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

Abstract

This study investigates $B^*_{s,d}\rightarrow \mu ^+\mu ^-$ in the dimuon distributions and the hadronic contribution $B_{s,d}\rightarrow B^*_{s,d}\gamma \rightarrow \mu ^+\mu ^-$. The $\mu ^+\mu ^-$ decay widths of the vector mesons $B^*_{s,d}$ are approximately a factor of 700 larger than the corresponding scalar mesons $B_{s,d}$. The ratio of the branching fractions obtained, $\frac{Br({B_{s,d}^*\rightarrow \mu ^+\mu ^-})}{{Br({B_{s,d}\rightarrow \mu ^+\mu ^-})}}$, is approximately $\frac{0.3\times \mathrm{eV}}{{\Gamma (B^*_{s,d}\rightarrow B_{s,d}\gamma )}}$. The hadronic contribution $B_{s,d}\rightarrow B^*_{s,d}\gamma \rightarrow \mu ^+\mu ^-$ is also estimated. The relative increase in the $B_{s,d}\rightarrow \mu ^+\mu ^-$ amplitude is approximately $(0.01\pm 0.006) \sqrt{\frac{{\Gamma (B^*_{s,d}\rightarrow B_{s,d} \gamma )}}{{100~\mathrm{eV}}}}$. If we select $\Gamma (B^*_{s,d}\rightarrow B_{s,d}\gamma )=2~$eV, then the branching fractions of the vector mesons to the lepton pair are $5.3\times 10^{-10}$ and $1.6\times 10^{-11}$ for $B^*_{s}$ and $B^*_{d}$, respectively. If we select $\Gamma (B^*_{s,d}\rightarrow B_{s,d}\gamma )=200~$eV, then the updated branching fractions of the scalar mesons to the 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. If we select the recent predicted M1 widths $\Gamma (B^*_{s,d}\rightarrow B_{s,d}\gamma )=313, 1230$ eV (arXiv:1607.02169), then the updated branching fractions are $(3.8\,\pm \,0.3)\times 10^{-9}$ and $(1.2\,\pm \,0.1)\times 10^{-10}$ for $B_{s}\rightarrow \mu ^+\mu ^-$ and $B_{d}\rightarrow \mu ^+\mu ^-$, respectively. Further studies on $B^*_{s,d}$, including those on dielectron decay and two-body decay with $J/\psi$, should be conducted.

1 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]. The leptonic decays are highly suppressed in the SM because flavor-changing neutral current decays are generated through W-box and Z-penguin diagrams. Furthermore, the branching fractions of the leptonic decays of scalar meson go through an additional helicity suppression factor by $m_\mu ^2/M_{S}^2$, where $m_\mu$ and $M_{S}$ denote the masses of the muon lepton and the scalar meson, respectively. The suppression can be removed in several NP models, such as the two-Higgs-doublet models [3], the minimal supersymmetric standard model [4], the next minimal supersymmetric standard model [5], the dark matter [6], the universal extra dimensional model [7], the lepton universality violation model [8], the fourth generation of fermions [9], and so on [10]. The branching fractions of $B_{s,d} \rightarrow \mu ^+ \mu ^-$ measured by the CMS and LHCb Collaborations [2], and predicted within the SM [1] with NNLO QCD [11] and NLO EW [12] corrections are presented in Table 1.

On one hand, the experimental branching fractions of $B_{s,d} \rightarrow \mu ^+ \mu ^-$ are measured from the dimuon distributions by the CMS and LHCb Collaborations [2]. Thus, the process $B^*_{s,d} \rightarrow \mu ^+ \mu ^-$  will enhance the dimuon distributions for mass splitting between $B_{s,d}$ and $B^*_{s,d}$ at approximately 45 MeV. On the other hand, the hadronic contribution $B_{s,d} \rightarrow B^*_{s,d} \gamma \rightarrow \mu ^+ \mu ^-$ is missing in the theoretical prediction [1]. Therefore, this study focuses on $B^*_{s,d} \rightarrow \mu ^+ \mu ^-$  and its impact on $B_{s,d} \rightarrow \mu ^+ \mu ^-$ within SM. The $B_s \rightarrow B_s^* \gamma \rightarrow \mu ^+ \mu ^- \gamma$ process was considered in Ref. [13]. $B^*_{s,d} \rightarrow \mu ^+ \mu ^-$ was recently considered in Refs. [14, 15]. Moreover, Refs. [16, 17]. also investigated the hadronic contribution of charmonium in $B \rightarrow K^{(*)} \ell ^{+} \ell ^{-}$ and $B \rightarrow X_s \gamma$.

