Exploring dark matter, neutrino mass and $R_{K^{(*)},\phi}$ anomalies in $L_{\mu}-L_{\tau}$ model

We investigate Majorana dark matter in a new variant of $U(1)_{L_{\mu}-L_{\tau}}$ gauge extension of Standard Model, where the scalar sector is enriched with an inert doublet and a $(\bar{3},1,1/3)$ scalar leptoquark. We compute the WIMP-nucleon cross section in leptoquark portal and the relic density mediated by inert doublet components, leptoquark and the new $Z^{\prime}$ boson. We constrain the parameter space consistent with Planck limit on relic density, PICO-60 and LUX bounds on spin-dependent direct detection cross section. Furthermore, we constrain the new couplings from the present experimental data on ${\rm Br}(\tau \to \mu \nu_\tau \bar \nu_\mu)$, ${\rm Br}( B \to X_s \gamma)$, ${\rm Br}( B^0 \to K^0 \mu^+ \mu^-)$, ${\rm Br}(B^+ \to K^+ \tau^+ \tau^-)$ and $B_s-\bar{B_s}$ mixing, which occur at one-loop level in the presence of $Z^\prime$ and leptoquark. Using the allowed parameter space, we estimate the form factor independent $P_{4,5}^\prime$ observables and the lepton non-universality parameters $R_{K}$, $R_{K^*}$ and $R_\phi$. We also briefly discuss about the neutrino mass generation at one-loop level and the viable parameter region to explain current neutrino oscillation data.


I. INTRODUCTION
Though the experimental measured values of various physical observables are in excellent agreement with the Standard Model (SM) predictions, there are many open unsolved problems like the matter-antimatter asymmetry, hierarchy problem and the dark matter (DM) content of the universe etc., which make ourselves believe that there is something beyond the SM. In this regard, the study of rare semileptonic B decay processes provide an ideal testing ground to critically test the SM and to look for possible extension of it. Although, so far we have not observed any clear indication of new physics (NP) in the B sector, there are several physical observables associated with flavor changing neutral current (FCNC) b → sl + l − processes which have (2 − 4)σ [1][2][3][4][5][6] discrepancies. Especially, the observation of 3σ anomaly in the P 5 angular observables [4] and the decay rate [5] of B → K * µ + µ − processes have attracted a lot of attention in recent times. The decay rate of B s → φµ + µ − has also 3σ deviation compared to its SM prediction [3]. Furthermore, the LHCb Collaboration has observed the violation of lepton universality in B + → K + l + l − process in the low which has a 2.6σ deviation from the corresponding SM result [7] R SM K = 1.0003 ± 0.0001.
To resolve the above b → sll anomalies, we extend the SM gauge group SU (3) C ×SU (2) L × U (1) Y with a local U (1) Lµ−Lτ symmetry. The anomaly free L µ − L τ gauge extensions [9,10] are captivating with minimal new particles and parameters, rich in phenomenological perspective. The model is quite simple in structure, suitable to study the phenomenology of DM, neutrino and also the flavor anomalies. It is well explored in dark matter context in literature [11][12][13][14], in the gauge and scalar portals. The approach of adding color triplet particles to shed light on the flavor sector thereby connecting with dark sector is interesting.
Leptoquarks (LQ) are not only advantageous in addressing the flavor anomalies, but also act as a mediator between the visible and dark sector. Few works were already done with this motivation [15][16][17][18].
Leptoquarks are hypothetical color triplet gauge particles, with either spin-0 (scalar) or spin-1 (vector), which connect the quark and lepton sectors and thus, carry both baryon and lepton numbers simultaneously. They can arise from various extended standard model scenarios [19][20][21][22][23][24][25][26][27][28][29][30], which treat quarks and leptons on equal footing, such as the grand unified theories (GUTs) [19][20][21][22], color SU (4) Pati-Salam model [23][24][25][26][27], extended technicolor model [28,29] and the composite models of quark and lepton [30]. In this article, we study a new version of U (1) Lµ−Lτ gauge extension of SM with a (3, 1, 1/3) scalar LQ (SLQ) and an inert doublet, to study the phenomenology of dark matter, neutrino mass generation and compute the flavor observables on a single platform. The SLQ mediates the annihilation channels contributing to relic density and also plays a crucial role in direct searches as well, providing a spin-dependent WIMP-nucleon cross section which is quite sensitive to the recent and ongoing direct detection experiments such as PICO-60 and LUX. The Z gauge boson of extended U (1) symmetry and the SLQ also play an important role in settling the known issues of flavor sector. In this regard, we would like to investigate whether the observed anomalies in the rare leptonic/semileptonic decay processes mediated by b → sl + l − transitions, can be explained in the present framework. We analyze the implications of the model on both the DM and flavor sectors, in particular on B → V l + l − (V = K * , φ) decay modes. In literature , there were many attempts being made to explain the observed anomalies of rare B decays in the scalar leptoquark model.
The paper is structured as follows. We describe the particle content, relevant Lagrangian and interaction terms, pattern of symmetry breaking in section-II. We derive the mass eigenstates of the new fermions and the scalar spectrum in section-III. We then provide a detailed study of DM phenomenology in prospects of relic density and direct detection observables in section-IV. Mechanism of generating light neutrino mass at one-loop level is illustrated in section-V. Section-VI contains the additional constraint on the new parameters obtained from the existing anomalies of the flavor sector, like Br(τ → µν τνµ ), Br(B → X s γ), Br(B → Kτ τ ), R K and B s −B s mixing. We then investigate the impact of additional U (1) Lµ−Lτ gauge symmetry on the R K * , R φ LNU parameters and optimized P 4,5 observables in section-VII. We summarize our findings in Section-VIII.
We study the well known anomaly free U (1) Lµ−Lτ extension of SM with three neutral fermions N e , N µ , N τ , with L µ − L τ charges 0, 1 and −1 respectively. A scalar singlet φ 2 , charged +2 under the new U (1) is added to spontaneously break the local U (1) Lµ−Lτ gauge symmetry. We also introduce an inert doublet (η) and a scalar leptoquark S 1 (3, 1, 1/3) with L µ − L τ charges 0 and −1 to the scalar content of the model. We impose an additional Z 2 symmetry under which all the new fermions, η and the leptoquark are odd and rest are even. The particle content and their corresponding charges are displayed in Table. I .
The Lagrangian of the present model can be written as where the scalar potential V is The gauge symmetry SU (2) L ×U (1) Y ×U (1) Lµ−Lτ is spontaneously broken to SU (2) L ×U (1) Y by assigning a VEV v 2 to the complex singlet φ 2 . Then the SM Higgs doublet breaks the The inert doublet is denoted by η = The masses of its charged

