Consistent Perturbative Fixed Point Calculations in QCD and SQCD

TL;DR

This paper presents a scheme-independent method to calculate the anomalous dimension at infrared fixed points in QCD and SQCD using finite order perturbation theory, demonstrating good agreement with known results and exploring the conformal window.

Contribution

It introduces a scheme-independent approach to compute the anomalous dimension at fixed points in asymptotically free theories using perturbation theory up to arbitrary order.

Findings

Perturbative calculations match exact results in supersymmetric theories.

The anomalous dimension is well approximated by few-loop calculations across the conformal window.

Small anomalous dimensions are found for a wide range of flavors in non-supersymmetric theories.

Abstract

We suggest how to consistently calculate the anomalous dimension $\gamma_*$ of the $\bar{\psi}\psi$ operator in finite order perturbation theory at an infrared fixed point for asymptotically free theories. If the $n+1$ loop beta function and $n$ loop anomalous dimension are known then $\gamma_*$ can be calculated exactly and fully scheme independently through $O(\Delta_f^n )$ where $\Delta_f = \bar{N_f} - N_f$ and $N_f$ is the number of flavors and $\bar{N}_f$ is the number of flavors above which asymptotic freedom is lost. For a supersymmetric theory the calculation preserves supersymmetry order by order in $\Delta_f$. We then compute $\gamma_*$ through $O(\Delta_f^2)$ for supersymmetric QCD in the $\overline{\text{DR}}$ scheme and find that it matches the exact known result. We find that $\gamma_*$ is astonishingly well described in perturbation theory already at the few loops level throughout the entire conformal window. We finally compute $\gamma_*$ through $O(\Delta_f^3)$ for QCD and a variety of other non-supersymmetric fermionic gauge theories. Small values of $\gamma_*$ are observed for a large range of flavors.