Landau gauge Yang-Mills correlation functions

We investigate Landau gauge $SU(3)$ Yang-Mills theory in a systematic vertex expansion scheme for the effective action with the functional renormalization group. Particular focus is put on the dynamical creation of the gluon mass gap at nonperturbative momenta and the consistent treatment of quadratic divergences. The nonperturbative ghost and transverse gluon propagators as well as the momentum-dependent ghost-gluon, three-gluon and four-gluon vertices are calculated self-consistently with the classical action as the only input. The apparent convergence of the expansion scheme is discussed and within the errors, our numerical results are in quantitative agreement with available lattice results.

Figure 1

Approximation for the effective action. Only classical tensor structures are included. See Fig. 2 for diagrams that contribute to the individual propagators and vertices.

Figure 2

Diagrams that contribute to the truncated flow of propagators and vertices. Wiggly (dotted) lines correspond to dressed gluon (ghost) propagators, filled circles denote dressed (1PI) vertices and regulator insertions are represented by crossed circles. Distinct permutations include not only (anti-)symmetric permutations of external legs but also permutations of the regulator insertions.

Figure 3

Gluon dressing $1/Z_A$ (left) and ghost dressing $1/Z_c$ (right) in comparison to the lattice results from [79]. The scale setting and normalisation procedures are described in Appendix pp6.

Figure 4

Left: Gluon propagator in comparison to the lattice results from [79]. Right: Effective running couplings defined in (20) as obtained from different Yang-Mills vertices as function of the momentum.

Figure 5

Ghost-gluon vertex (left) and three-gluon (right) vertex dressing functions $Z_{A\overline{c}c,\perp }(p,p,-\frac{1}{2})$ and $Z_{A^3,\perp }(p,p,-\frac{1}{2})$ in the symmetric point configuration. More momentum configurations and comparisons to Dyson-Schwinger and lattice results can be found in Figs. 11, 12, 13. In contrast to Fig. 3, the decoupling dressings are normalised to the scaling solution in the UV.

Figure 6

Left: Four-gluon vertex dressing function as defined in (10) at the symmetric point in comparison to Dyson-Schwinger computations [39]. We normalized all curves to match the scaling result at $p=2\mathrm{GeV}$. Right: Gluon propagator dressings obtained with different momentum approximations, see Sec. 4b for details.

Figure 7

Running couplings (20) in comparison with DSE running. The grey band gives the spread of vertex couplings from the FRG in the present work. The DSE results are shown rescaled to fit our ghost-gluon vertex running coupling at 10 GeV to facilitate the comparison. The inlay shows the unscaled couplings. Note that the FRG running couplings naturally lie on top of each other and are not depicted rescaled.

Figure 8

Ghost dressing functions $1/Z_c$ (left) and gluon propagators (right) for different values of the ultraviolet gluon mass parameter. Blue results correspond to the Higgs-type branch and red results to the confined branch. The solutions in all branches have been normalized to the scaling solution in the UV.

Figure 9

Left: Gluon mass gap as a function of the gluon mass parameter $m_{\mathrm{\Lambda }}^2-m_{\mathrm{\Lambda },\text{scaling}}^2$, where $m_{\mathrm{\Lambda },\text{scaling}}^2$ denotes the gluon mass parameter that yields the scaling solution. Right: Momentum value at which the gluon propagator assumes its maximum, as a function of the gluon mass parameter $m_{\mathrm{\Lambda }}^2-m_{\mathrm{\Lambda },\text{scaling}}^2$. The inlay exposes the power law behavior of the gluon propagator maximum in the vicinity of the transition region, see (33). Both plots were obtained from our numerically less-demanding 1D approximation. We have repeated this analysis in the transition regime from Higgs-type to confinement branch also with the best approximation and find the same behavior. The shaded area marks momentum scales that are not numerically resolved in the present work. The points in this region rely on a generic extrapolation.

Figure 10

Left: Gluon mass parameter $m_k^2={\mathrm{\Gamma }}_{AA,k}^{(2)}(p=0)$ over $k$. Right: Possible choices for the scaling prefactor in the gluon regulator: $Z_{A,k}(k)$ (black, solid), ${\overline{Z}}_{A,k}(k)$ (red, dashed), ${\widehat{Z}}_{A,k}(k)$ (blue, dot-dashed) and ${\tilde{Z}}_{A,k}$ (green, dotted) as defined in (e3) and (e5). Independence of the results from the above choice has been checked explicitly.

Figure 11

Left: Four-gluon vertex dressing function $Z_{A^4,\perp }(p,q,0)$ in the tadpole configuration. The angular dependence is small compared to the momentum dependence. Right: Four-gluon tadpole configuration evaluated in the symmetric point approximation, showing a quantitative and qualitative deviation from the full calculation.

Figure 12

Left: Ghost-gluon vertex dressing function $Z_{A\overline{c}c,\perp }(p,q,0)$. Right: Three-gluon vertex dressing function $Z_{A^3,\perp }(p,q,0)$.

Figure 13

Left: Ghost-gluon vertex dressing function $Z_{A\overline{c}c,\perp }(p,q,\cos \sphericalangle (p,q))$ in comparison to $SU(2)$ lattice [93, 94, 95] and DSE results [44, 89]. The lattice results are obtained from $N=32^4$ lattices. The magenta/orange/green points (color online) correspond to $\beta \in \{2.13,2.39,2.60\}$ and lattice spacing $a^{-1}\in \{0.8\mathrm{GeV},1.6\mathrm{GeV},3.2\mathrm{GeV}\}$, respectively. Right: Three-gluon vertex dressing function $Z_{A^3,\perp }(p,q,\cos \sphericalangle (p,q))$ compared with $SU(2)$ lattice [93, 94, 95] and Dyson-Schwinger [47] results. The colored lattice points are taken from [93, 94] and correspond to $\beta \in \{2.2,2.5\}$ and different lattice sizes $1.4\mathrm{fm}<L<4.7\mathrm{fm}$. The lattice results shown in black are based on [93, 94] but stem from [95]. These are gained from $N\in \{24^4,32^4\}$ lattices with $\beta \in \{2.13,2.39,2.60\}$ and lattice spacing $a^{-1}\in \{0.8\mathrm{GeV},1.6\mathrm{GeV},3.2\mathrm{GeV}\}$. The comparison with $SU(2)$ lattice simulations is justified since the propagators of $SU(2)$ and $SU(3)$ Yang-Mills theory agree well for a large range of momenta [96, 97] after a respective normalization in this regime. We rescaled all DSE results such that they match the scaling solution in the symmetric point configuration at $p=2\mathrm{GeV}$. Note that the scaling and decoupling solutions differ in the UV just due to the different field renormalizations, cf. Fig. 3. The physically relevant couplings, given by (20), agree in the UV.