Infrared analysis of Dyson-Schwinger equations taking into account the Gribov horizon in Landau gauge

The low momentum behavior of the Landau gauge Gribov-Zwanziger action is investigated using the respective Dyson-Schwinger equations. Because of the mixing of the gluon and the auxiliary fields four scenarios can be distinguished for the infrared behavior. Two of them lead to inconsistencies and can be discarded. Another one corresponds to the case where the auxiliary fields behave exactly like the Faddeev-Popov ghosts and the same scaling relation as in standard Landau gauge, ${\kappa }_A+2{\kappa }_c=0$, is valid. Even the parameter $\kappa$ is found to be the same, 0.595. The mixed propagators, which appear, are suppressed in all loops, and their anomalous infrared exponent can also be determined. A fourth case provides an even stricter scaling relation that includes also the mixed propagators, but possesses the same qualitative feature, i.e. the propagators of the Faddeev-Popov ghost and the auxiliary fields are infrared enhanced and the mixed and the gluon propagators are infrared suppressed. In this case the system of equations to obtain the parameter $\kappa$ is nonlinear in all variables.

Figure 1

The DSE of the gluon two-point function. A propagator with the exponent $-1$ denotes the two-point function. This convention, strictly speaking being mathematically incorrect, is chosen for the purpose of diagrammatic representation only. The propagators are labeled by the respective fields. The indices $i$ and $j$ denote the first and second fields of the depicted two-point function DSE.

Figure 2

The DSE of the $V$-field two-point function.

Figure 3

The DSEs of the $A\mathrm{\text{−}}V$-mixed two-point function. The first is the $AV$ DSE and the second one the $VA$ DSE.

Figure 4

The DSE of the $\eta$-field two-point function.

Figure 5

Diagrams of the $AA$ and $VV$ DSEs used explicitly in the text.

Figure 6

Diagrams of the $VA$ DSE used explicitly in the text.

Figure 7

The plot to determine solutions for ${\kappa }_{AV}$. On the $Y$ axis we plot the left- and right-hand sides of Eq. (68).

Figure 8

One specific diagram in the RGE of the $AVV$ vertex. Internal lines represent dressed propagators, and the black blobs dressed vertices. The gray blob is a regulator.