1. Introduction
In the standard model, flavor-changing neutral current (FCNC) induced processes are forbidden at tree level. They can only occur via loop diagrams. Meanwhile they are also sensitive to contributions of new physics. Particles of new physics may contribute via loop diagrams as ‘virtual particles’, thereby affecting the physical processes induced by FCNC. With continuous improvement of experimental accuracy, FCNC processes play an increasingly important role in the new physics research in heavy flavor physics. The most typical process is the one caused by $b\to {{sl}}^{+}{l}^{-}$ , such as the rare semileptonic decays of ${\bar{B}}_{s}^{0}\to \phi {l}^{+}{l}^{-}$ (l = e, μ, τ).
In the past two decades, the decays of ${\bar{B}}_{s}^{0}\to \phi {l}^{+}{l}^{-}$ (l = e, μ, τ) have been studied by using several different approaches such as lattice QCD [1], QCD light-cone sum rule (LCSR) [2, 3], constituent quark model (CQM) [4, 5], QCD sum rule [6], relativistic quark model (RQM) [7] and covariant quark model [8]. The method of QCD sum rule (SR) was originally developed by Shifman, Vainshtein and Zakharov in the late 1970s [9, 10], which was then widely applied to the calculation of hadronic physics [11]. Several years ago the translation form factors of ${B}_{s}^{0}\to \phi $ in ${\bar{B}}_{s}^{0}\to \phi {l}^{+}{l}^{-}$ (l = e, μ, τ) decays were calculated with QCD sum rule in [6]. Compared with other results, some form factors obtained in [6] are different by negative signs, which are not simply due to different definition for the form factors.
Experimentally, LHCb Collaboration updated the measurement of the branching ratio of ${\bar{B}}_{s}^{0}\to \phi {\mu }^{+}{\mu }^{-}$ recently [12] $ \begin{eqnarray}\begin{array}{l}{\rm{Br}}({\bar{B}}_{s}^{0}\to \phi {\mu }^{+}{\mu }^{-})\\ \quad =\,({7.97}_{-0.43}^{+0.45}\pm 0.22\pm 0.23\pm 0.60)\times {10}^{-7}.\end{array}\end{eqnarray}$ Hence, considering the status of theoretical calculation and the recent improvement in experimental measurement, we believe that it is valuable to re-consider the decays of ${\bar{B}}_{s}^{0}\to \phi {l}^{+}{l}^{-}$ (l = e, μ, τ) theoretically. In this work, we revisit the form factors in ${\bar{B}}_{s}^{0}\to \phi $ transition in QCD sum rule, and use these form factors to calculate the branching ratios of ${\bar{B}}_{s}^{0}\to \phi {e}^{+}{e}^{-}$ , ${\bar{B}}_{s}^{0}\to \phi {\mu }^{+}{\mu }^{-}$ and ${\bar{B}}_{s}^{0}\to \phi {\tau }^{+}{\tau }^{-}$ . Finally, we compare our results of form factors and branching ratios with previous theoretical works as well as the latest experimental data.
The paper is organized as followings. In section 2 , we present the effective Hamiltonian and effective amplitude of ${\bar{B}}_{s}^{0}\to \phi {l}^{+}{l}^{-}$ decay. Sections 3 and 4 are devoted to the calculation of the form factors in QCD sum rule method. Section 5 is for the numerical analysis and discussion. Finally, a brief summary is presented in section 6 .
2. Effective Hamiltonian
At quark level, the effective Hamiltonian of the rare semileptonic decay $b\to {{sl}}^{+}{l}^{-}$ can be written as [13] $ \begin{eqnarray}{{ \mathcal H }}_{\mathrm{eff}}=-\displaystyle \frac{{G}_{F}}{\sqrt{2}}{V}_{{tb}}{V}_{{ts}}^{* }\displaystyle \sum _{i=1}^{10}{C}_{i}(\mu ){O}_{i}(\mu ),\end{eqnarray}$ where ${V}_{{tb}}{V}_{{ts}}^{* }$ is the product of relevant CKM matrix elements. Ci denotes Wilson coefficient, and the operators Oi are
$ \begin{eqnarray*}\begin{array}{rcl}{Q}_{1} & = & {\left({\bar{s}}_{\alpha }{c}_{\beta }\right)}_{V-A}{\left({\bar{c}}_{\beta }{b}_{\alpha }\right)}_{V-A},\\ {Q}_{2} & = & {\left(\bar{s}c\right)}_{V-A}{\left(\bar{c}b\right)}_{V-A},\\ {Q}_{3} & = & {\left(\bar{s}b\right)}_{V-A}\displaystyle \sum _{q}{\left(\bar{q}q\right)}_{V-A},\\ {Q}_{4} & = & {\left({\bar{s}}_{\alpha }{b}_{\beta }\right)}_{V-A}\displaystyle \sum _{q}{\left({\bar{q}}_{\beta }{q}_{\alpha }\right)}_{V-A},\\ {Q}_{5} & = & {\left(\bar{s}b\right)}_{V-A}\displaystyle \sum _{q}{\left(\bar{q}q\right)}_{V+A},\\ {Q}_{6} & = & {\left({\bar{s}}_{\alpha }{b}_{\beta }\right)}_{V-A}\displaystyle \sum _{q}{\left({\bar{q}}_{\beta }{q}_{\alpha }\right)}_{V+A},\\ {Q}_{7} & = & \displaystyle \frac{{\alpha }_{e}}{2\pi }{m}_{b}{\bar{s}}_{\alpha }{\sigma }^{\mu \nu }(1+{\gamma }^{5}){b}_{\alpha }{F}_{\mu \nu },\\ {Q}_{8} & = & \displaystyle \frac{{\alpha }_{s}}{2\pi }{m}_{b}{\bar{s}}_{\alpha }{\sigma }^{\mu \nu }(1+{\gamma }^{5}){T}_{\alpha \beta }^{a}{b}_{\beta }{G}_{\mu \nu }^{a},\\ {Q}_{9} & = & \displaystyle \frac{\alpha }{2\pi }{\left(\bar{s}b\right)}_{V-A}{\left(\bar{l}l\right)}_{V},\\ \,{Q}_{10} & = & \displaystyle \frac{\alpha }{2\pi }{\left(\bar{s}b\right)}_{V-A}{\left(\bar{l}l\right)}_{A}.\end{array}\end{eqnarray*}$
