Eta ^{(prime )}-meson twist-2 distribution amplitude within QCD sum rule approach and its application to the semi-leptonic decay D_s^+ rightarrow 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 phi _{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 langle xi ^n_{2;eta ^{(prime )}}rangle |_{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 langle xi _{2;eta }^2rangle |_{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 '}^2rangle |_{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 the D_srightarrow 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 '}_+(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^+rightarrow eta ^{(prime )}ell ^+ nu _ell , i.e. {{mathcal {B}}}(D_s^+ rightarrow 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^+ rightarrow 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 ' with varphi and ratio for the different decay channels {{mathcal {R}}}_{eta '/eta }^ell are given, which show good agreement with the recent BESIII measurements.