A Finite Tension for the $\phi^4_4$ Domain Wall

TL;DR

This paper investigates the quantum states of domain wall solitons in a 3+1 dimensional $^4$ model, demonstrating that a simple deformation of coherent states can cancel divergences in energy density.

Contribution

It introduces a novel approach to construct finite-energy quantum states for domain walls in higher dimensions using a deformation of coherent states.

Findings

Deformation of coherent states cancels one-loop divergences.

Leading deformation is a squeeze operation.

Method applies to 3+1 dimensional $^4$ domain walls.

Abstract

In 1+1 dimensions, it is well known that the quantum states corresponding to solitons are well described by coherent states. In his 1975 Erice lectures, Coleman observed that this construction does not extend to higher dimensions, as the coherent states have infinite energy density. He challenged the students to construct the quantum states corresponding to solitons in higher dimensions, a problem which remains unsolved today. However, even in 1+1 dimensions, the correct quantum states are actually given by deformations of coherent states. In the 3+1 dimensional $\phi^4$ double-well model, we show that the leading deformation, which is just a squeeze, already cancels the one-loop divergence in the energy density of the domain wall soliton.