$U(1)_{B_3-3L_\mu}$ gauge symmetry as a simple description of $b\to s$ anomalies

We present a simple $U(1)_{B_3-3L_\mu}$ gauge Standard Model extension that can easily account for the anomalies in $R(K)$ and $R(K^*)$ reported by LHCb. The model is economical in its setup and particle content. Among the Standard Model fermions, only the third generation quark family and the second generation leptons transform non-trivially under the new $U(1)_{B_3-3L_\mu}$ symmetry. This leads to lepton non-universality and flavor changing neutral currents involving the second and third quark families. We discuss the relevant experimental constraints and some implications.


I. INTRODUCTION
A host of increasingly sophisticated experiments over several decades has been able to thoroughly verify various predictions of the Standard Model of particle physics. These culminated with the discovery of the 125 GeV Higgs boson at the Large Hadron Collider (LHC) in 2012. Despite its amazing success, there are good reasons to think that the Standard Model may not be the ultimate theory. Apart from many theoretical shortcomings, the existence of neutrino mass suggests the existence of new physics, possibly in the electroweak-TeV range. By making very precise measurements of decay rates and angular observables, the LHCb Collaboration looks for the effect of new particles in various hadronic processes. Of particular interest to LHCb are the processes which are either forbidden or are extremely rare within the Standard Model. Since such processes may be allowed in new physics models, these searches can probe new physics models with good sensitivity, sometimes higher than attainable at the ATLAS and CMS experiments. LHCb has recently announced anomalous measurements of b → sµ + µ − transitions [1]. They measured the ratio R K * ≡ B(B 0 → K * 0 µ + µ − )/B(B 0 → K * 0 e + e − ) as (1) These measurements involve two ranges of q 2 , the dilepton invariant squared mass. These numbers are very similar to the previous LHCb measurement of the ratio R K ≡ B(B + → K + µ + µ − )/B(B + → K + e + e − ) [2], These observations are also in tune with the so called P 5 anomaly observed in the angular variable P 5 of B → K * µ + µ − decays [3][4][5]. In addition to these, LHCb has also observed other several other anomalies all involving b → s transitions, such as B s → φµ + µ − [6]. Specially remarkable is the fact that, although each individual result is not specially significant, all of the anomalies observed in b → sµ + µ − transitions and form a coherent global picture [7][8][9]. At a basic level such rare transitions may be described by the effective Hamiltonian, where C Here each coefficient is separated into a Standard Model part (SM) and a new physics contribution (NP). The relevant semileptonic operators required to account for the observed b → sµ + µ − anomalies are of the restricted type, In this paper we propose a consistent gauge model, constructed from first principles, that induces just one of the Wilson operators, O 9 . Its strength parameter can describe the observed b → sµ + µ − anomalies in agreement with all existing experimental restrictions. It provides a minimal way to account for the b → sµ + µ − discrepancies, while adding as few new particles and symmetries as possible : just a new U(1) B3−3Lµ gauge symmetry. Unlike other alternative schemes, which typically require addition of new charged fermions [10][11][12][13][14][15][16][17][18][19][20][21][22][23][24][25][26][27][28] or leptoquarks [29][30][31][32][33][34][35][36][37][38], all anomalies induced by our U(1) B3−3Lµ gauge symmetry, including gravitational ones, cancel without need for adding any new charged states. The only new fermions present are right handed neutrinos, required in order to generate small neutrino masses through the type-I seesaw mechanism [39]. The only other new particles required are the Z boson, which mediates the b → sµ + µ − transitions, and new scalars involved in symmetry breaking, so as to generate mass for the Z boson and neutrinos.
The plan of the paper is as follows. In Section II we discuss the basics of the U(1) B3−3Lµ gauge model including the gauge and gauge-gravity anomaly cancellation conditions. We show that the model is free from anomalies without need for additional charged fermions. In Section III we summarize the main properties of the model and show that the flavor changing neutral currents (FCNC) mediated by the Z boson have all the essential features required in order to account for the observed b → sµ + µ − discrepancies. In Section IV we discuss the various constraints on our model coming from flavor, collider and precision physics. We show that, after taking all relevant constraints into account, the model still has enough freedom to account for the observed anomalies. Finally, results and conclusions are given in Figs. 1 and 2 and Secs.V and VI.
In this section we discuss the details of the model. The SU(3) C ⊗ SU(2) L ⊗ U(1) Y and U(1) B3−3Lµ charge assignments for the fermions and scalars are given in Table I, where i = 1, 2 labels the first two generations of quarks. Table I. Particle content and assignments. The singlet χν ensures a realistic pattern of neutrino oscillations [40].
Notice that we assume two SU(2) L doublet scalars denoted where Φ 2 is the "Standard Model like" doublet. The second doublet scalar Φ 1 , as well as the gauge singlet scalars χ, χ ν are all charged under the U(1) B3−3Lµ symmetry, carrying charges y 1 , y 2 and y 3 , respectively. Their charges will be fixed by other considerations, as we discuss shortly. Apart from the Standard Model fermions we also include three right handed neutrinos ν Ri (i = e, µ, τ ) to account for neutrino masses and oscillations. In order for U(1) B3−3Lµ to be a consistent gauge symmetry it is important to ensure that the model is anomaly free. The U(1) B3−3Lµ symmetry can potentially induce the following triangular anomalies: In addition we have anomaly conditions involving the U(1) B3−3Lµ just with itself and with gravity, [Gravity] where (X) i denote the U(1) B3−3Lµ fermion charges. Using the assignments in Table. I, we find that the first four anomalies, i.e. eqs. (4)- (7), cancel, irrespective of the charges of the right handed neutrinos. The anomaly cancellation conditions eqs. (8) and (9) give the following two conditions on the U(1) B3−3Lµ charge of the right handed neutrinos The only solution for eq. (11) is given by This implies that under the U(1) B3−3Lµ symmetry, one of the right handed neutrinos should transform as −3 while the others can carry any arbitrary equal and opposite charge. For definiteness and keeping in mind simple scenarios of neutrino mass generation, we choose to assign the following charges to the right handed neutrinos: Once the scalars get vacuum expectation values (vevs) the SU (3) Notice that, since only Φ 1 and the χ's are charged under the U(1) B3−3Lµ symmetry, the latter is broken only by the vevs of these scalars. Note also that both Standard Model doublet scalars Φ 1 and Φ 2 contribute in the breaking of the SU(3) C ⊗ SU(2) L ⊗ U(1) Y symmetry. The U(1) B3−3Lµ charges of Φ 1 and χ are not constrained by the anomaly cancellation conditions. However, in order to generate a realistic Cabbibo-Kobayashi-Masakawa (CKM) mixing matrix, we must choose the U(1) B3−3Lµ charge of Φ 1 to be the same as that of Q 3 , i.e.
Likewise, the U(1) B3−3Lµ charge of singlet scalar χ is also not fixed by anomaly cancellation conditions. However, if the U(1) B3−3Lµ charge of χ is not the same as the charge of Φ 1 , then the scalar potential will have a residual global U (1) symmetry leading to a massless Goldstone boson. This can be avoided by taking which provides a term in the scalar potential like , where κ is a dimensionful parameter. With these assignments for ν Ri , Φ 1 and χ plus a scalar singlet χ ν , transforming as χ ν ∼ 3, one has a realistic pattern of neutrino mass and mixing 1 .

