Short-time perturbation theory and nonrelativistic duality

Abstract

We give a simple proof of the nonrelativistic duality relation $〈{W}^{2}{\ensuremath{\sigma}}_{\mathrm{bound}}〉\ensuremath{\approx}〈{W}^{2}{\ensuremath{\sigma}}_{\mathrm{free}}〉$ for appropriate energy averages of the cross sections for ${e}^{+}{e}^{\ensuremath{-}}\ensuremath{\rightarrow}(q\overline{q} \mathrm{bound}\mathrm{states})$ and ${e}^{+}{e}^{\ensuremath{-}}\ensuremath{\rightarrow}(\mathrm{free} q\overline{q} \mathrm{pair})$, and calculate the corrections to the relation by relating ${W}^{2}\ensuremath{\sigma}$ to the Fourier transform of the Feynman propagation function and developing a short-time perturbation series for that function. We illustrate our results in detail for simple power-law potentials and potentials which involve combinations of powers.