Phenomenological study of a gauged ${L_\mu -L_\tau}$ model with a scalar leptoquark

A $Z'$ gauge boson with sub-GeV mass has acquired a significant interest in phenomenology, particularly in view of the muon $g-2$ anomaly and coherent elastic neutrino-nucleon scattering. The latter is challenged by the nuclear recoil energy of a few tens of keV but has been observed by the COHERENT experiment. To further reconcile the observed excesses in $R(D^{(*)})$ from semileptonic charmful $B$ decays and in the $W$ boson mass, we investigate a model with a gauged $U(1)_{L_\mu-L_\tau}$ symmetry and a scalar leptoquark. In contrast to the mechanism that involves kinetic mixing between the gauge bosons of $U(1)_{\rm em}$ and $U(1)_{L_\mu-L_\tau}$, we adopt a dynamical symmetry breaking of $U(1)_{L_\mu-L_\tau}$ by incorporating an additional Higgs doublet. Through mixing with the $U(1)_{L_\mu-L_\tau}$-charged Higgs doublet, new Higgs decay channels $h\to Z_1 Z_1/Z_1 Z_2$ occur at percent-level branching ratios, which could be accessible at the LHC. The $W$-mass anomaly observed by CDF II can be potentially resolved through the enhancement in the oblique parameter $T$. Due to the flavored gauge symmetry, the introduced scalar leptoquark $S^{\frac{1}{3}}=(\bar{3},1,2/3)$ exhibits a unique coupling to the $\tau$-lepton, offering an explanation for the excesses observed in $R(D^{(*)})$. Moreover, $\tau \to \mu (Z_1\to ) e^- e^+$ via the resonant light gauge boson decay can reach the sensitivity of Belle II at an integrated luminosity of 50 ab$^{-1}$.


I. INTRODUCTION
The possible existence of a sub-GeV Z ′ gauge boson has attracted much attention in recent years in addressing unresolved problems in particle physics phenomenology, particularly in flavor physics and dark matter (DM).The former includes the long-lasting puzzle in the anomalous magnetic dipole moment of muon (muon g − 2), while the latter could lead to establishing a portal between the visible and dark sectors.
Meanwhile, several deviations from the SM predictions have emerged in experiments, such as the muon g − 2, R(D ( * ) ) in the B → D ( * ) τ ν decays, and the mass of W gauge boson.
It would be intriguing to build a light Z ′ model that can not only resolve these observed anomalies but also have a significant impact on the CEνNS phenomenon.To fulfill these objectives within one coherent framework, we consider extending the SM with a new local U (1) gauge symmetry.Numerous potential candidates of such a U (1) gauge symmetry exist in the literature, including universal U (1), B − L, B y + L µ + L τ , B − 3L ℓ , B − L e − 2L µ , and 25], where B i and L ℓ denote the quantum numbers of quark and lepton, respectively.
Among these, we find that in addition to satisfying gauge anomaly-free conditions without introducing new chiral fermions, the gauged U (1) Lµ−Lτ ≡ U (1) µ−τ model can effectively address the above-mentioned concerns.
The observed anomalies in the ratio of branching ratios (BRs) in semileptonic charmed B decays are defined by with M = D, D * .The SM predictions are R(D) = 0.298 ± 0.004 and R(D * ) = 0.254 ± 0.005 [56][57][58][59][60][61][62][63] while the current experimental values are R(D) = 0.356 ± 0.029 and R(D * ) = 0.284 ± 0.013 [64].Recent measurements from LHCb have been included in the average [65,66].As seen, there is an overall 3.3σ deviation from the SM predictions [64].Because B → M τ ν is mediated by the tree-level charged weak currents in the SM, the required mechanism to enhance R(M ) should have non-universal lepton couplings and be induced at the tree level.Although R(J/Ψ) and R(Λ c ) have the potential to observe the breakdown of lepton universality as well, their statistical errors in the experimental data are still too large to be conclusive [67][68][69].Hence, we concentrate solely on R(D) and R(D * ) in this work.
Without further introducing a heavy charged gauge boson (e.g., W ′ ) or vector leptoquark (LQ) for the R(D ( * ) ) anomalies, the new mediating bosons of interest include a charged Higgs boson [70][71][72][73][74][75] and a scalar LQ [76][77][78][79][80][81][82][83] U (1) µ−τ gauge symmetry, there is a natural suppression in the LQ Yukawa couplings to the light leptons, while the τ -lepton and τ -neutrino respectively couple to up-and down-type quarks to resolve the observed anomaly in R(M ).It is worth mentioning that using the exclusive-and hadronic-tag approaches with 362 fb −1 of data, the Belle II Collaboration recently has observed the first evidence of B + → K + ν ν decay with a 2.7σ deviation from the SM prediction, and the measured result is given as B(B + → K + ν ν) = (2.3 ± 0.5 +0.5 −0.4 ) × 10 −5 [84].Applying the LQ S 1 3 in the model, the branching ratios for B → (K, K * )ν τ ντ can be significantly enhanced.A detailed phenomenological analysis of the neutrino-pair production in B and K meson decays can be found in Ref. [85].
In addition to the total cross section of CEνNS and R(D ( * ) ), we propose new observables sensitive to new physics as a function of incident neutrino energy for elastic neutrino-nucleus scattering and as a function of invariant mass-square q 2 of ℓν for semileptonic charmed B decays.We find that CEνNS mediated by the light physical Z 1 can deviate significantly from the SM in the low neutrino energy regime.Additionally, R(D) in the large q 2 regime is more sensitive to the leptoquark effects and can significantly differ from the SM.
and the oblique parameters.Constraints on the model parameters and detailed numerical analysis are presented in Sec.IV.A summary of our findings is given in Sec.V.

II. THE MODEL
We consider in this work a model that extends the SM gauge symmetry by the U (1) µ−τ gauge symmetry, under which only the µ and τ leptons in the SM are charged.Due to the opposite U (1) µ−τ charges within the second and third generations of leptons, it can be easily checked that the loop-induced triangle anomalies mediated by the muon and τ -lepton for  I.As we will see, such a model can simultaneously accommodate the measured lepton g − 2, R(D ( * ) ), and W mass anomalies while the cross section of the CEνNS process can be enhanced to deviate from the SM expectation by up to 25%.
In the following subsections, we analyze the spectra of Higgs and gauge bosons and determine their physical eigenstates.In addition, we also derive the gauge, Yukawa, and trilinear couplings of Higgs bosons, which are used for the phenomenological analysis presented in the paper.

