Study Standard Model and Majorana Neutrino Contributions to $B^{+} \to K^{(*)\pm}\mu^+\mu^{\mp}$

Lepton number violation processes can be induced by the Majorana neutrino exchange, which provide evidence for the Majorana nature of neutrinos. In addition to the natural explanation of the small neutrino masses, Type-I seesaw mechanism predicts the existence of Majorana neutrinos. The aim of this work is to study the B meson rare decays $B^{+} \to K^{(*)+}\mu^+\mu^-$ in the standard model and its extensions, and then to investigate the same-sign decay processes $B^{+}\to K^{(*)-}\mu^{+}\mu^+$. The corresponding dilepton invariant mass distributions are predicted. It is found that the dilepton angular distributions illustrate the properties of new interactions induced by the Majorana neutrinos.


I. INTRODUCTION
The discovery of the neutrino oscillation [1][2][3][4][5], confirming the existence of the massive neutrinos, well motivates the search for new physics beyond the SM. Along with the neutrino mass puzzle, it is a crucial question to explore the nature of the neutrino. If neutrinos are Dirac particles, lepton number is conserved. Otherwise, lepton number violating (LNV) processes, which are forbidden by SM, can be induced via a Majorana neutrino exchange.
Therefore the LNV process will be a promising signal for new physics beyond the SM. The aim of this paper is to study the B meson rare decays B + → K ( * )+ µ + µ − in the SM and its extensions, and then to investigate the LNV process B + → K ( * )− µ + µ + .
Type-I seesaw mechanism is one of the most natural ways to generate tiny neutrino masses among various new physics models. In this model, the right-handed SU(2) L × U(1) Y singlet neutrinos N R are introduced to extend the SM. Apart from Dirac mass M D , the right-handed neutrino singlets with Majorana mass matrix M R are allowed with the gauge invariance. As a result, the effective mass matrix for the light neutrinos can be expressed In terms of the neutrino mass eigenstate, the gauge interaction for the charged current has the formula of where P L = (1 − γ 5 )/2, ν m (m = 1, 2, 3) and N m ′ (m ′ = 4, · · · , 3 + n) are the mass eigenstates, is the mixing matrix element between the lepton flavor and light (heavy) neutrinos. Moreover, lots of proposals have been made to search for heavier Majorana neutrinos at e − e − (e + e − ), eγ, pp (pp) collider experiments [30][31][32][33] and also in top quark and W boson rare decays [24,34]. Recently, the differential branching ratios of B + → K + µ + µ − and B + → K * + µ + µ − decays have been reported by the LHCb collaboration with the integrated luminosity of 3 fb −1 and the integrated branching fractions are (4.29 ± 0.07(stat) ± 0.21(syst)) × 10 −7 and (9.24 ± 0.93(stat)±0.67(syst))×10 −7 [35], respectively. This is the most precise measurement so far.
Along with the development of the experiments, theoretical studies on the B → K ( * ) ℓ + ℓ − have been reported both in the SM [36][37][38][39][40][41][42] and new physics models [43][44][45]. These SM predictions for the branching ratios are comparable with the LHCb data, however, new physics contributions can not be excluded. In this paper, we first study the opposite-sign B meson dileptonic rare decays B + → K ( * )+ µ + µ − both in the SM and type-I seesaw model, then the the contributions from Majorana neutrino are investigated in the same-sign LNV This paper is organized as the follows. In Sec.2, the theoretical framework is introduced with the formulas of B meson rare decays B + → K ( * )+ ℓ + ℓ − in the SM and mediated by Majorana neutrino, as well as the same-sign LNV decays B + → K ( * )− ℓ + ℓ + . In Sec.3, we give the numerical results on the branching ratios, the dilepton invariant mass distributions and angular distributions of B + → K ( * )± µ + µ ∓ . The excluded regions of the Majorana neutrino mass and the mixing matrix element are given with the fitting results. Furthermore, the dilepton invariant mass distributions and the dilepton angular distributions of LNV processes are studied. Finally, we give a brief summary. We first study the B meson rare decays B + (p B ) → K ( * )+ (p K ( * ) )ℓ + (p 1 )ℓ − (p 2 ). Fig.1(a) is the Feynman diagram for B + → K ( * )+ ℓ + ℓ − in the SM. These processes are induced by the flavor changing neutral current b → sℓ + ℓ − , which can be described through the effective Hamiltonian [5]. The decay amplitude of b → sℓ + ℓ − can be written as [37,46]

