# Scheme-Independent Calculations of Physical Quantities in an ${\cal N}=1$ Supersymmetric Gauge Theory

## TL;DR

This paper provides exact, scheme-independent calculations of anomalous dimensions and beta function derivatives at the infrared fixed point in an ${
m N}=1$ supersymmetric gauge theory, demonstrating monotonic behavior and rapid convergence.

## Contribution

It introduces scheme-independent expansions for anomalous dimensions and beta function derivatives at the IR fixed point in supersymmetric gauge theories, with exact results and convergence analysis.

## Key findings

- Anomalous dimensions increase monotonically with decreasing $N_f$.
- Scheme-independent expansions converge rapidly to exact values.
- Calculated derivatives of the beta function at IR fixed point up to third or fourth order.

## Abstract

We consider an asymptotically free, vectorial, ${\cal N}=1$ supersymmetric gauge theory with gauge group $G$ and $N_f$ pairs of chiral superfields in the respective representations ${\cal R}$ and $\bar {\cal R}$ of $G$, having an infrared fixed point (IRFP) of the renormalization group at $\alpha_{IR}$. We present exact results for the anomalous dimensions of various (gauge-invariant) composite chiral superfields $\gamma_{{\Phi}_{\rm prod}}$ at the IRFP and prove that these increase monotonically with decreasing $N_f$ in the non-Abelian Coulomb phase of the theory and that scheme-independent expansions for these anomalous dimensions as powers of an $N_f$-dependent variable, $\Delta_f$, exhibit monotonic and rapid convergence to the exact $\gamma_{{\Phi}_{\rm prod}}$ throughout this phase. We also present a scheme-independent calculation of the derivative of the beta function, $d\beta/d\alpha |_{\alpha=\alpha_{IR}}$, denoted $\beta'_{IR}$, up to $O(\Delta_f^3)$ for general $G$ and ${\cal R}$, and, for the case $G={\rm SU}(N_c)$, ${\cal R}=F$, we give an analysis of the properties of $\beta'_{IR}$ calculated to $O(\Delta_f^4)$.