Vertex functions of Coulomb gauge Yang-Mills theory

The canonical recursive Dyson-Schwinger equations for the three-gluon and ghost-gluon vertices are solved numerically. The employed truncation includes several previously neglected diagrams and includes back-coupling effects. We find an infrared finite ghost-gluon vertex and an infrared diverging three-gluon vertex. We also compare our results with those obtained in previous calculations, where bare vertices were used in the loop diagrams.

Figure 1

Schematic representation of the four-gluon variational kernel ${\gamma }_4$ [Eq. (10)]. Empty boxes represent variational kernels, while the small empty dot stands for the trivial tensor structure Eq. (12). The double line represents the Coulomb propagator Eq. (14).

Figure 2

The CRDSE (17) for the ghost-gluon vertex, arising from the operator identity Eq. (16). Wiggly and dashed lines represent the bare gluon and ghost propagators, respectively. If these lines are augmented by a full dot they represent full propagators. Empty and full (fat) dots stand for bare and full (dressed, one-particle-irreducible) vertices.

Figure 3

Alternative form of the CRDSE for the ghost-gluon vertex, arising from the functional identity Eq. (7). For notation see caption of Fig. 2. Furthermore, empty boxes represent variational kernels.

Figure 4

CRDSE for the three-gluon vertex.

Figure 5

Dressing function of the ghost-gluon vertex. The antighost momentum is denoted by $p$, the gluon momentum by $k$. (Left panel) Antighost legs attached to bare vertices, see Fig. 2. (Right panel) Gluon legs attached to bare vertices, see Fig. 3.

Figure 6

Comparison of the results from the two different ghost-gluon vertex CRDSEs. Continuous/dashed lines are from the versions with the antighost/gluon legs attached to the bare vertices. (Left panel) Equal antighost and gluon momenta, different angles. (Right panel) Zero gluon momentum.

Figure 7

Ghost-gluon vertex dressing function for equal momenta and various angles.

Figure 8

Comparison to lattice results [21] at the symmetric point (left panel) and for vanishing gluon momentum (right panel). Different colors correspond to different lattice sizes $N\in \{40,60\}$ and values for $\beta \in \{4.2,6\}$; see Ref. [21] for details.

Figure 9

Comparison of the full nonperturbative calculation carried out in the present paper with the semiperturbative calculation [30] of the ghost-gluon vertex. (Left panel) Dressing function of the ghost-gluon vertex. (Right panel) Contributions from the Abelian and non-Abelian diagrams. The values for the non-Abelian diagram are larger than those in Fig. 2 of Ref. [30], where the coupling constant $g$ was factored out.

Figure 10

(Left panel) Dressing of the three-gluon vertex. (Right panel) The ratio of three-gluon vertex to the variational kernel. The deviation from Bose symmetry at the boundaries is a numerical artifact due to the smallness of ${\gamma }_3$ which enhances small numerical errors considerably.

Figure 11

(Left panel) Dressing function of the three-gluon vertex. (Right panel) Three-gluon vertex over variational kernel. Both plots for equal magnitude of two momenta with various angles between them.

Figure 12

The three-gluon vertex from the full calculation (continuous line), from a simplified four-gluon kernel (dashed line), from a triangles-only calculation (dot-dashed line) and from a ghost-triangle-only calculation (dotted line). (Left panel) Three-gluon vertex dressing function. (Right panel) Ratio of the three-gluon vertex to the variational kernel. Colors correspond to the same angles as in the left panel.

Figure 13

IR behavior of ghost triangle (green, upper line) and the IR dominant part of the dynamic swordfish (red, lower line) projected as in Eq. (26). The dashed blue line is shown to illustrate the power law $p^{-5}$.

Figure 14

Three-gluon vertex dressing function at the symmetric point calculated with the full ghost-gluon vertex obtained here (green, dashed line) compared to results calculated with a bare ghost-gluon vertex (red, continuous line). The right panel shows the region around the zero crossing.