$\eta^{(\prime)}$-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$

TL;DR

This paper determines the twist-2 distribution amplitudes of eta and eta' mesons using QCD sum rules, calculates the form factors for D_s decays into these mesons, and predicts branching ratios consistent with experimental data.

Contribution

It provides new calculations of eta and eta' meson distribution amplitudes and D_s decay form factors using QCD sum rules and light-cone sum rules, including NLO corrections.

Findings

First three moments of distribution amplitudes determined.

Predicted branching ratios agree with BESIII measurements.

Form factors at zero momentum transfer calculated.

Abstract

In this paper, we make a detailed discussion on the $\eta$ and $\eta'$-meson leading-twist light-cone distribution amplitude $\phi_{2;\eta^{(\prime)}}(u,\mu)$ 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 $\langle\xi^n_{2;\eta^{(\prime)}}\rangle|_{\mu_0} $ with $n = (2,4,6)$ at initial scale $\mu_0 = 1$ GeV can be determined. e.g. $\langle\xi_{2;\eta}^2\rangle|_{\mu_0} =0.231_{-0.013}^{+0.010}$, $\langle\xi_{2;\eta}^4 \rangle|_{\mu_0} =0.109_{-0.007}^{+0.007}$, and $\langle\xi_{2;\eta}^6 \rangle|_{\mu_0} =0.066_{-0.006}^{+0.006}$ for $\eta$-meson, $\langle\xi_{2;\eta'}^2\rangle|_{\mu_0} =0.211_{-0.017}^{+0.015}$, $\langle\xi_{2;\eta'}^4 \rangle|_{\mu_0} =0.093_{-0.009}^{+0.009}$, and $\langle\xi_{2;\eta'}^6 \rangle|_{\mu_0} =0.054_{-0.008}^{+0.008}$ for $\eta'$-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 $f^{\eta'}_+(0) = 0.544_{-0.042}^{+0.046}$. 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. ${\cal B}(D_s^+\to\eta^{(\prime)} e^+\nu_e)=2.346_{-0.331}^{+0.418}(0.792_{-0.118}^{+0.141})\times10^{-2}$ and ${\cal B}(D_s^+\to\eta^{(\prime)} \mu^+\nu_\mu)=2.320_{-0.327}^{+0.413}(0.773_{-0.115}^{+0.138})\times10^{-2}$ for $\ell = (e, \mu)$ channels respectively. And in addition to that, the mixing angle for $\eta-\eta'$ with $\varphi$ and ratio for the different decay channels ${\cal R}_{\eta'/\eta}^\ell$ are given, which show good agreement with the recent BESIII measurements.