Uncovering novel phase structures in $\Box^k$ scalar theories with the renormalization group

TL;DR

This paper explores the phase structures of higher derivative scalar theories using the renormalization group, revealing new fixed points and critical behaviors depending on the divisibility of parameters.

Contribution

It classifies scalar theories based on divisibility of parameters and derives their RG equations, providing new critical data and uncovering novel phase structures.

Findings

Existence of Wilson-Fisher type fixed points for coprime parameters.

Discovery of a new interacting structure when parameters share a common divisor.

Identification of an infrared fixed point with a pure derivative interaction in $ox^2$ theories.

Abstract

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.