Spectral functions in Minkowski quantum electrodynamics from neural reconstruction: Benchmarking against dispersive Dyson--Schwinger integral equations

TL;DR

This paper introduces a neural network approach to directly solve Minkowski-space Dyson--Schwinger equations in quantum electrodynamics, achieving accurate results across energy scales and enabling future complex extensions.

Contribution

It presents a novel Minkowskian physics-informed neural network method combined with dispersive techniques to solve QED Dyson--Schwinger equations directly in Minkowski space.

Findings

Quantitative agreement with dispersive solutions across energy scales.

Reproduces solutions in both on-shell and momentum-subtraction schemes.

Maintains computational efficiency and differentiability.

Abstract

A Minkowskian physics-informed neural network approach (M--PINN) is formulated to solve the Dyson--Schwinger integral equations (DSE) of quantum electrodynamics (QED) directly in Minkowski spacetime. Our novel strategy merges two complementary approaches: (i) a dispersive solver based on Lehmann representations and subtracted dispersion relations, and (ii) a M--PINN that learns the fermion mass function $B(p^2)$, under the same truncation and renormalization configuration (quenched, rainbow, Landau gauge) with the loss integrating the DSE residual with multi--scale regularization, and monotonicity/smoothing penalties in the spacelike branch in the same way as in our previous work in Euclidean space. The benchmarks show quantitative agreement from the infrared (IR) to the ultraviolet (UV) scales in both on-shell and momentum-subtraction schemes. In this controlled setting, our M--PINN reproduces the dispersive solution whilst remaining computationally compact and differentiable, paving the way for extensions with realistic vertices, unquenching effects, and uncertainty-aware variants.