Evidence for the unbinding of the \U0001d7194 kink\u2019s shape mode

Jarah Evslin

doi:10.1007/jhep09(2021)009

Abstract

The \U0001d7194 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 −0.115567λ/M in terms of the coupling λ/4 and the meson mass M evaluated at the minimum of the potential. On the other hand, the correction to the continuum threshold is −0.433λ/M. A naive extrapolation to finite coupling then suggests that the shape mode melts into the continuum at the modest coupling of λ/4 ∼ 0.106M2, where the ℤ2 symmetry is still broken.

Highlights

We find that the leading quantum correction to the energy needed to excite the shape mode is −0.115567λ/M in terms of the coupling λ/4 and the meson mass M evaluated at the minimum of the potential
The diagrams are similar to Feynman diagrams, which describe perturbation theory in the vacuum sector, except that momentum space p is replaced by a transform with respect to normal modes labeled by k, which runs over continuous and discrete values
Transforming into the normal mode basis, these lead to k-dependent kink-sector tadpoles which are not removed by ordinary counterterms, which are already fixed to remove tadpoles in the vacuum sector

Summary

Review of the φ4 kink

Where ::a is normal ordering with respect to the operators that create plane wave excitations of the φ field about a classical vacuum. It admits a classical kink solution φ(x, t) = f (x) = √M. Orthonormal perturbations about the kink solution with frequency ωk will all be called normal modes. They include a zero-mode proportional to f as well as continuum modes gk(x) and a shape mode gS(x) e−ikx gk(x).

Energy required to excite a shape mode

Continuum threshold

A Derivation of the excitation energy