A. Spectra of Higgs bosons and Higgs-related trilinear couplings
We first write down the scalar potential consistent with the SU (2 gauge symmetry as: Owing to the U (1) µ−τ symmetry, there is no so-called µ term that couples H 1,2 quadratically and all terms in Eq. ( 5) are self-Hermitian due to the U (1) µ−τ symmetry, rendering all the coefficients real.The components of two Higgs doublets can be parametrized as (i = 1, 2): Using the tadpole conditions ∂V /∂v 1,2 = 0, we obtain two equalities: with λ 34 ≡ λ 3 + λ 4 .To achieve spontaneous breakdown of the SU (2 gauge symmetry, we require µ 2 1,2 < 0. For the vacuum stability, where the scalar potential is bounded from below in all field configurations, the quartic couplings have to satisfy the criteria given by [86,87] Two neutral Goldstone bosons result from the mixing between the two CP-odd components: where β is defined by , c β ≡ cos β, and s β ≡ sin β.To obtain the states of charged Goldstone and charged Higgs bosons, we can use Eq. ( 9) by substituting (G ± , H ± ) and (ϕ + 1 , ϕ + 2 ) for (G 0 Z ′ , G 0 Z ) and (η 1 , η 2 ), respectively.As a result, the mass-squared of the charged Higgs boson is solely dependent on the parameter λ 4 as follows: Since the massive LQ is irrelevant to the EWSB, its mass-squared with the assumption that µ 2 S > 0 is found to be: and can be as heavy as O(TeV).
From the scalar potential in Eq. ( 5) and the tadpole conditions in Eq. ( 7), the mass terms for the CP -even scalars can be written as: Eq. ( 12) can be diagonalized by a 2 × 2 orthogonal matrix, and the resulting eigenstates of neutral Higgses can be parametrized using a mixing angle α as: where h is the 125-GeV SM-like Higgs boson, c α ≡ cos α, and s α ≡ sin α.In the following, we would focus on the scenario where the new CP -even state is lighter than the SM-like Higgs boson, i.e., m h > m H . Using the parameters λ i and v i , the masses of the h and H states, as well as the mixing angle between them, can be obtained as: The scalar potential in the model involves six parameters, namely, µ 2 1,2 and λ 1−4 .One can write them in terms of the physical parameters {m H ± ,h,H , v, α, β} as An important parameter of the scalar potential in the SM is the quartic coupling λ SM , which not only determines the mass of the SM Higgs boson via m 2 h = λ SM v 2 , but also controls the potential shape.Therefore, to probe the existence of extra scalars, it becomes crucial to precisely determine the Higgs self-coupling through the hh production that involves the Higgs trilinear coupling [88].In the 2HDM, the SM-like Higgs field is a linear combination of ϕ 0 1,2 and, instead of a factor of 3m 2 h /v for the SM, the Higgs self-coupling also involves the parameters β and α.Moreover, when m H < m h /2, the decay channel h → HH becomes accessible.Current measurements of Higgs decays can impose stringent constraints on the related parameters.To take these constraints into account, we present all the Higgs trilinear couplings as follows: Taking the limits of α → 0 and s β → 1, it can be seen that only the self-couplings of h and H remain.We note that the scalar couplings to the LQ are also included, which can be used to analyze the loop-induced Higgs boson decays.

B.
Z ′ − Z mixing and gauge couplings of scalars The masses of the gauge bosons and the gauge couplings of scalars are determined by the kinetic terms of H 1,2 , with the covariant derivatives given as: where g, g ′ , and g Z ′ denote the gauge couplings of SU (2) L , U (1) Y , and U (1) µ−τ , respectively, X = 2q X is the U (1) µ−τ charge of H 1 , and Q S = 1/3 is the electric charge of LQ.As in the conventional 2HDM, the tree-level W boson mass can be obtained as m W = gv/2.However, since H 1 carries the charges of both electroweak and U (1) µ−τ symmetries, its VEV breaks not only SU (2) L × U (1) Y but also U (1) µ−τ at the same time.As a result, the Z and Z ′ states are not physical and generally mix with each other.More explicitly, the mass-squared terms for Z and Z ′ are given by: where m 2 Z ′ , m 2 Z , and m 2 Z ′ Z are defined as: The states of the photon and Z boson fields are written, as in the SM, as: where c W ≡ cos θ W , s W ≡ sin θ W , and θ W is the weak mixing angle.The mass-squared matrix in Eq. ( 18) can be diagonalized using a 2 × 2 orthogonal matrix, parametrized by a mixing angle θ Z , in a fashion analogous to Eq. (13).Assuming that m Z ′ ≪ m Z and taking Z 1 and Z 2 as the physical states of the neutral gauge bosons, their mass-squares and mixing angle can be approximately obtained as follows: where sign(θ Z ) = ±1 represents the sign of the mixing angle.Apparently, the mixing angle To study the loop-induced processes or variables (e.g., lepton g − 2) mediated by the Z 1 boson, we also need the gauge couplings of scalars and LQ as follows:

C. Yukawa couplings of fermions
The Yukawa sector plays a crucial role in flavor physics as it governs the mass generation of the SM fermions and the couplings of scalars to fermions in the model.The Lagrangian of the Yukawa sector under SU (2) L × U (1) Y × U (1) µ−τ gauge symmetry can be written based on the quantum number assignments in Table I as: where, except for the Lµ H 1 τ R term, the flavor indices are all suppressed, Q T L = (u, d) T L and L T = (ν ℓ , ℓ) T L represent the quark and lepton doublets, respectively, ℓ R denotes the right-handed charged lepton, and F c = Cγ 0 F * for a fermion F with C being the charge conjugation operator.The U (1) µ−τ gauge symmetry restricts the 3 × 3 Yukawa matrix Y ℓ to be a diagonal matrix, i.e., Y ℓ = diag(y e , y µ , y τ ).We note that because H 1 and H 2 simultaneously couple to the charged leptons, the term Lµ H 1 τ R will induce flavor-changing neutral-currents (FCNCs) at tree level.After diagonalizing the quark mass matrices and using the physical states of scalars, the Yukawa couplings of quarks to h(H) and H ± are found to be the same as those in type-I 2HDM [89].Although the charged Higgs boson could in principle enhance the b → cτ ν transition [70][71][72][73][74][75], the involved Yukawa couplings in this model are suppressed by m b,c /(tan β v 2 1 + v 2 2 ) and are irrelevant for our later discussions.The explicit expressions of the couplings can be found in Ref. [89].
While the diagonal Y ℓ matrix contributes to the charged lepton masses, the Lµ H 1 τ R term induces flavor mixing between the µ and τ leptons.Thus, the electron mass is simply √ 2, and the mass matrix for the µ and τ leptons is expressed as: where mµ(τ) = y µ(τ ) v 2 / √ 2 and mµτ = y µτ v 1 / √ 2. The matrix M ℓ can be diagonalized through a bi-unitary transformation: Accordingly, the Yukawa couplings of the Higgs bosons to the leptons are found to be: where X ℓ is defined as: It is worth mentioning that X ℓ induces the tree-level FCNCs mediated by the Higgs bosons in the lepton sector.To see the decoupling and large tan β limits, it is useful to rewrite c α /s β and s α /s β as: When the lepton Yukawa couplings are real, we can obtain the 2×2 flavor mixing matrices V ℓ R,L using the identities: By parametrizing V ℓ R,L in the same form as U α in Eq. ( 13), we can obtain the mixing angles θ R,L as: In the limit when mµ mµτ / m2 τ is negligible, these mixing angles can be obtained to a good approximation as: As a free parameter with the mass dimension that appears only in the µ − τ element of X ℓ , mµτ can be parametrized in terms of a free parameter χ µτ as mµτ = χ µτ √ m µ m τ , where m µ,τ are the physical masses of µ and τ leptons.Using the approximate mixing angles in Eq. ( 30), we obtain We now discuss the LQ couplings to quarks and leptons.Since the Yukawa couplings y q L and y u R are free parameters, the up-type quark flavor mixings can be absorbed into these parameters.As such, the up-type quark fields appearing in the LQ couplings in Eq. ( 23) can be treated as the physical states.However, the same y q L also appears in the couplings to the down-type quarks.Therefore, in addition to V d L , the LQ couplings to the down-type quarks must include , we can express the Yukawa couplings of the LQ as:

