Relativistic three-particle quantization condition for nondegenerate scalars

TL;DR

This paper extends the relativistic three-particle quantization formalism to nondegenerate scalar particles, providing new quantization conditions and integral equations to connect finite-volume spectra with scattering amplitudes.

Contribution

The authors generalize the relativistic three-particle formalism to nondegenerate scalars, deriving multiple quantization conditions and integral equations for practical application.

Findings

Derived three versions of the quantization condition for nondegenerate particles.

Established integral equations relating the K matrix to the three-particle scattering amplitude.

Presented a practical Lorentz-invariant K matrix formulation.

Abstract

The formalism relating the relativistic three-particle infinite-volume scattering amplitude to the finite-volume spectrum has been developed thus far only for identical or degenerate particles. We provide the generalization to the case of three nondegenerate scalar particles with arbitrary masses. A key quantity in this formalism is the quantization condition, which relates the spectrum to an intermediate K matrix. We derive three versions of this quantization condition, each a natural generalization of the corresponding results for identical particles. In each case we also determine the integral equations relating the intermediate K matrix to the three-particle scattering amplitude, $\mathcal M_3$. The version that is likely to be most practical involves a single Lorentz-invariant intermediate K matrix, $\widetilde{\mathcal K}_{\rm df,3}$. The other versions involve a matrix of K matrices, with elements distinguished by the choice of which initial and final particles are the spectators. Our approach should allow a straightforward generalization of the relativistic approach to all other three-particle systems of interest.