Pseuduscalar Heavy Quarkonium Decays With Both Relativistic and QCD Radiative Corrections

Abstract

We estimate the decay rates of $η_c\rightarrow 2γ$, $η_c'\rightarrow 2γ$, and $J/ψ\rightarrow e^+ e^-$, $ψ^\prime\rightarrow e^+e^-$, by taking into account both relativistic and QCD radiative corrections. The decay amplitudes are derived in the Bethe-Salpeter formalism. The Bethe-Salpeter equation with a QCD-inspired interquark potential are used to calculate the wave functions and decay widths for these $c\bar{c}$ states. We find that the relativistic correction to the ratio $R\equiv Γ(η_c \rightarrow 2γ)/ Γ(J/ ψ\rightarrow e^+ e^-)$ is negative and tends to compensate the positive contribution from the QCD radiative correction. Our estimate gives $Γ(η_c \rightarrow 2γ)=(6-7) ~keV$ and $Γ(η_c^\prime \rightarrow 2γ)=2 ~keV$, which are smaller than their nonrelativistic values. The hadronic widths $Γ(η_c \rightarrow 2g)=(17-23) ~MeV$ and $Γ(η_c^\prime \rightarrow 2g)=(5-7)~MeV$ are then indicated accordingly to the first order QCD radiative correction, if $α_s(m_c)=0.26-0.29$. The decay widths for $b\bar b$ states are also estimated. We show that when making the assmption that the quarks are on their mass shells our expressions for the decay widths will become identical with that in the NRQCD theory to the next to leading order of $v^2$ and $α_s$.