Time-independant stochastic quantization, DS equations, and infrared critical exponents in QCD

Abstract

We derive the equations of time-independent stochastic quantization, without reference to an unphysical 5th time, from the principle of gauge equivalence. It asserts that probability distributions $P$ that give the same expectation values for gauge-invariant observables $<W>=\int dAWP$ are physically indistiguishable. This method escapes the Gribov critique. We derive an exact system of equations that closely resembles the Dyson-Schwinger equations of Faddeev-Popov theory, which we then solve non-perturbatively for the critical exponents that characterize the asymptotic form at $k\approx 0$ of the tranverse and longitudinal parts of the gluon propagator in Landau gauge, $D^T\sim (k^2{)}^{-1-{\text{a}}_T}$ and $D^L\sim a(k^2{)}^{-1-{\text{a}}_L}$, and obtain ${\text{a}}_T=-2{\text{a}}_L\approx -1.043$ (short range), and ${\text{a}}_L\approx 0.521$, (long range). Although the longitudinal part vanishes with the gauge parameter $a$ in the Landau gauge limit, $a\to 0$, there are vertices of order $a^{-1}$, so the longitudinal part of the gluon propagator contributes in internal lines, replacing the ghost that occurs in Faddeev-Popov theory. We compare our results with the corresponding results in Faddeev-Popov theory.

Comments: 50 pages, 2 figures