  We continue the study, initiated in arXiv:1404.1094, of the $O(N)$ symmetric
theory of $N+1$ massless scalar fields in $6-\epsilon$ dimensions. This theory
has cubic interaction terms $\frac{1}{2}g_1 \sigma (\phi^i)^2 + \frac{1}{6}g_2
\sigma^3$. We calculate the 3-loop beta functions for the two couplings and use
them to determine certain operator scaling dimensions at the IR stable fixed
point up to order $\epsilon^3$. We also use the beta functions to determine the
corrections to the critical value of $N$ below which there is no fixed point at
real couplings. The result suggests a very significant reduction in the
critical value as the dimension is decreased to $5$. We also study the theory
with $N=1$, which has a $Z_2$ symmetry under $\phi\rightarrow -\phi$. We show
that it possesses an IR stable fixed point at imaginary couplings which can be
reached by flow from a nearby fixed point describing a pair of $N=0$ theories.
We calculate certain operator scaling dimensions at the IR fixed point of the
$N=1$ theory and suggest that, upon continuation to two dimensions, it
describes a non-unitary conformal minimal model.
