Partial-wave projection of relativistic three-body amplitudes

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.