# RG Flows and Fixed Points of $O(N)^{r}$ Models

###### Abstract

By means of $\epsilon$ and large $N$ expansions, we study generalizations of
the $O(N)$ model where the fundamental fields are tensors of rank $r$ rather
than vectors, and where the global symmetry (up to additional discrete
symmetries and quotients) is $O(N)^{r}$, focusing on the cases $r\leq 5$.
Owing to the distinct ways of performing index contractions, these theories
contain multiple quartic operators, which mix under the RG flow. At all large
$N$ fixed points, melonic operators are absent and the leading Feynman
diagrams are bubble diagrams, so that all perturbative fixed points can be
readily matched to full large $N$ solutions obtained from Hubbard-Stratonovich
transformations. The family of fixed points we uncover extend to arbitrary
higher values of $r$, and as their number grows superexponentially with $r$,
these theories offer a vast generalization of the critical $O(N)$ model.

We also study sextic $O(N)^{r}$ theories, whose large $N$ limits are obscured
by the fact that the dominant Feynman diagrams are not restricted to melonic
or bubble diagrams. For these theories the large $N$ dynamics differ
qualitatively across different values of $r$, and we demonstrate that the RG
flows possess a numerous and diverse set of perturbative fixed points
beginning at rank four.

###### Contents

  1. 1 Introduction
    1. 1.1 Large $N$ limits
    2. 1.2 Small $\epsilon$ expansions
    3. 1.3 Overview of paper
  2. 2 Tensorial $\phi^{4}$ Theory
    1. 2.1 The $O(N)^{2}$ model
    2. 2.2 The $O(N)^{3}$ model
    3. 2.3 The $O(N)^{4}$ model
    4. 2.4 The $O(N)^{5}$ model
    5. 2.5 Large $N$ limit at general tensor rank
  3. 3 Tensorial $\phi^{6}$ Theory
    1. 3.1 The $O(N)^{2}$ model
    2. 3.2 The $O(N)^{3}$ model
    3. 3.3 The $O(N)^{4}\rtimes S_{4}$ model
    4. 3.4 Overview and discussion
  4. 4 Outlook
  5. A Optimal Scaling
    1. A.1 Matricial optimal scaling
    2. A.2 Three-tensor optimal scaling
    3. A.3 Optimal scaling at higher ranks
    4. A.4 Other scalings
    5. A.5 $U(N)^{5}$ theory with multiple optimal scalings
  6. B Quartic Large $N$ $O(N)^{r}$ Fixed Points at Higher Rank
    1. B.1 Types of operators
    2. B.2 Absence of melonic fixed points
    3. B.3 Invariance of the vectorial RG flow
    4. B.4 Diagonalizing the vectorial RG flow
    5. B.5 Large $N$ field content
  7. C Locations of Fixed Points in Sextic $O(N)^{4}\rtimes S_{4}$ theory
  8. D Gradient Flow and Linear Relations

##  1 Introduction

In the absence of supersymmetry, and setting aside low-dimensional toy models,
the means by which to gain an analytic handle on the observables of
interacting quantum field theories are preciously rare. The current methods
available to tackle such theories are well-illustrated by the family of
conformal field theories known as the critical $O(N)$ model, whose
experimental motivation derives from their utility in describing critical
phenomena in liquids, gases, and magnets, and which can be identified with the
non-trivial RG fixed point of a scalar QFT of vector fields $\phi^{a}$,
$a\in\\{1,2,...,N\\}$, subject to a quartic interaction,

$\displaystyle S=\int
d^{d}x\bigg{(}\frac{1}{2}(\partial_{i}\phi^{a})(\partial_{i}\phi^{a})+\frac{1}{4!}g\big{(}\phi^{a}\phi^{a}\big{)}^{2}\bigg{)}\,.$
(1)

Formally, the theories can be defined in a continuous range of spacetime
dimensions $2<d<4$ and admit perturbative treatments in the $\epsilon$
expansion in $d=4-\epsilon$ as well as in $d=2+\epsilon$ dimensions, in
addition to being analytically tractable order by order in $1/N$ across the
range of dimensions.111 In dimensions above four, the $O(N)$ model possesses a
UV stable fixed point at large $N$. It was pointed out by Parisi [1, 2] that
the operator $\phi^{i}\phi^{i}$ violates the unitarity bound for $d>4$ but
that the UV fixed point may be unitary between four and six dimension. More
recently, Ref. [3] matched the large $N$ UV fixed point to a $6-\epsilon$
expansion. Outside these limits, theoretical knowledge of this family of
theories derives from numerical studies using Monte Carlo methods and the
conformal bootstrap.222See Ref. [4] for a comprehensive review of the critical
$O(N)$ model.

Since their pioneering usage in the context of the $O(N)$ model, the small
$\epsilon$ and large $N$ expansions have been fruitfully brought to bear on a
host of other quantum theories, and in the present paper we shall apply these
methods to the generalizations of the $O(N)$ model obtained by advancing from
a single-index scalar $\phi^{a}$ to multi-index scalars $\phi^{ab}$,
$\phi^{abc}$, etc., where we take each index to transform under its own $O(N)$
group. These theories can be thought of as tensorial generalizations of
$\phi^{4}$ theory in the same sense as the $O(N)$ model can be viewed as the
vectorial generalization. While the $O(N)$ model at fixed $N$ has but a single
non-trivial fixed point in the $\epsilon$ expansion, the $O(N)^{r}$ models
generally have multiple fixed points whose number depends on $N$. But as we
let $N$ increase, the number of perturbative fixed points at fixed $r$
eventually stabilizes to a number whose value exhibits double exponential
growth with the tensor rank $r$. For the theories with $r\leq 5$ we will see
explicitly how all of the fixed points, exact in $N$ but perturbative in
$\epsilon$, can be matched to large $N$ solutions that are exact in $d$ but
perturbative in $1/N$. In the strict large $N$ limit, the fixed points all
look similar to the free or the interacting large $N$ fixed point of the
$O(N)$ model, but their $1/N$ corrections are vastly different and suggest
that the $O(N)^{r}$ models furnish a large set of distinct three-dimensional
CFTs.

For the sake of explicitness, let us take a look at the $O(N)^{3}$ model. The
possible types of index contractions among quartic operators in this theory
can be illustrated by the three diagrams shown below.

$\displaystyle\begin{matrix}\text{ \includegraphics[scale={1}]{threedia.pdf}
}\end{matrix}$ (2)

We can refer to the three types of interactions as “double-trace”,“pillow”,
and “tetrahedron” respectively. If we do not symmetrize with respect to the
three $O(N)$ groups of the model, there are actually three different pillow
operators, so that the RG flow, after tuning away the quadratic operator, is
five-dimensional. When $N\geq 29$, the model has 16 real fixed points in the
$\epsilon$ expansion. On scaling the coupling constants with appropriate
powers of $N$, one finds that the tetrahedron operator vanishes for all the 16
fixed points as $N$ is taken to infinity. Phrased differently: at large $N$,
fixed points with the tetrahedron present require the other coupling constants
to be complex. The 16 real fixed points, with only double-trace and pillow
operators present, can be matched to 16 large $N$ solutions obtained via
Hubbard-Stratonovich transformations. The double trace operator can be
integrated through the introduction of the standard $\sigma$ field as in the
$O(N)$ model, while the three pillow operators can be integrated out with
three fields $\chi^{i_{1}i_{2}}$ that each transforms as a traceless symmetric
matrix under one of the $O(N)$ groups of the model. The presence or absence of
the $\sigma$ field and the three $\chi$ fields results in the $2^{4}$ total
large $N$ solutions. The $O(N)^{4}$ and $O(N)^{5}$ models exhibit a similar,
simple type of behaviour at large $N$: at any fixed point, only such operators
are present as can be integrated out via the Hubbard-Stratonovich
transformation, so that leading anomalous dimensions can always be computed
exactly in $d$. To determine subleading $1/N$ corrections, however, it is
necessary to incorporate all the operators of the models, including, in the
rank three case, the tetrahedron. The type of fixed points just described can
be shown to exist in increasing numbers for the $O(N)^{r}$ models with $r>5$,
but other types of fixed points may also be present at higher even rank.

In addition to analyzing tensorial $\phi^{4}$ theory, we also present a study
of tensorial $\phi^{6}$ theory. In the vector case, ie. for sextic $O(N)$
theory, the beta functions become trivial in the strict large $N$ limit. A UV
fixed point does appear in three dimensions at sub-leading order in $1/N$ but
is situated in a regime where the theory is believed to be unstable at large
$N$ (although potentially not at finite $N$), and away from three dimensions
this large $N$ fixed point becomes complex. Tensorial $\phi^{6}$ theory,
however, exhibits a substantially different large $N$ limit from the vector
model, with non-trivial beta functions and fixed points present already to
leading order at large $N$. For tensors at at ranks two and three, a single
non-trivial large $N$ fixed point is present in the $3-\epsilon$ dimension,
but as we shall see, the number of fixed points increases drastically at rank
four.

In the following two subsections, we briefly review aspects of large $N$ and
small $\epsilon$ expansions relevant to this paper, before we proceed in
Subsection 1.3 to provide an overview of the paper and its results.

###  1.1 Large $N$ limits

Within the vast body of literature devoted to studying large $N$ physics,
there are three broad classes of known large $N$ limits among the quantum
theories with a single interaction term, each of which is dominated by its own
type of Feynman diagrams:333We refer the reader to Ref. [5] for a review and
comparison of the different types of large $N$ limits.

  * •

The vector model large $N$ limit, dominated by bubble diagrams

This limit, which is realized by the $O(N)$ model and also describes aspects
of polymer physics, is the most tractable of the three types of large $N$
limits. At large $N$ the only surviving Feynman diagrams in the $O(N)$ model
are bubble diagrams like the one shown below.

$\displaystyle\begin{matrix}\text{ \includegraphics[scale={1}]{bubbles.pdf}
}\end{matrix}$ (3)

The leading large $N$ approximation of the $O(N)$ model was obtain in Ref. [6]
through the use of a Hubbard-Stratonovich transformation that we discuss in
more detail later on (in 2.2.3). Higher-order corrections in $1/N$ were
subsequently computed in Refs. [7, 8, 9]. An important fact about the
vectorial large $N$ limit is that on renormalizing the Feynman diagram with a
single bubble, the diagrams with multiple bubbles are also rendered finite, in
consequence of which the large $N$ beta function is one-loop exact. Ref. [10]
provides a review of QFTs exhibiting the vector model large $N$ limit.

  * •

The matrix model large $N$ limit, dominated by planar diagrams.

This limit is possible in theories of tensors of rank two and higher. A simple
example of a theory that realizes the limit is the quartic matrix theory with
the following interaction:

$\displaystyle\begin{matrix}\text{ \includegraphics[scale={1}]{quartInt.pdf}
}\end{matrix}$ (4)

The Feynman diagrams that contribute to the four-point function at large $N$
are planar diagrams like the following:

$\displaystyle\begin{matrix}\text{ \includegraphics[scale={0.8}]{planar.pdf}
}\end{matrix}$ (5)

The set of planar diagrams is generally a large, unwieldy set, unlike the
subsets of planar diagrams that dominate in the other two types of large $N$
limits. For this reason, theories in the planar limit are usually much more
complicated than theories in the other types of limits, and only for a limited
number of low-dimensional examples of theories, such as $2d$ $SU(N)$ theory
coupled to quarks in the fundamental representation [11] or general zero-
dimensional QFTs [12], are large $N$ solutions known in the planar limit.
Conversely, however, matrix theories offer the benefit of describing an
extraordinarily rich range of phenomena, spanning QCD and quantum gravity.
Reviews of matrix models at large $N$ and their connections to 2D gravity can
be found in Refs. [13, 14].

  * •

The melonic large $N$ limit, dominated by so-called melon diagrams.

This limit was discovered and analyzed by Gurau and collaborators in Refs.
[15, 16, 17, 18, 19, 20] and can appear in theories of tensors of rank three
and higher. Foundational work on tensor models [21, 22] was motivated by
attempts to describe quantum gravity above two dimensions, to which continual
effort has been direct in the form of the tensor track [23], but other
physical applications have emerged later on. In particular, the melonic limit
provides a way to realize the Schwinger-Dyson equation of the SYK model [24,
25, 26] in genuine quantum mechanical models, rather than in disorder averages
of theories, as first shown by Witten [27] and also demonstrated for the
simple quantum mechanical theory put forward by Klebanov and Tarnopolsky
[28].444Although they share the same Schwinger-Dyson equation, there are also
important differences between the SYK and Klebanov-Tarnopolsky models. The
latter contains singlet operators not present in the former [29] as well as
additional modes in the fluctuations about the finite temperature saddle point
[30]. The simplest version of this theory contains a quartic fermionic
interaction:

$\displaystyle\begin{matrix}\text{ \includegraphics[scale={1}]{quartFerm.pdf}
}\end{matrix}$ (6)

To describe the class of diagrams that dominate at large $N$ one introduces
the melonic substitution of free propagator given by

$\displaystyle\begin{matrix}\text{ \includegraphics[scale={1}]{melSub.pdf}
}\end{matrix}$ (7)

In a simplified diagrammatic notation, the melonic substitution can be
expressed as

$\displaystyle\begin{matrix}\text{ \includegraphics[scale={1}]{melSubSim.pdf}
}\end{matrix}\hskip 34.1433pt$ (8)

The diagrams that contribute at large $N$, ie. the subset of planar diagrams
known as the melonic diagrams, are those generated by iteratively performing
the melonic substitution. In Figure 1 we display all melonic vacuum diagrams
up to eigth order in the coupling constant.

$\displaystyle\begin{matrix}\text{ \includegraphics[scale={0.9}]{melons.pdf}
}\end{matrix}$ Figure 1: Melonic vacuum diagrams to eigth order in the
coupling constant

The diagrams that contribute to the two-point function admit a re-summation
into a simple Schwinger-Dyson equation that can be solved in the IR, and the
melonic four-point diagrams posses a recursive structure in term of ladder
diagrams from which it is possible to derive the spectrum of bilinear
operators [28], while higher-point functions can be tackled using the
machinery developed in Ref. [31]. We emphasize that the interaction (6) is but
one example of a much wider class of interactions exhibiting a melonic pattern
of diagrams generated by iteratively replacing a free propagator with a
propagator containing two interaction vertices, and these interactions need be
neither fermionic nor quartic, see for example Refs. [32, 33, 34, 35] for
studies of higher-rank interactions. For more details on the melonic limit, we
refer the reader to Ref. [36] for a review of melonic tensor models, Ref. [37]
for a review of melonic tensor field theories in $d\geq 2$, and Ref. [38] for
a collection of reviews of random tensor models.555Recent work on melonic
theories include studies of spontaneous symmetry breaking in tensor models
[39, 40, 41, 42, 43], phase diagrams of melonic quantum mechanics [44, 45],
melonic theories defined over $p$-adic numbers [46, 33], supersymmetric tensor
models [47, 48, 49], Hagedorn temperature in fermionic tensor quantum
mechanics [50], and surface defects in tensor models [51].

While the above three limits have each been studied thoroughly, theories with
multiple competing interaction terms of different type are generally much less
understood than purely vectorial, matricial, or melonic theories. But this is
precisely the situation we face when we advance from the $O(N)$ theory of a
single-index scalar to theories of multi-index scalars: interactions of every
type emerge, arising from the distinct ways of contracting indices amongst the
fundamental fields, and renormalized perturbation theory dictates that we omit
no relevant operator compatible with the symmetry we choose to consider. While
some interactions generate more powers of $N$ than others, a full exploration
of the dynamics of a theory with multiple interactions requires that coupling
constants be scaled down by suitable powers of $N$ such that all operators
contribute on an equal footing. Theories with couplings scaled in this manner
are sometimes referred to and studied in the literature under the name of
_enhanced tensor theories_ [52, 53]666See Ref. [54] for recent work on quartic
theories of this kind., with the different scalings of couplings representing
the tensorial generalization of the familiar matrix theory fact that coupling
constants are scaled according to the number of traces in the associated
operators. For rank-three tensor models with $O(N)^{3}$ symmetry Ref. [55]
worked out the suitable scalings of the coupling constants and established the
large $N$ expansion, but in general no algorithm exists for determining large
$N$ scalings. Abstractly considered, a theory allows for a set of permitted
ranges of scalings that do not lead to a blow up of the theory as
$N\rightarrow\infty$, and among these permitted choices Ref. [34] introduced
the term _optimal scaling_ to denote the smallest set of scaling exponents
compatible with a well-defined $1/N$ expansion. For the specific multi-index
generalizations of the $O(N)$ model we study in this paper, it is not
difficult to determine the optimal scalings. We are then left with the
question, what are the large $N$ RG flows and fixed points under these
scalings? By invoking the aid of the $\epsilon$ expansion, we will argue that
the large $N$ landscape can be charted out with reasonable confidence.

