Multicritical hypercubic models

TL;DR

This paper investigates multicritical fixed points in hypercubic scalar field theories using the $ ext{epsilon}$-expansion, analyzing their fixed points, critical exponents, and large-$N$ behavior across various dimensions and interaction orders.

Contribution

It introduces a comprehensive analysis of hypercubic multicritical models, including explicit beta functions, fixed point structures, and the large-$N$ limit, extending previous studies to new dimensions and interaction types.

Findings

Identified families of multicritical hypercubic fixed points in various dimensions.

Derived explicit beta functions for three- and four-critical models.

Analyzed the large-$N$ limit and its relation to low-$N$ cases.

Abstract

We study renormalization group multicritical fixed points in the $\epsilon$-expansion of scalar field theories characterized by the symmetry of the (hyper)cubic point group $H_N$. After reviewing the algebra of $H_N$-invariant polynomials and arguing that there can be an entire family of multicritical (hyper)cubic solutions with $\phi^{2n}$ interactions in $d=\frac{2n}{n-1}-\epsilon$ dimensions, we use the general multicomponent beta functionals formalism to study the special cases $d = 3-\epsilon$ and $d =\frac{8}{3}-\epsilon$, deriving explicitly the beta functions describing the flow of three- and four-critical (hyper)cubic models. We perform a study of their fixed points, critical exponents and quadratic deformations for various values of $N$, including the limit $N=0$, that was reported in another paper in relation to the randomly diluted single-spin models, and an analysis of the large $N$ limit, which turns out to be particularly interesting since it depends on the specific multicriticality. We see that, in general, the continuation in $N$ of the random solutions is different from the continuation coming from large-$N$, and only the latter interpolates with the physically interesting cases of low-$N$ such as $N=3$. Finally, we also include an analysis of a theory with quintic interactions in $d =\frac{10}{3}-\epsilon$ and, for completeness, the NNLO computations in $d=4-\epsilon$.