Relativistic corrections to $Υ(1S,2S)\rightarrowγη^{(\prime)}$ in perturbative QCD

Abstract

We calculate the radiative decays $Υ(1S),Υ(2S)\rightarrowγη^{(\prime)}$ in perturbative quantum chromodynamics through order $q^{2}$ in the relativistic expansion, including the quark-content, two-gluon, and electromagnetic contributions. The quark contribution dominates, with substantial constructive electromagnetic interference and a strongly suppressed gluon contribution. The annihilation rates depend weakly on the tested light-cone distribution amplitudes and light-quark masses. Relativistic corrections reduce the $Υ(1S)$ rates by a few percent but enhance the $Υ(2S)$ rates by approximately $17\%$--$30\%$. The ratio of the $γη^{\prime}$ and $γη$ branching fractions is sensitive to the light-meson mixing parameters, with predictions differing by about a factor of $5$ between two representative schemes. At a scale of half the quarkonium mass, however, the $Υ(1S)\rightarrowγη^{\prime}$ central predictions in the two mixing schemes exceed the experimental upper limit by factors of approximately $5$ and $7$. We also examine the heavy-pseudoscalar admixtures in the light mesons. The $η_c$ contribution is small, whereas $η_b$ mixing can substantially modify the rates through interference. Destructive interference alleviates the discrepancy. With the scale set to the quarkonium mass, the prediction in the mixing scheme inferred from transition form factors is compatible with the upper limit within the input uncertainty.