III. THE MODEL
Having satisfied the anomaly cancellation conditions one can write the Yukawa and scalar sectors of the model in the standard way. After spontaneous symmetry breaking the resulting charged current weak interactions of quarks and leptons are completely standard, except for the presence of the extra Higgs doublet. Hence we focus here on the neutral current, mediated both by the Standard Model Z boson as well as the Z . Since Φ 1 and χ i are charged under U(1) B3−3Lµ , their vevs Φ 1 = v 1 / √ 2 and χ i = u i break the U(1) B3−3Lµ symmetry and contribute to the Z mass. Thus, in the limit in which χ i v 1 and negligible Z − Z mixing, the neutral gauge boson masses are given by, where g , g 1 and g 2 are the coupling constants of the U(1) B3−3Lµ , U(1) Y and SU(2) L symmetries, respectively. The vevs of the doublet scalars satisfy v 2 ≡ (246 GeV) 2 In what follows we focus on the main features of the resulting form of the neutral current weak interactions relevant in order to describe the observed b → sµ + µ − discrepancies.
The neutral current and the new Z boson The key part of the theory to explain the anomaly is the neutral current. Since we do not add any exotic quarks, such as vector-like quarks, the Standard Model part of the neutral current is canonical, obeying the Glashow-Iliopoulos-Maiani mechanism. However, by construction, the neutral current Lagrangian associated to the Z boson will give rise to flavor changing transitions, that is L Z = g Z µ J 0 µ , where the neutral current is given by After spontaneous symmetry breaking the part for down-type quark becomes where D L = V CKM and D R = I, similar to [14,15]. Then, the interactions between Z and the down-type quarks can be rewritten as follows where d = (d, s, b), We stress here that there are no FCNCs in the charged lepton sector, hence no tree level lepton flavour violation [14,15].

