Current Status of the Standard Model Prediction for the B_s → μ⁺μ⁻ Branching Ratio

Abstract

The rare decay $B_s\to {\mu }^{+}{\mu }^{-}$ provides an important constraint on possible deviations from the Standard Model in b-s-ℓ-ℓ interactions. The present weighted average of its branching ratio measurements amounts to $(3.34\pm 0.27)\times {10}^{-9}$, which remains in good agreement with the theoretical prediction of $(3.64\pm 0.12)\times {10}^{-9}$ within the Standard Model. In the present paper, we review calculations that have contributed to this prediction and discuss the associated uncertainties.

1. Introduction

The Higgs boson discovery [1,2] through direct production at the LHC completed the experimental search for the Standard Model (SM) particle content. Since then, no clear signal for Beyond-Standard Model (BSM) particle production has been seen at the high-energy frontier of experimental particle physics. Consequently, more and more focus is being shifted towards precise studies of rare processes that are sensitive to corrections from BSMs.

In particular, decays of the B meson mediated through Flavor-Changing Neutral Currents (FCNCs) have been a very active area of research. Theoretical studies of the B mesons are greatly aided by the framework of Heavy Quark Expansion which, in many cases, allows us to parameterize the effects of their hadronic structure through a series of non-perturbative matrix elements suppressed by powers of ${\mathsf{\Lambda }}_{QCD}/M_B$. Moreover, while the FCNC-mediated processes are loop-suppressed in the SM, they can receive sizable tree-level contributions from BSMs, which underlines them as primary candidates for observables where indirect signals from new particles may be detected [3]. The rich phenomenology of rare B meson decays is being actively studied by LHC experiments and Belle II that follow and extend past investigations at CLEO, Belle, and BABAR. To fully take advantage of the current and future experimental data, improvements in the precision of the SM predictions are often necessary.

One of the most interesting rare decays of the B meson and the focus of this review is the $B_s\to {\mu }^{+}{\mu }^{-}$ channel that was first observed over a decade ago [4]. Since then, the experimental precision for its average time-integrated branching ratio ${\overline{\mathcal{B}}}_{s\mu }$ has reached $\mathcal{O}(10\%)$ [5,6,7,8]. The current world average reads [9]

$${\overline{\mathcal{B}}}_{s\mu }^{\exp }=(3.34\pm 0.27)\times {10}^{-9}.$$

(1)

The similar $B_d\to {\mu }^{+}{\mu }^{-}$ channel is suppressed by a factor $|V_{td}/V_{ts}{|}^2\approx 0.04$, which has a negative effect on experimental precision due to significantly reduced statistics, as illustrated in Figure 1.

The SM description of $B_s\to {\mu }^{+}{\mu }^{-}$ is greatly simplified by a factorization of long- and short-distance Quantum Chromodynamics (QCD) effects. In contrast to many other B decays, the SM decay amplitude of $B_s\to {\mu }^{+}{\mu }^{-}$ depends, to a very good approximation, only on a single non-perturbative hadronic quantity, namely the $B_s$-meson decay constant $f_{B_s}$. Its determinations from lattice simulations (see below) are mature and precise. Moreover, the hard QCD and electroweak (EW) corrections are, to a very good approximation, contained within a single Wilson coefficient $C_A$, which can be calculated in a standard, perturbative matching procedure as a series in the strong and electromagnetic couplings ${\alpha }_s$ and ${\alpha }_e$. Thanks to these properties,  SM calculations have reached the accuracy of a few percent. The current SM prediction for the branching ratio reads

$${\overline{\mathcal{B}}}_{s\mu }^{SM}=(3.64\pm 0.12)\times {10}^{-9}.$$

(2)

We describe its evaluation in the next sections. The numerical value in Equation (2) has been obtained by updating the input parameters in the semi-numerical expressions of Ref. [10] and including the power-enhanced QED correction from Refs. [11,12] that amounts to around $-0.5\%$. A difference with respect to $(3.66\pm 0.14)\times {10}^{-9}$ in Ref. [12] is due to the parameter update. The above SM prediction is well in agreement with each individual measurement [5,6,7,8], as well as with their average in Equation (1).

In the SM, on top of the aforementioned FCNC loop suppression, ${\overline{\mathcal{B}}}_{s\mu }$ receives a helicity suppression by the mass-squared ratio $m_{\mu }^2/m_{B_s}^2$. One or both of these suppressions may be lifted in models with additional Higgs doublets or with a $Z^{\prime }$. In effect, one finds restrictions on allowed parameter spaces in, among others, Two Higgs Doublet Models (2HDMs) [13] and the Minimal Supersymmetric Standard Model [14]. In addition, several time-dependent observables in $B_s\to {\mu }^{+}{\mu }^{-}$ can be used to study potential BSM $\mathrm{CP}$-violation mechanisms [15].

