Time-independent stochastic quantization, Dyson-Schwinger equations, and infrared critical exponents in QCD

We derive the equations of time-independent stochastic quantization, without reference to an unphysical fifth time, from the principle of gauge equivalence. It asserts that probability distributions P that give the same expectation values for gauge-invariant observables $〈W〉=\int \mathrm{dAWP}$ are physically indistinguishable. This method escapes the Gribov critique. We derive an exact system of equations that closely resembles the Dyson-Schwinger equations of Faddeev-Popov theory. The system is truncated and solved nonperturbatively, by means of a power law ansatz, for the critical exponents that characterize the asymptotic form at $k\approx 0$ of the gluon propagator in Landau gauge. For the transverse and longitudinal parts, we find, respectively, $D^T\sim {(k}^2{)}^{-1-{\alpha }_T}\approx {(k}^2{)}^{0.043},$ suppressed and in fact vanishing, though weakly, and $D^L\sim {a(k}^2{)}^{-1-{\alpha }_L}\approx {a(k}^2{)}^{-1.521},$ enhanced, with ${\alpha }_T=-2{\alpha }_L.$ Although the longitudinal part vanishes with the gauge parameter a in the Landau-gauge limit $\overrightarrow{a}0$ there are vertices of order $a^{-1}$ so, counterintuitively, the longitudinal part of the gluon propagator does contribute in internal lines in the Landau gauge, replacing the ghost that occurs in Faddeev-Popov theory. We compare our results with the corresponding results in Faddeev-Popov theory.