Coulomb gauge ghost Dyson–Schwinger equation

A numerical study of the ghost Dyson–Schwinger equation in Coulomb gauge is performed and solutions for the ghost propagator found. As input, lattice results for the spatial gluon propagator are used. It is shown that in order to solve completely, the equation must be supplemented by a nonperturbative boundary condition (the value of the inverse ghost propagator dressing function at zero momentum), which determines if the solution is critical (zero value for the boundary condition) or subcritical (finite value). The various solutions exhibit a characteristic behavior where all curves follow the same (critical) solution when going from high to low momenta until forced to freeze out in the infrared to the value of the boundary condition. The renormalization is shown to be largely independent of the boundary condition. The boundary condition and the pattern of the solutions can be interpreted in terms of the Gribov gauge-fixing ambiguity. The connection to the temporal gluon propagator and the infrared slavery picture of confinement is explored.

Figure 1

The unrenormalized Dyson–Schwinger equation for the ghost two-point proper function, ${\Gamma }_{\overline{c}c}$. Filled blobs denote dressed two-point functions and empty circles denote dressed proper vertex functions. Wavy lines denote proper functions, springs denote connected (propagator) functions and dashed lines denote the ghost propagator.

Figure 2

Results for the ghost dressing functions $\Gamma (x)$ [left panel] and $D(x)$ [right panel]. See text for details.

Figure 3

Plot of the ghost renormalization coefficient $Z_c$ as a function of $\Gamma (0)$. See text for details.

Figure 4

Plot of the ghost propagator dressing function in the case $\Gamma (0)={10}^{-5}$ (indistinguishable from the powerlaw for the scales shown) compared with the fit formula Eq. (4.2). See text for details.

Figure 5

Plot of the ghost propagator dressing function in the case $\Gamma (0)={10}^{-4}$ (indistinguishable from $\Gamma (0)=0$ on the scales shown) and $\Gamma (0)=0.2$ compared with the powerlaws characterized by the exponents $\kappa =-{\gamma }_g$ (perturbative), $\kappa =0.2$ (illustrative lattice result), and $\kappa =1/2$ (infrared analysis). The scale $x=0.25$ roughly corresponds to the lowest lattice momenta. See text for details.