Explanations for anomalies of muon anomalous magnetic dipole moment, $b\to s \mu\bar\mu$ and radiative neutrino masses in a leptoquark model

We propose a leptoquark model simultaneously to explain anomalies of muon anomalous magnetic dipole moment and $b\to s \mu\bar\mu$ in light of experimental reports very recently. Here, we satisfy several stringent constraints such as $\mu\to e\gamma$ and meson mixings. In addition, we find these leptoquarks also play an role in generating tiny neutrino masses at one-loop level without introducing any additional symmetries. We have numerical analysis and show how degrees our parameter space is restricted.


I. INTRODUCTION
The anomalous magnetic dipole moment of muon (muon g − 2) is precisely predicted in the SM and the deviation from the prediction indicates new physics beyond the SM. The E821 experiment at Brookhaven National Lab (BNL) reported a 3.3σ deviation from the SM prediction that is written by [1][2][3] ∆a µ = a exp µ − a SM µ = (26.1 ± 7.9) × 10 −10 .
In this work we propose a model with three scalar leptoquarks. Combination of interactions among these leptoquarks and the SM fermions can explain muon g − 2, b → s anomaly and radiative neutrino mass generation at the same time. For neutrino mass generation, we adopt the mechanism in ref. [38] realized by two leptoquarks, and one SU (2) L doublet leptoquark is added to improve b → s ¯ explanation and to realize sizable muon g − 2. We then analyze our model to find a solution to explain these issues taking into account possible flavor constraints such as lepton flavor violating (LFV) decays of charged leptons ( → γ) and mixing between meson and anti-meson (M -M mixing). This paper is organized as follows. In Sec. II, we review our model and show relevant formulas for neutrino mass, Wilson coefficients for b → s ¯ decay, LFV branching ratios(BRs), The new field contents and their charges are shown in Table I. The relevant Lagrangian for the interactions of η and ∆ with fermions and the Higgs field is given by H is the SM Higgs field that develops a nonzero vacuum expectation value (VEV), which Although λ 0 induces the mixing between η 2/3 and η 2/3 , we neglect this term for simplicity. We work in the basis where all the coefficients are real and positive for simplicity. 2 The scalar fields can be parameterized as where the subscript of the fields represents the electric charge, v ≈ 246 GeV, and w ± and z are, respectively, the Nambu-Goldstone bosons, which are absorbed by the longitudinal components of the W and Z bosons. Due to the µ term in Eq. (II.2), the charged components with 1/3 and 2/3 electric charges mix each other, and their mixing matrices and mass eigenstates are defined as follows: where their mass eigenstates are denoted as m A i and m B i , respectively. Then whole the interaction in terms of the mass eigenstates can be written by In the following we summarize various phenomenological formulae derived from these inter-

actions.
A. Neutrino mixing In our model active neutrino mass can be generated at one loop level via interactions among leptons, quarks and leptoquarks where the mechanism is the same as the one in ref. [38]. Calculating one-loop diagram the active neutrino mass matrix m ν is given by where we assume m d i << m A 1 , m B 1 . (m ν ) ab is diagonalized by the Pontecorvo-Maki-Nakagawa-Sakata mixing matrix V MNS (PMNS) [54] as (m ν ) ab = (V MNS D ν V T MNS ) ab with D ν ≡ (m ν 1 , m ν 2 , m ν 3 ). Then, we rewrite g in terms of observables [55] and several input parameters as follows [38,56]: where A is an arbitrary three by three antisymmetric matrix with complex values, and perturbative limit g √ 4π has to be satisfied.
In our model b → s ¯ can be induced by leptoquark exchanging process at tree level. The effective Hamiltonian to estimate the b → s ¯ is given by is a scale factor from the SM effective Hamiltonian. C LQ,µ 10 also contributes to B s → µ + µ − whose experimental data is consistent with the SM prediction. In our numerical analysis we compare our results with best fit values of these Wilson coefficients obtained from global fit [27] that includes all the B meson decay data. The best fit values for new physics contributions are Among them cases, only the cases with C µ 9 , C µ 9 = −C µ 10 and |C 9 | = |C 10 | improve fit more than 5σ compared to the SM case. On the other hand C µ 9 = C µ 10 case less improve the fit.

