Главная Математика, химия, физика Phase transition for gluon field: a qualitative analysis
gap for glueball
Now we want to show how it is possible to calculate approximately mass gap for glueball. Shortly the idea is as follows (in details it is possible to familiarize with calculations in [6-7]). In the scalar model of glueball the Lagrangian of SU(3) non-Abelian gauge field is approximately represented in the form of Lagrangian of two scalar fields . One of these fields approximately describes the 2-point Green function of the gauge field and another scalar field, respectively, the 2-point Green function of the gauge field where is a coset. 4-point Green's function is a bilinear combination of 2-point Green's functions. Similar methods are used for the description of turbulent fluid flow (see, for example ). Thus these scalar fields describe the quantum fluctuations of the gluon field in the nonperturbative regime.
The Lagrangian for these fields is as follows
where g is the gauge coupling constant of SU(3) gauge field; and are constants. Corresponding field equations for the spherical symmetric case is given in Eq's (14)-(15) where the constant is chosen in such a way that at infinity the energy density would tend to zero. These equations are considered as a nonlinear eigenvalue problem for eigenfunctions and eigenvalue . In  a spherically symmetric solution describing the ball filled with quantum fluctuations of the gluon field is obtained. This solution is interpreted as a glueball.
Glueball energy is calculated as
Here are the dimensionless quantity; the dimension. The numerical calculation gives the following estimation . Expression (12) allows us to estimate the value of the dimensionless coupling constant that is required for this model scalar glueball if we know the glueball mass and its characteristic size. We will take (that is an expected value of the glueball) and characteristic dimensions of glueball of the same order as the size of a proton or neutron, namely . Then (12) gives us
That is in the excellent agreement with our statement that we are in the nonperturbative region.
Now we want to show that is indeed a mass gap. In the spherical symmetric case the field equations for scalar fields (that follows from the Lagrangian (11)) have the form
Here the prime is the differentiation with respect to x. The analysis in  shows that if we fix the parameters values and then the boundary conditions and are defined in a unique way so that the solution is regular (it means that it should has a finite energy). That is, the equations (14)-(15) can be considered as a nonlinear eigenvalue problem for the eigenvalues , and eigenfunctions . It means that our calculated value of the energy is the mass gap .