IV. DARK MATTER PHENOMENOLOGY
A. Relic abundance The model allows the dark matter (N − ) to have gauge and scalar mediated annihilation channels. The possible contributing diagrams are provided in Fig. 1 which are mediated by (H 1 , H 2 , η + , η 0 , S 1 , Z ). Majorana DM in H 1,2 portal (upper row in Fig. 1 ) has already been well explored in literature [52,53]. Here, we focus on (Z , S 1 , η)-mediated channels (middle and bottom rows in Fig. 1 ) contributing to DM observables, which we later make connection with radiative neutrino mass as well as flavor observables. The relic abundance of dark matter is computed by Here the Planck mass, M pl = 1.22 × 10 19 GeV and g * = 106.75 denotes the total number of effective relativistic degrees of freedom. The function J(x f ) reads as The thermally averaged annihilation cross section σv is given by the expression where K 1 , K 2 denote the modified Bessel functions and x = M D /T , where T is the temperature. The analytical expression for the freeze out parameter x f is Here g represents the number of degrees of freedom of the dark matter particle N − .
As seen from the left panel of Fig. 2, the relic density with s-channel contribution is featured to meet the PLANCK limit [54] near the resonance in propagator (H 1 , H 2 , Z ), i.e., near M − = Mprop 2 . We restrict our discussion to the mass region (in GeV), 100 ≤ M Z ≤ 1000, 80 ≤ M − ≤ 1000 and also H 2 is considered to be sufficiently large such that its resonance doesn't meet the PLANCK limit below 1 TeV region of DM mass. Now, in this mass range of DM, the channels mediated by (Z , η, S 1 ) drive the relic density observable, where the gauge coupling g µτ controls the s-channel contribution, while Y ll , y qR are relevant in t-channel contributions. The relevant parameters in our investigation are (M − , g µτ , M Z , Y ll , y qR ). The effect of these parameters on the relic abundance is made transparent in Fig. 2 , where we fixed Y ll ∼ 10 −2 , in order to explain neutrino mass at one loop level. Left panel shows the variation of relic density with varying gauge parameters g µτ and M Z , right panel depicts the behaviour with varying y qR parameter. No significant constraint on M Z , g µτ parameters is observed, however relic density has an appreciable footprint on M − − (y qR ) 2 parameter space, which will be discussed in the next section.

B. Direct searches
Moving to direct searches, the WIMP-nucleon cross section is insensitive to direct detection experiments as Z couples differently to Majorana fermion (axial-vector type) and quarks (vector type) [55,56]. The t-channel scalar (H 1 , H 2 ) exchange can give spin-independent contribution, but it doesn't help our purpose of study. In the scalar portal, one can obtain contribution from spin-dependent (SD) interaction mediated by SLQ, of the form The corresponding cross section is given by [55] where the angular momentum J N = 1 2 , M n 1 GeV for nucleon. The values of quark spin functions ∆ d,s are provided in [55]. Now, it is obvious that it can constrain the parameters M − and (y qR ) 2 . Fig. 3 left panel displays M − − (y 2 qR ) parameter space (green and red regions) remained after imposing PLANCK [54] 3σ limit on current relic density. Here, the region shown in green turns out to be excluded by most stringent PICO-60 [57] limit on SD WIMP-proton cross section, as seen from the right panel.

