The Euclidean loop expansion for massive λΦ 4 4 : Through one loop

Abstract

As an application of the theory of solutions of the classical, Euclidean field equation, we prove the existence of solutions to the renormalized functional field equation, for the λΦ4 interaction in four Euclidean space dimensions, with non-negative λ and nonzero mass, through orderℏc. That is, we prove that the functional derivative of the connected generating functional is in the Schwartz space Reℒ(R 4), when evaluated at external sources in Reℒ, through orderℏc. We also prove the existence of all functional derivatives of the connected generating functional through the same order. All quantities of interest are analytic in the coupling constant at 0≦λ<∞, and continuous in the external source.