D. Gauge couplings of fermions
Next, we consider the gauge couplings of the fermions.Since the U (1) µ−τ gauge symmetry does not affect the weak charged currents, they remain the same as those in the SM.Although quarks do not carry the U (1) µ−τ charge and thus do not directly couple to the Z ′ gauge boson, their couplings to the Z ′ boson can be induced through the mixing with the SM Z boson.Intriguingly, the distinct U (1) µ−τ charges carried by the muon and tau-lepton lead to a lepton FCNC in the interaction μL γ µ τ L Z ′ µ .Due to the Z ′ − Z mixing, they then result in Z-mediated lepton FCNCs although such effects are suppressed by s θ L s θ Z .Using the results shown in Eq. ( 21) and Eq. ( 30) for the lepton-flavor and Z ′ − Z mixings, respectively, we obtain the neutral gauge couplings to fermions as follows: where the coefficients C f iV,iA are explicitly given by: f given in terms of the weak isospin T 3 f and the electric charge Q f of the fermion f , and X f V = (0, 1/2, −1/2, 0, 1, −1, 0, 0) and X f A = (0, 1/2, −1/2, 0, 0, 0, 0, 0) for f = (ν e , ν µ , ν τ , e, µ, τ, u, d).Because only vector currents are involved in the Z ′ couplings to the charged leptons, X ℓ A = 0.However, X νµ,ντ A are nonvanishing because neutrinos are left-handed particles in the model.

III. PHENOMENOLOGY
In this section, we derive the formalisms for the processes studied in this work.These include the cross section for CEνNS via the Z ′ − Z mixing, the R(D) and R(D * ) from LQ interactions, new Higgs decay modes, lepton g − 2, and the effects on the oblique parameters and the W mass.
A. CEνNS through the Z ′ − Z mixing In the model, elastic electron-and muon-neutrino (including anti-neutrino) scatterings off a nucleus arise from gauge interactions with the neutral gauge bosons Z 1 and Z 2 .Using the gauge couplings given in Eq. ( 33), we can write the effective Hamiltonian for neutrino scattering at the quark level as: where the Kronecker delta δ ℓ µ indicates that only the muon-neutrino or anti-muon-neutrino contributes.The second line in Eq. ( 36) results from the limits of c θ Z ≃ 1, m Z 2 ≃ m Z and large t β .We will demonstrate that due to the h → HH and h → Z 1 Z 1 constraints, a large t β is required for the model.As a result, the electron-neutrino scattering becomes insignificant and negligible.Since the structure of the four-fermion interaction in Eq. ( 35) is the same as that in the SM, the new physics contribution can be obtained simply by replacing ) .In contrast to the effects induced by the kinetic mixing in the conventional U (1) µ−τ model, the g Z ′ dependence has been absorbed into m Z 1 .Thus, the new physics effect depends only on m Z 1 in the large-t β scheme.Because the LQ mass is of O(1) TeV, its contribution is negligible.As such, we skip the discussion related to the LQ effects.
The cross section for the elastic neutrino-nucleus scattering can be written as [13]: where m A is the nucleus mass, Z(N ) is the proton (neutron) number of the target nucleus, E ν is the incident neutrino energy, E r is the nuclear recoil energy, and q 2 ≃ 2m A E r .The couplings to the proton g p and the neutron g n are respectively given by: Since the contribution from the weak axial-vector currents is much smaller than that from the vector currents, we have ignored their effects in Eq. (37).To include the nuclear effects, we adopt the Klein-Nystrand approach [90] for F p/n (q 2 ), expressed as [13] where R A = 1.2A 1/3 with A being the mass number, j 1 is the spherical Bessel function of order one, and a K denotes the range of a short-range Yukawa potential.For a numerical estimate, we take a K = 0.7 fm.The neutrinos detected in the COHERENT experiment are produced by the stopped π + decay via π + → ν µ + µ + and by the subsequent µ + decay through µ + → e + + ν e + νµ .In this study, we assume that the shapes of neutrino fluxes are the same as their energy spectra, expressed as [4,91,92]: with N being a normalization factor.Hence, the total cross section averaged over the neutrino fluxes can be obtained as: where The model has two different mechanisms contributing to the b → cℓν process: one involves the charged Higgs boson, and the other is from the LQ.However, the effects of the charged Higgs are not significant as its couplings to quarks and leptons are suppressed by m b,c,ℓ /(vt β ).
We thus focus exclusively on the LQ contributions.Based on the Yukawa couplings of LQ in Eq. ( 32), the effective Hamiltonian for b → cℓν mediated by the W gauge boson and S 1 3 can be obtained as [80]: where the effective Wilson coefficients at the m S scale are given as: We note that since the electron does not mix with the µ and τ leptons, the b → ceν process for the B → Dℓν process as a function of the invariant mass q 2 of ℓν can be expressed as: where X ℓ +,0,S,T and λ M are defined as: The q 2 dependence of the form factors has been suppressed.
The B → D * ℓν ℓ decay involves D * polarizations, and the transition form factors are more complicated.Using the parametrization in Eq. (A2), the differential decay rate after summing all D * helicities is given by: where λ D * can be found in Eq. ( 46) and The quantities h ℓ 0 , h 0ℓ T , h ℓ ± , and h ±ℓ T are defined by: with X e,µ,τ V = (1, C µ V , 1+C τ V ), respectively.Based on Eqs. ( 45) and (47), R(M ) (M = D, D * ) can be calculated by:

C. New Higgs decays
Eq. (21) shows that utilizing an additional Higgs doublet to spontaneously break the U (1) µ−τ gauge symmetry leads to a strong correlation among m Z 1 , g Z ′ , and t β .As a result, several processes involving the same set of parameters exhibit distinct behaviors.In the following, we discuss these interesting processes.
With focus on the scenario with m H < m h /2 and m Z 1 < 200 MeV, the new Higgs decay channels h → HH and h → (Z 1 Z 1 , Z 1 Z 2 ) become kinematically accessible.Using the Higgs trilinear and gauge couplings given in Eqs. ( 16) and ( 22), the partial decay rates for these channels are obtained as: In the decoupling limit when s β−α → 1, as required by the current Higgs signal strength measurements, the processes h → (HH, Z 1 Z 1 ) can in principle have large decay rates.Hence, the observed Higgs width Γ h strongly constrains the values of t β and c β−α .Therefore, from Eq. ( 51), the condition of c β−α ∼ s β−α /t β ≪ 1 has to be satisfied, i.e., a large t β scheme is demanded by data in the model.Interestingly, when we use a large t β value, the same condition can be used to suppress the partial decay width of h → Z 1 Z 1 .Moreover, since h → Z 1 Z 2 does not depend on the t β parameter, we can use the limit of Γ(h as an independent constraint on c β−α .Although our analysis does not focus on the search for collider signals, the percent-level BR for h → Z 1 Z 2 with invisible Z 1 decay could be an interesting channel for detecting the new physics.We note that c β−α ∼ 0.1 is still permissible when considering the constraints from the current measurements of Higgs decays.We will see later that the BRs of new Higgs decay modes can reach the percent level with c β−α ∼ 0.05. In addition to the flavor-conserving Higgs Yukawa couplings, which are suppressed by m ℓ /v according to Eq. ( 25), there is a tree-level LFV Higgs coupling, i.e., hμ L τ R , where the strength of this LFV coupling is primarily determined by The partial decay rate for h → µτ can thus be written as: When c β−α and t β are determined from the processes h → HH/Z 1 Z 2 , the h → µτ decay rate then depends only on χ µτ .
From Eq. ( 33), it can be seen that the tree-level lepton FCNC arises not only from the Higgs couplings but also from the Z i couplings.For a light Z 1 gauge boson, the τ → µZ 1 decay can be induced at the tree level and the BR can be obtained as: where we have dropped the m µ,Z 1 /m τ factors because m µ,Z 1 ≪ m τ .The 1/m 2 Z 1 factor in the parentheses from the contribution of the longitudinal component of Z 1 will largely enhance the BR as m Z 1 is taken at the sub-GeV level.Since the BR of this decay is mainly determined by g Z ′ , m Z 1 , and s θ L , we can use τ → µZ 1 to constrain the θ L parameter when g Z ′ and m Z 1 are fixed by other processes.