###  1.2 Small $\epsilon$ expansions

The success of Wilson’s and Fisher’s outlandish idea [56] to use the number of
spacetime dimensions below four as an expansion parameter in $\phi^{4}$ theory
hinges on the fortuitous circumstance that the initial coefficients of the
expansions are sufficiently small and decreasing that even an expansion
parameter of unit size provides a reasonable approximation to critical
exponents in three dimensions. The idea has since proven practicable widely
beyond $\phi^{4}$ theory and spurred a slew of work charting out fixed points
in the $\epsilon$ expansion.777See Refs. [57] and [58] for careful reviews of
fixed points in the $4-\epsilon$ expansion, and Ref. [59] for early
foundational work in this direction. In the $4-\epsilon$ expansion, the fixed
points of four-component [60] and five-component [61] scalar theories have now
been completely classified, while scalars with six [62, 63, 64] and seven
components [65] have been thoroughly if not exhaustively explored, in addition
to which several infinite families are known. One of these families consists
of the fixed points of the $O(N)$ model itself, while another is the set of
bifundamental fixed points of $N_{1}\times N_{2}$ matrix fields $\phi^{ab}$
[66, 67, 68]. The type of fixed points relevant to the present paper are
further extensions of these two families into multifundamental fixed points of
tensor fields of rank $r$, where we limit the enormous parameter space by
taking the dimensions of all the indices to equal the same number:
$N_{1}=N_{2}=...=N_{r}\equiv N$.888There are no fixed points in the
bifundamental family with $N_{1}=N_{2}$ unless $N_{1}=N_{2}=2$, but the same
does not apply at higher rank. These $r$-fundamental fixed points exhibit an
$O(N)^{r}/(\mathbb{Z}_{2})^{r-1}$ symmetry, where the $\mathbb{Z}_{2}$
quotients owe to the fact each $O(N)$ group carries its own $\mathbb{Z}_{2}$
action, while the theory enjoys but a single overall $\mathbb{Z}_{2}$ symmetry
that sends the fundamental fields to minus themselves. Alternatively, for $N$
odd, the symmetry can be labelled as $\mathbb{Z}_{2}\times SO(N)^{r}$. For
general $r$, however, and for sufficiently large $N$, there exists not one but
numerous distinct multi-fundamental fixed points, and for most of them the
symmetry is enhanced to a larger group. Although in some cases the symmetry is
enhanced with an extra $S_{r}$ permutation symmetry, we emphasize that the
multi-fundamental fixed points are distinct from the so-called MN fixed points
[69, 70, 68], which possess $O(N)^{r}\rtimes S_{r}$ symmetry, but which are
fixed points of $(rN)$-component scalar theories, as opposed to the
$N^{r}$-component theories we will study. In uplifting the $O(N)$ model to
tensorial generalizations, our approach is ultimately complementary to the
$\epsilon$ expansion literature, which generally focuses on theories with
small field contents, while we will be dealing with theories with vast numbers
of field components. In the case of $r=3$, for example, the full set of tri-
fundamental fixed points do not appear until $N\geq 29$, ie. until the scalar
fields carry at least $24\,389$ components.

###  1.3 Overview of paper

Using large $N$ and small $\epsilon$ methods, this paper comprehensively
elucidates the structure of the RG flows and fixed points of quartic
$O(N)^{r}$ scalar theory with $r\leq 5$ and provides formulas for subleading
anomalous dimension at large $N$, which are simply related to known $O(N)$
model formulas. What ultimately renders these $O(N)^{r}$ models solvable at
large $N$ is the fact that the subset of operators that are vectorial, by
which we mean operators for which the dominant diagrams at large $N$ are
bubble diagrams, close on themselves under the RG flow at large $N$. This
means that the RG flow can be consistently truncated by omitting operators
that are not vectorial, and in the absence of these operators the large $N$
beta functions terminate at quadratic order and can be explicitly diagonalized
by a simple basis transformation. This truncation can also be performed at
higher rank $r>5$, and we will write down explicit closed-form expressions
that count the number of vectorial fixed points at large yet finite $N$ for
arbitrary rank $r$, but it is possible that additional fixed pointsk beyond
those in the vectorial class exist at higher even ranks. For the vectorial
fixed points, the large $N$ descriptions at any $r$ are obtained by
generalizations of the Hubbard-Stratonovich transformation, expressing
operators as integrals over intermediary fields. In addition to the standard
$\sigma$ field, the additional intermediary fields carry pairs of indices,
with each pair transforming as a traceless symmetric or antisymmetric matrix
under its own $O(N)$ group. For some fixed points, these intermediate fields
give rise to Feynman rules by which $1/N$ corrections to observables are
computable to arbitrary order, but for many of the fixed points non-vectorial
operators are present at suppressed order, so that no such simple Feynman
rules can be formulated. There generally exist numerous relevant deformations
of the $O(N)^{r}$ fixed points, unless particular symmetries are imposed to
exclude them. Either scenario renders the challenge of finding concrete real-
life applications of the $O(N)^{r}$ fixed points rather formidable, but it
will likely be less challenging to identify statistical mechanical systems
with fixed points that fit in this class.

As a contrast and supplement to our analysis of quartic $O(N)^{r}$ theory, we
also present a survey of sextic $O(N)^{r}$ models for $r\leq 4$ and the
challenges they afford. In contradistinction to the simple structure that
underlies quartic tensor theory, the space of RG fixed points for tensor
theories with sextic interactions is vastly more complicated. This is not
apparent from the sextic rank-two and rank-three theories, which have
previously been studied in the literature, and for which, in each case, there
exists but a single standard large $N$ fixed point in the $3-\epsilon$
expansion, but the complexity becomes discernible at rank four. For the sextic
$O(N)^{4}$ model we find that even the reduced system with additional $S_{4}$
permutation symmetry possesses 17 Wilson-Fisher fixed points. As the rank $r$
is taken to be even larger, simply determining the number of sextic $O(N)^{r}$
invariants quickly becomes a difficult task, posing a roadblock to a
classification of sextic fixed points. For the new rank-four fixed points we
uncover in the $\epsilon$ expansion, the determination of the leading large
$N$ behaviour is hindered by the presence of matricial type of operators that
contribute to the large $N$ dynamics alongside vectorial operators. Attempts
to match known melonic large $N$ solutions of sextic models to $\epsilon$
expansions in renormalized perturbation theory are faced with a separate set
of difficulties, as the $d$-dependence of these large $N$ solutions cannot be
matched by conventional Wilson-Fisher fixed points but require coupling
constants with a leading scaling $g\sim\sqrt{\epsilon}$, meaning that the
perturbative fixed points do not appear at quadratic order in beta functions,
but rely on a cancellation between linear and cubic terms; and the couplings
that scale as $\sqrt{\epsilon}$ feed into other beta functions and induce yet
more exotic scalings in other couplings, such as $\epsilon^{0}$ or
$\epsilon^{1/4}$.

The remainder of the paper is organized as follows: Section 2 analyses
tensorial generalizations of the quartic $O(N)$ model, first reviewing and
expanding on the literature on the matrix and three-index tensor
generalizations, before studying the models with
$O(N)^{4}/(\mathbb{Z}_{2})^{3}$ and $O(N)^{5}/(\mathbb{Z}_{2})^{4}$ symmetry
in the $\epsilon$ expansion and at large $N$, and finally discussing
properties of $O(N)^{r}/(\mathbb{Z}_{2})^{r-1}$ theory for general values of
$r$. Section 3 studies $O(N)^{r}/(\mathbb{Z}_{2})^{r-1}$ tensor theories with
sextic interactions, reviewing examples in the literature and examining the
particular example of the rank-four version of $\phi^{6}$ theory with global
$S_{4}\ltimes O(N)^{4}/(\mathbb{Z}_{2})^{3}$ symmetry, illustrating the wealth
of fixed points emerging at higher rank but also the obstacles to extracting
reliable information from sextic theories. Section 4 concludes the main
portion of the paper by outlining directions of future inquiry. Appendix A
contains a discussion of the suitable scalings of coupling constants with $N$
for general tensor operators, while Appendix B discusses large $N$ fixed
points of quartic $O(N)^{r}$ theory at general $r$, describing in detail the
properties of vectorial fixed points and presenting an argument that melonic
operators are always absent. Appendix C lists the explicit values of the
couplings for the fixed points of the sextic $S_{4}\ltimes
O(N)^{4}/(\mathbb{Z}_{2})^{3}$ model.

In submitting this paper to arXiv, we have included a supplementary
Mathematica notebook that contains the beta functions and anomalous dimensions
to third order in the coupling constants for all the theories studied in this
paper.

##  2 Tensorial $\phi^{4}$ Theory

In this section we study quartic scalar theories of tensors of rank $r$, where
each index transforms under a separate $O(N)$ group, endowing the theories
with global $O(N)^{r}/(\mathbb{Z})^{r-1}$ symmetry. The theories all contain a
relevant quartic deformation $\phi^{2}$, which we will tune away. In the
special case when $N=1$, every one of the models reduces to $\phi^{4}$ theory,
and in the case when $r=1$, the theories reduce to the $O(N)$ vector model.

In Subsections 2.1, 2.2, 2.3, and 2.4, we analyse in turn the theories with
$r\in\\{2,3,4,5\\}$ and their perturbative fixed points. The theories with
$r=2$ and $r=3$ were previously studied in Ref. [71], which pointed out that
they do not posses any melonic large $N$ fixed points with real-valued
couplings. We will find that the same applies for $r=4$ and $r=5$, in fact all
the real-valued large $N$ fixed points are dominated by operators that realize
a vectorial large $N$ limit. For this reason every one of the fixed points can
be understood by simple generalizations of vector model techniques. In the
last subsection, 2.5, we explain how these vectorial fixed points extend to
higher values of $r$.

Our survey of quartic theories focuses exclusively on dimensions below four:
$d=4-\epsilon$ with $\epsilon>0$. In four dimensions, $\phi^{4}$ theory is
well-known to flow to a free theory in the infrared, although it has been
recently proposed that the quartic vector model can be made to flow to an
interacting infrared theory by starting with a negative bare coupling [72,
73], see also [74] and [75].

###  2.1 The $O(N)^{2}$ model

The rank-two theory contains two quartic singlets:

$\displaystyle\begin{matrix}\text{
\includegraphics[scale={1}]{quarticRank2ops.pdf} }\end{matrix}$ (9)

At large $N$, the coupling constants $g_{1}$ and $g_{2}$ of these operators
have the optimal scalings $g_{1}\sim N^{-2}$ and $g_{2}\sim N^{-1}$. In
addition to a global $O(N)^{2}/\mathbb{Z}_{2}$ symmetry, the model also has an
$S_{2}$ symmetry under interchange of the two matrix indices, owing to the
fact that the two operators (9) are symmetric with respect to the two colours.
At higher ranks $r$, such permutation symmetries are not automatically present
for generic values of couplings, but only exist in subsectors of the full
$O(N)^{r}$ theories.

On deforming the free matrix theory by $\mathcal{O}_{1}$ and
$\mathcal{O}_{2}$, renormalized perturbation theory in the $\epsilon$
expansion reveals that, unless $N=2$, the two-loop beta functions have but one
non-trivial RG fixed-point, and at this fixed point the coupling of the
single-trace operator is zero. The fixed point is therefore just the vector
model fixed point of an $O(N^{2})$ model. For $N=2$ there is one additional
fixed point with both couplings non-zero.999It generally happens in tensor
theories that there are linear relations among the operators at low values of
$N$. However, for the tensorial $\phi^{4}$ theories we are studying in this
section, it can be checked that all the quartic operators are linearly
independent already at $N=2$.

A straightforward generalization of the $O(N)^{2}$ model is obtained by
allowing the two indices to assume different numbers of values $N_{1}$ and
$N_{2}$. Without loss of generality, it may be assumed that $N_{2}\geq N_{1}$.
It can then be checked that fixed points with non-zero single-trace coupling
exist only when $N_{1}=N_{2}=2$ or when

$\displaystyle N_{2}\geq 22\,,\hskip 42.67912pt2\leq N_{1}\leq
5N_{2}+2-2\sqrt{6(N_{2}+2)(N_{2}-1)}\,.$ (10)

The fixed points with non-zero value of $g_{2}$ that exist in these cases
exhibit $O(N_{1})\times O(N_{2})\,/\,\mathbb{Z}_{2}$ symmetry and are known in
the literature as the bifundamental fixed points [66, 67, 68, 58].

###  2.2 The $O(N)^{3}$ model

Turning to the rank-three theory, there are five quartic singlets:

$\displaystyle\begin{matrix}\text{
\includegraphics[scale={1}]{quarticRank3ops.pdf} }\end{matrix}$ (11)

and the action of the theory can be written as101010 See Refs. [76, 77, 78,
79] for studies of variants of this theory with a non-local kinetic term,
including indications of large $N$ unitarity and an explicit check of the
$F$-theorem. For the generalization of this model to the theory with
$O(N_{1})\times O(N_{2})\times O(N_{3})$ symmetry, see Ref. [80], and see Ref.
[81] for the quartic rank-three model of fermionic tensors with $Sp(N)$
symmetry and Refs. [42, 82, 83] for studies of quartic tensors with three or
more copies of $U(N)$ symmetry. See Ref. [84] for details on the diagrammatics
of $O(N)^{3}$ theory, and Ref. [85] for a study of the beta functions at
higher loops.

$\displaystyle S=\int
d^{4-\epsilon}x\bigg{(}\frac{1}{2}(\partial_{i}\phi^{abc})(\partial_{i}\phi^{abc})+\frac{1}{4!}\sum_{n=1}^{5}g_{n}\,\mathcal{O}_{n}\bigg{)}\,.$
(12)

####  2.2.1 Small $\epsilon$ expansion

Expanding the beta functions of the theory in a perturbative expansion,

$\displaystyle\beta_{g_{i}}=-\epsilon
g_{i}+\beta^{(2)}_{g_{i}}+\beta^{(3)}_{g_{i}}+\mathcal{O}(g^{4})\,,$ (13)

the quadratic pieces are given by

$\displaystyle\beta_{g_{1}}^{(2)}=\frac{1}{48\pi^{2}}\Big{(}$
$\displaystyle(8+N^{3})g_{1}^{2}+2(1+N+N^{2})g_{1}g_{2}+3g_{2}^{2}+2(1+N+N^{2})g_{1}g_{3}+2Ng_{2}g_{3}+3g_{3}^{2}$
$\displaystyle+2(1+N+N^{2})g_{1}g_{4}+2Ng_{2}g_{4}+2Ng_{3}g_{4}+3g_{4}^{2}+6Ng_{1}g_{5}+2g_{2}g_{5}+2g_{3}g_{5}$
$\displaystyle+2g_{4}g_{5}\Big{)}\,,$ (14)
$\displaystyle\beta_{g_{2}}^{(2)}=\frac{1}{48\pi^{2}}\Big{(}$ $\displaystyle
12g_{1}g_{2}+(4+N+N^{2})g_{2}^{2}+2(1+N)g_{2}g_{3}+2(1+N)g_{2}g_{4}+4g_{3}g_{4}+4Ng_{2}g_{5}$
$\displaystyle+4g_{3}g_{5}+4g_{4}g_{5}+(2+N)g_{5}^{2}\Big{)}\,,$ (15)
$\displaystyle\beta_{g_{3}}^{(2)}=\frac{1}{48\pi^{2}}\Big{(}$ $\displaystyle
12g_{1}g_{3}+2(1+N)g_{2}g_{3}+(4+N+N^{2})g_{3}^{2}+4g_{2}g_{4}+2(1+N)g_{3}g_{4}+4g_{2}g_{5}$
$\displaystyle+4Ng_{3}g_{5}+4g_{4}g_{5}+(2+N)g_{5}^{2}\Big{)}\,,$ (16)
$\displaystyle\beta_{g_{4}}^{(2)}=\frac{1}{48\pi^{2}}\Big{(}$ $\displaystyle
4g_{2}g_{3}+12g_{1}g_{4}+2(1+N)g_{2}g_{4}+2(1+N)g_{3}g_{4}+(4+N+N^{2})g_{4}^{2}+4g_{2}g_{5}$
$\displaystyle+4g_{3}g_{5}+4Ng_{4}g_{5}+(2+N)g_{5}^{2}\Big{)}\,,$ (17)
$\displaystyle\beta_{g_{5}}^{(2)}=\frac{1}{48\pi^{2}}\Big{(}$ $\displaystyle
4g_{2}g_{3}+4g_{2}g_{4}+4g_{3}g_{4}+12g_{1}g_{5}+2(1+N)g_{2}g_{5}+2(1+N)g_{3}g_{5}$
$\displaystyle+2(1+N)g_{4}g_{5}\Big{)}\,.$ (18)

The cubic pieces are listed in the supplementary Mathematica file. Setting to
zero the quadratic order beta functions, we can determine the perturbative
fixed points to leading order in $\epsilon$, and we can then use the cubic
order beta functions to compute the subleading corrections to these. The
number of fixed points (including the trivial one) depends on $N$ as
follows:111111We have chosen to include the trivial fixed point so that the
total number of large $N$ fixed points, 16, equals a power of 2, which will
turn out to be a general fact.