Table 1 The branching fractions of $B_{s,d} \rightarrow \mu ^+ \mu ^-$ measured by the CMS and LHCb Collaborations [2] and predicted within the SM [1] with NNLO QCD [11] and NLO EW [12] corrections included

2 The Decay of $B_s^* (B_d^*) \rightarrow \mu ^+\mu ^-$

An effective Lagrangian related to $b \bar{s} \rightarrow \mu ^+ \mu ^-$ within the SM is given in Refs. [18–20]

$$\begin{aligned} \mathscr {L} = \mathscr {N}\biggl [C_7 ^\mathrm{eff}(\mu _f) \mathscr {O}^\gamma _7 + C_9^\mathrm{eff} (\mu _f) \mathscr {O}^V_9+ C_{10} (\mu _f)\mathscr {O}^A_{10}\biggl ],\nonumber \\ \end{aligned}$$

(1)

where $\mathscr {N}=\frac{G_F}{\sqrt{2}} V_{tb} V^{*}_{ts}\frac{e^2}{4\pi ^2}$, and the operators $\mathscr {O}_{7,9,10}$ read as follows:

$$\begin{aligned} \mathscr {O}^\gamma _7= & {} - \frac{2im_b (p_\mu ^\nu +p_{\bar{\mu }}^\nu )}{(p_\mu +p_{\bar{\mu }})^2} (\bar{s}\sigma _{\rho \nu }P_R b)(\bar{\mu }\gamma ^{\rho } \mu ), \end{aligned}$$

(2)

$$\begin{aligned} \mathscr {O}^V_9= & {} (\bar{s}\gamma _\rho P_Lb)(\bar{\mu }\gamma ^{\rho } \mu ), \end{aligned}$$

(3)

$$\begin{aligned} \mathscr {O}^A_{10}= & {} (\bar{s}\gamma _\rho P_Lb)(\bar{\mu }\gamma ^{\rho }\gamma _5 \mu ), \end{aligned}$$

(4)

where $P_L=(1-\gamma _5)/2$ , $P_R=(1+\gamma _5)/2$. The Wilson coefficients are $C^\mathrm{eff}_{7, 9, 10}(\mu _f)=(-0.316, 4.403-0.47i, -4.493)$ at $\mu _f=m_b=4.5$ GeV [15]. The superscripts $\gamma$, V, and A denote the contributions from photon, vector current, and axial vector current, respectively.

The relationships between the quark level operators and the meson are described as follows:

$$\begin{aligned}&\langle 0|\bar{s}\gamma ^\mu b|B_s^*(q,\varepsilon )\rangle = m_{B_s^*}\,f_{B_s^*}\,\varepsilon ^\mu , \end{aligned}$$

(5)

$$\begin{aligned}&\langle 0|\bar{s}\sigma ^{\mu \nu } b|B_s^*(q,\varepsilon )\rangle = -i\,f_{B_s^*}^T(q^\mu \varepsilon ^\nu -\varepsilon ^\mu q^\nu ), \end{aligned}$$

(6)

$$\begin{aligned}&\langle 0|\bar{s}\gamma ^\mu \gamma _5 b|B_s(q)\rangle = if_{B_s}\,q^\mu , \end{aligned}$$

(7)

where the three decay constants $f_{B_s},~ f_{B_s^*}$, and $f_{B_s^*}^T$ depend on the renormalization scale, whose relationships have been investigated in the heavy-quark limit of the Ref.  [21]. Ignoring the mass difference between $B_s$ and $B_s^*$ and the high-order QCD corrections, we derive the following expression:

$$\begin{aligned} f_{B_s^*}=f_{B_s^*}^T=\,f_{B_s}. \end{aligned}$$

(8)

