Eur. Phys. J. C (2024) 84: 15
https://doi.org/10.1140/epjc/s10052-023-12333-w

Regular Article - Theoretical Physics

Properties of the ${\eta }_q$ leading-twist distribution amplitude and its effects to the $B/D^{+}\to {\eta }^{(\prime )}{\ell }^{+}{\nu }_{\ell }$ decays

¹ Department of Physics, Chongqing Key Laboratory for Strongly Coupled Physics, Chongqing University, 401331, Chongqing, People’s Republic of China
² Department of Physics, Guizhou Minzu University, 550025, Guiyang, People’s Republic of China

^b wuxg@cqu.edu.cn

Received: 30 October 2023
Accepted: 4 December 2023
Published online: 8 January 2024

Abstract

The ${\eta }^{(\prime )}$-mesons in the quark-flavor basis are mixtures of two mesonic states $|{\eta }_q⟩=|\overline{u}u+\overline{d}d⟩/\sqrt{2}$ and $|{\eta }_s⟩=|\overline{s}s⟩.$ In previous work, we have made a detailed study on the ${\eta }_s$ leading-twist distribution amplitude by using the $D_s^{+}$ meson semileptonic decays. As a sequential work, in the present paper, we fix the ${\eta }_q$ leading-twist distribution amplitude by using the light-cone harmonic oscillator model for its wave function and by using the QCD sum rules within the QCD background field to calculate its moments. The input parameters of ${\eta }_q$ leading-twist distribution amplitude ${\varphi }_{2;{\eta }_q}$ at the initial scale ${\mu }_0\sim 1$ GeV are fixed by using those moments. The QCD sum rules for the $0_{\text{th}}$-order moment can also be used to fix the magnitude of ${\eta }_q$ decay constant, giving $f_{{\eta }_q}=0.141\pm 0.005$ GeV. As an application of ${\varphi }_{2;{\eta }_q},$ we calculate the transition form factors $B{(D)}^{+}\to {\eta }^{(\prime )}$ by using the QCD light-cone sum rules up to twist-4 accuracy and by including the next-to-leading order QCD corrections to the leading-twist part, and then fix the related CKM matrix element and the decay width for the semi-leptonic decays $B{(D)}^{+}\to {\eta }^{(\prime )}{\ell }^{+}{\nu }_{\ell }.$