  * •

For $N$ = 2 there are 16 perturbative fixed points.

  * •

For $3\leq N\leq 10$ there are two perturbative fixed points: the trivial
fixed point and the $O(N^{3})$ model fixed point with only $g_{1}$ non-zero.
These fixed points are present for all $N$.

  * •

For $11\leq N\leq 19$ there are eight fixed points.

  * •

For $20\leq N\leq 28$ there are 14 fixed points.

  * •

For $29\leq N$ there are 16 fixed points.

The above tally of fixed points includes only those present already at
quadratic order in the beta functions. At higher order, additional fixed
points emerge, with couplings that do not scale with integer powers of
$\epsilon$, including a melonic fixed point, which, however, is complex [71].
Generally, the additional fixed points are not perturbatively reliable.
Indeed, the act of setting beta functions to zero produces a set of polynomial
equations in the coupling constants that can be solved to search for RG fixed
points; going to higher perturbative order means increasing the order of the
polynomials; by mathematical necessity, higher-order polynomial equations have
more solutions, but this fact does not convey physical information about CFTs.

$\displaystyle\begin{matrix}\text{ {
\fcolorbox{gray}{white}{\includegraphics[scale=0.65]{rank3plot-1.png}} }
}\end{matrix}$ Figure 2: Perturbative fixed points for quartic
$O(N)^{3}/(\mathbb{Z}_{2})^{2}$ theory in $4-\epsilon$ dimensions. The fixed
points are labelled according to their large $N$ field content in terms of
Hubbard-Stratonovich fields $\sigma$ and $\chi$. For each point on the curves
marked with
${\color[rgb]{.5,0,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,0,.5}\chi}$,
${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}2\chi}$,
${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\sigma+\chi}$,
and
${\color[rgb]{0,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,1,1}\sigma+2\chi}$
there are actually three separate RG fixed points, related by permutations of
the values for the coupling constants $g_{2}$, $g_{3}$, and $g_{4}$.

Figure 2 displays a plot of the values of $g_{1}$ at the perturbative fixed
points as functions $N$. Several fixed points are related by permutations of
indices. Fixed points related in this manner assume the same value for $g_{1}$
as well as for anomalous dimensions of $O(N)^{3}$ singlets, and for this
reason there are not 16 separate fixed points visible for $N\geq 29$ in Figure
2, but only eight. These eight families of fixed points correspond to orbits
under the action of an $S_{3}$ group that permutes the $O(N)$ groups of the
model. We will label these orbits with roman numerals (i) to (viii), where
(i) is the orbit containing only the trivial fixed point with all couplings
equal to zero.
(ii) is the orbit containing only the vector model fixed. In the space of
couplings, this fixed point is situated at

$\displaystyle
g_{1}^{\text{(ii)}}=\frac{48\pi^{2}}{8+N^{3}}\epsilon+\frac{144(14+3N^{3})\pi^{2}}{(8+N^{3})^{3}}\epsilon^{2}+\mathcal{O}(\epsilon^{3})\,,\hskip
42.67912ptg^{\text{(ii)}}_{2}=g^{\text{(ii)}}_{3}=g^{\text{(ii)}}_{4}=g^{\text{(ii)}}_{5}=0\,.$
(19)

The field and mass anomalous dimensions at this fixed point are given by

$\displaystyle\gamma_{\phi}^{\text{(ii)}}=\,$
$\displaystyle\frac{2+N^{3}}{4(8+N^{3})^{2}}\epsilon^{2}-\frac{(2+N^{3})(N^{6}-56N^{3}-272)}{16(8+N^{3})^{4}}\epsilon^{3}+\mathcal{O}(\epsilon^{4})\,,$
(20) $\displaystyle\gamma^{\text{(ii)}}_{\phi^{2}}=\,$
$\displaystyle\frac{2+N^{3}}{8+N^{3}}\epsilon+\frac{(2+N^{3})(44+13N^{3})}{2(8+N^{3})^{3}}\epsilon^{2}+\mathcal{O}(\epsilon^{3})\,.$
(21)

At the vector model fixed point, the $O(N)^{3}$ symmetry of the theory is
enhanced to $O(N^{3})$ symmetry. The enhanced symmetry is present everywhere
along the RG flow whenever $g_{1}$ is the only non-zero coupling. The operator
$\mathcal{O}_{1}$ alone can never generate $\mathcal{O}_{2}$ to
$\mathcal{O}_{5}$.
(iii) and (iv) are orbits containing three fixed points each. They are
characterized by $g_{1}$ and either $g_{2}$, $g_{3}$, or $g_{4}$ being non-
zero, while the other three couplings vanishing, which implies that the fixed
points in these orbits are members of the family of bifundamental fixed
points. Without loss of generality we can take $g_{1}$ and $g_{2}$ to be the
non-vanishing couplings, which assume the values

$\displaystyle
g_{1}^{\text{(iii)}}=\text{\scalebox{0.83}{$\frac{80-2N-3N^{2}-2N^{3}-N^{4}-(4+N+N^{2})\sqrt{52-4N-3N^{2}-10N^{3}+N^{4}}}{464-56N-72N^{2}-16N^{3}-8N^{4}+9N^{5}+2N^{6}+N^{7}}$}}24\pi^{2}\epsilon+\mathcal{O}(\epsilon^{2})\,,$
$\displaystyle
g_{1}^{\text{(iv)}}=\text{\scalebox{0.83}{$\frac{80-2N-3N^{2}-2N^{3}-N^{4}+(4+N+N^{2})\sqrt{52-4N-3N^{2}-10N^{3}+N^{4}}}{464-56N-72N^{2}-16N^{3}-8N^{4}+9N^{5}+2N^{6}+N^{7}}$}}24\pi^{2}\epsilon+\mathcal{O}(\epsilon^{2})\,,$
$\displaystyle
g_{2}^{\text{(iii)}}=\text{\scalebox{0.83}{$\frac{-4-10N-10N^{2}+4N^{3}+N^{4}+N^{5}+6\sqrt{52-4N-3N^{2}-10N^{3}+N^{4}}}{464-56N-72N^{2}-16N^{3}-8N^{4}+9N^{5}+2N^{6}+N^{7}}$}}48\pi^{2}\epsilon+\mathcal{O}(\epsilon^{2})\,,$
$\displaystyle
g_{2}^{\text{(iv)}}=\text{\scalebox{0.83}{$\frac{-4-10N-10N^{2}+4N^{3}+N^{4}+N^{5}-6\sqrt{52-4N-3N^{2}-10N^{3}+N^{4}}}{464-56N-72N^{2}-16N^{3}-8N^{4}+9N^{5}+2N^{6}+N^{7}}$}}48\pi^{2}\epsilon+\mathcal{O}(\epsilon^{2})\,.$

We refrain from writing down the order $\epsilon^{2}$ terms as they are quite
lengthy. Similarly, lengthy closed-form expressions are available for
$\gamma_{\phi}$ to order $\epsilon^{3}$ and for $\gamma_{\phi^{2}}$ to order
$\epsilon^{2}$. From the radicands in the above expressions, one sees that the
(iii) and (iv) fixed points collide and become complex for $N<N_{\ast}$ where

$\displaystyle
N_{\ast}=\frac{1}{2}\Big{(}5+4\sqrt{3}+\sqrt{33+24\sqrt{3}}\Big{)}\approx
10.2818\,,$ (23)

up to order $O(\epsilon)$ corrections. We observe that the (iii) and (iv)
fixed points exhibit an $O(N^{2})\times O(N)/\mathbb{Z}_{2}$ symmetry that
rotates the $a$ and $b$ indices of $\phi^{abc}$ together but the $c$ index
separately. This symmetry extends beyond the fixed points and is preserved
everywhere within a two-dimensional invariant subspace of the full five-
dimensional RG flow. When only $\mathcal{O}_{1}$ and $\mathcal{O}_{2}$ are
present, $\mathcal{O}_{3}$, $\mathcal{O}_{4}$, and $\mathcal{O}_{5}$ will
never be generated.
The orbits (v) and (vi) also contain three fixed points each. For each of
these fixed points all couplings are non-zero and two of the couplings
$g_{2}$, $g_{3}$, and $g_{4}$ assume identical values. Each of the fixed
points in these two orbits is situated on a four-dimensional invariant
manifold characterized by an $S_{2}$ permutation symmetry acting on two of the
indices of the fundamental fields, meaning that the manifolds and fixed points
enjoy an $O(N)^{3}\rtimes S_{2}$ symmetry. This symmetry reflects the fact
that if, e.g., $g_{2}$ equals $g_{3}$ at some scale along an RG flow, then
$g_{2}$ equals $g_{3}$ everywhere along the flowline. The leading values of
the couplings at small $\epsilon$ are given by roots of four coupled quadratic
equations, which do not generally admit closed form solutions as functions of
$N$, although for a given $N$ the leading and sub-leading values for the
couplings and anomalous dimensions can easily be evaluated numerically to high
precision. The fixed points in these two orbits merge and become complex for
$N<N_{\ast\ast}$ with $N_{\ast\ast}\approx 19.3955$.
Orbits (vii) and (viii) contain a single fixed point each, for which $g_{2}$,
$g_{3}$, and $g_{4}$ assume the same non-zero value, while $g_{1}$ and $g_{5}$
assume two other non-zero values. These fixed points are situated on the
three-dimensional invariant manifold of coupling configurations where the
theory exhibits an $S_{3}$ permutation symmetry under interchanges of the
three tensor indices: if the theory starts out being symmetric with respect to
the three indices, then renormalization will not spoil this property. The
leading values of the couplings as functions of $N$ are given in terms of
roots of sixth order polynomials. The two fixed points in these orbits merge
and become complex for $N<N_{\ast\ast\ast}$ with $N_{\ast\ast\ast}\approx
28.0935$.

####  2.2.2 Large $N$ expansion

The operators (11) have have an optimal large $N$ scaling, explicitly worked
out by Carrozza and Tanasa [55] and given by

$\displaystyle g_{1}=\frac{\lambda_{1}}{N^{3}}\,,\hskip
34.1433ptg_{2}=\frac{\lambda_{2}}{N^{2}}\,,\hskip
34.1433ptg_{3}=\frac{\lambda_{3}}{N^{2}}\,,\hskip
34.1433ptg_{4}=\frac{\lambda_{4}}{N^{2}}\,,\hskip
34.1433ptg_{5}=\frac{\lambda_{5}}{N^{3/2}}\,.$ (24)

The fact that the last operator has the smallest scaling exponent can be
interpreted to mean that this operator dominates at large $N$ unless its
coupling constant is suppressed. Ref. [71] demonstrated how the large $N$
theory of this operator is melonic and solvable, but also pointed out that
this melonic theory requires the four other operators to have complex
couplings.121212 Specifically, $\beta_{\lambda_{5}}$ has a fixed point at
$\lambda_{5}=\pm 48\pi^{2}\sqrt{\epsilon}$. But at this value of
$\lambda_{5}$, the fixed point values for $\beta_{\lambda_{2}}$,
$\beta_{\lambda_{3}}$, and $\beta_{\lambda_{4}}$ are given by $\lambda_{2}$,
$\lambda_{3}$, and $\lambda_{4}$ equal to
$48\pi^{2}(\epsilon\pm\sqrt{\epsilon^{2}-2\epsilon})$, which are complex for
small $\epsilon$. Incidentally, the fixed point values become real right in
two dimensions:  $\displaystyle\epsilon=2:\hskip 42.67912pt\lambda_{1}=\pm
96\sqrt{6}\pi^{2}\,,\hskip
42.67912pt\lambda_{2}=\lambda_{3}=\lambda_{4}=96\pi^{2}\,,\hskip
42.67912pt\lambda_{5}=\pm 96\pi^{2}\,.$ (25) The signs of $\lambda_{1}$ and
$\lambda_{5}$ can be chosen independently, so that there are a total of four
real fixed points. But because of the finite value of $\epsilon$, higher-order
corrections will not be parametrically small, and so we cannot trust these
fixed points.  To third order in the couplings, the beta functions in the
$N\rightarrow\infty$ limit are given by

$\displaystyle\beta_{\lambda_{1}}=-\epsilon\lambda_{1}+\frac{1}{48\pi^{2}}\Big{(}\lambda_{1}^{2}+2\lambda_{1}\lambda_{2}+2\lambda_{1}\lambda_{3}+2\lambda_{2}\lambda_{3}+2\lambda_{1}\lambda_{4}+2\lambda_{2}\lambda_{4}+2\lambda_{3}\lambda_{4}\Big{)}$
$\displaystyle\hskip
28.45274pt-\frac{1}{(48\pi^{2})^{2}}\Big{(}\frac{5}{2}\lambda_{1}\lambda_{5}^{2}+2\lambda_{2}\lambda_{5}^{2}+2\lambda_{3}\lambda_{5}^{2}+2\lambda_{4}\lambda_{5}^{2}\Big{)}+\mathcal{O}(\lambda^{4})\,,$
$\displaystyle\beta_{\lambda_{2}}=-\epsilon\lambda_{2}+\frac{\lambda_{2}^{2}+\lambda_{5}^{2}}{48\pi^{2}}-\frac{\lambda_{2}\lambda_{5}^{2}}{2(48\pi^{2})^{2}}+\mathcal{O}(\lambda^{4})\,,$
$\displaystyle\beta_{\lambda_{3}}=-\epsilon\lambda_{3}+\frac{\lambda_{3}^{2}+\lambda_{5}^{2}}{48\pi^{2}}-\frac{\lambda_{3}\lambda_{5}^{2}}{2(48\pi^{2})^{2}}+\mathcal{O}(\lambda^{4})\,,$
(26)
$\displaystyle\beta_{\lambda_{4}}=-\epsilon\lambda_{2}+\frac{\lambda_{4}^{2}+\lambda_{5}^{2}}{48\pi^{2}}-\frac{\lambda_{4}\lambda_{5}^{2}}{2(48\pi^{2})^{2}}+\mathcal{O}(\lambda^{4})\,,$
$\displaystyle\beta_{\lambda_{5}}=-\epsilon\lambda_{5}+\frac{\lambda_{5}^{3}}{2(48\pi^{2})^{2}}+\mathcal{O}(\lambda^{4})\,.$

These beta functions have 16 real fixed points towards which the 16 finite $N$
fixed points that exist for $N\geq 29$ converge as $N$ goes to infinity. For
all the large $N$ fixed points, the melonic operator is absent or suppressed
below its combinatorial scaling. The explicit locations of these fixed points
in the space of $\lambda$-couplings are listed below, where it is to be
understood that unspecified couplings are zero:

  1. (i)

one fixed point: all $\lambda_{i}$ equal to zero,

  2. (ii)

one fixed point: $\lambda_{1}=48\pi^{2}\epsilon$,

  3. (iii)

three fixed points: $\lambda_{1}=-48\pi^{2}\epsilon$, one of $\lambda_{2}$,
$\lambda_{3}$, and $\lambda_{4}$ equal to $48\pi^{2}\epsilon$,

  4. (iv)

three fixed points: one of $\lambda_{2}$, $\lambda_{3}$, and $\lambda_{4}$
equal to $48\pi^{2}\epsilon$,

  5. (v)

three fixed points: $\lambda_{1}=-96\pi^{2}\epsilon$, two of $\lambda_{2}$,
$\lambda_{3}$, and $\lambda_{4}$ equal to $48\pi^{2}\epsilon$,

  6. (vi)

three fixed points: $\lambda_{1}=-48\pi^{2}\epsilon$, two of $\lambda_{2}$,
$\lambda_{3}$, and $\lambda_{4}$ equal to $48\pi^{2}\epsilon$,

  7. (vii)

one fixed point: $\lambda_{1}=-144\pi^{2}\epsilon$,
$\lambda_{2}=\lambda_{3}=\lambda_{4}=48\pi^{2}\epsilon$,

  8. (viii)

one fixed point: $\lambda_{1}=-96\pi^{2}\epsilon$,
$\lambda_{2}=\lambda_{3}=\lambda_{4}=48\pi^{2}\epsilon$.