Afterward, the $B_s^* (B_s) \rightarrow \mu ^+\mu ^-$  amplitudes are expressed as follows  [13]:

$$\begin{aligned} \mathscr {M}({B_s^*\rightarrow \mu ^+\mu ^-})= & {} f_{B_s^*}\frac{\mathscr {N}}{2} m_{B_s^*}\bar{\mu }\slash \!\!\!{\varepsilon }\Big [C_V^\mathrm{eff}+C_{10}\gamma _5 \Big ] \mu , \nonumber \\ \mathscr {M}({B_s\rightarrow \mu ^+\mu ^-})= & {} if_{B_s}\mathscr {N}C_{10} m_\mu \bar{\mu }\gamma _5 \mu , \end{aligned}$$

(9)

where

$$\begin{aligned} C_V^\mathrm{eff}=C_9^\mathrm{eff}+2\frac{m_b}{m_{B_s^*}} C_7^\mathrm{eff}. \end{aligned}$$

(10)

The helicity suppression factor $m_\mu ^2/m_M^2$ in the decay width is removed in the vector meson decay. Then the $B_s^* (B_s) \rightarrow \mu ^+\mu ^-$  decay widths are obtained:

$$\begin{aligned} \Gamma ({B_s^* \rightarrow \mu ^+\mu ^-})= & {} \frac{G_f^2\alpha _{em}^2}{96\pi ^3}\left| V_{tb}V_{ts}^*\right| ^2\Big (\left| C_{10}\right| ^2 +\left| C_{V}^\mathrm{eff}\right| ^2) \nonumber \\&\times m_{B_s^*}^3f_{B_s^*}^2\Big (1+\mathcal{{O}}(m_\mu ^2/m_{B_s}^2)\Big ), \nonumber \\ \Gamma ({B_s \rightarrow \mu ^+\mu ^-})= & {} \frac{G_f^2\alpha _{em}^2}{16\pi ^3}\left| V_{tb}V_{ts}^*\right| ^2\left| C_{10}\right| ^2\nonumber \\&\times m_\mu ^2m_{B_s}f_{B_s}^2\Big (1+\mathcal{{O}}(m_\mu ^2/m_{B_s}^2)\Big ) ,\nonumber \\ \frac{ \Gamma ({B_s^* \rightarrow \mu ^+\mu ^-})}{\Gamma ({B_s \rightarrow \mu ^+\mu ^-})}= & {} \frac{\left( \left| C_{10}\right| ^2 +\left| C_{V}^\mathrm{eff}\right| ^2 \right) m_{B_s^*}^3f_{B_s^*}^2}{6\left| C_{10}\right| ^2 m_\mu ^2m_{B_s}f_{B_s}^2}\nonumber \\&\times \Big (1 + \mathcal{{O}}(m_\mu ^2/m_{B_s}^2)\Big ). \end{aligned}$$

(11)

The decay width ratio is approximately 700 for $B_s^{(*)}$ and $B_d^{(*)}$ both.

Fig. 1

Feynman diagrams of $B_{s,d} \rightarrow B^*_{s,d} \gamma \rightarrow \mu ^+ \mu ^-$

3 The impact of $B_s^* (B_d^*) \rightarrow \mu ^+\mu ^-$ on $B_s (B_d) \rightarrow B_s^* (B_d^*) \gamma \rightarrow \mu ^+\mu ^-$

Furthermore, $B^*_{s,d}$ will impact on the $B_{s,d}$  leptonic decay through the loop contribution  $B_{s,d} \rightarrow B^*_{s,d} \gamma \rightarrow \mu ^+ \mu ^-$. The Feynman diagrams are shown in Fig. 1. This calculation is a part of EM corrections to $B_{s,d} \rightarrow \mu ^+ \mu ^-$. The NLO EW corrections of $B_{s,d} \rightarrow \mu ^+ \mu ^-$ within the SM have been calculated in Refs. [1, 12]. The hadronic contribution of $B_{s,d} \rightarrow B^*_{s,d} \gamma _\mathrm{soft} \rightarrow \mu ^+ \mu ^- +\gamma _\mathrm{soft}$ has been calculated in Ref. [13]. However, the contribution of $B_{s,d} \rightarrow B^*_{s,d} \gamma \rightarrow \mu ^+ \mu ^-$ is missing in the previous calculation. This calculation is incomplete. For instance, there is double counting between the NLO EW corrections $B_{s} \rightarrow b+ s + \gamma \rightarrow \mu ^+ \mu ^-$ [12] and this calculation $B_{s} \rightarrow B^*_{s} \gamma \rightarrow \mu ^+ \mu ^-$. We considered that the contribution of $B_{s} \rightarrow B^*_{s} \gamma \rightarrow \mu ^+ \mu ^-$ will be retained only when the $B^*_s$ is nearly on-shell. If $B^*_s$ is far away from mass shell, $B_{s} \rightarrow b+ s + \gamma \rightarrow \mu ^+ \mu ^-$ is dominant. As is well known, the propagator of hadron will be modified duo to the off-shell of hadron [22], and the Wilson coefficients $C^\mathrm{eff}_{7, 9, 10}$ will be modified too [17, 19]. Therefore, this treatment may be regarded as a crude estimate, and the error may be large in this treatment.

