Systematic analysis of the transition form factors of $B_c$ to $D$-wave charmonia and corresponding semileptonic decays

TL;DR

The paper develops three-point QCD sum rules to compute the vector, axial-vector and tensor transition form factors for $B_c$ decays to $D$-wave charmonia ($\psi_J$ with $J=1,2,3$ and $\eta_{c2}$). By forming appropriate correlators with tailored interpolating currents and performing the operator product expansion including gluon condensates, the authors extract the form factors in the space-like region and then extrapolate to the full kinematic range using a $z$-series expansion. With these form factors, they calculate the widths and branching ratios of the semileptonic decays $B_c\to X\,l\bar{\nu}_l$, finding a clear suppression as the final-state angular momentum increases, and they provide lepton-universality and cross-channel ratios as benchmarks. The results offer theoretical guidance for upcoming LHCb measurements and deepen the understanding of heavy-flavor dynamics and $D$-wave charmonia.

Abstract

In this article, the vector, axial vector and tensor form factors of $B_c$ to $D$-wave charmonia $ψ_J$ ($J=1,2,3$) and $η_{c2}$ are analyzed within the framework of three-point QCD sum rules. With the estimated vector and axial vector form factors, we study the decay widths and branching ratios of semileptonic decays $B_c\toψ_Jl\bar{ν_l}$ and $B_c\toη_{c2}l\bar{ν_l}$. Our results indicate that the semileptonic decay branching ratios of $B_c$ to $D$-wave charmonia decreases as the total angler momentum of the final state $D$-wave charmonia increases. For decay processes $B_c\to ψ_1l\barν_l$, $B_c\to ψ_2[η_{c2}]l\barν_l$ and $B_c\to ψ_3l\barν_l$, the branching ratios can reach the order $10^{-3}$, $10^{-4}$ and $10^{-5}$, respectively, which can provide a valuable reference for future experimental measurements for $B_c$ meson by LHCb collaboration. Furthermore, these results also can provide useful information to study the properties of $B_c$ meson and $D$-wave charmonia.

Figures (5)

• Figure 1: The Feynman diagrams for the perturbative part (a) and gluon condensate (b)-(g), where the blue and red solid lines denote the $b$ and $c$ quark lines, respectively. The black loop lines are the gluon lines
• Figure 2: Feynman diagram of semileptonic decay $B_c\to\psi_{J}[\eta_{c2}]l\bar{\nu}_l$.
• Figure 3: The pole and continuum contributions (a) and the contributions of perturbative part and gluon condensates (b) of form factor $g$ for $B_c\to\psi_1$ transition. The grey bounds are the Borel platform.
• Figure 4: Fitting results of all form factors.
• Figure 5: The differential decay width $d\Gamma/dq^2$ with variations of $q^2$ for semileptonic decay processes $B_c\to\psi_{J}[\eta_{c2}]l\bar{\nu}_l$.