Heavy Pseudoscalar Twist-3 Distribution Amplitudes within QCD Theory in Background Fields

In this paper, we study the properties of the twist-3 distribution amplitude (DA) of the heavy pseudo-scalars such as $\eta_c$, $B_c$ and $\eta_b$. New sum rules for the twist-3 DA moments $\left<\xi^n_P\right>_{\rm HP}$ and $\left<\xi^n_\sigma\right>_{\rm HP}$ up to sixth orders and up to dimension-six condensates are deduced under the framework of the background field theory. Based on the sum rules for the twist-3 DA moments, we construct a new model for the two twist-3 DAs of the heavy pseudo-scalar with the help of the Brodsky-Huang-Lepage prescription. Furthermore, we apply them to the $B_c\to\eta_c$ transition form factor ($f^{B_c\to\eta_c}_+(q^2)$) within the light-cone sum rules approach, and the results are comparable with other approaches. It has been found that the twist-3 DAs $\phi^P_{3;\eta_c}$ and $\phi^\sigma_{3;\eta_c}$ are important for a reliable prediction of $f^{B_c\to\eta_c}_+(q^2)$. For example, at the maximum recoil region, we have $f^{B_c\to\eta_c}_+(0) = 0.674 \pm 0.066$, in which those two twist-3 terms provide $\sim33\%$ and $\sim22\%$ contributions. Also we calculate the branching ratio of the semi-leptonic decay $B_c \to\eta_c l\nu$ as $Br(B_c \to\eta_c l\nu) = \left( 9.31^{+2.27}_{-2.01} \right) \times 10^{-3}$.