The leading anomalous dimensions for $\phi$ and $\phi^{2}$ at the fixed points
are listed in Table 1.

| # of copies | $\gamma_{\phi}-\mathcal{O}(\epsilon^{4})$ | $\gamma_{\phi^{2}}-\mathcal{O}(\epsilon^{3})$
---|---|---|---
(i) | 1 | 0 | 0
(ii) | 1 | $\frac{1}{N^{3}}(\frac{1}{4}\epsilon^{2}-\frac{1}{16}\epsilon^{3})+\mathcal{O}(\frac{1}{N^{6}})$ | $\epsilon+\frac{1}{N^{3}}(-6\epsilon+\frac{13}{2}\epsilon^{2})+\mathcal{O}(\frac{1}{N^{6}})$
(iii) | 3 | $\frac{1}{N}(\frac{1}{8}\epsilon^{2}-\frac{1}{32}\epsilon^{3})+\mathcal{O}(\frac{1}{N^{2}})$ | $\frac{1}{N}(3\epsilon-\frac{13}{4}\epsilon^{2})+\mathcal{O}(\frac{1}{N^{2}})$
(iv) | 3 | $\frac{1}{N}(\frac{1}{8}\epsilon^{2}-\frac{1}{32}\epsilon^{3})+\mathcal{O}(\frac{1}{N^{2}})$ | $\epsilon+\frac{1}{N}(-3\epsilon+\frac{13}{4}\epsilon^{2})+\mathcal{O}(\frac{1}{N^{2}})$
(v) | 3 | $\frac{1}{N}(\frac{1}{4}\epsilon^{2}-\frac{1}{16}\epsilon^{3})+\mathcal{O}(\frac{1}{N^{2}})$ | $\frac{1}{N}(6\epsilon-\frac{13}{2}\epsilon^{2})+\mathcal{O}(\frac{1}{N^{2}})$
(vi) | 3 | $\frac{1}{N}(\frac{1}{4}\epsilon^{2}-\frac{1}{16}\epsilon^{3})+\mathcal{O}(\frac{1}{N^{2}})$ | $\epsilon+\frac{1}{N}(-6\epsilon+\frac{13}{2}\epsilon^{2})+\mathcal{O}(\frac{1}{N^{2}})$
(vii) | 1 | $\frac{1}{N}(\frac{3}{8}\epsilon^{2}-\frac{3}{32}\epsilon^{3})+\mathcal{O}(\frac{1}{N^{2}})$ | $\frac{1}{N}(9\epsilon-\frac{39}{4}\epsilon^{2})+\mathcal{O}(\frac{1}{N^{2}})$
(viii) | 1 | $\frac{1}{N}(\frac{3}{8}\epsilon^{2}-\frac{3}{32}\epsilon^{3})+\mathcal{O}(\frac{1}{N^{2}})$ | $\epsilon+\frac{1}{N}(-9\epsilon+\frac{39}{4}\epsilon^{2})+\mathcal{O}(\frac{1}{N^{2}})$

Table 1: Leading anomalous dimensions at the large $N$ fixed points of quartic
$O(N)^{3}$ theory

There is a pattern to these anomalous dimensions, which can be understood by
working out the large $N$ limit of the model outside of perturbation theory in
$\epsilon$. To describe the pattern, we first review how to solve for the
vector model fixed point at large $N$, before turning to the other fixed
points.

####  2.2.3 Vector model fixed point

The vector model fixed point (ii) is the large $N$ limit of the fixed point in
(19). The anomalous dimensions for the large $N$ vector model were computed in
Refs. [7, 8], and the large $N$ solution was also nicely reviewed more
recently in Ref. [3]. For small $\epsilon$, the large $N$ anomalous dimensions
can be found by simply expanding (20) and (21):

$\displaystyle\gamma_{\phi}^{\text{(ii)}}=\,$
$\displaystyle\frac{\epsilon^{2}}{N^{3}}\Big{(}\frac{1}{4}-\frac{7}{2N^{3}}+\mathcal{O}(\frac{1}{N^{6}})\Big{)}+\frac{\epsilon^{3}}{N^{3}}\Big{(}-\frac{1}{16}+\frac{43}{8N^{3}}+\mathcal{O}(\frac{1}{N^{6}})\Big{)}+\mathcal{O}(\epsilon^{4})\,,$
(27) $\displaystyle\gamma^{\text{(ii)}}_{\phi^{2}}=\,$
$\displaystyle\epsilon\Big{(}1-\frac{6}{N^{3}}+\frac{48}{N^{6}}+\mathcal{O}(\frac{1}{N^{6}})\Big{)}+\frac{\epsilon^{2}}{N^{3}}\Big{(}\frac{13}{2}-\frac{121}{N^{3}}+\mathcal{O}(\frac{1}{N^{6}})\Big{)}+\mathcal{O}(\epsilon^{3})\,.$
(28)

But by leveraging the power of large $N$, these anomalous dimensions can be
determined for any $d$ between 2 and 4. A standard way to obtain the large $N$
solution of an $O(N)$ or $O(N^{r})$ model, dating back to the work of Coleman,
Jackiw, and Politzer [6], is through the use of the Hubbard-Stratonovich
transformation. Without changing the path integral of the theory, we can
introduce an integral over an auxiliary $\sigma$ field and perform the
following substitution in the action:

$\displaystyle\frac{g_{1}}{4!}(\phi^{abc}\phi^{abc})^{2}\rightarrow\,\,$
$\displaystyle\frac{g_{1}}{4!}(\phi^{abc}\phi^{abc})^{2}-\frac{3}{2g_{1}N^{3}}\Big{(}\sigma-\frac{g_{1}N^{3/2}}{6}\phi^{abc}\phi^{abc}\Big{)}^{2}$
$\displaystyle=\frac{1}{2N^{3/2}}\sigma\phi^{abc}\phi^{abc}-\frac{3}{2g_{1}N^{3}}\sigma^{2}\,.$

On flowing to a CFT the $\sigma^{2}$ term goes away, leaving us with an
effective action

$\displaystyle S^{\text{(ii)}}=\int
d^{d}x\bigg{(}\frac{1}{2}(\partial_{i}\phi^{abc})(\partial_{i}\phi^{abc})+\frac{1}{2N^{3/2}}\sigma\phi^{abc}\phi^{abc}\bigg{)}\,.$
(30)

The dimension $\Delta_{\phi^{2}}$ can be determined from the expectation value
of $\sigma$:

$\displaystyle\left<\sigma(k)\sigma(p)\right>=\delta^{(d)}(k+p)\frac{C_{\sigma}}{|p|^{d-2\Delta_{\phi^{2}}}}\,,$
(31)

where $C_{\sigma}$ is $d$-dependent constant. The leading value in $1/N^{3}$
of this correlator, which gives us the effective non-local propagator for
$\sigma$, can be determined by integrating out the fundamental fields
$\phi^{abc}$:

$\displaystyle\text{ \leavevmode\hbox to58.11pt{\vbox
to3.65pt{\pgfpicture\makeatletter\hbox{\hskip 29.05275pt\lower-0.2pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{{}}{}
{}{}\pgfsys@beginscope\pgfsys@invoke{
}\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@invoke{
}\pgfsys@color@gray@fill{1}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{1,1,1}{}\pgfsys@moveto{-2.84544pt}{0.0pt}\pgfsys@lineto{2.84544pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}
{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{1.2pt}\pgfsys@invoke{
}\pgfsys@setdash{3.0pt,3.0pt}{0.0pt}\pgfsys@invoke{
}{}\pgfsys@moveto{-28.45276pt}{2.84544pt}\pgfsys@lineto{28.45276pt}{2.84544pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}} }\hskip
8.53581pt\left<\sigma(k)\sigma(p)\right>_{0}=\delta^{(d)}(k+p)\frac{2^{d+1}(4\pi)^{\frac{d-3}{2}}\Gamma(\frac{d-1}{2})\sin(\frac{\pi
d}{2})}{|p|^{d-4}}\,.$ (32)

Comparing the above two equations we see that
$\Delta_{\phi^{2}}=2+\mathcal{O}(1/N^{3})$. Subtracting off the engineering
dimension $[\phi^{2}]=d-2$ from this value, we obtain the anomalous dimension:

$\displaystyle\gamma_{\phi^{2}}^{\text{(ii)}}=\Delta_{\phi^{2}}-[\phi^{2}]=4-d+\mathcal{O}(\frac{1}{N^{3}})\,.$
(33)

The anomalous dimension for $\phi$ is determined to leading order in $1/N^{3}$
by the sunset diagram:

$\displaystyle\begin{matrix}\text{ \includegraphics[scale={1}]{sunset.pdf}
}\end{matrix}$ (34)

where the solid line indicates the propagator for $\phi^{abcd}$, and
$I_{\text{sunset}}$ is the loop integral associated to the diagram. By
carrying out the integral, one finds that

$\displaystyle\gamma_{\phi}^{\text{(ii)}}=\frac{A_{\phi}(d)}{N^{3}}+\mathcal{O}(\frac{1}{N^{6}})\,,\hskip
36.98857pt\text{where}\hskip
39.83385ptA_{\phi}(d)\equiv\frac{2\Gamma(d-2)\sin(\frac{\pi
d}{2})}{\pi\Gamma(\frac{d}{2}-2)\Gamma(\frac{d}{2}+1)}\,.$ (35)

The $1/N^{3}$ term in $\gamma_{\phi^{2}}$ is determined from the following
three diagrams:

$\displaystyle\begin{matrix}\text{ \includegraphics[scale={1}]{diagrams.pdf}
}\end{matrix}$ (36)

Through evaluation of these diagrams, one finds that

$\displaystyle\gamma_{\phi^{2}}^{\text{(ii)}}=4-d+\frac{B_{\phi^{2}}(d)}{N^{3}}+\mathcal{O}(\frac{1}{N^{6}})\,,\hskip
36.98857pt\text{where}\hskip
39.83385ptB_{\phi^{2}}(d)\equiv\frac{4\Gamma(d)\sin(\frac{\pi
d}{2})}{\pi\Gamma(\frac{d}{2}-1)\Gamma(\frac{d}{2}+1)}\,.$ (37)

Expanding (35) and (37) in $\epsilon=4-d$, we recover the values in row (ii)
of Table 1 To understand the other rows, we need to introduce additional field
content to the large $N$ effective theory.

####  2.2.4 Other fixed points

We now turn to the three large $N$ fixed points in orbit (iii). Without loss
of generality, we can study the fixed point with
$\lambda_{1}=-48\pi^{2}\epsilon$, $\lambda_{2}=48\pi^{2}\epsilon$, and
$\lambda_{3}=\lambda_{4}=\lambda_{5}=0$. We note that despite the negative
value of $\lambda_{1}$, the potential is in fact bounded below at this fixed
point, as can be seen by applying the Cauchy-Schwarz inequality to
$N^{2}$-component vectors $v$ and $w$ given by
$v^{cc^{\prime}}=\phi^{abc}\phi^{abc^{\prime}}$ and
$w^{cc^{\prime}}=\delta^{cc^{\prime}}$.

One could attempt to apply the Hubbard-Stranovich transformation to the
operator $\mathcal{O}_{2}$, but it is not the best way to proceed in the
present case. We should pick the large $N$ field content to transform under
irreducible representations of the $O(N)$ group.131313Insofar as the Hubbard-
Stratonovich transformation is a mathematical re-writing of the path integral,
one can perform the transform whichever way one pleases. The issue, we
believe, in using an intermediate field, call it
$\widetilde{\chi}^{c_{1}c_{2}}$, that does not transform as an irrep is that
one can no longer simply drop the square term $\widetilde{\chi}^{2}$ at a CFT
since this term will not just be a mass term, but will also contain cross
terms of fields that are irreps. This means we should apply the Hubbard-
Stratonovich transformation not to $\mathcal{O}_{2}$ alone but rather to a
specific linear combination of $\mathcal{O}_{1}$ and $\mathcal{O}_{2}$:

$\displaystyle\frac{g_{2}}{4!}\big{(}\mathcal{O}_{2}-\frac{1}{N}\mathcal{O}_{1}\big{)}=\,$
$\displaystyle\frac{3}{2g_{2}N^{3}}\bigg{(}\chi_{3}^{c_{1}c_{2}}-\frac{N^{3/2}g_{2}}{6}\Big{(}\phi^{abc_{1}}\phi^{abc_{2}}-\frac{1}{N}\delta^{c_{1}c_{2}}\phi^{abc}\phi^{abc}\Big{)}\bigg{)}^{2}$
$\displaystyle-\frac{3}{2g_{2}N^{3}}(\chi_{3}^{c_{1}c_{2}})^{2}+\frac{1}{2N^{3/2}}\chi_{3}^{c_{1}c_{2}}\Big{(}\phi^{abc_{1}}\phi^{abc_{2}}-\frac{\delta^{c_{1}c_{2}}}{N}\phi^{abc}\phi^{abc}\Big{)}\,.$

This particular way of re-expressing $\mathcal{O}_{1}$ and $\mathcal{O}_{2}$
has been chosen such that the new auxiliary field $\chi_{3}^{c_{1}c_{2}}$ in
the last line multiplies a combination of the fundamental fields that is
traceless and symmetric in the $c$ indices.

Taking the $(\chi_{3}^{c_{1}c_{2}})^{2}$ term to go away at criticality, we
are left with the following effective action:

$\displaystyle S^{\text{(iii)}}=\int
d^{d}x\bigg{(}\frac{1}{2}(\partial_{i}\phi^{abc})(\partial_{i}\phi^{abc})+\frac{\chi_{3}^{c_{1}c_{2}}}{2N^{3/2}}\Big{(}\phi^{abc_{1}}\phi^{abc_{2}}-\frac{\delta^{c_{1}c_{2}}}{N}\phi^{abc}\phi^{abc}\Big{)}\bigg{)}\,.$
(39)

The cubic interaction for $\chi_{3}^{c_{1}c_{2}}$ induces the following
propagator:

$\displaystyle\left<\chi_{3}^{c_{1}c_{2}}(k)\chi_{3}^{c_{3}c_{4}}(p)\right>_{0}=\delta^{(d)}(k+p)G^{c_{1}c_{2}c_{3}c_{4}}\frac{C_{\sigma}}{|p|^{d-4}}\,,$
(40) $\displaystyle\text{where}\hskip
17.07164ptG^{c_{1}c_{2}c_{3}c_{4}}=\frac{\delta^{c_{1}c_{3}}\delta^{c_{2}c_{4}}+\delta^{c_{1}c_{4}}\delta^{c_{2}c_{3}}}{2}-\frac{\delta^{c_{1}c_{2}}\delta^{c_{3}c_{4}}}{N}\,,\hskip
25.60747pt{\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@color<EMAIL_ADDRESS>(41)

so that $\chi_{3}^{c_{1}c_{2}}$ behaves as a traceless symmetric matrix field.
In the presence of such a field, the anomalous dimension for the fundamental
field recieves a contribution from the sunset diagram (34) with the $\sigma$
propagator replaced with the $\chi_{3}^{c_{1}d_{2}}$ propagator:

$\displaystyle D_{\text{sunset}}\rightarrow\,$
$\displaystyle\delta^{a_{1}a_{2}}\delta^{b_{1}b_{2}}G^{c_{1}c_{3}c_{2}c_{3}}\frac{1}{2}\bigg{(}\frac{\sigma}{N^{3/2}}\bigg{)}^{2}\,I_{\text{sunset}}$
(42)
$\displaystyle=\delta^{a_{1}a_{2}}\delta^{b_{1}b_{2}}\frac{N^{2}+N-2}{2N}\delta^{c_{1}c_{2}}\frac{1}{2}\bigg{(}\frac{\sigma}{N^{3/2}}\bigg{)}^{2}\,I_{\text{sunset}}=\frac{N^{2}}{2}D_{\text{sunset}}\Big{(}1+\mathcal{O}(\frac{1}{N})\Big{)}\,.$

The Hubbard-Stranovich transformation we just applied to $\mathcal{O}_{2}$ can
of course equally well be applied to $\mathcal{O}_{3}$ and $\mathcal{O}_{4}$,
by respectively introducing traceless symmetric matrices
$\chi_{2}^{b_{1}b_{2}}$ and $\chi_{1}^{a_{1}a_{2}}$.

