Perturbative Beta Function of N=2 Super Yang-Mills Theories

TL;DR

The paper presents a purely algebraic, regularization-free proof that the perturbative $\beta$-function of $N=2$ Super Yang–Mills theory is one-loop exact. By employing a Witten twist, it relates the action to the gauge-invariant operator $\mathrm{Tr}\phi^{2}$ and shows this operator has zero anomalous dimension through scalar supersymmetry Ward identities. Using the Callan–Symanzik equation and a BRST–exact deformation, the authors derive the differential constraint $\frac{3}{g}\beta_g+\partial_g\beta_g=0$, implying $\beta_g = K g^{3}$ in an appropriate scheme. This work reinforces the nonrenormalization theorem for the $N=2$ beta function and connects perturbative finiteness to the topological structure of the twisted theory.

Abstract

An algebraic proof of the nonrenormalization theorem for the perturbative beta function of the coupling constant of N=2 Super Yang-Mills theory is provided. The proof relies on a fundamental relationship between the N=2 Yang-Mills action and the local gauge invariant polynomial Tr phi^2, phi(x) being the scalar field of the N=2 vector gauge multiplet. The nonrenormalization theorem for the beta function follows from the vanishing of the anomalous dimension of Tr phi^2.