  We present a detailed version of our recent work on the renormalization group
approach to multicritical scalar theories with higher derivative kinetic term
of the form $\phi(-\Box)^k\phi$ and upper critical dimension $d_c = 2nk/(n-1)$.
Depending on whether the numbers $k$ and $n$ have a common divisor two classes
of theories have been distinguished which show qualitatively different
features. For coprime $k$ and $n-1$ the theory admits a Wilson-Fisher type
fixed point with a marginal interaction $\phi^{2n}$. We derive in this case the
renormalization group equations of the potential at the functional level and
compute the scaling dimensions and some OPE coefficients, mostly at leading
order in $\epsilon$. While giving new results, the critical data we provide are
compared, when possible, and accord with a recent alternative approach using
the analytic structure of conformal blocks. Instead when $k$ and $n-1$ have a
common divisor we unveil a novel interacting structure at criticality. In this
case the phase diagram is more involved as other operators come into play at
the scale invariant point. $\Box^2$ theories with odd $n$, which fall in this
class, are analyzed in detail. Using the RG flows that are derived at quadratic
level in the couplings it is shown that a derivative interaction is unavoidable
at the critical point. In particular there is an infrared fixed point with a
pure derivative interaction at which we compute the scaling dimensions. For the
particular example of $\Box^2$ theory in $d_c=6$ we include some cubic
corrections to the flow of the potential which enable us to compute some OPE
coefficients as well.
