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Regularization Schemes and Higher Order Corrections

William B. Kilgore

TL;DR

This study systematically compares four regularization schemes (CDR, DRED, FDH, and HV variants) in a high-order QCD calculation. By computing vacuum polarization and related observables up to NNLO and partial N^3LO, it shows that CDR and DRED yield consistent, unitary results, with DRED requiring careful treatment of evanescent degrees of freedom but ultimately equivalent to CDR. In contrast, the FDH scheme fails to maintain unitarity beyond NLO in nonsupersymmetric theories, due to improper handling of evanescent states and couplings. The results provide a strong validation for preferring CDR or properly renormalized DRED in high-precision QCD calculations, while cautioning against unrevised FDH usage at higher orders.

Abstract

I apply commonly used regularization schemes to a multi-loop calculation to examine the properties of the schemes at higher orders. I find complete consistency between the conventional dimensional regularization scheme and dimensional reduction, but I find that the four dimensional helicity scheme produces incorrect results at next-to-next-to-leading order and singular results at next-to-next-to-next-to-leading order. It is not, therefore, a unitary regularization scheme.

Regularization Schemes and Higher Order Corrections

TL;DR

This study systematically compares four regularization schemes (CDR, DRED, FDH, and HV variants) in a high-order QCD calculation. By computing vacuum polarization and related observables up to NNLO and partial N^3LO, it shows that CDR and DRED yield consistent, unitary results, with DRED requiring careful treatment of evanescent degrees of freedom but ultimately equivalent to CDR. In contrast, the FDH scheme fails to maintain unitarity beyond NLO in nonsupersymmetric theories, due to improper handling of evanescent states and couplings. The results provide a strong validation for preferring CDR or properly renormalized DRED in high-precision QCD calculations, while cautioning against unrevised FDH usage at higher orders.

Abstract

I apply commonly used regularization schemes to a multi-loop calculation to examine the properties of the schemes at higher orders. I find complete consistency between the conventional dimensional regularization scheme and dimensional reduction, but I find that the four dimensional helicity scheme produces incorrect results at next-to-next-to-leading order and singular results at next-to-next-to-next-to-leading order. It is not, therefore, a unitary regularization scheme.

Paper Structure

This paper contains 23 sections, 62 equations, 3 figures.

Table of Contents

  1. Introduction
  2. The Test Environment
  3. Methods
  4. Notation
  5. Conventional Dimensional Regularization
  6. Renormalization
  7. Vacuum polarization in the CDR scheme
  8. Total Decay rate and annihilation cross section in the CDR scheme
  9. Dimensional Reduction
  10. Renormalization
  11. Vacuum polarization in the DRED scheme
  12. Total Decay rate and annihilation cross section in the DRED scheme
  13. The Four-Dimensional Helicity Scheme
  14. Renormalization
  15. Vacuum polarization in the FDH scheme
  16. ...and 8 more sections

Figures (3)

  • Figure 1: Sample diagrams of one-, two- and three-loop contributions to the vacuum polarization of $V$.
  • Figure 2: Master integrals for the evaluation of vacuum polarization at a) one loop, b) two loops and c) three loops.
  • Figure 3: Four-loop diagrams that contribute to the $N_f^2$ term at N${}^3$LO.