II. THEORETICAL FRAMEWORK
Here, α is the fine-structure constant and V q 1 q 2 is the CKM matrix element. All the have the same analytic expressions as those used in the b → s transition processes [47], while the C ef f 9 can be found in [48] with the next-to-leading order approximation. Generally, exclusive decays B → K ( * ) ℓ + ℓ − are described by matrix  . elements of the quark operators over meson states, and the matrix elements are ulteriorly be parameterized in terms of B → K ( * ) form factors [37]. For the pseudoscalar K meson, B → K form factors are defined as follows, Here, f B→K are vector, scalar and tensor form factors of B → K transition, respectively. For the vector meson K * with four-momenta p K * and polarization vector ǫ µ , the semileptonic form factors of V − A current and the penguin form factors can be defined as with Using the above definition of the form factors, the decay amplitudes for B + → K ( * )+ ℓ + ℓ − corresponding to Fig.1(a) can be obtained, with for the pseudoscalar meson K, and for the vector meson K * . Here, The contributions from the light and heavy neutrinos are shown in the Feynman diagrams of Fig.1(b) and Fig.1(c), respectively. For simplification, we suppose that only one heavy Majorana neutrino exists in type-I seesaw model. The decay amplitudes for B + → K ( * )+ ℓ + ℓ − corresponding to Fig.1(b) and Fig.1(c) can be expressed as where meson. p ν and p N stand for the four-momentum of the light neutrino ν and the heavy one N, respectively. Γ N ≈ 2 ℓ |V ℓN | 2 (m N /m τ ) 5 × Γ τ represents the total decay width of the Majorana neutrino with V ℓN denoting the mixing matrix element between ℓ and N [19]. The decay rates can be written as with the total decay amplitude M = M a + M b + M c , where p B 1 ( p * 2 ) and dΩ B 1 (dΩ * 2 ) denote the 3-momentum and solid angle of charged lepton ℓ + (ℓ − ) in the rest frame of B meson (ℓ − K ( * ) system), respectively. It is found that the contribution from Fig.1(b) and interference terms is about five orders less than that from SM and can be neglected.
The same-sign ∆L = 2 LNV processes B + (p B ) → K ( * )− (p K ( * ) )ℓ + (p 1 )ℓ + (p 2 ) is more sensitive to the new physics models. These decay channels may occur via Majorana neutrino exchange, especially provide evident signal if the mediate Majorana neutrino is on-shell. The dominant contribution is from the Feynman diagram of Fig.2(a), while the contribution from Fig.2(b) is small enough to be neglected as concluded in [21,22]. The corresponding decay Then branching ratios of these same-sign charged dilepton decays can be readily obtained by the same way as in eq.(13).

III. NUMERICAL ANALYSIS
The B → K form factors are parameterized by the following formulae [49], where m B * s (1 − ) = 5.413 GeV is the mass of the B * s (1 − ). The values of other parameters r 1(2) and m 2 fit are collected in Table II. The parameters for B → K * form factors are obtained from the calculation of AdS/QCD at low-to-intermediate q 2 and the lattice data at high q 2 [41]. The seven independent form factors can be expressed as the formula of where F stands for A 0 , A 1 , A 2 , T 1 , T 2 , T 3 , V . The corresponding values of F (0), a, b are listed in Table III.  [49].
The SM dilepton invariant mass distributions dB(B + → K ( * )+ µ + µ − )/dq 2 are shown in Fig.3(a). It leads to different distributions for dB(B + → K + µ + µ − )/dq 2 and dB(B + → K * + µ + µ − )/dq 2 because the form factors and the amplitudes are different for scalar and vector mesons. The integrated branching ratios are where the dominant uncertainty comes from the CKM matrix elements and the renormalization scale variation. Experimentally, the most precise measurements on the differential branching fractions of B + → K ( * )+ µ + µ − have been performed using a data set with 3 fb −1 of integrated luminosity collected by the LHCb detector [35].
It turns out our SM predictions for B + → K ( * )+ µ + µ − are roughly consistent with the most recent LHCb data within the range of experimental and theoretical errors, and comparable with the other SM calculation results [36,37,40,46]. The angular distributions of the opposite-sign leptons are plotted in Fig.3

53],
B(B + → K * − µ + µ + ) < 5.9 × 10 −7 at 90% C.L. , which can be used to constrain the parameter space between the mixing matrix element As shown in this figure, the LNV rare decay channel B + → π − µ + µ + provides more rigorous constraint than the LNV B → K channel because the B + → π − µ + µ + process is much less suppressed by CKM factors than B + → K ( * )− µ + µ + processes. However, the bounds for the new physics parameters are loosen with the B + → K * − µ + µ + process until more precise experimental data are released.
The best fitting values of |V µN | 2 and m N obtained from our previous work [22] on B + → π − µ + µ + agree with the LHCb upper limit [10]. As a result, we listed the branching ratios for LNV processes B + → K ( * )− µ + µ + with the best fitting values of |V µN | 2 and m N in Table V. There are three cases corresponding to the form factors with the heavy quark symmetry and lattice QCD method (HQS+LQCD), perturbative QCD method (PQCD) and light cone QCD sum rule method (LCSR) [22]. Here, we choose three typical values of m N in each case to estimate the branching ratios. It shows that the branching ratios of LNV processes B + → K ( * )− µ + µ + induced by Majorana neutrino exchange are in agreement with TABLE V. Majorana neutrino contributions to the branching ratios of B + → K ( * )− µ + µ + . The m N and |V µN | 2 are the best fits corresponding to heavy quark symmetry and lattice QCD method (HQS+LQCD), perturbative QCD method (PQCD) and light cone QCD sum rule method (LCSR) referred as [22]. processes B + → K ( * )− µ + µ + which are a few orders less than the SM contributions. Once the LNV processes are observed, the differential distributions are necessary to be studied. In ∆L = 2 semileptonic decay processes have been elaborately calculated for the hints of new physics. We study the B + → K ( * )+ µ + µ − both in SM and type-I seesaw model. The decay branching ratios are roughly consistent with the experimental measurements. We also investigate the LNV decays B + → K ( * )− µ + µ + . Parameter constraints on m N and |V µN | 2 , obtained with the experimental upper limits for B + → K ( * )− µ + µ + processes, is less strict than the constraints from the B → π process. Thus, utilizing the best fitting for Majorana neutrino mass m N and the mixing matrix elements |V µN | 2 obtained in our previous work, we give the branching ratios of the same-sign charged dilepton processes B + → K ( * )− µ + µ +