Complete Monopole Dominance of the Yang-Mills Confining Potential
We continue our investigation of quark confinement using a particular variant of the Cho-Duan-Ge gauge independent Abelian decomposition. The decomposition splits the gauge field into a restricted Abelian part and a coloured part in a way that preserves gauge covariance. The restricted part of the gauge field can be divided into a Maxwell term and a topological term. Previously, we showed that by a particular choice of this decomposition we could fully describe the confining potential using only the restricted gauge field. We proposed that various topological objects (a form of magnetic monopole) could arise in the restricted field which would drive confinement. Our mechanism does not explicitly refer to a dual Meissner effect, nor does it use centre vortices. We did not need to gauge fix or introduce any new dynamical fields. We show that if we do gauge fix as well as performing the Abelian decomposition then it is possible to ensure that the topological part of the restricted field fully accounts for the confining potential. Our relationship is exact: there is no approximation or model involved. This isolates the objects responsible for confinement from non-confining contributions to the gauge field, allowing a direct search for our proposed topological objects. Using numerical studies in SU(2), we confirm that our proposed monopoles are present in the field, and the winding number associated with these monopoles is a key factor driving quark confinement. In SU(2), our monopoles are described by two parameters. We show that it is possible to re-parametrise the Yang Mills action and the functional integration measure in terms of these variables (plus the necessary additional parameters). We can thus treat the monopoles as dynamical variables in the functional integral. This might be the first step in a future analytical computation to complement our numerical results.