Abstract
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 $Br(B_c \to\eta_c l\nu) = \left( 9.31^{+2.27}_{-2.01} \right) \times 10^{-3}$.
Highlights
Due to non-relativistic nature of the heavy constituent quark/antiquark, the leading-twist distribution amplitude (DA) of the heavy pseudoscalar (HP) can be roughly treated as a δ-function [1], i.e. in the leading order of the expansion over the relative velocities, the quark and the antiquark in the HP DA share the momentum of the meson according to their masses
A comparison of various ηc twist-2 DA has been done in Ref. [10], where it has pointed out that by using a proper ηc twist-2 DA, one may resolve the disagreement between the experimental observations and the NRQCD prediction on the production cross-section of e+e− → J/ + ηc
In Ref. [15], by using the Shifman– Vainshtein–Zakharov (SVZ) sum rules within the framework of background field theory (BFT), we have made a detailed study of the properties of the HP twist-2 DA and have constructed a new model for the twist-2 DA
Summary
Due to non-relativistic nature of the heavy constituent quark/antiquark, the leading-twist DA (or the twist-2 DA) of the HP can be roughly treated as a δ-function [1], i.e. in the leading order of the expansion over the relative velocities, the quark and the antiquark in the HP DA share the momentum of the meson according to their masses. [15], by using the SVZ sum rules within the framework of BFT, we have made a detailed study of the properties of the HP twist-2 DA and have constructed a new model for the twist-2 DA. We shall construct a model for φ3P;HP and φ3σ;HP with the help of the BHL-prescription. In this way the HP twist-3 DAs with a better end-point behavior can be achieved. 3. As an application of the suggested HP twist-3 DA model, we give the light-cone sum rules (LCSR) for f+Bc→ηc (q2) by using the conventional correlator, and a comparison of f+Bc→ηc (q2) with other approaches is given in Sect. As an application of the suggested HP twist-3 DA model, we give the light-cone sum rules (LCSR) for f+Bc→ηc (q2) by using the conventional correlator, and a comparison of f+Bc→ηc (q2) with other approaches is given in Sect. 4; Section 5 is reserved for a summary
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