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