The above considerations present a clear way of understanding the 16 large $N$
fixed points of $O(N)^{3}$ theory and the values for the anomalous dimension
$\gamma_{\phi}$ in Table 1. The possible field content of each large $N$ fixed
point is an $O(N)$ scalar $\sigma$ and up to three traceless symmetric
matrices $\chi$. Whenever the $\sigma$ field is present, it contributes a term
$\frac{A_{\phi}}{N^{3}}$ to the anomalous dimension $\gamma_{\phi}$. The
substitution (42) shows that for each traceless symmetric matrix present in
the effective large $N$ theory, the anomalous dimension receives a
contribution of $\frac{A_{\phi}}{2N}$. As for the anomalous dimension for
$\phi^{2}$, the $\sigma$ field makes a contribution of
$4-d+\frac{B_{\phi^{2}}}{N^{3}}$, and by replacing the internal $\sigma$
propagators in the diagrams (36) with traceless symmetric matrix propagators,
it is a simple exercise to show that each traceless symmetric matrix makes a
contribution $-\frac{B_{\phi^{2}}}{2N^{2}}$ to the anomalous dimension for
$\sigma$. For the large $N$ fixed points without $\sigma$ the $\epsilon$ term
in $\gamma_{\phi^{2}}$ should not be present. The relative sign flip in the
$1/N$ corrections to $\gamma_{\phi^{2}}$ may be viewed as ensuing from the
fact that, in the absence of $\sigma$, $\Delta_{\phi^{2}}$ is determined from
the correlators of $\phi^{abc}\phi^{abc}$ rather than $\sigma$. Since $\sigma$
couples to $\phi^{abc}\phi^{abc}$, their anomalous dimensions are opposite.

The rows in Table 1 where $\gamma_{\phi^{2}}$ does not contain an
$\mathcal{O}(N^{0})$ term $\epsilon$ correspond to fixed points without
$\sigma$ present, and these are precisely the fixed points for which the
coupling constants satisfy the relation
$\lambda_{1}=\lambda_{2}+\lambda_{3}+\lambda_{4}$, so that in these cases, on
eliminating $\mathcal{O}_{2}$, $\mathcal{O}_{3}$, $\mathcal{O}_{4}$ via
Hubbard-Stratonovich transformations in $\chi$ fields as in
(LABEL:introducingChi), $\mathcal{O}_{1}$ precisely gets eliminated as well.

The number of fixed points in each of the orbits (i) to (viii) is simply a
binomial coefficient associated to choices among the three traceless symmetric
matrices. Concretely, the fixed points have the following large $N$ field
content, which all interact with the fundamental fields $\phi^{abcd}$ via
cubic interactions, along with the following values for $\gamma_{\phi}$ and
$\gamma_{\phi^{2}}$:

  1. (i)

One fixed point: no $\sigma$ or $\chi$ field present, $O(N^{3})$ symmetry,

$\displaystyle\gamma_{\phi}^{\text{(i)}}=0\,,\hskip
156.49014pt\gamma_{\phi^{2}}^{\text{(i)}}=0\,.$ (43)

  2. (ii)

One fixed point: only $\sigma$ present, $O(N^{3})$ symmetry,

$\displaystyle\gamma_{\phi}^{\text{(ii)}}=\frac{A_{\phi}(d)}{N^{3}}+\mathcal{O}(\frac{1}{N^{6}})\,,\hskip
76.82243pt\gamma_{\phi^{2}}^{\text{(ii)}}=4-d+\frac{B_{\phi^{2}}(d)}{N^{3}}+\mathcal{O}(\frac{1}{N^{6}})\,.$
(44)

  3. (iii)

Three fixed points: one $\chi$ present, $\sigma$ absent,
$\displaystyle\frac{O(N^{2})\times O(N)}{\mathbb{Z}_{2}}$ symmetry,

$\displaystyle\gamma_{\phi}^{\text{(iii)}}=\frac{A_{\phi}(d)}{2N}+\mathcal{O}(\frac{1}{N^{2}})\,,\hskip
71.13188pt\gamma_{\phi^{2}}^{\text{(iii)}}=-\frac{B_{\phi^{2}}(d)}{2N}+\mathcal{O}(\frac{1}{N^{2}})\,.$
(45)

  4. (iv)

Three fixed points: one $\chi$ present, $\sigma$ present,
$\displaystyle\frac{O(N^{2})\times O(N)}{\mathbb{Z}_{2}}$ symmetry,

$\displaystyle\gamma_{\phi}^{\text{(iv)}}=\frac{A_{\phi}(d)}{2N}+\mathcal{O}(\frac{1}{N^{2}})\,,\hskip
71.13188pt\gamma_{\phi^{2}}^{\text{(iv)}}=4-d+\frac{B_{\phi^{2}}(d)}{2N}+\mathcal{O}(\frac{1}{N^{2}})\,.$
(46)

  5. (v)

Three fixed points: two $\chi$ present, $\sigma$ absent,
$\displaystyle\frac{O(N)^{3}\rtimes S_{2}}{(\mathbb{Z}_{2})^{2}}$ symmetry,

$\displaystyle\gamma_{\phi}^{\text{(v)}}=\frac{A_{\phi}(d)}{N}+\mathcal{O}(\frac{1}{N^{2}})\,,\hskip
73.97716pt\gamma_{\phi^{2}}^{\text{(v)}}=-\frac{B_{\phi^{2}}(d)}{N}+\mathcal{O}(\frac{1}{N^{2}})\,.$
(47)

  6. (vi)

Three fixed points: two $\chi$ present, $\sigma$ present,
$\displaystyle\frac{O(N)^{3}\rtimes S_{2}}{(\mathbb{Z}_{2})^{2}}$ symmetry,

$\displaystyle\gamma_{\phi}^{\text{(vi)}}=\frac{A_{\phi}(d)}{N}+\mathcal{O}(\frac{1}{N^{2}})\,,\hskip
68.2866pt\gamma_{\phi^{2}}^{\text{(vi)}}=4-d+\frac{B_{\phi^{2}}(d)}{N}+\mathcal{O}(\frac{1}{N^{2}})\,.$
(48)

  7. (vii)

One fixed point: all three $\chi$ present, $\sigma$ absent,
$\displaystyle\frac{O(N)^{3}\rtimes S_{3}}{(\mathbb{Z}_{2})^{2}}$ symmetry,

$\displaystyle\gamma_{\phi}^{\text{(vii)}}=\frac{3A_{\phi}(d)}{2N}+\mathcal{O}(\frac{1}{N^{2}})\,,\hskip
59.75078pt\gamma_{\phi^{2}}^{\text{(vii)}}=-\frac{3B_{\phi^{2}}(d)}{2N}+\mathcal{O}(\frac{1}{N^{2}})\,.$
(49)

  8. (viii)

One fixed point: all three $\chi$ and $\sigma$ present,
$\displaystyle\frac{O(N)^{3}\rtimes S_{3}}{(\mathbb{Z}_{2})^{2}}$ symmetry,

$\displaystyle\gamma_{\phi}^{\text{(viii)}}=\frac{3A_{\phi}(d)}{2N}+\mathcal{O}(\frac{1}{N^{2}})\,,\hskip
56.9055pt\gamma_{\phi^{2}}^{\text{(viii)}}=4-d+\frac{3B_{\phi^{2}}(d)}{2N}+\mathcal{O}(\frac{1}{N^{2}})\,.$
(50)

For the vector model fixed point, the effective action (30) can be used to
compute not just the leading $1/N$ corrections we have discussed here, but
also higher-order corrections to arbitrary order. We expect the same to apply
for the action (39), as well as for the effective actions for the fixed points
in orbit (iv), e.g.

$\displaystyle S^{\text{(iv)}}=\int
d^{d}x\bigg{(}\frac{1}{2}(\partial_{i}\phi^{abc})(\partial_{i}\phi^{abc})+\frac{1}{2N^{3/2}}\sigma\phi^{abc}\phi^{abc}+\frac{\chi_{3}^{c_{1}c_{2}}}{2N^{3/2}}\Big{(}\phi^{abc_{1}}\phi^{abc_{2}}-\frac{\delta^{c_{1}c_{2}}}{N}\phi^{abc}\phi^{abc}\Big{)}\bigg{)}\,.$
(51)

But while the fixed points in orbits (iii) and (iv) exhibit an $O(N^{2})$
symmetry that enforces $g_{5}$ to be zero even at finite $N$, the fixed points
in orbits (v) to (viii) enjoy no such symmetry, and for these fixed points
$\lambda_{5}$ is non-vanishing at subleading order in $1/N$. To capture
higher-order $1/N$ corrections, it will therefore be necessary to take the
melonic operator $\mathcal{O}_{5}$ into account.

The separate possibilities for the $\sigma$ and $\chi$ fields to be present or
absent at the fixed points can be made manifest in the large $N$ beta
functions by switching operator basis. First set $\lambda_{5}=0$ so as to only
study the RG flow among the couplings $\lambda_{1}$ to $\lambda_{4}$. For this
starting configuration, $\lambda_{5}$ will remain zero to leading order in
$1/N$ throughout the RG flow. Next, change operator basis from
$\mathcal{O}_{n}$, $n\in\\{1,2,3,4\\}$, to a basis given by

$\displaystyle\mathcal{O}_{1}\,,\hskip
28.45274pt\mathcal{O}_{n}-\mathcal{O}_{1}\hskip 8.53581pt\text{for
}n\in\\{2,3,4\\}\,,$ (52)

which corresponds to changing coupling constants from $\lambda_{n}$ to $h_{n}$
given by

$\displaystyle h_{1}=\lambda_{1}+\lambda_{2}+\lambda_{3}+\lambda_{4}\,,\hskip
42.67912pth_{n}=\lambda_{n}\hskip 11.38109pt\text{for }n\in\\{2,3,4\\}\,.$
(53)

This new basis diagonalizes the large $N$ beta functions:

$\displaystyle\beta_{h_{n}}=-\epsilon\,h_{n}+\frac{h_{n}^{2}}{48\pi^{2}}\hskip
14.22636pt\text{for }n\in\\{1,2,3,4\\}\,.$ (54)

Since the operators $\mathcal{O}_{1}$ to $\mathcal{O}_{4}$ are all vectorial,
meaning that the Feynman diagrams constructed out of them are dominated by
bubble diagrams at large $N$, the beta functions for the $h$ couplings
truncate at quadratic order, so that equation (54) is valid to all orders in
perturbation theory. The 16 fixed points correspond to the $2^{4}$ possible
options of independently choosing $h_{n}=0$ or $h_{n}=48\pi^{2}\epsilon$ for
each $n\in\\{1,2,3,4\\}$.

The above diagonalization clarifies not only the number of fixed points but
also their respective levels of stability under
$O(N)^{r}/(\mathbb{Z}_{2})^{r-1}$ symmetric deformations since $h_{n}=0$ is an
unstable fixed point to $\beta_{h_{n}}$, while $h_{n}=48\pi^{2}$ is stable.
This means that for each of the Hubbard-Stratonovich fields $\sigma$,
$\chi_{1}$, $\chi_{2}$, $\chi_{3}$ present at a fixed point there is a stable
direction in the RG flow. The most stable fixed point is therefore fixed point
(viii), which is stable along all four $h$-directions. But even this fixed
point has one unstable direction as all the fixed points are unstable under
deformations wrt. $\mathcal{O}_{5}$. Therefore, generic configurations of
couplings do not flow to a perturbative fixed point, even after tuning away
the quadratic operator $\phi^{abc}\phi^{abc}$. At minimum one must also tune
$g_{5}$. Note however that fixed point (ii) and the three fixed points (iv)
are all stable under quartic deformations that respect the symmetries present
at those fixed points.

###  2.3 The $O(N)^{4}$ model

We now turn to the quartic scalar theory of rank-four tensors. We take the
action to be

$\displaystyle S=\int
d^{4-\epsilon}x\bigg{(}\frac{1}{2}(\partial_{i}\phi^{abcd})(\partial_{i}\phi^{abcd})+\frac{1}{4!}\sum_{n=1}^{14}g_{n}\,\mathcal{O}_{n}\bigg{)}\,,$
(55)

where the 14 operators in the potential are given by

$\displaystyle\mathcal{O}_{1}=\Big{(}\phi^{abcd}\phi^{abcd}\Big{)}^{2}\,,$
(56)
$\displaystyle\mathcal{O}_{2}=\phi^{a_{1}b_{1}c_{1}d_{1}}\phi^{a_{1}b_{1}c_{1}d_{2}}\phi^{a_{2}b_{2}c_{2}d_{1}}\phi^{a_{2}b_{2}c_{2}d_{2}}\,,$
$\displaystyle\mathcal{O}_{3}=\phi^{a_{1}b_{1}c_{1}d_{1}}\phi^{a_{1}b_{1}c_{2}d_{1}}\phi^{a_{2}b_{2}c_{1}d_{2}}\phi^{a_{2}b_{2}c_{2}d_{2}}\,,$
$\displaystyle\mathcal{O}_{4}=\phi^{a_{1}b_{1}c_{1}d_{1}}\phi^{a_{1}b_{2}c_{1}d_{1}}\phi^{a_{2}b_{1}c_{2}d_{2}}\phi^{a_{2}b_{2}c_{2}d_{2}}\,,$
$\displaystyle\mathcal{O}_{5}=\phi^{a_{1}b_{1}c_{1}d_{1}}\phi^{a_{2}b_{1}c_{1}d_{1}}\phi^{a_{1}b_{2}c_{2}d_{2}}\phi^{a_{2}b_{2}c_{2}d_{2}}\,,$
$\displaystyle\mathcal{O}_{6}=\phi^{a_{1}b_{1}c_{1}d_{1}}\phi^{a_{1}b_{1}c_{2}d_{2}}\phi^{a_{2}b_{2}c_{1}d_{1}}\phi^{a_{2}b_{2}c_{2}d_{2}}\,,$
$\displaystyle\mathcal{O}_{7}=\phi^{a_{1}b_{1}c_{1}d_{1}}\phi^{a_{1}b_{2}c_{1}d_{2}}\phi^{a_{2}b_{1}c_{2}d_{1}}\phi^{a_{2}b_{2}c_{2}d_{2}}\,,$
(58)
$\displaystyle\mathcal{O}_{8}=\phi^{a_{1}b_{1}c_{1}d_{1}}\phi^{a_{2}b_{1}c_{1}d_{2}}\phi^{a_{1}b_{2}c_{2}d_{1}}\phi^{a_{2}b_{2}c_{2}d_{2}}\,,$
$\displaystyle\mathcal{O}_{9}=\phi^{a_{1}b_{1}c_{1}d_{1}}\phi^{a_{1}b_{1}c_{2}d_{2}}\phi^{a_{2}b_{2}c_{1}d_{2}}\phi^{a_{2}b_{2}c_{2}d_{1}}\,,$
$\displaystyle\mathcal{O}_{10}=\phi^{a_{1}b_{1}c_{1}d_{1}}\phi^{a_{1}b_{2}c_{1}d_{2}}\phi^{a_{2}b_{1}c_{2}d_{2}}\phi^{a_{2}b_{2}c_{2}d_{1}}\,,$
$\displaystyle\mathcal{O}_{11}=\phi^{a_{1}b_{1}c_{1}d_{1}}\phi^{a_{1}b_{2}c_{2}d_{1}}\phi^{a_{2}b_{1}c_{2}d_{2}}\phi^{a_{2}b_{2}c_{1}d_{2}}\,,$
$\displaystyle\mathcal{O}_{12}=\phi^{a_{1}b_{1}c_{1}d_{1}}\phi^{a_{2}b_{1}c_{1}d_{2}}\phi^{a_{1}b_{2}c_{2}d_{2}}\phi^{a_{2}b_{2}c_{2}d_{1}}\,,$
$\displaystyle\mathcal{O}_{13}=\phi^{a_{1}b_{1}c_{1}d_{1}}\phi^{a_{2}b_{1}c_{2}d_{1}}\phi^{a_{1}b_{2}c_{2}d_{2}}\phi^{a_{2}b_{2}c_{1}d_{2}}\,,$
$\displaystyle\mathcal{O}_{14}=\phi^{a_{1}b_{1}c_{1}d_{1}}\phi^{a_{2}b_{2}c_{1}d_{1}}\phi^{a_{1}b_{2}c_{2}d_{2}}\phi^{a_{2}b_{1}c_{2}d_{2}}\,.$

Diagrammatically, we can depict these operators as shown in Figure 3.

