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 ${\mathit{M}}_{\mathrm{\mu }}^{\mathit{T}\mathit{s}\mathit{T}\mathit{t}\mathit{s}}$ and a two-u–two-t special amplitude ${\mathit{M}}_{\mathrm{\mu }}^{\mathit{T}\mathit{u}\mathit{T}\mathit{t}\mathit{s}}$, where s, t, and u are the Mandelstam variables. ${\mathit{M}}_{\mathrm{\mu }}^{\mathit{T}\mathit{s}\mathit{T}\mathit{t}\mathit{s}}$ 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 (∂T/∂s and/or ∂T/∂t). Likewise ${\mathit{M}}_{\mathrm{\mu }}^{\mathit{T}\mathit{u}\mathit{T}\mathit{t}\mathit{s}}$ depends only on the elastic T matrix evaluated at four sets of (u,t) also fixed by the requirement that the amplitude ${\mathit{M}}_{\mathrm{\mu }}^{\mathit{T}\mathit{u}\mathit{T}\mathit{t}\mathit{s}}$ be free of derivatives (∂T/∂u and/or ∂T/∂t). In deriving these two amplitudes, we imposed the condition that ${\mathit{M}}_{\mathrm{\mu }}^{\mathit{T}\mathit{s}\mathit{T}\mathit{t}\mathit{s}}$ and ${\mathit{M}}_{\mathrm{\mu }}^{\mathit{T}\mathit{u}\mathit{T}\mathit{t}\mathit{s}}$ reduce to M${\mathrm{¯}}_{\mathrm{\mu }}^{\mathit{T}\mathit{sTts}}$ and M${\mathrm{¯}}_{\mathrm{\mu }}^{\mathit{T}\mathit{u}\mathit{T}\mathit{t}\mathit{s}}$, respectively, their tree-level approximations.