Two-point correlation functions of QCD in the Landau gauge

We investigate the gluon, ghost and quark propagators in the Landau gauge with dynamic quarks. We perform a one-loop calculation in a model where the standard Faddeev-Popov Lagrangian is complemented by a mass term for the gluons which is seen as a minimal way of taking into account the effect of the Gribov copies. The analytic results are compared with lattice data obtained in four dimensions and for two, three and four quark flavors. The gluon and ghost propagators are reproduced with a few percent accuracy in the whole range of accessible momenta. The scalar part of the quark propagator is found to be in good agreement with the lattice data. However, the quark renormalization is poorly described. We attribute this discrepancy to the fact that the one-loop corrections to this quantity are unusually small so that the two-loop contribution cannot be discarded. The results are expressed in terms of the coupling, the gluon mass and the light quark mass at 1 GeV.

Figure 1

First line: four diagrams contributing to the gluon self-energy. Second line: diagram contributing to the ghost self-energy.

Figure 2

Diagram contributing to the quark two-point vertex.

Figure 3

Gluon dressing function (top) and ghost dressing function (bottom) as a function of momentum in $d=4$ for $N_f=2$. The points are lattice data of [44].

Figure 4

Gluon dressing function (top) and ghost dressing function (bottom) as a function of momentum in $d=4$ for $N_f=2+1+1$. The points are lattice data of [45].

Figure 5

Contour levels for the quantities ${\chi }_{AA}$ (left) and ${\chi }_{c\overline{c}}$ (right) for $d=4$, both for $N_f=2$ (above) and $N_f=2+1+1$ (below). The contour lines correspond to 8%, 10% and 12% (top left), 1.5%, 2% and 2.2% (top right), 3%, 6% and 10% (bottom left), 1%, 2% and 4% (bottom right). The quality of the fits is almost insensitive to the value of the quark mass.

Figure 6

Gluon dressing function (top) and ghost dressing function (bottom) for different number of flavors in $d=4$.

Figure 7

Gluon dressing function (top) and quark mass $M_{u,d}(p)$ (bottom) in $d=4$ for $N_f=2+1$. The parameters are $g=4.8$, $m=0.42\mathrm{GeV}$ and $M=0.08$. The points are lattice data of [15, 48].

Figure 8

Contour levels for the quantity ${\chi }_M$. The contour lines correspond to 10%, 20% and 30%.

Figure 9

Quark mass $M_{u,d}(p)$ in $d=4$ for $N_f=2+1$. The parameters are $g=9.6$, $m=1.4\mathrm{GeV}$ and $M=0.13\mathrm{GeV}$. The points are lattice data of [15, 48].

Figure 10

Quark masses of the light (full line, red) and heavy (dashed, black) quark.

Figure 11

$Z_{u,d}(p)$ in $d=4$. The points are lattice data of [48].

Figure 12

Gluon dressing function (top), quark mass $M_{u,d}(p)$ (middle) and $Z_{u,d}(p)$ (bottom) as a function of momentum in $d=4$. In red the hybrid calculation results (see text) and in black the one-loop calculation and the symmetrized one-loop results with respect to hybrid calculation. The points are lattice data of [44] and [15, 48].