Soluble model of four-fermion interactions in de Sitter space.

We consider the theory of four-fermion interactions with N-component fermions in de Sitter space. It is found that the effective potential for a composite operator in the theory is calculable in the leading order of the 1/N expansion. The resulting effective potential is analyzed by varying both the four-fermion coupling constant and the curvature of the space-time. The critical curvature at which the dynamically generated fermion mass disappears is found to exist and is calculated analytically. The dynamical fermion mass is expressed as a function of the space-time curvature. e-mail : [email protected] Supported in part by Monbusho Grant-in-Aid for Scientific Research (C) under contract No.04640301. 1 According to the well-accepted scenario of early universe it is believed that the GUT phase is broken down to the QCD and electroweak phase through the symmetry breaking due to the Higgs mechanism. At this era the quantum gravity plays a minor role while the space-time curvature due to the external strong gravity is important in triggering the phase transition. Thus it is of interest to consider quantum field theory in curved space-time in connection with physics in the very early universe. On the other hand the Higgs mechanism is often explained as a dynamical effect due to the emergence of the composite Higgs field. A typical example of such a model is the technicolor model. In this regard it is interesting to deal with a quantum field theory with the composite Higgs field. The four-fermion interaction theory is one of the prototype models of the composite Higgs theory. Under these circumstances we find it useful to discuss the phase structure of the fourfermion theory in curved space-time. Since the phase transition is a nonperturbative phenomenon, we need to use the method free from the perturbative approach and also to avoid any approximation in dealing with the space-time curvature, e. g. a weak curvature approximation in which we rely on an expansion in powers of the curvature. In the present paper we adopt the 1/N expansion method as a nonperturvative approach and try to find the effective potential without making any approximation in the space-time curvature. For this purpose we restrict ourselves to the specific space-time, i. e. the de Sitter space-time, and calculate the effective potential for the composite operator made of a fermion-antifermion pair. By the use of the effective potential we shall argue the symmetry breaking caused by the curvature effect. We consider the theory in the curved space-time defined by the action, S = ∫