Pion-proton bremsstrahlung calculation and the ‘‘experimental’’ magnetic moment ofΔ++(1232)

A bremsstrahlung amplitude in the special two-energy-two-angle (TETAS) approximation, which is relativistic, gauge invariant, and consistent with the soft-photon theorem, is derived for the pion-proton bremsstrahlung (${\mathrm{\pi }}^{+}$pγ) process near the ${\mathrm{\Delta }}^{++}$(1232) resonance. In order to take into account bremsstrahlung emission from an internal ${\mathrm{\Delta }}^{++}$ line with both charge and the anomalous magnetic moment ${\lambda }_{\mathrm{\Delta }}$, we have applied a radiation decomposition identity to modify Low’s standard prescription for constructing a soft-photon amplitude. This modified procedure is very general; it can be used to derive the TETAS amplitude for any bremsstrahlung process with resonance. The derived TETAS amplitude is applied to calculate all ${\mathrm{\pi }}^{+}$pγ cross sections which can be compared with the experimental data. Treating ${\lambda }_{\mathrm{\Delta }}$ as a free parameter in these calculations, we extract the ‘‘experimental’’ magnetic moment of the ${\mathrm{\Delta }}^{++}$, ${\mathrm{\mu }}_{\mathrm{\Delta }}$, from recent data. The extracted values of ${\mathrm{\mu }}_{\mathrm{\Delta }}$ are (3.7–4.2)e/(2${\mathit{m}}_{\mathit{p}}$) from the University of California, Los Angeles data and (4.6–4.9)e/(2${\mathit{m}}_{\mathit{p}}$) from the Paul Scherrer Institute data. Here, ${\mathit{m}}_{\mathit{p}}$ is the proton mass.