Introducing vortices in the continuum using direct and indirect methods

TL;DR

This paper demonstrates how vortices and chains emerge in the continuum QCD vacuum through direct and indirect gauge transformation methods, connecting lattice findings with continuum theory.

Contribution

It introduces a continuum formalism for vortices using gauge transformations, linking lattice methods to continuum QCD and revealing vortex and chain structures.

Findings

Vortices appear in the continuum QCD vacuum under certain gauge transformations.

Chains containing monopoles and vortices emerge via the indirect method.

The formalism connects lattice vortex observations with continuum gauge theories.

Abstract

Inspired by direct and indirect maximal center gauge methods which confirm the existence of vortices in lattice calculations and by using the connection formalism, we show that under some appropriate gauge transformations vortices and chains appear in the QCD vacuum of the continuum limit. In the direct method, by applying center gauge transformation and \textquotedblleft center projection,\textquotedblright QCD is reduced to a gauge theory including vortices, which corresponds to the non-trivial first homotopy group $\Pi_1\left( \text{SO}(3)\right) =Z_2.$ On the other hand, using the indirect method, in addition to the center gauge transformation and \textquotedblleft center projection,\textquotedblright an initial step called Abelian gauge transformation and then Abelian projection are applied. Therefore, instead of single vortices, chains that contain monopoles and vortices appear in the theory.