Nonperturbative quantization ‘a la Heisenberg for non-Abelian gauge theories: two-equation approximation. Applications: a flux tube solution and a scalar model of glueball

The nonperturbative quantization technique à la Heisenberg is applied for the SU(3) gauge theory. The operator Yang-Mills equation and corresponding infinite set of equations for all Green’s functions are considered. Gauge degrees of freedom are splitted into two groups: (1) Aμa ∈ SU (2) × U(1) ⊂ SU(3); (2) coset degrees of freedom SU(3)/SU(2) × U(1). Using some assumptions about 2- and 4-point Green’s functions, the infinite set of equations is truncated to two equations. The first equation is the SU(2) × U(1) Yang-Mills equation, and the second equation describes a gluon condensate formed by coset fields. A flux tube solution describing longitudinal color electric fields stretched between quark and antiquark located at the ± infinities is obtained. It is shown that the dual Meissner effect appears in this solution: the electric field is pushed out from the gluon condensate