A direct test of the integral Yang-Mills equations through SU(2) monopoles

TL;DR

This paper tests the integral Yang-Mills equations using SU(2) monopole configurations, demonstrating their validity up to second order and highlighting the potential significance of these equations for understanding non-abelian gauge theories.

Contribution

It provides a direct verification of the integral Yang-Mills equations using monopole solutions, revealing their consistency and suggesting further exploration of their physical implications.

Findings

Monopole configurations satisfy the integral equations up to second order.

The integral equations involve parameters not present in differential equations.

Results indicate the potential importance of these equations for non-abelian gauge theories.

Abstract

We use the SU(2) 't Hooft-Polyakov monopole configuration, and its BPS version, to test the integral equations of the Yang-Mills theory. Those integral equations involve two (complex) parameters which do not appear in the differential Yang-Mills equations, and if they are considered to be arbitrary it then implies that non-abelian gauge theories (but not abelian ones) possess an infinity of integral equations. For static monopole configurations only one of those parameters is relevant. We expand the integral Yang-Mills equation in a power series of that parameter and show that the 't Hooft-Polyakov monopole and its BPS version satisfy the integral equations obtained in first and second order of that expansion. Our results points to the importance of exploring the physical consequences of such an infinity of integral equations on the global properties of the Yang-Mills theory.