Ward-Takahashi Identity in Denominator Regularization at One Loop

TL;DR

This work tests Denominator Regularization (DEN REG) as a gauge-preserving regularization scheme for quantum electrodynamics by establishing a precise correspondence with dimensional regularization and applying it to one-loop electron self-energy and vertex corrections. Through this correspondence, the authors fix the necessary coefficient functions and compute the UV pole and finite parts of the self-energy and vertex form factors, finding that the Dirac form factor $F_1$ contains a $1/\epsilon$ pole while the Pauli form factor $F_2$ remains finite in the $\epsilon\to 0$ limit. They explicitly verify the Ward-Takahashi identity on-shell, showing that $F_1(0)$ is tied to the derivative of the self-energy, thereby confirming gauge invariance is preserved within DEN REG at this order. The results support DEN REG as a viable, symmetry-preserving regularization approach with potential applicability to more complex processes and higher-loop calculations.

Abstract

Explicit analytic expressions for the electron self-energy and the vertex correction in quantum electrodynamics are derived at one loop using the recently proposed regularization scheme known as denominator regularization, assisted by its correspondence with dimensional regularization to determine the coefficient functions, which are a specific ingredient of this approach. We then show that the regularized amplitudes satisfy the Ward-Takahashi identity, thereby ensuring that gauge symmetry is preserved after regularization.

Figures (2)

• Figure 1: NLO corrections in QED: (a) self-energy loop modifying the electron propagator, (b) vertex loop modifying the electron-photon coupling.
• Figure 2: WTI diagram