# Asymptotic freedom in certain $SO(N)$ and $SU(N)$ models

## TL;DR

This paper recalculates the beta functions for certain $SO(N)$ and $SU(N)$ gauge theories with scalars, refining previous results and analyzing the conditions for asymptotic freedom across different N values.

## Contribution

It provides corrected beta functions for $SO(N)$ and $SU(N)$ models and explores the N-dependent constraints for asymptotic freedom, including large N behavior.

## Key findings

- Corrected beta functions for $SO(N)$ and $SU(N)$ gauge theories.
- Identified larger minimum N values for asymptotic freedom than previous studies.
- Large N limit shows extensive regions of total asymptotic freedom.

## Abstract

We calculate the $\beta$-functions for $SO(N)$ and $SU(N)$ gauge theories coupled to adjoint and fundamental scalar representations, correcting long-standing, previous results. We explore the constraints on $N$ resulting from requiring asymptotic freedom for all couplings. When we take into account the actual allowed behavior of the gauge coupling, the minimum value of $N$ in both cases turns out to be larger than realized in earlier treatments. We also show that in the large $N$ limit, both models have large regions of parameter space corresponding to total asymptotic freedom.