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

Rodrigo Carmo Terin

TL;DR

This work tackles nonperturbative Dyson-Schwinger equations in Minkowski spacetime for QED by combining two complementary approaches: a dispersive solver based on Lehmann spectral representations and subtracted dispersion relations, and a physics-informed neural network (M-PINN) that learns the fermion mass function. Under identical truncation and renormalization (quenched, rainbow, Landau gauge), the authors demonstrate quantitative agreement between the dispersive solution and the M-PINN across infrared to ultraviolet scales in both on-shell and momentum-subtraction schemes, while delivering a differentiable and compact representation of the real-axis solution. The study highlights multi-scale loss design, timelike Fourier features, and monotonicity/smoothing constraints that stabilize learning and reproduce the expected analytic structure. These results lay the groundwork for extensions to dressed vertices, unquenching effects, and uncertainty-aware variants, potentially scaling to QCD and bound-state problems through a gauge-consistent, real-axis neural solver framework.

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.

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

TL;DR

This work tackles nonperturbative Dyson-Schwinger equations in Minkowski spacetime for QED by combining two complementary approaches: a dispersive solver based on Lehmann spectral representations and subtracted dispersion relations, and a physics-informed neural network (M-PINN) that learns the fermion mass function. Under identical truncation and renormalization (quenched, rainbow, Landau gauge), the authors demonstrate quantitative agreement between the dispersive solution and the M-PINN across infrared to ultraviolet scales in both on-shell and momentum-subtraction schemes, while delivering a differentiable and compact representation of the real-axis solution. The study highlights multi-scale loss design, timelike Fourier features, and monotonicity/smoothing constraints that stabilize learning and reproduce the expected analytic structure. These results lay the groundwork for extensions to dressed vertices, unquenching effects, and uncertainty-aware variants, potentially scaling to QCD and bound-state problems through a gauge-consistent, real-axis neural solver framework.

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 , 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.

Paper Structure

This paper contains 4 sections, 18 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Background and Spectral Formulation
  3. Traditional and PINN methods for Minkowskian space-time
  4. Conclusions and Outlook

Figures (2)

  • Figure 1: Comparison among PINNs (left) and the traditional dispersive Dyson--Schwinger solver (right) for the scalar dressing $B(p^2)$ in the spacelike regime. All curves use $\mu^2=m^2$ and the normalization $B(p^2)$.
  • Figure 2: Comparison between M-PINNs and the traditional dispersive Dyson--Schwinger solver for the real part of the mass function $B(p^2)$ in the timelike regime. All curves use $\mu^2=m^2$ and the normalization $B(p^2)$.