The present review is organized as follows. In Section 2, we present the effective Lagrangian and the branching ratio formula for $B_s\to {\mu }^{+}{\mu }^{-}$ in the SM. Section 3 and Section 4 are devoted to describing, respectively, the perturbative QCD and EW corrections to the Wilson coefficient $C_A$. At the end of Section 4, the power-enhanced QED correction to ${\overline{\mathcal{B}}}_{s\mu }$ is discussed. In Section 5, we summarize the current parameter update and evaluate the SM prediction for ${\overline{\mathcal{B}}}_{s\mu }$ (2) together with the corresponding uncertainty. We conclude in Section 6. In Appendix A, we recall the derivation of the branching ratio formula. In Appendix B, we present the current SM prediction for ${\overline{\mathcal{B}}}_{d\mu }$, i.e., the average time-integrated branching ratio of $B_d\to {\mu }^{+}{\mu }^{-}$.

2. The Effective Lagrangian and the Branching Ratio Formula

The effective Lagrangian used to describe $B_s\to l^{+}{l}^{-},$ $l\in \{e,\mu ,\tau \}$ in the SM is obtained through simultaneous integrating out of all fields heavier than the b quark at the scale ${\mu }_0=\mathcal{O}\left(m_t\right)$. It has the form

$$\mathcal{L}={\mathcal{L}}_{\mathrm{QCD}\times \mathrm{QED}}\left(\mathrm{fields}\mathrm{lighter}\mathrm{than}W\right)+\left[N\sum _n{C}_n{Q}_n+\mathrm{h}.\mathrm{c}.\right],$$

(3)

where

$$N\equiv {\displaystyle \frac{V_{tb}^{\ast }{V}_{ts}{G}_F^2{M}_W^2}{{\pi }^2}},$$

(4)

$V_{ij}$ are the Cabibbo–Kobayashi–Maskawa (CKM) matrix elements, $G_F$ is the Fermi constant, while $M_W$ is the W-boson mass defined in the on-shell renormalization scheme. The local operators $Q_n$ are polynomials in the light fields and their derivatives. The Wilson coefficients $C_n$ can be treated as real-valued (up to negligible corrections) once the global normalization factor N is set as in Equation (4). The operators $Q_n$ are of mass-dimension 5 or higher and have to be suppressed by powers of $1/M_W$ necessary to make the overall mass-dimension of $\mathcal{L}$ equal to 4. At the leading order in $1/M_W$ and ${\alpha }_e$, it is sufficient to consider the operators $Q_n$, where a $\mathsf{\Delta }B=-\mathsf{\Delta }S=-1$ flavor-changing quark current multiplies a lepton current. Moreover, the quark current must violate parity to annihilate the pseudoscalar $B_s$ meson. Once the Lorentz invariance is imposed, one is left with the following four operators only (no tensor current  $\overline{b}{\sigma }^{\alpha \beta }{\gamma }_{5}s$  needs to be considered because no antisymmetric tensor could be formed from $p^{\alpha }$ alone in the corresponding matrix element, analogous to the one in Equation (7)):

$$\begin{array}{cc}\hfill Q_A& \equiv \left[\overline{l}{\gamma }_{\alpha }{\gamma }_{5}l\right]\left[\overline{b}{\gamma }^{\alpha }{\gamma }_{5}s\right]\equiv \left[\overline{l}{\gamma }_{\alpha }{\gamma }_{5}l\right]j_A^{\alpha },\hfill \\ \hfill Q_V& \equiv \left[\overline{l}{\gamma }_{\alpha }l\right]j_A^{\alpha },\hfill \\ \hfill Q_P& \equiv \left[\overline{l}{\gamma }_{5}l\right]\left[\overline{b}{\gamma }_{5}s\right]\equiv \left[\overline{l}{\gamma }_{5}l\right]j_P,\hfill \\ \hfill Q_S& \equiv \left[\overline{l}l\right]j_P.\hfill \end{array}$$

(5)

