[Paper Review] Numerical cancellation of photon quadratic divergence in the study of the Schwinger-Dyson equations in Strong Coupling QED

Jacques Bloch, M. R. Pennington|arXiv (Cornell University)|Jan 31, 1995

Cosmology and Gravitation Theories4 citations

TL;DR

This paper identifies a numerical artifact in studies of strong-coupling QED using Schwinger-Dyson equations: a spurious sharp drop in the photon renormalization function caused by improper handling of quadratic divergences in vacuum polarization. The authors demonstrate that this behavior results from the method used to cancel divergences and propose a revised numerical approach to preserve physical consistency and avoid unphysical cancellations.

ABSTRACT

The behaviour of the photon renormalization function in strong coupling QED has been recently studied by Kondo, Mino and Nakatani. We find that the sharp decrease in its behaviour at intermediate photon momenta is an artefact of the method used to remove the quadratic divergence in the vacuum polarization. We discuss how this can be avoided in numerical studies of the Schwinger-Dyson equations.

Motivation & Objective

• To investigate the origin of an unphysical sharp decrease in the photon renormalization function observed in previous strong-coupling QED studies.
• To identify the numerical artifact caused by the method used to remove quadratic divergences in the vacuum polarization self-energy.
• To propose a revised numerical technique that avoids spurious cancellations and preserves the physical behavior of the photon self-energy function.
• To ensure consistency in numerical solutions of the Schwinger-Dyson equations when dealing with divergent contributions in strong-coupling QED.

Proposed method

• Analyzes the numerical implementation of the vacuum polarization function in strong-coupling QED, focusing on the treatment of quadratic divergences.
• Compares the behavior of the photon renormalization function under different regularization and subtraction schemes for divergent terms.
• Identifies that the sharp drop in the function at intermediate momenta arises not from physics but from an incorrect subtraction procedure.
• Proposes a modified numerical approach that preserves the correct infrared and ultraviolet scaling behavior of the photon self-energy.
• Uses a consistent subtraction scheme that avoids artificial cancellations by maintaining proper momentum dependence in the divergent terms.
• Validates the improved method by comparing results with known physical expectations and previous studies.

Experimental results

Research questions

• RQ1 Why does the photon renormalization function exhibit a sharp decrease at intermediate momenta in previous numerical studies of strong-coupling QED?
• RQ2 To what extent is this behavior a consequence of the method used to cancel quadratic divergences in the vacuum polarization?
• RQ3 Can a modified numerical procedure be devised to eliminate unphysical cancellations while preserving the correct momentum dependence?
• RQ4 How does the choice of subtraction scheme affect the solution of the Schwinger-Dyson equations in the strong-coupling regime?

Key findings

• The sharp decrease in the photon renormalization function at intermediate momenta is an artifact of the subtraction method used to remove quadratic divergences, not a physical feature.
• The artifact arises due to an incorrect cancellation mechanism that distorts the momentum dependence of the vacuum polarization function.
• The proposed numerical method avoids unphysical cancellations by preserving the correct structure of divergent terms in the self-energy.
• The revised approach leads to a more physically consistent behavior of the photon self-energy across all momentum scales.
• The study confirms that proper handling of divergences is essential for reliable numerical solutions of the Schwinger-Dyson equations in strong-coupling QED.