Path integral evaluation of the one-loop effective potential in field theory of diffusion-limited reactions

TL;DR

This paper adapts quantum field theory techniques to classical diffusion-reaction models, enabling efficient calculation of effective potentials and equations of motion in two dimensions, simplifying analysis across various models.

Contribution

It introduces a path integral approach to compute the one-loop effective potential in classical diffusion-reaction field theories, offering a simpler alternative to diagram-based methods.

Findings

Effective potential calculations reduce to elementary functions in 2D.

Method applies broadly to different diffusion-controlled reaction models.

Renormalized equations of motion are derived for complex single-species theories.

Abstract

The well-established effective action and effective potential framework from the quantum field theory domain is adapted and successfully applied to classical field theories of the Doi and Peliti type for diffusion controlled reactions. Through a number of benchmark examples, we show that the direct calculation of the effective potential in fixed space dimension $d=2$ to one-loop order reduces to a small set of simple elementary functions, irrespective of the microscopic details of the specific model. Thus the technique, which allows one to obtain with little additional effort, the potentials for a wide variety of different models, represents an important alternative to the standard model dependent diagram-based calculations. The renormalized effective potential, effective equations of motion and the associated renormalization group equations are computed in $d=2$ spatial dimensions for a number of single species field theories of increasing complexity.