Partial-wave projection of relativistic three-body amplitudes
Raúl A. Briceño, Caroline S. R. Costa, Andrew W. Jackura
Abstract
We derive the integral equations for partial-wave projected three-body scattering amplitudes, starting from the integral equations for three-body amplitudes developed for lattice QCD analyses. The results, which hold for generic three-body systems of spinless particles, build upon the recently derived partial-wave projected one-particle exchange, a primary component of the relativistic framework proven to satisfy <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"> <a:mi>S</a:mi> </a:math> matrix unitarity. We derive simplified expressions for factorizable short-distance interactions, <c:math xmlns:c="http://www.w3.org/1998/Math/MathML" display="inline"> <c:msub> <c:mi mathvariant="script">K</c:mi> <c:mn>3</c:mn> </c:msub> </c:math> , in two equivalent formalisms—one symmetric under particle interchange and one asymmetric. For the asymmetric case, we offer parametrizations useful for amplitude analysis. Finally, we examine toy models for <f:math xmlns:f="http://www.w3.org/1998/Math/MathML" display="inline"> <f:mn>3</f:mn> <f:mi>π</f:mi> </f:math> systems at unphysically heavy pion masses with total isospins 0, 1, and 2.