Feynman Gauge on the Lattice: New Results and Perspectives

Abstract

Abstract.We have recently introduced a new implementation of the Feynman gauge on the lattice, based on a minimizing functional that extends in a natural way the Landau-gauge case, while preserving all the properties of the continuum formulation. The only remaining difficulty with our approach is that, using the standard (compact) discretization, the gluon field is bounded, while its four-divergence satisfies a Gaussian distribution, i.e. it is unbounded. This can give rise to convergence problems when a numerical implementation is attempted. In order to overcome this problem, one can use different discretizations for the gluon field, or consider an SU(N c ) group with sufficiently large N c . Here we discuss these two possible solutions.

On the gauge-invariant operatorAmin2in Euclidean Yang-Mills theories

“…Until now, this behaviour is in very good agreement with the most recent lattice numerical simulations [46][47][48][49]. The generalization of these results to the linear covariant gauges has been worked out recently and can be found in [50][51][52][53][54][55][56][57][58][59].…”

Lattice Landau Gauge and Algebraic Geometry

“…It would be important to compute the number of solutions for different random orbits for the 3 × 3 lattice with the modified LLG using the NPHC method. Efforts to solving the corresponding gauge-fixing equations for the modified LLG for compact U(1) and also the linear covariant gauge-fixing equations for compact U(1), in the spirit of [40,41], are in progress. It would also be very interesting to solve the Langevin dynamics equations for the XY model and other models [39] using algebraic geometry methods.…”

Lattice gluon propagator in renormalizableξgauges

“…While this class of gauges is the only one completely under control at the perturbative level, its lattice implementation has nevertheless proven to be quite complicated due to poor numerical convergence of the corresponding GF algorithm [36][37][38][39][40][41]. A GF procedure with an improved convergence rate was finally implemented in [42]; however, one still encountered significant problems, which unfortunately become more severe as the GF parameter ξ and/or the lattice volume become larger, and the number of colors N c and/or the lattice coupling β become smaller [43][44][45]. As a result, there have been only preliminary studies of the R ξ gluon propagator [41,42,45].…”