Angular observables and branching ratio for \(B_s\to \phi \ell^+ \ell^-\) decay
October 04, 2025
Abstract
In this paper, an analysis of the \(B_s\to \phi \ell^+ \ell^-\) rare decay is presented within the framework of the covariant confined quark model. The \(B_s\to \phi\) transition form factors are calculated and then used to compute the branching fractions and angular observables in various \(q^2\) bins, including the forward-backward asymmetry \(A_{FB}\), the longitudinal polarization \(F_L\), and the optimized observables \(A_i\) and \(S_i\). The results show agreement with the latest experimental data given by LHCb collaboration and compared with available theoretical predictions.
1 Introduction↩︎
Among the \(b\to s\ell^+\ell^-\) transitions, the rare decays \(B\to K^\ast(\to K\pi)\mu^+\mu^-\) and \(B_s\to \phi(\to K^+K^-)\mu^+\mu^-\) are the most prominent and extensively studied. The first angular analysis of \(B^0_s\to \phi \mu^+\mu^-\) decay by LHCb [1] found results largely consistent with the Standard Model (SM), though the measured decay width in one bin was more than 3 \(\sigma\) below SM predictions. These measurements were subsequently refined in 2021, with a new analysis reporting updated branching ratios, branching fractions, and angular observables in bins [2].
\[\begin{align} \frac{ {\cal B}(B_{s}^{0} \to \phi \mu^{+} \mu^{-}) }{ {\cal B}(B_{s}^{0} \to J/\psi \phi) } & = (8.00 \pm 0.21 \pm 0.16 \pm 0.03) \times 10^{-4}, \nonumber\\ {\mathcal{B}} (B_{s}^{0} \to \phi \mu^{+} \mu^{-}) & = (8.14 \pm 0.21 \pm 0.16 \pm 0.03 \pm 0.39)\times 10^{-7}. \nonumber\\ \end{align}\]
The experimental investigation of the \(B_s \to \phi \mu^+\mu^-\) decay began with its observation by the CDF collaboration [3] and has since been extensively refined by LHCb [1], [2], [4]. These studies have progressed beyond simple branching ratio measurements to a detailed analysis of the decay’s full kinematic phenomenology.
A wide range of theoretical studies have been performed on \(B_s \to \phi \mu^+\mu^-\) exclusive decays. Within the Standard Model, these include the covariant and light-front quark models [5]–[7], QCD factorization [8], and light-cone sum rules [9]–[13]. New physics scenarios, such as universal extra dimensions [14] and supersymmetry [15], have also been investigated. Form factors obtained from lattice QCD [16] were used in [17], [18] to calculate the decay widths of \(B\to K^\ast\mu^+\mu^-\) and \(B_s\to \phi \mu^+\mu^-\). PQCD factorization approach with lattice QCD input was used in [19].
2 ↩︎
We express the matrix element through dimensionless form factors [6], [20] as: \[\begin{align} && \langle \phi(p_2,\epsilon_2)\, |\,\bar s\, O^{\,\mu}\,b\, |\,B_{s}(p_1) \rangle \,=\, \nonumber\\ &=& N_c\, g_{B_s}\,g_\phi \!\! \int\!\! \frac{d^4k}{ (2\pi)^4 i}\, \widetilde{\Phi}_{B_s}\Big(-(k+w_{13} p_1)^2\Big)\, \widetilde{\Phi}_\phi\Big(-(k+w_{23} p_2)^2\Big) \nonumber\\ &\times& {\rm tr} \biggl[ O^{\,\mu} \,S_b(k+p_1)\,\gamma^5\, S_s(k) \not\!\epsilon_2^{\,\,\dagger} \, S_s(k+p_2)\, \biggr] \nonumber\\ & = & \frac{\epsilon^{\,\dagger}_{\,\nu}}{m_1+m_2}\, \Big( - g^{\mu\nu}\,P\cdot q\,A_0(q^2) + P^{\,\mu}\,P^{\,\nu}\,A_+(q^2) + q^{\,\mu}\,P^{\,\nu}\,A_-(q^2) \nonumber\\ && + i\,\varepsilon^{\mu\nu\alpha\beta}\,P_\alpha\,q_\beta\,V(q^2)\Big), \label{eq:PV} \end{align}\tag{1}\] \[\begin{align} && \langle \phi(p_2,\epsilon_2)\, |\,\bar s\, (\sigma^{\,\mu\nu}q_\nu(1+\gamma^5))\,b\, |\,B_{s}(p_1) \rangle \,=\, \nonumber\\ &=& N_c\, g_{B_s}\,g_\phi \!\! \int\!\! \frac{d^4k}{ (2\pi)^4 i}\, \widetilde{\Phi}_{B_s}\Big(-(k+w_{13} p_1)^2\Big)\, \widetilde{\Phi}_\phi\Big(-(k+w_{23} p_2)^2\Big) \nonumber\\ &\times& {\rm tr} \biggl[ (\sigma^{\,\mu\nu}q_\nu(1+\gamma^5)) \,S_b(k+p_1)\,\gamma^5\, S_s(k) \not\!\epsilon_2^{\,\,\dagger} \,S_s(k+p_2)\, \biggr] \nonumber\\ & = & \epsilon^{\,\dagger}_{\,\nu}\, \Big( - (g^{\mu\nu}-q^{\,\mu}q^{\,\nu}/q^2)\,P\cdot q\,a_0(q^2) + (P^{\,\mu}\,P^{\,\nu}-q^{\,\mu}\,P^{\,\nu}\,P\cdot q/q^2)\,a_+(q^2) \nonumber\\ && + i\,\varepsilon^{\mu\nu\alpha\beta}\,P_\alpha\,q_\beta\,g(q^2)\Big). \label{eq:PVT} \end{align}\tag{2}\]
Here \(P=p_1+p_2\), \(q=p_1-p_2\), \(\epsilon_2^\dagger\cdot p_2=0\), \(p_1^2=m_1^2\equiv m^2_{B_s}\), \(p_2^2=m_2^2\equiv m^2_\phi\) and the weak matrix \(O^{\,\mu} = \gamma^{\,\mu}(1-\gamma^5)\). \(w_{ij}=m_{q_j}/(m_{q_i}+m_{q_j})\) \((i,j=1,2,3)\), and \(w_{ij}+w_{ji}=1\). The reason given for the two-index notation was that three quarks were involved in the process.
