Energy-Momentum Structure Form Factors of Particles

By emphasizing the analogy between mass and charge as sources of fields, we are led to examine the mechanical structure of a particle in terms of the matrix elements $〈{\mathrm{p}}^{\ensuremath{'}}|{\ensuremath{\theta}}_{\ensuremath{\alpha}\ensuremath{\beta}}(0)|\mathrm{p}〉$, where ${\ensuremath{\theta}}_{\ensuremath{\alpha}\ensuremath{\beta}}(x)$ is the total energy-momentum tensor, just as the matrix elements $〈{\mathrm{p}}^{\ensuremath{'}}|{j}_{\ensuremath{\alpha}}(0)|\mathrm{p}〉$ of the current operator ${j}_{\ensuremath{\alpha}}(x)$ define its electromagnetic structure. Although the off-diagonal matrix elements $〈{\mathrm{p}}^{\ensuremath{'}}|{\ensuremath{\theta}}_{\ensuremath{\alpha}\ensuremath{\beta}}(0)|\mathrm{p}〉$ are not accessible to direct experimental observation, the diagonal element $〈\mathrm{p}|{\ensuremath{\theta}}_{\ensuremath{\alpha}\ensuremath{\beta}}(0)|\mathrm{p}〉$ is just proportional to the total mass. Consequently, we can study the contributions to the total mass in terms of vertex functions instead of propagators and, using the techniques of dispersion theory, relate the contribution to the total mass to integrals over physical scattering processes. We examine electrodynamics and the pion-nucleon interaction in perturbation theory and show how the mass divergences emerge as a consequence of the high-energy behavior of the Coulomb amplitude and the nucleon-nucleon scattering amplitude. Finally, using elastic unitarity, we can relate mass splittings in a multiplet to integrals over the differences in $S$-wave phase shifts.