V. RADIATIVE NEUTRINO MASS
To generate light neutrino mass at one-loop level, we can write the interaction term using The corresponding diagram is shown in Fig. 4 . Assuming m 2 0 = (M 2 ηe + M 2 ηo )/2 is much greater than M 2 ηe − M 2 ηo = λ Hη v 2 , the expression for the radiatively generated neutrino mass [59] is given by  Here

VI. FLAVOR PHENOMENOLOGY
The general effective Hamiltonian responsible for the quark level transition b → sl + l − is given by [60,61] where where α em denotes the fine-structure constant and P L,R = (1∓γ 5 )/2 are the chiral operators.
The primed operators are absent in the SM, but can exist in the proposed L µ − L τ model.
The previous section has discussed the available new parameter space consistent with the DM observables which are within their respective experimental limits. However, these parameters can be further constrained from the quark and lepton sectors, to be presented in the subsequent sections.
where λ t = V tb V * ts , η B is the QCD correction factor and S 0 (x t ) is the loop function [63] with x t = m 2 t /M 2 W . Using Eqn. (22), the B s −B s mass difference in the SM is given as The SM predicted value of ∆M s by using the input parameters from [64,65] is and the corresponding experimental value is [64] ∆M Expt s = 17.761 ± 0.022 ps −1 .
Even though the theoretical prediction is in good agreement with the experimental B s −B s oscillation data, it does not completely rule out the possibility of new physics. where with χ ∓ = M 2 ∓ /M 2 S 1 and k (χ ± , χ ∓ , 1) , j (χ ± , χ ∓ , 1) are the loop functions [15]. Using Eqn. (26), the mass difference of B s −B s mixing due to the exchange of S 1 and N ± is found to be Including the NP contribution arising due to the SLQ exchange, the total mass difference can be written as Using Eqns. (24) and (25) in (29), one can put bound on (y qR ) 2 and M − parameters.
The rare semileptonic B → Kl + l − process is mediated via b → sl + l − quark level transitions. In the current framework, the b → sl + l − transitions can occur via the Z exchanging one-loop penguin diagrams shown in Fig. 6 .
where p B (p K ) and M B (M K ) denote the 4-momenta and mass of the B (K) meson and q 2 is the momentum transfer. By using Eqn. (30), the transition amplitude of B → Kµ + µ − process is given by where p 1 and p 2 are the four momenta of charged leptons and V sb (χ − , χ + ) is the loop function [15,67]. Now comparing this amplitude (31) with the amplitude obtained from the effective Hamiltonian (20), we obtain a new Wilson coefficient associated with the right-handed semileptonic electroweak penguin operator O 9 as The differential branching ratio of B → Kll process with respect to q 2 is given by where with and λ(a, b, c) = a 2 + b 2 + c 2 − 2(ab + bc + ca), For numerical estimation, we have used the lifetime and masses of particles from [64] and the form factors are taken from [68]. The upper limit on the branching ratio of B + → K + τ + τ − process is [64] Br while its predicted value in the SM is Since Z doesn't couple to electron, the branching ratio of B + → K + e + e − process is considered to be SM like. The anomalies of b → sll decay modes can put constraint on all the four parameters, i.e., (y qR ) 2 , g µτ , M Z and M − .

