Partial Differential Equations with Respect to Coupling Constants: Electromagnetic Mass Difference of Hadrons

Abstract

The partial differential equations with respect to strong and electromagnetic coupling constants are established for the $S$ matrix, Heisenberg operators, and "out" field operators. The macrocausality relations-vanishing of derivatives with respect to coupling constants for free "in" field operators-are important in the formalism. As a consequence of macrocausality relations, observable masses do not depend on strong and electromagnetic coupling constants. This makes the observable $n\ensuremath{-}p$ mass difference uncomputable. Thus, our approach goes along the lines of divergent (but renormalizable) field theory. We are assuming (although not discussing) the existence of nonelectromagnetic interactions which make ${m}_{n}\ensuremath{-}{m}_{p}\ensuremath{\ne}0$ when $e=0$. The partial differential equations with respect to strong and electromagnetic coupling constants are derived for bare $n$ and $p$ masses. In order to integrate these differential equations, the observable $n$ and $p$ must be known a priori. After integrating these differential equations, a formal expression for the observable $n\ensuremath{-}p$ mass difference is obtained. This expression, compared with the usual expression from the literature, besides containing the difference of $n$ and $p$ mass shifts due to the electromagnetic interactions ("renormalized" by strong interactions), also contains the bare $n\ensuremath{-}p$ mass difference and the difference of $n$ and $p$ mass shifts due to strong interactions. One cannot "recover" the usual expression for the $n\ensuremath{-}p$ mass difference since our expression, as far as the observable $n\ensuremath{-}p$ mass difference is concerned, is an identity and not the relation from which it can be computed.