New Rigorous Upper Bound for the Renormalization Constant of the Nucleon

Abstract

From the analyticity properties of the nucleon propagator and the improper $\ensuremath{\pi}\mathrm{NN}$ vertex function a new upper bound is established for ${Z}_{2}$, the wave-function renormalization constant of the nucleon, in terms of the pion-nucleon coupling constant $g$ and the ${P}_{11}$ and ${S}_{11}$ elastic $\ensuremath{\pi}N$ phase shifts. The result is $Z_{2}^{}{}_{}{}^{\ensuremath{-}1}\ensuremath{-}1\ensuremath{\ge}0.096(\frac{{g}^{2}}{4\ensuremath{\pi}})$, ${Z}_{2}\ensuremath{\le}0.42$, representing improvements by factors of 8 and 2, respectively, over a previous bound obtained by Drell, Finn, and Hearn. In addition the phase of the one-nucleon-irreducible ${P}_{11} \ensuremath{\pi}N$ partial-wave amplitude, in the elastic region, is calculated in an $\frac{N}{D}$ approximation. The result of this calculation strongly suggests the existence of a zero of the nucleon propagator function, the possibility of which has been widely discussed in connection with the validity of an upper bound on $\frac{{g}^{2}}{4\ensuremath{\pi}}$.