The Lagrangian (3) can be used to derive the formula for ${\overline{\mathcal{B}}}_{s\mu }$. A sketch of the derivation is given in Appendix A. While evaluating the contribution of $Q_A$ there, it becomes clear that $Q_V$ does not contribute at the leading order in ${\alpha }_e$. As far as $Q_P$ and $Q_S$ are concerned, their Wilson coefficients are computed in a matching to full SM amplitudes with the quark and lepton currents exchanging Higgs bosons. Such contributions are suppressed by $m_b^2/M_W^2$ and can be neglected as being of the same order as dimension-8 operator effects. Hence, neglecting these tiny effects, ${\overline{\mathcal{B}}}_{s\mu }$ in the SM depends on $C_A$ only. The explicit expression reads

$${\overline{\mathcal{B}}}_{s\mu }^{SM}={\displaystyle \frac{{\left|N\right|}^2{M}_{B_s}^3{f}_{B_s}^2}{8\pi {\mathsf{\Gamma }}_H^s}}\beta r^2{\left|C_A\right|}^2+\mathcal{O}\left({\alpha }_e,{\displaystyle \frac{m_b^2}{M_W^2}}\right),$$

(6)

where $r\equiv 2m_{\mu }/M_{B_s}$, $\beta \equiv \sqrt{1-r^2}$, while ${\mathsf{\Gamma }}_H^s$ is the heavier mass eigenstate width in the $B_s$–${\overline{B}}_s$ system. Finally, the $B_s$ decay constant $f_{B_s}$ is defined through the relation

$$\langle 0|j_A^{\alpha }\left(x\right)|B_s\left(p\right)\rangle \equiv i{p}^{\alpha }{f}_{B_s}{e}^{-ipx}.$$

(7)

It is calculated using lattice QCD methods with errors at a sub-percent level. The current world average based on $2+1+1$ simulations [16,17,18,19] alone amounts to [20]

$$f_{B_s}=(230.3\pm 1.3)\mathrm{MeV}.$$

(8)

In several popular BSMs, the Wilson coefficients $C_S$ and $C_P$ can become comparable in size to $r{C}_A$. For example, in the 2HDM-II with large $tan{\beta }_H\equiv v_2/v_1$ (the ratio of the vacuum expectation values of the two Higgs doublets), one finds [21],

$$C_S\simeq C_P\simeq {\displaystyle \frac{ln\rho }{\rho -1}}{\displaystyle \frac{m_{\mu }{m}_b}{4M_W^2}}{tan}^2{\beta }_H,$$

(9)

where $\rho \equiv M_{H^{+}}^2/m_t^2$. The SM suppression factor of $m_b^2/M_W^2$ can be compensated by values of $tan{\beta }_H=\mathcal{O}\left(50\right)$ or larger. The branching ratio formula in the 2HDM becomes (see Appendix A)

$${\overline{\mathcal{B}}}_{s\mu }^{2HDM}={\displaystyle \frac{{\left|N\right|}^2{M}_{B_s}^3{f}_{B_s}^2}{8\pi {\mathsf{\Gamma }}_H^s}}\beta \left[|r{C}_A-u{C}_P{|}^2+{\displaystyle \frac{{\mathsf{\Gamma }}_H^s}{{\mathsf{\Gamma }}_L^s}}{\left|u\beta C_S\right|}^2\right]+\mathcal{O}\left({\alpha }_e\right),$$

(10)

where $u\equiv M_{B_s}/(m_b+m_s)$, and ${\mathsf{\Gamma }}_L^s$ is the lighter mass eigenstate width in the $B_s$–${\overline{B}}_s$ system.

Coming back to the SM expression for ${\overline{\mathcal{B}}}_{s\mu }$ in Equation (6), several comments concerning the $\mathcal{O}\left({\alpha }_e\right)$ terms are necessary. First, all the corrections of this order to $C_A$ are already known and included in the numerical result in Equation (2). They will be discussed in Section 4.1. Some of them are enhanced by $1/{sin}^2{\theta }_W$, $m_t^2/M_W^2$ or ${ln}^2{M}_W^2/m_b^2$, where ${\theta }_W$ is the weak mixing angle. Second, some of the remaining electromagnetic corrections in the $\mathcal{O}\left({\alpha }_e\right)$ term in Equation (6) receive a power enhancement by $M_{B_s}/{\mathsf{\Lambda }}_{\mathrm{QCD}}$ [11]. They come from virtual photons emitted by the s quark and absorbed by the leptons. We shall comment on them in Section 4.2. Their evaluation requires extending the operator basis to include the effects of other dimension-6 operators, not present in Equation (5), such as

$$Q_2^{†}\equiv \left[\overline{b}{\gamma }_{\alpha }{P}_{L}c\right]\left[\overline{c}{\gamma }^{\alpha }{P}_{L}s\right]\mathrm{or}{Q}_7^{†}\equiv {\displaystyle \frac{e{m}_b}{16{\pi }^2}}\left[\overline{b}{\sigma }^{\alpha \beta }{P}_{L}s\right]F_{\alpha \beta },$$

(11)

where $P_L=(1-{\gamma }_5)/2$. Third, the dependence of $C_A$ on the renormalization scale ${\mu }_b$ (that arises due to QED effects only) induces around $\mathcal{O}(0.3\%)$ uncertainty in ${\overline{\mathcal{B}}}_{s\mu }^{SM}$ [10]. Such a dependence must be compensated by the yet unknown $\mathcal{O}\left({\alpha }_e\right)$ corrections that receive no extra enhancement factors.

