Classical Yang Mills equations with sources: consequences of specific scalar potentials

TL;DR

This paper investigates classical Yang-Mills equations with specific scalar potentials, revealing that only Abelian solutions are spherically symmetric, and explores non-spherical configurations that could influence confinement mechanisms.

Contribution

It analyzes well-known scalar potentials within SU(2) Yang-Mills theory, deriving solutions and highlighting the potential role of non-spherical configurations in confinement.

Findings

Spherically symmetric solutions only exist in the Abelian limit.

Non-spherical solutions exhibit strong deviations from symmetry.

Non-spherical configurations may contribute to confinement mechanisms.

Abstract

Some well known gauge scalar potential very often considered or used in the literature are investigated by means of the classical Yang Mills equations for the $SU(2)$ subgroups of $N_c=3$. By fixing a particular shape for the scalar potential, the resulting vector potentials and the corresponding color-charges sources are found. By adopting the spherical coordinate system, it is shown that spherically symmetric solutions, only dependent on the radial coordinate, are only possible for the Abelian limit, otherwise, there must have angle-dependent component(s). The following solutions for the scalar potential are investigated: the Coulomb potential and a non-spherically symmetric generalization, a linear potential $A_0 (\vec{r}) \sim (\kappa r)$, a Yukawa-type potential $A_0 (\vec{r}) \sim (C e^{-r/r_0}/r)$ and finite spatial regions in which the scalar potential assumes constant values. The corresponding chromo-electric and chromo-magnetic fields, as well as the color-charge densities, are found to have strong deviations from the spherical symmetric configurations. We speculate these types of non-spherically symmetric configurations may contribute (or favor) for the (anisotropic) confinement mechanism since they should favor color charge-anti-charge (or three-color-charge) bound states that are intrinsically non spherically symmetric with (asymmetric) confinement of fluxes. Specific conditions and relations between the parameters of the solutions are also presented.