$b\rightarrow s \mu^+ \mu^-$ anomalies and related phenomenology in $U(1)_{B_3 - x_\mu L_\mu - x_\tau L_\tau}$ flavor gauge models

We propose a generation dependent lepton/baryon gauge symmetry, $U(1)_{B_3 - x_\mu L_\mu - x_\tau L_\tau} \equiv U(1)_X$ (with $x_\mu + x_\tau =1$ for anomaly cancellation), as a possible solution for the $b \rightarrow s \mu^+ \mu^-$ anomalies. By introducing two Higgs doublet fields, we can reproduce the observed CKM matrix, and generate flavor changing $Z'$ interactions in the quark sector. Thus one can explain observed anomalies in $b \to s \ell^+ \ell^-$ decay with the lepton non-universal $U(1)_X$ charge assignments. We show the minimal setup explaining $b \to s \ell^+ \ell^-$ anomalies, neutrino masses and mixings and dark matter candidate, taking into account experimental constraints of flavor physics such as charged lepton flavor violations and the $B_s$--$\bar B_s$ mixing. Finally we discuss collider physics focusing on $Z'$ production at the Large Hadron Collider and relic density of our dark matter candidate.


Introduction
Although the standard model (SM) of particle physics is very successful we still do not have clear understanding of the physics regarding the flavors; namely the origin of fermion masses and mixing patterns. Then it is interesting to construct a model describing flavor physics with some symmetry as a guiding principle. One of the attractive possibility is an introduction of flavor dependent U (1) gauge symmetry which can constrain structure of Yukawa couplings generating masses for quarks, charged leptons and neutrinos. In this kind of approaches to the flavor problem, these models may generate flavor changing neutral current (FCNC) processes through Z boson exchange, which will induce rich phenomenology.
Recently there have been some indication of anomalies in B physics measurements for b → s + − process. The angular observable P 5 in decay of B meson, B → K * µ + µ − [1], indicates 3.4σ deviations from the data with integrated luminosity of 3.0 fb −1 at the LHCb [2], confirming an earlier observation with 3.7σ deviations [3]. In addition, 2.1σ deviations were anomalies are cancelled when the U (1) X charges of fermions satisfy the condition which we will always assume in the following. In Sec. 2.1, we first discuss the case with general x µ,τ and investigate an explanation of b → s + − anomalies via flavor-changing Z interactions. Then the minimal model with x µ = −1/3 is constructed in Sec. 2.2, taking into account the generation of active neutrino masses and mixings via Type-I seesaw mechanism.
2.1 Discussion for general (x µ , x τ ) case Firstly we consider quark sector which does not depend on our choice of x µ and x τ = 1−x µ . In this model we have to introduce at least two Higgs doublets in order to induce the realistic CKM mixing matrix: Then the Yukawa couplings for quarks are given by where i = 1, 2 andΦ i = iσ 2 Φ * i . Φ 2 is the Higgs doublet with vanishing U (1) X charge, and is the SM-like Higgs doublet. After two Higgs doublet fields get the non-zero vacuum expectation values (VEVs) Φ 1,2 = (0 v 1,2 / √ 2) T , we obtain the following forms of quark mass matrices: Note that the matrices (ξ u,d ) ij ≡ỹ u,d ij v 1 / √ 2 have the same structure as those discussed in Ref. [14]. We shall assume the second terms with ξ u,d are small perturbation effects generating realistic 3×3 CKM mixing matrix where the (33) 2m t(b) following the discussion in Ref. [14].
As in the SM, the quark mass matrices are diagonalized by unitary matrices U L,R and D L,R which change quark fields from interaction basis to mass basis: . Then the CKM matrix is given by V CKM = U † L D L . Thus we obtain relation between mass matrices M u,d and diagonalized ones as follows: where diagonal mass matrices are given by The structures of mass matrices in Eq. (2.4) indicate that the off-diagonal elements associated with 3rd generations are more suppressed for More specifically, we find that where {i, j, k} = 1, 2. Then we can approximate U L and D R to be close to unity matrix since they are associated with diagonalizaition of M u (M u ) † and (M d ) † M d , respectively, where mixing angles in D R (U L ) generated by ξ parameters are suppressed by m d,s(u,c) /m b(t) to those in D L (U R ). Therefore CKM matrix can be approximated as V CKM D L , and D R 1, as obtained in Ref. [14]. Taking D L = V CKM , we can obtain sizes of (ξ d ) 13 and (ξ d ) 23 from Eq. (2.6) applying mass eigenvalues of down-type quarks. We thus obtain GeV. Therefore we can reconstruct mass eigenvalues of downtype quarks with D L V CKM taking these values for ξ d (values of y ij are chosen to fit m d and m s ). In addition, the values of ξ u tend to be smaller than ξ d due to mass relation m b m t .

