# Dyon in the $SU(2)$ Yang--Mills theory with a gauge-invariant gluon mass toward quark confinement

## TL;DR

This paper constructs and analyzes a novel gauge-invariant dyon solution in massive SU(2) Yang--Mills theory, exploring its properties and implications for quark confinement and phase transitions.

## Contribution

It extends previous monopole work to include dyons with electric charge, providing a new gauge-invariant solution relevant for understanding confinement.

## Key findings

- Identified a gauge-invariant dyon solution with minimal magnetic charge.
- Compared the new dyon with Julia--Zee and Wu--Yang dyons.
- Proposed the dyon as a tool to study confinement/deconfinement transitions.

## Abstract

In the previous paper, we have shown the existence of magnetic monopoles in the pure $SU(2)$ Yang--Mills theory with a gauge-invariant mass term for the gluon field being introduced. In this paper, we extend our previous construction of magnetic monopoles to obtain dyons with both magnetic and electric charges. In fact, we solve under the static and spherically symmetric ansatz the field equations of the $SU(2)$ "complementary" gauge-scalar model, which is the $SU(2)$ Yang--Mills theory coupled to a single adjoint scalar field whose radial degree of freedom is eliminated. We show that the novel dyon solution can be identified with the gauge field configuration of a dyon with a minimum magnetic charge in the massive Yang--Mills theory. Moreover, we compare the dyon of the massive Yang--Mills theory obtained in this way with the Julia--Zee dyon in the Georgi--Glashow gauge-Higgs scalar model and the dyonic extension of the Wu--Yang magnetic monopole in the pure Yang--Mills theory. Finally, we identify the novel dyon solution found in this paper with a dyon configuration on $S^1 \times \mathbb{R}^3$ space with nontrivial holonomy and propose to use it to understand the confinement/deconfinement phase transition in the Yang--Mills theory at finite temperature, instead of using the dyons constituting the Kraan--van Baal--Lee--Lu caloron.