Anatomy of the Soft-Photon Approximation in Hadron-Hadron Bremsstrahlung

Abstract

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 $M^{TsTts}_μ$ and a two-u-two-t special amplitude $M^{TuTts}_μ$, where s, t and u are the Mandelstam variables. $M^{TsTts}_μ$ 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 ($\partial$T/$\partial$s and /or $\partial$T/$\partial t$). Likewise $M^{TuTts}_μ$ depends only on the elastic T-matrix evaluated at four sets of (u,t). In deriving these amplitudes, we impose the condition that $M^{TsTts}_μ$ and $M^{TuTts}_μ$ reduce to $\bar{M}^{TsTts}_μ$ and $\bar{M}^{TuTts}_μ$, respectively, their tree level approximations. The amplitude $\bar{M}^{TsTts}_μ$ represents photon emission from a sum of one-particle t-channel exchange diagrams and one-particle s-channel exchange diagrams, while the amplitude $\bar{M}^{TuTts} _μ$ represents photon emission from a sum of one-particle t-channel exchange diagrams and one-particle u-channel exchange diagrams. The precise expressions for $\bar{M}^{TsTts}_μ$ and $\bar{M}^{TuTts}_μ$ are determined by using the radiation decomposition identities of Brodsky and Brown. We point out that it is theoretically impossible to describe all bremsstrahlung processes by using only a single class of soft-photon amplitudes. At least two different classes are required: the amplitudes which depend on s and t or the amplitudes which depend on u and t. When resonance effects are important, the amplitude $M^{TsTts}_μ$, not $M^{Low(st)}_μ$, should be used. For processes with strong u-channel exchange effects, the amplitude $M^{TuTts}_μ$ should be the first choice.