[Paper Review] Structure of quantum corrections in ${\cal N}=1$ supersymmetric gauge theories

K. V. Stepanyantz|arXiv (Cornell University)|Nov 25, 2017

Black Holes and Theoretical Physics4 references3 citations

TL;DR

This paper establishes that in ${\cal N}=1$ supersymmetric gauge theories regularized by higher covariant derivatives, the NSVZ beta-function relation holds universally in all perturbative orders. The derivation relies on the finiteness of three-point ghost-gauge vertices and a specific subtraction scheme—minimal subtraction of logarithms—that enforces the NSVZ scheme. The key result is the exact NSVZ relation between the beta-function and anomalous dimensions, confirmed by explicit three-loop calculations.

ABSTRACT

Some recent research of quantum corrections in ${\cal N}=1$ supersymmetric theories is briefly reviewed. The most attention is paid to the theories regularized by higher covariant derivatives. In particular, we discuss, how the NSVZ and NSVZ-like relations appear with this regularization and how one can construct the NSVZ scheme in all orders.

Motivation & Objective

• To establish the all-order validity of the NSVZ beta-function relation in ${\cal N}=1$ supersymmetric gauge theories.
• To resolve the scheme dependence of the NSVZ relation by identifying a unique subtraction scheme in higher derivative regularization.
• To demonstrate that the NSVZ scheme emerges naturally from minimal subtraction of logarithmic divergences in the higher derivative regularization framework.
• To confirm the NSVZ relation through explicit three-loop perturbative calculations using this regularization.

Proposed method

• Utilizes higher covariant derivative regularization, which preserves manifest ${\cal N}=1$ supersymmetry and is mathematically consistent.
• Introduces Pauli–Villars determinants to regularize one-loop divergences while removing higher-loop divergences via higher-derivative terms.
• Applies minimal subtraction of logarithmic divergences (logarithmic terms in $\ln\Lambda/\mu$) to define the renormalization constants.
• Imposes boundary conditions on renormalization constants to enforce the NSVZ scheme, ensuring the beta-function matches the exact NSVZ form.
• Uses the finiteness of three-point ghost-gauge vertices as a key ingredient to rewrite the NSVZ equation in a scheme-independent form.
• Performs explicit three-loop calculations to verify the NSVZ relation in the non-Abelian case, confirming agreement with the theoretical prediction.

Experimental results

Research questions

• RQ1 Does the NSVZ beta-function relation hold in all perturbative orders when using higher covariant derivative regularization in ${\cal N}=1$ supersymmetric gauge theories?
• RQ2 How can a consistent subtraction scheme be defined that realizes the NSVZ relation universally, independent of the renormalization scheme?
• RQ3 What role does the finiteness of three-point ghost-gauge vertices play in deriving the NSVZ relation in non-Abelian theories?
• RQ4 Can the NSVZ scheme be uniquely identified within higher derivative regularization through a minimal subtraction of logarithmic terms?
• RQ5 Do explicit three-loop calculations in this framework confirm the NSVZ relation for the beta-function in terms of the bare couplings?

Key findings

• The NSVZ beta-function relation is confirmed to hold in all loops for ${\cal N}=1$ supersymmetric gauge theories regularized by higher covariant derivatives.
• The NSVZ scheme is uniquely realized through minimal subtraction of logarithmic divergences, which corresponds to imposing specific boundary conditions on renormalization constants.
• The finiteness of three-point ghost-gauge vertices is essential for the derivation and allows the NSVZ equation to be rewritten in a scheme-independent form.
• Explicit three-loop calculations show perfect agreement with the NSVZ relation, confirming its validity beyond the two-loop level.
• The relation $\widetilde{\beta}(\alpha,\lambda)/\alpha^2 = -\frac{1}{2\pi}(3C_2 - T(R)) - \frac{1}{2\pi r}C(R)^i_j\widetilde{\gamma}_\phi^j_i$ holds in the NSVZ scheme, with no scheme-dependent corrections.
• The scheme dependence of the NSVZ relation is resolved by fixing the subtraction scheme via $g_1 = b_2 = -x_0$, ensuring $b_2 - g_1 = 0$ and restoring the NSVZ form.