Covariant variational approach to Yang-Mills theory: Thermodynamics

The thermodynamics of $SU(2)$ Yang-Mills theory in the covariant variational approach is studied by relating the free action density in the background of a nontrivial Polyakov loop to the pressure of the gluon plasma. The Poisson resummed expression for the pressure can be evaluated analytically in limiting cases, and shows the correct Stefan-Boltzmann limit at $T\to \infty$, while the limit $T\to 0$ is afflicted by artifacts due to massless modes in the confined phase. Several remedies to remove these artifacts are discussed. Using the numerical $T=0$ solutions for the ghost and gluon propagators in the covariant variational approach as input, the pressure, energy density and interaction strength are calculated and compared to lattice data.

Figure 1

The zero-temperature gluon propagator (left) and the ghost form factor (right) for the decoupling type of solution, compared to high-precision lattice data [25].

Figure 2

Left: The effective potential $V_{\mathrm{eff}}(x)$ for the Polyakov loop on the $SU(2)$ Weyl alcove $x\in [0,1]$ defined in Eq. (22). Right: The Polyakov loop as a function of temperature, extracted from the minimum of $V_{\mathrm{eff}}$.

Figure 3

Left: The transversal gluon form factor $k^2/\omega (k)$ of our variational solution, compared to the best fit to the form factor of a massive propagator. Right: The integrand $[{\varphi }^{\prime }(k)+k{\varphi }^{\prime \prime }(k)]$ of the Hankel transform Eq. (30) without the oscillating Bessel functions, again compared to the best massive fit to the form factor.

Figure 4

Left: The full integrand for the Hankel transform in Eq. (30) for $\lambda =100$. Right: The Hankel transform $h(\lambda )$ from Eq. (30) for the transversal modes in our variational solution, compared to the best massive fit, for which $h(\lambda )$ can be calculated analytically.

Figure 5

Left: The free energy $u(x,\beta )={\beta }^4{f}_{\beta }(x)$ from Eq. (28) as a function of the Polyakov loop background for various temperatures. The asymptotic limits at $T\to 0$ and $T\to \infty$ are taken from Eqs. (33) and (34), respectively.

Figure 6

Left: The dimensionless pressure $p(T)/T^4$ of the variational solution, normalized to the Stefan-Boltzmann limit. In the left panel, our solution is compared to $SU(2)$ lattice data from Ref. [31], while the right panel compares with a model of free massive bosons.

Figure 7

The energy density $\epsilon (T)/T^4$ (left) and the interaction strength $\mathrm{\Delta }$ (right) of our variational approach, compared to $SU(2)$ lattice data from Ref. [31].

Figure 8

Left: The dimensionless pressure $p(T)/T^4$ normalized to the Stefan-Boltzmann limit, compared to a model in which the transversal gluon form factor has been replaced by a mass fit, but longitudinal gluons, ghosts and curvature are fully retained. Right: The pressure $P(T)/T^4$ with manual removal of (spurios) massless modes, compared to lattice data [31].