Nonperturbative Landau gauge and infrared critical exponents inQCD

We discuss Faddeev-Popov quantization at the nonperturbative level and show that Gribov’s prescription of cutting off the functional integral at the Gribov horizon does not change the Schwinger-Dyson equations, but rather resolves an ambiguity in the solution of these equations. We note that Gribov’s prescription is not exact, and we therefore turn to the method of stochastic quantization in its time-independent formulation, and recall the proof that it is correct at the nonperturbative level. The nonperturbative Landau gauge is derived as a limiting case, and it is found that it yields the Faddeev-Popov method in the Landau gauge with a cutoff at the Gribov horizon, plus a novel term that corrects for overcounting of Gribov copies inside the Gribov horizon. Nonperturbative but truncated coupled Schwinger-Dyson equations for the gluon and ghost propagators $D\left(k\right)\mathrm{}$ and $G\left(k\right)\mathrm{}$ in the Landau gauge are solved asymptotically in the infrared region. The infrared critical exponents or anomalous dimensions, defined by $D\left(k\right)\mathrm{}\sim {1/(k}^2{)}^{{1+a}_D}$ and $G\left(k\right)\mathrm{}\sim {1/(k}^2{)}^{{1+a}_G},$ are obtained in space-time dimensions $d=2,$ 3, 4. Two possible solutions are obtained with the values, in $d=4\mathrm{}$ dimensions, $a_G=1,$ $a_D=-2,$ or $a_G=\left(93\mathrm{}-\sqrt{1201}\right)/98\mathrm{}\approx 0.595353,$ $a_D=-{2a}_G.$