The results of our numerical calculations are approximated by the dipole parametrization \[F(q^2)=\frac{F(0)}{1-a s+b s^2}\,, \qquad s=\frac{q^2}{m_1^2}\,, \label{eq:ff95approx}\tag{3}\] the relative error is less than 1\(\%\). The values of \(F(0)\), \(a\), and \(b\) are for the \(B_s\to \phi\) transition within the covariant confined quark model (CCQM) are listed in Table 1 and same as in our previous work[6].
| \(A_0\) | \(A_+\) | \(A_-\) | \(V\) | \(a_0\) | \(a_+\) | \(g\) | |
|---|---|---|---|---|---|---|---|
| \(F(0)\) | 0.40 | 0.27 | \(-0.29\) | 0.31 | 0.27 | 0.27 | 0.27 |
| \(a\) | 0.62 | 1.41 | 1.48 | 1.51 | 0.66 | 1.41 | 1.52 |
| \(b\) | \(-0.30\) | 0.38 | 0.45 | 0.47 | \(-0.26\) | 0.39 | 0.49 |
Table 2 presents the form factor values from CCQM as BSW form factors (see eqs.23-24 in [6]), and compares them with other theoretical approaches. The superscript on the form factors is omitted when making comparisons with other approaches.
| Model | \(A_0(0)\) | \(A_1(0)\) | \(A_2(0)\) |
|---|---|---|---|
| CCQM | \(0.28\pm 0.03\) | \(0.27\pm 0.03\) | \(0.27\pm 0.03\) |
| LCSR [9] | \(0.474\pm0.037\) | 0.311\(\pm\)0.029 | 0.234\(\pm\)0.028 |
| LCSR [11] | \(0.271\pm0.014\) | \(0.212\pm0.011\) | |
| PQCD [19] | 0.262 | 0.247 | 0.239 |
| CQM [20] | 0.29 | 0.28 | |
| RQM [21] | \(0.322\pm 0.016\) | \(0.320\pm 0.016\) | \(0.318\pm 0.016\) |
| PQCD [22] | \(0.30\pm 0.05\) | \(0.19\pm0.04\) | |
| CQM [23] | 0.42 | 0.34 | 0.31 |
| PQCD [24] | \(0.31\pm 0.07\) | \(0.18^{+0.06}_{-0.05}\) | \(0.12\pm0.03\) |
| LCUM [25] | 0.279 | 0.232 | 0.210 |
| Model | \(V(0)\) | \(T_1(0)\) | \(T_3(0)\) |
| CCQM | \(0.31\pm 0.03\) | \(0.27\pm 0.03\) | \(0.18\pm 0.02\) |
| LCSR [9] | 0.434\(\pm\)0.035 | 0.349\(\pm\)0.033 | \(0.175\pm0.018\) |
| LCSR [11] | \(0.339\pm0.017\) | \(0.299\pm0.016\) | \(0.191 \pm 0.010\) |
| PQCD [19] | 0.311 | 0.264 | 0.196 |
| CQM [20] | 0.32 | 0.28 | |
| RQM [21] | \(0.406\pm 0.020\) | \(0.275\pm0.014\) | \(0.133\pm 0.006\) |
| PQCD [22] | \(0.25\pm0.05\) | ||
| CQM [23] | \(0.44\) | 0.38 | 0.26 |
| PQCD [24] | \(0.26\pm0.07\) | \(0.23^{+0.06}_{-0.05}\) | \(0.19\pm 0.05\) |
| LCUM [25] | 0.329 | 0.276 | 0.170 |
3 Effective Hamiltonian↩︎
The rare decay \(b \to s \ell^+ \ell^-\) is described in terms of the effective Hamiltonian[26] as \[{\mathcal{H}}_{\rm eff} = - \frac{4G_F}{\sqrt{2}} \lambda_t \sum_{i=1}^{10} C_i(\mu) \mathcal{O}_i(\mu) , \label{eq:effHam}\tag{4}\] where \(C_i(\mu)\) – the Wilson coefficients and \(\mathcal{O}_i(\mu)\) – local operators. \(\lambda_t~=~|V_{tb}V_{ts}^\ast|\) is the product of the elements of the Cabibbo-Kobayashi-Maskawa (CKM) matrix. It should be noted that small corrections proportional to \(\lambda_u = |V_{ub}V_{us}^\ast|\) are discarded due to the relatively small values. The standard set of local operators[26] obtained within the SM framework for \(b \to s l^+ l^-\) transition is written as \[\begin{align} \begin{array}{ll} \mathcal{O}_1 = (\bar{s}_{a_1}\gamma^\mu P_L c_{a_2}) (\bar{c}_{a_2}\gamma_\mu P_L b_{a_1}), & \mathcal{O}_2 = (\bar{s}\gamma^\mu P_L c) (\bar{c}\gamma_\mu P_L b), \\[2ex] \mathcal{O}_3 = (\bar{s}\gamma^\mu P_L b) \sum_q(\bar{q}\gamma_\mu P_L q), & \mathcal{O}_4 = (\bar{s}_{a_1}\gamma^\mu P_L b_{a_2}) \sum_q (\bar{q}_{a_2}\gamma_\mu P_L q_{a_1}), \\[2ex] \mathcal{O}_5 = (\bar{s}\gamma^\mu P_L b) \sum_q(\bar{q}\gamma_\mu P_R q), & \mathcal{O}_6 = (\bar{s}_{a_1}\gamma^\mu P_L b_{a_2 }) \sum_q (\bar{q}_{a_2} \gamma_\mu P_R q_{a_1}), \\[2ex] \mathcal{O}_7 = \frac{e}{16\pi^2} \bar m_b\, (\bar{s} \sigma^{\mu\nu} P_R b) F_{\mu\nu}, & \mathcal{O}_8 = \frac{g}{16\pi^2} \bar m_b\, (\bar{s}_{a_1} \sigma^{\mu\nu} P_R {\boldsymbol{T}}_{a_1a_2} b_{a_2}) {\boldsymbol{G}}_{\mu\nu}, \\[2ex] \mathcal{O}_9 = \frac{e^2}{16\pi^2} (\bar{s} \gamma^\mu P_L b) (\bar\ell\gamma_\mu \ell), & \mathcal{O}_{10} = \frac{e^2}{16\pi^2} (\bar{s} \gamma^\mu P_L b) (\bar\ell\gamma_\mu\gamma_5 \ell), \end{array} \label{eq:operators} \end{align}\tag{5}\] where \({\boldsymbol{G}}_{\mu\nu}\) – gluon field strength and \(F_{\mu\nu}\)0 – photon field strengths; \({\boldsymbol{T}}_{a_1a_2}\) are the generators of the \(SU(3)\) color group; \(a_1\) and \(a_2\) denote the color indices, and are omitted in the color singlet currents. The chirality projection operators are \(P_{L,R} = (1 \mp \gamma_5)/2\), and \(\mu\) is the renormalization scale. \(\mathcal{O}_{1,2}\) – current-current operators, \(\mathcal{O}_{3-6}\) – QCD penguin operators, \(\mathcal{O}_{7,8}\) – "magnetic penguin" operators, and \(\mathcal{O}_{9,10}\) – the semileptonic electroweak penguin operators. The masses of QCD quarks are denoted by a bar to distinguish them from the masses of the constituent quarks used in CCQM.
