Nonperturbative Faddeev-Popov formula and the infrared limit of QCD

We show that an exact nonperturbative quantization of continuum gauge theory is provided by the Faddeev-Popov formula in the Landau gauge, $\delta \left(\partial \cdot A\right)\det \left[-\partial \cdot D\right(A\left)\mathrm{}\right]\exp \left[-S_{\mathrm{YM}}\mathrm{}\right(A\left)\mathrm{}\right],$ restricted to the region where the Faddeev-Popov operator is positive $-\partial \cdot D\left(A\right)>0\mathrm{}$ (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 $P\left(A\right)\mathrm{}$ by a method that is free of the Gribov critique. In the Landau-gauge limit the support of $P\left(A\right)\mathrm{}$ shrinks down to the Gribov region with Faddeev-Popov weight. The cutoff of the resulting functional integral on the boundary of the Gribov region does not change the form of the Dyson-Schwinger (DS) equations because $\det \left[-\partial \cdot D\right(A\left)\mathrm{}\right]$ vanishes on the boundary, so there is no boundary contribution. However this cutoff does provide 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 nonperturbative phase. The infrared asymptotic limit of the DS equations of QCD is obtained by neglecting the Yang-Mills action $S_{\mathrm{YM}}.$ 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.