Predictions of Angular Observables for $\bar{B}_s\rightarrow K^{*}\ell \ell$ and $\bar{B}\rightarrow \rho \ell \ell$ in Standard Model

Abstract

Exclusive semileptonic decays based on $b\rightarrow s$ transitions have been attracting a lot of attention as some angular observables deviate significantly from the Standard Model (SM) predictions in specific $q^2$ bins. B meson decays induced by other Flavor Changing Neutral Current (FCNC), $b\rightarrow d$, can also offer a probe to new physics with an additional sensitivity to the weak phase in Cabibbo–Kobayashi–Masakawa (CKM) matrix. We provide predictions for angular observables for $b\rightarrow d$ semileptonic transitions, namely $\bar{B}_s\rightarrow K^{*}\ell ^+\ell ^-$, $\bar{B}^0\rightarrow \rho ^0 \ell ^+\ell ^-$, and their CP-conjugated modes including various non-factorizable corrections.

7.1 Introduction

Experimental evidence of new physics has been found in the channels involving FCNC $b\rightarrow s\ell ^+\ell ^-$ and charged current $b\rightarrow c\ell \nu$. However, the $b\rightarrow d$ counterpart of the weak decay, i.e., $b\rightarrow d\ell ^+\ell ^-$, has not caught much attention perhaps because of low branching ratio. The weak phases incorporate CKM matrix elements $\xi _{q}^i=V_{qi}^* V_{qb}$, where $q\in \{u,c,t\}$ and $i\in \{s,d\}$. For $b\rightarrow s \ell \ell$ transition, $\xi ^s_{c,t}\sim \lambda ^2$ and $\xi ^s_{u}\sim \lambda ^4$ where $\lambda =0.22$. Since $u\bar{u}$ contribution introduces CKM phase which is negligible for $b\rightarrow s \ell \ell$, CP violating quantities are very small in SM. On the other hand, since $\xi ^d_u\sim \xi ^d_c\sim \xi ^d_t\sim \lambda ^4$ for $b\rightarrow d\ell \ell$, the B decays mediated through this transition allow for large CP violating quantities. Also, leading order contribution in this case is smaller than the leading contribution in $b\rightarrow s \ell \ell$ which makes it more sensitive to new particles and interactions. In this work, we focus on two such decay channels, $B_s\rightarrow \bar{K}^{*}\ell ^+\ell ^-$ and $B\rightarrow \rho \ell ^+\ell ^-$[1].

7.2 Decay Amplitude

We follow the effective Hamiltonian approach as used in [2] to write the Hamiltonian and decay amplitude. Th amplitude is written as a product of short-distance contributions through Wilson coefficients and long-distance contribution which is further expressed in terms of form factors,

$$\begin{aligned} \mathcal {M}=\,&\frac{G_F \alpha }{\sqrt{2}\pi }V_{tb}V_{td}^* \Big \{ \Big [\langle V|\bar{d}\gamma ^{\mu }(C_9^{\text {eff}}P_L)b|P \rangle -\frac{2m_b}{q^2}\langle V|\bar{d}~i~ \sigma ^{\mu \nu }q_{\nu }(C_{7}^{\text {eff}}P_R)b|P\rangle \Big ](\bar{\ell }\gamma _{\mu }\ell )\nonumber \\&+\langle V|\bar{d}\gamma ^{\mu }(C_{10}^{\text {eff}}P_L)b|P\rangle (\bar{\ell }\gamma _{\mu }\gamma _5\ell )-16 \pi ^2\frac{\bar{\ell }\gamma ^{\mu }\ell }{q^2} \mathcal {H}_{\mu }^{\text {non-fac}}\Big \} . \end{aligned}$$

(7.1)

Wilson coefficients ($C_i's$) are computed upto next-to-next-to leading order (NNLO) [3] and form factors are computed using the method of Light Cone Sum Rules (LCSR) and QCD lattice calculation [4]. $\mathcal {H}_{\mu }^{\text {non-fac}}$ represents the non-factorizable contribution of non-local hadronic matrix element. This results from four quark and chromomagnetic operators combined with virtual photon emission which then decays to lepton pair through electromagnetic interaction. These corrections are given in terms of hard-scattering kernels ($\mathcal {T}_a^q s$), where $a\in \{\perp ,\parallel \}$ and $q\in \{u,c\}$, which are convoluted with B(B$_S$)-meson and $\rho (\bar{K^{*}})$ distribution amplitudes. The non-factorizable corrections included here are spectator scattering $\mathcal {T}_a^{q,\text {spec}}$, weak annihilation $\mathcal {T}_a^{q,\text {WA}}$, and soft-gluon emission $\varDelta C_9^{q,\text {soft}}$. These corrections have been computed in [5,6,7] except charm loop corrections corresponding to up quark in the loop. For present work, we are assuming that its contribution is less than $10\%$ of $C_9$: $\varDelta C_{9,u}^{\text {soft}}= a e^{i \theta }; |a|\in \{0,0.5\}, \theta \in \{0,\pi \}$.

