Two-loop calculation of Yang-Mills propagators in the Curci-Ferrari model

The Landau-gauge gluon and ghost correlation functions obtained in lattice simulations can be reproduced qualitatively and, to a certain extent, quantitatively if a gluon mass is added to the standard Faddeev-Popov action. This has been tested extensively at one loop, for the two and three point correlation functions of the gluons, ghosts and quarks. In this article, we push the comparison to two loops for the gluon and ghost propagators. The agreement between lattice results and the perturbative calculation considerably improves. This validates the Curci-Ferrari action as a good phenomenological model for describing the correlation functions of Yang-Mills theory in the Landau gauge. It also indicates that the perturbation theory converges fairly well, in the infrared regime.

Figure 1

One- and two-loop integral families.

Figure 2

Additional one- and two-loop master topologies.

Figure 3

Different trajectories of the IS-flow in the plane $(\tilde{m}\equiv m/\mu ,\lambda )$ including two-loop corrections. The infrared fixed point is located at $({\lambda }^{*},{\tilde{m}}^{*})\sim (1.64,0.6)$. Green trajectories (left side) present a Landau pole, while blue trajectories (right side) are infrared safe. The dotted line represents the position for the extrema of the gluon propagator. The orange (dashed) curve is the one used to reproduce lattice data in $SU(3)$.

Figure 4

Gluon propagator for different initial conditions of the flow.

Figure 5

Level curves for the error $\chi$ for the two-loop correction (left) and one-loop correction (right) in $SU(3)$ (top) and $SU(2)$ (bottom) all in $d=4$ and renormalized using the IS scheme. From dark to light, the different regions correspond to 4%, 5% and 6% (top left); 7%, 8%, 9% and 10% (top right); 6%, 7% and 8% and 10% (bottom left); 7%, 8%, 9% and 10% (bottom right).

Figure 6

Comparison with lattice data from [55] for the gluon propagator (top), gluon dressing function (middle) and with lattice data from [56] for the ghost dressing function (bottom), in four dimensions and for the SU(3) gauge group.

Figure 7

Renormalization-group flow of the coupling constant (top) and mass (bottom) for the $SU(3)$ gauge group, in the IS scheme.

Figure 8

Expansion parameter $\tilde{\lambda }(\mu )$ [see Eq. (33)] as a function of the renormalization-group scale for $SU(3)$.

Figure 9

The strong coupling constant in the Taylor scheme. The data points are those of Ref. [57]. The dashed and plain lines correspond to our one-loop and two-loop results, respectively. In the lower plot, a correction factor is applied to the value of ${\alpha }_S$ in the UV; see the main text for more explanations.

Figure 10

Expansion parameter $\tilde{\lambda }(\mu )$ [see Eq. (33)] as a function of the renormalization-group scale for $SU(2)$.

Figure 11

Comparison with lattice data from [31] for the gluon propagator (top), gluon dressing function (middle) and ghost dressing function (bottom) in four dimensions and $SU(2)$.

Figure 12

Comparison with lattice data from [56] for the gluon dressing function in four dimensions and $SU(3)$ using different schemes at two loops (top) and one loop (bottom).