The $ϕ^4$ Kink Mass at Two Loops

Abstract

The two-loop correction to the mass of the φ 4 kink is 0.0126λ/m in terms of the coupling λ 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.

The Two-Loop $ϕ^4$ Kink Mass

“…Using (2.14) and (3.10) we obtain [23] the following contributions to the kink mass, including error bars only on dominant sources of error, This result is positive, which is consistent with the studies cited in the Introduction except for Ref. [9].…”

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

“…This occurs at smaller coupling than other expected features of the model, such as the restoration of the φ → −φ symmetry at λ/m 2 ∼ 1.2(1) found in Ref. [13], or the point at which the meson mass exceeds twice [14,15] or π times [16] the semiclassical [17,18] kink mass and one expects mesons to drop out of the spectrum.…”

Cut-off kinks

“…In refs. [21,22], the calculation of γ1 was reviewed and it played an essential role in the elimination of IR divergences in the calculation of the two-loop kink mass. The shifted counterterm and all successive terms are finite at Λ → ∞, with the first few given in eq.…”