C. LFVs and muon g − 2
Yukawa interactions associated with leptoquarks induce LFV decay of → γ at oneloop level. We then estimate the branching rations calculating relevant one-loop diagrams propagating leptoquarks. Branching ratio of a → b γ is given by where m a(b) , (a(b) = 1, 2, 3) is the mass for the initial(final) eigenstate of charged-lepton; 1 ≡ e, 2 ≡ µ, 3 ≡ τ , (C 21 , C 31 , C 32 ) = (1, 0.1784, 0.1736). a R is given by where m t is the mass of top quark, and a L is obtained by m a ↔ m b in the last line. Clearly, the first term is dominant. The current experimental upper bounds are given by [59,60] B(µ → eγ) This estimation can be achieved by effective operators [62], and each of their formulae is given by where F M (m 1 , m 2 ) is given by The current constraints are found as follows [3,63]:

III. NUMERICAL ANALYSIS
In this section, we carry out numerical analysis scanning free parameters of the model searching for solutions to fit neutrino oscillation data and to explain B anomalies and muon g − 2, taking into account all the flavor constraints discussed above. Firstly, we assume almost degenerate leptoquark masses M LQ ≡ m δ4 = m η 2,5 = m A 1,2 = m B 2 for simplicity; we only take m B 2 = M LQ ± 10 GeV to avoid too small R i in Eq. (II.6). In fact degeneracy of masses for the components of η ( ) and ∆ is motivated to suppress the oblique S-and T -parameters 3 . We then scan our parameters with the following ranges: Here we choose smaller values of Yukawa coupling associated with first generation to avoid strong flavor constraints. The Yukawa couplings g ij are obtained from Eq. (II.7) to fit neutrino oscillation data. Note that we select smaller values for Yukawa coupling f ij so that g ij can be sizable to fit neutrino data. As a result C LQ, 9,10 is suppressed and we do not show these values explicitly. Then we impose constraints from BR( → γ) and M -M mixing discussed in previous section, and obtain the possible values of Wilson coefficient C LQ,µ 9,10 and ∆a µ for allowed parameter sets.
In Fig. 1, we show our results for C LQ,µ 9 and C LQ,µ 10 values where we impose ∆a µ > 0 in addition to the other constraints. We find that many parameter sets give C LQ,µ 9 ∼ ±C LQ,µ

10
. Note that we have less points for same sign case C LQ,µ 9 ∼ C LQ,µ 10 compared to opposite sign case since same sign contribution comes from Yukawa coupling g that is more constrained by neutrino data. Although some tunings of parameters are required, we can obtain the case of |C LQ,µ 10 | = |C LQ,µ 9 | due to cancellation between diagrams. Thus we have more freedom to obtain {C 9 , C 10 } values thanks to the contributions from two different leptoquarks. In principle we can obtain all the best fit values in ref. [27] summarized by Eqs. (II.11)-(II. 15) and that for C LQ,µ 9 = −C LQ,µ 10 case can be most easily realized. 3 We thus do not explicitly consider these oblique parameters making them to be small by degenerate masses. In Fig. 2, we show the values of C LQ,µ 9(10) and ∆a µ that is compared with the new muon g − 2 result Eq. (I.3) in the left(right) figure. As we see, it is possible to obtain the observed value of muon g − 2 and C LQ,µ 9 ∼ −1 at the same time; we also find both sign of C LQ,µ 10 is achieved. Note that sizable contribution is obtained due to enhancement by m t in Eq. (II.17).
Therefore we have parameter sets which explain both b → s ¯ and muon g − 2 anomalies.
Note that we don't find clear dependence on LQ masses and any values in [1.0, 5.0] TeV are allowed.

IV. CONCLUSIONS
We have discussed a model with three leptoquarks {∆, η, η } to explain b → s ¯ anomalies, muon g − 2 and neutrino masses, motivated by recent experimental results. Active neutrino mass is generated at one-loop level via interactions among ∆, η and SM fermions.
The leptoquarks ∆ and η contribute to Wilson coefficient C 9,10 at tree level. Interestingly combination of two leptoquark contributions can provide a case of |C LQ,µ 9 | = |C LQ,µ 10 | in contrast to one leptquark scenario. In addition all three leptoquarks can provide contribution to muon g − 2, and induce LFV decay processes and M -M mixing which are taken into account as constraints. We then have carried out numerical analysis to explore C LQ,µ 9,10 and muon g − 2 values when neutrino data and relevant flavor constraints are accommodated.
Our finding are as follows: 1. We can obtain sizable muon g − 2 from η loop due to m t enhancement. Then we can Thus combination of leptoquarks is attractive scenario to explain b → s anomaly, muon g − 2 and neutrino mass at the same time.