The $\phi^4$ Kink Mass at Two Loops

TL;DR

This paper calculates the two-loop correction to the mass of the $^4$ kink using an alternative method, addressing divergences and performing analytical and numerical integrations.

Contribution

It introduces a new approach to compute two-loop kink mass corrections, handling IR divergences via energy density subtraction and combining analytical and numerical methods.

Findings

Two-loop correction is 0.0126λ/m.

IR divergences are canceled by subtracting energy densities.

All spatial integrals are performed analytically, with momentum integrals numerically evaluated.

Abstract

The two-loop correction to the mass of the $\phi^4$ kink is $0.0126\lambda/m$ in terms of the coupling $\lambda$ and the meson mass $m$ evaluated at the minimum of the potential. This is calculated using a recently proposed alternative to collective coordinates. Both the kink energy and the vacuum energy are IR divergent at this order. To cancel the divergence, the two energy densities are subtracted before integrating over space, or equivalently a finite counterterm is added to the Hamiltonian density to cancel the vacuum energy density. All spatial integrals are performed analytically. However in the last step of our calculation, integrals over virtual momenta are performed numerically.