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 $M_{\mu }^{TsTts}$ 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 ($\partial$T/$\partial$s and /or $\partial$T/$\partial t$). Likewise $M_{\mu }^{TuTts}$ depends only on the elastic T-matrix evaluated at four sets of (u,t). In deriving these amplitudes, we impose the condition that $M_{\mu }^{TsTts}$ and $M_{\mu }^{TuTts}$ reduce to ${\bar{M}}_{\mu }^{TsTts}$ and ${\bar{M}}_{\mu }^{TuTts}$, respectively, their tree level approximations. The amplitude ${\bar{M}}_{\mu }^{TsTts}$ represents photon emission from a sum of one-particle t-channel exchange diagrams and one-particle s-channel exchange diagrams, while the amplitude ${\bar{M}}_{\mu }^{TuTts}$ represents photon emission from a sum of one-particle t-channel exchange diagrams and one-particle u-channel exchange diagrams. The precise expressions for ${\bar{M}}_{\mu }^{TsTts}$ and ${\bar{M}}_{\mu }^{TuTts}$ 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 $M_{\mu }^{TsTts}$, 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.

Comments: 49 pages report # LA-UR-92-2702