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We review our lattice results concerning the Gribov-Zwanziger confinement mechanism in Coulomb gauge. In particular, we verify the validity of Gribovs IR divergence condition for the Coulomb ghost form factor. We also show how the quark self-energy is, like that of the transverse gluon, IR divergent, thus effectively extending the Gribov-Zwanziger scenario to full QCD.
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We discuss the gluon propagator in 3- and 4-dimensional Yang-Mills theories in Coulomb gauge and compare it with the corresponding Landau gauge propagator, showing that for both the relevant IR mass scale coincides. We also report preliminary results on Coulomb gauge ghost form factor and quark propagators and give a comment on the gluon propagators strong coupling limit.
We calculate the lattice quark propagator in Coulomb gauge both from dynamical and quenched configurations. We show that in the continuum limit both the static and full quark propagator are multiplicatively renormalizable. From the propagator we extract the quark renormalization function Z(|p|) and the running mass M(|p|) and extrapolate the latter to the chiral limit. We find that M(|p|) practically coincides with the corresponding Landau gauge function for small momenta. The computation of M(|p|) can be however made more efficient in Coulomb gauge; this can lead to a better determination of the chiral mass and the quark anomalous dimension. Moreover from the structure of the full propagator we can read off an expression for the dispersion relation of quarks, compatible with an IR divergent effective energy. If confirmed on larger volumes this finding would allow to extend the Gribov-Zwanziger confinement mechanism to the fermionic sector of QCD.
We show that in the lattice Hamiltonian limit all Coulomb gauge propagators are consistent with the Gribov-Zwanziger confinement mechanism, with an IR enhanced effective energy for quarks and gluons and a diverging ghost form factor compatible with a dual-superconducting vacuum. Multiplicative renormalizability is ensured for all static correlators, while for non-static ones their energy dependence plays a crucial role in this respect. Moreover, from the Coulomb potential we can extract the Coulomb string tension sigma_C ~ 2 sigma.
I will review essential features of the Hamiltonian approach to QCD in Coulomb gauge showing that Gribovs confinement scenario is realized in this gauge. For this purpose I will discuss in detail the emergence of the horizon condition and the Coulomb string tension. I will show that both are induced by center vortex gauge field configurations, which establish the connection between Gribovs confinement scenario and the center vortex picture of confinement. I will then extend the Hamiltonian approach to QCD in Coulomb gauge to finite temperatures, first by the usual grand canonical ensemble and second by the compactification of a spatial dimension. I will present results for the pressure, energy density and interaction measure as well as for the Polyakov loop.
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 non-trivial 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 e.g.~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.
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