Nonperturbative functional renormalization group for Higgs-singlet models with physics-informed neural networks

Abstract

We develop a nonperturbative functional renormalization group framework within the LPA' to solve the Wetterich flow equation for the $Z_2$-symmetric real singlet extension of the Standard Model at zero and finite temperature, without a polynomial expansion of the scale-dependent effective potential, using a physics-informed neural network (PINN) representation. In contrast to conventional truncations based on low-order field expansions, our hybrid tree-level--plus--neural-network ansatz yields a continuous, mesh-free description of the effective potential over the full relevant field and scale range. As a proof of concept, we apply the framework to the finite-temperature effective potential relevant to the electroweak phase transition in this model: one-dimensional field-space slices at two benchmark temperatures, and a two-dimensional reconstruction at $T=100$ GeV from which a two-step bounce action is extracted. The renormalization group flow is implemented in a multi-domain setup in scale and field space with derivative matching conditions that ensure smoothness and numerical stability. Gauge and Yukawa sectors are incorporated via independently computed perturbative two-loop running couplings, and anomalous dimensions are included in the flow at the LPA' level. We benchmark the network against resummed perturbation theory and against a grid-based relaxation solver of the same equation. Finally, we introduce a soft consistency constraint that keeps the solution close, in sign and order of magnitude, to perturbation theory across the field-space domain; we find it necessary, rather than merely helpful, for selecting a physically sensible solution of the flow equation, and find that the converged result retains a residual dependence on this guidance -- through hand-tuned weights and an analytic thermal target -- which we identify as the central open problem.