Z interactions with SM fermions
The Z couplings to the SM fermions are written as where g X is the gauge coupling constant associated with the U (1) X and the lepton sector is given in the flavor basis here. The coupling matrices Γ d R and Γ d L for down-type quarks are given approximately by where V qq 's are the CKM matrix elements. We have applied the relation V CKM D L , as we discussed above. In our model the Z mass, m Z , is dominantly given by the VEV of SM singlet scalar field as discussed below. At this point, x µ is an arbitrary parameter requiring only anomaly cancellation condition Eq. (2.1). This value will be fixed to obtain negative ∆C µ 9 and to realize minimal scalar sector. The mass of Z can be a free parameter since it is given by new gauge coupling g X and scalar singlet VEV where we have freedom to chose the VEV even if the gauge coupling is fixed.

Effective interaction for
where G F is the Fermi constant and α em is the electromagnetic fine structure constant.
We thus obtain the Z contribution to Wilson coefficient ∆C µ 9 as (2.12) In order to obtain ∆C µ 9 ∼ −1, x µ should be negative and g X is required to be ∼ 0.6 for m Z = 1.5 TeV and x µ = − 1 3 . Figure 1 shows the contour of ∆C µ 9 in the (m Z , g X ) plane where we took x µ = − 1 3 where the yellow(light-yellow) region corresponds to 1σ (2σ) region from global fit in Ref. [11].

Minimal model
Here we consider the minimal cases for choosing U (1) X charges of leptons as In this case we add two SU (2) L singlet scalar fields: where yellow(light-yellow) region corresponds to 1σ (2σ) region from global fit in Ref. [11]. Table 2. Scalar fields and extra fermion χ in the minimal model and their representation under where ϕ 1 is also necessary to induce Φ † 1 Φ 2 terms 2 , while ϕ 2 is added for generating the 23(32) element of Majorana mass matrix of right-handed neutrino. Note that we obtain a massless Goldstone boson from two Higgs doublet sector without ϕ 1 due to an additional global symmetry. In addition we introduce additional Dirac fermion χ of mass m X with U (1) X charge 5/6, which can be our DM candidate since its stability is guaranteed due to fractional charge assignment under U (1) X . Note that the stability of Dirac fermion DM χ is guaranteed by remnant Z 2 symmetry after U (1) X symmetry breaking: particles with U (1) X charge 2n/6 (n is integer) are Z 2 even and those with U (1) X charge (2n + 1)/6 are Z 2 odd, since U (1) X symmetry is broken by VEVs of scalar fields ϕ 1 , ϕ 2 and Φ 1 whose charges correspond to 2n/6 [47]. We summarize the charge assignment of scalar fields and new fermion in Table 2. In the later analysis, we will adopt this minimal setting.
In our set up, the full scalar potential for scalar fields in our model is given by where we assumed all the coupling constants are real for simplicity. The VEVs of singlet scalar fields are written by 1,2 and U (1) X symmetry is spontaneously broken at a scale higher than the electroweak scale. We then approximately obtain VEVs of ϕ 1,2 from the condition where the above assumption for VEV hierarchy can be consistent requiring λ X v 2 Then the mass of the Z boson is approximately given by Then a typical value of the ϕ 1 VEV is v ϕ 1 7.5 × (m Z /1.5 TeV)(0.6/g X ) TeV in our scenario. Note that the Z-Z mass mixing is highly suppressed by v 2 1 /v 2 ϕ 1 factor which is ∼ 10 −5 for tan β = v 2 /v 1 = 10 and v ϕ 1 = 7.5 TeV. Thus we will ignore this effect in our analysis 3 .
(2.20) 3 The Z-Z mixing effect is constrained by precision measurements of ZfSM fSM coupling at the LEP experiments where the upper bound of the mixing θ ZZ is around ∼ 10 −3 − 10 −4 [45,46]. Thus our mixing angle is sufficiently smaller than the bound. 4 Here we do not consider scalar bosons from ϕ1,2 since they are assumed to be much heavier than those from Higgs doublets and mixing among singlet and doublet scalars will be small.
As in the two-Higgs-doublet model (THDM), we obtain mass eigenstate {H, h, A, H ± } in the two Higgs doublet sector: is a Nambu-Goldstone boson (NG) absorbed by the Z(W + ) boson, and h is the SM-like Higgs boson. The masses of H ± and A are given as in THDM: Mass eigenvalues of CP-even scalar bosons are also obtained by Thus Z can decay into HA, hA and H + H − pair.
Here we briefly comment on deviation in the couplings of the SM-like Higgs h and constraint in the scalar sector in the model. The Yukawa interactions with h are given by Eq. (5.1) in the Appendix. In particular, we have flavor violating interaction associated with ξ u,d coupling. In our analysis, we assume the interactions are SM-like that can be realized taking large tan β and alignment limit of cos(α − β) 0. Note also that new scalar bosons do not contribute to explanation of b → sµ + µ − anomalies in our scenario except for relaxing the constraint from B s -B s mixing as we discuss below; we can fit the data with the mass value of ∼ 500 to ∼ 1000 GeV for exotic scalar bosons from two-Higgs doublet sector. In such a mass region, we can find a parameter to avoid collider constraints for exotic scalar production like that of charged scalar bosons [48]. We thus just assume new scalar bosons are sufficiently heavy and we can avoid constraints from scalar boson search at the LHC. Discussion of scalar sector can be referred to, for example, Refs. [14,34].