D. Lepton (g − 2)'s
Our model makes additional contributions to the lepton (g − 2)'s through the mediations of Z 1 , H, and LQ at the one-loop level.One can neglect the contribution from LQ as it is suppressed by m 2 µ /m 2 S .Based on the gauge couplings given in Eq. (33), although the LFV coupling µτ Z 1 can contribute to the muon and tau (g − 2)'s, its effect is negligible as the coupling is proportional to g Z ′ s θ L , where g Z ′ is of O(10 −4 ) and s θ L is highly constrained by the τ → µZ 1 decay, as argued at the end of last subsection.On the contrary, the contribution from the light H is through the LFV coupling µτ H. From Eq. ( 25), it can be seen that although this coupling is suppressed by a factor of m ℓ /v, the factor 1/c β can enhance the lepton (g − 2)'s in the regime of large t β and small m H .
The explicit expressions of the Z 1 and H contributions to the lepton (g − 2)'s are respectively given by: with ℓ = (e, µ, τ ) and ℓ ′ = (µ, τ ).Although the couplings of Z 2 , excluding the SM part, can contribute to the lepton (g − 2)'s, the suppression factors of (s θ Z , g Z ′ )m 2 ℓ /m 2 Z 2 make the effects negligible.We, therefore, disregard the new physics contribution from Z 2 .

E. Oblique parameters and the W mass
An important set of precision measurements for constraining new physics comprises the oblique parameters denoted by S, T , and U .These quantities are related to the loopinduced vacuum polarizations of vector gauge bosons, and their detailed definitions can be found in Refs.[96,97].In our model, in addition to the SM-like Higgs doublet H 2 , the oblique parameters receive effects from the extra SU (2) Higgs doublet H 1 and the new gauge coupling to Z ′ .Since we will focus on g Z ′ ∼ O(10 −4 ), we ignore the Z ′ contribution and take m Z 2 ≃ m Z in the analysis.However, a distinctive difference is that the pseudoscalar in the conventional 2HDM becomes the longitudinal component of Z ′ .Thus, the main contributions running in the loops to the oblique parameters are from H ± , h, and H.
To calculate the S, T , and U parameters in the model, we use the results obtained in Ref. [98], where the resulting oblique parameters are suitable for the multi-Higgs-doublet models, and even for the models with new singlet charged scalars.Except for the absence of pseudoscalar contributions, the effects from H ± , h, and H are similar to the conventional 2HDM.The detailed expressions for the S, T , and U parameters as functions of the scalar masses and couplings are given in Appendix B.
Using the obtained oblique parameters, the W mass under the influence of new radiative corrections can be expressed as [97,99,100]: where m SM W denotes the W mass in the SM, and its relationship with m Z is defined to be the same as that in the SM, i.e., m SM W = m Z c W .It is worth mentioning that the tree-level Z ′ − Z mixing can affect the oblique parameters and modify the relation between m SM W and m Z [100].However, since the mixing angle θ Z in the model is of O(10 −5 ) in our study, the effects can be safely ignored.

IV. NUMERICAL ANALYSIS AND DISCUSSIONS
Before conducting a numerical analysis of the physical processes studied in this work, we should first find the viable ranges of new physics parameters in the U (1) µ−τ -extended model.
For example, as alluded to before, the most influential parameter for the CEνNS is m Z 1 and its cross section can be potentially enhanced by a larger value of m Z 1 .The magnitude of m Z 1 , on the other hand, is proportional to g Z ′ whose value can be constrained by, e.g., the observed muon g − 2. In the following, we start by setting bounds on the parameter space and then make predictions for the CEνNS cross section, R(D ( * ) ), and the oblique parameters and W boson mass.We will also study the decays of the Z 1 and H bosons in the model.

