Nonperturbative finite-temperature Yang-Mills theory

We present nonperturbative correlation functions in Landau-gauge Yang-Mills theory at finite temperature. The results are obtained from the functional renormalisation group within a self-consistent approximation scheme. In particular, we compute the magnetic and electric components of the gluon propagator, and the three- and four-gluon vertices. We also show the ghost propagator and the ghost-gluon vertex at finite temperature. Our results for the propagators are confronted with lattice simulations and our Debye mass is compared to hard thermal loop perturbation theory.

Figure 1

Constituents of our vertex expansion. We use the classical tensors that are present in the bare action and attach magnetic (blue) and electric (red) projection operators to the gluon legs. Missing combinations, e.g., vertices with one electric leg, vanish if the Matsubara modes are set to zero and are not computed in our truncation.

Figure 2

Expectation value of $⟨L[A_0]⟩$ versus $L[⟨{\overline{A}}_0⟩]$ from [79]. Both observables are order parameters for the confinement-deconfinement phase transition. Moreover, $L[⟨{\overline{A}}_0⟩]=1$ entails $⟨{\overline{A}}_0⟩=0$.

Figure 3

Debye screening mass $m_s$; see Fig. 18 for the fits of (27) to $G_T(x)$. (a) Screening mass $m_s$ in units of GeV at low temperatures. (b) Dimensionless Debye screening $m_s/T$ mass at high temperatures in comparison with leading order perturbation theory (28) and the Arnold-Yaffe prescription (29) for accommodating beyond leading order effects [92].

Figure 4

Graphical representation of the Wetterich equation. Wiggly (dotted) lines represent the dressed gluon (ghost) propagators. The cross in the circle denotes the regulator insertion ${\partial }_t{R}_k$ of the corresponding field type.

Figure 5

Truncated equations of all computed $n$-point functions. Permutations of external legs and regulator insertions are omitted in the vertex equations. The gluon lines represent the sum over electric and the magnetic propagators, cf. (9). They are connected to the corresponding magnetic and electric components of the vertices; see also Fig. 1.

Figure 6

Magnetic gluon propagator dressing, (9).

Figure 7

Electric gluon propagator dressing, (9).

Figure 8

Ghost propagator dressing, (9), compared to $SU(2)$ lattice results [69, 123] and ghost-gluon vertex, (b1).

Figure 9

Temperature dependence of the three-gluon vertex dressing, (b2).

Figure 10

Temperature dependence of the four-gluon vertex dressing, (b3).

Figure 11

Temperature dependence of the magnetic three-gluon vertex zero crossing ${\lambda }_{A^3}^{\mathrm{M}}(p_0)=0$.

Figure 12

Gluon propagator dressing obtained with the exponential regulator in comparison with dressings calculated with the flat regulator in [2] and $SU(3)$ lattice data [124]. The lattice results are renormalized as in [2]. Newer lattice results [125] agree with [124] if the largest physical volumes are compared.

Figure 13

Four-gluon vertex vacuum flows from different projection operators, (d3) and (d4), and their differences at the RG scale $k=2\mathrm{GeV}$.

Figure 14

Relative deviations, e.g., $({\lambda }_{A^4}^{\mathrm{M}}({\lambda }_c=1)-{\lambda }_{A^4}^{\mathrm{M}}({\lambda }_c=2))/{\lambda }_{A^4}^{\mathrm{M}}({\lambda }_c=1)$, of the four-gluon vertex dressings, (b3), calculated with different parameters in the smoothed theta function (d9). Depending on the temperature, the dressings depend either on ${\lambda }_c$ or ${\mathrm{\Lambda }}_c$; see (d10).

Figure 15

Vacuum limit and initial scale independence of gluon propagator, (9).

Figure 16

Electric and magnetic gluon propagators with and without mass subtraction procedure, (g2).

Figure 17

Magnetic $SU(2)$ lattice propagator [69, 123] over FRG propagator; see Fig. 6 for the color coding.

Figure 18

Exponential tail of the Fourier transformed electric propagator, $G_T^{\mathrm{E}}(x)$; see (26). The dashed lines are fits of (i1) with $c_a=0$, i.e., (27), to the large points. Small points show $G_T^{\mathrm{E}}(x)$ beyond the fit regions. The fitted screening masses as a function of temperature are shown in Fig. 3.