Alternative derivation of the relativistic three-particle quantization condition

Abstract

We present a simplified derivation of the relativistic three-particle quantization condition for identical, spinless particles described by a generic relativistic field theory satisfying a ${\mathbb{Z}}_2$ symmetry. The simplification is afforded by using a three-particle quasilocal K matrix that is not fully symmetrized, ${\tilde{\mathcal{K}}}_{\mathrm{df},3}^{(u,u)}$, and makes extensive use of time-ordered perturbation theory (TOPT). We obtain a new form of the quantization condition. This new form can then be related algebraically to the standard quantization condition, which depends on a fully symmetric three-particle K matrix, ${\mathcal{K}}_{\mathrm{df},3}$. The new derivation is fully explicit, allowing, for example, a closed-form expression for ${\mathcal{K}}_{\mathrm{df},3}$ to be given in terms of TOPT amplitudes. The new form of the quantization condition is similar in structure to that obtained in the "finite-volume unitarity" approach, and in a companion paper we make this connection concrete. Our simplified approach should also allow a more straightforward generalization of the quantization condition to nondegenerate particles, and perhaps also to more than three particles.

39 pages, 7 appendices, 11 figures; v2--added clarifications, corrected typos; v3--corrected additional typos, results unchanged; matches published version

References in corpus (13)

• A relativistic, model-independent, three-particle quantization condition
• Three particle quantization condition in a finite volume: 2. general formalism and the analysis of data
• Three-body Unitarity in the Finite Volume
• Three-particle quantization condition in a finite volume: 1. The role of the three-particle force
• Multi-Pion States in Lattice QCD and the Charged-Pion Condensate
• n-Boson Energies at Finite Volume and Three-Boson Interactions
• Implementing the three-particle quantization condition including higher partial waves
• Three-boson problem at low energy and Implications for dilute Bose-Einstein condensates
• Three-body Unitarity with Isobars Revisited
• The Energy of n Identical Bosons in a Finite Volume at O(L^{-7})
• Applying the relativistic quantization condition to a three-particle bound state in a periodic box
• Lattice QCD study of mixed systems of pions and kaons
• Finite-volume matrix elements in multi-boson states

Cited by in corpus (4)

• Relativistic three-particle quantization condition for nondegenerate scalars
• Finite-volume energy shift of the three-pion ground state
• Decay amplitudes to three hadrons from finite-volume matrix elements
• On the three-particle analog of the Lellouch-Lüscher formula