The vertex of $B_{s,d} \rightarrow B^*_{s,d} \gamma$ is expressed as the following operator [23, 24]:

$$\begin{aligned} \mathscr {M}_{B_s B_s^{*} \gamma }= & {} \sum _{q=s,b}<B_s^{*} \gamma | i e e_q \bar{q} (p_{\bar{q}}) \gamma _\mu q (p_q)| B_s> \\= & {} \sum _{q=s,b}\varepsilon ^{\mu }_\gamma p_\gamma ^\nu <B_s^{*} | iee_q \bar{q} (p_{\bar{q}}) \frac{i \sigma _{\mu \nu }}{2m_q} q (p_q)| B_s> . \end{aligned}$$

We can simplify the matrix element $<B_s^{*} | \bar{q} (p) \sigma _{\mu \nu } q (p)| B_s\!>$ with the procedure in Refs. [25, 26],¹

$$\begin{aligned} \mathscr {M}_{B_s B_s^{*} \gamma }= & {} \sum _{q=s,b}\frac{-e e_q}{2 m_q}\varepsilon ^{\mu }_\gamma p_\gamma ^\nu <B_s^{*} | \bar{q} (p) \sigma _{\mu \nu } q (p)| B_s> \nonumber \\= & {} i \epsilon _{\mu \nu \alpha \beta } \varepsilon ^{\mu }_\gamma p^{\nu }_\gamma \varepsilon ^{\alpha }_{B^*_s } p^{\beta }_{B^*_s }\sum _{q=s,b}\left( \frac{e e_q}{ m_q}\right) \mathcal{{I}}. \end{aligned}$$

(12)

$\mathcal {I}$ is related to the wave functions of $B^*_s$ and $B_s$ [25], which is $\mathcal{{I}}=<B_s| j_0(p_\gamma r)| B_s^{*}>\sim 1$ [27, 28]. We can rewrite Eq. (12) as follows:

$$\begin{aligned} \mathscr {M}_{B_s B_s^{*} \gamma }= & {} i \frac{g_{B_s B^*_s \gamma }}{m_{B_s^*}} \epsilon _{\mu \nu \alpha \beta } \varepsilon ^{\mu }_\gamma p^{\nu }_\gamma \varepsilon ^{\alpha }_{B^*_s }p^{\beta }_{B^*_s }. \end{aligned}$$

(13)

Here the dimensionless vector–scalar–photon coupling constant $g_{B_s B^*_s \gamma }$ is related to the magnetic moments of b and s quarks; and the phase factor i is consistent with the amplitude of $\gamma ^*\rightarrow VP$ in Ref. [29].

Ultraviolet (UV) logarithmic divergences are observed in the evaluation of loop integrals. In the NLO EW corrections of $B_{s,d} \rightarrow \mu ^+ \mu ^-$ [12], the UV divergences are canceled by the renormalization of $C_{10}$. Just as $R-$ value of hadron production in $e^+e^-$ annihilation, the hadronic contributions will return the quark contributions if the $B^*_{s}$ is far away from the mass shell. So that the UV part of loop integral will be suppressed due to avoid the double counting. We introduce a cutoff regularization scheme for the following UV divergence integral:

$$\begin{aligned}&\int \frac{\mathrm{d}^4 q}{(2\pi )^4}\frac{1}{\left( q_i^2-m_i^2\right) \left( q_j^2-m_j^2\right) } \nonumber \\&\quad \rightarrow \int \frac{\mathrm{d}^4 q}{(2\pi )^4}\left[ \frac{1}{\left( q_i^2-m_i^2\right) \left( q_j^2-m_j^2\right) }\right. \nonumber \\&\left. -\frac{1}{\left( q_i^2-(m_i+\Lambda )^2\right) \left( q_j^2-(m_j+\Lambda )^2\right) }\right] , \end{aligned}$$

