Gribov horizon and Gribov copies effect in lattice Coulomb gauge

Following a recent proposal by Cooper and Zwanziger, we investigate via $SU(2)$ lattice simulations the effect on the Coulomb gauge propagators and on the Gribov-Zwanziger confinement mechanism of selecting the Gribov copy with the smallest nontrivial eigenvalue of the Faddeev-Popov operator, i.e., the one closest to the Gribov horizon. Although such choice of gauge drives the ghost propagator towards the prediction of continuum calculations, we find that it actually overshoots the goal. With increasing computer time, we observe that Gribov copies with arbitrarily small eigenvalues can be found. For such a method to work, one would therefore need further restrictions on the gauge condition to isolate the physically relevant copies, since, for example, the Coulomb potential $V_C$ defined through the Faddeev-Popov operator becomes otherwise physically meaningless. Interestingly, the Coulomb potential alternatively defined through temporal link correlators is only marginally affected by the smallness of the eigenvalues.

Figure 1

Gauge precision $\theta$ over the number of over-relaxation steps. Left panel: Five gauge copies of the same configuration. The light blue (top) curve is with simple relaxation; the other lines correspond to over-relaxation ($\omega =1.7$). The pink line with the smallest slope corresponds to a significantly smaller value (compared to the other copies) of the first nontrivial eigenvalue ${\lambda }_1$ of the FP operator. Right panel: Red (top) and green (bottom) line correspond to over-relaxation ($\omega =1.7$) without (red) and with (green) simulated annealing preconditioning. As can be seen, the preconditioning removes the first phase where the algorithm tries to locate a maximum, while the slope of the second phase (the eventual convergence speed) is unchanged. The blue curve in the middle employs preconditioning with simple relaxation ($\omega =1$) and shows no fluctuating first phase but a much smaller convergence speed. All three lines converge towards the same Gribov copy, as was confirmed by identical functional values and an identical first nontrivial FP eigenvalue.

Figure 2

Smallest eigenvalue ${\lambda }_1$ of the Faddeev–Popov operator as a function of the number of gauge-fixing iterations. From each set A1, A2, and A3, we used 10 configurations and calculated 1000 gauge copies. The data points which correspond to fewer iterations (left) are from runs with $\omega =1.9$; for the points with more iterations (right), we used $\omega =1$.

Figure 3

Smallest eigenvalue vs functional value for 1000 copies from four arbitrary configurations of the $16^4$ lattices A1, A2, A3 (from top to bottom). The number of distinct Gribov copies decreases with finer lattices.

Figure 4

Best approximation of the FMR, i.e., the b -approach, and the Gribov horizon, i.e., the lc approach. Full light-colored symbols denote 1000 trials; the empty, dark-colored symbols denote 10,000 gauge copies. Lattices: B1 (left), B3 (right). There is no configuration where $bc=lc$.

Figure 5

Number of distinct Gribov copies vs the number of gauge copies for the $16^4$ (left) and the $24^4$ (right) lattices. Only for the fines $16^4$ lattice a saturation is observed. The Gribov copy is identified only by the value of the gauge fixing functional.

Figure 6

The gluon propagator for the B1 lattice with the bc and lc approaches from 1000 trials. The solid line is a fit to the Gribov formula [15, 16]. The choice of Gribov copy apparently makes no visible difference.

Figure 7

The ghost form factor after gauge fixing to the lowest-eigenvalue copy with increasing number of trials from 10 to 10,000 on $24^4$ lattices at $\beta =2.2$ (B1, l.h.s.) and $\beta =2.4$ (B3, r.h.s.).

Figure 8

The ghost form factor after gauge fixing to the best functional copy with increasing number of trials from 10 to 1000 on $24^4$ lattices at $\beta =2.2$ (B1, lhs) and $\beta =2.4$ (B3, rhs). The data points for 10,000 copies are omitted since no better copy is found, compare Fig. 4.

Figure 9

The ghost form factor from the lattices B1 and B3 after 10,000 copies of bc and lc strategies.

Figure 10

Fits of $d(p)$ to an IR power law for the D1 lattices. The continuous lines are the fits to Eq. (9); the dotted lines just fit the last three points to a power law. The respective $\kappa$ values are given in the legend.

Figure 11

Coulomb potential in the bc approach as a function of $N_r$. The data with $N_r=10$, 000 are omitted, since they show no difference as compared to $N_r=1,000$, see Fig. 4.

Figure 12

The Coulomb potential from the lattices B1 and B3 after 10,000 copies.

Figure 13

Coulomb potential in position space from the $⟨U_0{U}_0^{†}⟩$ correlator in Eq. (12) (D1 lattice). The bc configurations are preconditioned with simulated annealing.