3. The QCD Corrections to ${\mathit{C}}_{\mathit{A}}$

As described in the previous section, at the leading order in ${\alpha }_e$ and $m_b^2/M_W^2$, the only significant short-distance parameter entering the SM branching ratio formula is the $C_A\left({\mu }_b\right)$ Wilson coefficient, where we specifically indicate its dependence on the low-energy renormalization scale ${\mu }_b=\mathcal{O}\left(m_b\right)$. It is evaluated by demanding the equality of corresponding Green’s functions in the SM and the effective theory (3) at the renormalization scale ${\mu }_0=\mathcal{O}\left(m_t\right)$ at which the electroweak degrees of freedom are decoupled. Computationally, the simplest choice of a Green’s function for the extraction of $C_A$ is $G[\overline{b},s,\overline{l},l]$ with vanishing external momenta:

$$G_{\mathrm{S}M}[\overline{b},s,\overline{l},l]\left({\mu }_0\right){{|}_{p_i=0}=G_{\mathrm{eff}}[\overline{b},s,\overline{l},l]\left({\mu }_0\right)|}_{p_i=0}.$$

(12)

At the matching scale ${\mu }_0$, QCD on both sides is treated perturbatively. This equation is then solved for $C_A\left({\mu }_0\right)$ order-by-order in ${\alpha }_s$ and ${\alpha }_e$, resulting in a double series

$$C_A\left({\mu }_0\right)=\sum _{m,n=0}^{\infty }{\tilde{\alpha }}_s^m{\tilde{\alpha }}_e^n{C}_A^{(m,n)}\left({\mu }_0\right),$$

(13)

where ${\alpha }_i\left({\mu }_0\right)\equiv 4\pi {\tilde{\alpha }}_i$ are the running couplings renormalized in the $\overline{\mathrm{MS}}$ scheme at the scale ${\mu }_0$. In this section, we focus on the leading terms in ${\alpha }_e$, namely $C_A^{(m,0)}$.

As we work at the leading order in $1/M_W$, all light masses in the matching Equation (12) can be set to 0. In dimensional regularization, all scaleless loop integrals vanish. In consequence, on the effective theory side, one is left only with tree diagrams, including the UV-counterterm ones. On the SM side, partially massive tadpoles have to be calculated. Removing light masses from propagators in the SM Green’s function leads to spurious infrared divergences in loop integrals evaluated in $d=4-2ϵ$ dimensions. The resulting additional $ϵ$-poles are not removed by the renormalization constants of the SM. Instead, they cancel in the matching Equation (12) against the tree-level UV counterterms in the effective theory.

Once such a procedure is applied in our case, one has to supplement the effective Lagrangian with additional operators that vanish when $ϵ\to 0$ due to the Dirac algebra identities. Such operators are called evanescent. The UV-counterterms with these operators cancel against some of the spurious infrared divergence effects on the SM side. Moreover, their Wilson coefficients evaluated at lower orders affect the physical operator Wilson coefficients at higher orders. For the $C_A^{(m,0)}$ terms, it is sufficient to introduce only one evanescent operator [22]:

$$Q_A^E=\left[\overline{b}{\gamma }_{\alpha }{\gamma }_{\beta }{\gamma }_{\delta }s\right]\left[\overline{l}{\gamma }^{\delta }{\gamma }^{\beta }{\gamma }^{\alpha }l\right]-4Q_A.$$

(14)

The bare fields, couplings, and Wilson coefficients on the effective side are replaced by the renormalized ones, with mixing occurring between the physical and evanescent operators:

$$C_A^{\left(b\right)}{Q}_A^{\left(b\right)}+C_A^{E\left(b\right)}{Q}_A^{E\left(b\right)}=Z_q{Z}_l(C_A{Z}_{NN}{Q}_A+C_A{Z}_{NE}{Q}_A^E+C_A^E{Z}_{EN}{Q}_A+C_A^E{Z}_{EE}{Q}_A^E),$$

