Covariant variational approach to Yang-Mills theory at finite temperatures

We extend the covariant variational approach for $SU(N)$ Yang-Mills theory in Landau gauge to nonzero temperatures. The renormalization of the zero-temperature case is revisited and it is shown that the same counterterms are sufficient to render the low-order Green’s function finite at nonzero temperature. We compute the ghost and gluon propagator numerically and show that it agrees in all qualitative respects with the results of high-precision lattice calculations.

Figure 1

The ghost form factor $\eta (p^2)$ (top) and the gluon propagator $D(p^2)$ (bottom) in Landau gauge at zero temperature. Lattice data is taken from Ref. [13], and the gluon mass renormalization parameter is $M_A\approx 0.48\mathrm{GeV}$.

Figure 2

The contribution ${\mathrm{\Delta }}_n(k)$ of the $n$th Matsubara mode to the gluon form factor $\eta (0,k)$ at external momentum $k$. The left panel shows data for $k=0.5\mu$ and various temperatures on a logarithmic plot, while the right panel shows the same for $k=10\mu$. The contributions are multiplied by $n^3$ to better show their asymptotics.

Figure 3

The contribution ${\mathrm{\Delta }}_n(k)$ of the $n$th Matsubara mode to the gluon form factor $\eta (0,k)$ at external momentum $k$. The left panel shows data for a large momentum $k=10\mu$ and two different temperatures on a linear plot, while the right panel displays data for a very low temperature $\beta =100/\mu$ at two different momenta $k$. Again, the contributions are multiplied by $n^3$ to better display their asymptotics.

Figure 4

The renormalized ghost form factor at various temperatures, as a function of the spatial momentum $k=|\mathbf{k}|$ with frequency $k_0=0$. Lattice data is taken from Ref. [13].

Figure 5

The renormalized gluon propagator at various temperatures, as a function of the spatial momentum $k=|\mathbf{k}|$ with frequency $k_0=0$. The left panel shows the transversal component $D_{\perp }(0,k)={\overline{\omega }}_{\perp }(0,k{)}^{-1}$, while the right panel shows the longitudinal component $D_{\parallel }(0,k)={\overline{\omega }}_{\parallel }(0,k{)}^{-1}$. Lattice data is taken from Ref. [13].