$\displaystyle\begin{matrix}\text{
\includegraphics[scale={1}]{quarticRank4ops.pdf}}\end{matrix}$ Figure 3:
Quartic singlet operators in $O(N)^{4}$ theory

By standard Feynman diagrammatic methods, one can compute the perturbative
beta functions and anomalous dimensions of this model. The expressions one
arrives at are rather lengthy owing to the large number of operators that mix
in this theory, but we obtain a significant simplification by taking the large
$N$ limit of the theory in the next subsection. Another means to simplify the
model is to focus on the sector of the theory that is symmetric with respect
to the four tensor indices, which we will do in Subsection 2.3.2.

####  2.3.1 Large $N$ limit

The first step to studying the large $N$ limit of the model is to set down the
appropriate scalings of the couplings constants. For this model there exists a
unique optimal scaling

$\displaystyle g_{i}=\frac{\lambda_{i}}{N^{e_{i}}}\,,$ (60)

where the exponents are given by

$\displaystyle e=\\{4,\hskip 5.69054pt3,3,3,3,\hskip 5.69054pt2,2,2,\hskip
5.69054pt2,2,2,2,2,2\\}\,.$ (61)

In the supplementary Mathematica notebook we list the large $N$ beta functions
and anomalous dimensions obtained from this scaling to third order in the
coupling constants. By setting to zero the quadratic beta functions, we
determine the Wilson-Fisher fixed points with coupling constants proportional
to $\epsilon$. One can subsequently use the cubic terms in the beta functions
to compute order $\epsilon^{2}$ corrections. In total the model has 32 large
$N$ fixed points including the trivial one, and they can be grouped into ten
orbits according to the action of an $S_{4}$ group that permutes the four
$O(N)$ groups of the model:

  1. (i)

one fixed point: The trivial fixed point.

  2. (ii)

one fixed point: $\lambda_{1}=48\pi^{2}\epsilon$. This is the familiar vector
model fixed point.

  3. (iii)

four fixed points: $\lambda_{1}=-48\pi^{2}\epsilon$, one of $\lambda_{2}$ to
$\lambda_{5}$ equal to $48\pi^{2}\epsilon$,

  4. (iv)

four fixed points: one of $\lambda_{2}$ to $\lambda_{5}$ equal to
$48\pi^{2}\epsilon$,

  5. (v)

six fixed points: $\lambda_{1}=-96\pi^{2}\epsilon$, $\lambda_{2}$ to
$\lambda_{5}$ all equal to $48\pi^{2}\epsilon$ except for any two of them,

  6. (vi)

six fixed points: $\lambda_{1}=-48\pi^{2}\epsilon$, $\lambda_{2}$ to
$\lambda_{5}$ all equal to $48\pi^{2}\epsilon$ except for any two of them,

  7. (vii)

four fixed points: $\lambda_{1}=-144\pi^{2}\epsilon$, $\lambda_{2}$ to
$\lambda_{5}$ all equal to $48\pi^{2}\epsilon$ except for any one of them,

  8. (viii)

four fixed points: $\lambda_{1}=-96\pi^{2}\epsilon$, $\lambda_{2}$ to
$\lambda_{5}$ all equal to $48\pi^{2}\epsilon$ except for any one of them,

  9. (ix)

one fixed point: $\lambda_{1}=-192\pi^{2}\epsilon$,
$\lambda_{2}=\lambda_{3}=\lambda_{4}=\lambda_{5}=48\pi^{2}\epsilon$,

  10. (x)

one fixed point: $\lambda_{1}=-144\pi^{2}\epsilon$,
$\lambda_{2}=\lambda_{3}=\lambda_{4}=\lambda_{5}=48\pi^{2}\epsilon$.

For each fixed point, the unspecified couplings are zero. Note in particular
that the couplings $\lambda_{6}$ to $\lambda_{14}$ are all zero to leading
order in $1/N$ at all the fixed points. The leading anomalous dimensions for
$\phi$ and $\phi^{2}$ at each of the fixed points are listed in Table 2.

| $\gamma_{\phi}-\mathcal{O}(\epsilon^{4})$ | $\gamma_{\phi^{2}}-\mathcal{O}(\epsilon^{3})$
---|---|---
(i) | 0 | 0
(ii) | $\frac{1}{N^{4}}(\frac{1}{4}\epsilon^{2}-\frac{1}{16}\epsilon^{3})+\mathcal{O}(\frac{1}{N^{8}})$ | $\epsilon+\frac{1}{N^{4}}(-6\epsilon+\frac{13}{2}\epsilon^{2})+\mathcal{O}(\frac{1}{N^{8}})$
(iii) |  $\frac{1}{N^{2}}(\frac{1}{8}\epsilon^{2}-\frac{1}{32}\epsilon^{3})+\mathcal{O}(\frac{1}{N^{3}})$ |  $\frac{1}{N^{2}}(3\epsilon-\frac{13}{4}\epsilon^{2})+\mathcal{O}(\frac{1}{N^{3}})$
(vi) | $\frac{1}{N^{2}}(\frac{1}{8}\epsilon^{2}-\frac{1}{32}\epsilon^{3})+\mathcal{O}(\frac{1}{N^{3}})$ | $\epsilon+\frac{1}{N^{2}}(-3\epsilon+\frac{13}{4}\epsilon^{2})+\mathcal{O}(\frac{1}{N^{3}})$
(v) |  $\frac{1}{N^{2}}(\frac{1}{4}\epsilon^{2}-\frac{1}{16}\epsilon^{3})+\mathcal{O}(\frac{1}{N^{3}})$ |  $\frac{1}{N^{2}}(6\epsilon-\frac{13}{2}\epsilon^{2})+\mathcal{O}(\frac{1}{N^{3}})$
(vi) | $\frac{1}{N^{2}}(\frac{1}{4}\epsilon^{2}-\frac{1}{16}\epsilon^{3})+\mathcal{O}(\frac{1}{N^{3}})$ | $\epsilon+\frac{1}{N^{2}}(-6\epsilon+\frac{13}{2}\epsilon^{2})+\mathcal{O}(\frac{1}{N^{3}})$
(vii) |  $\frac{1}{N^{2}}(\frac{3}{8}\epsilon^{2}-\frac{3}{32}\epsilon^{3})+\mathcal{O}(\frac{1}{N^{3}})$ |  $\frac{1}{N^{2}}(9\epsilon-\frac{39}{4}\epsilon^{2})+\mathcal{O}(\frac{1}{N^{3}})$
(viii) | $\frac{1}{N^{2}}(\frac{3}{8}\epsilon^{2}-\frac{3}{32}\epsilon^{3})+\mathcal{O}(\frac{1}{N^{3}})$ | $\epsilon+\frac{1}{N^{2}}(-9\epsilon+\frac{39}{4}\epsilon^{2})+\mathcal{O}(\frac{1}{N^{3}})$
(ix) |  $\frac{1}{N^{2}}(\frac{1}{2}\epsilon^{2}-\frac{1}{8}\epsilon^{3})+\mathcal{O}(\frac{1}{N^{3}})$ |  $\frac{1}{N^{2}}(12\epsilon-13\epsilon^{2})+\mathcal{O}(\frac{1}{N^{3}})$
(x) | $\frac{1}{N^{2}}(\frac{1}{2}\epsilon^{2}-\frac{1}{8}\epsilon^{3})+\mathcal{O}(\frac{1}{N^{3}})$ | $\epsilon+\frac{1}{N^{2}}(-12\epsilon+13\epsilon^{2})+\mathcal{O}(\frac{1}{N^{3}})$

Table 2: Leading anomalous dimensions at the large $N$ fixed points of quartic
$O(N)^{4}$ theory

Since we know the perturbative fixed points exactly in $N$, it is
straightforward to expand the values in the table to higher orders in $1/N$ if
desired, as long as we remain at the same order in $\epsilon$. Meanwhile, the
exact $d$ dependencies of the leading large $N$ values be determined by a
simple adjustment of the analysis we performed on the rank-three model: only
the vectorial operators $\mathcal{O}_{1}$ to $\mathcal{O}_{5}$ are present at
large $N$, and by introducing four traceless symmetric fields
$\chi_{1}^{a_{1}a_{2}}$, $\chi_{2}^{b_{1}b_{2}}$, $\chi_{3}^{c_{1}c_{2}}$, and
$\chi_{4}^{d_{1}d_{2}}$ we can perform Hubbard-Stratonovich transformations on
the linear combinations $\mathcal{O}_{i}-\frac{1}{N}\mathcal{O}_{1}$ for
$i\in\\{2,3,4,5\\}$. For the fixed points satisfying the condition
$\lambda_{1}=\lambda_{2}+\lambda_{3}+\lambda_{4}+\lambda_{5}$, the Hubbard-
Stratonovich transformations in $\chi$ eliminate $\mathcal{O}_{1}$ as well,
whereas otherwise we need to also introduce a $\sigma$ field. Each $\chi$
field makes a contribution of $\frac{A_{\phi}(d)}{2N^{2}}$ to $\gamma_{\phi}$,
and a contribution of $\frac{B_{\phi^{2}}(d)}{2N^{2}}$ to $\gamma_{\phi^{2}}$
if $\sigma$ is present and of $\frac{-B_{\phi^{2}}(d)}{2N^{2}}$ if $\sigma$ is
absent, where $A_{\phi}(d)$ and $B_{\phi^{2}}(d)$ are given in equations (35)
and (37) respectively. The possible inclusion or exclusion of each of the five
fields $\sigma$, $\chi_{1}$, $\chi_{2}$, $\chi_{3}$, $\chi_{4}$, yields up the
total $2^{5}$ fixed points. The number of fixed points in each of the orbits
(i) to (ix) are the binomial coefficients associated to choices among the four
$\chi$ fields.

Concretely, the fixed points have the following large $N$ field content, which
all interact with the fundamental fields $\phi^{abcd}$ via cubic interactions,
along with the following anomalous dimensions:

  1. (i)

One fixed point: no $\sigma$ or $\chi$ present, $O(N^{4})$ symmetry,

$\displaystyle\gamma_{\phi}^{\text{(i)}}=0\,,\hskip
150.79959pt\gamma_{\phi^{2}}^{\text{(i)}}=0\,.$ (62)

  2. (ii)

One fixed point: only $\sigma$ present, $O(N^{4})$ symmetry,

$\displaystyle\gamma_{\phi}^{\text{(ii)}}=\frac{A_{\phi}(d)}{N^{4}}+\mathcal{O}(\frac{1}{N^{8}})\,,\hskip
73.97716pt\gamma_{\phi^{2}}^{\text{(ii)}}=d-4+\frac{B_{\phi^{2}}(d)}{N^{4}}+\mathcal{O}(\frac{1}{N^{8}})\,.$
(63)

  3. (iii)

Four fixed points: one $\chi$ present, $\sigma$ absent,
$\displaystyle\frac{O(N^{3})\times O(N)}{\mathbb{Z}_{2}}$ symmetry,

$\displaystyle\gamma_{\phi}^{\text{(iii)}}=\frac{A_{\phi}(d)}{2N^{2}}+\mathcal{O}(\frac{1}{N^{3}})\,,\hskip
71.13188pt\gamma_{\phi^{2}}^{\text{(iii)}}=-\frac{B_{\phi^{2}}(d)}{2N^{2}}+\mathcal{O}(\frac{1}{N^{3}})\,.$
(64)

  4. (iv)

Four fixed points: one $\chi$ present, $\sigma$ present,
$\displaystyle\frac{O(N^{3})\times O(N)}{\mathbb{Z}_{2}}$ symmetry,

$\displaystyle\gamma_{\phi}^{\text{(iv)}}=\frac{A_{\phi}(d)}{2N^{2}}+\mathcal{O}(\frac{1}{N^{3}})\,,\hskip
68.2866pt\gamma_{\phi^{2}}^{\text{(iv)}}=4-d+\frac{B_{\phi^{2}}(d)}{2N^{2}}+\mathcal{O}(\frac{1}{N^{3}})\,.$
(65)

  5. (v)

Six fixed points: two $\chi$ present, $\sigma$ absent,
$\displaystyle\frac{O(N^{2})\times O(N)^{2}}{(\mathbb{Z}_{2})^{2}}\rtimes
S_{2}$ symmetry,

$\displaystyle\gamma_{\phi}^{\text{(v)}}=\frac{A_{\phi}(d)}{N^{2}}+\mathcal{O}(\frac{1}{N^{3}})\,,\hskip
71.13188pt\gamma_{\phi^{2}}^{\text{(v)}}=-\frac{B_{\phi^{2}}(d)}{N^{2}}+\mathcal{O}(\frac{1}{N^{3}})\,.$
(66)

  6. (vi)

Six fixed points: two $\chi$ present, $\sigma$ present,
$\displaystyle\frac{O(N^{2})\times O(N)^{2}}{(\mathbb{Z}_{2})^{2}}\rtimes
S_{2}$ symmetry,

$\displaystyle\gamma_{\phi}^{\text{(vi)}}=\frac{A_{\phi}(d)}{N^{2}}+\mathcal{O}(\frac{1}{N^{3}})\,,\hskip
65.44133pt\gamma_{\phi^{2}}^{\text{(vi)}}=4-d+\frac{B_{\phi^{2}}(d)}{N^{2}}+\mathcal{O}(\frac{1}{N^{3}})\,.$
(67)

  7. (vii)

Four fixed points: three $\chi$ present, $\sigma$ absent,
$\displaystyle\frac{O(N)^{4}\rtimes S_{3}}{(\mathbb{Z}_{2})^{3}}$ symmetry,

$\displaystyle\gamma_{\phi}^{\text{(vii)}}=\frac{3A_{\phi}(d)}{2N^{2}}+\mathcal{O}(\frac{1}{N^{3}})\,,\hskip
59.75078pt\gamma_{\phi^{2}}^{\text{(vii)}}=-\frac{3B_{\phi^{2}}(d)}{2N^{2}}+\mathcal{O}(\frac{1}{N^{3}})\,.$
(68)

  8. (viii)

Four fixed points: three $\chi$ present, $\sigma$ present,
$\displaystyle\frac{O(N)^{4}\rtimes S_{3}}{(\mathbb{Z}_{2})^{3}}$ symmetry,

$\displaystyle\gamma_{\phi}^{\text{(viii)}}=\frac{3A_{\phi}(d)}{2N^{2}}+\mathcal{O}(\frac{1}{N^{3}})\,,\hskip
56.9055pt\gamma_{\phi^{2}}^{\text{(viii)}}=4-d+\frac{3B_{\phi^{2}}(d)}{2N^{2}}+\mathcal{O}(\frac{1}{N^{3}})\,.$
(69)

  9. (ix)

One fixed point: all four $\chi$ present, $\sigma$ absent,
$\displaystyle\frac{O(N)^{4}\rtimes S_{4}}{(\mathbb{Z}_{2})^{3}}$ symmetry,

$\displaystyle\gamma_{\phi}^{\text{(ix)}}=\frac{2A_{\phi}(d)}{N^{2}}+\mathcal{O}(\frac{1}{N^{3}})\,,\hskip
59.75078pt\gamma_{\phi^{2}}^{\text{(ix)}}=-\frac{2B_{\phi^{2}}(d)}{N^{2}}+\mathcal{O}(\frac{1}{N^{3}})\,.$
(70)

  10. (x)

One fixed point: all four $\chi$ and $\sigma$ present,
$\displaystyle\frac{O(N)^{4}\rtimes S_{4}}{(\mathbb{Z}_{2})^{3}}$ symmetry,

$\displaystyle\gamma_{\phi}^{\text{(x)}}=\frac{2A_{\phi}(d)}{N^{2}}+\mathcal{O}(\frac{1}{N^{3}})\,,\hskip
62.59605pt\gamma_{\phi^{2}}^{\text{(x)}}=4-d+\frac{2B_{\phi^{2}}(d)}{N^{2}}+\mathcal{O}(\frac{1}{N^{3}})\,.$
(71)

For the fixed points in the first four orbits, the presence of an $O(N^{3})$
symmetry bars the operators $\mathcal{O}_{6}$ to $\mathcal{O}_{14}$ from being
generated by renormalization, even at finite $N$. For this reason, Feynman
rules derived from the rank-four analogs of the actions (39) and (51) should
correctly capture $1/N$ corrections to arbitrary order. The $O(N^{2})$
symmetry of the fixed points in orbits (v) and (vi) prevents the melonic
operators $\mathcal{O}_{9}$ to $\mathcal{O}_{14}$ from being generated, but
the matricial operators $\mathcal{O}_{6}$, $\mathcal{O}_{7}$, and
$\mathcal{O}_{8}$ are present at subleading order in $1/N$. For the fixed
points in orbits (vii) to (x), all of the operators $\mathcal{O}_{6}$ to
$\mathcal{O}_{14}$ are present at subleading order. Because of the presence of
matricial operators, the task of developing a $1/N$ expansion to arbitrary
order for the fixed points in orbits (v) to (x) is a difficult endeavour,
certainly more so than the same task for orbits (v) to (viii) in the rank-
three case, where only the contributions of a melonic operator has to be
incorporated.