A. Constraints of parameters
The free parameters considered in this study are m H , m Z 1 , m S , g Z ′ , χ µτ , c β−α , and t β , where χ µτ parametrizes the µ − τ mixing effect through s θ L ≃ χ µτ m µ /m τ and the Z ′ − Z mixing is determined by m Z 1 and t β .Based on the constraints from the neutrino trident process [101], measured by CCFR [102], and the 4µ final states in the BaBar experiment [103], we can conservatively take the bounds of g Z ′ q X ≲ 1.3×10 −3 and m Z 1 < 200 MeV.According to Eq. ( 55), the Z 1 boson makes an important contribution to the muon g − 2. Therefore, we show in Fig. 1 the CCFR bound [101] and the ±3σ contours (blue dot-dashed curves) of the measured muon g − 2 in the m Z 1 -g Z ′ q X plane, where the shaded region above the red dashed curve is ruled out by the CCFR experiment.Although ∆a Z 1 µ depends on t β via the Z ′ − Z mixing, its effect is negligibly small because s θ ∼ O(10 −5 ) in the considered range of m Z 1 .As a result, the electron g − 2 mediated by Z 1 and induced through Z ′ − Z mixing is estimated to be ∆a Z 1 e ≈ −1.4 × 10 −16 , completely negligible.We will show later that due to the small lepton flavor mixing, as constrained by other processes, the effect mediated by H for the lepton g − 2 is also highly suppressed.In the model, m Z 1 and g Z ′ q X are not independent parameters and are related by m Z 1 = 2vt β g Z ′ q X /(1 + t 2 β ).In Fig. 1, we also show contours of t β using solid lines.The large t β scheme, as required to restrict the h → HH and h → Z 1 Z 1 rates, to be discussed in more detail below, further narrows down the preferred m Z 1 range.The SM prediction for the Higgs boson width is Γ SM h ≈ 4.1 MeV [104], while the current measurement gives Γ exp h = 3.2 +2.8 −2.2 MeV [105].As an illustrated example, we assume that each new Higgs decay channel in the model contributes less than 5% of Γ SM h , i.e., Γ NP h ≤ 0.20 MeV.This assumption is consistent with the current upper limit on the Higgs invisible decays, BR(h → invisible) < 0.19 [105].To fit the observed Higgs signal strengths, the Higgs couplings to the fermions and the W ± and Z gauge bosons should have s β−α ≈ 1.
We now use Γ(h → HH) to bound c β−α and t β .Since the h → HH process depends on m H , we show the upper bound on ξ, defined in Eq. ( 51), for some benchmarks of m H : To illustrate the dependence of ξ on c β−α and t β , we show in Fig. 2  In the large t β scheme, Γ(h → Z 1 Z 2 ) only depends on c β−α .With m h = 125 GeV and m Z 2 = 91.187GeV, the limit of c β−α can be determined as: The assumption of Γ(h → Z 1 Z 2 ) ≤ 0.20 MeV then translates into the dashed line in Fig. 2.
According to the result in Eq. ( 21), we can estimate the Z ′ − Z mixing angle to have: Clearly, s θ Z can be larger than the loop-induced kinetic mixing between Z ′ and γ, characterized by the mixing parameter [28,106]: Consequently, we concentrate on the contributions from the Z ′ − Z mixing in this study.
The χ µτ parameter contributes to h → µτ , τ → µZ 1 , and ∆a H ℓ .Since the τ → µZ 1 process is strongly enhanced by the factor of m 2 τ /m 2 Z 1 , its measurement will put a strict constraint on χ µτ .To bound the χ µτ parameter using available data, we can use the upper limit of the process τ → µ+light boson as an estimate, where the current data give BR(τ → µ + light boson) < 5 × 10 −3 [105,107].With c θ L ≈ c θ Z ≈ 1 and the result in Eq. ( 54), we obtain an upper bound on χ µτ as: The resulting BR(h → µτ ) and ∆a H µ are then less than O(10 −11 ) and O(10 −16 ), respectively.The primary purpose of introducing the scalar LQ, S 1 3 , in the model is to address the R(D ( * ) ) anomalies.Along with the mass of LQ, the related parameters are y q L3,L2 , y u R2 , and V ℓ Lℓτ .Due to the τ → µZ 1 constraint, the lepton flavor mixing matrix can be approximated as V ℓ L ≈ 1, allowing us to ignore its contribution to the muon mode.Consequently, the LQ only couples to the third-generation leptons.According to Eq. ( 32), unlike the independent couplings to the different up-type quarks, the LQ couplings to the different down-type quarks are related by the CKM matrix and can be written as where λ ≈ 0.2257 is a Wolfenstein parameter and been applied.To suppress the LQ couplings to the first-and second-generation quarks so as to satisfy constraints from low-energy physics, such as P − P mixing and q i → q j f ′ f ′ , where P and f ′ are respectively possible neutral mesons and leptons, we require the Yukawa couplings to have the hierarchy: If cancellations are allowed in the terms of (V T CKM y q L ) d,s , small LQ couplings to the first two generations of down-type quarks can be easily achieved in the model.Although D − D mixing can constrain y u R2 y u R1 , we can take a small y u R1 to avoid this constraint on |y u R2 |, for which we need y u R2 ∼ O(0.5) to enhance R(D ( * ) ).In this model, the LQ couplings to the third-generation quarks are dominant.Both CMS [108] and ATLAS [109] have searched for the scalar LQ with an electric charge of e/3 using the tτ and bν production channels.ATLAS has placed a stronger upper bound on the LQ mass when BR(S −1/3 → tτ ) = 1/2, obtaining m S ≥ 1.22 TeV.If we set y u R3 = 0, the ATLAS measurement can be directly applied to our model, and tτ and bν τ thus become the dominant decays of the LQ.To be more conservative, we use m S = 1.5 TeV in our numerical calculations.

B. Phenomenological analysis
Here, we present the numerical results of the observables discussed in Sec.III and highlight their features while taking into account the constrained parameter space obtained in Sec.IV A.