Then the effective Hamiltonian above leads to the following decay amplitude of ${\bar{B}}_{s}^{0}\to \phi {l}^{+}{l}^{-}$ [13] $ \begin{eqnarray}\begin{array}{l}{ \mathcal M }({\bar{B}}_{s}^{0}\to \phi {l}^{+}{l}^{-})\\ =\,\displaystyle \frac{{G}_{F}\alpha }{2\sqrt{2}\pi }{V}_{{tb}}{V}_{{ts}}^{* }\left[{C}_{9}^{\mathrm{eff}}\langle \phi (\varepsilon ,{p}_{2})| \bar{s}{\gamma }_{\nu }(1-{\gamma }_{5})b| {\bar{B}}_{s}^{0}({p}_{1})\rangle \bar{{\ell }}{\gamma }^{\nu }{\ell }\right.\\ \ \,+\,{C}_{10}\langle \phi (\varepsilon ,{p}_{2})| \bar{s}{\gamma }_{\nu }(1-{\gamma }_{5})b| {\bar{B}}_{s}^{0}({p}_{1})\rangle \bar{{\ell }}{\gamma }^{\nu }{\gamma }_{5}{\ell }\\ \ \,\left.-\,2{C}_{7}^{\mathrm{eff}}{m}_{b}\displaystyle \frac{{\rm{i}}}{{q}^{2}}\langle \phi (\varepsilon ,{p}_{2})| \bar{s}{\sigma }_{\nu \lambda }{q}^{\lambda }(1+{\gamma }_{5})b| {\bar{B}}_{s}^{0}({p}_{1})\rangle \bar{{\ell }}{\gamma }^{\nu }{\ell }\right],\end{array}\end{eqnarray}$ where p1 and p2 are momenta of ${\bar{B}}_{s}^{0}$ and φ mesons, respectively. q is the momentum transfer $q={p}_{1}-{p}_{2}$ . ${C}_{9}^{\mathrm{eff}}$ and ${C}_{7}^{\mathrm{eff}}$ are two effective Wilson coefficients, with ${C}_{7}^{\mathrm{eff}}={C}_{7}-{C}_{5}/3-{C}_{6}$ . As for the effective Wilson coefficient ${C}_{9}^{\mathrm{eff}}$ , we take the expression in [13], which is given as followings $ \begin{eqnarray}\begin{array}{rcl}{C}_{9}^{\mathrm{eff}} & = & {C}_{9}+{C}_{0}\left[h({\hat{m}}_{c},\hat{s})\right.\\ & & \left.+\displaystyle \frac{3\pi \kappa }{{\alpha }^{2}}\displaystyle \sum _{{V}_{i}=\psi (1s;2s)}\displaystyle \frac{{\rm{\Gamma }}({V}_{i}\to {l}^{+}{l}^{-}){m}_{{V}_{i}}}{{m}_{{V}_{i}}^{2}-{q}^{2}-{\rm{i}}{m}_{{V}_{i}}{{\rm{\Gamma }}}_{{V}_{i}}}\right]\\ & & -\displaystyle \frac{1}{2}h(1,\hat{s})(4{C}_{3}+4{C}_{4}+3{C}_{5}+{C}_{6})\\ & & -\displaystyle \frac{1}{2}h(0,\hat{s})({C}_{3}+3{C}_{4})+\displaystyle \frac{2}{9}(3{C}_{3}+{C}_{4}+3{C}_{5}+{C}_{6}),\end{array}\end{eqnarray}$ where we define
$ \begin{eqnarray*}{C}_{0}=3{C}_{1}+{C}_{2}+3{C}_{3}+{C}_{4}+3{C}_{5}+{C}_{6},\end{eqnarray*}$
$ \begin{eqnarray*}h(0,\hat{s})=\displaystyle \frac{8}{27}-\displaystyle \frac{8}{9}\mathrm{ln}\displaystyle \frac{{m}_{b}}{\mu }-\displaystyle \frac{4}{9}\mathrm{ln}\hat{s}+{\rm{i}}\pi \displaystyle \frac{4}{9},\end{eqnarray*}$ and
$ \begin{eqnarray*}\begin{array}{l}h({\hat{m}}_{c},\hat{s})=-\displaystyle \frac{8}{9}\mathrm{ln}\displaystyle \frac{{m}_{b}}{\mu }-\displaystyle \frac{8}{9}\mathrm{ln}{\hat{m}}_{c}+\displaystyle \frac{8}{27}+\displaystyle \frac{4}{9}x\\ \quad -\,\displaystyle \frac{2}{9}(2+x)| 1-x{| }^{\tfrac{1}{2}}\left\{\begin{array}{l}\left(\mathrm{ln}| \tfrac{\sqrt{1-x}+1}{\sqrt{1-x}-1}| -{\rm{i}}\pi \right),\,x\lt 1\\ 2\arctan \tfrac{1}{\sqrt{x-1}}\,\,\,\,\,,\,x\gt 1\end{array}\right.,\end{array}\end{eqnarray*}$
with $x=4{\hat{m}}_{c}^{2}/\hat{s}$ , ${\hat{m}}_{c}={m}_{c}/{m}_{{B}_{s}}$ , $\hat{s}={q}^{2}/{m}_{{B}_{s}}^{2}$ , $\kappa =1/{C}_{0}$ and μ = mb.
3. Form factors from QCD sum rule
We have calculated the hadronic matrix elements $\langle \phi (\varepsilon ,{p}_{2})| \bar{s}{\gamma }_{\nu }(1-{\gamma }_{5})b| {\bar{B}}_{s}^{0}({p}_{1})\rangle $ in the decay amplitude given in equation (3 ) in our previous work [14]. So in this work, we need only to deal with the other hadronic matrix element $\langle \phi (\varepsilon ,{p}_{2})| \bar{s}{\sigma }_{\nu \lambda }{q}^{\lambda }(1+{\gamma }_{5})b| {\bar{B}}_{s}^{0}({p}_{1})\rangle $ in equation (3 ).
Similarly the hadronic matrix element $\langle \phi | \bar{s}{\sigma }_{\nu \lambda }{q}^{\lambda }(1+{\gamma }_{5})b| {\bar{B}}_{s}^{0}\rangle $ can be decomposed as [15] $ \begin{eqnarray}\begin{array}{l}\langle \phi (\varepsilon ,{p}_{2})| \bar{s}{\sigma }_{\nu \lambda }{q}^{\lambda }(1+{\gamma }_{5})b| {\bar{B}}_{s}^{0}({p}_{1})\rangle \\ \quad =\,2{\rm{i}}{\varepsilon }_{\nu \rho \alpha \beta }{\varepsilon }^{* \rho }{p}_{1}^{\alpha }{p}_{2}^{\beta }{T}_{1}({q}^{2})\\ \qquad +\,\left[{\varepsilon }_{\nu }^{* }({m}_{{B}_{s}}^{2}-{m}_{\phi }^{2})-({\varepsilon }^{* }\cdot q){\left({p}_{1}+{p}_{2}\right)}_{\nu }\right]{T}_{2}({q}^{2})\\ \qquad +\,({\varepsilon }^{* }\cdot q)\left[{q}_{\nu }-\displaystyle \frac{{q}^{2}}{{m}_{{B}_{s}}^{2}-{m}_{\phi }^{2}}{\left({p}_{1}+{p}_{2}\right)}_{\nu }\right]{T}_{3}({q}^{2}),\end{array}\end{eqnarray}$ where T1, T2 and T3 are the transition form factors associated with the current of ${j}_{\nu }^{T}(0)=\bar{s}{\sigma }_{\nu \lambda }{q}^{\lambda }(1+{\gamma }_{5})b$ .