(14)

where $i,j= B_s^*,~\gamma$ or $\mu$, and $q_i$ corresponds to the i momentum in the loop. $\Lambda \ll M_W$ for the amplitude is UV finite when W boson is involved. The hadronic contribution will be suppressed when $\sqrt{q_j^2}-m_j\gg \Lambda _{QCD}$, where $\Lambda$ is approximately several $\Lambda _{QCD}$. The cutoff regularization scheme is similar to Pauli–Villars regularization scheme; however, the cutoff regularization scheme acts on two propagators. The Pauli–Villars regularization scheme of the UV divergence integral is the same as the form factor $\mathcal{F}$ introduced in the $B_s B^*_s \gamma$ vertex in Ref. [30],

$$\begin{aligned} \mathcal{F} = \left( \frac{\Lambda ^2 - m_{B^*_s}^2}{\Lambda ^2-q_{B^*_s}^2}\right) , \end{aligned}$$

(15)

for

$$\begin{aligned} \frac{1}{q_{B^*_s}^2-m_{B^*_s}^2}\mathcal{F}= \frac{1}{q_{B^*_s}^2-m_{B^*_s}^2}-\frac{1}{q_{B^*_s}^2-\Lambda ^2}. \end{aligned}$$

(16)

However, the cutoff regularization scheme acts on the UV divergence term, as well as the two propagators. Then the soft contribution will be maintained in our calculation.

The amplitude from $B_s \rightarrow B^*_s \gamma \rightarrow \mu ^+ \mu ^-$ can be written as follows:

$$\begin{aligned}&\mathscr {M}({B_s\rightarrow B^*_s \gamma \rightarrow \mu ^+\mu ^-})\nonumber \\&\quad =i e \mathscr {N} g_{B_s B^*_s \gamma } R(\Lambda ) C_V^\mathrm{eff}f_{B_s^*}m_\mu \bar{\mu }\gamma _5 \mu , \end{aligned}$$

(17)

where $C_V^\mathrm{eff}=C_9^\mathrm{eff}+2m_b/m_{B_s^*} C_7^\mathrm{eff}$ and considered as a constant. $m_\mu$ reappears in the amplitude of the leptonic decay of the scalar mesons. The $R(\Lambda )$ factor serves as a function of the high energy cut, as shown in Fig. 2. Detailed information on the $R(\Lambda )$ factor is provided in the appendix.

Fig. 2

$R(\Lambda )$ of $B_{s} \rightarrow \mu ^+ \mu ^-$ defined in Eq. (17) as a function of the cut off energy

Compared with Eq. (9), the previous amplitude added the factor F,

$$\begin{aligned} F({B_s^*})= & {} \frac{\mathscr {M}({B_s\rightarrow B^*_s \gamma \rightarrow \mu ^+\mu ^-})}{\mathscr {M}({B_s\rightarrow \mu ^+\mu ^-})}\nonumber \\= & {} \frac{C_V^\mathrm{eff}f_{B_s^*}}{C_{10}f_{B_s}} {e g_{B_s B^*_s \gamma }} R(\Lambda ). \end{aligned}$$

(18)

We can estimate $g_{B_s B^*_s \gamma }$ in several ways, including the heavy-quark and chiral effective theories [31, 32] with the radiative and pion transition widths of $D^{*+}$, the light cone QCD sum rules [33, 34], and the radiative M1 decay widths of $B^*_s \rightarrow B_s \gamma$ from the potential model [27, 35].

