We now turn to the inequality (10). After the substitution of the values of all variables and taking and we obtain the following lower bound for the transition temperature

. (16)

At the LHC experiments is succeeded the temperature of. Comparing this experimental value with the value calculated in (16) shows that in calculating the average energy of (6) in the nonperturbative regime should either consider the energy with or the expression (7) to estimate the phase transition temperature is too rough.

Thus, we have qualitatively investigated how there is a phase transition in a gluon plasma due the temperature change. At low temperatures the partition function is the sum over the discrete energy states. These states are correlated quantum states describing a nonperturbatively quantized SU(3) non-Abelian gauge field. At high temperatures the partition function is determined by the gas of interacting gluons. The phase transition for SU(3) non-Abelian gauge field means that there is a transition from the description of quantum gluon field on the language of nonperturbative correlated quantum states to the description of this quantum field on the language of perturbative interacting gluons. Such phase transition is similar to the transition from a liquid to a gas. It is crucial that in the nonperturbative case the correlated quantum states are not a set of quanta. For the calculations we have considered a glueball that is in a thermal equilibrium with the thermostat. The lower bound of the phase transition temperature has been performed by comparing the average energy glueball (which is in thermal equilibrium with the thermostat) calculated in the perturbative and nonperturbative regimes: . The actual value of this temperature is above this lower bound approximately in two orders of magnitude. For a more accurate estimation of the phase transition temperature it is necessary: (a) to do a nonperturbative calculations of the next values of energy of correlated quantum states; (b) take into account the interaction between gluons. It is shown that this lower bound with the mass gap is connected.

We have considered a scalar model of glueball in which quantum fluctuations of SU(3) gauge field are considered in a nonperturbative way and are described by two scalar fields. The corresponding equations for these two scalar fields are considered as a nonlinear eigenvalue problem. The eigenvalues are the boundary conditions ,for these fields and the eigenfunctions are . It means that the regular solutions do not exist for other values of ,. Physically, the regular solution (solution with finite energy) describes the distribution of quantum fluctuations of the gluon field. The energy of such distribution of the quantum fluctuations of gluon field is a mass gap since the solutions with other boundary conditions does not exist. The resulting expression for the glueball energy is compared with conventional energy value of glueball . The comparison shows that the agreement is reached in the case that the coupling constant is in a nonperturbative regime . That is in excellent agreement with our statement that we are working in the nonperturbative region.

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