All orders factorization and the Coulomb problem
R. Hill, Ryan Plestid
Abstract
In the limit of large nuclear charge, <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"><a:mi>Z</a:mi><a:mo>≫</a:mo><a:mn>1</a:mn></a:math>, or small lepton velocity, <c:math xmlns:c="http://www.w3.org/1998/Math/MathML" display="inline"><c:mi>β</c:mi><c:mo>≪</c:mo><c:mn>1</c:mn></c:math>, Coulomb corrections to nuclear beta decay and related processes are enhanced as <e:math xmlns:e="http://www.w3.org/1998/Math/MathML" display="inline"><e:mi>Z</e:mi><e:mi>α</e:mi><e:mo>/</e:mo><e:mi>β</e:mi></e:math> and become large or even nonperturbative (with <g:math xmlns:g="http://www.w3.org/1998/Math/MathML" display="inline"><g:mi>α</g:mi></g:math> the QED fine structure constant). We provide a constructive demonstration of factorization to all orders in perturbation theory for these processes and compute the all-orders hard and soft functions appearing in the factorization formula. We clarify the relationship between effective field theory amplitudes and historical treatments of beta decay in terms of a Fermi function. Published by the American Physical Society 2024