Multicritical scalar theories with higher-derivative kinetic terms: A perturbative RG approach with the $\epsilon$-expansion

We employ perturbative renormalization group and $\epsilon$-expansion to study multicritical single-scalar field theories with higher derivative kinetic terms of the form $\varphi (-\square {)}^k\varphi$. We focus on those with a ${\mathbb{Z}}_2$-symmetric critical point which are characterized by an upper critical dimension $d_c=2nk/(n-1)$ 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 critical theory involves a marginal powerlike interaction ${\varphi }^{2n}$ and the deformations admit a derivative expansion that at leading order involves only the potential. In this case we present the beta functional of the potential and use this to calculate some anomalous dimensions and operator product expansion coefficients. These confirm some conformal field theory 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 marginal derivative operators at criticality. 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 a purely derivative interaction.

Figure 1

Phase diagrams of ${\mathbb{Z}}_2$ symmetric ${\square }^2$ theories for $\epsilon =0.1$. Left panel: for $n=2$ one has the Gaussian and the generalized Wilson-Fisher fixed point with ${\varphi }^4$ interaction. Right panel: for $n=3$ one has a Gaussian fixed point, a fixed point with a derivative interaction ${\varphi }^2(\partial \varphi {)}^2$, and two fixed points with a mixture of ${\varphi }^2(\partial \varphi {)}^2$ and ${\varphi }^6$ interactions. Figures qualitatively represent phase diagram for all $n$.