Neutrino mass and flavor constraints
In this section we formulate neutrino mass matrices (both Dirac and Majorana mass matrices), and explore constraints from flavor physics such as µ → eγ, µ → e conversion and B s -B s mixing.

Neutrino mass matrices
The Yukawa interactions for leptons are given by where a = 1, 2, 3 and Y ab = Y ba . After the symmetry breaking, Dirac and Majorana mass matrices for neutrinos have the structure of where the elements of the mass matrices are given by The active neutrino mass matrix is given by type-I seesaw mechanism: Note that our neutrino mass matrix does not have zero structure and neutrino oscillation data can be easily fit. Here we do not carry out further analysis of the neutrino phenomenology in this paper.

Charged lepton mass matrices
The charged lepton mass matrix is given by

Charged lepton flavor violation
Here we consider charged lepton flavor violation (cLFV) in the model associated with Z . The Z gauge interactions for mass eigenstates of charged leptons are given by where the flavor violating structure for left-handed charged lepton currents is given by (3.9) Thus we have LFV interaction for e and µ. Then we first consider µ → eγ process induced by Z loop in Fig. 2 where the left diagram gives dominant contribution due to suppression by . Estimating the loop diagram we obtain dominant contribution to the decay width for the µ → eγ process such that (3.11) -10 -  Branching ratio for the LFV process is given by where G F 1.17 × 10 −5 GeV −2 is the Fermi constant and α 1/137 is the fine structure constant. In Fig. 3, we show BR(µ → eγ) on {g X , log | |} plane fixing m Z = 1.5(2.0) TeV where the shaded regions are excluded by the current constraint BR(µ → eγ) 4.2×10 −13 by the MEG experiment [49]. Further parameter region will be explored in future with improved sensitivity [50].
Here we also discuss µ → e conversion via Z exchange. In our case, the relevant effective Lagrangian for the process is derived as follows [51][52][53] where the corresponding coefficients are given by (3.14) Then we obtain the spin-independent contribution to the BR for µ → e conversion on a nucleus such that  Table 3. A summary of parameters for the µ − e conversion formula for 27 13 Al and 197 79 Au nuclei [52,54].  where Γ cap is the rate for the muon to transform to a neutrino by capture on the nucleus, and V (p,n) is the integral over the nucleus for lepton wavefunctions with corresponding nucleon density. The values of Γ cap and V (n,p) depend on target nucleus and those for 197 79 Au and 27 13 Al are given in Table. 3 [52,54]. In Fig. 4, we show BR(µ → e) for 27 13 Al on {g X , log | |} plane fixing m Z = 1.5(2.0) TeV in left(right)-panel where gray(light-gray) shaded region is excluded by current µ → eγ BR (µ → e BR on 197 79 Au [55]) constraints. We find that large parameter region can be explored by µ → e conversion measurement since its sensitivity will reach ∼ 10 −16 on 27 13 Al nucleus in future experiments [56,57]. We next consider the LFV B decay B s → µ ± e ∓ which is related to C µ 9 above. It is because that the process is induced from C µe 10 which is obtained as C µe 10 = − ∆C µ 9 in the model. The branching ratio can be given by 16) where we used C SM 10 (µ b ) −4.2 and BR(B s → µ + µ − ) SM = (3.65 ± 0.23) × 10 −9 is the SM prediction for the BR of B s → µ + µ − . We find that BR(B s → µe) < 10 −11 in the parameter region satisfying the constraint from BR(µ → eγ) which is well below the current constraint.
