  We employ perturbative RG and $\epsilon$-expansion to study multi-critical
single-scalar field theories with higher derivative kinetic terms of the form
$\phi(-\Box)^k\phi$. We focus on those with a $\mathbb{Z}_2$-symmetric critical
point which are characterized by an upper critical dimension $d_c=2 n k/(n-1)$
accumulating at even integers. We distinguish two types of theories depending
on whether or not the numbers $k$ and $n-1$ are relatively prime. When they
are, the theory admits a local potential approximation. In this case we present
the beta functional of the potential and use this to calculate some anomalous
dimensions and OPE coefficients. These confirm some CFT data obtained using
conformal block techniques, while giving new results. In the second case where
$k$ and $n-1$ have a common divisor, the theories show a much richer structure
induced by the presence of derivative operators. We study the case $k=2$ with
odd values of $n$, which fall in the second class, and calculate the functional
flows and spectrum. These theories have a phase diagram characterized at
leading order in $\epsilon$ by four fixed points which apart from the Gaussian
UV fixed point include an IR fixed point with purely derivative interactions.
