Effective potential for local composite operators and the problem of factorization of multi-field condensates

Abstract

A new formulation of effective potential for local composite operators is given. The two-fermion condensate\(\langle \bar \Psi \Psi \rangle \) and four-fermion condensate\(\langle \bar \Psi \Psi \bar \Psi \Psi \rangle \) are calculated simultaneously in the Gross-Neveu model up to next-to-the-leading order in 1/N expansion. It is shown that factorization\(\langle \bar \Psi \Psi \bar \Psi \Psi \rangle = C_1 \langle \bar \Psi \Psi \rangle ^2 \) holds only in theN→∞ limit and the non-factorized part of\(\langle \bar \Psi \Psi \bar \Psi \Psi \rangle \) contributed by the order-1/N terms is comparable to\(C_1 \langle \bar \Psi \Psi \rangle ^2 \) when takingN=3.