Electromagnetic Corrections to Nucleon Transitions of the Vector and Axial-Vector Currents

We examine the matrix elements of the divergences of the vector and axial-vector current between nucleon states. These matrix elements are related to the nucleon mass difference and the corrections to the Goldberger-Treiman relation, respectively. For the nucleon mass difference we indicate that for the sign of this quantity to be understood in terms of the electromagnetic interaction requires (i) comparable longitudinal and transverse virtual photon-nucleon cross sections, or (ii) ${{\sigma }_t}^P(q^2,\nu )-{{\sigma }_t}^n(q^2,\nu )<\mathrm{}0\mathrm{}$ over a large region of the ($\left|q^2\right|,\nu$) plane, where $q^2$ is the spacelike virtual-photon mass and $\nu$ is the photon energy (this requirement is contraindicated by experimental data at $q^2=0\mathrm{}$), or (iii) fixed $J$-plane poles at $J=0\mathrm{}$, $I=1\mathrm{}$ in the virtual Compton amplitude. We also estimate the electromagnetic correction to the Goldberger-Treiman relation, and it is shown to be very small.