Different versions of soft-photon theorems exemplified at leading and next-to-leading terms for pion-pion and pion-proton scattering

We investigate the photon emission in pion-pion and pion-proton scattering in the soft-photon limit where the photon energy <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"><a:mi>ω</a:mi><a:mo stretchy="false">→</a:mo><a:mn>0</a:mn></a:math>. The expansions of the <d:math xmlns:d="http://www.w3.org/1998/Math/MathML" display="inline"><d:msup><d:mi>π</d:mi><d:mo>−</d:mo></d:msup><d:msup><d:mi>π</d:mi><d:mn>0</d:mn></d:msup><d:mo stretchy="false">→</d:mo><d:msup><d:mi>π</d:mi><d:mo>−</d:mo></d:msup><d:msup><d:mi>π</d:mi><d:mn>0</d:mn></d:msup><d:mi>γ</d:mi></d:math> and the <g:math xmlns:g="http://www.w3.org/1998/Math/MathML" display="inline"><g:msup><g:mi>π</g:mi><g:mo>±</g:mo></g:msup><g:mi>p</g:mi><g:mo stretchy="false">→</g:mo><g:msup><g:mi>π</g:mi><g:mo>±</g:mo></g:msup><g:mi>p</g:mi><g:mi>γ</g:mi></g:math> amplitudes, satisfying the energy-momentum relations, to the orders <j:math xmlns:j="http://www.w3.org/1998/Math/MathML" display="inline"><j:msup><j:mi>ω</j:mi><j:mrow><j:mo>−</j:mo><j:mn>1</j:mn></j:mrow></j:msup></j:math> and <l:math xmlns:l="http://www.w3.org/1998/Math/MathML" display="inline"><l:msup><l:mi>ω</l:mi><l:mn>0</l:mn></l:msup></l:math> are derived. We show that these terms can be expressed completely in terms of the on-shell amplitudes for <n:math xmlns:n="http://www.w3.org/1998/Math/MathML" display="inline"><n:msup><n:mi>π</n:mi><n:mo>−</n:mo></n:msup><n:msup><n:mi>π</n:mi><n:mn>0</n:mn></n:msup><n:mo stretchy="false">→</n:mo><n:msup><n:mi>π</n:mi><n:mo>−</n:mo></n:msup><n:msup><n:mi>π</n:mi><n:mn>0</n:mn></n:msup></n:math> and <q:math xmlns:q="http://www.w3.org/1998/Math/MathML" display="inline"><q:msup><q:mi>π</q:mi><q:mo>±</q:mo></q:msup><q:mi>p</q:mi><q:mo stretchy="false">→</q:mo><q:msup><q:mi>π</q:mi><q:mo>±</q:mo></q:msup><q:mi>p</q:mi></q:math>, respectively, and their partial derivatives with respect to <t:math xmlns:t="http://www.w3.org/1998/Math/MathML" display="inline"><t:mi>s</t:mi></t:math> and <v:math xmlns:v="http://www.w3.org/1998/Math/MathML" display="inline"><v:mi>t</v:mi></v:math>. The structure term which is nonsingular for <x:math xmlns:x="http://www.w3.org/1998/Math/MathML" display="inline"><x:mi>ω</x:mi><x:mo stretchy="false">→</x:mo><x:mn>0</x:mn></x:math> is determined to the order <ab:math xmlns:ab="http://www.w3.org/1998/Math/MathML" display="inline"><ab:msup><ab:mi>ω</ab:mi><ab:mn>0</ab:mn></ab:msup></ab:math> from the gauge-invariance constraint using the generalized Ward identities for pions and the proton. For the reaction <cb:math xmlns:cb="http://www.w3.org/1998/Math/MathML" display="inline"><cb:msup><cb:mi>π</cb:mi><cb:mo>−</cb:mo></cb:msup><cb:msup><cb:mi>π</cb:mi><cb:mn>0</cb:mn></cb:msup><cb:mo stretchy="false">→</cb:mo><cb:msup><cb:mi>π</cb:mi><cb:mo>−</cb:mo></cb:msup><cb:msup><cb:mi>π</cb:mi><cb:mn>0</cb:mn></cb:msup><cb:mi>γ</cb:mi></cb:math> we discuss in detail the soft-photon theorems in the versions of both Low and Weinberg. We show that these two versions are different and must not be confounded. Weinberg’s version gives the pole term of a Laurent expansion in <fb:math xmlns:fb="http://www.w3.org/1998/Math/MathML" display="inline"><fb:mi>ω</fb:mi></fb:math> of the amplitude for <hb:math xmlns:hb="http://www.w3.org/1998/Math/MathML" display="inline"><hb:msup><hb:mi>π</hb:mi><hb:mo>−</hb:mo></hb:msup><hb:msup><hb:mi>π</hb:mi><hb:mn>0</hb:mn></hb:msup><hb:mo stretchy="false">→</hb:mo><hb:msup><hb:mi>π</hb:mi><hb:mo>−</hb:mo></hb:msup><hb:msup><hb:mi>π</hb:mi><hb:mn>0</hb:mn></hb:msup><hb:mi>γ</hb:mi></hb:math> around the phase-space point of zero radiation. Low’s version gives an approximate expression for the above amplitude at a fixed phase-space point, corresponding to nonzero radiation. Clearly, the leading and next-to-leading terms in theses two approaches must be, and are indeed, different. We show their relation. We also discuss the expansions of differential cross sections for <kb:math xmlns:kb="http://www.w3.org/1998/Math/MathML" display="inline"><kb:msup><kb:mi>π</kb:mi><kb:mo>−</kb:mo></kb:msup><kb:msup><kb:mi>π</kb:mi><kb:mn>0</kb:mn></kb:msup><kb:mo stretchy="false">→</kb:mo><kb:msup><kb:mi>π</kb:mi><kb:mo>−</kb:mo></kb:msup><kb:msup><kb:mi>π</kb:mi><kb:mn>0</kb:mn></kb:msup><kb:mi>γ</kb:mi></kb:math> with respect to <nb:math xmlns:nb="http://www.w3.org/1998/Math/MathML" display="inline"><nb:mi>ω</nb:mi></nb:math> for <pb:math xmlns:pb="http://www.w3.org/1998/Math/MathML" display="inline"><pb:mi>ω</pb:mi><pb:mo stretchy="false">→</pb:mo><pb:mn>0</pb:mn></pb:math>.