IV. EXPERIMENTAL CONSTRAINTS
Having shown that the flavor changing neutral current (FCNC) mediated by the Z boson has the right form, we now turn to the experimental constraints on the two parameters, the Z mass and coupling strength, relevant for describing in our model the anomalies recently observed in rare B decays.

A. B meson mixing
The exchange of the Z boson induces B s −B s mixing at tree-level. This leads to a constraint on the Z boson that can be quantified from [10,[41][42][43] where by S 0 2.3 we denote the Standard Model loop-function [44,45].

B. Neutrino trident production
The production of a µ + µ − pair in the scattering of a muon neutrino in the Coulomb field of a heavy nucleus, i.e. ν µ N → ν µ N µ + µ − , provides a sensitive probe of a Z boson. In the model the correction to the trident cross section can be written as [10] where v = 246 GeV and s W = sin θ W . The measurement of the trident cross section by the CCFR collaboration is [46] σ CCFR σ SM = 0.82 ± 0.28.

C. Lepton Flavor Universality
The presence of Z µµ and Z ν µ ν µ couplings will break lepton flavor universality (LFU). This is manifest in the Z boson couplings to muons and muon neutrinos through loop effects. The corrections to the vector couplings of Z boson to muons and muon neutrinos are given by [10] where κ(m Z ) is the loop factor associated with the Z loop [47]. The experimental constraints from LEP and SLC [48] set very stringent upper limits on g and m Z . We use these constraints to find out 95% CL upper limits as shown by the red dashed line in Figs. 1 and 2.
The ATLAS [49] and CMS [50] collaborations both have set upper limits on the branching fraction of the Z boson decay to four charged leptons. In particular ATLAS has set an upper limit on Br(Z → 4µ) < (4.2 ± 0.4) × 10 −6 with the combined 7 TeV and 8 TeV dataset [49]. In our model the Z → 4 decay will receive contributions from processes involving the Z as the intermediate state. This process will receive a significant contribution from Z → µ + µ − Z followed by Z → µ + µ − for the case where m Z < m Z . We utilize this upper limit to extract a constraint on g and m Z in our model. To determine the upper limit, the cross sections are generated in the Monte Carlo event generator MadGraph5 aMC@NLO [51], interfaced to PYTHIA 6.4 [52] for hadronization and showering and finally fed into fast detector simulator Delphes 3.3.3 [53] to incorporate detector effects. In our analysis we adopt the PDF set NN23LO1 PDF [54]. The effective Lagrangians written in eq. (17) and eq. (18) are implemented in FeynRules 2.0 [55].
We have closely followed the ATLAS analysis in our paper. Selected events are required to contain four isolated muons with two opposite sign same flavor dimuon pairs. The four muons in the quadruplet are required to be separated by ∆R = ∆η 2 + ∆φ 2 > 0.1, with each of them having maximum pseudo-rapidity |η| < 2.5. The three leading muons in an event should have transverse momenta p T > 20 GeV, 15 GeV and 8 GeV respectively. The four muons will constitute two same flavor oppositely charged muon pairs in an event. The pair closest to Z boson mass must have invariant mass > 20 GeV and the other > 5 GeV. We finally impose an invariant mass cut 76 GeV < m 4µ < 106 GeV on the four muons in the event. Assuming Z → µ + µ − is the only new physics contribution and 2σ error on the observed Br(Z → 4 ) we find the upper limit on g and m Z shown in Figs. 1 and 2.
The Z boson will be produced at LHC predominantly via flavor violating bs → Z (and its conjugate process) and the flavor conserving bb → Z processes. Hence, search of heavy resonance in the dimuon final state by ATLAS and CMS will constrain the parameter space of our model. In particular ATLAS [56] has set a 95% CL (confidence level) upper limit on σ(pp → Z + X)Br(Z → µ + µ − ) in the 150 GeV m Z 5 TeV mass range, with the 13 TeV and ∼ 13 fb −1 dataset. CMS [57] has also searched for heavy resonances decaying to dimuon pair in the mass range 400 GeV m Z 4.5 TeV with 13 TeV and ∼ 13 fb −1 dataset, setting a 95% CL upper limit on R σ defined as: We reinterpret R σ and extract σ(pp → Z + X)Br(Z → µ + µ − ) by multiplying with the Standard Model prediction of σ(pp → Z + X)Br(Z → µ + µ − ) = 1928.0 pb [58].
To determine the upper limit, we generate matrix element (ME) of the pp → Z process up to two additional jets in the final state to include inclusive contributions. The ME is then merged and matched with parton shower (PS) following the MLM [59] matching prescription. We restrict ourselves up to two additional jets due to computational limitations. It should be noted that we have not used any K factor in our analysis. We finally convert the observed ATLAS (CMS) 95% CL upper limit on σ(pp → Z + X)Br(Z → µ + µ − ) in the mass range 150 GeV < m Z < 5 TeV (400 GeV < m Z < 4.5 TeV). This way we constrain g and m Z as shown in Figs

