evaluation of the phase transition temperature
Thus we assume that in the strong coupling regime, i.e. when the coupling constant , the quantum SU(3) gauge field forms a kind of correlated distribution of the fluctuating field in space. Such distributions form a discrete spectrum. The discreteness means that the energy spectrum of such distributions is discrete one. Note the following features of this point of view: the energy density of such distribution of quantum field has to tend fast enough to zero at infinity that the energy of the field in whole space would be finite. It means that such fields are finite in space. For example, it may be a glueball. In this case above mentioned discrete spectrum of energy means that there is a mass spectrum of the glueball. The lowest energy will be called a mass gap.
If bring the glueball in thermal contact with the thermostat then the glueball can be considered as a statistical system described by a certain temperature and the fluctuating energy.
As it is known the problem of phase transition in the gluon plasma is that the behavior of the plasma for small and large temperatures is essentially different. General expectation is that this difference is due to the fact that at low temperatures the quantum description of the gluon field should be carried out by a non-perturbative way while at high temperatures the description of quantum gluon field should be carried out by the perturbative way.
For statistical calculations at high temperatures we will apply the statistics of gas of interacting gluons. In the limit we have the partition function
Where is the energy of noninteracting gluons. After that by the standard way we obtain the Planck's formula for energy density (as for photons)
At small temperature the statistical sum looks as follows
Here is the energy of n-th quantum correlated state; is a mass gap. The calculation of is a problem of nonperturbative quantization of strongly nonlinear fields (in our case it is SU(3) non-Abelian gauge field). This problem is extremely difficult. Below we will give an approximate calculation of the bound state of glueball using a scalar approximation for 2nd and 4th Green's functions.
Thus at low temperatures we should use the expression (3) but for high temperatures the expression (1). The phenomenon of phase transition is that at some temperature the mean values of energy corresponding to statistical sums (1) and (3) are of the same order of magnitude. It means that there is a transition from the description of a quantum field on nonperturbative language to the description of a quantum field on perturbative language. It is essentially important to note that cannot be calculated using Feynman diagram technique.
Let us consider glueball being in thermal equilibrium with the thermostat. To describe the phenomenon of phase transition in this thermodynamic system we will consider two limiting cases. In the 1st case we have the glueball filled with weakly interacting gluons (perturbative case). The situation in this case is completely similar to statistics of the photons filling a certain box. With one exception: photons do not interact with each other. In the 2nd case the glueball can be in one of energy levels. The probability to be in one of is defined by standard methods of statistical mechanics.
The calculation of the mean value of energy density for weakly interacting gluon gas in the limit when the coupling constant could be made on the usual way (i.e. perturbative way) and gives us the standard formula for the total density of the Planck energy
The total energy of gluon gas in such glueball is
Where is the characteristic size of the glueball, and is the phase transition temperature. We assume that the characteristic size of the glueball coincides with the characteristic size of the nucleus .
The calculation of statistical mean value of energy for glueball filled with a strongly interacting gluon field gives us
The phase transition temperature separating perturbative physics from nonperturbative physics can be estimated as follows
Unfortunately we can not calculatebut since the energy spectrum
we can take the lower bound of this temperature as follows
where is the mass gap. Thus we obtain the lower bound