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

Abstract

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 $〈W〉=\ensuremath{\int}\mathrm{dAWP}$ 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 $k\ensuremath{\approx}0$ of the gluon propagator in Landau gauge. For the transverse and longitudinal parts, we find, respectively, ${D}^{T}\ensuremath{\sim}{(k}^{2}{)}^{\ensuremath{-}1\ensuremath{-}{\ensuremath{\alpha}}_{T}}\ensuremath{\approx}{(k}^{2}{)}^{0.043},$ suppressed and in fact vanishing, though weakly, and ${D}^{L}\ensuremath{\sim}{a(k}^{2}{)}^{\ensuremath{-}1\ensuremath{-}{\ensuremath{\alpha}}_{L}}\ensuremath{\approx}{a(k}^{2}{)}^{\ensuremath{-}1.521},$ enhanced, with ${\ensuremath{\alpha}}_{T}=\ensuremath{-}2{\ensuremath{\alpha}}_{L}.$ Although the longitudinal part vanishes with the gauge parameter a in the Landau-gauge limit $\stackrel{\ensuremath{\rightarrow}}{a}0$ there are vertices of order ${a}^{\ensuremath{-}1}$ 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.