We can clarify the above considerations by performing a change of operator
basis. First we set $\lambda_{6}=\lambda_{7}=...=\lambda_{14}=0$ to study only
the flow among the couplings $\lambda_{1}$ to $\lambda_{5}$. It can be checked
from the large $N$ beta functions that for this starting configuration,
$\lambda_{6}$ to $\lambda_{14}$ will remain zero throughout the RG flow. Next
we change operator basis from $\mathcal{O}_{n}$, $n\in\\{1,2,3,4,5\\}$, to a
basis given by

$\displaystyle\mathcal{O}_{1}\,,\hskip
28.45274pt\mathcal{O}_{n}-\mathcal{O}_{1}\hskip 8.53581pt\text{for
}n\in\\{2,3,4,5\\}\,,$ (72)

which corresponds to the following change of coupling constants:

$\displaystyle
h_{1}=\lambda_{1}+\lambda_{2}+\lambda_{3}+\lambda_{4}+\lambda_{5}\,,\hskip
28.45274pth_{n}=\lambda_{n}\hskip 8.53581pt\text{for }n\in\\{2,3,4,5\\}\,.$
(73)

This new basis diagonalizes the large $N$ beta functions:

$\displaystyle\beta_{h_{n}}=-\epsilon h_{n}+\frac{h_{n}^{2}}{48\pi^{2}}\hskip
14.22636pt\text{for }i\in\\{1,2,3,4,5\\}\,.$ (74)

The five quadratic equations (74) are true only at leading order in $1/N$ but
to all orders in $h_{n}$.

All of the fixed points are unstable under deformations with respect to any of
the nine non-vectorial operators $\mathcal{O}_{6}$ to $\mathcal{O}_{14}$, so a
tremendous amount of tuning is required to reach any of the fixed points if we
only impose $O(N)^{4}/(\mathbb{Z}_{2})^{3}$ symmetry. But the fixed point (ii)
and the four fixed points (iv) are stable under quartic deformations that
respect their respective symmetries, and the other fixed points too have
lessened numbers of relevant deformations under their own symmetries.

$g_{a}/\pi^{2}-\mathcal{O}(\epsilon^{3})$ | $g_{b}/\pi^{2}-\mathcal{O}(\epsilon^{3})$ | $g_{c}/\pi^{2}-\mathcal{O}(\epsilon^{3})$ | $g_{d}/\pi^{2}-\mathcal{O}(\epsilon^{3})$
---|---|---|---
$2\epsilon+\frac{31}{48}\epsilon^{2}$ | 0 | 0 | 0
$\frac{14}{3}\epsilon+\frac{35}{27}\epsilon^{2}$ | $-16\epsilon-8\epsilon^{2}$ | $12\epsilon+6\epsilon^{2}$ | 0
$\frac{9}{2}\epsilon+\frac{513}{256}\epsilon^{2}$ | $-12\epsilon-\frac{171}{32}\epsilon^{2}$ | $9\epsilon+\frac{513}{128}\epsilon^{2}$ | 0
$\frac{23}{7}\epsilon-\frac{929}{1029}\epsilon^{2}$ | $-\frac{72}{7}\epsilon+\frac{984}{343}\epsilon^{2}$ | $\frac{90}{7}\epsilon+\frac{2046}{343}\epsilon^{2}$ | $-\frac{36}{7}\epsilon-\frac{2784}{343}\epsilon^{2}$
$\frac{27}{8}\epsilon-\frac{621}{4096}\epsilon^{2}$ | $-9\epsilon+\frac{207}{512}\epsilon^{2}$ | $\frac{45}{4}\epsilon+\frac{13\,941}{2048}\epsilon^{2}$ | $-\frac{9}{2}\epsilon-\frac{7281}{1024}\epsilon^{2}$
$\frac{9}{5}\epsilon+\frac{27}{25}\epsilon^{2}$ | $-\frac{24}{5}\epsilon-\frac{72}{25}\epsilon^{2}$ | $\frac{54}{5}\epsilon+\frac{162}{25}\epsilon^{2}$ | $-\frac{36}{5}\epsilon-\frac{108}{25}\epsilon^{2}$
$\frac{13}{6}\epsilon+\frac{1241}{1728}\epsilon^{2}$ | $-4\epsilon-\frac{95}{24}\epsilon^{2}$ | $9\epsilon+\frac{285}{32}\epsilon^{2}$ | $-6\epsilon-\frac{95}{16}\epsilon^{2}$

Table 3: Locations of peturbative fixed points in quartic $O(2)^{4}\rtimes
S_{4}$ theory

####  2.3.2 Reduced System with $S_{4}$ theory

$\displaystyle\begin{matrix}\scalebox{1.11}{ \text{ {
\fcolorbox{gray}{white}{\includegraphics[scale=0.7]{rank4plot-1.png}} }
}}\end{matrix}$ Figure 4: Values of the coupling $g_{a}$ at Wilson-Fisher
fixed points in rank-four scalar theory with $S_{4}\ltimes
O(N)^{4}/(\mathbb{Z}_{2})^{3}$ symmetry in $d=4-\epsilon$ dimensions for
different values of $N$. The dotted lines indicate the asymptotic values for
$g_{a}$ at the large $N$ fixed points.

The 16-dimensional RG flow of the quartic model (55) contains a four-
dimensional invariant subspace, owing to the fact any configuration of
coupling constants that is symmetric with respect to a global $S_{4}$ symmetry
that permutes indices will remain symmetric under the RG flow. The flow of
this subspace is associated to the following four operators:

$\displaystyle\mathcal{O}_{a}=\,$ $\displaystyle\mathcal{O}_{1}\,,$
$\displaystyle\mathcal{O}_{b}=\,$
$\displaystyle\frac{1}{4}\Big{(}\mathcal{O}_{2}+\mathcal{O}_{3}+\mathcal{O}_{4}+\mathcal{O}_{5}\Big{)}\,,$
$\displaystyle\mathcal{O}_{c}=\,$
$\displaystyle\frac{1}{3}\Big{(}\mathcal{O}_{6}+\mathcal{O}_{7}+\mathcal{O}_{8}\Big{)}\,,$
$\displaystyle\mathcal{O}_{d}=\,$
$\displaystyle\frac{1}{6}\Big{(}\mathcal{O}_{9}+\mathcal{O}_{10}+\mathcal{O}_{11}+\mathcal{O}_{12}+\mathcal{O}_{13}+\mathcal{O}_{14}\Big{)}\,.$

The theory living in the invariant subspace is described by the action

$\displaystyle S=\int
d^{4-\epsilon}x\bigg{(}\frac{1}{2}(\partial_{i}\phi^{abcd})(\partial_{i}\phi^{abcd})+\frac{1}{4!}\Big{(}g_{a}\,\mathcal{O}_{a}+g_{b}\,\mathcal{O}_{a}+g_{c}\,\mathcal{O}_{a}+g_{d}\,\mathcal{O}_{a}\Big{)}\bigg{)}\,.$
(76)

Rescaling the coupling constants to obtain a smooth large $N$ limit,

$\displaystyle g_{a}=\frac{\lambda_{a}}{N^{4}}\,,\hskip
42.67912ptg_{b}=\frac{\lambda_{b}}{N^{3}}\,,\hskip
42.67912ptg_{c}=\frac{\lambda_{c}}{N^{2}}\,,\hskip
42.67912ptg_{d}=\frac{\lambda_{d}}{N^{2}}\,,$ (77)

the perturbative fixed points of the reduced system at large $N$ are precisely
the four fixed points of the full system that are symmetric under the global
$S_{4}$ symmetry:

(i) $\displaystyle\lambda_{a}=\lambda_{b}=\lambda_{c}=\lambda_{d}=0\,,$ (ii)
$\displaystyle\lambda_{a}=48\pi^{2}\epsilon\,,\hskip
44.10176pt\lambda_{b}=\lambda_{c}=\lambda_{d}=0\,,$ (ix)
$\displaystyle\lambda_{a}=-192\pi^{2}\epsilon\,,\hskip
28.45274pt\lambda_{b}=192\pi^{2}\epsilon\,,\hskip
28.45274pt\lambda_{c}=\lambda_{d}=0\,,$ (x)
$\displaystyle\lambda_{a}=-144\pi^{2}\epsilon\,,\hskip
28.45274pt\lambda_{b}=192\pi^{2}\epsilon\,,\hskip
28.45274pt\lambda_{c}=\lambda_{d}=0\,.$

Because of the relative simplicity of the reduced system compared to the full
system, it is not difficult to also chart out the finite $N$ fixed points of
the model. Studying the beta functions at quadratic order in the coupling
constants but exactly in $N$, one finds that the three non-trivial large $N$
fixed points extend down to finite $N$ values for all $N\geq 7$, but that the
fixed points (ix) and (x) merge and disappear in a saddle-node bifurcation at
$N\approx 6.247$. For $3\leq N\leq 6$, only the trivial fixed point and the
vector model fixed point are present. At $N=2$, seven non-trivial fixed points
are present, situated at the values of coupling constants listed in Table 3.
Figure 4 displays the perturbative fixed points of the model as a function of
$N$.

###  2.4 The $O(N)^{5}$ model

We turn now to the quartic theory of rank-five tensors, with action given by

$\displaystyle S=\int
d^{4-\epsilon}x\bigg{(}\frac{1}{2}(\partial_{i}\phi^{abcde})(\partial_{i}\phi^{abcde})+\frac{1}{4!}\sum_{n}g_{n}\,\mathcal{O}_{n}\bigg{)}\,.$
(79)

The sum over $n$ runs over 41 distinct quartic operators and their associated
couplings. If we do not distinguish between indices, there are five possible
ways of performing index contractions, and they are depicted diagrammatically
below, along with the number of operators each of them gives rise to.

$\displaystyle\begin{matrix}\text{
\includegraphics[scale={1}]{quarticRank5opTypes.pdf} }\end{matrix}$

The number of operators associated to a diagram equals the number of distinct
ways of colouring the graph with five colours. It is not important for the
reader to know the precise manner in which we associate an operator to a
colouring, but for the sake of clarity and completeness, we nonetheless
provide the identifications in Figure 5.

$\displaystyle\begin{matrix}\text{
\includegraphics[scale={0.68}]{quarticRank5figs.pdf} }\end{matrix}$ Figure 5:
Quartic singlet operators in $O(N)^{5}$ theory

####  2.4.1 Large $N$ limit

The optimal large $N$ scalings for the operators are given by

$\displaystyle g_{1}=\frac{\lambda_{1}}{N^{5}}\,,\hskip
22.76219ptg_{2,i}=\frac{\lambda_{2,i}}{N^{4}}\,,\hskip
22.76219ptg_{3,ij}=\frac{\lambda_{3,ij}}{N^{3}}\,,\hskip
22.76219ptg_{4,ij}=\frac{\lambda_{4,ij}}{N^{3}}\,,\hskip
22.76219ptg_{5,i,jk}=\frac{\lambda_{5,i,jk}}{N^{5/2}}\,.$ (80)

The number of large $N$ Wilson-Fisher fixed points increases rather
dramatically compared with the rank-four theory. The rank-five model has a
total of $10^{26}=67\,108\,864$ perturbative fixed points in the $\epsilon$
expansion including the trivial fixed point. The couplings $\lambda_{5,i,jk}$
vanish on all these fixed points. The set of fixed points and the values of
their non-vanishing couplings admit a simple explanation in terms of the
following 26 Hubbard-Stratonovich fields:

  * •

One $O(N)$ singlet $\sigma$, arising from integrating out $\mathcal{O}_{1}$.

  * •

Five traceless symmetric fields $\chi_{i}^{i_{1}i_{2}}$, arising from
integrating out $\mathcal{O}_{2,i}-\frac{1}{N}\mathcal{O}_{1}$ where
$i\in\\{a,b,c,d,e\\}$.

  * •

Ten doubly antisymmetric fields $\tau_{ij}^{i_{1}i_{2}\,j_{1}j_{2}}$, where
$i_{1}$ and $i_{2}$ transform antisymmetrically under one $O(N)$ group, while
$j_{1}$ and $j_{2}$ transform asymmetrically under another $O(N)$ group. The
ten versions of these fields correspond to the $\binom{5}{2}$ choices of two
out of the total five $O(N)$ groups. The $\tau_{ij}$ field integrates out the
linear combination $\mathcal{O}_{3,ij}-\mathcal{O}_{4,ij}$. For example,
consider the two operators

$\displaystyle\mathcal{O}_{3,de}=\,$
$\displaystyle\phi^{a_{1}b_{1}c_{1}d_{1}e_{1}}\phi^{a_{1}b_{1}c_{1}d_{2}e_{2}}\phi^{a_{2}b_{2}c_{2}d_{1}e_{1}}\phi^{a_{2}b_{2}c_{2}d_{2}e_{2}}\,,$
$\displaystyle\mathcal{O}_{4,de}=\,$
$\displaystyle\phi^{a_{1}b_{1}c_{1}d_{1}e_{1}}\phi^{a_{1}b_{1}c_{1}d_{2}e_{2}}\phi^{a_{2}b_{2}c_{2}d_{1}e_{2}}\phi^{a_{2}b_{2}c_{2}d_{2}e_{1}}\,.$

The field $\tau_{de}$ is generated by integrating out the linear combination
$\mathcal{O}_{3,de}-\mathcal{O}_{4,de}$ through use of the identity

$\displaystyle\hskip
51.21495pt\frac{1}{2}\Big{(}\tau_{de}^{d_{1}d_{2}\,e_{1}e_{2}}-\phi^{abcd_{1}e_{1}}\phi^{abcd_{2}e_{2}}+\phi^{abcd_{1}e_{2}}\phi^{abcd_{2}e_{1}}\Big{)}^{2}=$
$\displaystyle\frac{1}{2}(\tau_{de}^{d_{1}d_{2}\,e_{1}e_{2}})^{2}+\mathcal{O}_{3,de}-\mathcal{O}_{4,de}+\tau_{de}^{d_{1}d_{2}\,e_{1}e_{2}}\Big{(}\phi^{abcd_{1}e_{2}}\phi^{abcd_{2}e_{1}}-\phi^{abcd_{1}e_{1}}\phi^{abcd_{2}e_{2}}\Big{)}\,.$

