Three-particle scattering amplitudes from a finite volume formalism

TL;DR

This paper derives a finite volume quantization condition for three-boson systems, connecting finite volume spectra with infinite volume scattering amplitudes, including cases with bound states and resonances.

Contribution

It introduces a new integral equation-based quantization condition for three-particle systems that accounts for bound states, resonances, and volume corrections, extending previous Luscher formulas.

Findings

Reduces to Luscher formula for bound states below breakup energy.

Includes nonperturbative power-law volume corrections for scattering states.

Provides a numerical framework for extracting infinite volume amplitudes from finite volume spectra.

Abstract

We present a quantization condition for the spectrum of a system composed of three identical bosons in a finite volume with periodic boundary conditions. This condition gives a relation between the finite volume spectrum and infinite volume scattering amplitudes. The quantization condition presented is an integral equation that in general must be solved numerically. However, for systems with an attractive two-body force that supports a two-body bound-state, a diboson, and for energies below the diboson breakup, the quantization condition reduces to the well-known Luscher formula with exponential corrections in volume that scale with the diboson binding momentum. To accurately determine infinite volume phase shifts, it is necessary to extrapolate the phase shifts obtained from the Luscher formula for the boson-diboson system to the infinite volume limit. For energies above the breakup threshold, or for systems with no two-body bound-state (with only scattering states and resonances) the Luscher formula gets power-law volume corrections and consequently fails to describe the three-particle system. These corrections are nonperturbatively included in the quantization condition presented.