The matrix element of the exclusive transition \(B_s\to \phi \ell^+ \ell^-\) can be written by using the effective Hamiltonian defined by eq.(4 ) as \[\begin{align} {\mathcal{M}} & = & \frac{G_F}{\sqrt{2}}\cdot\frac{ \alpha\lambda_t}{\pi} \cdot \Big\{ C_9^{\rm eff}\,<\phi\,|\,\bar{s}\,\gamma^\mu\, P_L\, b\,|\,B_s> \left( \bar \ell \gamma_\mu \ell \right) \nonumber\\ &-& \frac{2\bar m_b}{q^2}\,C_7^{\rm eff}\, <\phi\,|\, \bar{s}\,i\sigma^{\mu \nu} q_\nu \,P_R\, b\, |\,B_s> \left( \bar \ell \gamma_\mu \ell \right) \nonumber\\ &+& C_{10}\, <\phi\,|\,\bar{s}\,\gamma^\mu P_L \, b\,|\,B_s> \left(\bar \ell \gamma_\mu \gamma_5 \ell\right) \Big\}, \label{eq:matrix-elem} \end{align}\tag{6}\] where \(C_7^{\rm eff}= C_7 -C_5/3 -C_6\). It should be noted that the matrix element in equation (6 ) contains both the free quark decay amplitude coming from the operators \(\mathcal{O}_7\), \(\mathcal{O}_9\), and \(\mathcal{O}_{10}\) (gluon magnetic penquin \(\mathcal{O}_{8}\) does not contribute), and, in addition, certain long-range effects from the matrix elements of the four-quark operators \(\mathcal{O}_i\,\,(i=1,\ldots,6)\), which are usually absorbed by the redefinition of the Wilson coefficients at short distances. The Wilson coefficient \(C_9^{\rm eff}\) effectively takes into account, firstly, the contributions from the four-quark operators \(\mathcal{O}_i\) (\(i=1,...,6\)) and, secondly, the nonperturbative effects arising from the \(c\bar c\)-resonance contributions, which, as usual, are parameterized by the Breit-Wigner ansatz [27]. In addition to the perturbative charm-loop contribution, two-loop contributions were calculated in [6], [28], [29]. They effectively modify the Wilson coefficients as \[\begin{align} C_7^{\rm eff} &\to& C_7^{\rm eff} - \frac{\alpha_S}{4\pi}\Big( C_1 F_1^{(7)} + C_2 F_2^{(7)} \Big)\,, \nonumber\\ C_9^{\rm eff} &\to& C_9^{\rm eff} - \frac{\alpha_S}{4\pi}\Big( C_1 F_1^{(9)} + C_2 F_2^{(9)} \Big)\, \label{eq:Greub} \end{align}\tag{7}\] where the two-loop form factors \(F_{1,2}^{(7,9)}\) are available in [29] as Mathematica files.
The SM Wilson coefficients are taken from the paper [30]. They were calculated at the matching scale \(\mu_0=2 M_W\) and reduced to the hadronic scale \(\mu_b= 4.8\) GeV. The evolution of the couplings and current quark masses occurs similarly.
A analysis of the \(b\to s\ell\ell\) decays was performed in [13] with Next-to-Next-to-Leading Logarithmic corrections included. They were shown to be up to 15\(\%\).
The values of the model-independent input parameters and Wilson coefficients are given in the Table 3.
| \(m_W\) | \(\sin^2\theta_W\) | \(\alpha(M_Z)\) | \(\bar m_c\) | \(\bar m_b\) | \(\bar m_t\) | \(\lambda_t\) | ||
|---|---|---|---|---|---|---|---|---|
| \(80.41\) | \(0.2313\) | \(1/128.94\) | \(1.27\) | \(4.68\) | \(173.3\) | 0.041 | ||
| \(C_1\) | \(C_2\) | \(C_3\) | \(C_4\) | \(C_5\) | \(C_6\) | \(C^{\rm eff}_7\) | \(C_9\) | \(C_{10}\) |
| \(-0.2632\) | \(1.0111\) | \(-0.0055\) | \(-0.0806\) | 0.0004 | 0.0009 | \(-0.2923\) | 4.0749 | \(-4.3085\) |
4 Numerical results↩︎
The fourfold distribution in the \(B\to\phi(\to K^+K^-)\bar \ell \ell\) cascade decay allows us to define a number of physical observables that can be measured experimentally. The observables available in the \(B_s\to\phi\mu^+\mu^-\) decay [1], [2] are the CP-averaged differential branching ratio \(d{\cal B}/dq^2\), the CP-averaged \(\phi\) longitudinal polarization fraction \(F_L\), the forward-backward asymmetry \(A_{FB}\) and the CP-averaged angular observables \(S_{3,4,7}\) which can be related to the optimized observables \(P_i\) [13]. The CP asymmetries of \(A_{5,6,8,9}\) [8] in the SM are induced by the weak phase from the CKM matrix. For \(b\to s\) transitions, the CP asymmetries are proportional to \({\rm Im}(\hat{\lambda}_u)\equiv {\rm Im}(V_{ub}V^\ast_{us}/V_{tb}V^\ast_{ts})\) which is of the order of \(10^{-2}\) [8].