Cross sections of CEνNS on Ar and CsI targets
Since the targets of the measured CEνNS in the COHERENT experiment are Ar and CsI, we focus on both targets in the following numerical analysis.Because CsI is a compound of cesium and iodide, the fraction of each nucleus contributing to the cross section is defined by f i = A i /(A Cs + A Ar ) [93].Based on COHERENT's best-fit results for ⟨σ⟩ e and ⟨σ⟩ µ+μ [3], where the resulting ⟨σ⟩ µ+μ is noticeably smaller than the SM prediction, we choose to present the numerical results with sign(θ Z ) = −1.
To calculate the cross section of CEνNS for Ar and CsI, the quantities involved in Eq. (42) are taken as follows: The weak mixing angle is s 2 W = 0.23112, the number of the protons and neutrons in 40   (20, 6.5) keV [94].Using E ν ≈ m T E r /2, the minimum neutrino energy of producing the threshold RE for Ar and CsI can be estimated to be E min ν ∼ 19 MeV.If we apply this E min ν to Eq. ( 42), it is found that the total cross section of CEνNS will be reduced by ∼ 2.4%, which is the same as the uncertainty from the nuclear form factor. Due to the fact that E νµ ≃ 29.80 MeV, the kinematic cut of E min ν ∼ 19 MeV does not influence the ν µ scattering.Additionally, according to neutrino fluxes shown in Eq. ( 41), E ν ≲ 19 MeV locates at the front tail of the ν e and νµ fluxes, where the contributions from this region are much smaller than those from 19 MeV to 52.80 MeV.Since our purpose is to demonstrate the sensitivity to the new physics effects, for simplicity, we do not consider the kinematic cut based on the experimental conditions.The detailed event analysis based on the experimental setup can be found in Ref. [25].
Using Eq. ( 42), we show the total cross section of CEνNS for Ar (solid) and CsI (dashed) as a function of m Z 1 in Fig. 3(a).We estimate the SM results for Ar and CsI to be 18.2×10 −40 cm 2 and 183.12×10 −40 cm 2 , respectively.Since the cross section is plotted in the logarithmic scale, the sensitivity in m Z 1 is not obvious.To illustrate the new physics effects, we show the deviation from the SM result, defined by (⟨σ NP+SM ⟩ ϕ − ⟨σ SM ⟩ ϕ )/⟨σ SM ⟩ ϕ , in Fig. 3(b).It can be seen that the influence of new physics can exceed 10% when m Z 1 ≳ 12 MeV, with a slightly larger influence on CsI than on Ar.
In addition to the total cross section of CEνNS, the cross section at specific incident neutrino energy E ν serves as another useful physical observable for probing the new physics effects.For clarity, we define the averaged total cross section as a function of E ν as follows: In Fig. 4 U (1) gauge symmetries.The gauged U (1) symmetries can be classified as U (1 where X q,ℓ denote the U (1) charges of quark and lepton, respectively.Since the experiments from the searches of visible dark photons place strict constraints on g Z ′ and m Z ′ , not all U (1) models are of interest in the study.To illustrate the contributions from different gauged U (1) symmetries to CEνNS, we consider the potential models, including universal, B − L, B − L e − 2L µ , and L µ − L τ with kinetic mixing, from the model listed in Ref. [25], where the charge assignments of the selected U (1)'s are given in Table II.Using the central value of data along with 1σ errors as the upper bound for CEνNS, the flux-averaged cross section ⟨σ⟩ ϕ for the selected U (1) models as a function of g Z ′ and m Z ′ is shown in Fig. 5, where the solid, long dashed, dotted, dashed, and dot-dashed curves represent the results from our model, universal, B − L, B − L e − 2L µ , and L µ − L τ with kinetic mixing, respectively.It can be seen that in the mass region of 10 ≤ m Z ′ ≤ 100 MeV, our model can fit better the constraint from CCFR and the observed muon g − 2.
TABLE II: Charge assignments of the selected new U (1) gauged models [25].
CCFR Δa μ 3 σ transitions.In this study, we use the form factors given in Ref. [59], obtained using the heavy quark effective theory (HQET).With the input values of m B + = 5.28 GeV, m D 0 = 1.864GeV, m D 0 * = 2.007 GeV, τ B − = 2.450 × 10 12 GeV −1 , and V ub = 0.0395, the BRs for B + → (D 0 , D 0 * )ℓν are found to be consistent with current experimental data, as shown in Table III.Using the formulas presented in Sec.III B, we obtain for the SM that:
TABLE III: Branching ratios of the B − → D 0( * ) ℓν decays in the SM and their experimental measurements.
Exp [105] (2.30 ± 0.09)% (7.7 ± 2.5) × 10 −3 (5.58 ± 0.22)% (1.88 ± 0.20)% The parameters involved in the b → cτ ν transition mediated by the LQ appear in the combinations of y q L3 y q L2 /m 2 S and y q L3 y u R2 /m 2 S .For the numerical analysis, we fix m S = 1.5 TeV.From Eq. ( 63), we see that y q L2 ∼ O(λ 2 ) ≪ y q L3 , indicating that the dominant effect on R(D) and R(D * ) comes from the combination y q L3 y u R2 .To simplify the analysis, we take the assumption that y q L2 = 0, in which case R(D ( * ) ) is found to deviate from that with y q L2 = 0.04 by only ∼ 2%.We present the contours of R(D) and R(D * ) in the y q L3 -y u R2 plane in the left plot of Fig. 6, with the shaded areas (light-green and grey, respectively) covering the 2σ ranges of their world averages.It is seen that the low boundaries of R(D) and R(D * ) match exactly, while the upper boundary for R(D) = 0.414 is close to the contour of R(D * ) = 0.297.This illustrates that an accurate measurement of R(D) can indirectly constrain the value of R(D * ), and vice versa.The right plot of Fig. 6 shows the dependence of R(D ( * ) ) on the product y q L3 y u R2 .To explain the R(D) and R(D * ) anomalies, we need −1 < y q L3 y u R2 < 0 for m LQ = 1.5 TeV.It is observed that R(D) is more sensitive to the S In addition to the ratio of the BR for τ ν to that for ℓν, other physical observables may be sensitive to the new physics, such as the forward-backward asymmetry of the charged lepton, τ polarization [73,75], and q 2 -dependent differential decay rates.The BR is sensitive to the CKM matrix elements and the form factors of the B → (D, D * ) transitions.To eliminate these factors, we propose the ratio of the q 2 -dependent differential decay rates, defined to be: where H(x) is the Heaviside step function, and dΓ ℓ ′ M /dq 2 is the average of the electron and muon modes.Because the threshold invariant mass-squared of τ ν in the B → M τ ν decay is τ , we thus require that the denominator dΓ ℓ ′ M /dq 2 also starts from the same invariant mass-squared.To appreciate the benefit of considering the observable defined in Eq. ( 67), we first show the q 2 -dependent BRs for B − → (D 0 , D 0 * )ℓ ′′ ν (ℓ ′′ = ℓ ′ , τ ) in the SM in Figs.7(a) and (b), respectively.Plot (a) shows that when q 2 ≳ 8 GeV 2 , the decay B − → D 0 τ ν becomes larger than the light lepton mode, and it is expected that R D (q 2 ) > 1 in this region.D * is a vector meson and has longitudinal (P L ) and transverse (P T ) components.To exhibit their contributions, we separately show P L and P T in Fig. 7 (b).The results indicate that P T becomes larger than P L at somewhat large q 2 regions in both light lepton and τ modes.In contrast to the B − → D 0 ℓ ′′ ν decay, dΓ ℓ ′ D * /dq 2 is always larger than dΓ τ D * /dq 2 in the allowed kinematic region, thus, it is expected that R D * (q 2 ) < 1.  where the longitudinal and transverse polarizations of D * are illudtrated separately.The ratios R D (q 2 ) (c) and R D * (q 2 ) (d) as functions of q 2 , where the solid, dashed, and dot-dashed curves are plotted for y q L3 y u R2 = 0, −0.5, and −1, respectively.
The q 2 -dependence of R D (q 2 ) and R D * (q 2 ) in the SM is shown in Figs.7(c) and (d), respectively, using the solid curves.It is confirmed that R D (q 2 ) ≳ 1 at q 2 ≳ 8 GeV, while R D * (q 2 ) < 1 in the physical kinematic region.Additionally, we find that R M (q 2 ) increases monotonically with q 2 .This means that the decreasing rate of dΓ ℓ ′ M /dq 2 in q 2 is faster than that of dΓ τ M /dq 2 .To see how sensitive R M (q 2 ) is to new physics effects, we show the results using benchmarks of y q L3 y u R2 = −0.5 (dashed) and y q L3 y u R2 = −1 (dotdashed) for R D (q 2 ) and R D * (q 2 ) in the corresponding plots.We also consider the quantity (R NP M (q 2 ) − R SM (q 2 ))/R SM M (q 2 ) to exhibit the deviation caused by the new physics effects in R M (q 2 ) from the SM prediction, and the results are shown in Fig. 8.The variations of these curves show that R D (q 2 ) is more sensitive to new physics than R D * (q 2 ) in the model.
We can use these results to constrain the free parameters in the model.Based on Eqs.(B1), (B3), and (B4), the oblique parameters have a quadratic dependence on c β−α .However, c β−α ≲ O(0.04) as previously discussed, meaning that its effects on the S, T , and U parameters are negligible.Therefore, these parameters can be approximated for the model as follows: In this simplified form, the oblique parameters depend only on the ratio m H + /m H .The contours for T (solid) and S (dashed) in the plane of m H + and m H for the model are drawn in Fig. 9

