[Paper Review] Higher derivative regularization and quantum corrections in N=1 supersymmetric theories

A. B. Pimenov, E. S. Shevtsova+2 more|ArXiv.org|Dec 11, 2007

Black Holes and Theoretical Physics4 citations

TL;DR

This paper demonstrates that in N=1 supersymmetric theories, all quantum corrections contributing to the Gell-Mann–Low function are integrals of total derivatives when using higher covariant derivative regularization. As a result, the exact NSVZ beta-function is derived directly from perturbation theory without scheme tuning, revealing a new identity for Green functions not implied by known symmetries.

ABSTRACT

We review some results of applying the higher covariant derivative regularization to the investigation of quantum corrections structure in N=1 supersymmetric theories. In particular, we demonstrate that all integrals, defining the Gell-Mann--Low function in supersymmetric theories, are integrals of total derivatives. As a consequence, there is an identity for Green functions, which does not follow from any known symmetry of the theory, in N=1 supersymmetric theories. We also discuss how to derive the exact $β$-function by methods of the perturbation theory.

Motivation & Objective

• To investigate quantum corrections in N=1 supersymmetric theories using higher covariant derivative regularization, which preserves supersymmetry.
• To determine whether the exact NSVZ beta-function can be derived directly from perturbation theory without tuning the renormalization scheme.
• To uncover new identities for Green functions that do not follow from known symmetries of the theory.
• To explore the implications of these identities for finite N=1 supersymmetric theories and potential underlying symmetries.

Proposed method

• Application of higher covariant derivative regularization to N=1 supersymmetric Yang-Mills theory and supersymmetric electrodynamics, preserving supersymmetry at the quantum level.
• Use of the background field method to compute quantum corrections in a manifestly supersymmetric way.
• Explicit calculation of Feynman diagrams up to four-loop order in Abelian and three-loop in non-Abelian cases using supergraph techniques.
• Derivation of the Gell-Mann–Low function by showing all relevant integrals are total derivatives, simplifying their evaluation.
• Solution of Schwinger–Dyson equations and Slavnov–Taylor identities to constrain the structure of quantum corrections.
• Verification of the new Green function identity in both Abelian and non-Abelian theories through explicit loop calculations.

Experimental results

Research questions

• RQ1 Can the exact NSVZ beta-function be derived directly from perturbation theory using higher covariant derivative regularization?
• RQ2 Do quantum corrections in N=1 supersymmetric theories always reduce to integrals of total derivatives under this regularization?
• RQ3 Is there a new identity for Green functions in N=1 supersymmetric theories that does not follow from any known symmetry?
• RQ4 Can the structure of the anomalous dimension in finite N=1 theories also be expressed as a total derivative?
• RQ5 What is the role of the higher derivative regularization in revealing nontrivial quantum identities in supersymmetric theories?

Key findings

• All integrals contributing to the Gell-Mann–Low function in N=1 supersymmetric theories are integrals of total derivatives under higher covariant derivative regularization.
• The Gell-Mann–Low function computed via this method directly yields the exact NSVZ beta-function without requiring scheme tuning.
• A new identity for Green functions is found that does not follow from any known symmetry of the theory, suggesting possible hidden invariances.
• The identity is verified explicitly in the four-loop approximation for the Abelian case and the three-loop approximation for the non-Abelian case.
• The renormalized action in this regularization is exhausted at one-loop order, indicating that divergences only appear at one-loop level.
• The result supports the idea that the Wilsonian effective action may be complete at one-loop in N=1 supersymmetric theories.