Here we also discuss the branching ratio for B → K ( * ) µe through lepton flavor violating Z coupling. It is suppressed compared to BR(B → K ( * ) µµ) by a factor of | ∆C µ 9 /C µ 9 | 2 ∼ 10 −3 for ∆C µ 9 = −1 and = 0.1. Thus the BR is small as order of 10 −10 − 10 −9 and it is well below current bound and challenging to search for the signal at the future experiments such as (upgraded) LHCb [58] and Belle II [59].
3.4 Constraint from neutrino trident process and Z contribution to muon g − 2 U (1) X gauge coupling and Z mass are constrained by the neutrino trident process νN → νN µ + µ − where N is a nucleon [60]. The bound is approximately given by m Z /g X 550 GeV for m Z > 1 GeV. We then consider parameter region of {m Z , g X } satisfying this bound.
The observed muon magnetic dipole moment is deviated from the SM prediction as ∆a µ = (26.1 ± 8.0) × 10 −10 [61] (muon g − 2). The Z boson can contribute to muon g − 2 at one loop level as where r ≡ (m µ /M Z ) 2 . We find that the Z contribution is small for the parameter region providing ∆C 9 ∼ −1; for example ∆a Z µ ∼ 1.7 × 10 −12 with m Z = 1500 GeV and g X = 0.6.

Constraint from B s -B s mixing
In our model, Z and neutral scalar bosons induce flavor changing neutral current (FCNC) interactions. Here we consider constraints from B s -B s mixing where other ∆F = 2 processes are more suppressed by CKM factors.
The effective Hamiltonian for the B s -B s mixing is given by The relevant Wilson coefficients are where Γ η qq is couplings for ηqq interactions (η = h, H, A), the explicit expressions of which are given in the Appendix. Using these Wilson coefficients we obtain ratio between ∆m Bs in our model and the SM prediction ∆m SM Bs , under large tan β and small α, such that where the first and second terms in the right-hand side corresponds to contributions from Z and scalars, respectively [14,62,63]. The allowed range of R Bs is estimated by [62,63] 0.83 < R Bs < 0.99. (3.21) We find that R Bs will be deviated from the allowed range by Z contribution when ∆C µ 9 −1 is required. Thus cancellation between Z and scalar contribution is necessary to satisfy the experimental constraint 5 . Here we derive allowed parameter region on {m H , m A −m H } plane satisfying B s -B s constraints when we fit C µ 9 to explain b → s + − anomalies choosing tan β = 10 and cos(α−β) ∼ 0 as reference values. In Fig. 5, we show the allowed parameter region where the yellow(light yellow) region corresponds to that in Fig. 1.

Collider physics and dark matter
In this section we explore collider physics focusing on Z production at the LHC and estimate relic density of our DM candidate searching for parameter region providing observed value.
Here we set masses of H, A and H ± as 400 GeV and apply tan β = 10 and cos(α − β) = 0 where the effects of the Z decays into scalar bosons are small. Also right-handed neutrinos and DM χ are taken to be heavier than m Z /2 so that Z does not decay into on-shell righthanded neutrinos and DM.
Our Z boson also decays into neutrinos with BR value of BR(Z → ν τντ ) = 16BR(Z → ν µνµ ) 0.25. Thus we can also test our model by pp → Z g → ννg process at the LHC experiments searching for signal with mono-jet plus missing transverse momentum. The cross section of pp → Z g → ννg process is, for example, ∼ 1 fb with g X = 0.6 and m Z = 1500 GeV estimated by CalcHEP with p T > 25 GeV cut. We thus need large integrated luminosity to analyze the signal [74] and it will be tested in future LHC experiments.  Figure 6. Left(right) plot: σ(pp → Z )BR(Z → + − (τ + τ − )) with = e, µ for several values of g X compared with LHC limit; from Refs. [72] and [73] for + − and τ + τ − modes.

