The Hagedorn-type structure of the non-perturbative gluon pressure within the mass gap approach to QCD

We have shown in detail that the low-temperature expansion for the non-perturbative gluon pressure has the Hagedorn-type structure. Its exponential spectrum of all the effective gluonic excitations are expressed in terms of the mass gap. It is this which is responsible for the large-scale dynamical structure of the QCD ground state. The non-perturbative gluon pressure properly scaled has a maximum at some characteristic temperature $T=T_c=266.5\text{MeV}$, separating the low- and high temperature regions. It is exponentially suppressed in the $T\to 0$ limit. In the $T \rightarrow T_c$ limit it demonstrates an exponential rise in the number of dynamical degrees of freedom. Its exponential increase behavior with temperature is valid only up to $T_c$. This makes it possible to identify $T_c$ with the Hagedorn-type transition temperature $T_h$, i.e., to put $T_h=T_c$ within the mass gap approach to QCD at finite temperature. The non-perturbative gluon pressure has a complicated dependence on the mass gap and temperature near $T_c$ and up to approximately $(4-5)T_c$. In the limit of very high temperatures $T\to \mathrm{\infty }$ its polynomial character is confirmed, containing the terms proportional to $T^2$ and $T$, multiplied by the corresponding powers of the mass gap. \end{abstract}