C. B → X s γ process
The B → X s γ process involves b → sγ quark level transition, the experimental limit on the corresponding branching ratio is given by [69] Br(B → X s γ)
In the presence of Z boson, the τ → µν τνµ process can occur via box diagram as shown in Fig. 8 . There are four possible one-loop box diagrams with the Z connected to the lepton legs. The total branching ratio of this process is given by [71] where the branching ratio in the SM is given by [71] Br(τ → µν τνµ ) SM = (17.29 ± 0.032)%. (44) Now comparing the theoretical result with the experimental measured value [64] Br(τ → µν τνµ ) Expt = (17.39 ± 0.04)%, (45) one can put bounds on M Z − g µτ parameter space. parameter space. Since Z does not couple to quarks, these gauge parameters couldn't be constrained from b → sγ decay modes and B s −B s oscillation data. The constraint on M − − (y qR ) 2 parameter space is obtained from R K , Br(B + → K + τ + τ − ), Br(B → X s γ) and B s −B s mixing results. In addition, the branching ratio of rare semileptonic B → Kν lνl process can play a vital role in restricting these parameters. Though the proposed model can allow b → sν lνl decay modes, but the contributions of µ and τ leptons cancel with each other in the leading order of NP due to their equal and opposite L µ − L τ charges. Since there is no Z µτ coupling, the neutral and charged lepton flavor violating decay processes like B → K ( * ) µ ∓ τ ± , τ − → µ − γ, τ → µµµ do not play any role. In this analysis, we consider that the y qR coupling is perturbative, i.e., |y qR | √ 4π. Left (right) panel in Fig.   9 denotes the parameter space in the plane of M Z − g µτ (M − − (y qR ) 2 ) consistent with DM and flavor studies. From left panel, one can obtain the lower limit on the ratio M Z /g µτ around 4615 GeV, which is far more stringent than the lower limit imposed by neutrino trident production [72,73], i.e., 540 GeV. It is also noted that the allowed region favored by the (g − 2) µ anomaly is completely excluded by the constraint from the neutrino trident production [71]. In the right panel of Fig. 9 , we redisplay M − − (y qR ) 2 parameter space of where q 2 is the momentum transfer between the B and V mesons, i.e., q µ = p µ − k µ and µ is the polarization vector of the V meson. The full angular differential decay distribution in terms of q 2 , θ l , θ V and φ variables is given as [76][77][78] where θ l is the angle between l − and B in the dilepton frame, θ V is defined as the angle between K − and B in the K − π + (K − K + ) frame, the angle between the normals of the K − π + (K − K + ) and the dilepton planes is given by φ. The complete expression for J (q 2 , θ l , θ V , φ) as a function of transversity amplitudes can be found in the Ref. [79]. The transversity amplitudes written in terms of the form factors and Wilson coefficients are as follows [79] A ⊥L,R = N √ 2λ (C eff 9 + C NP 9 ) ∓ C 10 where The dilepton invariant mass spectrum for B → V l + l − decay after integration over all angles [76] is given by where J i = 2J s i + J c i . The most interesting observables in these decay modes are the lepton non-universality parameter defined as the form factor independent (FFI) observables [80]  figures, the blue dashed lines stand for the SM contribution, the orange bands are due to the allowed region of parameters shown in Table II  implying the presence of lepton universality violation in the B s → φµ + µ − process. In Table   III , we present our predicted values of R K * and R φ for different bins. The q 2 variation of famous optimized observables −P 4 (top-left panel) and P 5 (top-right panel) of B → K * µ + µ − process are depicted in Fig. 11 . The bottom panel of this figure describes analogous plots for B s → φµ + µ − process in both the high and low recoil limit. It should be noted that P 4 LHCb = −P 4 . In the low q 2 region, our predictions on −P 4 observable of B → K * µ + µ − process is in very good agreement with the LHCb data. For B → K * µ + µ − decay mode, we are able to explain the P 5 observable within 1σ of the experimental limit in the full q 2 region (excluding the intermediate resonance regions). We notice profound deviation between the results of SM and the presented L µ − L τ model on the P 4,5 observables for B s → φµ + µ − decay modes. The numerical values of all these observables are given in Table III . We found that our results on the angular observables of B → V ll process, obtained from DM-I parameter space are almost consistent with the corresponding measured experimental data.   only leptoquark portal channels contribute to spin-dependent WIMP-nucleon cross section.
Imposing PLANCK limit on relic density and well known PICO-60, LUX bounds on spindependent cross section, we have constrained the new parameters of the model. We have also showed the mechanism of generating light neutrino mass radiatively using the inert doublet.
We have further restricted the new parameters from quark and lepton sectors i.e., by comparing the theoretical predictions of Br(τ → µν τνµ ), Br(B → X s γ), Br(B + → K + τ + τ − ), R K and B s −B s mixing with their corresponding 3σ experimental data. The neutral and charged lepton flavor violating decay processes are absent due to zero Z τ µ coupling. And also the vanishing Z qq coupling restricts the involvement of Z in B s −B s mixing, b → sγ processes at one-loop level. We have then investigated the implication on P 4,5 , R K * and R φ observables of B (s) → K * (φ)l + l − decay modes in the full kinematically allowed q 2 region for two cases i.e., dark matter and flavor allowed, only dark matter allowed parameter space.
We found that the R K * observable obtained from the parameter space consistent with only dark matter (M − ≤ 560 GeV) is within its 1σ, only dark matter (M − > 560 GeV) and both dark matter and flavor is within 2σ experimental limit. In the presence of new physics, the violation of lepton universality is observed in B s → φµ + µ − process, thus, can be probed in LHCb experiment. We noticed that the proposed L µ − L τ model is also able to explain the LHCb experimental data of the famous optimized P 4,5 observables of B → K * l + l − process in the high recoil limit. We also perceived that the form factor independent observables for