Sphalerons and Other Saddles from Cooling

TL;DR

This paper introduces a new cooling algorithm for SU(2) lattice gauge theory capable of identifying all critical points, including sphalerons, and provides detailed analysis of their properties and energies under different boundary conditions.

Contribution

It presents a novel cooling method that finds all critical points of the energy functional, including sphalerons with multiple unstable modes, and analyzes their characteristics in lattice gauge theory.

Findings

Identified sphaleron energies for twisted and periodic boundary conditions.

Demonstrated the convergence properties of the new cooling algorithm.

Showed that the magnetic field in the periodic sphaleron has equal trace for all components.

Abstract

We describe a new cooling algorithm for SU(2) lattice gauge theory. It has any critical point of the energy or action functional as a fixed point. In particular, any number of unstable modes may occur. We also provide insight in the convergence of the cooling algorithms. A number of solutions will be discussed, in particular the sphalerons for twisted and periodic boundary conditions which are important for the low-energy dynamics of gauge theories. For a unit cubic volume we find a sphaleron energy of resp. $\cE_s=34.148(2)$ and $\cE_s=72.605(2)$ for the twisted and periodic case. Remarkably, the magnetic field for the periodic sphaleron satisfies at all points $\Tr B_x^2=\Tr B_y^2=\Tr B_z^2$.