Finite size vertex corrections to the three-gluon decay widths of $J/\psi$ and ${\rm Y}$ and a redetermination of ${\alpha }_s(\mu )$ at $\mu =m_c$ and $m_b$

Abstract

We calculate the corrections to the three-gluon decay widths ${\Gamma }_{3g}$ 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 $\gamma$ where $\gamma =0.31$ for the $J/\psi$ and $\gamma =0.69$ for the ${\rm Y}$. These large corrections necessitate a redetermination of the values ${\alpha }_s(\mu )$ extracted from ${\Gamma }_{3g}$. We find ${\alpha }_s(m_c)=0.28\pm .01$ and ${\alpha }_s(m_b)=0.20\pm .01$.

Comments: 11 pages, HD-TVP-93-04