(15)

where $Z_q$ and $Z_l$ are the quark- and lepton-field $\overline{\mathrm{MS}}$ renormalization constants, respectively, with $Z_l=1+\mathcal{O}\left({\alpha }_e\right)$. To fix the renormalization constants $Z_{ij}$, one demands that in the renormalized Green’s functions, the terms proportional to $C_A$ are finite when $ϵ\to 0$, while those proportional to $C_A^E$ vanish in this limit [23]. Examples of diagrams contributing to $Z_{ij}$ at ${\alpha }_s$ and ${\alpha }_s^2$ are shown in Figure 2. Results for the relevant $Z_{ij}$ up to $\mathcal{O}\left({\alpha }_e^0{\alpha }_s^2\right)$ are given in Equation (15) of Ref. [24].

It is important to emphasize that $Z_{NN}=1$ to all orders in QCD, once higher-dimensional operators and $\mathcal{O}\left({\alpha }_e\right)$ effects are neglected. Therefore, the Renormalization Group Equation (RGE) for $C_A$ is trivial at this level:

$$\mu {\displaystyle \frac{d}{d\mu }}{C}_A=\mathcal{O}\left({\alpha }_e\right).$$

(16)

Consequently, there is no RG evolution of $C_A\left(\mu \right)$ at the leading order in ${\alpha }_e$:

$$C_A\left({\mu }_b\right)=C_A\left({\mu }_0\right)+\mathcal{O}\left({\alpha }_e\right).$$

(17)

The SM Green’s function in Equation (12) receives QCD corrections from two classes of diagrams: W-boxes and Z-penguins, shown in Figure 3. Such bare diagrams are renormalized using lower-loop SM diagrams with counterterms. The QCD coupling constant renormalization $Z_g$ has to be modified by an appropriate threshold correction to match the coupling constant of the effective theory with only five active flavors (see Section 3 in Ref. [25]). Similarly, the light-field wave-function renormalization constants have to be shifted to account for the decoupling threshold (see Section 4 in Ref. [26]). Heavy fields and masses are renormalized in the $\overline{\mathrm{MS}}$ scheme.

Once both sides of the matching Equation (12) have been properly renormalized, the value of $C_A\left({\mu }_0\right)$ can be extracted. All the Dirac structures appearing on the SM side are mapped onto either $Q_A$ or $Q_A^E$ and their coefficients compared to the ones on the effective side.

The procedure described in this section was completed in Ref. [27] at the leading order, followed by Refs. [22,28,29] and [24] for the $\mathcal{O}\left({\alpha }_s^1\right)$ and $\mathcal{O}\left({\alpha }_s^2\right)$ corrections, respectively. In the latter case, high-order expansions in $y\equiv M_W/m_t$ and $w\equiv 1-M_W^2/m_t^2$ were computed, and their combination was used to obtain a numerical result at the physical value of $M_W/m_t$.

4. The Electroweak Corrections

4.1. The $C_A^{(0,1)}$ Correction

The next-to-leading order EW correction $C_A^{(0,1)}\left({\mu }_0\right)$ was first obtained in Ref. [30] (Let us note that it was called $c_{10}^{(2,2)}$ in that paper due to different notational conventions). Its calculation follows a similar pattern to the one described in the previous section. The main difference comes from additional Dirac structures appearing in the $\mathcal{O}\left({\alpha }_e\right)$ corrections on the SM side of the matching equation, which necessitate including additional operators in the effective Lagrangian. For the complete matching at the scale ${\mu }_0$, at the leading order in $1/M_W$ and including $\mathcal{O}\left({\alpha }_e\right)$ terms, one has to retain all the operators from Equation (5), and supplement them with $Q_2^{†}$ defined in Equation (11). The cancellation of spurious infrared divergence effects requires also the inclusion of additional evanescent operators (see Appendix A of Ref. [30]). The renormalization constants of all resulting Wilson coefficients are then calculated in a way analogous to the $C_A^{(m,0)}$ calculation.

Examples of diagrams contributing to $C_A^{(0,1)}\left({\mu }_0\right)$ on the SM side are shown in Figure 4. Before this correction can be extracted, the UV divergences on the SM side have to be renormalized. For a discussion of different renormalization schemes for the electroweak boson and top-quark masses, we refer to the original article [30]. Here, we will continue the analysis in the OS-2 scheme defined therein, as it was used in the subsequent phenomenological analysis [10]. In this scheme, all the QCD corrections to mass renormalization constants are defined in the $\overline{\mathrm{MS}}$, but the $\mathcal{O}\left({\alpha }_e\right)$ corrections to $Z_{m_t}$, $Z_{M_W}$, and $Z_{M_Z}$ are defined on-shell. In practice, the calculation is first performed fully in the $\overline{\mathrm{MS}}$ scheme, and the finite terms in these three renormalization constants are subsequently added to the renormalized results.