These corrections are then added to transversity amplitudes in the following way:

$$\begin{aligned} A_{\perp L,R}(q^2)&= \sqrt{2 \lambda } ~N\big [2\frac{m_b}{q^2}(C_7^{\text {eff}}T_1(q^2)+\varDelta T_{\perp }) +(C_9^{\text {eff}}\mp C_{10}+ \varDelta C_9^1(q^2))\frac{V(q^2)}{M_B+M_V} \big ]\end{aligned}$$

(7.2)

$$\begin{aligned} A_{\parallel L,R}(q^2)&= -\sqrt{2}N(M_B^2-M_V^2)\big [2\frac{m_b}{q^2}(C_7^{\text {eff}}T_2(q^2)+2 \frac{E(q^2)}{M_B}\varDelta T_{\perp })+ \nonumber \\&(C_9^{\text {eff}}\mp C_{10}+\varDelta C_9^2(q^2))\frac{A_1(q^2)}{M_B-M_V}\big ]\end{aligned}$$

(7.3)

$$\begin{aligned} A_{0 L,R}(q^2)&=-\frac{N}{2M_V \sqrt{q^2}}\big [2 m_b\big ((M_B^2+3M_V^2-q^2)(C_7^{\text {eff}}T_2(q^2)\big )\nonumber \\&-\frac{\lambda }{M_B^2-M_v^2}(C_7^{\text {eff}}T_3(q^2)+\varDelta T_{\parallel }))+(C_9^{\text {eff}}\mp C_{10}+\varDelta C_9^3)\nonumber \\&\big ((M_B^2+M_V^2-q^2)(M_B+M_V)A_1(q^2)-\frac{\lambda }{M_B+M_V}A_2(q^2)\big )\big ]\end{aligned}$$

(7.4)

$$\begin{aligned} A_t(q^2)=\,&\frac{N}{\sqrt{s}}~\sqrt{\lambda }2 ~C_{10}~A_0(q^2) \end{aligned}$$

(7.5)

where,

$$\begin{aligned} \varDelta T_{\perp }&=\frac{\pi ^2}{N_c} \frac{f_{P}f_{V,\perp }}{M_B}\frac{\alpha _sC_F}{4\pi }\int \frac{d\omega }{\omega }\varPhi _{P,-}(\omega )\int _0^1du~ \varPhi _{V,\perp }(u)(T_{\perp }^{c,\text {spec}}+\frac{\xi _u}{\xi _t}(T_{\perp }^{u,\text {spec}}))\end{aligned}$$

(7.6)

$$\begin{aligned} \varDelta T_{\parallel }&=\frac{\pi ^2}{N_c} \frac{f_{P}f_{V,\parallel }}{M_B}\frac{M_V}{E}\sum _{\pm }\int \frac{d\omega }{\omega }\varPhi _{P}(\omega )\int _0^1du~ \varPhi _{V,\parallel }(u)\big [T_{\parallel }^{c,WA}+\frac{\xi _u}{\xi _t}T_{\parallel }^{u,WA}\nonumber \\&\frac{\alpha _sC_F}{4\pi }(T_{\parallel }^{c,\text {spec}}+\frac{\xi _u}{\xi _t}T_{\parallel }^{,\text {spec}})\big ]\end{aligned}$$

(7.7)

$$\begin{aligned} \varDelta C_9^i&=\varDelta C_{9,c}^{i,soft}+\varDelta C_{9,u}^{i,soft} \end{aligned}$$

(7.8)

7.3 Observables

The angular decay distribution of $B\rightarrow V(\rightarrow M_1M_2)\ell ^+\ell ^-$ is given in terms of angular functions ($I_i(q^2,\theta _V,\theta _l,\phi$), the value of which can be obtained by integrating data over specific values of the parameters. We consider an optimized set of observables constricted choosing specific combinations of these angular functions. The observables considered here are

• Form Factor Dependent observables.

