Infrared finite scattering theory: Amplitudes and soft theorems

Any nontrivial scattering with massless fields in four spacetime dimensions will generically produce an out-state with memory. Scattering with any massless fields violates the standard assumption of asymptotic completeness—that all “in” and “out” states lie in the standard (zero-memory) Fock space—and therefore leads to infrared divergences in the standard <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"><a:mi>S</a:mi></a:math>-matrix amplitudes. In this paper, we define an infrared finite scattering theory which assumes only (1) the existence of in-/out-algebras and (2) that Heisenberg evolution is an automorphism of these algebras. The resulting “superscattering” map <c:math xmlns:c="http://www.w3.org/1998/Math/MathML" display="inline"><c:mi>$</c:mi></c:math> allows for transitions between different in/out memory states and agrees with the standard <e:math xmlns:e="http://www.w3.org/1998/Math/MathML" display="inline"><e:mi>S</e:mi></e:math> matrix when it is defined. We construct <g:math xmlns:g="http://www.w3.org/1998/Math/MathML" display="inline"><g:mi>$</g:mi></g:math> amplitudes by defining (3) a “generalized asymptotic completeness” which accommodates states with memory in the space of asymptotic states and (4) a complete basis of improper states that generalize the usual <i:math xmlns:i="http://www.w3.org/1998/Math/MathML" display="inline"><i:mi>n</i:mi></i:math>-particle momentum basis to account for states with memory. Using only general properties of <k:math xmlns:k="http://www.w3.org/1998/Math/MathML" display="inline"><k:mi>$</k:mi></k:math>, we prove an analog of the Weinberg soft theorems in quantum gravity and QED which imply that all <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" display="inline"><m:mi>$</m:mi></m:math> amplitudes are well defined in the infrared. We comment on how one must generalize this framework to consider <o:math xmlns:o="http://www.w3.org/1998/Math/MathML" display="inline"><o:mi>$</o:mi></o:math> amplitudes for theories with collinear divergences (e.g., massless QED and Yang-Mills theories). Published by the American Physical Society 2024