Z 1 and H decays
Finally, let's discuss possible decays of the light Z 1 and H.Because the mass of the light gauge boson is limited in the region of m Z 1 ∈ (10, 100) MeV, it can only decay dominantly into on-shell light leptons through two-body decays.The Z 1 partial decay rate for possible final leptons is given by: where , f denotes the possible light leptons (such as the three active neutrinos and the electron), and m 2 f /m 2 Z 1 ≈ 0 is applied.The effective couplings of C f R,L for each involved f are given as follows: Although Z ′ does not couple to the first-generation leptons, the physical Z 1 can decay to them via Z ′ − Z mixing.
If s θ Z were not significantly smaller than g Z ′ , the decay rates for Z 1 → (ν e ν e , e − e + ) could be sizable compared to the Z 1 → νℓ ′ ν ℓ ′ decays.However, due to the large t β enhancement in the Z 1 gauge coupling to ν µ,τ , the dominant decay channels are Z 1 → ν µ νµ /ν τ ντ , with estimated BRs of approximately 50.5% and 49.5%, respectively.The BRs for ν e νe and e − e + as functions of t β are presented in Fig. 10(a).It is found that the BRs are more sensitive to t β and less sensitive to m Z 1 .Because Z 1 can be produced in the τ → µZ 1 decay, which depends on the lepton flavor mixing θ L , a significant BR(Z 1 → e − e + ) thus implies a large BR for the LFV process τ → µZ 1 → µe − e + , where the current upper limit is BR(τ → µe − e + ) < 1.8×10 −8 [105].Our estimate of BR(τ → µe − e + ) is shown in Fig. 10(b), where χ µτ = 10 −5 is used.Since τ → µZ 1 is also not sensitive to m Z 1 , the dependence of m Z 1 in BR(τ → µe − e + ) is not manifest.Assuming the integrated luminosity of 50 ab −1 , Belle II will be capable of probing the LFV process BRs down to the level of 10 −10 − 10 −9 [111].The BR of O(10 −9 ) for τ → µe − e + predicted in this model can thus be probed at Belle II.
As discussed earlier, when m H < m h /2, H can be produced through the h → HH decay.The partial decay width of this process can provide a strict limit on the t β and c β−α parameters.In the following, we concentrate on this scenario, even though H generally can be heavier.
For two-body decays, H should decay into a pair of fermions, as long as the phase space permits.From Eq. ( 25), its Yukawa couplings to fermions are suppressed by m f /v and (c β−α − s β−α /t β ), with no other factors that can enhance the partial decay width.As a result, Γ(H → f f ) is small and negligible.However, even though suppressed by m 2 Z 1 /v from the gauge coupling as shown in the H → Z 1 Z 1 decay rate can be enhanced by the longitudinal component, which is proportional to 1/m Z 1 .This leads to a partial decay width, The original suppression factor from the gauge coupling is seen to be canceled by the longitudinal effect of 1/m 2 Z 1 from each Z 1 boson.With m H = 50 GeV and t β = 20, we obtain Γ(H → Z 1 Z 1 ) ≈ 8.2 GeV.The other decay processes are subdominant.For example, the H → Z 1 Z * 2 → Z 1 f f decay has additional suppression factors due to the phase space and 1/m 2 Z 2 .An explicit estimate shows that the partial width for H → Z 1 Z * 2 is of O(10 −5 ) GeV.According to the earlier analysis, Z 1 → ν ν is the dominant decay channel.Consequently, H predominantly decays into invisible neutrinos and becomes missing energy in the detector.
We now turn to the production of H at the LHC.First, H could be singly produced according to Eq. ( 22) via the vector boson fusion (VBF) process.But the W − W + (ZZ)H coupling is suppressed by c β−α .Additionally, the Yukawa coupling for the bremsstrahlung production of H with the top quark is determined by (m t /v)(c β−α −s β−α /t β ) and is also suppressed.However, H can be pair-produced more copiously through the hHH and W − H + H couplings.In the former case, the H pair is produced by the on-shell Higgs boson, i.e., pp → h → HH.From Eq. ( 51), although Γ(h → HH) is associated with the small factor ξ, its BR can still be at the percent level.This amounts to the invisible decay of the Higgs boson [112].In the latter case, the W − H + H coupling, as given in Eq. ( 22), is determined by the gauge coupling g with s β−α ≈ 1.When H + is taken as an intermediate state in the t-channel scattering, H pair production occurs via the VBF channel, i.e., pp → HH + forward jets.
We may probe such an effect via the search for invisible decays of the new Higgs boson H [112].

V. SUMMARY
A sub-GeV Z ′ gauge boson has received much attention recently in the literature due to its distinctive characteristics, which could potentially resolve the observed anomalies, such as the muon g − 2, and serve as a messenger between visible and dark sectors.Additionally, a light Z ′ gauge boson can make a significant contribution to CEνNS, as recently observed by the COHERENT experiment.Accordingly, we investigate the phenomenological impacts on flavor physics when the light Z ′ gauge boson originates from the local U (1) Lµ−Lτ gauge symmetry.
We have found that when a second Higgs doublet carrying the U (1) Lµ−Lτ charge is introduced to spontaneously break the U (1) Lµ−Lτ gauge symmetry, the new neutral and charged scalars can result in a larger W mass.Moreover, when a scalar leptoquark S In addition to explaining the observed excesses in R(D) and R(D * ) using the introduced leptoquark, we have proposed a q 2 -dependent ratio of dΓ/dq 2 (B → M τ ν) to the avreaged differential decay rate of the light leptons dΓ/dq 2 (B → M ℓ ′ ν), denoted by R M (q 2 ).Our results show that in the high q 2 region, R D (q 2 ) is more sensitive to the new physics effects and exhibits a significant deviation from the SM.
We have also studied the impact of the two-Higgs-doublet model on the oblique parameters and their relations to the W boson mass.With the approximation that cos(β − α) ≪ 1, the parameters involved in the oblique parameters are m H and m H + .We find a significant space in the m H -m H + plane that allows an enhancement of m W up to the value observed by CDF II.Finally, we have discussed the possible decay channels for Z 1 and H in the scenario where m Z 1 ∈ (10, 100) MeV and m H < m h /2.The analysis shows that Z 1 → ν µ νµ /ν τ ντ and H → Z 1 Z 1 are the dominant decay channels.
The w-dependent functions C Γ i can be found in Ref. [114], and the sub-leading Isgur-Wise functions are [115]: The form factor parametrizations in Eqs.(A1) and (A2), using which we formulate the BRs, and in Eqs.(A4) and (A5), for which we evaluate within the framework of the HQET, are related as follows: The relations for the form factors arising from the pseudoscalar, vector, and axial-vector currents for the B → D * transitions are found to be: Finally, the tensor form factors for the B → D * transitions are related by: Appendix B: Oblique parameters in the model To calculate the S, T , and U parameters in the model, we apply the results obtained in Ref. [98].Using the mixing matrices of Goldstone and scalar bosons shown in Eqs. ( 9) and ( 13), the resulting T parameter subtracting the SM result is expressed as: where α em = e 2 /4π is the fine structure constant of QED, and the function F is defined as: In the limit of s β−α → 1, the H ± -and H-mediated loop effects are the most dominant.
The S and U parameters are respectively given by: and where the functions of G and G are given by: G(m 2 a , m 2 b , m 2 c ) = − . The flavored U (1) µ−τ symmetry strictly limits the Yukawa couplings to different lepton flavors, resulting in a suppressed contribution of the charged Higgs to R(M ) by m b m τ /v 2 in this model, where v is the combined vacuum expectation value (VEV) of the introduced Higgs doublets.Hence, the introduction of scalar LQ emerges as a more apposite solution.We find that among various LQ representations, the simplest choice to explain the observed excess in R(M ) is the S 1 3 = ( 3, 1, 2/3) representation under the SU (3) C × SU (2) L × U (1) Y gauge symmetries.Additionally, based on the flavored This paper is organized as follows: In Sec.II, we formulate the model and derive the spectrum of scalar bosons and various new couplings.The Z ′ − Z mixing and lepton flavor mixing are also discussed in detail.With the new interactions, Sec.III discusses the new physics effects on various phenomena, including the cross section of CEνNS, values of R(D) and R(D * ), new Higgs decay channels h and gravity 2 -U (1) µ−τ cancel out automatically without the need of introducing extra chiral fermions.As a result, the gauged U (1) µ−τ symmetry model stands free from gauge anomalies.In addition to the SM Higgs doublet, denoted by H 2 , whose neutral component has a VEV, v 2 , to spontaneously break SU (2) L ×U (1) Y , we introduce an additional Higgs doublet, denoted by H 1 , which carries not only the U (1) µ−τ charge, twice that of µ, but also the weak isospin and U (1) Y hypercharge.The new Higgs doublet is assumed to also develop a VEV, v 1 , in its neutral component to break U (1) µ−τ besides SU (2) L ×U (1) Y , resulting in a massive Z ′ boson.Therefore, unlike the conventional two-Higgs-doublet model (2HDM), the model has one charged Higgs and two CP-even Higgs bosons but has no CP-odd Higgs boson, as it has become the longitudinal component of Z ′ .Finally, we include an SU (2) L -singlet scalar LQ with hypercharge Y = 2/3 that also has the same U (1) µ−τ charge as µ.The quantum number assignments of the leptons, the Higgs doublets, and the LQ are given in Table