The heavy-quark and chiral effective theories yield the following expression [13–15]:

$$\begin{aligned} g_{B_d B^*_d \gamma }= & {} -1.7\pm 0.2,\nonumber \\ g_{B_s B^*_s \gamma }= & {} -1.2\pm 0.2. \end{aligned}$$

(19)

The light cone QCD sum rules yield the following expression [33, 34]:

$$\begin{aligned} g_{B_d B^*_d \gamma }= & {} -2.3\pm 0.3,\nonumber \\ g_{B_s B^*_s \gamma }= & {} -1.5\pm 0.2. \end{aligned}$$

(20)

The radiative M1 decay width of $B^*_s \rightarrow B_s \gamma$ is derived as follows:

$$\begin{aligned} g_{B_s B^*_s \gamma }=-m_{ B^*_s}\left( \frac{12 \pi }{E_\gamma ^3} \Gamma (B^*_s \rightarrow B_s \gamma )\right) ^{1/2}. \end{aligned}$$

(21)

The predicted M1 widths are 0.15–400  eV and 10–300  eV for $B^*_s \rightarrow B_s \gamma$ and $B^*_d \rightarrow B_d \gamma$, respectively [24, 26, 27, 35, 36]. Recently new predicted M1 widths were given in Ref. [28]:

$$\begin{aligned} \Gamma _{B_s^* \rightarrow B_s \gamma }= & {} 0.313~ \mathrm{KeV},\nonumber \\ \Gamma _{B^*_d \rightarrow B_d \gamma }= & {} 1.23~ \mathrm{KeV}. \end{aligned}$$

(22)

Then we can get the following value:

$$\begin{aligned} g_{B_d B^*_d \gamma }= & {} -3.8,\nonumber \\ g_{B_s B^*_s \gamma }= & {} -2.0. \end{aligned}$$

(23)

4 Numerical result

The parameters in the numerical calculation are selected as follows [37]:

$$\begin{aligned}&\Lambda =1.2 ~\mathrm{GeV},\nonumber \\&m_b=4.2 ~\mathrm{GeV},\nonumber \\&\alpha _{em}=1/137. \end{aligned}$$

(24)

In the branch fraction of $B^*_{s,d}$, the weak decay is less than the M1 decay, $\Gamma _\mathrm{tot}(B^*_{s,d})\approx \Gamma (B^*_{s,d} \rightarrow B_{s,d} \gamma )$. The following ratio is obtained:

$$\begin{aligned} \frac{ Br({B_s^*\rightarrow \mu ^+\mu ^-})}{Br({B_s\rightarrow \mu ^+\mu ^-})}= & {} (0.34 \pm 0.03) \times \frac{\mathrm eV}{\Gamma (B^*_{s} \rightarrow B_{s} \gamma )}, \nonumber \\ \frac{ Br({B_d^*\rightarrow \mu ^+\mu ^-})}{Br({B_d\rightarrow \mu ^+\mu ^-})}= & {} (0.33 \pm 0.03)\times \frac{\mathrm eV}{\Gamma (B^*_{d} \rightarrow B_{d} \gamma )}.\nonumber \\ \end{aligned}$$

(25)

The main uncertainty is derived from the $f_{B_{s,d}}^*$ value. The dimuon invariant mass distribution measured by the CMS and LHCb Collaborations in Ref.  [2] should include the $B^*_{s,d}\rightarrow \mu ^+ \mu ^-$ contributions. If $\Gamma (B^*_{s} \rightarrow B_{s} \gamma )=1~$eV  [27], we can get $\frac{ Br({B_s^*\rightarrow \ell ^+\ell ^-})}{Br({B_s\rightarrow \mu ^+\mu ^-})}=0.34$ for $\ell =e,~\mu$. Afterward, $B_s^*\rightarrow e^+e^-$ may be searched by the CMS and LHCb experiments with larger data samples.

