Narrow Results

We continue our research${}^{1\text{–}4}$ and extend the class of finite BRST–anti-BRST transformations with odd-valued parameters ${\lambda }_a$, $a=1,2$, introduced in these works. In doing so, we evaluate the Jacobians induced by finite BRST–anti-BRST transformations linear in functionally-dependent parameters, as well as those induced by finite BRST–anti-BRST transformations with arbitrary functional parameters. The calculations cover the cases of gauge theories with a closed algebra, dynamical systems with first-class constraints, and general gauge theories. The resulting Jacobians in the case of linearized transformations are different from those in the case of polynomial dependence on the parameters. Finite BRST–anti-BRST transformations with arbitrary parameters induce an extra contribution to the quantum action, which cannot be absorbed into a change of the gauge. These transformations include an extended case of functionally-dependent parameters that implies a modified compensation equation, which admits nontrivial solutions leading to a Jacobian equal to unity. Finite BRST–anti-BRST transformations with functionally-dependent parameters are applied to the Standard Model, and an explicit form of functionally-dependent parameters ${\lambda }_a$ is obtained, providing the equivalence of path integrals in any 3-parameter $R_{\xi }$-like gauges. The Gribov–Zwanziger theory is extended to the case of the Standard Model, and a form of the Gribov horizon functional is suggested in the Landau gauge, as well as in $R_{\xi }$-like gauges, in a gauge-independent way using field-dependent BRST–anti-BRST transformations, and in $R_{\xi }$-like gauges using transverse-like non-Abelian gauge fields.

The quantization of Yang–Mills theories relies on the gauge-fixing procedure. However, in the non-Abelian case this procedure leads to the well-known Gribov ambiguity. In order to solve the ambiguity a modification of the functional integral formula must be introduced. As a consequence of this, the Green functions get deep modifications in the infrared. We consider, in particular, the SU(N) case and show that in the pure gauge case the ghost propagator is enhanced, while the gluon propagator is suppressed in this limit, therefore the study of the Gribov ambiguity may shed some light on the mass gap problem and on color confinement. We discuss some recent developments on the subject in the case of a curved background. We argue that the concurrent presence of a spacetime curvature and the Gribov ambiguity may introduce further modifications to the Green functions in the infrared.

After reviewing Gribov ambiguity of non-Abelian gauge theories, a phenomenon related to the topology of the bundle of gauge connections, we show that there is a similar feature for noncommutative QED over Moyal space, despite the structure group being Abelian, and we exhibit an infinite number of solutions for the equation of Gribov copies. This is a genuine effect of noncommutative geometry which disappears when the noncommutative parameter vanishes.