Yang-Mills correlators across the deconfinement phase transition

We compute the finite temperature ghost and gluon propagators of Yang-Mills theory in the Landau-DeWitt gauge. The background field that enters the definition of the latter is intimately related with the (gauge-invariant) Polyakov loop and serves as an equivalent order parameter for the deconfinement transition. We use an effective gauge-fixed description where the nonperturbative infrared dynamics of the theory is parametrized by a gluon mass which, as argued elsewhere, may originate from the Gribov ambiguity. In this scheme, one can perform consistent perturbative calculations down to infrared momenta, which have been shown to correctly describe the phase diagram of Yang-Mills theories in four dimensions as well as the zero-temperature correlators computed in lattice simulations. In this article, we provide the one-loop expressions of the finite temperature Landau-DeWitt ghost and gluon propagators for a large class of gauge groups and present explicit results for the SU(2) case. These are substantially different from those previously obtained in the Landau gauge, which corresponds to a vanishing background field. The nonanalyticity of the order parameter across the transition is directly imprinted onto the propagators in the various color modes. In the SU(2) case, this leads, for instance, to a cusp in the electric and magnetic gluon susceptibilities as well as similar signatures in the ghost sector. We mention the possibility that such distinctive features of the transition could be measured in lattice simulations in the background field gauge studied here.

Figure 1

Diagrammatic representation of the ghost (dashed) and gluon (curly) propagators for momentum $K$ and color charge $\kappa$. The common orientation of the flow of momentum and color charge is arbitrary.

Figure 2

Diagrammatic representation of the cubic vertices.

Figure 3

Diagrammatic representation of the four-gluon vertex.

Figure 4

One-loop contribution to the ghost self-energy. Momenta and color charges are all either incoming or outgoing at the vertices.

Figure 5

One-loop diagrams for the gluon self-energy.

Figure 6

The physical background $r_{\min }(T)$ for the SU(2) theory as a function of the temperature, obtained from the minimization of the potential (15) at two-loop order (top panel). For $T<T_c$, the minimum sits at the ${\mathbb{Z}}_2$-symmetric point $r=\pi$. The symmetry is spontaneously broken for $T>T_c$ and the transition is continuous. The bottom plot shows the dimensionful background $r_{\min }(T)T\propto {\overline{A}}_{\min }(T)$.

Figure 7

The gauge-invariant Polyakov loop (13) at next-to-leading order as a function of the temperature with the parameters of Fig. 6.

Figure 8

Temperature dependence of the electric and magnetic inverse square masses in the neutral mode normalized to their common zero temperature value. To emphasize the effect of the Polyakov loop, we compare with the corresponding results at vanishing background field ($r=0$), which corresponds to the Landau gauge.

Figure 9

Same as Fig. 8 for the charged gluon modes, where the corresponding square masses are defined in Eqs. (65) and (66).

Figure 10

The one-loop electric and magnetic propagators in the neutral sector at vanishing frequency as functions of the spatial momentum $k$ for various temperatures.

Figure 11

The one-loop electric and magnetic propagators of the charged gluon at vanishing frequency as functions of the spatial momentum $k$ for various temperatures.

Figure 12

Normalized inverse of the zero-momentum gluon square mass in any of the charged modes (this differs from the charged susceptibilities defined above).

Figure 13

The ghost dressing function (93) at vanishing background field as a function of the momentum $k$ for various temperatures. As discussed in [51], for temperature-independent parameters, at sufficiently high temperature, the ghost dressing function develops a pole.

Figure 14

The dressing function (93) of the neutral ghost color mode as a function of the momentum $k$ for various temperatures. The pole of the vanishing-background case is absent.

Figure 15

The ghost propagator at vanishing frequency in the charged color sector as a function of momentum $k$ for various temperatures.

Figure 16

The (normalized) ghost propagator for charged color modes at vanishing frequency and zero momentum as a function of temperature.

Figure 17

The electric (top) and magnetic (bottom) square masses in the neutral sector as functions of $T/m$ compared to the corresponding asymptotic expressions (e26) and (e27) with $r\to r_{\infty }$ and $g=5$. One clearly observes the negative logarithmic behavior which drives the magnetic square mass to negative values at large $T/m$. The dashed curve illustrates how this artifact is cured by the running of the coupling: We plot the magnetic square mass computed with a running coupling put by hand as $1/g^2(T)=1/g^2+\frac{11}{24{\pi }^2}\ln (\frac{T^2+m^2}{{\mu }^2})$, with $m=0.75\mathrm{GeV}$, $g=5$, and $\mu =1\mathrm{GeV}$.