Conformal Invariance for Non-Relativistic Field Theory

TL;DR

This paper derives Ward identities for non-relativistic conformal field theories, revealing how conformal invariance constrains Green's functions and scattering amplitudes, especially in the infinite scattering length limit.

Contribution

It provides explicit Ward identities for non-relativistic conformal theories and analyzes their implications for scattering amplitudes and Green's functions.

Findings

Scale invariant 2-to-2 scattering amplitudes are conformally invariant.

Conformal invariance constrains off-shell Green's functions and three-particle amplitudes.

Explicit example: non-relativistic particles at infinite scattering length exhibit conformal invariance.

Abstract

Momentum space Ward identities are derived for the amputated n-point Green's functions in 3+1 dimensional non-relativistic conformal field theory. For n=4 and 6 the implications for scattering amplitudes (i.e. on-shell amputated Green's functions) are considered. Any scale invariant 2-to-2 scattering amplitude is also conformally invariant. However, conformal invariance imposes constraints on off-shell Green's functions and the three particle scattering amplitude which are not automatically satisfied if they are scale invariant. As an explicit example of a conformally invariant theory we consider non-relativistic particles in the infinite scattering length limit.