As what we did in [14], at first we consider a three-point correlation function that is defined as $ \begin{eqnarray}{{\rm{\Pi }}}_{\mu \nu }={{\rm{i}}}^{2}\int {{\rm{d}}}^{4}x{{\rm{d}}}^{4}y{{\rm{e}}}^{{\rm{i}}{p}_{2}\cdot x-{\rm{i}}{p}_{1}\cdot y}\langle 0| T\{{j}_{\mu }^{\phi }(x){j}_{\nu }^{T}(0){j}_{5}(y)\}| 0\rangle ,\end{eqnarray}$ where ${j}_{\mu }^{\phi }(x)=\bar{s}(x){\gamma }_{\mu }s(x)$ , ${j}_{\nu }^{T}(0)=\bar{s}{\sigma }_{\nu \lambda }{q}^{\lambda }(1+{\gamma }_{5})b$ and ${j}_{5}(y)=\bar{b}(y){\rm{i}}{\gamma }_{5}s(y)$ , which are the current of φ channel, the current of weak transition and the current of ${\bar{B}}_{s}^{0}$ channel, respectively.
Next we reexpress the correlation function by using the double dispersion relation $ \begin{eqnarray}{{\rm{\Pi }}}_{\mu \nu }=\int {\rm{d}}{s}_{1}{\rm{d}}{s}_{2}\displaystyle \frac{\rho ({s}_{1},{s}_{2},{q}^{2})}{({s}_{1}-{p}_{1}^{2})({s}_{2}-{p}_{2}^{2})},\end{eqnarray}$ where the spectral density function ρ (s1, s2, q2) can be expressed as the form containing a full set of intermediate hadronic states as shown below $ \begin{eqnarray}\begin{array}{rcl}\rho ({s}_{1},{s}_{2},{q}^{2}) & = & \displaystyle \sum _{X}\displaystyle \sum _{Y}\langle 0| {j}_{\mu }^{\phi }| X\rangle \langle X| {j}_{\nu }^{T}| Y\rangle \langle Y| {j}_{5}| 0\rangle \delta \\ & & ({s}_{1}-{m}_{Y}^{2})\delta ({s}_{2}-{m}_{X}^{2})\theta ({p}_{X}^{0})\theta ({p}_{Y}^{0}),\end{array}\end{eqnarray}$ where X and Y denote the full set of hadronic states of φ and ${\bar{B}}_{s}^{0}$ channels, respectively. According to equations (7 ) and (8 ), we can integrate over s1 and s2, then separate the ground states, excited states and continuum states, the correlation function can be expressed as $ \begin{eqnarray}\begin{array}{rcl}{{\rm{\Pi }}}_{\mu \nu } & = & \displaystyle \frac{{m}_{\phi }{f}_{\phi }{\varepsilon }_{\mu }^{(\lambda )}\langle \phi ({\varepsilon }_{\mu }^{(\lambda )},{p}_{2})| {j}_{\nu }^{T}| {\bar{B}}_{s}^{0}({p}_{1})\rangle {f}_{{B}_{s}}{m}_{{B}_{s}}^{2}}{({m}_{{B}_{s}}^{2}-{p}_{1}^{2})({m}_{\phi }^{2}-{p}_{2}^{2})({m}_{b}+{m}_{s})}\\ & & +{\rm{excited}}\,{\rm{and}}\,{\rm{continuum}}\,{\rm{states}}.\end{array}\end{eqnarray}$ In the above equation, we have used the following definition of relevant matrix elements $ \begin{eqnarray}\begin{array}{rcl} & & \langle 0| \bar{s}{\gamma }_{\mu }s| \phi \rangle ={m}_{\phi }{f}_{\phi }{\varepsilon }_{\mu }^{(\lambda )},\\ & & \langle 0| \bar{s}{\rm{i}}{\gamma }_{5}b| {\bar{B}}_{s}^{0}\rangle =\displaystyle \frac{{f}_{{B}_{s}}{m}_{{B}_{s}}^{2}}{{m}_{b}+{m}_{s}},\end{array}\end{eqnarray}$ where fφ and ${f}_{{B}_{s}}$ are decay constants of the relevant mesons. In principle, φ and ω can mix via strong interaction, the mixing angle δ between nonstrange and strange quark wave function has been analyzed to be $\delta =-{(3.34\pm 0.17)}^{\circ }$ [16–20], which shows that φ meson is dominated by component $s\bar{s}$ . Therefore, we can safely drop the mixing effect of ω − φ in ${D}_{s}\to \phi $ transition process, and φ meson is treated as $s\bar{s}$ component, which is referred to as ideal mixing.
