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 ${\mathit{M}}_{\mathrm{\ensuremath{\mu}}}^{\mathit{T}\mathit{s}\mathit{T}\mathit{t}\mathit{s}}$ and a two-u--two-t special amplitude ${\mathit{M}}_{\mathrm{\ensuremath{\mu}}}^{\mathit{T}\mathit{u}\mathit{T}\mathit{t}\mathit{s}}$, where s, t, and u are the Mandelstam variables. ${\mathit{M}}_{\mathrm{\ensuremath{\mu}}}^{\mathit{T}\mathit{s}\mathit{T}\mathit{t}\mathit{s}}$ 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 (\ensuremath{\partial}T/\ensuremath{\partial}s and/or \ensuremath{\partial}T/\ensuremath{\partial}t). Likewise ${\mathit{M}}_{\mathrm{\ensuremath{\mu}}}^{\mathit{T}\mathit{u}\mathit{T}\mathit{t}\mathit{s}}$ depends only on the elastic T matrix evaluated at four sets of (u,t) also fixed by the requirement that the amplitude ${\mathit{M}}_{\mathrm{\ensuremath{\mu}}}^{\mathit{T}\mathit{u}\mathit{T}\mathit{t}\mathit{s}}$ be free of derivatives (\ensuremath{\partial}T/\ensuremath{\partial}u and/or \ensuremath{\partial}T/\ensuremath{\partial}t). In deriving these two amplitudes, we imposed the condition that ${\mathit{M}}_{\mathrm{\ensuremath{\mu}}}^{\mathit{T}\mathit{s}\mathit{T}\mathit{t}\mathit{s}}$ and ${\mathit{M}}_{\mathrm{\ensuremath{\mu}}}^{\mathit{T}\mathit{u}\mathit{T}\mathit{t}\mathit{s}}$ reduce to M${\mathrm{\ifmmode\bar\else\textasciimacron\fi{}}}_{\mathrm{\ensuremath{\mu}}}^{\mathit{T}\mathit{sTts}}$ and M${\mathrm{\ifmmode\bar\else\textasciimacron\fi{}}}_{\mathrm{\ensuremath{\mu}}}^{\mathit{T}\mathit{u}\mathit{T}\mathit{t}\mathit{s}}$, respectively, their tree-level approximations.The amplitude M${\mathrm{\ifmmode\bar\else\textasciimacron\fi{}}}_{\mathrm{\ensuremath{\mu}}}^{\mathit{TsTts}}$ represents photon emission from a sum of one-particle t-channel exchange diagrams and one-particle s-channel exchange diagrams, while the amplitude M${\mathrm{\ifmmode\bar\else\textasciimacron\fi{}}}_{\mathrm{\ensuremath{\mu}}}^{\mathit{T}\mathit{u}\mathit{T}\mathit{t}\mathit{s}}$ 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\ifmmode\bar\else\textasciimacron\fi{} $_{\mathrm{\ensuremath{\mu}}}^{\mathit{T}\mathit{s}\mathit{T}\mathit{t}\mathit{s}}$ and M${\mathrm{\ifmmode\bar\else\textasciimacron\fi{}}}_{\mathrm{\ensuremath{\mu}}}^{\mathit{T}\mathit{u}\mathit{T}\mathit{t}\mathit{s}}$ are determined by using the radiation decomposition identities of Brodsky and Brown. We also demonstrate that two Low amplitudes ${\mathit{M}}_{\mathrm{\ensuremath{\mu}}}^{\mathrm{L}\mathrm{o}\mathrm{w}(\mathit{s}\mathit{t})}$ and ${\mathit{M}}_{\mathrm{\ensuremath{\mu}}}^{\mathrm{L}\mathrm{o}\mathrm{w}(\mathit{u}\mathit{t})}$, derived using Low's standard procedure, can be obtained from ${\mathit{M}}_{\mathrm{\ensuremath{\mu}}}^{\mathit{T}\mathit{s}\mathit{T}\mathit{t}\mathit{s}}$ and ${\mathit{M}}_{\mathrm{\ensuremath{\mu}}}^{\mathit{T}\mathit{u}\mathit{T}\mathit{t}\mathit{s}}$, 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 amplutides (such as ${\mathit{M}}_{\mathrm{\ensuremath{\mu}}}^{\mathit{T}\mathit{s}\mathit{T}\mathit{t}\mathit{s}}$, ${\mathit{M}}_{\mathrm{\ensuremath{\mu}}}^{\mathrm{L}\mathrm{o}\mathrm{w}(\mathit{s}\mathit{t})}$, and M${\mathrm{\ifmmode\bar\else\textasciimacron\fi{}}}_{\mathrm{\ensuremath{\mu}}}^{\mathit{T}\mathit{s}\mathit{T}\mathit{t}\mathit{s}}$), which depend on s and t, and the amplitudes (such as ${\mathit{M}}_{\mathrm{\ensuremath{\mu}}}^{\mathit{T}\mathit{u}\mathit{T}\mathit{t}\mathit{s}}$, ${\mathit{M}}_{\mathrm{\ensuremath{\mu}}}^{\mathrm{L}\mathrm{o}\mathrm{w}(\mathit{u}\mathit{t})}$, and M${\mathrm{\ifmmode\bar\else\textasciimacron\fi{}}}_{\mathrm{\ensuremath{\mu}}}^{\mathit{T}\mathit{u}\mathit{T}\mathit{t}\mathit{s}}$), which depend on u and t. When resonance effects are important, the amplitude ${\mathit{M}}_{\mathrm{\ensuremath{\mu}}}^{\mathit{T}\mathit{s}\mathit{T}\mathit{t}\mathit{s}}$, not ${\mathit{M}}_{\mathrm{\ensuremath{\mu}}}^{\mathrm{L}\mathrm{o}\mathrm{w}(\mathit{s}\mathit{t})}$, should be used. For processes with strong u-channel exchange effects, the amplitude ${\mathit{M}}_{\mathrm{\ensuremath{\mu}}}^{\mathit{T}\mathit{u}\mathit{T}\mathit{t}\mathit{s}}$ 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.