Nonperturbative solutions in a field theory withSp 6 symmetry

Abstract

AnSp6-invariant nonlinear spinor equation is introduced and discussed. The covariant Fock equations are considered by using nonperturbative methods based on appropriate decompositions of many-point amplitudes. In the first part of the paper an exactSp6 symmetry is assumed taking into account the possible spontaneous breakdown of theγ5-invariance. In order to avoid the divergence difficulties, some self-consistent boundary conditions are imposed to be satisfied by the two- and three-point amplitudes for vanishing relative co-ordinates. These self-consistent boundary conditions cannot be satisfied in a trivial way due to the well-known singularities of the relativistic interactions. A kind of eigenvalue problem is obtained for the couplings and masses. The results indicate that an acceptable solution of the problem is not excluded. This possibility is connected with the simultaneous occurence of the interactions generated by trion-antitrion pairs characterized by spin zero and one. The coupling generated by pseudoscalar pairs is a derivative one. For pseudoscalar trion-antitrion amplitudes the fulfilment of the self-consistent boundary conditions is discussed in detail in the framework of the nonperturbative methods introduced in this paper. The self-consistent boundary conditions can be satisfied if the bare coupling constantg0(11) characterizing the coupling of trions toSp6 singlet trion-antitrion pairs with spin one is fixed by (g0(11))2=12π2. The coupling of trions toSp6 singlet pseudoscalar pairs has to satisfy a simple self-consistent subsidiary condition. The other couplings of trions vanish. No conflict, with the general framework of relativistic quantum field theory appears; the usual finite cut-off is not introduced. Only the action principle is rejected. In the second part of the paper the possibility of the spontaneous medium-strong breakdown of theSp6-invariance is discussed in detail.