Table 2 The branching fractions of $B_{s,d} \rightarrow \mu ^+ \mu ^-$ measured by the CMS and LHCb Collaborations [2] and updated SM prediction with $\Gamma (B^*_{s,d} \rightarrow B_{s,d} \gamma )=313, 1230$ eV [28]

Moreover, we observe that the amplitude of $B_{s,d}\rightarrow \mu ^+\mu ^-$ is modified by the contributions of $B_{s,d}^*$ with a factor

$$\begin{aligned} F({B_{s,d}^*})= & {} (0.011\pm 0.006) \sqrt{\frac{\Gamma (B^*_{s,d} \rightarrow B_{s,d} \gamma )}{100 \mathrm{eV}}} , \end{aligned}$$

(26)

if $\Gamma (B^*_{s,d} \rightarrow B_{s,d} \gamma )\sim 100$ eV. The main uncertainty is caused by the $\Lambda$ value. The new predictions of $\Gamma (B_{s,d} \rightarrow \mu ^+\mu ^-)$ are provided as follows:

$$\begin{aligned}&Br(B_{s} \rightarrow \mu ^+ \mu ^-)=(36.6 \pm 2.3 ) \times \\&\quad \times \left( 1 + ( 0.023 \pm 0.012) \sqrt{\frac{\Gamma _\mathrm{tot}(B^*_{s} )}{100 \mathrm{eV}}}\right) \times 10^{-10},\\&Br(B_{d} \rightarrow \mu ^+ \mu ^-) =(10.6 \pm 0.9) \times \nonumber \\&\quad \times \left( 1 + ( 0.023 \pm 0.012)\sqrt{\frac{\Gamma _\mathrm{tot}(B^*_{d} )}{100 \mathrm{eV}}}\right) \times 10^{-11}. \end{aligned}$$

If $\Gamma (B^*_{s,d} \rightarrow B_{s,d} \gamma )=200$ eV, then this factor will increase the $\Gamma (B_{s,d} \rightarrow \mu ^+\mu ^-)$ decay width by a factor of $(3.3 \pm 1.7)\%$, which is approximately a factor 10 times larger than the neglect NLO EW correction factor $0.3\%$ at the decay width in Ref. [1]. In addition, the corresponding $g_{B_{s,d} B^*_{s,d} \gamma }=-1.5$ about a factor 15 times larger than $e_q e=-1/3\sqrt{4\pi \alpha _{em}}=-0.10$. $\Gamma (B^*_{s,d} \rightarrow B_{s,d} \gamma )$ may be measured through two-body decay $B^*_{s,d}\rightarrow J/\psi +M$ by CMS and LHCb.

If $\Gamma (B^*_{s,d} \rightarrow B_{s,d} \gamma )=313, 1230$ eV [28], then we can get

$$\begin{aligned} \frac{ Br({B_s^*\rightarrow \mu ^+\mu ^-})}{Br({B_s\rightarrow \mu ^+\mu ^-})}= & {} (1.1 \pm 0.1) \times 10^{-3}, \nonumber \\ \frac{ Br({B_d^*\rightarrow \mu ^+\mu ^-})}{Br({B_d\rightarrow \mu ^+\mu ^-})}= & {} (2.7 \pm 0.3)\times 10^{-4}. \end{aligned}$$

(27)

New predictions of $\Gamma (B_{s,d} \rightarrow \mu ^+\mu ^-)$ are provided as follows:

$$\begin{aligned}&Br(B_{s} \rightarrow \mu ^+ \mu ^-)=(3.8 \pm 0.3 )\times 10^{-9}\nonumber ,\\&Br(B_{d} \rightarrow \mu ^+ \mu ^-) =(1.2 \pm 0.1) \times 10^{-10},\nonumber \\&\frac{Br(B_{d} \rightarrow \mu ^+ \mu ^-)}{Br(B_{s} \rightarrow \mu ^+ \mu ^-)}=0.031\pm 0.0036. \end{aligned}$$

(28)

The numerical results are shown in Table 2 too.

