# Higher-Order Scheme-Independent Calculations of Physical Quantities in the Conformal Phase of a Gauge Theory

## TL;DR

This paper develops scheme-independent, higher-order calculations of key physical quantities at the infrared fixed point in certain gauge theories, improving accuracy and consistency with known results.

## Contribution

It introduces a scheme-independent expansion method for calculating anomalous dimensions and beta function derivatives at the IR fixed point, extending previous lower-order results.

## Key findings

- Calculated $eta'_{IR}$ to $O(\Delta_f^5)$
- Computed $\gamma_{ar\psi\psi,IR}$ to $O(\Delta_f^4)$
- Confirmed accuracy through supersymmetric QCD comparison

## Abstract

We consider an asymptotically free vectorial SU($N_c$) gauge theory with $N_f$ massless fermions in a representation $R$, having an infrared fixed point (IRFP) of the renormalization group at $\alpha_{IR}$ in the conformal non-Abelian Coulomb phase. The cases with $R$ equal to the fundamental, adjoint, and symmetric rank-2 tensor representation are considered. We present scheme-independent calculations of the anomalous dimension $\gamma_{\bar\psi\psi,IR}$ to $O(\Delta_f^4)$ and $\beta'_{IR}$ to $O(\Delta_f^5)$ at this IRFP, where $\Delta_f$ is an $N_f$-dependent expansion parameter. Comparisons are made with conventional $n$-loop calculations and lattice measurements. As a test of the accuracy of the $\Delta_f$ expansion, we calculate $\gamma_{\bar\psi\psi,IR}$ to $O(\Delta_f^3)$ in ${\cal N}=1$ SU($N_c$) supersymmetric quantum chromodynamics and find complete agreement, to this order, with the exactly known expression. The $\Delta_f$ expansion also avoids a problem in which an IRFP may not be manifest as an IR zero of a higher $n$-loop beta function.