Relativistic, model-independent, three-particle quantization condition

Abstract

We present a generalization of Lüscher's relation between the finite-volume spectrum and scattering amplitudes to the case of three particles. We consider a relativistic scalar field theory in which the couplings are arbitrary aside from a Z2 symmetry that removes vertices with an odd number of particles. The theory is assumed to have two-particle phase shifts that are bounded by π/2 in the regime of elastic scattering. We determine the spectrum of the finite-volume theory from the poles in the odd-particle-number finite-volume correlator, which we analyze to all orders in perturbation theory. We show that it depends on the infinite-volume two-to-two K-matrix as well as a nonstandard infinite-volume three-to-three K-matrix. A key feature of our result is the need to subtract physical singularities in the three-to-three amplitude and thus deal with a divergence-free quantity. This allows our initial, formal result to be truncated to a finite dimensional determinant equation. At present, the relation of the three-to-three K-matrix to the corresponding scattering amplitude is not known, although previous results in the non-relativistic limit suggest that such a relation exists.

S-waveKK*interactions in a finite volume and thefGeng1, Ren2, Zhou3 et al. 2015Phys. Rev. D292330View full textAdd to dashboardCite

“…The formalism to deal with three body-systems in finite volume is also available in Refs. [48][49][50]. For two-body systems with one unstable particle, one can use a formalism in which the self-energy of the unstable particle is discretized in the moving frame [11].…”

Axial resonances a 1(1260), b 1(1235) and their decays from the lattice

“…Proper treatment of this problem appears considerably more challenging. An analytic framework for the scattering of unstable particles in finite volume has been recently proposed in [30] while the related problem of three particles where two of them can resonate was considered in [31][32][33][34]. These approaches can serve as a guideline for future lattice simulations.…”

“…Let us point out that analytical frameworks for rigorous multi-channel treatment on the lattice have been proposed, but their realization in practice remains a serious challenge. Analytic approaches for the scattering of unstable particles (with some emphasis on swave scattering) in the finite volume were recently formulated in [30], while the related challenging problem of three particles in a finite box was analytically considered in [31][32][33][34]. In ref.…”

Resonance Extraction from the Finite Volume

“…Theoretical developments addressing finite-volume effects for three-body systems are pioneered in Refs. [48][49][50][51][52][53][54], although much conceptual work, including three-body unitarity constraints, remains to be done.…”