Relativistic three-particle quantization condition for nondegenerate scalars

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, ${\tilde{\mathcal{K}}}_{\mathrm{df},3}$. 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.

arXiv comment34 pages, 7 figures; v2: added section 2 providing general recipe for developing formalism; fixed typos; matches with published version

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