By taking the operator product expansion (OPE) for the time-ordered current operator in equation (6 ), we can get another expression for the correlation function in terms of Wilson coefficients and condensates of local operators $ \begin{eqnarray}\begin{array}{rcl}{{\rm{\Pi }}}_{\mu \nu } & = & {{\rm{i}}}^{2}\displaystyle \int {{\rm{d}}}^{4}x{{\rm{d}}}^{4}y{{\rm{e}}}^{{\rm{i}}{p}_{2}\cdot x-{\rm{i}}{p}_{1}\cdot y}\langle 0| T\{{j}_{\mu }^{\phi }(x){j}_{\nu }^{T}(0){j}_{5}(y)\}| 0\rangle \\ & = & {C}_{0\mu \nu }I+{C}_{3\mu \nu }\langle 0| \bar{{\rm{\Psi }}}{\rm{\Psi }}| 0\rangle +{C}_{4\mu \nu }\langle 0| {G}_{\alpha \beta }^{a}{G}^{a\alpha \beta }| 0\rangle \\ & & +{C}_{5\mu \nu }\langle 0| \bar{{\rm{\Psi }}}{\sigma }_{\alpha \beta }{T}^{a}{G}^{a\alpha \beta }{\rm{\Psi }}| 0\rangle \\ & & +{C}_{6\mu \nu }\langle 0| \bar{{\rm{\Psi }}}{\rm{\Gamma }}{\rm{\Psi }}\bar{{\rm{\Psi }}}{{\rm{\Gamma }}}^{{\prime} }{\rm{\Psi }}| 0\rangle +\ \cdots ,\end{array}\end{eqnarray}$ where ${C}_{i\mu \nu }$ denotes Wilson coefficients. I, $\bar{{\rm{\Psi }}}{\rm{\Psi }}$ and ${G}_{\alpha \beta }^{a}$ are the unit operator, the local fermion field operator of light quarks and the gluon strength tensor, respectively. Γ and ${{\rm{\Gamma }}}^{{\prime} }$ are the matrices that appear in the calculation of Wilson’s coefficients. From the Lorentz structure of the correlation function, we can know that equation (11 ) can be rewritten as $ \begin{eqnarray}\begin{array}{rcl}{{\rm{\Pi }}}_{\mu \nu } & = & {\rm{i}}{\kappa }_{0}{\varepsilon }_{\mu \nu \alpha \beta }{p}_{1}^{\alpha }{p}_{2}^{\beta }+({\kappa }_{1}{p}_{1\mu }{p}_{1\nu }+{\kappa }_{2}{p}_{2\mu }{p}_{2\nu }\\ & & +{\kappa }_{3}{p}_{1\mu }{p}_{2\nu }+{\kappa }_{4}{p}_{1\nu }{p}_{2\mu }+{\kappa }_{5}{g}_{\mu \nu }).\end{array}\end{eqnarray}$ The coefficients κi's contain perturbative and condensate contributions $ \begin{eqnarray}{\kappa }_{i}={\kappa }_{i}^{\mathrm{pert}}+{\kappa }_{i}^{(3)}+{\kappa }_{i}^{(4)}+{\kappa }_{i}^{(5)}+{\kappa }_{i}^{(6)}+\ \cdots ,\end{eqnarray}$ where ${\kappa }_{i}^{\mathrm{pert}}$ is the perturbative contribution, and ${\kappa }_{i}^{(3)}$ , ${\kappa }_{i}^{(4)}$ , ${\kappa }_{i}^{(5)}$ , ${\kappa }_{i}^{(6)}$ , ⋯ are contributions of condensates of operators with increasing dimension in OPE.
Since the perturbative contribution and gluon-condensate contribution contain the loop integral of momentum, we can obtain the dispersion integrals of ${\kappa }_{i}^{\mathrm{pert}}$ and ${\kappa }_{i}^{(4)}$ , which can be expressed as $ \begin{eqnarray}\begin{array}{rcl}{\kappa }_{i}^{\mathrm{pert}} & = & {\displaystyle \int }_{{s}_{1}^{L}}^{\infty }{\rm{d}}{s}_{1}{\displaystyle \int }_{{s}_{2}^{L}}^{\infty }{\rm{d}}{s}_{2}\displaystyle \frac{{\rho }_{i}^{\mathrm{pert}}({s}_{1},{s}_{2},{q}^{2})}{({s}_{1}-{p}_{1}^{2})({s}_{2}-{p}_{2}^{2})},\\ {\kappa }_{i}^{(4)} & = & {\displaystyle \int }_{{s}_{1}^{L}}^{\infty }{\rm{d}}{s}_{1}{\displaystyle \int }_{{s}_{2}^{L}}^{\infty }{\rm{d}}{s}_{2}\displaystyle \frac{{\rho }_{i}^{(4)}({s}_{1},{s}_{2},{q}^{2})}{({s}_{1}-{p}_{1}^{2})({s}_{2}-{p}_{2}^{2})},\end{array}\end{eqnarray}$ where ${s}_{1}^{L}$ and ${s}_{2}^{L}$ are the lower limits of s1 and s2, respectively, which can be found in appendix A . In principle equations (9 ) and (12 ) should be equivalent to each other, because they are two different expressions for the same correlation function ${{\rm{\Pi }}}_{\mu \nu }$ . By using the assumption of quark-hadron duality [9, 10], one can approximate the contribution of the higher excited and continuum states in ${{\rm{\Pi }}}_{\mu \nu }$ in equation (9 ) as the integration of $\int {{\rm{d}}{s}}_{1}{{\rm{d}}{s}}_{2}$ in equation (14 ) over some thresholds ${s}_{1}^{0}$ and ${s}_{2}^{0}$ . Then one can get rid of the contribution of the higher excited and continuum states in equation (9 ), and obtain an equation for the form factors by equating equations (9 ) and (12 ), where equation (14 ) should be replaced as $ \begin{eqnarray}\begin{array}{rcl}{\kappa }_{i}^{\mathrm{pert}} & = & {\displaystyle \int }_{{s}_{1}^{L}}^{{s}_{1}^{0}}{\rm{d}}{s}_{1}{\displaystyle \int }_{{s}_{2}^{L}}^{{s}_{2}^{0}}{\rm{d}}{s}_{2}\displaystyle \frac{{\rho }_{i}^{\mathrm{pert}}({s}_{1},{s}_{2},{q}^{2})}{({s}_{1}-{p}_{1}^{2})({s}_{2}-{p}_{2}^{2})},\\ {\kappa }_{i}^{(4)} & = & {\displaystyle \int }_{{s}_{1}^{L}}^{{s}_{1}^{0}}{\rm{d}}{s}_{1}{\displaystyle \int }_{{s}_{2}^{L}}^{{s}_{2}^{0}}{\rm{d}}{s}_{2}\displaystyle \frac{{\rho }_{i}^{(4)}({s}_{1},{s}_{2},{q}^{2})}{({s}_{1}-{p}_{1}^{2})({s}_{2}-{p}_{2}^{2})}.\end{array}\end{eqnarray}$ In order to improve the equation, Borel transformation needs to be introduced, that is, for any function f(x2),