The width of \(B_s\to \phi\bar \ell \ell\) decay is calculated by integrating the \(q^2\)-differential distribution \[\begin{align} && \frac{d\Gamma(B\to \phi\bar \ell \ell)}{dq^2} =\, \frac{G^2_F}{(2\pi)^3}\, \left(\frac{\alpha \lambda_t}{2\pi}\right)^2 \frac{|{\boldsymbol{p}_2}|\,q^2\,\beta_\ell}{12\,m_1^2} {\cal H}_{\rm tot}\,, \nonumber\\[1.2ex] && {\cal H}_{\rm tot} = \frac{1}{2}\left( {\cal H}^{11}_U + {\cal H}^{22}_U + {\cal H}^{11}_L + {\cal H}^{22}_L \right ) + \delta_{\ell\ell}\, \left[\,\frac{1}{2} {\cal H}^{11}_U - {\cal H}^{22}_U + \frac{1}{2} {\cal H}^{11}_L - {\cal H}^{22}_L + \frac{3}{2}\, {\cal H}^{22}_S \right]. \label{eq:distr1} \end{align}\tag{8}\] Below, we use the shorthand notation \(m_1=m_{B_s}\), \(m_2=m_\phi\), \(\beta_\ell=\sqrt{1-4m_\ell^2/q^2}\), \(\delta_{\ell\ell} = 2m^2_\ell/q^2\). Then \(|{\boldsymbol{p}_2}|=\lambda^{1/2}(m_1^2,m_2^2,q^2)/(2\,m_1)\) is the momentum of the \(\phi\)-meson, specified in the rest frame of \(B_s\). Bilinear combinations of the helicity amplitudes \({\cal H}\) are defined as (see the papers [6], [31] for details): \[\begin{align} {\cal H}^{ii}_U &=& |H^i_{+1 +1}|^2 + |H^i_{-1 -1}|^2, \qquad {\cal H}^{ii}_L = |H^i_{00}|^2, \qquad {\cal H}^{ii}_S = |H^i_{t0}|^2 , \label{eq:bilinear} \end{align}\tag{9}\]
The differential rate of the decay \(B_s\to\phi\nu\bar\nu\) is calculated according to
\[\frac{d\Gamma(B_s\to\phi\nu\bar\nu)}{dq^2} = \frac{G_F^2}{(2\pi)^3} \Big(\frac{\alpha\lambda_t}{2\pi}\Big)^2 \Big[\frac{D_\nu(x_t)}{\sin^2\theta_W}\Big]^2 \frac{|{\boldsymbol{p}_2}|\, q^2}{4m_1^2}\cdot (H_U+H_L)\,,\] where \(x_t=\bar m_t^2/m_W^2\) and the function \(D_\nu\) is given by \[D_\nu(x) = \frac{x}{8}\left(\frac{2+x}{x-1}+\frac{3x-6}{(x-1)^2}\,\ln x\right).\] The relevant bilinear helicity combinations are defined as \[\begin{align} {\cal H}_U &=& |H_{+1 +1}|^2 + |H_{-1 -1}|^2, \qquad {\cal H}_L = |H_{00}|^2, \nonumber\\[1.2ex] H_{\pm1\pm1} &=& \frac{1}{m_1+m_2}\left(-Pq\, A_0\pm 2\,m_1\,|{\boldsymbol{p}_2}|\, V \right), \nonumber\\[1.2ex] H_{00} &=& \frac{1}{m_1+m_2}\frac{1}{2\,m_2\sqrt{q^2}} \left(-Pq\,(m_1^2 - m_2^2 - q^2)\, A_0 + 4\,m_1^2\,|{\boldsymbol{p}_2}|^2\, A_+\right). \label{eq:bilinear-2} \end{align}\tag{10}\]
The width of the color-suppressed nonleptonic decay \(B_s\to J/\psi\phi\) is determined by the formula[20] \[\begin{align} \Gamma(B_s\to J/\psi\phi ) &=& \frac{G_F^2}{16\pi}\frac{|{\boldsymbol{p}_{\,2}}|}{m^2_{1}} |V_{cb}V_{cs}|^2 \left(C^{\,\rm eff}_1+ C^{\,\rm eff}_5\right)^2 \left( m_{J/\psi}\,f_{J/\psi} \right)^2\,(H_U+H_L) \label{eq:BsJpsiPhi} \end{align}\tag{11}\] where the square of the transferred momentum is taken from the mass \(J/\psi\), that is \(q^2=m^2_{J/\psi}\), \(V_{cb}~=~0.406\), \(V_{cs}=0.975\) and \(f_{J/\psi}=415\) MeV. The Wilson coefficients are combined as \(C^{\,\rm eff}_{1}=C_1+\xi\, C_2+C_3+\xi\, C_4\) and \(C^{\,\rm eff}_{5}=C_5+\xi\, C_6\) according to the naive factorization. Terms multiplied by the color factor \(\xi=1/N_c\) will be discarded in numerical calculations in accordance with the \(1/N_c\) expansion.
Finally, the radiative decay width \(B_s\to\phi\gamma\) is defined as \[\Gamma(B_s\to\phi\gamma) = \frac{G_F^2\alpha\lambda_t^2}{32\pi^4} \bar m_b^2 m_1^3\Big(1-\frac{m_2^2}{m_1^2}\Big)^3\,|C^{\rm eff}_7|^2\, g^2(0)\,.\]
Table 4 presents theoretical predictions and experimental data [1], [2], [32] for branching decays \(B_s\to \phi\mu^+\mu^-\), \(B_s\to \phi\tau^+\tau^-\), \({\cal B}(B_s\to\phi \nu\bar\nu)\), \(B_s\to \phi\gamma\) and \(B_s\to \phi J/ \psi\) .
| CCQM | LFQM [7] | PQCD [19] | RQM [21] | LHCb [1], [32] | LHCb [2] | |
|---|---|---|---|---|---|---|
| \(10^7 {\cal B}(B_s\to\phi \mu^+\mu^-)\) | \(9.11\pm 1.82\) | 16.4 | \(7.07\pm 2.81\) | \(11.1\pm 1.1\) | \(7.97 \pm 0.77\) | \(8.14 \pm 0.472\) |
| \(10^7 {\cal B}(B_s\to\phi \tau^+\tau^-)\) | \(1.5\pm 0.2\) | 1.60 | \(0.81\pm 0.35\) | \(1.03\pm 0.20\) | ||
| \(10^5 {\cal B}(B_s\to\phi \nu\bar\nu)\) | \(0.84\pm 0.16\) | 1.165 | \(0.796\pm 0.080\) | \(< 540\) | ||
| \(10^5 {\cal B}(B_s\to\phi\gamma)\) | \(2.39\pm 0.48\) | \(3.8\pm 0.4\) | \(3.52\pm 0.34\) | |||
| \(10^2 {\cal B}(B_s\to J/\psi \phi)\) | \(0.16\pm 0.03\) | \(0.113\pm 0.016\) | \(0.108\pm 0.009\) | \(0.102\pm 0.010\) |
As detailed in [31], the full angular decay distribution for this rare \(B\) decay is formulated using helicity amplitudes and incorporates lepton mass effects. This distribution, which depends on three angles and the invariant mass \(q^2\) of the lepton pair, enables the definition of multiple physical observables that are experimentally accessible. Notable among these observables are the branching ratio, the longitudinal polarization fraction of the \(\phi\) meson, and the forward-backward asymmetry. The branching ratio is determined by integrating the complete fourfold angular decay distribution over the entire angular phase space. This results in the explicit expression provided in equation (8 ), which is formulated using helicity amplitudes. The required ratio of these helicity amplitudes to the transverse amplitudes was previously derived in [6].
