Meson-Exchange Effects in Neutron-Proton Bremsstrahlung

A prescription for including exchange effects in neutron-proton bremsstrahlung ($\mathrm{np}\gamma$) by requiring gauge invariance of the complete $\mathrm{np}\gamma$ amplitude is given entirely within a potential framework. To lowest order in the photon momentum ($K^0$) this prescription is unambiguous and it includes, as well as exchange effects, any other nonlocal effects in the nuclear potential. The lowest-order contribution is written, with the use of the Schrödinger equation, so as to eliminate the necessity of intergrating over the nuclear potential. The higher-order exchange contributions are treated through order $K^2$ using the one-pion exchange (OPE) only. (Because the effective expansion is $\frac{K}{\mu }$, where $\mu$ is the mass of the exchanged meson, the pion is expected to dominate.) These higher-order terms are the same as the Feynman-diagram prescription would give for $\mathrm{np}\gamma$ due to OPE. Calculations with the Hamada-Johnston (HJ) and Bryan-Scott III (BS) potentials are compared to the experimental results at 208 MeV by Brady et al. and at 130 MeV by Edgington et al. This inclusion of the exchange bremsstrahlung (order $K^0$) increases the cross section by roughly a factor of 2, providing generally good agreement with both experiments. An estimate of the order-$K^2$ terms indicates that they contribute $K$ contributions are discussed. Contributions arising from nonlocal terms other than exchange such as momentum-dependent and spin-orbit effects amount to about 1%. There is little difference between the $\mathrm{np}\gamma$ predictions for the HJ and BS potentials. Our $\mathrm{np}\gamma$ coplanar results for the HJ potential are compared to those obtained in a calculation, which uses the low-energy theorem for internal radiation, by Celenza et al.