5 Summary

In summary, this study investigated $B^*_{s,d}\rightarrow \mu ^+ \mu ^-$ in the dimuon distributions and the hadronic contribution $B_{s,d} \rightarrow B^*_{s,d} \gamma \rightarrow \mu ^+ \mu ^-$. The $\mu ^+ \mu ^-$ decay widths of the vector mesons $B^*_{s,d}$ are approximately a factor of 700 larger than the corresponding scalar mesons $B_{s,d}$. The obtained ratio of the branching fractions $\frac{ Br({B_{s,d}^*\rightarrow \mu ^+\mu ^-})}{{Br({B_{s,d}\rightarrow \mu ^+\mu ^-})}}$ is approximately $\frac{0.3 \times \mathrm{eV}}{{\Gamma (B^*_{s,d} \rightarrow B_{s,d} \gamma )}}$. The hadronic contribution $B_{s,d} \rightarrow B^*_{s,d} \gamma \rightarrow \mu ^+ \mu ^-$ is also estimated. The relative increase in the $B_{s,d}\rightarrow \mu ^+\mu ^-$ amplitude is approximately $(0.01\pm 0.006) \sqrt{\frac{{\Gamma (B^*_{s,d} \rightarrow B_{s,d} \gamma )}}{{100~ \mathrm{eV}}}}$. If we select $\Gamma (B^*_{s,d} \rightarrow B_{s,d} \gamma )=2~$eV, then the branching fractions of the vector mesons to the lepton pair are $5.3 \times 10^{-10}$ and $1.6 \times 10^{-11}$ for $B^*_{s}$ and $B^*_{d}$, respectively. If we select $\Gamma (B^*_{s,d} \rightarrow B_{s,d} \gamma )=200~$eV, then the updated branching fractions of the scalar mesons to the 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. If we select the recent predicted M1 widths $\Gamma (B^*_{s,d} \rightarrow B_{s,d} \gamma )=313, 1230$ eV [28], then the updated branching fractions are $(3.8\pm 0.3 ) \times 10^{-9}$ and $(1.2 \pm 0.1) \times 10^{-10}$ for $B_{s}\rightarrow \mu ^+\mu ^-$ and $B_{d}\rightarrow \mu ^+\mu ^-$, respectively. Further studies on $B^*_{s,d}$, including those on dielectron decay and two-body decay with $J/\psi$ should be conducted.

Appendix A: Appendix \(R(\Lambda )\)

$R(\Lambda )$ of $B_{s} \rightarrow B^*_{s} \gamma \rightarrow \mu ^+ \mu ^-$ defined in Eq. (17) is obtained as follows:

$$\begin{aligned} R(\Lambda )= & {} \frac{1}{32 \pi ^2 {m_{B_s}}^2 m_\mu ^2}\Bigg \{ 3 {m_{B_s}}^2 m_\mu ^2 -2m_\mu ^2 \left( {m_{B_s}}^2- {m_{B^*_s}}^2\right) ^2\nonumber \\&\left. \times \,{C_0}\left( {m_{B_s}}^2,m_\mu ^2,m_\mu ^ 2,{m_{B^*_s}}^2,0,m_\mu ^2\right) \right. \nonumber \\&\left. +\,{m_{B_s}}^2(m_\mu ^2-{m_{B^*_s}}^2)\left( {B_0}\left( 0,m_\mu ^2,{m_{B^*_s}}^2\right) \nonumber \right. \right. \\&\left. \left. -\,{B_0}\left( 0,({\Lambda }+m_\mu )^2,( {\Lambda }+{m_{B^*_s}})^2\right) \right) \right. \nonumber \\&\left. +\,\left( m_{B_s}^2 \left( 2m_\mu ^2 +{m_{B^*_s}}^2\right) +2 m_\mu ^2{m_{B^*_s}}^2\right) \right. \nonumber \\&\left. \left( {B_0}\left( m_\mu ^2,m_\mu ^2,{m_{B^*_s}}^2\right) \right. \right. \nonumber \\&\left. \left. -\,{B_0}\left( m_\mu ^2, (\Lambda +m_\mu )^2,({\Lambda }+{m_{B^*_s}})^2\right) \right) \right. \nonumber \\&\left. +\,m_\mu ^2 \left( 3 {m_{B_s}}^2-2{m_{B^*_s}}^2\right) \left( {B_0}\left( m_\mu ^2,0,m_\mu ^2\right) \right. \right. \nonumber \\&\left. -\,{B_0}\left( m_\mu ^2,{\Lambda }^2,(\Lambda +m_\mu )^2\right) \right) \Bigg \}. \end{aligned}$$

(A.1)

The scalar functions ${B_0}$ and ${C_0}$ are given in Refs. [38–40]. As a numerical fit between $0.5-2$ GeV, $R(\Lambda )$ is obtained as follows:

$$\begin{aligned} R(\Lambda )=0.022+0.062\times ln\left( \frac{\Lambda +m_{B_s}}{m_{B_s}}\right) . \end{aligned}$$

(A.2)