Evidence for the Unbinding of the $\phi^4$ Kink's Shape Mode

TL;DR

This paper investigates the quantum corrections to the shape mode of the $^4$ kink, revealing that it unbinds into the continuum at a relatively low coupling, affecting the kink's phenomenology.

Contribution

It provides the first calculation of the quantum correction to the shape mode energy and shows its unbinding from the continuum at finite coupling.

Findings

Quantum correction to shape mode energy is approximately -0.116λ/m.

Correction to the continuum threshold is approximately -0.434λ/m.

Shape mode unbinds into the continuum at coupling around 0.106 m^2.

Abstract

The $\phi^4$ double-well theory admits a kink solution, whose rich phenomenology is strongly affected by the existence of a single bound excitation called the shape mode. We find that the leading quantum correction to the energy needed to excite the shape mode is $-0.115567\lambda/m$ in terms of the coupling $\lambda/4$ and the meson mass $m$ evaluated at the minimum of the potential. On the other hand, the correction to the continuum threshold is $-0.433\lambda/m$. A naive extrapolation to finite coupling then suggests that the shape mode melts into the continuum at the modest coupling of $\lambda/4\sim 0.106 m^2$, where the $\mathbb{Z}_2$ symmetry is still broken.