$ \begin{eqnarray*}{\hat{B}}_{\left|\tfrac{}{}\right.{x}^{2},{M}^{2}}f({x}^{2})=\mathop{\mathrm{lim}}\limits_{\begin{array}{cc} k\to \infty ,{x}^{2}\to -\infty \\ -{x}^{2}/k={M}^{2}\end{array}}\displaystyle \frac{{\left(-{x}^{2}\right)}^{k}}{(k-1)!}\displaystyle \frac{{\partial }^{k}}{\partial {\left({x}^{2}\right)}^{k}}f({x}^{2}).\end{eqnarray*}$ Borel transformation can suppress both the contribution of higher excited states and contributions of operators of higher dimension in OPE. Then matching these two forms of the correlation function in equations (9 ) and (12 ), and performing Borel transformation for both variables ${p}_{1}^{2}$ and ${p}_{2}^{2}$ , QCD sum rules for these three form factors related to matrix hadronic element $\langle \phi (\varepsilon ,{p}_{2})| \bar{s}{\sigma }_{\nu \lambda }{q}^{\lambda }(1+{\gamma }_{5})b| {\bar{B}}_{s}^{0}({p}_{1})\rangle $ can be obtained $ \begin{eqnarray}\begin{array}{l}{T}_{1}({q}^{2})\\ =\,\displaystyle \frac{({m}_{b}+{m}_{s})}{2{m}_{\phi }{f}_{\phi }{f}_{{B}_{s}}{m}_{{B}_{s}}^{2}}{{\rm{e}}}^{{m}_{{B}_{s}}^{2}/{M}_{1}^{2}}{{\rm{e}}}^{{m}_{\phi }^{2}/{M}_{2}^{2}}{M}_{1}^{2}{M}_{2}^{2}\cdot \hat{B}{\kappa }_{0},\\ {T}_{2}({q}^{2})\\ =\,-\displaystyle \frac{({m}_{b}+{m}_{s})}{{m}_{\phi }{f}_{\phi }{f}_{{B}_{s}}{m}_{{B}_{s}}^{2}({m}_{{B}_{s}}^{2}-{m}_{\phi }^{2})}{{\rm{e}}}^{{m}_{{B}_{s}}^{2}/{M}_{1}^{2}}{{\rm{e}}}^{{m}_{\phi }^{2}/{M}_{2}^{2}}{M}_{1}^{2}{M}_{2}^{2}\cdot \hat{B}{\kappa }_{5},\\ {T}_{3}({q}^{2})\\ =\,-\displaystyle \frac{({m}_{b}+{m}_{s})}{{m}_{\phi }{f}_{\phi }{f}_{{B}_{s}}{m}_{{B}_{s}}^{2}}{{\rm{e}}}^{{m}_{{B}_{s}}^{2}/{M}_{1}^{2}}{{\rm{e}}}^{{m}_{\phi }^{2}/{M}_{2}^{2}}{M}_{1}^{2}{M}_{2}^{2}\cdot \displaystyle \frac{1}{2}\hat{B}({\kappa }_{1}-{\kappa }_{3}),\end{array}\end{eqnarray}$ where $\hat{B}{\kappa }_{i}$ denotes Borel transformation of κi for both variables ${p}_{1}^{2}$ and ${p}_{2}^{2}$ . M1 and M2 are Borel parameters.
4. The calculation of the Wilson coefficients
In this section, we discuss the calculation of Wilson coefficients in the OPE. The diagrams to be considered here are similar to that used in our previous work in [14]. The difference is that the weak transition current ${j}_{\nu }(0)\,=\bar{s}{\gamma }_{\nu }(1-{\gamma }_{5})b$ is replaced by the tensor current ${j}_{\nu }^{T}(0)\,=\bar{s}{\sigma }_{\nu \lambda }{q}^{\lambda }(1+{\gamma }_{5})b$ appearing in equation (3 ).
Here we only depict the diagrams for contributions of gluon–gluon operator in figure 1, because our calculation shows that the contribution of gluon–gluon operator does not completely cancel out for the tensor current, which is different from the case of V−A current. But the contributions of these diagrams are very small compared with other operators. Different from the treatment in [6], we do not ignore these contributions in the following calculations.
Figure 1. Diagrams for contributions of gluon–gluon operator. |
The cancellation of the contribution of gluon–gluon operator for the case of V–A current seems not because of any symmetry principle. It is only only because, in the fixed-point gauge the color field can be expanded as ${A}_{\mu }^{a}(z)\,={\int }_{0}^{1}{\rm{d}}\beta \beta {z}^{\rho }{G}_{\rho \mu }^{a}(\beta z)$ $=\,\tfrac{1}{2}{z}^{\rho }{G}_{\rho \mu }^{a}(0)+\cdots $ at leading order, only at leading order the contribution of gluon–gluon operator vanish. If the higher order in the expansion ${A}_{\mu }^{a}(z)=\tfrac{1}{2}{z}^{\rho }{G}_{\rho \mu }^{a}(0)$ $+\,\tfrac{1}{3}{z}^{\alpha }{z}^{\rho }{\hat{D}}_{\alpha }{G}_{\rho \mu }^{a}(0)+\cdots \,$ is considered, the contribution may not vanish for the case of V–A current, but it must be small because of the short-distance nature of Wilson coefficients.
The final results of Borel transformed coefficients $\hat{B}{\kappa }_{0}$ , $\hat{B}({\kappa }_{1}-{\kappa }_{3})$ and $\hat{B}{\kappa }_{5}$ in equation (16 ) are given in appendix A .
5. Numerical analysis and discussion
The input parameters required for numerical calculation are taken as followings [9–11]: $ \begin{eqnarray}\begin{array}{l}\langle \bar{q}q\rangle =-{\left(0.24\pm 0.01{\rm{GeV}}\right)}^{3},\,\,\,\,\langle \bar{s}s\rangle =(0.8\pm 0.2)\langle \bar{q}q\rangle ,\\ g\langle \bar{{\rm{\Psi }}}\sigma {TG}{\rm{\Psi }}\rangle ={m}_{0}^{2}\langle \bar{{\rm{\Psi }}}{\rm{\Psi }}\rangle ,\,\,{\alpha }_{s}\langle \bar{{\rm{\Psi }}}{\rm{\Psi }}{\rangle }^{2}=6.0\times {10}^{-5}\,{{\rm{GeV}}}^{6},\\ {\alpha }_{s}\langle {GG}\rangle =0.038\,{{\rm{GeV}}}^{4},\,\,{m}_{0}^{2}=0.8\pm 0.2\,{{\rm{GeV}}}^{2}.\end{array}\end{eqnarray}$
The standard values of the condensates above at the renormalization point μ = 1 GeV are from [9–11], and the relevant mass parameters and decay constants are [21, 22] $ \begin{eqnarray}\begin{array}{rcl} & & {m}_{s}=95\,{\rm{MeV}},\,\,\,\,\,\,\,\,\,\,{m}_{b}=4.18\,{\rm{GeV}},\\ & & {m}_{e}=0.511\,{\rm{MeV}},\\ & & {m}_{\mu }=0.106\,{\rm{GeV}},\,\,\,\,\,\,\,{m}_{\tau }=1.777\,{\rm{GeV}},\\ & & {m}_{\phi }=1.02\,{\rm{GeV}},\\ & & {m}_{{B}_{s}}=5.367\,{\rm{GeV}},\,\,\,\,\,{m}_{J/\psi }=3.097\,{\rm{GeV}},\\ & & {m}_{{\psi }^{{\prime} }}=3.686\,{\rm{GeV}},\\ & & {f}_{{B}_{s}}=0.266\pm 0.019\,{\rm{GeV}},\,\,\,\,\,\,\,{f}_{\phi }=0.228\,{\rm{GeV}}.\end{array}\end{eqnarray}$ Other parameters to be used include [21]: $ \begin{eqnarray}\begin{array}{rcl}{G}_{F} & = & 1.1663787\times {10}^{-5}\,{{\rm{GeV}}}^{-2},\\ \alpha & = & 7.297\times {10}^{-3},\,\,\,| {V}_{{ts}}^{* }{V}_{{tb}}| =0.039741,\end{array}\end{eqnarray}$ and the threshold parameters ${s}_{1}^{0}$ and ${s}_{2}^{0}$ for ${\bar{B}}_{s}^{0}$ and φ mesons are $ \begin{eqnarray}{s}_{1}^{0}=34.9\sim 35.9\,{{\rm{GeV}}}^{2},\,\,\,\,{s}_{2}^{0}=1.9\sim 2.1\,{{\rm{GeV}}}^{2}.\end{eqnarray}$
Table 1. Wilson coefficients (at renormalization scale μ = mb). |
| C1 | C2 | C3 | C4 | C5 | C6 | Ceff7 | C9 | C10 |
|---|---|---|---|---|---|---|---|---|
| −0.176 | 1.078 | 0.014 | −0.034 | 0.008 | −0.039 | −0.313 | 4.344 | −4.669 |
Next we need to select the appropriate regions for Borel parameters M1 and M2. In our previous works [14, 25, 26], we have discussed the selection of Borel parameters in detail. So we do not repeat the details in this paper. The requirements to select Borel Parameters are directly given in table 2, and the selected two-dimensional region for M1 and M2 are depicted in figure 2.
