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Analytical derivation and numerical calculation of the $α_s$-order correction to the non-perturbative equation of state for SU(3) gluon matter

In our previous works the effective potential approach for composite operators has been generalized to non-zero temperature in order to derive the analytical equation of state for pure SU(3) Yang-Mills fields without quark degrees of freedom. 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. 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 dynamical structure of the QCD ground state. The perturbative part of the analytical equation of state does depend on the QCD fine-structure coupling constant $α_s$. Here we develop the analytical formalism, incorporating the perturbative part in a self-consistent way. It makes it possible to calculate the PT contributions to the equation of state in terms of the convergent series in integer powers of a small $α_s$. We also explicitly derive and numerically calculate the first perturbative contribution of the $α_s$-order to the non-perturbative part of the equation of state derived and calculated previously. The analytical equation of state or, equivalently, the gluon pressure is exponentially suppressed at low temperatures, while at temperature $T=T_c = 266.5 \ \MeV$ it has a maximum, if divided by $T^4/3$. It demonstrates a highly non-trivial dependence on the mass gap and the temperature near to $T_c$ and up to approximately $(3-4)T_c$. At very high temperatures its polynomial character is confirmed, containing the terms proportional to $T^2$ and $T$ with a non-analytical dependence on the mass gap.