  $$\begin{aligned} \frac{d\varGamma }{dq^2}&=\frac{1}{4}(3I_1^c+6I_1^s-I_2^c-2I_2^s)&A_{FB}(q^2)&=\frac{-3I_6^s}{3I_1^c+6I_1^s-I_2^c-2I_2^s}\nonumber \\ F_L(q^2)&=\frac{3I_1^c-I_2^c}{3I_1^c+6I_1^s-I_2^c-2I_2^s}&\end{aligned}$$

  (7.9)

• Form Factor independent observables.

  $$\begin{aligned} P_1&=\frac{I_3}{2 I_{2}^s},~ P_2=\beta _l\frac{I_6^s}{8I_2^s},~P_3=-\frac{I_9}{4I_2^s},~ P_4^{\prime }=\frac{I_4}{\sqrt{-I_2^cI_2^s}}\nonumber \\ P_5^{\prime }&=\frac{I_5}{2\sqrt{-I_2^cI_2^s}},~ P_6^{\prime }=-\frac{I_7}{2\sqrt{-I_2^cI_2^s}},~P_8^{\prime }=-\frac{I_8}{2\sqrt{-I_2^cI_2^s}} \end{aligned}$$

  (7.10)

• Lepton Flavor Universality violating observables.

  $$\begin{aligned} R_{K^{*}}^{B_s}=\frac{\left[ \mathcal {BR}(B_s\rightarrow \bar{K}^{*}\mu ^+\mu ^-)\right] _{q^2\in \{q_1^2,q_2^2\}}}{\left[ \mathcal {BR}(B_s\rightarrow \bar{K}^{*}e^+e^-\right] _{q^2\in \{q_1^2,q_2^2\}}} \end{aligned}$$

  (7.11)

These observables are valid for $B_s\rightarrow \bar{K}^{*}\ell \ell$. For the CP-conjugate process, the $I_i$ are replaced by $\tilde{I}_i\equiv \xi _i\bar{I_i}$, where $\bar{I}_i$ are $I_s$ only with weak phase conjugated and $\xi _i=1$ for $i=\{1,2,3,4,7\}$ and $-1$ for $i=\{5,6,8,9\}$. For$B\rightarrow \rho \ell \ell$, angular functions are replaced with time-dependent angular functions, since the final state in this case is self conjugate [1]. Thus, observables are sensitive to $B^0-\bar{B}^0$ oscillations in this case and the $I_i's$ are replaced by $J_i's$ in the definition of observables, where $J_i's$ are given as [9],

$$\begin{aligned} J_i(t)+\tilde{J}_i(t)=e^{-\varGamma t}[(I_i+\bar{I}_i) \text {cosh}(y\varGamma t)-h_i \text {sinh}(y\varGamma t)]\end{aligned}$$

(7.12)

$$\begin{aligned} J_i(t)-\tilde{J}_i(t)=e^{-\varGamma t}[(I_i-\bar{I}_i) \text {cosh}(y\varGamma t)-s_i \text {sinh}(y\varGamma t)] \end{aligned}$$

(7.13)

where $x=\varDelta m/\varGamma ,$ $y=\varDelta \varGamma /\varGamma ,$ and $\tilde{J}_i\equiv \xi _i\bar{J}_i$. The extra terms $h_i$ and $s_i$ are the cross terms because of meson mixing [9]. These are time-dependent angular functions. To construct time-independent observables, these are integrated over a range of time which is $t\in \{-\infty ,\infty \}$ in the case of LHCb and $t\in [0,\infty \}$ in case of Belle. Because of this difference, the integrated angular functions are slightly different for Belle and LHCb. We have taken this into account and given the prediction of angular observables separately.

Table 7.1 Observables for $\bar{B_s}\rightarrow K^{*}\mu ^+\mu ^-$ and $B_s\rightarrow \bar{K^{*}}\mu ^+\mu ^-$ using form factors based on LCSR and QCD lattice calculation

7.4 Results

The binned values for the decay modes in study are listed in Tables 7.1 and 7.2, where the first uncertainty is due to form factors and scond uncertainty is due to soft-gluon emission from up quark. Moreover, the full branching ratio for $B_s\rightarrow \bar{K}^{*}\ell \ell$ is $(3.356\pm 0.814 )\times 10^{-8}$ which is consistent with the recent measurement [8].

Table 7.2 Binned values of observables for the process $B\rightarrow \rho \mu ^+\mu ^-$ and $\bar{B}\rightarrow \rho \mu ^+\mu ^-$ for tagged events to be measured at Belle. Form factors are based on LCSR form factors