The longitudinal polarization fraction and the forward-backward asymmetry are defined in terms of helicity amplitudes as \[\begin{align} F_L &=& \frac{1}{2} \beta_\ell^2 \frac{ {\cal H}_L^{11} + {\cal H}_L^{22}}{ {\cal H}_{\rm tot} }, \tag{12}\\[2ex] F_T &=& \frac{1}{2} \beta_\ell^2 \frac{ {\cal H}_U^{11} + {\cal H}_U^{22}}{ {\cal H}_{\rm tot} }, \tag{13}\\[2ex] A_{\rm FB} &=& \frac{1}{d\Gamma/dq^2} \left[ \int\limits_0^1 - \int\limits_{-1}^0 \right] d\!\cos\theta\, \frac{d^2\Gamma}{dq^2 d\!\cos\theta} = -\frac{3}{4}\beta_\ell \frac{ {\cal H}_P^{12}}{ {\cal H}_{\rm tot} }\,, \tag{14} \end{align}\] where \(\theta\) denotes the polar angle between the \(\ell^+\ell^-\) plane and the \(z\)-axis. As these quantities are defined as ratios of hadronic amplitudes, \(A_{\rm FB}\) and \(F_L\) are expected to exhibit a reduced dependence on theoretical uncertainties.
The behavior of the differential branching fraction \(d {\cal B}/dq^2\), forward-backward asymmetry \(A_{FB}\) and longitudinal polarization \(F_L\) is shown in Figs. 1, 2 and 3, respectively.
The behavior of the differential branching \({\cal B}(B_s\to\phi \nu\bar\nu)\) is shown in Figs. 4
To minimize hadronic uncertainties, a set of optimized observables \(P_i\) was constructed by taking appropriate ratios of form factors [30]. This construction aimed to reduce form factor dependence, maximize the sensitivity to New Physics beyond the SM, and ensure the observables could be measured experimentally. Despite these advantages, they are more difficult to assign a clear physical meaning, in contrast to observables such as \(A_{FB}\) and \(F_L\).
The optimized observables were not explicitly stated in [1]. Their numerical values were obtained in [13] by transforming the results for the CP averages \(S_{3,4,7}\) into optimized observables.
For comparison, the values of \(S_{3,4,7}\) were calculated using the relations from [33]. \[S_3 = \frac{1}{2} F_T P_1, \qquad S_4 = \frac{1}{2} \sqrt{F_T F_L} P'_4, \qquad S_7 = - \sqrt{F_T F_L} P'_6 .\] The \(q^2\)-dependence of the \(A_{i}\) and \(S_{i}\) for \(B_s\to\phi \mu^+\mu^-\) decay is displayed in Fig.5.
The \(q^2\)-dependence of the \(A_{i}\) and \(S_{i}\) for \(B_s\to\phi \tau^+\tau^-\) decay is shown in Fig.6.
The average values of the angular observables, integrated over the full \(q^2\) range, are shown in Table 5. The table compares our results from the CCQM with the PQCD predictions from [19].
| \(B_s\to\phi \mu^+\mu^-\) | ||||||
|---|---|---|---|---|---|---|
| \(<A_{FB}>\) | \(<F_L>\) | \(<P_1>\) | \(<P_4'>\) | \(<S_3>\) | \(<S_4>\) | |
| CCQM | \(-0.24\pm 0.05\) | \(0.45\pm 0.09\) | \(-0.52\pm 0.1\) | \(1.05\pm 0.21\) | \(-0.14\pm 0.03\) | \(0.26\pm 0.05\) |
| PQCD [19] | \(-0.233\pm 0.004\) | \(0.454 ^{+0.006}_{-0.007}\) | \(-0.555 ^{+0.010}_{-0.012}\) | \(1.111 ^{+0.011}_{-0.012}\) | \(-0.144\pm 0.005\) | \(0.258^{+0.003}_{-0.004}\) |
| \(B_s\to\phi \tau^+\tau^-\) | ||||||
| \(<A_{FB}>\) | \(<F_L>\) | \(<P_1>\) | \(<P_4'>\) | \(<S_3>\) | \(<S_4>\) | |
| CCQM | \(-0.18\pm 0.04\) | \(0.090\pm 0.02\) | \(-0.76\pm 0.15\) | \(1.33\pm 0.27\) | \(-0.067\pm 0.013\) | \(0.083\pm 0.017\) |
| PQCD [19] | \(-0.171\pm 0.002\) | \(0.396^{+0.002}_{-0.003}\) | \(-0.795^{+0.007}_{-0.004}\) | \(1.33\pm 0.002\) | \(-0.080\pm 0.001\) | \(0.100\pm 0.001\) |
The preceding calculations employed Wilson coefficients at the Next-to-Leading Logarithmic (NLL) order. At this order, only the coefficient \(C_9^{\rm eff}\) possesses an imaginary part. Given that the calculated form factors are real, the optimized observable \(S_7\) is identically zero. Furthermore, the optimized observable \(P_1\) remains small across a large range of values for any choice of Wilson coefficients. It is easy to verify that \(P_1\propto A_0(0)-V(0)\) for \(q^2=0\). In CCQM, \(A_0(0)=0.40\) and \(V(0)=0.31\), so this leads to a truly small value of \(P_1\). Note that \(A_0(0)=V(0)\) in the heavy quark limit.
The calculation of two-loop corrections for the \(b\to s\ell^+\ell^-\) decay has been addressed in distinct \(q^2\) regions. For the low-\(q^2\) regime, [28] provided results as an expansion in the small parameters \(\hat{s}=q^2/\bar m_b^2\) and \(z=\bar m_c^2/\bar m_b^2\), considering the range \(0.05 \leq \hat{s} \leq 0.25\). For the high-\(q^2\) region above the charm threshold (\(q^2>4\bar m_c^2\)), the NNLO corrections were derived in [29]. The high-\(q^2\) region was limited to the range \(0.4 \le \hat{s} \le 1.0\). Using the QCD bottom quark value \(\bar m_b=4.68\) GeV from Table 3, we can obtain the \(q^2\) regions where the two-loop corrections are valid: \[1.1 \le q^2 \le 5.5 \, \, \text{GeV}^2 \quad \text{(small region)} \quad \text{and} \quad 8.8 \le q^2 \le 22 \, \, \text{GeV}^2 \quad \text{(large region)}. \label{eq:NNLL}\tag{15}\] The two-loop calculation for low \(q^2\) [28] is valid under the conditions that \(q^2/\bar m_b^2 \ll 1\) and \(q^2/(4\bar m_c^2) \ll 1\), making its results reliable for \(0.1 \le q^2 \le 6\) GeV\(^2\). In contrast, the high-\(q^2\) expansion [29] requires \(q^2/\bar m_b^2 > 0.4\). Due to this constraint, the intervals \([5,8]\) and \([6,8]\) GeV\(^2\) are omitted from the consideration of two-loop corrections.
The inclusion of NNLL corrections results in non-zero values for the observables \(P'_6\) and \(S_7\). These corrections are significant, contributing up to 20\(\%\) at low squared momentum transfer (\(q^2 \le 6\) GeV\(^2\)), but their impact becomes truly negligible at high \(q^2\).
