Relativistic duality, and relativistic and radiative corrections for heavy-quark systems

Abstract

We give a JWKB proof of a relativistic duality relation which relates an appropriate energy average of the physical cross section for $e^{+}{e}^{-}\to q\overline{q}$ bound states → hadrons to the same energy average of the perturbative cross section for $e^{+}{e}^{-}\to q\overline{q}$. We show that the duality relation can be used effectively to estimate relativistic and radiative corrections for bound-quark systems to order ${{\alpha }_s}^2$. We also present a formula which relates the square of the "large" ${}_{}{}^3{}_{}{}^{}S_1^{}{}_{}{}^{}$ Salpeter-Bethe-Schwinger wave function for zero space-time separation of the quarks to the square of the nonrelativistic Schrödinger wave function at the origin for an effective potential which reproduces the relativistic spectrum. This formula allows one to use the nonrelativistic wave functions obtained in potential models fitted to the $\psi$ and $\Upsilon$ spectra to calculate relativistic leptonic widths for $q\overline{q}$ states via a relativistic version of the Van Royen - Weisskopf formula.