Non-perturbative Faddeev-Popov formula and infrared limit of QCD

Abstract

We show that an exact non-perturbative quantization of continuum gauge theory is provided by the Faddeev-Popov formula in Landau gauge, $\d(\p \cdot A) \det[-\p \cdot D(A)] \exp[-S_{\rm YM}(A)]$, restricted to the region where the Faddeev-Popov operator is positive $-\p \cdot D(A) > 0$ (Gribov region). Although there are Gribov copies inside this region, they have no influence on expectation-values. The starting point of the derivation is stochastic quantization which determines the Euclidean probability distribution $P(A)$ by a method that is free of the Gribov critique. In the Landau-gauge limit the support of $P(A)$ shrinks down to the Gribov region with Faddeev-Popov weight. The cut-off of the resulting functional integral on the boundary of the Gribov region does not change the {\it form} of the Dyson-Schwinger equations, because $\det[-\p \cdot D(A)]$ vanishes on the boundary, so there is no boundary contribution. However this cut-off does provide {\it supplementary conditions} that govern the choice of solution of the DS equations. In particular the "horizon condition", though consistent with the perturbative renormalization group, puts QCD into a non-perturbative phase. The infrared asymptotic limit of the DS equations of QCD is obtained by neglecting the Yang-Mills action $S_{\rm YM}$. We sketch the extension to a BRST-invariant formulation. In the infrared asymptotic limit, the BRST-invariant action becomes BRST-exact, and defines a topological quantum field theory with an infinite mass gap. Confinement of quarks is discussed briefly.