$η_c$ leading-twist distribution amplitude and the $B_c \to η_c\ell\barν_\ell$ semileptonic decays using QCD Sum Rules

Abstract

In this paper, we investigate the semileptonic decays $B_c \to η_c\ell\barν_\ell$ 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 $ξ$-moments $\langleξ_{2;η_c}^{n}\rangle$ of the $η_c$-meson leading-twist distribution amplitude $ϕ_{2;η_c}$ using the QCD sum rules within the background field theory. Considering contributions from the vacuum condensates up to dimension-six, the first two nonzero $ξ$-moments at the scale of $4$ GeV are found to be $\langleξ_{2;η_c}^{2}\rangle = 0.103^{+0.009}_{-0.009}$ and $\langleξ_{2;η_c}^{4}\rangle = 0.031^{+0.003}_{-0.003}$. Using these moments, we then fix the Gegenbauer expansion series of $ϕ_{2;η_c}$ and apply it to compute the $B_c \to η_c$ 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 $R_{η_c}|_{\rm SM} = 0.308^{+0.084}_{-0.062}$. Furthermore, we explore the potential impacts of various NP scenarios on $R_{η_c}$. Specifically, we compute the forward-backward asymmetry $\mathcal{A}_{\rm FB}({q^2})$, the convexity parameter $\mathcal{C}_F^τ({q^2})$, and the longitudinal and transverse polarizations $\mathcal{P}_L ({q^2})$ and $\mathcal{P}_T ({q^2})$ for $B_c \to η_c$ 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.