$\eta_c$ leading-twist distribution amplitude and the $B_c \to \eta_c\ell\bar\nu_\ell$ semileptonic decays using QCD Sum Rules

Abstract

In this paper, we investigate the semileptonic decays $B_c \to \eta_c\ell\bar\nu_\ell$ using the quantum chromodynamics(QCD) sum rules within the framework of Standard Model (SM). We further explore the potential to probe signatures of new Physics (NP) beyond the SM through these decays. First, we derive the $\xi$-moments $\langle\xi_{2;\eta_c}^{n}\rangle$ of the $\eta_c$-meson leading-twist distribution amplitude $\phi_{2;\eta_c}$ using the QCD sum rules within the background field theory. Considering contributions from the vacuum condensates up to dimension-six, the first two nonzero $\xi$-moments at the scale of $4$ GeV are found to be $\langle\xi_{2;\eta_c}^{2}\rangle = 0.103^{+0.009}_{-0.009}$ and $\langle\xi_{2;\eta_c}^{4}\rangle = 0.031^{+0.003}_{-0.003}$. Using these moments, we then fix the Gegenbauer expansion series of $\phi_{2;\eta_c}$ and apply it to compute the $B_c \to \eta_c$ transition form factors (TFFs) using QCD light cone sum rules. Second, we extrapolate those TFFs to physically allowable $q^2$-range via a simplified series expansion, and we obtain $R_{\eta_c}|_{\rm SM} = 0.308^{+0.084}_{-0.062}$. Furthermore, we explore the potential impacts of various NP scenarios on $R_{\eta_c}$. Specifically, we compute the forward-backward asymmetry $\mathcal{A}_{\rm FB}({q^2})$, the convexity parameter $\mathcal{C}_F^\tau ({q^2})$, and the longitudinal and transverse polarizations $\mathcal{P}_L ({q^2})$ and $\mathcal{P}_T ({q^2})$ for $B_c \to \eta_c$ transitions within both the SM and two types of NP scenarios. Our results contribute to a deeper understanding of $B_c$-meson semileptonic decays and provide insights into the search for the NP beyond the SM.