Connection of relativistic and nonrelativistic wave functions in the calculation of leptonic widths

Abstract

We generalize our previous JWKB relations between the relativistic $q\bar{q}$ wave function at the origin and (a) the inverse density of states of the $q\bar{q}$ system and (b) the nonrelativistic $q\bar{q}$ wave function at the origin, to the case of potentials with a Coulomb singularity. We show that the square of the Bethe-Salpeter wave function at the the origin is given approximately for $1^{-}$ states by $\{{\left|{\chi }_{\mathrm{nS}}\mathrm{}\right(0,0\left)\mathrm{}\right|}^2,\simeq F(v^{\mathrm{rel}}\left)v^{\mathrm{rel}}\frac{{M_n}^2}{16{\pi }^2}\frac{d{M}_n}{\mathrm{dn}}\left[1-\frac{16{\alpha }_s}{3\pi }g\left(v^{\mathrm{rel}}\right)+O\left({{\alpha }_s}^2\right)\right]\right\}\left\{\simeq \frac{F\left(v^{\mathrm{rel}}\right)}{F\left(v^{\mathrm{nonrel}}\right)}\frac{{M_n}^2}{4{m_q}^2}\frac{v^{\mathrm{rel}}}{v^{\mathrm{nonrel}}}{\left|{\psi }_{\mathrm{nS}}^{\mathrm{nonrel}}\mathrm{}\right(0\left)\right|}^2\left[1-\frac{16\alpha }{3\pi }g\left(v^{\mathrm{rel}}\right)+O\left({{\alpha }_s}^2\right)\right]\right\}$ for $M_n>2\mathrm{}{m}_q$, where $F\left(v\right)=\left(\frac{4\pi {\alpha }_s}{3v}\right){[1-\exp (-\frac{4\pi {\alpha }_s}{3v}\left)\right]}^{-1\mathrm{}}$ is the usual Coulomb factor and $g\left(v\right)\mathrm{}\simeq 1$ is associated with the lowest-order gluonic radiative corrections. We present numerical evidence for the remarkable accuracy of these relations, which have important implications for the use of nonrelativistic potential models to describe quarkonium systems. We also discuss some subtleties in the $v$ and ${\alpha }_s$ dependence of corrections to leptonic widths.