Integration Theory for Kinks and Sphalerons in One Dimension

TL;DR

This paper develops an integration framework for kink, sphaleron, and kink chain solutions in one-dimensional scalar field theories, clarifying BPS and semi-BPS cases through first-order ODEs and complex analysis.

Contribution

It introduces a unified integration approach for static solutions in scalar field models, distinguishing BPS and semi-BPS cases via contour integrals in the complex plane.

Findings

First-order ODEs characterize kink, sphaleron, and chain solutions.

BPS solutions correspond to contour integrals in the $\

Semi-BPS solutions involve more complex Riemann surface integrals.

Abstract

The static kink, sphaleron and kink chain solutions for a single scalar field $\phi$ in one spatial dimension are reconsidered. By integration of the Euler--Lagrange equation, or through the Bogomolny argument, one finds that each of these solutions obeys a first-order field equation, an autonomous ODE that can always be formally integrated. We distinguish the BPS case, where the required integral is along a contour in the $\phi$-plane, from the semi-BPS case, where the integral is along a contour in the Riemann surface double-covering the $\phi$-plane, and is generally more complicated.