The Non-Perturbative Analytical Equation of State for $SU(3)$ Gluon Plasma

The effective potential approach for composite operators has been generalized to non-zero temperatures in order to derive the analytical equation of state for pure SU(3) Yang-Mills fields from first principles. In the absence of external sources this is nothing but the vacuum energy density. The key element of this derivation is the introduction of a temperature dependence into the expression for the bag constant evaluated in the previous publications. The non-perturbative part of the analytical equation of state does not depend on the coupling constant, but instead introduces a dependence on the mass gap. This is responsible for the large-scale structure of the QCD ground state. Its perturbative part does analytically depend on the fine-structure constant of strong interactions as well. As it follows from our equation of state, the two massive gluonic excitations with the effective masses m′eff = 1.17 GeV and meff = 0.585 GeV, as well as the different types of massless gluonic excitations, are present in the SU(3) gluon plasma. Important thermodynamic quantities such as the pressure, energy and entropy densities, etc., have been calculated. We show explicitly that the pressure may continuously change around Tc = 266.5 MeV in order to achieve its Stefan-Boltzmann limit at high temperatures. All other thermodynamic quantities change drastically at this point. The entropy and energy densities have jump discontinuities at Tc. This is a firm evidence of the first-order phase transition in SU(3) pure gluon plasma. Our value for the latent heat is ǫLH = 1.54 (in dimensionless units). The heat capacity has a δ-type singularity (an essential discontinuity) at Tc, so that the speed of light squared becomes zero at this point. The proposed NP analytical approach makes it possible to control for the first time the thermodynamics of the gluon plasma at low temperatures, below Tc. We have also calculated the gluon condensate, and hence the trace anomaly relation, as a function of temperature. Properly scaled they decrease not as 1/T 4 but as 1/T 2 at high temperatures due to the explicit presence of the mass gap in the equation of state. All our numerical results are in very good agreement with corresponding lattice data at T ≥ 2Tc. PACS numbers: 11.10.Wx, 12.38.Mh, 12.38.Lg, 12.38.Aw The Non-Perturbative Analytical Equation of State for SU(3) Gluon Plasma 2