A geometric phase approach to quark confinement from stochastic gauge-geometry flows

Abstract

We apply a stochastic version of the geometric (Ricci) flow, complemented with the stochastic flow of the gauge Yang--Mills sector, in order to seed the chromo-magnetic and chromo-electric vortices that source the area-law for QCD confinement. The area-law is the key signature of quark confinement in Yang--Mills gauge theories with a non-trivial center symmetry. In particular, chromo-magnetic vortices enclosed within the chromo-electric Wilson loops instantiate the area-law asymptotic behaviour of the Wilson loop vacuum expectation values. The stochastic gauge-geometry flow is responsible for the topology changes that induce the appearance of the vortices. When vortices vanish, due to topology changes in the manifolds associated to the hadronic ground states, the evaluation of the Wilson loop yields a dependence on the length of the path, hence reproducing the perimeter law of the hadronic (Higgs) phase of real QCD. Confinement, instead, is naturally achieved within this context as a by-product of the topology change of the manifold over which the dynamics of the Yang--Mills fields is defined. It is then provided by the Aharonov--Bohm effect induced by the concatenation of the compact chromo-electric and chromo-magnetic fluxes originated by the topology changes. The stochastic gauge-geometry flow naturally accomplishes a treatment of the emergence of the vortices and the generation of turbulence effects. Braiding and knotting, resulting from topology changes, namely stochastic fluctuations, stabilize the chromo-magnetic vortices. Finally, we observe that dimensional transmutation for the Yang-Mills fields can be derived from the scaling property of the geometric part of the stochastic flow. Specifically, a relation that involves the infrared equilibrium limit of the Planck constant can be derived that yields the correct order of magnitude for $\Lambda_{\rm QCD}$.