Asymptotic behavior of Nambu-Bethe-Salpeter wave functions for multi-particles in quantum field theories

TL;DR

This paper derives the asymptotic behavior of multi-particle Nambu-Bethe-Salpeter wave functions in quantum field theories, linking them to the on-shell T-matrix and supporting the HAL QCD scheme for defining multi-nucleon potentials.

Contribution

It provides a theoretical foundation for understanding multi-particle NBS wave functions and their relation to scattering parameters, extending the HAL QCD approach to systems with more than two particles.

Findings

NBS wave functions encode T-matrix information such as phase shifts.

Asymptotic behavior expressed in terms of low-energy T-matrix parameters.

Supports the use of NBS wave functions to define multi-nucleon potentials in QCD.

Abstract

We derive asymptotic behaviors of the Nambu-Bethe-Salpeter (NBS) wave function at large space separations for systems with more than 2 particles in quantum field theories. To deal with $n$-particles in the center of mass flame coherently, we introduce the Jacob coordinates of $n$ particles and then combine their $3(n-1)$ coordinates into the one spherical coordinate in $D=3(n-1)$ dimensions. We parametrize on-shell $T$-matrix for $n$-particle system of scalar fields at low energy, using the unitarity constraint of the $S$-matrix. We then express asymptotic behaviors of the NBS wave function for $n$ particles at low energy, in terms of parameters of $T$-matrix, and show that the NBS wave function carry the information of $T$-matrix such as phase shifts and mixing angles of the $n$-particle system in its own asymptotic behavior, so that the NBS wave function can be considered as the scattering wave of $n$-particles in quantum mechanics. This property is one of the essential ingredients of the HAL QCD scheme to define "potential" from the NBS wave function in quantum field theories such as QCD. Our results, together with an extension to systems with spin 1/2 particles, justify the HAL QCD's definition of potentials for 3 or more nucleons(baryons) in terms the NBS wave functions.