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

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.