ANATOMY OF THE SOFT PHOTON APPROXIMATION IN HADRON-HADRON BREMSSTRAHLUNG

A modified Low procedure for constructing soft-photon amplitudes has been used to derive two general soft-photon amplitudes, a two-s-two-t special amplitude M(mu)TsTts and a two-u-two-t special amplitude M(mu)TuTts, where s, t, and u are the Mandelstam variables. M(mu)TsTts depends only on the elastic T matrix evaluated at four sets of (s, t) fixed by the requirement that the amplitude be free of derivatives (partial derivative T/partial derivative s and/or partial derivative T/partial derivative t). Likewise M(mu)TuTts depends only on the elastic T matrix evaluated at four sets of (u, t) also fixed by the requirement that the amplitude M(mu)TuTts be free of derivatives (partial derivative T/partial derivative u and/or partial derivative T/partial derivative t). In deriving these two amplitudes, we imposed the condition that M(mu)TsTts and M(mu)TuTts reduce to M(mu)TsTtsBar and M(mu)TuTtsBAR, respectively, their tree-level approximations. The amplitude M(mu)TsTtsBAR represents photon emission from a sum of one-particle t-channel exchange diagrams and one-particle s-channel exchange diagrams, while the amplitude M(mu)TuTts represents photon emission from a sum of one-particle t-channel exchange diagrams and one-particle u-channel exchange diagrams. The precise expressions for M(mu)TuTts and M(mu)TuTts are determined by using the radiation decomposition identities of Brodsky and Brown. We also demonstrate that two Low amplitudes M(mu)low(st) and M(mu)Low(ut), derived using Low's standard procedure, can be obtained from M(mu)TsTts and M(mu)TuTts, respectively, as an expansion in powers of K (photon energy) when terms of order K and higher are neglected. We point out that it is theoretically impossible to describe all nuclear bremsstrahlung processes by using only a single class of soft-photon amplitudes. At least two different classes are required: the amplitudes (such as M(mu)TsTts, M(mu)Low(st), and M(mu)TsTtsBar), which depend on s and t, and the amplitudes (such as M(mu)TuTts, M(mu)Low(ut), and M(mu)TuTtsBAr), which depend on u and t. When resonance effects are important, the amplitude M(mu)TsTts, not M(mu)Low(st), should be used. For processes with strong u-channel exchange effects, the amplitude M(mu)TuTts should be the first choice. As for those processes which exhibit neither resonance effects nor u-channel exchange effects, all amplitudes converge essentially to the same description. Finally, we discuss the relationship between the two classes.