The value of $C_A$ extracted at the scale ${\mu }_0$ must then be RG-evolved down to the ${\mu }_b$ scale of the $B_s\to {\mu }^{+}{\mu }^{-}$ decay. The renormalized Wilson coefficients of the extended set of effective operators are related to the bare ones through a relation similar to Equation (15):

$$C_j^{\left(b\right)}{Q}_j^{\left(b\right)}=Z_q{Z}_l\sum _k{C}_k{Z}_{kj}{Q}_j⟹C_j^{\left(b\right)}=\sum _k{C}_k{Z}_{kj}.$$

(18)

The one-loop RGE for Wilson coefficients $\overrightarrow{C}$ has a general form

$$\mu {\displaystyle \frac{d}{d\mu }}\overrightarrow{C}={\left({\tilde{\alpha }}_s{\gamma }_s^{\left(0\right)}+{\tilde{\alpha }}_e{\gamma }_e^{\left(0\right)}\right)}^T\overrightarrow{C},$$

(19)

where the Anomalous Dimension Matrices (ADMs) ${\gamma }_s^{\left(0\right)}$ and ${\gamma }_e^{\left(0\right)}$ are obtained from the renormalization constants of Wilson coefficients. Once we restrict to non-evanescent operators only, the necessary MS-scheme relation reads

$$Z_{kj}={\delta }_{kj}+{\displaystyle \frac{1}{2ϵ}}{\left[{\tilde{\alpha }}_s{\gamma }_s^{\left(0\right)}+{\tilde{\alpha }}_e{\gamma }_e^{\left(0\right)}\right]}_{kj}+\mathcal{O}\left({\tilde{\alpha }}_s^2,{\tilde{\alpha }}_e^2,{\tilde{\alpha }}_s{\tilde{\alpha }}_e\right).$$

(20)

For the RGE (19) to close, one has to extend the set of operators in the effective Lagrangian further (see the discussion under Equation (15) in Ref. [30]). Analytical expressions for the evolution operator associated with Equation (19) can be found, e.g., in Ref. [31]. The details and results of the numerical solution can be found in Appendix B of Ref. [30].

4.2. Power-Enhanced QED Corrections

Within the effective theory framework, the S-matrix element corresponding to the $B_s\to {\mu }^{+}{\mu }^{-}$ decay receives $\mathcal{O}\left({\alpha }_e\right)$ contributions from diagrams with tree-level Wilson coefficients and a virtual photon exchanged between the fermions. In particular, there is a class of diagrams with a photon emitted from the spectator s quark and absorbed by one of the outgoing muons (see Figure 5). The leading terms of the power series in $\{m_s,m_{\mu },{\mathsf{\Lambda }}_{\mathrm{QCD}}\}/m_b$ of this correction were computed in Ref. [11]. It was observed that its helicity suppression by the factor of $r^2$ in the tree-level branching ratio (6) is partially lifted due to a relative enhancement by $M_{B_s}/{\mathsf{\Lambda }}_{\mathrm{QCD}}$.

Effects of hard virtual photons with energies and virtualities above the ${\mu }_b$ scale are contained in the Wilson coefficients $C_i$. The remaining virtual photons need to be taken into account in the physical matrix elements that are evaluated at the scale ${\mu }_b$. In particular, virtual photons exchanged in the diagrams in Figure 5 probe the inner structure of the $B_s$ meson, smearing the annihilation point of the valence quarks over a distance $\mathcal{O}({\left(M_{B_s}{\mathsf{\Lambda }}_{\mathrm{QCD}}\right)}^{-{\displaystyle \frac{1}{2}}})$, which corresponds to inverse virtuality of the off-shell s quark. Such long-distance QCD effects cannot be parameterized solely by $f_{B_s}$, as at the leading order. Instead, one has to estimate the effects of matrix elements like

$$\langle 0|\int d^{d}x\mathbf{T}\left[\left[\overline{\mu }{\gamma }^{\alpha }\mu \right]\left(x\right)Q_A\left(0\right)\right]|B_s\rangle ,$$

(21)

where $\mathbf{T}$ is the time-ordering operator. They involve the B meson light-cone distribution amplitude [32] and its first logarithmic moments.

