The Non-Perturbative Analytical Equation of State for SU(3) Gauge Theory

The effective potential approach for composite operators is generalized to non-zero temperature in order to derive the non-perturbative analytical equation of state for pure SU(3) Yang-Mills fields valid in the whole temperature range. Adjusting our parametrization of the gluon plasma pressure to the lattice pressure at high temperature for SU(3) Yang-Mills case, we have reproduced well our analytical curves and numbers not only for the pressure but for all other independent thermodynamic quantities as well in the whole temperature range $[0, \infty)$. We explicitly show that the pressure is a continuous function of the temperature across a phase transition at $T_c = 266.5 \MeV$. The entropy and energy densities have finite jump discontinuities at $T_c$ with latent heat $\epsilon_{LH}= 1.414$. This is a firm evidence of the first-order phase transition in SU(3) pure gluon plasma. The heat capacity has a $\delta$-type singularity (an essential discontinuity) at $T_c$, so that the velocity of sound squared becomes zero at this point. All the independent thermodynamic quantities are exponentially suppressed below $T_c$ and rather slowly approach their respective Stefan-Boltzmann limits at high temperatures. Those thermodynamic quantities which are the ratios of their independent counterparts such as conformity, conformality and the velocity of sound squared approach their Stefan-Boltzmann limit at high temperatures rather rapidly and demonstrate the non-trivial dependence on the temperature below $T_c$. We predict the existence of the three massive and the two massless excitations, all of non-perturbative dynamical origin. One of the massive excitations has an effective mass $1.17 \GeV$ and the two others have the same effective mass $0.585 \GeV$, but are propagating in different ways.