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.

These values are smaller than the value 5.58e/(2${\mathit{m}}_{\mathit{p}}$), the ‘‘bare’’ magnetic moment predicted by the SU(6) model or the quark model, but they are close to the value 4.25e/(2${\mathit{m}}_{\mathit{p}}$) predicted by the modified SU(6) model of Beg and Pais and to the value (4.41–4.89)e/(2${\mathit{m}}_{\mathit{p}}$) predicted by the corrected bag-model of Brown, Rho, and Vento. Using the extracted ${\mathrm{\mu }}_{\mathrm{\Delta }}$ as an input for calculating ${\mathrm{\pi }}^{+}$pγ cross sections, we show that the overall agreement between the theoretical predictions calculated with the extracted ${\mathrm{\mu }}_{\mathrm{\Delta }}$ and the experimental measurements is excellent. This agreement demonstrates that the TETAS amplitude can be used to describe almost all the available ${\mathrm{\pi }}^{+}$pγ data. Finally, we also treat ${\lambda }_{\mathrm{\Delta }}$ as a complex quantity, ${\lambda }_{\mathrm{\Delta }}$=${\lambda }_{\mathit{R}}$+i${\lambda }_{\mathit{I}}$, in order to estimate the contribution from the imaginary part ${\lambda }_{\mathit{I}}$. The best fit to the data gives ${\lambda }_{\mathit{I}}$≊0, independent of the choice of ${\lambda }_{\mathit{R}}$. This fact implies that further dynamical corrections to the TETAS amplitude from the open pion-proton channel are small. Therefore, there is a good reason to believe that the ‘‘experimental’’ magnetic moment, which is very close to the ‘‘bare’’ magnetic moment predicted by the modified SU(6) or the quark model with corrections, should be nearly equal to the ‘‘effective’’ magnetic moment.

• Received 21 May 1991

DOI:https://doi.org/10.1103/PhysRevC.44.1819