Ghost-gluon vertex in the presence of the Gribov horizon

We consider Yang-Mills theories quantized in the Landau gauge in the presence of the Gribov horizon via the refined Gribov-Zwanziger (RGZ) framework. As the restriction of the gauge path integral to the Gribov region is taken into account, the resulting gauge field propagators display a nontrivial infrared behavior, being very close to the ones observed in lattice gauge field theory simulations. In this work, we explore a higher correlation function in the refined Gribov-Zwanziger theory: the ghost-gluon interaction vertex, at one-loop level. We show explicit compatibility with kinematical constraints, as required by the Ward identities of the theory, and obtain analytical expressions in the limit of vanishing gluon momentum. We find that the RGZ results are nontrivial in the infrared regime, being compatible with lattice Yang-Mills simulations in both SU(2) and SU(3), as well as with solutions from Schwinger-Dyson equations in different truncation schemes, Functional Renormalization Group analysis, and the renormalization group-improved Curci-Ferrari model.

Figure 1

Feynman diagram expansion up to one-loop order for the ghost-gluon vertex in the refined Gribov-Zwanziger theory. Dashed lines represent ghosts and antighosts, while the curly lines stand for gluons. Full lines, that only appear in mixed propagators, correspond to the auxiliary fields $\phi ,\overline{\phi }$. The roman numbers identifying the one-loop diagrams will be used as reference in the appendices.

Figure 2

The form factor ${\mathrm{\Gamma }}_{A\overline{c}c}(p)$ of the SU(2) ghost-gluon vertex in the soft-gluon limit as a function of the antighost momentum for $d=4$ is compared to lattice simulations. The solid lines represent our results for different values of the coupling: $g=2$, 2.5, 3, from the bottom to the top curve, respectively. Data points correspond to lattice results from Ref. [43].

Figure 3

The scalar function ${\mathrm{\Gamma }}_{A\overline{c}c}(p)$ of the SU(2) ghost-gluon vertex in the soft-gluon limit as a function of momentum for $d=4$. The solid lines represent our results for different values of the coupling: $g=2$, 2.5, 3, from the bottom to the top curve, respectively. Perturbative calculations with RG improvement in the Curci-Ferrari model [44] are displayed as the dashed and the dotted lines, which correspond to different renormalization schemes. Lattice data points from Ref. [43].

Figure 4

The form factor ${\mathrm{\Gamma }}_{A\overline{c}c}(p)$ of the SU(3) ghost-gluon vertex in the soft-gluon limit as a function of the antighost momentum for $d=4$ is compared to lattice simulations. The solid lines represent our results for different values of the strong coupling: $\alpha =\frac{g^2}{4\pi }=0.23$, 0.3, 0.42, from the bottom to the top curve, respectively. Data points correspond to lattice results from Ref. [46].

Figure 5

The scalar function ${\mathrm{\Gamma }}_{A\overline{c}c}(p)$ of the SU(2) ghost-gluon vertex in the soft-gluon limit as a function of momentum for $d=4$. Solid thick (black) lines represent our results for different values of the strong coupling: $\alpha =\frac{g^2}{4\pi }=0.18$ (bottom) and $\alpha =0.3$ (top), while the dashed (red) lines include a one-loop perturbative running coupling with different renormalization conditions: $\alpha (\mu =4.3\mathrm{GeV})=0.15$, 0.18, 0.22, from the bottom to the top curve, respectively. A DSE result from Ref. [47] is represented as the thin (blue) line.