Some more details of minimal-Landau-gauge SU(2) Yang-Mills propagators

The propagators of the elementary degrees of freedom of (minimal-)Landau-gauge Yang-Mills theory have been a useful tool in various investigations. However, in lattice calculations they show severe dependencies on lattice artifacts. This problem has been addressed for various subsets of lattice artifacts and various subsets of propagators over time. Here, an extended study of all propagators in momentum space, and for the gluon also in position space, as well as derived quantities like the running coupling, is provided simultaneously for two, three, and four dimensions over one or more orders of magnitude in both physical volume and lattice spacing, in lower dimensions also over more than 2 orders of magnitude for the gauge group SU(2). Most of the known qualitative results are confirmed, but two quantities also indicate a slight, but possibly interesting deviation.

Figure 1

The lattice settings employed, in physical units, for two dimensions (top panel), three dimensions (middle panel), and four dimensions (bottom panel).

Figure 2

The average number of gauge-fixing sweeps against lattice extension and lattice spacing, for two dimensions (top panel), three dimensions (middle panel), and four dimensions (bottom panel).

Figure 3

The impact of the selection of various center sectors on the gluon propagator (top panel) and the ghost dressing function (bottom panel). All momenta are along the $x$-axis. $1\sigma$-error bars here and hereafter are always shown, but may be smaller than the symbols. Results are from three dimensions.

Figure 4

The impact of the violation of rotational symmetry in two dimensions. The top panel shows the gluon dressing function, and the bottom panel the ghost dressing function. Closed symbols are along the $x$-axis, and open symbols are along the $xy$-axis. Note that for better visibility the results for different lattice spacings have been rescaled by constant factors.

Figure 5

The impact of the violation of rotational symmetry in three dimensions. The top panel shows the gluon dressing function, and the bottom panel the ghost dressing function. Closed symbols are along the $x$-axis, and open symbols are along the $xyz$-axis. Note that for better visibility the results for different lattice spacings have been rescaled by constant factors.

Figure 6

The impact of the violation of rotational symmetry in four dimensions. The top panel shows the gluon dressing function, and the bottom panel the ghost dressing function. Closed symbols are along the $x$-axis, and open symbols are along the $xyzt$-axis. Note that for better visibility the results for different lattice spacings have been rescaled by constant factors, and the results have been renormalized at 2 GeV.

Figure 7

The dependence of the gluon asymptotics, as isolated from (3)–(5), for the smallest volume and the different discretizations, in two (top panel), three (middle panel), and four (bottom panel) dimensions.

Figure 8

The gluon dressing function in two dimensions at large momenta along the space-time diagonal, compared to the leading-order behavior (3) for $c{g}^2=-0.60_{-25}^{40}$. On the right-hand side, the leading asymptotic has been isolated.

Figure 9

The gluon dressing function in three dimensions at large momenta along the space-time diagonal, compared to the leading-order behavior (4) for $g^2=2.05(5)$. On the right-hand side, the leading asymptotic has been isolated.

Figure 10

The gluon dressing function in four dimensions at large momenta along all the space diagonals for all possible energy values, compared to the leading-order behavior (5) for $g=4.8(1)$. On the right-hand side, the leading asymptotic has been isolated. The dressing function was renormalized at $\mu =3\mathrm{GeV}$.

Figure 11

The gluon dressing function (top panel) and propagator (bottom panel) at small momenta along the $x$-axis in two dimensions. The symbols have the same meaning as in Fig. 8. The function shown is $4.5p^{2.8}$ and $4.5p^{0.8}$ for the dressing function and propagator, respectively.

Figure 12

The gluon dressing function (top panel) and propagator (bottom panel) at small momenta along the $x$-axis in three dimensions. The symbols have the same meaning as in Fig. 9.

Figure 13

The gluon dressing function (top panel) and propagator (bottom panel) at small momenta along the $x$-axis in four dimensions. The symbols have the same meaning as in Fig. 10. The renormalization was done as in Fig. 10.