Virtual photon exchanges leading to power-enhanced QED corrections were thoroughly studied in the formalism of the Soft-Collinear Effective Theory by Beneke, Bobeth, and Szafron in Ref. [12]. Their effect can be included in the SM prediction through a replacement

$${\overline{\mathcal{B}}}_{s\mu }\to {\eta }_{BBS}{\overline{\mathcal{B}}}_{s\mu },$$

(22)

with

$${\eta }_{BBS}=0.{995}_{-0.005}^{+0.003}.$$

(23)

The main uncertainty in ${\eta }_{BBS}$ comes from poorly known values of hadronic parameters [33]. We have extracted the numerical value of ${\eta }_{BBS}$ as well as its uncertainty in Equation (23) from Equations (8.8) and (8.10) of Ref. [12]. One can observe there that the relatively small central value of the correction ($-0.5\%$) arises as an effect of partial cancellation between potentially unrelated contributions from the $Q_V$ and $Q_7^{†}$ operators. Thanks to this cancellation, the overall effect remains below the $\pm 1.5\%$ non-parametric uncertainty estimated in Ref. [10]. That uncertainty was primarily due to unknown $\mathcal{O}\left({\alpha }_e\right)$ effects, although the possibility of their power enhancement remained unknown at the time.

The contribution of $Q_7^{†}$ to the considered corrections was reanalyzed in Ref. [34]. Despite several differences in the analytical expressions with respect to the earlier analysis of Ref. [12], the numerical results of the two papers remain in qualitative agreement, and no modification of the factor ${\eta }_{BBS}$ in Equation (23) is necessary.

5. Numerical Analysis

In this section, we update the SM prediction for ${\overline{\mathcal{B}}}_{s\mu }$ based on Equation (6), including the complete $\mathcal{O}\left({\alpha }_s^2\right)$ and $\mathcal{O}\left({\alpha }_e\right)$ corrections to $C_A$, as well as the QED correction factor in Equation (23). In practice, it is sufficient to use the semi-numerical expressions from Equation (6) of Ref. [10], which gives

$${\overline{\mathcal{B}}}_{s\mu }^{SM}\times {10}^9=(3.65\pm 0.06){\left({\displaystyle \frac{M_t\left[\mathrm{GeV}\right]}{173.1}}\right)}^{3.06}{\left({\displaystyle \frac{{\alpha }_s\left(M_Z\right)}{0.1184}}\right)}^{-0.18}{R}_s{\eta }_{BBS},$$

(24)

where

$$R_s={\left({\displaystyle \frac{f_{B_s}\left[\mathrm{MeV}\right]}{227.7}}\right)}^2{\left({\displaystyle \frac{|V_{cb}|}{0.0424}}\right)}^2{\left({\displaystyle \frac{|V_{tb}^{\ast }{V}_{ts}/V_{cb}|}{0.980}}\right)}^2{\displaystyle \frac{{\tau }_H^s\left[\mathrm{ps}\right]}{1.615}}.$$

(25)

In the above expressions, all the explicitly displayed input parameters are normalized to their 2013 central values. In 

Table 1. Numerical values of the updated input parameters.

Parameter	Value	Unit	Refs.
$f_{B_s}$	230.3 (1.3)	MeV	[16,17,18,19,20]
$|V_{cb}|\times {10}^3$	41.97 (48)	-	[35]
$|V_{tb}^{\ast }{V}_{ts}/V_{cb}|$	0.9820 (4)	-	derived from Ref. [36]
${\tau }_H^s$	1.622 (8)	ps	[37]
$M_t$	172.57 (29)	GeV	[9]
${\alpha }_s\left(M_Z\right)$	0.1180 (9)	-	[9]

, we update these central values together with the corresponding uncertainties. All the remaining parameters are retained unaltered with respect to Table I of Ref. [10], as their update would not affect ${\overline{\mathcal{B}}}_{s\mu }$ in a noticeable manner—they are either very precisely measured or have little effect on ${\overline{\mathcal{B}}}_{s\mu }$.

As already mentioned in Equation (2), we find ${\overline{\mathcal{B}}}_{s\mu }^{SM}=(3.64\pm 0.12)\times {10}^{-9}$. The overall uncertainty is now almost a factor of two smaller than that found in Ref. [10], while the central value remains almost unchanged. The latter fact can be attributed to an approximate cancellation of shifts stemming from the parameter updates and ${\eta }_{BBS}$, as in the following sum:

$$+2.3\%\left(f_{B_s}\right)-1.6\%\left(CKM\right)+0.4\%\left({\tau }_H^s\right)-0.9\%\left(M_t\right)+0.1\%\left({\alpha }_s\right)-0.5\%\left({\eta }_{BBS}\right)\simeq -0.3\%.$$

(26)

As far as the uncertainty breakdown is concerned, its current version is compared to the 2013 one [10] in

Table 2. The current uncertainty breakdown in ${\overline{\mathcal{B}}}_{s\mu }^{SM}$, as compared to the 2013 one.

