Correlation functions of three-dimensional Yang-Mills theory from Dyson-Schwinger equations

The two- and three-point functions and the four-gluon vertex of three-dimensional Yang-Mills theory are calculated from their Dyson-Schwinger equations and the three-particle irreducible effective action. Within a self-contained truncation, various effects of truncating Dyson-Schwinger equations are studied. Estimates for the errors induced by truncations are derived from comparisons between results from different equations, comparisons with lattice results, and varying higher Green functions. The results indicate that the two-loop diagrams are important in the gluon propagator, where they are explicitly calculated, but not for the vertices. Furthermore, the influence of the four-gluon vertex on lower Green functions is found to be small.

Figure 1

The ghost propagator Dyson-Schwinger equation. Here and in all other figures, internal propagators are dressed, and thick blobs denote dressed vertices, wiggly lines gluons, and dashed ones ghosts.

Figure 2

The gluon propagator Dyson-Schwinger equations. The loop diagrams in the gluon propagator DSE are called the tadpole, gluon loop, ghost loop, sunset, and squint.

Figure 3

Top: Truncated $c$-DSE of the ghost-gluon vertex. Bottom: Truncated $A$-DSE of the ghost-gluon vertex. The full DSEs can be found, for example, in Ref. [3]. The loop diagrams are called Abelian and non-Abelian triangles.

Figure 4

Truncated DSE of the three-gluon vertex. The full one can be found, e.g., in Ref. [35]. The loop diagrams are called the ghost triangle, gluon triangle, static swordfish, and dynamic swordfish. The dynamic swordfish is required only once (with a modified symmetry factor) because of the symmetrization applied to the results.

Figure 5

Truncated DSE of the four-gluon vertex. The loop diagrams are called the ghost box, gluon box, swordfish, static triangle, and dynamic triangle.

Figure 6

Equation of motion from the 3PI effective action for the ghost-gluon vertex.

Figure 7

Equation of motion from the 3PI effective action for the three-gluon vertex.

Figure 8

Ghost-gluon vertex dressing function calculated with fixed input from different equations. Dark/light lines correspond to $x=0/x=p^2$.

Figure 9

Contributions of the Abelian (top) and non-Abelian (bottom) diagrams of the ghost-gluon vertex DSE calculated with fixed input from different equations. Dark/light lines corresponds to $x=0/x=p^2$.

Figure 10

Comparison of results from the DSE and the 3PI systems for the gluon dressing function.

Figure 11

Comparison of results from the DSE and the 3PI systems for the ghost dressing function.

Figure 12

Comparison of results from the DSE and the 3PI systems for the ghost-gluon vertex and three-gluon vertex dressing functions.

Figure 13

Gluon dressing function from the full system with a bare (green, dashed line) and a dynamic four-gluon vertex (red, continuous line).

Figure 14

Ghost dressing from the full system with a bare (green, dashed line) and a dynamic four-gluon vertex (red, continuous line).

Figure 15

Ghost-gluon vertex (top) and three-gluon vertex (bottom) dressing from the full system with a bare (green, dashed line) and a dynamic four-gluon vertex (red, continuous line).

Figure 16

Gluon dressing function (left) and gluon propagator (right) from the full system in comparison to lattice results [54]. Lattice momenta are along the $x$ axis. The shown lattice results correspond to lattice sizes between $N=68$ and $N=88$ and $\beta$ values between 3.48 and 19.2. The band is obtained by varying the maximum of the gluon dressing function between 922 and 1282 MeV.

Figure 17

Ratio of the gluon propagator from the lattice [53] and the full DSE system. Lattice data are for $\beta =10.21$ and $N=96$. The band is obtained by varying the maximum of the gluon dressing function between 922 and 1282 MeV. The red (lower) and green (upper) branches of the lattice data correspond to using the best and the first Gribov copy in the gauge fixing algorithm, respectively.

Figure 18

Ghost dressing from the full system in comparison to lattice results [54]. The parameters of the lattice setup are the same as in Fig. 16.

Figure 19

Ratio of the ghost propagator from the lattice [54] and the full DSE system. The parameters of the lattice setup are $N=68$ with $\beta =9.21$ and $N=74$ with $\beta =4.15$.

Figure 20

Coupling calculated from the propagators.

Figure 21

Contributions of individual diagrams to the gluon propagator DSE. Continuous/dotted lines denote positive/negative values.

Figure 22

Ghost-gluon vertex dressing from the full system in comparison with lattice results [78] (top) and the contributions from single diagrams (bottom). The shown lattice results correspond to lattice size $N=60$ and $\beta =3.18$, 5.61, and 10.5.

Figure 23

Ratio of the ghost-gluon vertex dressing from the lattice ($N=60$, $\beta =10.5$) [78] over the DSE result. The red (inner) lines correspond to the results obtained from varying the scale, and the green (outer) lines represent the error from the lattice calculation.

Figure 24

Three-gluon vertex dressing from the full system in comparison with lattice results [40] (top) and the contributions from single diagrams (bottom). The shown lattice results correspond to lattice sizes $N=40$ and 60 and $\beta =4.1$ and 6.

Figure 25

Ratio of the three-gluon vertex dressing from the lattice ($N=60$, $\beta =6$) [40] over the DSE result. The connected points correspond to the results obtained from varying the scale. The error bars from the lattice are added on top of that.

Figure 26

Tree-level dressing of the four-gluon vertex for different kinematic configurations (top left) and the individual contributions of single diagrams (top right and bottom).

Figure 27

Various dressing functions of the four-gluon vertex (top left) and the individual contributions of single diagrams (top right and bottom).

Figure 28

Left: Gluon dressing function calculated from the system of ghost and gluon propagators with a bare ghost-gluon vertex and a modeled three-gluon vertex with the correct and a rescaled value for $C_{\mathrm{sub}}$. Right: Cutoff dependence of the right-hand side of the gluon propagator DSE. The dots correspond to calculated values, and the line corresponds to the fit function.