Validity of Approximations Used in Mesonic Atom Calculations

Abstract

The Deser-Goldberger-Baumann-Thirring (DGBT) formula [$a=\frac{1}{4}B(\frac{\ensuremath{\Delta}{E}_{1S}}{|{E}_{1S}|})$] and other various approximations commonly used to extract meson-nucleus scattering lengths from energy-level measurements on mesonic atoms are studied by means of a model calculation and are shown to be inadequate for most of the ${\ensuremath{\pi}}^{\ensuremath{-}}$-mesonic atoms commonly considered. A simple empirical formula is obtained in place of the DGBT formula for the ${\ensuremath{\pi}}^{\ensuremath{-}}$-mesonic atoms of the light nuclei. The DGBT formula is also examined for the case of the ${K}^{\ensuremath{-}}\ensuremath{-}{\mathrm{He}}^{4}$ atom and is found to be a good approximation.