Special two-energy-two-angle approximation for bremsstrahlung

Two bremsstrahlung amplitudes in the special two-energy-two-angle approximation, which are relativistic, gauge invariant, and consistent with the soft-photon theorem, are derived for bremsstrahlung processes with or without scattering resonances. These two amplitudes include the first two terms in the series expansion of the bremsstrahlung amplitude in powers of the photon energy but they are independent of any derivative of the elastic T matrix with respect to the total energy squared s or the momentum transfer squared t. It is found that the special two-energy-two-angle approximation provides an excellent description of almost all the available ${\pi }^{\pm }$pγ and $p^{12}$Cγ data.