Finite size vertex corrections to the three-gluon decay widths of $J/ψ$ and $Υ$ and a redetermination of $α_s(μ)$ at $μ=m_c$ and $m_b$

Abstract

We calculate the corrections to the three-gluon decay widths $Γ_{3g}$ of $c\bar c$ and $b \bar b$ quarkonia due to the finite extension of the $Q\bar Q\to 3g$ vertex function. The widths computed with zero range vertex are reduced by a factor $γ$ where $γ=0.31$ for the $J/ψ$ and $γ=0.69$ for the $Υ$. These large corrections necessitate a redetermination of the values $α_s(μ)$ extracted from $Γ_{3g}$. We find $α_s(m_c)=0.28\pm.01$ and $α_s(m_b)=0.20\pm.01$.