Dark matter
We consider a Dirac fermion χ as our DM candidate, and the relic density is determined by the DM annihilation process χχ → Z → f SMfSM /HA/H + H − where f SM is a SM fermion and/or χχ → Z Z depending on kinematic condition. Then we estimate relic density of our DM using micrOMEGAs 4.3.5 [75] implementing relevant interactions. Fig. 7 shows the relic density Ωh 2 as a function of DM mass m X where we apply several values of g X and m Z = 1.5 TeV as reference values, and indicate observed Ωh 2 value by horizontal dashed line [76]. We see that the relic density drops at around m Z ∼ 2m X due to resonant enhancement of the annihilation cross section.
In addition, we scan parameters in the range of m X ∈ [200, 3100] GeV, m Z ∈ [500, 7000] GeV, g X ∈ [0.01, 1.5], (4.1) with assuming that tan β = 10 and cos(α − β) = 0 as reference values. We note that the effects of scalar bosons are subdominant. The left panel of Fig. 8 shows the parameter region which accommodates the observed relic density of DM, Ωh 2 = 0.1206 ± 0.0063, taking 3σ range of observed value by the Planck collaboration [76]. Moreover the right panel of the figure indicates the region in which both observed relic density and b → s + − anomalies are explained within 2σ. Notice that the allowed region with m Z < m X is partly excluded by or close to LHC constraint shown in Fig. 6 and will be explored in future LHC experiments. In addition DM-nucleon scattering cross section by Z exchange is suppressed by CKM factor and the allowed region is not constrained by the DM direct detection experiments. Before closing this section we discuss possibility of indirect detection of our DM. In this model DM pair annihilates mainly through χχ → Z → τ + τ − and/or χχ → Z Z → 2τ + τ −  and gamma-ray search gives the strongest constraint on the annihilation cross section by Fermi-LAT observation [77,78]. In our parameter region of m Z > 500 GeV, DM annihilation cross section explaining the relic density is well below the constraint for the τ + τ − dominant case [77,78] unless there is large enhancement factor; constraint on cross section for four τ mode would be similar. Thus our model is safe from indirect detection cross section and will be tested with larger amount of data in future.

Summary and discussions
We have discussed a flavor model based on U (1) B 3 −xµLµ−xτ Lτ (≡ U (1) X ) gauge symmetry in which two Higgs doublet fields are introduced to obtain the observed CKM matrix. Flavor changing Z interactions with the SM quarks are obtained after diagonalizing quark mass matrix, and b → s + − anomalies can be explained due to lepton flavor non-universal charge assignment when x µ is taken to be negative value. Then we have considered minimal set up explaining b → s + − anomalies and generated neutrino mass matrix where two SM singlet scalar fields and Dirac fermionic DM candidate are introduced.
We have computed the Z contribution to the Wilson coefficient C µ 9 relevant for b → sµ + µ − , as wel as neutrino mass matrices, charged lepton flavor violations and the B s -B s mixing, including the relevant experimental constraints. We have found that cancellation between Z and scalar bosons contributions to B s -B s is required to satisfy experimental constraint, while explaining b → s + − anomalies. In addition, we have shown constraints from lepton flavor violation process µ → eγ and future prospects for µ → e conversion measurements.
Then collider physics regarding Z production at the LHC and relic density of DM are explored. We have shown cross sections for the DY processes, pp → Z → µ + µ − (τ + τ − ), where constraints on the {m Z , g X } parameter space dominantly come from the data of dimuon resonance search at the LHC. The relic density of DM further constrains {m Z , g X } parameter space since the relic density is determined by DM pair annihilation process via Z interactions. The preferred parameter region can be further tested in future LHC experiments and observations for flavor physics such as LFVs. Under the approximation V D L and D R 1, we obtaiñ