Figure 2. Selected stability regions of |
Table 2. Requirements to select Borel parameters |
| Form factors | Contribution of condensate | Continuum of | Continuum of φ channel |
|---|---|---|---|
| T1(0) | ≤54.4% | ≤15.5% | ≤56% |
| T2(0) | | ≤12% | ≤56% |
| T3(0) | ≤55.4% | ≤41.2% | ≤56.8% |
After numerical analysis, the final results for the form factors at q2 = 0 are $ \begin{eqnarray}\begin{array}{rcl}{T}_{1}(0) & = & 0.33\pm 0.07,\\ {T}_{2}(0) & = & 0.33\pm 0.07,\\ {T}_{3}(0) & = & 0.22\pm 0.05,\end{array}\end{eqnarray}$ where the errors are estimated by the uncertainty of the standard values of the condensates, the variation of the threshold parameters ${s}_{1}^{0}$ and ${s}_{2}^{0}$ , the variation of Borel parameters, and the variation of the other input parameters. The error caused by the uncertainty of the condensates is about 25% of the central value of the form factors, the error caused by the variation of the threshold parameters ${s}_{1,2}^{0}$ is about 5% of the central value, the error caused by the variation of Borel parameters is about 6% of the central value, and the error caused by the uncertainty of the other input parameters is less than a few percent. All the errors are added quadratically. In addition, the b quark mass given by [21] is ${m}_{b}={4.18}_{-0.03}^{+0.04}$ . The error caused by the uncertainty of b quark mass is about 0.8%, which is much smaller than the errors caused by the other sources.
The comparison of the form factors obtained in this work in equation (21 ) with other theoretical results calculated by LCSR in [2], CQM in [4], RQM in [7], and also in QCD sum rule in [6] are shown in table 3. Some of the form factors obtained in [6] are different from others by a negative sign. This will affect the physical results of the differential decay width of ${\bar{B}}_{s}^{0}\to \phi {l}^{+}{l}^{-}$ . By comparison, we find that the results of T1(0), T2(0) and T3(0) in our work, especially the value of T3(0), are more consistent with the results obtained by LCSR method in [2] within the range of uncertainty. Comparing the OPE coefficients in [6] with the relevant coefficients in this work, we find that the reason for the difference is that there is no contribution of ${m}_{b}/{M}_{1}^{2}{M}_{2}^{2}$ and ${m}_{s}/{M}_{1}^{2}{M}_{2}^{2}$ in [6]. The contribution of these two types of terms comes from the first term in the right side of equation (22 ) [11, 25], which gives the main contribution in our calculation $ \begin{eqnarray}\begin{array}{l}\langle 0| {\bar{{\rm{\Psi }}}}_{\alpha }^{a}(x){{\rm{\Psi }}}_{\beta }^{b}(y)| 0\rangle ={\delta }_{{ab}}\left[\langle \bar{{\rm{\Psi }}}{\rm{\Psi }}\rangle \left(\displaystyle \frac{1}{12}{\delta }_{\beta \alpha }\right.\right.\\ \quad +\,{\rm{i}}\displaystyle \frac{m}{48}{\left({/}\!\!\!\!{x}-{/}\!\!\!\!{y}\right)}_{\beta \alpha }-\displaystyle \frac{{m}^{2}}{96}{\left(x-y\right)}^{2}{\delta }_{\beta \alpha }\\ \quad \left.-\displaystyle \frac{{\rm{i}}}{3!}\displaystyle \frac{{m}^{3}}{96}{\left(x-y\right)}^{2}{\left({/}\!\!\!\!{x}-{/}\!\!\!\!{y}\right)}_{\beta \alpha }\right)\\ \quad +\,g\langle \bar{{\rm{\Psi }}}\sigma {TG}{\rm{\Psi }}\rangle \left(\displaystyle \frac{1}{192}{\left(x-y\right)}^{2}{\delta }_{\beta \alpha }\right.\\ \quad \left.+\,\displaystyle \frac{{\rm{i}}}{3!}\displaystyle \frac{m}{192}{\left(x-y\right)}^{2}{\left({/}\!\!\!\!{x}-{/}\!\!\!\!{y}\right)}_{\beta \alpha }\right)\\ \quad -\,\displaystyle \frac{{\rm{i}}}{3!}\displaystyle \frac{{g}^{2}}{{3}^{4}\times {2}^{4}}\langle \bar{{\rm{\Psi }}}{\rm{\Psi }}{\rangle }^{2}{\left(x-y\right)}^{2}{\left({/}\!\!\!\!{x}-{/}\!\!\!\!{y}\right)}_{\beta \alpha }\left.+\,\cdots \right].\end{array}\end{eqnarray}$ Moreover, the contribution of the operator of dimension-5 is greater than that of the operator of dimension-3 in [6], which is also different from our calculation.