The CCQM results for the branching fraction \({\cal B}(B_s\to \phi\mu^+\mu^-)\) and the angular observables are summarized in Table 6. The calculations include NNLL corrections to the Wilson coefficients [28], [29], leading to effective modifications of \(C_7^{\rm eff}\) and \(C_9^{\rm eff}\). Thus, all values are derived from two-loop calculations with the exception of the \([5,8]\) and \([6,8]\) GeV\(^2\) bins (which use one-loop calculations).
| \(10^7 {\cal B}\) | CCQM | PQCD [19] | LHCb [1] | LHCb [2] |
| \([0.1,0.98]\) | \(0.66\pm 0.13\) | \(0.25^{+0.10}_{-0.07}\) | \(0.68 \pm 0.06\) | |
| \([0.1,2]\) | \(0.99\pm 0.2\) | \(0.40^{+0.16}_{-0.11}\) | \(1.11 \pm 0.16\) | |
| \([1.1,2.5]\) | \(0.42\pm 0.08\) | \(0.21^{+0.07}_{-0.05}\) | \(0.44 \pm 0.05\) | |
| \([1,6]\) | \(1.56\pm 0.31\) | \(1.10^{+0.34}_{-0.25}\) | \(1.29 \pm 0.19\) | |
| \([1.1,6]\) | \(1.52\pm 0.30\) | \(0.98^{+0.30}_{-0.22}\) | \(1.41 \pm 0.10\) | |
| \([2.5,4]\) | \(0.46\pm 0.09\) | \(0.27^{+0.09}_{-0.06}\) | \(0.35 \pm 0.04\) | |
| \([2,5]\) | \(0.90\pm 0.18\) | \(0.58^{+0.18}_{-0.13}\) | \(0.77\pm 0.14\) | |
| \([4,6]\) | \(0.65\pm 0.13\) | \(0.50^{+0.14}_{-0.11}\) | \(0.62\pm 0.06\) | |
| \([5,8]\) | \(1.25\pm 0.25\) | \(0.96^{+0.25}_{-0.19}\) | \(0.96\pm 0.15\) | |
| \([6,8]\) | \(0.87\pm 0.17\) | \(0.69^{+0.18}_{-0.13}\) | \(0.63\pm 0.06\) | |
| \([11,12.5]\) | \(0.84\pm 0.17\) | \(0.76^{+0.14}_{-0.11}\) | \(0.71 \pm 0.12\) | \(0.72 \pm 0.06\) |
| \([15,17]\) | \(1.15\pm 0.23\) | \(0.99^{+0.11}_{-0.10}\) | \(0.90 \pm 0.13\) | \(1.05 \pm 0.08\) |
| \([17,18.9]\) | \(0.75\pm 0.15\) | \(0.67^{+0.05}_{-0.05}\) | \(0.75 \pm 0.13\) | \(0.82 \pm 0.07\) |
| \([15,18.9]\) | \(1.89\pm 0.28\) | \(1.60^{+0.16}_{-0.16}\) | \(1.62\pm 0.20\) | \(1.85\pm 0.13\) |
| \(F_L\) | CCQM | PQCD [19] | LHCb [1] | LHCb [2] |
| \([0.1,0.98]\) | \(0.23\pm 0.05\) | \(0.254 \pm 0.048\) | ||
| \([0.1,2]\) | \(0.37\pm 0.07\) | \(0.472^{+0.011}_{-0.012}\) | \(0.20\pm 0.09\) | |
| \([1.1,4]\) | \(0.72\pm 0.14\) | \(0.723 \pm 0.055\) | ||
| \([2,5]\) | \(0.72\pm 0.14\) | \(0.796^{+0.007}_{-0.007}\) | \(0.68\pm 0.15\) | |
| \([4,6]\) | \(0.65\pm 0.13\) | \(0.701 \pm 0.05\) | ||
| \([5,8]\) | \(0.57 \pm 0.11\) | \(0.682^{+0.010}_{-0.008}\) | \(0.54\pm 0.10\) | \(0.54\pm 0.10\) |
| \([6,8]\) | \(0.55\pm 0.11\) | \(0.624 \pm 0.052\) | ||
| \([11,12.5]\) | \(0.40\pm 0.08\) | \(0.524^{+0.008}_{-0.008}\) | \(0.29 \pm 0.11\) | \(0.35 \pm 0.046\) |
| \([15,17]\) | \(0.34\pm 0.07\) | \(0.412^{+0.004}_{-0.004}\) | \(0.23 \pm 0.096\) | |
| \([17,18.9]\) | \(0.33\pm 0.06\) | \(0.363^{+0.001}_{-0.001}\) | \(0.40 \pm 0.15\) | \(0.4 \pm 0.14\) |
| \([1,6]\) | \(0.69\pm 0.14\) | \(0.777^{+0.008}_{-0.006}\) | \(0.63 \pm 0.09\) | |
| \([1.1,6]\) | \(0.69\pm 0.14\) | \(0.715 \pm 0.04\) | ||
| \([15,18.9]\) | \(0.34\pm 0.07\) | \(0.394^{+0.003}_{-0.003}\) | \(0.29\pm 0.07\) | \(0.35\pm 0.031\) |
| \(A_{FB}\) | CCQM | PQCD [19] | LHCb [1] | LHCb [2] |
| \([0.1,0.98]\) | \(0.11\pm 0.02\) | \(0.068 \pm 0.0065\) | ||
| \([0.1,2]\) | \(0.14\pm 0.02\) | \(0.122^{+0.003}_{-0.002}\) | ||
| \([1.1,4]\) | \(0.12\pm 0.02\) | \(0.023 \pm 0.054\) | ||
| \([2,5]\) | \(0.038\pm 0.008\) | \(-0.038^{+0.001}_{-0.002}\) | ||
| \([4,6]\) | \(-0.085\pm 0.017\) | \(-0.030 \pm 0.051\) | ||
| \([5,8]\) | \(-0.24 \pm 0.05\) | \(-0.200^{+0.006}_{-0.006}\) | ||
| \([6,8]\) | \(-0.26\pm 0.05\) | \(0.32 \pm 0.049\) | ||
| \([11,12.5]\) | \(-0.35\pm 0.07\) | \(-0.298^{+0.005}_{-0.005}\) | \(0.034 \pm 0.048\) | |
| \([15,17]\) | \(-0.33\pm 0.06\) | \(-0.290^{+0.005}_{-0.005}\) | ||
| \([17,18.9]\) | \(-0.24\pm 0.05\) | \(-0.208^{+0.004}_{-0.003}\) | ||
| \([1,6]\) | \(0.69\pm 0.14\) | \(0.777^{+0.008}_{-0.006}\) | ||
| \([1.1,6]\) | \(0.034\pm 0.006\) | \(0.006 \pm 0.036\) | ||
| \([15,18.9]\) | \(-0.29\pm 0.06\) | \(0.394^{+0.003}_{-0.003}\) | \(-0.011\pm 0.033\) | |
