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Auxiliary Field Loop Expansion for the Effective Action for Stochastic Partial Differential Equations I

Using a path integral formulation for correlation functions of stochastic partial differential equations based on the Onsager-Machlup approach, we show how, by introducing a composite auxiliary field one can generate an auxiliary field loop expansion for the correlation functions which is similar to the one used in the $1/N$ expansion for an $O(N)$ scalar quantum field theory. We apply this formalism to the Kardar Parisi Zhang (KPZ) equation, and introduce the composite field $σ= \fracλ{2} \nabla ϕ\cdot \nabla ϕ$ by inserting a representation of the unit operator into the path integral which enforces this constraint. In leading order we obtain a self-consistent mean field approximation for the effective action similar to that used for the Bardeen-Cooper-Schrieffer (BCS) and Bose-Einstein Condensate (BEC) theories of dilute Fermi and Bose gases. This approximation, though related to a self-consistent Gaussian approximation, preserves all symmetries and broken symmetries. We derive the leading order in the auxiliary field (LOAF) effective potential and compare our results to the one loop in the fluctuation strength ${\cal A}$ approximation. We find, contrary to what is found in the one loop and self-consistent Gaussian approximation schemes that in the LOAF approximation there is no fluctuation induced symmetry breaking as a function of the coupling constant in any dimension $d$.