The physical region for q2 in ${\bar{B}}_{s}^{0}\to \phi {l}^{+}{l}^{-}$ decay is: ${(2{m}_{l})}^{2}\leqslant {q}^{2}\leqslant {({m}_{{B}_{s}}-{m}_{\phi })}^{2}$ . The q2-dependence of the form factors within this range is shown in figure 3 using the central values of the input parameters. We can find that the q2-dependence of T1(q2) calculated in QCD sum rule can be well fitted by the single-pole model $ \begin{eqnarray}{T}_{1}({q}^{2})=\displaystyle \frac{{T}_{1}(0)}{1-{q}^{2}/{\left({m}_{\mathrm{pole}}^{{T}_{1}}\right)}^{2}},\end{eqnarray}$ while the q2-dependences of ${T}_{2}({q}^{2})$ and ${T}_{3}({q}^{2})$ are very weak, so we can take ${T}_{2}({q}^{2})={T}_{2}(0)$ , ${T}_{3}({q}^{2})={T}_{3}(0)$ as approximations. The weak dependence of ${T}_{2}({q}^{2})$ and ${T}_{3}({q}^{2})$ on q2 stems from the mutual cancellation of the perturbative contribution and the condensate contribution. For T2(q2), the perturbative contribution increases as q2 being large, while the contribution of condensates decreases, and as a sum the q2-dependence cancel mostly. For T3(q2), the perurbative contribution decreases while the condensates contribution increases as q2 being large. This is similar to the behavior of the form factors for D decays found in [27]. The weak dependence of T2,3(q2) on q2 calculated from QCD sum rule implies that the assumption of single-pole behavior for form factors is not always appropriate.
Figure 3. q2-dependence of the form factors from QCD sum rule. The solid curve is for T1(q2), the dashed curve for T2(q2), and the dotted curve for |
The pole mass in the expression of T1(q2) above obtained by fitting the results calculated by QCD sum rule is $ \begin{eqnarray}{m}_{\mathrm{pole}}^{{T}_{1}}=5.38\pm 0.23\,\,{\rm{GeV}}.\end{eqnarray}$
Table 4. Form factors related to |
| | | | | |
|---|---|---|---|---|
| q2 = 0 | 0.30 ± 0.25 | 0.32 ± 0.07 | 0.30 ± 0.07 | 0.45 ± 0.10 |
| | | A1(0) | | |
| mpole | 5.62 ± 2.38 GeV | − | 9.20 ± 0.40 GeV | 5.59 ± 0.27 GeV |
Next we shall use all of the ${\bar{B}}_{s}^{0}\to \phi $ transition form factors V, A0, A1, A2 and T1, T2, T3 calculated by QCD sum rules to investigate the differential decay widths and branching ratios of ${\bar{B}}_{s}^{0}\to \phi {l}^{+}{l}^{-}$ decays. The expression of differential decay width is given as [24] $ \begin{eqnarray}\begin{array}{l}\displaystyle \frac{{\rm{d}}{\rm{\Gamma }}({\bar{B}}_{s}^{0}\to \phi {l}^{+}{l}^{-})}{{\rm{d}}\hat{s}}=\displaystyle \frac{{G}_{F}^{2}{\alpha }^{2}{m}_{{B}_{s}}^{5}}{{2}^{10}{\pi }^{5}}| {V}_{{ts}}^{* }{V}_{{tb}}{| }^{2}\,\hat{u}(\hat{s})\\ \quad \times \,\left\{\displaystyle \frac{| A{| }^{2}}{3}\hat{s}\lambda (1+2\displaystyle \frac{{\hat{m}}_{l}^{2}}{\hat{s}})+| E{| }^{2}\hat{s}\displaystyle \frac{\hat{u}{\left(\hat{s}\right)}^{2}}{3}\right.\\ \quad +\,\displaystyle \frac{1}{4{\hat{m}}_{\phi }^{2}}\left[| B{| }^{2}\left(\lambda -\displaystyle \frac{\hat{u}{\left(\hat{s}\right)}^{2}}{3}+8{\hat{m}}_{\phi }^{2}(\hat{s}+2{\hat{m}}_{l}^{2})\right)\right.\\ \quad \left.+\,| F{| }^{2}\left(\lambda -\displaystyle \frac{\hat{u}{\left(\hat{s}\right)}^{2}}{3}+8{\hat{m}}_{\phi }^{2}(\hat{s}-4{\hat{m}}_{l}^{2})\right)\right]\\ \quad +\,\displaystyle \frac{\lambda }{4{\hat{m}}_{\phi }^{2}}\left[| C{| }^{2}\left(\lambda -\displaystyle \frac{\hat{u}{\left(\hat{s}\right)}^{2}}{3}\right)\right.\\ \quad \left.+\,| G{| }^{2}\left(\lambda -\displaystyle \frac{\hat{u}{\left(\hat{s}\right)}^{2}}{3}+4{\hat{m}}_{l}^{2}(2+2{\hat{m}}_{\phi }^{2}-\hat{s})\right)\right]\\ \quad -\,\displaystyle \frac{1}{2{\hat{m}}_{\phi }^{2}}\left[{\rm{Re}}({{BC}}^{* })\left(\lambda -\displaystyle \frac{\hat{u}{\left(\hat{s}\right)}^{2}}{3}\right)(1-{\hat{m}}_{\phi }^{2}-\hat{s})\right.\\ \quad \left.+\,{\rm{Re}}({{FG}}^{* })\left(\left(\lambda -\displaystyle \frac{\hat{u}{\left(\hat{s}\right)}^{2}}{3}\right)(1-{\hat{m}}_{\phi }^{2}-\hat{s})+4{\hat{m}}_{l}^{2}\lambda \right)\right]\\ \quad -\,2\displaystyle \frac{{\hat{m}}_{l}^{2}}{{\hat{m}}_{\phi }^{2}}\lambda \left[{\rm{Re}}({{FH}}^{* })-{\rm{Re}}({{GH}}^{* })(1-{\hat{m}}_{\phi }^{2})\right]\\ \quad \left.+\,\displaystyle \frac{{\hat{m}}_{l}^{2}}{{\hat{m}}_{\phi }^{2}}\hat{s}\lambda | H{| }^{2}\right\},\end{array}\end{eqnarray}$ where s = q2, $\hat{s}=s/{m}_{{B}_{s}}^{2}$ , ${\hat{m}}_{q}={m}_{q}/{m}_{{B}_{s}}$ , $\hat{u}(\hat{s})\,=\sqrt{\lambda (1-4{\hat{m}}_{l}^{2}/\hat{s})}$ , $\lambda \equiv \lambda (1,{\hat{m}}_{\phi }^{2},\hat{s})=1+{\hat{m}}_{\phi }^{4}+{\hat{s}}^{2}-2\hat{s}\,-2{\hat{m}}_{\phi }^{2}(1+\hat{s})$ , and the specific expressions of $A(\hat{s})\sim H(\hat{s})$ can be found in [24], which are not listed here for brevity.
