Weak Annihilation Contribution to Angular Observables in $B_{c}^+\to D^{\ast+}\ell^{+}\ell^{-}$ Decays

TL;DR

This paper develops a detailed Standard Model analysis of the rare decay $B_c^+ \to D^{*+}(\to P_1P_2)\ell^+\ell^-$, incorporating penguin-box, weak annihilation (WA), and long-distance resonance effects within an effective Hamiltonian framework. Using covariant confined quark model form factors for $B_c^+ \to D^{*}$ and WA form factors from prior work, the authors compute differential rates, forward-backward asymmetries, longitudinal $D^{*}$ polarization, and normalized angular observables $\langle I_i\rangle$ across low and high $q^2$ regions for $\ell=\mu,\tau$. They find WA effects are sizable at low $q^2$, significantly altering several observables and shifting zero-crossings, while LD resonances dominate at high $q^2$, reducing theoretical reliability in that region; nonetheless, clean windows such as $[0.2,7]\,\text{GeV}^2$ and high-$q^2$ resonance-free intervals $[10.5,13]$ and $[14, q_{\max}^2]$ GeV$^2$ emerge for NP sensitivity. The study establishes a comprehensive SM baseline that is essential for precise predictions and for isolating potential New Physics contributions in $B_c$ semileptonic decays, and it underscores the need for improved hadronic form factors and LD treatment. Overall, WA must be included to accurately interpret observables like $d\mathcal{B}/dq^2$, $A_{FB}$, and $f_L$ in future experimental analyses.

Abstract

We analyze the rare semileptonic decays $B_{c}^+ \to D^{\ast+}(\to P_1 P_2)\ell^{+}\ell^{-}$, with $P_1 P_2 = D^+ π^0$ or $D^0 π^+$, and $\ell=μ, τ$. We focus on the impact of weak annihilation contributions alongside penguin, box, and long-distance effects. Using the effective Hamiltonian for $b \to d \ell^+ \ell^-$ transitions and $B_c \to D^{*}$ form factors from covariant confined quark model inputs, we compute the differential branching ratios, forward-backward asymmetry, longitudinal helicity fraction of the $D^{\ast}$, and various normalized angular coefficients. The results of the observables show that weak annihilation effects are sizable, particularly at low $q^2$, significantly modifying several observables and shifting zero-crossings. Resonance effects dominate at high $q^2$, restricting reliable analysis windows. We conclude that the inclusion of weak annihilation is essential for precise Standard Model predictions and for isolating possible New Physics effects in $B_c^+ \to D^{*+} \ell^+ \ell^-$ decays.

Figures (8)

• Figure 1: Feynman diagrams contributing to the process.
• Figure 2: Kinematics of the $B_{c}\to D^{\ast}(\to P_1 P_2)l^{+}l^{-}$ decay.
• Figure 3: Branching ratios of the two channels $B_c \to D^*(\to P_1 P_2)\ell^+\ell^-$ decays in the PB, PB+LD, PB+WA and PB+WA+LD contributions. The first and the second columns show the differential branching ratios $\mathrm{d}B/\mathrm{d}q^2$, with $\ell= \mu$, as a function of the squared dilepton mass $q^2$, in the lower and the higher $q^2$ regions, respectively, While the third column displays the differential branching ratios with final stage particle $\ell = \tau$, in the kinematical region $q^2=[15, q_{\text{max}}^2]~\text{GeV}^2$.
• Figure 4: The first and the second columns show (from top to bottom) the $A_{\mathrm{FB}}$ and $f_L$ in the PB, PB+LD, PB+WA and PB+WA+LD contributions in the lower and the higher for $\ell = \mu$, as functions of the squared dilepton mass $q^2$$(\text{GeV}^2)$ respectively. While the third column displays the same observables with final stage particles $\ell = \tau$ in the higher bin only. Legends are similar as described in FIG. \ref{['Fig2']}.
• Figure 5: Angular observables of $B_c \to D^*(\to P_1 P_2)\ell^+\ell^-$ decay in the PB, PB+LD, PB+WA and PB+WA+LD contributions. The first and the second columns show the average values of angular observables $\langle I_{(1s,1c,2s,2c)} \rangle$ in the lower and the upper bins for $\ell= \mu$ as functions of the squared dilepton mass $q^2$$(\text{GeV}^2)$, respectively. While the third column displays the same observables with final stage particles $\ell= \tau$ in the higher bin.