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

Understand

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.