Eur. Phys. J. C (2022) 82: 12
https://doi.org/10.1140/epjc/s10052-021-09958-0

Regular Article - Theoretical Physics

${\eta }^{(\prime )}$-meson twist-2 distribution amplitude within QCD sum rule approach and its application to the semi-leptonic decay $D_s^{+}\to {\eta }^{(\prime )}{\ell }^{+}{\nu }_{\ell }$

¹ Department of Physics, Guizhou Minzu University, 550025, Guiyang, People’s Republic of China
² Department of Physics, Chongqing University, 401331, Chongqing, People’s Republic of China
³ Chongqing Key Laboratory for Strongly Coupled Physics, 401331, Chongqing, People’s Republic of China
⁴ School of Science, Chongqing University of Posts and Telecommunications, 400065, Chongqing, People’s Republic of China

^b fuhb@cqu.edu.cn

Received: 30 September 2021
Accepted: 17 December 2021
Published online: 7 January 2022

Abstract

In this paper, we make a detailed discussion on the $\eta$ and ${\eta }^{\prime }$-meson leading-twist light-cone distribution amplitude ${\varphi }_{2;{\eta }^{(\prime )}}(u,\mu )$ by using the QCD sum rules approach under the background field theory. Taking both the non-perturbative condensates up to dimension-six and the next-to-leading order (NLO) QCD corrections to the perturbative part, its first three moments $⟨{\xi }_{2;{\eta }^{(\prime )}}^n⟩{|}_{{\mu }_0}$ with $n=(2,4,6)$ can be determined, where the initial scale ${\mu }_0$ is set as the usual choice of 1 GeV. Numerically, we obtain $⟨{\xi }_{2;\eta }^2⟩{|}_{{\mu }_0}=0.{231}_{-0.013}^{+0.010}$, $⟨{\xi }_{2;\eta }^4⟩{|}_{{\mu }_0}=0.{109}_{-0.007}^{+0.007}$, and $⟨{\xi }_{2;\eta }^6⟩{|}_{{\mu }_0}=0.{066}_{-0.006}^{+0.006}$ for $\eta$-meson, $⟨{\xi }_{2;{\eta }^{\prime }}^2⟩{|}_{{\mu }_0}=0.{211}_{-0.017}^{+0.015}$, $⟨{\xi }_{2;{\eta }^{\prime }}^4⟩{|}_{{\mu }_0}=0.{093}_{-0.009}^{+0.009}$, and $⟨{\xi }_{2;{\eta }^{\prime }}^6⟩{|}_{{\mu }_0}=0.{054}_{-0.008}^{+0.008}$ for ${\eta }^{\prime }$-meson. Next, we calculate the $D_s\to {\eta }^{(\prime )}$ transition form factors (TFFs) $f_{+}^{{\eta }^{(\prime )}}(q^2)$ within QCD light-cone sum rules approach up to NLO level. The values at large recoil region are $f_{+}^{\eta }(0)=0.{476}_{-0.036}^{+0.040}$ and $f_{+}^{{\eta }^{\prime }}(0)=0.{544}_{-0.042}^{+0.046}$. After extrapolating TFFs to the allowable physical regions within the series expansion, we obtain the branching fractions of the semi-leptonic decay, i.e. $D_s^{+}\to {\eta }^{(\prime )}{\ell }^{+}{\nu }_{\ell }$, i.e. $\mathcal{B}(D_s^{+}\to {\eta }^{(\prime )}{e}^{+}{\nu }_e)=2.{346}_{-0.331}^{+0.418}(0.{792}_{-0.118}^{+0.141})\times {10}^{-2}$ and $\mathcal{B}(D_s^{+}\to {\eta }^{(\prime )}{\mu }^{+}{\nu }_{\mu })=2.{320}_{-0.327}^{+0.413}(0.{773}_{-0.115}^{+0.138})\times {10}^{-2}$ for $\ell =(e,\mu )$ channels respectively. And in addition to that, the mixing angle for $\eta -{\eta }^{\prime }$ with $\phi$ and ratio for the different decay channels ${\mathcal{R}}_{{\eta }^{\prime }/\eta }^{\ell }$ are given, which show good agreement with the recent BESIII measurements.