| \(S_3\) | CCQM | PQCD [19] | LHCb [1] | LHCb [2] |
| \([0.1,0.98]\) | \(0.004\pm 0.001\) | \(-0.004 \pm 0.069\) | ||
| \([0.1,2]\) | \(0.0031\pm 0.0006\) | \(0.002^{+0.000}_{-0.001}\) | \(-0.05 \pm 0.13\) | |
| \([1.1,4]\) | \(-0.017\pm 0.003\) | \(-0.030 \pm 0.057\) | ||
| \([2,5]\) | \(-0.035\pm 0.007\) | \(-0.021^{+0.001}_{-0.001}\) | \(-0.06\pm 0.021\) | |
| \([4,6]\) | \(-0.059\pm 0.001\) | \(-0.162 \pm 0.068\) | ||
| \([5,8]\) | \(-0.082\pm 0.016\) | \(-0.050^{+0.004}_{-0.003}\) | \(-0.10\pm 0.27\) | \(-0.10 \pm 0.25\) |
| \([6,8]\) | \(-0.088\pm 0.016\) | \(0.013 \pm 0.081\) | ||
| \([11,12.5]\) | \(-0.15\pm 0.03\) | \(-0.115^{+0.002}_{-0.001}\) | \(-0.19 \pm 0.21\) | \(-0.138 \pm 0.072\) |
| \([15,17]\) | \(-0.23\pm 0.05\) | \(-0.213^{+0.001}_{-0.001}\) | \(-0.06 \pm 0.18\) | |
| \([17,18.9]\) | \(-0.29\pm 0.06\) | \(-0.281^{+0.001}_{-0.001}\) | \(-0.07 \pm 0.25\) | |
| \([1,6]\) | \(-0.034\pm 0.007\) | \(-0.023^{+0.001}_{-0.001}\) | \(-0.02 \pm 0.13\) | |
| \([1.1,6]\) | \(-0.035\pm 0.007\) | \(-0.083 \pm 0.048\) | ||
| \([15,18.9]\) | \(-0.25\pm 0.05\) | \(-0.239^{+0.001}_{-0.001}\) | \(-0.09\pm 0.12\) | \(-0.247\pm 0.044\) |
| \(S_4\) | CCQM | PQCD [19] | LHCb [1] | LHCb [2] |
| \([0.1,0.98]\) | \(-0.062\pm 0.012\) | \(-0.213 \pm 0.082\) | ||
| \([0.1,2]\) | \(-0.038\pm 0.008\) | \(-0.053^{+0.000}_{-0.000}\) | \(-0.27 \pm 0.23\) | |
| \([1.1,4]\) | \(0.114\pm 0.023\) | \(0.110 \pm 0.079\) | ||
| \([2,5]\) | \(0.19\pm 0.04\) | \(0.177^{+0.003}_{-0.003}\) | \(0.47\pm 0.37\) | |
| \([4,6]\) | \(0.253\pm 0.051\) | \(0.222 \pm 0.092\) | ||
| \([5,8]\) | \(0.28\pm 0.06\) | \(0.259^{+0.003}_{-0.003}\) | \(0.10\pm 0.17\) | |
| \([6,8]\) | \(0.284\pm 0.057\) | \(0.176 \pm 0.078\) | ||
| \([11,12.5]\) | \(0.30\pm 0.06\) | \(0.303^{+0.001}_{-0.001}\) | \(0.47 \pm 0.25\) | \(0.319 \pm 0.061\) |
| \([15,17]\) | \(0.31\pm 0.06\) | \(0.323^{+0.001}_{-0.001}\) | \(0.03 \pm 0.15\) | |
| \([17,18.9]\) | \(0.32\pm 0.06\) | \(0.329^{+0.001}_{-0.001}\) | \(0.39 \pm 0.3\) | |
| \([1,6]\) | \(0.17\pm 0.03\) | \(0.169^{+0.001}_{-0.001}\) | \(0.19 \pm 0.14\) | |
| \([1.1,6]\) | \(0.174\pm 0.03\) | \(0.155 \pm 0.058\) | ||
| \([15,18.9]\) | \(0.31\pm 0.06\) | \(0.325^{+0.001}_{-0.001}\) | \(0.14\pm 0.11\) | \(0.208\pm 0.047\) |
| \(S_7\) | CCQM | PQCD [19] | LHCb [1] | LHCb [2] |
| \([0.1,0.98]\) | \(0.0056\pm 0.011\) | \(-0.178 \pm 0.072\) | ||
| \([0.1,2]\) | \(0.0065\pm 0.0013\) | \(1.551^{+0.003}_{-0.006}\) | \(0.04\pm 0.12\) | |
| \([1.1,4]\) | \(0.0075\pm 0.0015\) | \(-0.101 \pm 0.075\) | ||
| \([2,5]\) | \(0.0065\pm 0.0013\) | \(0.979^{+0.017}_{-0.018}\) | \(-0.03\pm 0.21\) | |
| \([4,6]\) | \(0.0051\pm 0.0010\) | \(0.175 \pm 0.089\) | ||
| \([5,8]\) | \(0.453^{+0.003}_{-0.003}\) | \(0.04\pm 0.18\) | ||
| \([6,8]\) | \(0.453^{+0.003}_{-0.003}\) | \(0.033\pm 0.081\) | ||
| \([11,12.5]\) | \(0.0021\pm 0.0004\) | \(0.185^{+0.001}_{-0.001}\) | \(0.00 \pm 0.16\) | \(-0.170 \pm 0.069\) |
| \([15,17]\) | \(0.00087\pm 0.0002\) | \(0.071^{+0.001}_{-0.001}\) | \(0.12 \pm 0.15\) | |
| \([17,18.9]\) | \(0.00034\pm 0.00007\) | \(0.027^{+0.001}_{-0.001}\) | \(0.20 \pm 0.26\) | |
| \([1,6]\) | \(0.0065\pm 0.0013\) | \(0.985^{+0.020}_{-0.022}\) | \(-0.03 \pm 0.14\) | |
| \([1.1,6]\) | \(0.0064\pm 0.0013\) | \(0.020\pm 0.059\) | ||
| \([15,18.9]\) | \(0.00066\pm 0.00013\) | \(0.054^{+0.001}_{-0.001}\) | \(0.13\pm 0.11\) | \(0.003\pm 0.046\) |
| \(A_5\) | CCQM | PQCD [19] | LHCb [1] | LHCb [2] |
| \([0.1,0.98]\) | \(0.268\pm 0.054\) | \(0.043 \pm 0.067\) | ||
| \([0.1,2]\) | \(0.0031\pm 0.0006\) | \(-0.02 \pm 0.13\) | ||
| \([1.1,4]\) | \(0.0004\pm 0.0001\) | \(0.026 \pm 0.067\) | ||
| \([2,5]\) | \(-0.035\pm 0.007\) | \(0.09\pm 0.25\) | ||
| \([4,6]\) | \(-0.252\pm 0.050\) | \(-0.084 \pm 0.084\) | ||
| \([5,8]\) | \(-0.3461\pm 0.0692\) | \(0.04\pm 0.17\) | ||
| \([6,8]\) | \(-0.3538\pm 0.0708\) | \(-0.022 \pm 0.082\) | ||