	${\mathit{f}}_{{\mathit{B}}_{\mathit{s}}}$	CKM	${\mathit{\tau }}_{\mathit{H}}^{\mathit{s}}$	${\mathit{M}}_{\mathit{t}}$	${\mathit{\alpha }}_{\mathit{s}}$	${\mathit{\eta }}_{\mathbf{BBS}}$	Other	Non-Parametric	∑
2024 [this paper]	$1.1$%	$2.3\%$	$0.5$%	$0.5$%	$0.1$%	$0.5$%	<0.1%	$1.5$%	$3.2$%
2013 [10]	$4.0$%	$4.3\%$	$1.3$%	$1.6$%	$0.1$%	$0.0$%	<0.1%	$1.5$%	$6.4$%

. In its last column, the uncertainties are combined in quadrature.

One can observe a significant improvement in the first four columns where the dominant parametric uncertainties originate from, in particular in the case of $f_{B_s}$, which is determined on the lattice. As far as the top-quark mass $M_t$ is concerned, let us recall that the PDG [9] value is treated as the on-shell mass, which is not strictly correct. However, the overall non-parametric uncertainty of $\pm 1.5\%$ is understood to contain a contribution from such an approximation. As is evident from Table 1 and Table 2, a $300\mathrm{MeV}$ shift in $M_t$ implies a $0.5\%$ shift in ${\overline{\mathcal{B}}}_{s\mu }^{SM}$.

At present, the most important uncertainty originates from $|V_{cb}|$, in which case we use the inclusive determination only [35]. A combination with exclusive determinations would not lead to an improvement, given the persistent discrepancy between the inclusive and exclusive results [9]. Our preference is the same as in Ref. [10], i.e., we treat the inclusive determination as theoretically cleaner and more reliable.

As already mentioned in Ref. [10], one can get rid of $|V_{cb}|$ in the ratio of ${\overline{\mathcal{B}}}_{s\mu }$ to the measured $B_s^{\left(H\right)}$-$B_s^{\left(L\right)}$ mass difference, at the cost of introducing an extra uncertainty from lattice determinations of the “bag parameter” $B_{B_s}$. Such an approach is likely to become relevant once the experimental accuracy in ${\overline{\mathcal{B}}}_{s\mu }$ (currently $8.1\%$ in Equation (1)) becomes closer to the theoretical one.

6. Conclusions

In this review, we analyzed the current SM prediction for the branching ratio of the rare $B_s\to {\mu }^{+}{\mu }^{-}$ decay. This channel continues to be among the most promising candidates for detecting BSM physics without the direct production of new particles, due to its SM suppression and possible BSM enhancements.

The SM analysis is, to a very good approximation, contained in the perturbative calculation of the Wilson coefficient $C_A$ and the lattice calculation of the long-distance QCD parameter $f_{B_s}$. The perturbative part was already complete up to and including next-to-next-to-leading QCD and next-to-leading EW effects.

The currently dominant theoretical uncertainty originates from the CKM-matrix element $|V_{cb}|$. The next-to-dominant uncertainty is already non-parametric, stemming mainly from the unknown higher-order electromagnetic corrections at the scale ${\mu }_b$. They depend on non-perturbative effects that are not contained in $f_{Bs}$. The current result for ${\overline{\mathcal{B}}}_{s\mu }^{SM}$ changes by around $0.3\%$ when ${\mu }_b$ is varied between $m_b/2$ and $2m_b$. However, since it provides a lower bound only on the possible size of unknown electromagnetic effects, the actual uncertainty estimate should be somewhat more conservative. Here, we have retained the overall non-parametric uncertainty at the same level as in Ref. [10], namely $\pm 1.5\%$.

The experimental error is currently much larger, around 8%, which sets a limit on the power of $B_s\to {\mu }^{+}{\mu }^{-}$ as a means for testing various BSM theories. The situation is expected to improve with time, when higher statistics are collected at the LHC and future experiments.

A final message that we would like to share with the reader is as follows. Our numerical result in Equation (2) will become outdated as soon as any of the input parameters are determined in a new analysis. Performing another update of ${\overline{\mathcal{B}}}_{s\mu }^{SM}$ does not require being an expert. It is sufficient to substitute new inputs into the simple formula (24).