Figure 14

The dependence of the gluon propagator at zero momentum as a function of lattice spacing and extension in two (top panel), three (middle panel), and four (bottom panel) dimensions. The results in two and three dimensions have been tadpole corrected [12, 47], and in four dimensions a renormalization with $\mu =3\mathrm{GeV}$ has been performed. The symbols indicate the same volumes as in Figs. 8, 9, 10.

Figure 15

The dependence of the effective gluon exponent (6) as a function of lattice spacing and extension in two (top panel), three (middle panel), and four (bottom panel) dimensions. The line in two dimensions is the expected value of $t=0.4$. Note the inverted scale in four dimensions for better visibility. The symbols indicate the same volumes as in Figs. 8, 9, 10.

Figure 16

The dependence on the lattice spacing of the absolute value of the gluon Schwinger functions, normalized to its value at time zero. The top panel shows a fixed volume of $(9.6\mathrm{fm}{)}^2$ in two dimensions, the middle panel a fixed volume of $(7.5\mathrm{fm}{)}^3$ in three dimensions, and the bottom panel a fixed volume of $(4.3\mathrm{fm}{)}^4$ in four dimensions. Points with a relative statistical error larger than 100% have been omitted.

Figure 17

The absolute value of the gluon Schwinger functions for different volumes, normalized to its value at time zero. The top panel shows the results in two dimensions, the middle panel in three dimensions, and the bottom panel in four dimensions. Points with a relative statistical error larger than 100% have been omitted. Symbols have the same meaning as in Figs. 8, 9, 10.

Figure 18

The ghost dressing function at large momenta along the space-time-diagonal (in four dimensions all possible space diagonals are shown), compared to the leading-order behavior (7)–(9). The values for the parameter bands are $f{g}^2=-0.32_{-17}^{+13}$, $g^2=2.05(5)$, and $g=4.8(1)$ in two, three, and four dimensions, respectively. On the right-hand side, the leading asymptotic has been isolated. The symbols have the same meaning as in Figs. 8, 9, 10. Note the different scales on the right-hand side. In four dimensions the dressing function was renormalized at $\mu =3\mathrm{GeV}$. Top panels are two dimensions, middle panels three dimensions, and bottom panels four dimensions.

Figure 19

The dependence of the ghost asymptotics, as isolated from (7)–(9) in two (top panel), three (middle panel), and four (bottom panel) dimensions. See Fig. 18 for details. The symbols have the same meaning as in Figs. 8, 9, 10.

Figure 20

The ghost dressing function at small momenta along the $x$-axis. Symbols are as in Figs. 8, 9, 10. In four dimensions the dressing function was renormalized with the same renormalization factors as in Fig. 18. The top panel shows two dimensions, the middle panel three dimensions, and the bottom panel four dimensions. The function shown for two dimensions is $1.1p^{0.4}$.

Figure 21

The dependence of the effective ghost exponent (10) as a function of lattice spacing and extension in two (top panel), three (middle panel), and four (bottom panel) dimensions. The symbols denote the same volumes as in Figs. 8, 9, 10.

Figure 22

The volume dependence of the effective exponents for the ghost (left panel) and the gluon (right panel) in two dimensions for the respective finest discretization for every volume.

Figure 23

The running coupling in two dimensions is shown in the top panel. The bottom panel shows the running coupling divided by $p^2$, to obtain a dimensionless quantity. All momenta are along the $x$-axis. The band is the perturbative result (12). Points with relative errors larger than 10% have been suppressed. Symbols have the same meaning as in Fig. 8.

Figure 24

The running coupling in three dimensions is shown in the top panel. The bottom panel shows the running coupling divided by $p$, to obtain a dimensionless quantity. All momenta are along the $x$-axis. The band is the perturbative result (13). Symbols have the same meaning as in Fig. 9.

Figure 25

The running coupling in four dimensions is shown in the top panel and the corresponding $\beta$-function (15) in the bottom panel. All momenta are along the $x$-axis. The band is the perturbative result (14). Symbols have the same meaning as in Fig. 10.