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

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 QCD sum rules approach under the background field theory. Taking both the non-perturbative condensates up to dimension-six and NLO QCD corrections to the perturbative part, its first three moments $⟨{\xi }_{2;{\eta }^{(\prime )}}^n⟩{|}_{{\mu }_0}$ with $n=(2,4,6)$ at initial scale ${\mu }_0=1$ GeV can be determined. e.g. $⟨{\xi }_{2;\eta }^2⟩{|}_{{\mu }_0}=0.231_{-0.013}^{+0.010}$, $⟨{\xi }_{2;\eta }^4⟩{|}_{{\mu }_0}=0.109_{-0.007}^{+0.007}$, 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 $D_s\to {\eta }^{(\prime )}$ 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.

52 pages, 11 figures, version to be published in EPJC

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