Heavy Pseudoscalar Leading-Twist Distribution Amplitudes within QCD Theory in Background Fields

In this paper, we study the leading-twist distribution amplitude (DA) of the heavy pseudoscalars (HPs), such as $\eta_c$, $\eta_b$ and $B_c$, within the QCD theory in the background fields. New sum rules up to dimension-six condensates for both the HP decay constants and their leading-twist DA moments are presented. From the sum rules for the HP decay constants, we obtain $f_{\eta_c} = 453 \pm 4 \textrm{MeV}$, $f_{B_c} = 498 \pm 14 \textrm{MeV}$, and $f_{\eta_b} = 811 \pm 34 \textrm{MeV}$. Basing on the sum rules for the HPs' leading-twist DA moments, we construct a new model for the $\eta_c$, $\eta_b$ and $B_c$ leading-twist DAs. Our present HP DA model can also be adaptable for the light pseudo-scalar DAs, such as the pion and kaon DAs. Thus, it shall be applicable for a wide range of QCD exclusive processes. As an application, we apply the $\eta_c$ leading-twist DA to calculate the $B_c \to \eta_c$ transition form factor $f_+^{B_c \to \eta_c}(q^2)$. At the maximum recoil region, we obtain $f_+^{B_c \to \eta_c}(0) = 0.612^{+0.053}_{-0.052}$. After further extrapolating the TFF $f_+^{B_c \to \eta_c}(q^2)$ to its allowable $q^2$ region, we predict the branching ratio for the semi-leptonic decay $B_c \to \eta_c l \nu$. We obtain ${\cal B}(B_c \to \eta_c l \nu)=\left(7.70^{+1.65}_{-1.48}\right) \times 10^{-3}$ for massless leptons, which is consistent with the LCSRs estimation obtained in the literature.