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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 $η_c$, $B_c$ and $η_b$. New sum rules for the twist-3 DA moments $\left<ξ^n_P\right>_{\rm HP}$ and $\left<ξ^n_σ\right>_{\rm HP}$ 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 $B_c\toη_c$ transition form factor ($f^{B_c\toη_c}_+(q^2)$) within the light-cone sum rules approach, and the results are comparable with other approaches. It has been found that the twist-3 DAs $ϕ^P_{3;η_c}$ and $ϕ^σ_{3;η_c}$ are important for a reliable prediction of $f^{B_c\toη_c}_+(q^2)$. For example, at the maximum recoil region, we have $f^{B_c\toη_c}_+(0) = 0.674 \pm 0.066$, in which those two twist-3 terms provide $\sim33\%$ and $\sim22\%$ contributions. Also we calculate the branching ratio of the semi-leptonic decay $B_c \toη_c lν$ as $Br(B_c \toη_c lν) = \left( 9.31^{+2.27}_{-2.01} \right) \times 10^{-3}$.