of gRµµ of O(0.1) orhigher, for which the 1σ range of GRbsµ in the theoretically clean fit becomes more challengingto fit, leading to constraints over the parameter space larger than the present limits ofBs − Bs mixing. Instead, if we consider the global fit that includes angular observables of

– 17 –

Δaμ 1σ

bL→sLμLμL (Clean fit) 1σ

bL→sLμLμL (Global fit) 1σ

10-4 0.001 0.010 0.100 110-4

0.001

0.010

0.100

1

gμμL

gbb

Bs mix. Avg.'19Bs mix. FLAG'19

ATLAS

MZ'= 1000 GeV

gμμR = 0.0001

M4C= 580 GeV

(a)

Δaμ 1σ

bL→sLμLμL (Clean fit) 1σ

bL→sLμLμL (Global fit) 1σ

10-4 0.001 0.010 0.100 110-4

0.001

0.010

0.100

1

gμμL

gbb

Bs mix. Avg.'19Bs mix. FLAG'19

ATLAS

MZ'= 2000 GeV

gμμR = 0.001

M4C= 580 GeV

(b)

Δaμ 1σ

bL→sLμLμL (Clean fit) 1σ

bL→sLμLμL (Global fit) 1σ

10-4 0.001 0.010 0.100 110-4

0.001

0.010

0.100

1

gμμL

gbb

Bs mix. Avg.'19Bs mix. FLAG'19

bL→sLμ RμR (Clean Fit) 1σ limit

ATLAS

MZ'= 1000 GeV

gμμR = 0.001

M4C= 200 GeV

(c)

Δaμ 1σ

bL→sLμLμL (Clean fit) 1σ

bL→sLμLμL (Global fit) 1σ

10-4 0.001 0.010 0.100 110-4

0.001

0.010

0.100

1

gμμL

gbb

Bs mix. Avg.'19Bs mix. FLAG'19

bL→sLμ RμR (Clean Fit) 1σ limit ATLAS

MZ'= 2000 GeV

gμμR = 0.01

M4C= 200 GeV

(d)

Figure 6: Bounds on the parameter space in the (gLµµ, gbb) plane for heavy fixed Z ′

masses: 1000 and 2000 GeV, as indicated on each panel. Each panel also displays theconsidered MC

4 and gRµµ, while the propagating mass of the VL lepton is always kept asmE ' ML

4 = 5 TeV. The green region explains ∆aµ up to 1σ. The yellow and pinkregions fit the Wilson coefficient GLbsµ (4.5) up to 1σ for the theoretically clean fit and theglobal fit [24], respectively. The red horizontal line shows the limit of the 1σ region for theWilson coefficient GRbsµ (4.6) in the more restrictive theoretically clean fit, in such a waythat the parameter space above the red line is excluded. The blue and orange areas showthe Bs − Bs mixing [74] and neutrino trident [75] exclusions, respectively, while the purpleregion is excluded by LHC dimuon resonance searches [76].

B → K∗µµ data, here GRbsµ is compatible with larger positive values and hence also largergRµµ are accessible, in such a way that explaining both anomalies with heavier masses of Z ′

is possible. However, Figs. 5 and 6 also show that collider constraints coming from dimuonresonance searches by ATLAS [76] are very constraining for MZ′ > 500 GeV. Although inevery case we can still find good points that simultaneously explain both RK(∗) and ∆aµwhile avoiding the ATLAS constraint, such points could be ruled out in the future by theupcoming LHC run 3 starting in 2022.

– 18 –

6 Conclusions

We have shown in this article that both muon anomalies RK(∗) and (g − 2)µ can be si-multaneously addressed by our fermiophobic Z ′ model with 75 GeV > MZ′ > 2 TeV. Theexplanation of (g − 2)µ in this model requires non-vanishing couplings of Z ′ to left-handedand right-handed muons gLµµ, gRµµ, along with a non-vanishing chirality-flipping mass MC

4

obtained from the coupling of a fourth vector-like lepton to the SM Higgs. The explanationof RK(∗) also requires a coupling of Z ′ to bs-quarks. Such Z ′ couplings are obtained inthis model through mixing of muons and bottom quarks with a fourth vector-like fermionfamily. The Z ′ coupling to bs is hence CKM suppressed as gbbVts. This way, we haveconsidered here the minimal mixing framework where only three mixing parameters areinvolved, while other similar Z ′ models that address both muon anomalies are either notfermiophobic [62, 63], consider extra symmetries [61] or consider a more general mixingframework with more parameters involved [64–66] where the parameter space is more dif-ficult to explore. In the minimal mixing framework that we have considered here, we havebeen able to systematically explore the parameter space.

The fact that gRµµ 6= 0 implies that we cannot do a purely left-handed explanation ofRK(∗) anymore like in previous studies [29, 30], instead we need to fit both the LH andRH Wilson coefficients of the effective operators, GLbsµ and GRbsµ, within the 1σ range thatexplains RK(∗) according to the latest global fits [24]. Because of this, explaining bothanomalies for MZ′ > 2 TeV becomes challenging if we consider the theoretically clean fitwhere the positive 1σ region of GRbsµ is small: larger gRµµ are required to explain (g − 2)µ,but then this implies a smaller gbb to keep GRbsµ within the 1σ range of the theoretically cleanfit, while at the same time a big gbb is required to fit GLbsµ for these heavier masses of Z ′.Moreover, the heavier the Z ′ boson is, the largerMC

4 is also required to explain (g − 2)µ, butMC

4 is bounded from above by perturbation theory toMC4 >

√4πv/

√2 ≈ 618 GeV. On the

other hand, if we consider the global fit that includes angular observables of B → K∗µµ

data, here GRbsµ is compatible with larger positive values and hence also larger gRµµ areallowed, but the perturbation theory constraint over MC

4 remains.

Finally, we have studied the impact of collider searches over this model: constraintscoming from experimental measurements of Z → 4µ can be avoided by keepingMZ′ > 70 GeV

[30, 79, 82, 83]. However, dimuon resonance searches by ATLAS [76] are already very con-straining for MZ′ > 500 GeV, in such a way that the good results of this model for heavyZ ′ masses could be ruled out by the the upcoming LHC run 3 starting in 2022.

Acknowledgements

This project has received funding from the European Union’s Horizon 2020 Research andInnovation programme under Marie Skłodowska-Curie grant agreement HIDDeN EuropeanITN project (H2020-MSCA-ITN-2019//860881-HIDDeN). SFK acknowledges the STFCConsolidated Grant ST/L000296/1.

– 19 –

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