$B^*_{s,d} \to \mu^+ \mu^-$ and its impact on $B_{s,d} \to \mu^+ \mu^-$

The decay of $B^*_{s,d} \to \mu^+ \mu^-$~ and its impact on $B_{s,d} \to \mu^+ \mu^-$ is studied here. The $ \mu^+ \mu^-$ decay widths of vector mesons $B^*_{s,d}$ are about a factor of 700 larger than the corresponding scalar mesons $B_{s,d}$. The obtained ratio of the branching fractions $Br({B_{s,d}^*\to \mu^+\mu^-})/{Br({B_{s,d}\to\mu^+\mu^-})}$ is about $\frac{0.3 \times {\rm eV}}{{\Gamma(B^*_{s,d} \to B_{s,d} \gamma)}}$. At the same time, the hadronic contribution $B_{s,d} \to B^*_{s,d} \gamma \to \mu^+ \mu^-$ is estimated too. The relative increase of the amplitude of $B_{s,d}\to \mu^+\mu^-$ is about $(0.01\pm 0.006) \sqrt{\frac{{\Gamma(B^*_{s,d} \to B_{s,d} \gamma)}}{{100~ {\rm eV}}}}$. If we choose $\Gamma(B^*_{s,d} \to B_{s,d} \gamma)=2~$eV, the branching fractions of the vector mesons to lepton pair are $(6.2 \pm 0.6) \times 10^{-10}$ and $(1.7 \pm 0.2) \times 10^{-11}$ for $B^*_{s}$ and $B^*_{d}$ respectively. If we choose $\Gamma(B^*_{s,d} \to B_{s,d} \gamma)=200~$eV, the updated branching fractions of the scalar mesons to muon pair are $(3.78 \pm 0.25)\times 10^{-9}$ and $(1.09 \pm 0.09)\times 10^{-10}$ for $B_{s}$ and $B_{d}$ respectively. Further studies on $B^*_{s,d}$ are usefully here, including dielectron decay, two-body decay with $J/\psi$, and so on.


Introduction
The leptonic decays of the B s,d mesons play an important role in the standard model (SM) and the new physics (NP) [1,2]. They are highly suppressed in the SM since the flavor changing neutral current decays are generated through W-box and Z-penguin diagrams. Furthermore, the branching fractions of leptonic decays of scalar meson undergo an additional helicity suppression factor by m 2 µ /M 2 S , where m µ and M S denote masses of the muon lepton and the scalar meson, respectively. The suppression factor can be removed in some NP models, such as the two-Higgs doublet models [3], the minimal supersymmetric standard model (MSSM) [4], the next minimal supersymmetric standard model (NMSSM) [5], the dark matter [6], the universal extra dimensional model [7], lepton universality violation model [8], the four generations fermion [9], and so on [10]. The branching fractions of B s,d → µ + µ − measured by the CMS and LHCb Collaborations [2] and predicted within the SM [1] with NNLO QCD [11] and NLO EW [12] corrections included are collected in Table 1.
Since the experimental branching fractions of B s,d → µ + µ − are measured from the dimuon distributions by the CMS and LHCb Collaborations [2], the process B * s,d → µ + µ − will enhance the dimuon distributions for the mass splitting are about 45 MeV between B s,d and B * s,d . In another hand, the hadronic contribution B s,d → B * s,d γ → µ + µ − are missed in the theoretical prediction [1]. So we study B * s,d → µ + µ − and its impact on B s,d → µ + µ − within SM here. The process B s → B * s γ → µ + µ − γ was considered in Ref. [13]. Recently, B * s,d → µ + µ − are considered in Ref. [14,15]. And hadronic contribution from charmonium in B → K ( * ) ℓ + ℓ − and B → X s γ had been studied in Refs. [16,17].  [2] and predicted within the SM [1] with NNLO QCD [11] and NLO EW [12] corrections included.
EX [2] SM [1] Deviations Within the SM, effective Lagrangian related with bs → µ + µ − is given in Ref. [18,19] 4π 2 , and the operators O 7,9,10 read as  [15]. The superscript γ, V , and A denote the contributions from photon, vector current, and axial vector current, respectively. The relations between the quark level operators and the meson are described as where the three decay constants f B s , f B * s and f T B * s depend on the renormalization scale. Their relations have been studied in Ref. [20] in the heavy-quark limit. Ignoring the mass difference between B s and B * s and the high order QCD corrections, we have where The helicity suppression factor m 2 µ /m 2 M in the decay width is removed in the vector meson decay. Then we can get the decay widths of The ratio of decay width is about 700 for B ( * ) s and B ( * ) d both.

