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Functional renormalization group flows as diffusive Hamilton-Jacobi-type equations

Adrian Koenigstein1, Martin J. Steil2, and Stefan Floerchinger1

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Phys. Rev. D 114, 036004 – Published 4 August, 2026

DOI: https://doi.org/10.1103/h75y-5t1k

Abstract

In order to find reliable and efficient numerical approximation schemes, we suggest to identify the functional renormalization group flow equations of one-particle irreducible two-point functions as Hamilton-Jacobi(-Bellman)-type partial differential equations. Based on this reformulation and reinterpretation we adopt a numerical scheme for the solution of field-dependent flow equations as nonlinear partial differential equations. We demonstrate this novel approach by first applying it to a simple fermion-boson system in zero spacetime dimensions—which itself presents as an interesting playground for method development. Afterwards, we show how the gained insights can be transferred to more interesting problems: one is the bosonic Z2-symmetric model in three Euclidean dimensions within a truncation that involves the field-dependent effective potential and field-dependent wave function renormalization. The other example is the (1+1)-dimensional Gross-Neveu model within a truncation that involves a field-dependent potential and a field-dependent fermion mass/Yukawa coupling at nonzero temperature, chemical potential, and finite fermion number.

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Phys. Rev. D 114, 036004– Published 4 August, 2026
  • Received 30 December 2025
  • Accepted 6 July 2026
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