(ψ¯ψ)2Ninteraction in four dimensions: Nonlocal theory

Abstract

The Gross-Neveu version of the Thirring model is analyzed in four-dimensional space-time. In 2 + epsilon dimensions this model is nonrenormalizable by power counting; however, it is known that up to4-epsilon dimensions the 1/N summation produces the improved propagator of the collective psi-barpsi field so that the theory requires no more subtractions than the theory in two dimensions. In four dimensions the situation deteriorates: There are two induced couplings of arbitrary strength, and the elastic unitarity is violated because the collective propagator decreases too fast. We show that the arbitrary coupling constant can be determined from the requirement that the effective potential has a minimum for zero values of the classical fermion and collective fields. One more reason for inconsistency is that a tachyon pole comes about even if one expands around the minimum of the potential. It is argued that one can get rid of inconsistencies by allowing the theory to be nonlocal. We construct the nonlocal form factor of the collective field from the requirements of unitarity, microcausality, correct spectral properties, and several assumptions about regularity properties of the nonlocal form factor.