| \([11,12.5]\) | \(-0.33\pm 0.07\) | \(0.08 \pm 0.21\) | \(0.035 \pm 0.063\) | |
| \([15,17]\) | \(-0.25\pm 0.05\) | \(0.02 \pm 0.14\) | ||
| \([17,18.9]\) | \(-0.16\pm 0.03\) | \(0.13 \pm 0.28\) | ||
| \([1,6]\) | \(-0.034\pm 0.007\) | \(0.20 \pm 0.13\) | ||
| \([1.1,6]\) | \(-0.108\pm 0.022\) | \(-0.007 \pm 0.051\) | ||
| \([15,18.9]\) | \(-0.21\pm 0.04\) | \(0.11\pm 0.10\) | \(-0.025\pm 0.043\) | |
| \(A_8\) | CCQM | PQCD [19] | LHCb [1] | LHCb [2] |
| \([0.1,0.98]\) | \(0.0043\pm 0.0008\) | \(-0.007 \pm 0.073\) | ||
| \([0.1,2]\) | \(-0.038\pm 0.008\) | \(-0.10 \pm 0.14\) | ||
| \([1.1,4]\) | \(0.0043\pm 0.0008\) | \(0.038 \pm 0.082\) | ||
| \([2,5]\) | \(0.19\pm 0.04\) | \(0.19\pm 0.24\) | ||
| \([4,6]\) | \(0.0017\pm 0.0003\) | \(0.012 \pm 0.090\) | ||
| \([5,8]\) | \(0.0026\pm 0.0005\) | \(-0.12\pm 0.18\) | ||
| \([6,8]\) | \(0.0025\pm 0.0005\) | \(-0.170 \pm 0.080\) | ||
| \([11,12.5]\) | \(-0.0007\pm 0.0001\) | \(-0.01 \pm 0.15\) | \(0.046 \pm 0.070\) | |
| \([15,17]\) | \(-0.0003\pm 0.0001\) | \(0.08 \pm 0.17\) | ||
| \([17,18.9]\) | \(-0.0002\pm 0.0001\) | \(-0.16 \pm 0.27\) | ||
| \([1,6]\) | \(0.17\pm 0.03\) | \(0.00 \pm 0.16\) | ||
| \([1.1,6]\) | \(0.0032\pm 0.0001\) | \(0.016 \pm 0.062\) | ||
| \([15,18.9]\) | \(-0.0003\pm 0.0001\) | \(0.03\pm 0.12\) | \(0.072\pm 0.051\) | |
| \(A_9\) | CCQM | PQCD [19] | LHCb [1] | LHCb [2] |
| \([0.1,0.98]\) | \(-0.0003\pm 0.0001\) | \(-0.030 \pm 0.079\) | ||
| \([0.1,2]\) | \(0.0065\pm 0.0013\) | \(0.03\pm 0.14\) | ||
| \([1.1,4]\) | \(-0.0007\pm 0.0001\) | \(0.020 \pm 0.068\) | ||
| \([2,5]\) | \(0.0065\pm 0.0013\) | \(-0.13\pm 0.27\) | ||
| \([4,6]\) | \(-0.0005\pm 0.0001\) | \(-0.008 \pm 0.061\) | ||
| \([5,8]\) | \(-0.0010\pm 0.0002\) | \(-0.03\pm 0.17\) | ||
| \([6,8]\) | \(-0.010 \pm 0.0002\) | \(-0.012\pm 0.090\) | ||
| \([11,12.5]\) | \(0.0005\pm 0.0001\) | \(-0.02 \pm 0.16\) | \(0.017 \pm 0.071\) | |
| \([15,17]\) | \(0.0004\pm 0.0001\) | \(0.21 \pm 0.15\) | ||
| \([17,18.9]\) | \(0.0002\pm 0.0001\) | \(-0.02 \pm 0.19\) | ||
| \([1,6]\) | \(0.0065\pm 0.0013\) | \(-0.01 \pm 0.13\) | ||
| \([1.1,6]\) | \(-0.0006\pm 0.0001\) | \(0.009\pm 0.046\) | ||
| \([15,18.9]\) | \(0.0003\pm 0.0001\) | \(0.12\pm 0.10\) | \(0.021\pm 0.042\) |
5 Discussion and Conclusion↩︎
We have calculated the angular observables \(<A_{FB}>\), \(<F_L>\), \(<P_1>\), \(<P_4'>\), \(<S_3>\), \(<S_4>\) integrated over the full \(q^2\) range for the \(\mu\) and \(\tau\) modes, finding them to be in complete agreement with the corresponding PQCD predictions [19].
Our results for the branching fractions and angular observables are generally consistent with experimental data from LHCb. One can see better agreement between CCQM predictions for branching fraction and \(F_L\) with the latest LHCb data [2]. This is especially evident for bins [11, 12.5], [15, 17], [17, 18.9], and [15, 18.9], which were measured in 2015 [1] and 2021 [2]. Notably, the inclusion of Next-to-next-to-leading logarithmic (NNLL) corrections to the Wilson coefficients \(C_{7}^{eff}\) and \(C_{9}^{eff}\) was essential for achieving agreement with data, particularly in the low-\(q^2\) region. These corrections also lead to non-zero values for observables such as \(S_7\) , which vanish at leading order. It should be noted that predictions of CCQM in the Table 6 for bins \([5,8]\) and \([6,8]\) are given for the one-loop calculations. The largest discrepancies with [2] are found in the forward-backward asymmetry \(A_{FB}\) in the bin [0.1, 0.98] and in the angular observable \(S_3\) in the bins [1.1, 4] and [4, 6]. It should be noted that our predictions of \(S_3\) and \(S_4\) for large-\(q^2\) region describe the experiment well.
6 ACKNOWLEDGEMENTS↩︎
This research has been funded by the Committee of Science of the Ministry of Science and Higher Education of the Republic of Kazakhstan (Grant No. AP19680084).