Considering the possible long-distance (LD) effects and to avoid the contributions of resonances, some cuts around the resonances of J/ψ and ${\psi }^{{\prime} }$ are taken in the physical distribution of q2. We use the same cuts as that used by LHCb Collaboration in [12]. There are three regions for ${\bar{B}}_{s}^{0}\to \phi {e}^{+}{e}^{-}$ and ${\bar{B}}_{s}^{0}\to \phi {\mu }^{+}{\mu }^{-}$ decays: $ \begin{eqnarray}\begin{array}{rcl}{\rm{i}}\,: & & \,0.1\,{\mathrm{GeV}}^{2}\leqslant {q}^{2}\leqslant 8.0\,{\mathrm{GeV}}^{2}\,;\\ \mathrm{ii}\,: & & \,11.0\,{\mathrm{GeV}}^{2}\leqslant {q}^{2}\leqslant 12.5\,{\mathrm{GeV}}^{2}\,;\\ \mathrm{iii}\,: & & \,15.0\,{\mathrm{GeV}}^{2}\leqslant {q}^{2}\leqslant 19.0\,{\mathrm{GeV}}^{2}\,.\end{array}\end{eqnarray}$ and two regions for ${\bar{B}}_{s}^{0}\to \phi {\tau }^{+}{\tau }^{-}$ decay: $ \begin{eqnarray}\begin{array}{rcl}{\rm{i}}\,: & & \,11.0\,{\mathrm{GeV}}^{2}\leqslant {q}^{2}\leqslant 12.5\,{\mathrm{GeV}}^{2}\,;\\ \mathrm{ii}\,: & & \,15.0\,{\mathrm{GeV}}^{2}\leqslant {q}^{2}\leqslant 19.0\,{\mathrm{GeV}}^{2}\,.\end{array}\end{eqnarray}$
The q2-dependence of differential decay widths with LD effects are shown in figure 4, where the gray bands denote the relevant uncertainties. Integrating the differential decay width in equation (25 ) with respect to q2 within the relevant region, we can obtain the value of integrated decay width ${\rm{\Gamma }}({\bar{B}}_{s}^{0}\to \phi {l}^{+}{l}^{-})$ . According to the definition of decay branching ratio $ \begin{eqnarray}{\rm{Br}}({\bar{B}}_{s}^{0}\to \phi {l}^{+}{l}^{-})=\displaystyle \frac{{\rm{\Gamma }}({\bar{B}}_{s}^{0}\to \phi {l}^{+}{l}^{-})}{{{\rm{\Gamma }}}_{\mathrm{total}}({\bar{B}}_{s}^{0})},\end{eqnarray}$ and the total decay width of ${\bar{B}}_{s}^{0}$ meson: ${{\rm{\Gamma }}}_{\mathrm{total}}({\bar{B}}_{s}^{0})=4.362\,\times {10}^{-13}\,{\rm{GeV}}$ [21], we can get the branching ratios of the three semileptonic decay channels of ${\bar{B}}_{s}^{0}\to \phi {l}^{+}{l}^{-}$ (l = e, μ, τ), $ \begin{eqnarray}{\rm{Br}}({\bar{B}}_{s}^{0}\to \phi {e}^{+}{e}^{-})=(7.12\pm 1.40)\times {10}^{-7},\end{eqnarray}$ $ \begin{eqnarray}{\rm{Br}}({\bar{B}}_{s}^{0}\to \phi {\mu }^{+}{\mu }^{-})=(7.06\pm 1.59)\times {10}^{-7},\end{eqnarray}$ $ \begin{eqnarray}{\rm{Br}}({\bar{B}}_{s}^{0}\to \phi {\tau }^{+}{\tau }^{-})=(3.49\pm 1.69)\times {10}^{-8}.\end{eqnarray}$ The experimental result of the total branching ratio of ${\bar{B}}_{s}^{0}\to \phi {\mu }^{+}{\mu }^{-}$ is [12] $ \begin{eqnarray}\begin{array}{l}{\rm{Br}}({\bar{B}}_{s}^{0}\to \phi {\mu }^{+}{\mu }^{-})\\ \quad =\,({7.97}_{-0.43}^{+0.45}\pm 0.22\pm 0.23\pm 0.60)\times {10}^{-7}.\end{array}\end{eqnarray}$ We find agreement between our predictions and the experimental data within uncertainties.
Figure 4. The differential decay widths of |
Furthermore, in order to show the physical effects caused by the sign of the form factors, we change the sign of the form factors V, A1, T1 and T3 as that of [6] to calculate the branching ratio of ${\bar{B}}_{s}^{0}\to \phi {\mu }^{+}{\mu }^{-}$ again, and obtain the central value of the branching ratio of as follows $ \begin{eqnarray}{\rm{Br}}({\bar{B}}_{s}^{0}\to \phi {\mu }^{+}{\mu }^{-})=6.14\times {10}^{-6}.\end{eqnarray}$ From equation (33 ) we can find that the branching ratio of ${\bar{B}}_{s}^{0}\to \phi {\mu }^{+}{\mu }^{-}$ calculated in this way is nearly an order of magnitude larger than the experimental data in equation (32 ). So the physical effect of the sign of the form factors are crucial.
6. Summary
We revisit the semileptonic decays of ${\bar{B}}_{s}^{0}\to \phi {l}^{+}{l}^{-}$ (l = e, μ, τ) with QCD sum rule method. The ${\bar{B}}_{s}^{0}\to \phi $ transition form factors V, A0, A1, A2 [14] and T1, T2, T3 are calculated, then they are used to obtain the branching ratios of ${\bar{B}}_{s}^{0}\to \phi {e}^{+}{e}^{-}$ , ${\bar{B}}_{s}^{0}\to \phi {\mu }^{+}{\mu }^{-}$ and ${\bar{B}}_{s}^{0}\to \phi {\tau }^{+}{\tau }^{-}$ respectively. For the measured decay channel ${\bar{B}}_{s}^{0}\to \phi {\mu }^{+}{\mu }^{-}$ , our theoretical result is ${\rm{Br}}({\bar{B}}_{s}^{0}\to \phi {\mu }^{+}{\mu }^{-})\,=(7.06\pm 1.59)\times {10}^{-7}$ , which is well consistent with the latest experimental data ${\rm{Br}}({\bar{B}}_{s}^{0}\to \phi {\mu }^{+}{\mu }^{-})=({7.97}_{-0.43}^{+0.45}\pm 0.22\,\pm 0.23\pm 0.60)\times {10}^{-7}$ from LHCb Collaboration within uncertainties. For the unmeasured decay channels: ${\bar{B}}_{s}^{0}\to \phi {e}^{+}{e}^{-}$ and ${\bar{B}}_{s}^{0}$ → $\phi {\tau }^{+}{\tau }^{-}$ , we hope that our theoretical predictions are useful for experimental test in the future.