r
denotes the nuclear threshold recoil energy, and E min ν is the minimum incident neutrino energy required to reach E min r .B. R(D) and R(D * ) only arises from the SM contribution.In addition, because the LQ contribution to b → cℓν only involves the tau-neutrino, the induced b → cµν τ decay does not interfere with the SM contribution.The effective couplings C ℓ S and C ℓ T at the m b scale can be obtained from the LQ mass scale via the renormalization group (RG) equations.Following the results in Ref. [95], we obtain C ℓ S (m b ) ≈ 1.57C ℓ S (m S ) and C ℓ T (m b ) = 0.86 C ℓ T (m S ).To calculate the BRs for the B → (D, D * )ℓν decays, one requires the hadronic effects for the B → D ( * ) transitions.The parametrization of form factors for different weak currents can be found in Appendix A 1. By utilizing these form factors, the differential decay rate

FIG. 1 :
FIG. 1: Parameter space preferred by the muon g − 2 (shaded region bounded by blue dot-dashed curves) and ruled out by the CCFR experiment (shaded region above the red dashed curve).The solid lines represent the contours for t β .

FIG. 3 :
FIG. 3: (a) Cross section averaged by neutrino fluxes for Ar and CsI targets as a function of m Z 1 , where the points for m Z 1 = 0 correspond to the SM results.(b) Fractional deviation on the total cross section ⟨σ⟩ ϕ from its SM value as a function of m Z 1 .In both plots, the solid and dashed curves represent the results for Ar and CsI targets, respectively.

FIG. 6 :
FIG. 6: Left: Contours of R(D) and R(D * ) in the y q L3 -y u R2 plane.The solid (darker-green) and dashed (red) lines cover the 2σ range of the world-averaged R(D) and R(D * ), respectively.Right: Dependence of R(D ( * ) ) on y q L3 y u R2 .The light-green [pink] shaded region represents the 2σ range of the world-averaged R(D) [R(D * )].
(a), where s β−α ≈ 1 is taken in the estimates.Due to the fact that U ≪ T , we do not show the results of U in the plot.The values of S and U in the model can only be up to the percent level and can be neglected in the numerical estimates for further phenomenological analyses.Thus, using the obtained T parameter, the loop-corrected W mass in the model is shown in Fig.9(b), where the contours correspond to the central value, ±2σ and ±5σ of the world average of m W = 80.4133 ± 0.0080[110].We observe that m W increases with m H + for a given m H , while a lower m H is needed to increase m W when m H + is fixed.For instance, m W ≈ 80.43 GeV can be achieved for m H ≈ 50 GeV and m H + ≈ 150 GeV.

FIG. 9 :
FIG. 9: (a) Contours of the oblique parameters, S and T , in the m H -m H + plane.(b) Contours of m W in the m H -m H + plane.

1 3 =
( 3, 1, 2/3) is added to the model, it would couple to the third-generation leptons in a unique way due to the U (1) Lµ−Lτ symmetry, so that the branching ratios of B → (D, D * )τ ν τ are enhanced, thus solving the R(D) and R(D * ) anomalies.With the new Higgs doublet, the mixing between the new scalar boson and the SM-like Higgs leads to new decay channels for the Higgs boson, including h → µτ /Z 1 Z 1 /Z 1 Z 2 (and h → HH when m H < m h /2).It is found that due to the enhancement of 1/m 2 Z 1 , the τ → µZ 1 decay strictly constrains the µ − τ flavor mixing, resulting in a highly suppressed h → µτ decay.By assuming proper partial widths to the new Higgs decay channels, the tan β and cos(β − α) parameters are limited and the large tan β scheme is favored.Although the µ − τ flavor-changing coupling is restricted to be small, the τ → µZ 1 → µe − e + decay, induced through the Z − Z ′ mixing, can still reach the sensitivity of O(10 −9 ) at Belle II.Taking into account all potential constraints, we have found that the cross section of CEνNS induced by the Z ′ −Z mixing depends solely on the light gauge boson mass, m Z 1 .The mass region of m Z 1 that is used to fit the CEνNS cross section, measured by COHERENT using the CsI target [3], can also explain the muon g −2 anomaly within 3σ.To demonstrate the sensitivity of new physics to CEνNS in the model, we propose to study the cross section as a function of the incident neutrino energy.Our results show that in the low energy region, such as E ν ∼ 10 MeV, the deviation from the SM can exceed 15%, depending on the value of m Z 1 .To compare with results from other U (1) gauge symmetries, we have examined the influence on the CEνNS cross section from selected U (1) gauged models, such as the universal, B − L, B − 3L µ , and L µ − L τ with kinetic mixing.It has been found that only the model with dynamical U (1) Lµ−Lτ breaking can explain the anomaly of muon g − 2 when the 1σ upper limits of the COHERENT data are imposed.

TABLE I :
Quantum numbers of the leptons, Higgs doublets, and scalar leptoquark.
(18,22) I, and133Cs are set to be (Z, N ) Ar =(18,22), (Z, N ) I = (53, 74), and (Z, N ) Cs = (55, 75), respectively, and the masses of the nuclei are m Ar = 37.20 GeV, m I = 118.24 GeV and m Cs = 123.86 GeVThe energy of the prompt ν µ is determined from the π + decay at rest.With m µ = 105.65MeV and m π = 139.57MeV, we obtain E νµ ≃ 29.80 MeV.By neglecting the electron mass, the maximum energy of ν e and νµ from the µ + decay is E max νe,νµ = m µ /2 ≃ 52.8 MeV.As mentioned in the Introduction, the difficulty in measuring the CEνNS is due to the small nuclear recoil energy (RE).We can estimate the maximum RE of the nuclear targets, argon, iodine, and cesium, by incident ν µ with the energy of 29.80 MeV = (47.66,15.01, 14.33) keV, respectively.The maximum RE of (Ar, I, Cs) from νµ or ν e with the maximum incident energy of 52.8 MeV is given by E