V. RESULTS
We now summarize the constraints on the coupling g and mass m Z of the Z in our U(1) B3−3Lµ model as shown in Figs. 1 and 2. The region of coupling-mass (g −m Z ) allowed in our model by the various flavor and collider constraints are divided into two ranges, for light (m Z < 1 TeV) and heavy (1 TeV < m Z < 10 TeV) Z , corresponding to Figs. 1 and 2, respectively. The allowed region of our model which produces correct 2σ band of C 9 indicated by a global fit of all the observed b → s transitions [8] is shown in green. The best fit value of C 9 = −1.12 is also shown within the allowed 2σ band by dashed black line. Although, we have only shown the allowed range of global fit values from [8], similar results have been obtained by other groups [7,9]. They result in very similar allowed bands for our model, with a significant overlap region. This allowed region in our model also explains the P 5 anomaly, as well as all other b → sµ + µ − anomalies mentioned earlier [3,4,6].
We have also studied all relevant constraints from flavor physics as well as from LEP precision [48] as well as LHC direct searches [50,56]. The most stringent LEP precision constraints on our model comes from the Z contribution to the Z → µµ vertex and Z → νν decays. The LEP constraint from these decays is shown by red dashed line. The region above the red dashed line is ruled out these constraints. The ATLAS collaboration [49] has also looked for the decay Z → 4µ which in our model can be facilitated by the Z . This places a constraint on our model parameter space which is shown by the solid red line (the region above the line is ruled out).
The ATLAS and CMS collaboration has also looked for direct decay of a heavy Z → µµ [56,57]. This also constrains significantly our parameter space. The black shaded region in the plot is ruled out by ATLAS, while the blue shaded region is ruled out by CMS. One sees that the direct search limits from ATLAS and CMS in fact rule out a simultaneous explanation of all b → s transition anomalies for Z masses in the range from 150 GeV up to ∼ 1.3 TeV. Low energy physics also puts severe constraints on the model. For example, the neutrino trident production puts a constraint on our model which is shown by the solid blue line. The region above the blue line is ruled out by neutrino trident constraints. [10,46]. Finally, the most stringent constraint on our model comes from B S mixing [42,43]: the pink shaded region in figures is ruled out by the B S mixing constraints.

VI. CONCLUSIONS
In this letter we have presented a minimal anomaly free U(1) B3−3Lµ Standard Model extension that can account for the recent anomalies in R(K) and R(K * ) reported by the LHCb collaboration. The model is minimalistic in that it has only the standard charged fermions, with the third quark family and the muon transforming non-trivially. This leads to a very simple pattern for lepton non-universality and flavor changing neutral currents involving the second and third generation of quarks which reproduces the LHCb findings in a way consistent with all the relevant experimental constraints, except for the range from 150 GeV up to ∼ 1.3 TeV, where the understanding of the R(K) and R(K * ) reported by LHCb would clash in our model with the ATLAS and CMS direct searches in the dimuon channel. One should also stress, as seen in Fig. 1, that the Z associated to our U(1) B3−3Lµ symmetry can be as light as 10 GeV, in contrast to Z associated to B-L such as [60,61] or 331 theories [62,63]. As a last comment we mention that throughout the paper we assumed dominance of the vector boson mediated neutral current contribution, neglecting all the scalars. We checked that, indeed, there is a realistic limit in parameters space where this can be achieved.