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Large corrections to asymptotic $F_{η_c γ}$ and $F_{η_b γ}$ in the light-cone perturbative QCD

The large-$Q^2$ behavior of $η_c$-$γ$ and $η_b$-$γ$ transition form factors, $F_{η_cγ}(Q^2)$ and $F_{η_bγ}(Q^2)$ are analyzed in the framework of light-cone perturbative QCD with the heavy quark ($c$ and $b$) mass effect, the parton's transverse momentum dependence and the higher helicity components in the light-cone wave function are respected. It is pointed out that the quark mass effect brings significant modifications to the asymptotic predictions of the transition form factors in a rather broad energy region, and this modification is much severer for $F_{η_bγ}(Q^2)$ than that for $F_{η_cγ}(Q^2)$ due to the $b$-quark being heavier than the $c$-quark. The parton's transverse momentum and the higher helicity components are another two factors which decrease the perturbative predictions. For the transition form factor $F_{η_cγ}(Q^2)$, they bring sizable corrections in the present experimentally accessible energy region ($Q^2 \leq 10 GeV^2$). For the transition form factor $F_{η_bγ}(Q^2)$, the corrections coming from these two factors are negligible since the $b$-quark mass is much larger than the parton's average transverse momentum. The coming $e^+ e^-$ collider (LEP2) will provide the opportunity to examine these theoretical predictions.