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• W1965518181 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 ${mathit{M}}_{mathrm{ensuremath{mu}}}^{mathit{T}mathit{s}mathit{T}mathit{t}mathit{s}}$ 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${mathrm{ifmmodebarelsetextasciimacronfi{}}}_{mathrm{ensuremath{mu}}}^{mathit{T}mathit{sTts}}$ and M${mathrm{ifmmodebarelsetextasciimacronfi{}}}_{mathrm{ensuremath{mu}}}^{mathit{T}mathit{u}mathit{T}mathit{t}mathit{s}}$, respectively, their tree-level approximations.The amplitude M${mathrm{ifmmodebarelsetextasciimacronfi{}}}_{mathrm{ensuremath{mu}}}^{mathit{TsTts}}$ represents photon emission from a sum of one-particle t-channel exchange diagrams and one-particle s-channel exchange diagrams, while the amplitude M${mathrm{ifmmodebarelsetextasciimacronfi{}}}_{mathrm{ensuremath{mu}}}^{mathit{T}mathit{u}mathit{T}mathit{t}mathit{s}}$ represents photon emission from a sum of one-particle t-channel exchange diagrams and one-particle u-channel exchange diagrams. The precise expressions for Mifmmodebarelsetextasciimacronfi{} $_{mathrm{ensuremath{mu}}}^{mathit{T}mathit{s}mathit{T}mathit{t}mathit{s}}$ and M${mathrm{ifmmodebarelsetextasciimacronfi{}}}_{mathrm{ensuremath{mu}}}^{mathit{T}mathit{u}mathit{T}mathit{t}mathit{s}}$ 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${mathrm{ifmmodebarelsetextasciimacronfi{}}}_{mathrm{ensuremath{mu}}}^{mathit{T}mathit{s}mathit{T}mathit{t}mathit{s}}$), 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${mathrm{ifmmodebarelsetextasciimacronfi{}}}_{mathrm{ensuremath{mu}}}^{mathit{T}mathit{u}mathit{T}mathit{t}mathit{s}}$), 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." @default.
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• W1965518181 date "1993-03-01" @default.
• W1965518181 modified "2023-09-26" @default.
• W1965518181 title "Anatomy of the soft photon approximation in hadron-hadron bremsstrahlung" @default.
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• W1965518181 doi "https://doi.org/10.1103/physrevc.47.973" @default.
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