Symmetrizing relativistic three-body partial wave amplitudes

TL;DR

This paper develops a method to construct symmetric three-particle scattering amplitudes from asymmetric ones, ensuring consistency with S matrix principles and symmetries, with applications to lattice QCD and Dalitz plot analysis.

Contribution

It introduces a systematic way to symmetrize three-body partial wave amplitudes using recoupling coefficients for arbitrary angular momentum and isospin, applicable to spinless particles with SU(2) symmetry.

Findings

Derived recoupling coefficients for arbitrary angular momentum and isospin.

Proposed an intensity observable for three-body dynamics visualization.

Numerical examples show the symmetrization aligns with Dalitz plot symmetries.

Abstract

S matrix principles and symmetries impose constraints on three-particle scattering amplitudes, which can be formulated as a class of integral equations for their partial wave projections. However, these amplitudes are typically expressed in an asymmetric basis, where one of the initial and final state particles is singled out, and all quantum numbers are defined relative to this spectator. In this work, we show how to construct symmetric partial wave amplitudes, which have been symmetrized over all possible spectator combinations, using their asymmetric counterparts and sets of recoupling coefficients. We derive these recoupling coefficients for arbitrary angular momentum and isospin for arbitrary systems of spinless particles with SU(2) flavor symmetry. We propose a simple intensity observable suitable for visualizing the structure of three-body dynamics in Dalitz distributions. Finally, we provide some numerical examples of Dalitz distributions relevant for future lattice QCD calculations of $3\pi$ systems and provide numerical evidence that the symmetrization procedure is consistent with expected symmetries of Dalitz plots.