Fractional soft limits

TL;DR

This paper reveals that phonon scattering amplitudes in solids can vanish with fractional powers of soft momentum, challenging the traditional integer power expectation due to Lorentz symmetry breaking and collinear kinematics.

Contribution

It demonstrates that soft phonon scattering amplitudes can exhibit fractional power scaling, extending the understanding of soft limits in non-relativistic shift-invariant theories.

Findings

Scattering amplitudes may vanish as fractional powers of soft momentum.

The fractional scaling depends on phonon helicities and kinematic configurations.

Results apply broadly to non-relativistic shift-invariant vector field theories.

Abstract

It is a common lore that the amplitude for a scattering process involving one soft Nambu--Goldstone boson should scale like an integer power of the soft momentum. We revisit this expectation by considering the $2 \to 2$ scattering of phonons in solids. We show that, depending on the helicities of the phonons involved in the scattering process, the scattering amplitude may in fact vanish like a fractional power of the soft momentum. This is a peculiarity of the 4-point amplitude, which can be traced back to (1) the (spontaneous or explicit) breaking of Lorentz invariance, and (2) the approximately collinear kinematics arising when one of the phonons becomes soft. Our results extend to the general class of non-relativistic shift-invariant theories of a vector field.