The impact of
The Feynman diagrams are given in Fig. 1 We can simplify the matrix element < B * s |q(p)σ µν q(p)|B s > with the procedure in Refs. [23,24] 1 , I is related with the wave functions of B * s and B s [23], and it is I =< B s | j 0 (p γ r )|B * s >∼ 1 in the non-relativistic limit [25]. We can rewrite Eq. (3.1) with Where the dimensionless vector-scalar-photon coupling constant g B s B * s γ is related with the magnetic moments of b and s quarks. And the phase factor i is consistent with the amplitude of γ * → V P in Ref. [26].
There are ultraviolet (UV) logarithmical divergences in the evaluation of loop integrals. Then we introduce a cut off regularization scheme for the UV divergence integral where i , j = B * s , γ or µ, and q i corresponds the momentum of i in loop. Λ ≪ M W for the amplitude is UV finite when W boson are involved. Since the hadronic contribution will be suppressed when q 2 j − m j ≫ Λ QC D , Λ is about several Λ QC D . The cut off regularization scheme is similar Pauli-Villars regularization scheme but acts on two propagators. The Pauli-Villars regularization scheme of the UV divergence integral is the same with the form factor F which is introduced in the B s B * s γ vertex in Ref. [27] But here it acts on the UV divergence term only and the two propagators. Then the soft contribution will be maintained in our calculation. The amplitude from B s → B * s γ * → µ + µ − can be written as m µ reappears in the amplitude of leptonic decay of scalar meson. The factor R(Λ) serves as a function of the high energy cut is shown in Fig.2. More information about the factor R(Λ) are given in the Appendix.
Only the C 10 term is taken into account in the calculation of the NNLO QCD [11] and NLO EW [12] corrections of B s,d → µ + µ − within the SM [1]. But the contribution from B * s is missed in the previous calculation, which is also related with C 7 and C 9 terms.
Compared with Eq.(2.9), the previous amplitude is added a factor F , We can estimate g B s B * s γ in several ways, including the heavy-quark and chiral effective theories [28,29] with the radiative and pion transition widthes of D * + , light cone QCD sum rules [30,31], and the radiative M1 decay widths of B * s → B s γ from potential model [25,32]. The radiative M1 decay width of B * s → B s γ is The predicted M1 widths are 0.15−400 eV and 10−300 eV for B * s → B s γ and B * d → B d γ, respectively [22,24,25,32,33].

Numerical Result
The parameters in the numerical calculation are chosen as [34] Λ = 1.2 GeV, The branch fraction of B * s,d weak decay is much less than the M1 decay, d γ). We can get the ratio , . (4. 2) The main uncertainty resulting from the value of f * B s,d . 3) The main uncertainty resulting from the value of Λ. Then the new predictions of Γ(B s,d → µ + µ − ) are If Γ(B * s,d → B s,d γ) = 200 eV, this factor will increase the decay width Γ(B s,d → µ + µ − ) by a factor (3.3 ± 1.7)%, which is about a factor of 10 larger than the neglect NLO EW correction factor 0.3% at the decay width in Ref. [1]. And the corresponded g B s,d B * s,d γ = −1.5, about a factor of 15 larger than the e q e = −1/3 4πα em = −0. 10  The scalar functions B 0 and C 0 are given in Ref. [35][36][37]. As a numerical fit between 0.5 − 2 GeV, we can get R(Λ) = 0.022 + 0.062 × l n( Λ + m B s m B s ). (6.2)