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We employ perturbative RG and $\epsilon$-expansion to study multi-criticalsingle-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 criticalpoint 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 dependingon whether or not the numbers $k$ and $n-1$ are relatively prime. When theyare, the theory admits a local potential approximation. In this case we presentthe beta functional of the potential and use this to calculate some anomalousdimensions and OPE coefficients. These confirm some CFT data obtained usingconformal 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 structureinduced by the presence of derivative operators. We study the case $k=2$ withodd values of $n$, which fall in the second class, and calculate the functionalflows and spectrum. These theories have a phase diagram characterized atleading order in $\epsilon$ by four fixed points which apart from the GaussianUV fixed point include an IR fixed point with purely derivative interactions.

Multi-critical $\square^k$ scalar theories: A perturbative RG approach with $\epsilon$-expansion