On Consistent Lagrangian Quantization of Yang--Mills Theories without Gribov Copies

TL;DR

This paper reviews a Lagrangian quantization approach for Yang--Mills theories that avoids Gribov copies by using finite BRST and BRST-antiBRST transformations, ensuring gauge independence and addressing the Gribov problem.

Contribution

It introduces a method using finite BRST and BRST-antiBRST transformations to maintain gauge invariance and solve the Gribov problem in Yang--Mills theories.

Findings

Finite BRST and BRST-antiBRST transformations form Abelian supergroups.

Explicit superdeterminants enable precise gauge-fixing changes.

The approach restores gauge independence even with symmetry-breaking terms.

Abstract

We review the results of our research [A.A. Reshetnyak, IJMPA 29 (2014) 1450184; P.Yu. Moshin, A.A. Reshetnyak, Nucl. Phys. B 888 (2014) 92; P.Yu. Moshin, A.A. Reshetnyak, Phys. Lett. B 739 (2014) 110; P.Yu. Moshin, A.A. Reshetnyak, arXiv:1406.5086[hep-th]], devoted to Lagrangian quantization for gauge theories with soft BRST symmetry breaking, in particular, for various descriptions of the YM theory without Gribov copies. The cited works rely on finite BRST and BRST-antiBRST transformations, respectively, with a singlet $\Lambda$ and a doublet $\lambda_{a}$, $a=1,2$, of anticommuting Grassmann parameters, both global and field-dependent. It turns out that global finite BRST and BRST-antiBRST transformations form a 1-parametric and a 2-parametric Abelian supergroup, respectively. Explicit superdeterminants corresponding to these changes of variables in the partition function allow one to calculate precise changes of the respective gauge-fixing functional. These facts provide the basis for a proof of gauge independence of the corresponding path integral under respective BRST and BRST-antiBRST transformations. It is shown that the gauge independence becomes restored for path integrals with soft BRST and BRST-antiBRST symmetry breaking terms. In this case, the form of transformation parameters is found to induce a precise change of the gauge in the path integral, thus connecting two arbitrary $R_{\xi}$-like gauges in the average effective action. Finite field-dependent BRST-antiBRST transformations are used to solve (perturbatively) the Gribov problem in the Gribov--Zwanziger approach. A modification of the path integral for theories with a gauge group, being consistent with gauge invariance and providing a restriction of the integration measure to the first Gribov region with a non-vanishing Faddeev--Popov determinant, is suggested.