Cancellation of the divergence of the wave function at the origin in leptonic decay rates

The wave function at the origin is an important factor in the leptonic decay rate of (QQ¯) states, since the rate is proportional to the square of its value. However, relativistic wave functions for the color-Coulomb potential diverge at the origin. In this paper, we consider the QED version of this problem: the O(α) corrections to the decay &→${\mathrm{e}}^{+}$ ${\mathrm{e}}^{\mathrm{-}}$. After renormalization, the divergence of the wave function at the origin is shown to cancel via the one-photon annihilation graph. With all two-loop diagrams included, the decay rate of the nS state is 1/$n^3$ that of the 1S state. This simple result is modified at the three-loop level by electron vacuum-polarization (VP) effects, which are numerically large. With these VP effects included, the decay rate of the 1 ${}_{}{}^3{}_{}{}^{}S_1^{}{}_{}{}^{}$ state is predicted to be 5.571×${10}^{11}$ ${\sec }^{\mathrm{-}1}$.