  * •

Ten doubly traceless symmetric fields $\rho_{ij}^{i_{1}i_{2}\,j_{1}j_{2}}$,
where $i_{1}$ and $i_{2}$ transform as a traceless symmetric matrix under one
$O(N)$ group, while $j_{1}$ and $j_{2}$ transform as a traceless symmetric
matrix under another $O(N)$ group. The field $\rho_{ij}$ integrates out the
linear combination
$\mathcal{O}_{3,ij}+\mathcal{O}_{4,ij}-\frac{2}{N}(\mathcal{O}_{i}+\mathcal{O}_{j})+\frac{2}{N^{2}}\mathcal{O}_{1}$.
Take for example the field $\rho_{de}$. Consider the operators

$\displaystyle\mathcal{O}_{2,d}=\,$
$\displaystyle\phi^{a_{1}b_{1}c_{1}d_{1}e_{1}}\phi^{a_{1}b_{1}c_{1}d_{2}e_{1}}\phi^{a_{2}b_{2}c_{2}d_{1}e_{2}}\phi^{a_{2}b_{2}c_{2}d_{2}e_{2}}\,,$
$\displaystyle\mathcal{O}_{2,e}=\,$
$\displaystyle\phi^{a_{1}b_{1}c_{1}d_{1}e_{1}}\phi^{a_{1}b_{1}c_{1}d_{1}e_{2}}\phi^{a_{2}b_{2}c_{2}d_{2}e_{1}}\phi^{a_{2}b_{2}c_{2}d_{2}e_{2}}\,,$

and consider also the following tensor:

$\displaystyle\Phi^{d_{1}d_{2}\,e_{1}e_{2}}\equiv\,$
$\displaystyle\phi^{abcd_{1}e_{1}}\phi^{abcd_{2}e_{2}}+\phi^{abcd_{1}e_{2}}\phi^{abcd_{2}e_{1}}$
(84)
$\displaystyle-\frac{2}{N}\delta^{d_{1}d_{2}}\phi^{abcde_{1}}\phi^{abcde_{2}}-\frac{2}{N}\delta^{e_{1}e_{2}}\phi^{abcd_{1}e}\phi^{abcd_{2}e}+\frac{2}{N^{2}}\delta^{d_{1}d_{2}}\delta^{e_{1}e_{2}}\phi^{abcde}\phi^{abcde}\,.$

The field $\rho_{de}$ is generated by integrating out the following linear
combination of operators:

$\displaystyle\hskip
42.67912pt\mathcal{O}_{3,de}+\mathcal{O}_{4,de}-\frac{2}{N}(\mathcal{O}_{2,d}+\mathcal{O}_{2,e})+\frac{2}{N^{2}}\mathcal{O}_{1}=$
$\displaystyle\frac{1}{2}(\rho_{de}^{d_{1}d_{2}\,e_{1}e_{2}}-\Phi^{d_{1}d_{2}\,e_{1}e_{2}})^{2}-\frac{1}{2}(\rho_{de}^{d_{1}d_{2}\,e_{1}e_{2}})^{2}+2\rho_{de}^{d_{1}d_{2}\,e_{1}e_{2}}\,\Phi^{d_{1}d_{2}\,e_{1}e_{2}}\,.$

The total number of perturbative fixed points equals the $2^{26}$ possible
choices of including or not including each of the fields in the above list of
possible large $N$ field content. This can be made explicit by switching to
the operator basis

$\displaystyle\mathcal{O}_{1}\,,$
$\displaystyle\mathcal{O}_{2,i}-\frac{1}{N}\mathcal{O}_{1}\,,$
$\displaystyle\mathcal{O}_{3,ij}-\mathcal{O}_{4,ij}\,,$
$\displaystyle\mathcal{O}_{3,ij}+\mathcal{O}_{4,ij}-\frac{2}{N}(\mathcal{O}_{2,i}+\mathcal{O}_{2,j})+\frac{2}{N^{2}}\mathcal{O}_{1}\,,$

which amounts to studying the flows of couplings $h_{n}$ related to the
couplings $\lambda_{n}$ via

$\displaystyle h_{1}=\,$
$\displaystyle\lambda_{1}+\sum_{i}\lambda_{2,i}+\sum_{i<j}(\lambda_{3,ij}+\lambda_{4,ij})\,,$
$\displaystyle h_{2,i}=\,$
$\displaystyle\lambda_{2,i}+\sum_{j}(\lambda_{3,ij}+\lambda_{4,ij})\,,$
$\displaystyle h_{3,ij}=\,$
$\displaystyle\frac{1}{2}(\lambda_{3,ij}-\lambda_{4,ij})\,,$ $\displaystyle
h_{4,ij}=\,$ $\displaystyle\frac{1}{2}(\lambda_{3,ij}+\lambda_{4,ij})\,.$

In the new basis the beta functions are diagonalized as follows:

$\displaystyle\beta_{h_{1}}\big{|}_{\lambda_{5,i,jk}=0}\,=$
$\displaystyle-\epsilon\beta_{h_{1}}+\frac{h_{1}^{2}}{48\pi^{2}}\,,$
$\displaystyle\beta_{h_{2,i}}\big{|}_{\lambda_{5,i,jk}=0}\,=$
$\displaystyle-\epsilon\beta_{h_{2,i}}+\frac{h_{2,i}^{2}}{48\pi^{2}}\,,$
$\displaystyle\beta_{h_{3,ij}}\big{|}_{\lambda_{5,i,jk}=0}\,=$
$\displaystyle-\epsilon\beta_{h_{3,ij}}+\frac{h_{3,ij}^{2}}{24\pi^{2}}\,,$
$\displaystyle\beta_{h_{4,ij}}\big{|}_{\lambda_{5,i,jk}=0}\,=$
$\displaystyle-\epsilon\beta_{h_{4,ij}}+\frac{h_{4,ij}^{2}}{24\pi^{2}}\,.$

Of the vast number of large $N$ fixed points of this model, many are related
by permutations of tensor indices and so cannot be associated to distinct
CFTs. The number of non-isomorphic fixed points is given by the number of
orbits of the $2^{26}$ fixed points under the action of $S_{5}$. By careful
counting, we find that this number equals 612 032.

large $N$ field content | $\gamma_{\phi}-\mathcal{O}(\epsilon^{4})$ | $\gamma_{\phi^{2}}-\mathcal{O}(\epsilon^{3})$
---|---|---
$\sigma$ | $\frac{1}{N^{5}}(\frac{1}{4}\epsilon^{2}-\frac{1}{16}\epsilon^{3})+\mathcal{O}(\frac{1}{N^{10}})$ |  $\epsilon+\frac{1}{N^{5}}(-6\epsilon+\frac{13}{2}\epsilon^{2})+\mathcal{O}(\frac{1}{N^{10}})$
$\chi$ | $\frac{1}{N^{3}}(\frac{5}{8}\epsilon^{2}-\frac{5}{32}\epsilon^{3})+\mathcal{O}(\frac{1}{N^{4}})$ | $\frac{1}{N^{3}}(15\epsilon-\frac{65}{4}\epsilon^{2})+\mathcal{O}(\frac{1}{N^{4}})$
$\sigma$, $\chi$ |  $\frac{1}{N^{3}}(\frac{5}{8}\epsilon^{2}-\frac{5}{32}\epsilon^{3})+\mathcal{O}(\frac{1}{N^{4}})$ | $\epsilon+\frac{1}{N^{3}}(-15\epsilon+\frac{65}{4}\epsilon^{2})+\mathcal{O}(\frac{1}{N^{4}})$
$\tau$ | $\frac{1}{N}(\frac{5}{8}\epsilon^{2}-\frac{5}{32}\epsilon^{3})+\mathcal{O}(\frac{1}{N^{2}})$ | $\frac{1}{N}(15\epsilon-\frac{65}{4}\epsilon^{2})+\mathcal{O}(\frac{1}{N^{2}})$
$\sigma$, $\tau$ |  $\frac{1}{N}(\frac{5}{8}\epsilon^{2}-\frac{5}{32}\epsilon^{3})+\mathcal{O}(\frac{1}{N^{2}})$ | $\epsilon+\frac{1}{N}(-15\epsilon+\frac{65}{4}\epsilon^{2})+\mathcal{O}(\frac{1}{N^{2}})$
$\chi$, $\tau$ | $\frac{1}{N}(\frac{5}{8}\epsilon^{2}-\frac{5}{32}\epsilon^{3})+\mathcal{O}(\frac{1}{N^{2}})$ | $\frac{1}{N}(15\epsilon-\frac{65}{4}\epsilon^{2})+\mathcal{O}(\frac{1}{N^{2}})$
$\sigma$, $\chi$, $\tau$ |  $\frac{1}{N}(\frac{5}{8}\epsilon^{2}-\frac{5}{32}\epsilon^{3})+\mathcal{O}(\frac{1}{N^{2}})$ | $\epsilon+\frac{1}{N}(-15\epsilon+\frac{65}{8}\epsilon^{2})+\mathcal{O}(\frac{1}{N^{2}})$
$\rho$ | $\frac{1}{N}(\frac{5}{8}\epsilon^{2}-\frac{5}{32}\epsilon^{3})+\mathcal{O}(\frac{1}{N^{2}})$ | $\frac{1}{N}(15\epsilon-\frac{65}{4}\epsilon^{2})+\mathcal{O}(\frac{1}{N^{2}})$
$\sigma$, $\rho$ |  $\frac{1}{N}(\frac{5}{8}\epsilon^{2}-\frac{5}{32}\epsilon^{3})+\mathcal{O}(\frac{1}{N^{2}})$ | $\epsilon+\frac{1}{N}(-15\epsilon+\frac{65}{4}\epsilon^{2})+\mathcal{O}(\frac{1}{N^{2}})$
$\chi$, $\rho$ | $\frac{1}{N}(\frac{5}{8}\epsilon^{2}-\frac{5}{32}\epsilon^{3})+\mathcal{O}(\frac{1}{N^{2}})$ | $\frac{1}{N}(15\epsilon-\frac{65}{4}\epsilon^{2})+\mathcal{O}(\frac{1}{N^{2}})$
$\sigma$, $\chi$, $\rho$ |  $\frac{1}{N}(\frac{5}{8}\epsilon^{2}-\frac{5}{32}\epsilon^{3})+\mathcal{O}(\frac{1}{N^{2}})$ | $\epsilon+\frac{1}{N}(-15\epsilon+\frac{65}{4}\epsilon^{2})+\mathcal{O}(\frac{1}{N^{2}})$
$\tau$, $\rho$ | $\frac{1}{N}(\frac{5}{4}\epsilon^{2}-\frac{5}{16}\epsilon^{3})+\mathcal{O}(\frac{1}{N^{2}})$ | $\frac{1}{N}(30\epsilon-\frac{65}{2}\epsilon^{2})+\mathcal{O}(\frac{1}{N^{2}})$
$\sigma$, $\tau$, $\rho$ |  $\frac{1}{N}(\frac{5}{4}\epsilon^{2}-\frac{5}{16}\epsilon^{3})+\mathcal{O}(\frac{1}{N^{2}})$ | $\epsilon+\frac{1}{N}(-30\epsilon+\frac{65}{2}\epsilon^{2})+\mathcal{O}(\frac{1}{N^{2}})$
$\chi$, $\tau$, $\rho$ | $\frac{1}{N}(\frac{5}{4}\epsilon^{2}-\frac{5}{16}\epsilon^{3})+\mathcal{O}(\frac{1}{N^{2}})$ | $\frac{1}{N}(30\epsilon-\frac{65}{2}\epsilon^{2})+\mathcal{O}(\frac{1}{N^{2}})$
$\sigma$, $\chi$, $\tau$, $\rho$ |  $\frac{1}{N}(\frac{5}{4}\epsilon^{2}-\frac{5}{16}\epsilon^{3})+\mathcal{O}(\frac{1}{N^{2}})$ | $\epsilon+\frac{1}{N}(-30\epsilon+\frac{65}{2}\epsilon^{2})+\mathcal{O}(\frac{1}{N^{2}})$

Table 4: Leading anomalous dimensions at the large $N$ fixed points of quartic
$O(N)^{5}\rtimes S_{5}$ theory

####  2.4.2 Reduced System with $S_{5}$ symmetry

If we study the subsector of the theory that is symmetric under permutations
of the tensor indices and so exhibits a global $\frac{O(N)^{5}\rtimes
S_{5}}{(\mathbb{Z}_{2})^{4}}$ symmetry, the RG flow reduces to a five-
dimensional dynamical system. The fixed points of this flow are the fixed
points of the full system that respect the $S_{5}$ permutation symmetry, and
these can be counted by the choices of large $N$ field content— $\sigma$ or no
$\sigma$, all $\chi$ fields or no $\chi$ fields, all $\tau$ fields or no
$\tau$ fields, and all $\rho$ fields or no $\rho$ fields— entailing that there
are $2^{4}=16$ fixed points, all of which are non-isomorphic. The leading
anomalous dimensions for $\phi$ and $\phi^{2}$ are listed in Table 4, while
Figure 6 shows how the large $N$ fixed points appear at finite values of $N$
and approach their asymptotic values.

$\displaystyle\begin{matrix}\text{\scalebox{1.1}{ {
\fcolorbox{gray}{white}{\includegraphics[scale=0.51]{rank5plot-1.png}} }}
}\end{matrix}$ Figure 6: Values of the coupling $g_{a}$ at Wilson-Fisher fixed
points in rank-five scalar theory with $S_{5}\ltimes
O(N)^{5}/(\mathbb{Z}_{2})^{4}$ symmetry in $d=4-\epsilon$ dimensions for
different values of $N$. The dotted lines indicate the asymptotic values for
$g_{a}$ at the large $N$ fixed points.

It is a simple exercise to work out the general $d$ dependence of the leading
$1/N$ corrections listed above by a slight modification of the large $N$
treatment presented in Subsection 2.2 where the fields $\tau$ and $\rho$
propagate along the dashed lines in diagrams (34) and (36). Loop integrals are
unchanged, but additional index contraction is involved, which explains why
the columnar values are related by simple fractions and powers of $N$, seeing
as the full $1/N$ corrections each equal $A_{\phi}(d)$ or $B_{\phi^{2}}(d)$
times a rational number and a negative power of $N$.

###  2.5 Large $N$ limit at general tensor rank

For quartic field theories of rank-$r$ tensors with
$O(N)^{r}/(\mathbb{Z}_{2})^{r-1}$ or $S_{r}\ltimes
O(N)^{r}/(\mathbb{Z}_{2})^{r-1}$ symmetry, some of the key characteristics are

$\displaystyle n_{\text{quartic}}:$ the number of quartic $O(N)^{r}$ singlets,
$\displaystyle n_{\text{fixed points}}:$ the number of large $N$ fixed points,
and $\displaystyle n_{\text{non-isomorphic}}:$ the number of non-isomorphic
large $N$ fixed points, ie. fixed points not related by a permutation of
tensor indices.

For the theories studied in the previous sections, with tensor ranks less than
or equal to five, this data can be summarized in Table 5. We now turn to the
question of how the RG flows and fixed points of tensorial $\phi^{4}$ theory
behave for general $r$ and to the question of how the above data extends
beyond $r=5$.

####  2.5.1 Number of operators

For models with $O(N)^{r}/(\mathbb{Z}_{2})^{r-1}$ symmetry, the counting of
quartic singlets can be performed in generality, and there is a simple
argument in [86] establishing that

$\displaystyle\frac{O(N)^{r}}{(\mathbb{Z}_{2})^{r-1}}:\hskip
128.0374ptn_{\text{quartic}}(r)=\frac{3+3^{r}}{6}\,.$ (89)

For models with $S_{r}\ltimes O(N)^{r}/(\mathbb{Z}_{2})^{r-1}$ symmetry, the
data in Table 5 might tempt the reader to think that $n_{\text{quartic}}$
simply equals $r$, but this is not the case. In the case of $S_{6}\ltimes
O(N)^{6}/(\mathbb{Z}_{2})^{5}$ symmetry one finds that $n_{\text{quartic}}=7$:

$\displaystyle\begin{matrix}\text{ \includegraphics[scale={1}]{rank6ops.pdf}
}\end{matrix}$ (90)

global symmetry $O(N)$ $\frac{O(N)^{2}\rtimes S_{2}}{\mathbb{Z}_{2}}$
$\frac{O(N)^{3}}{(\mathbb{Z}_{2})^{2}}$ $\frac{O(N)^{3}\rtimes
S_{3}}{(\mathbb{Z}_{2})^{2}}$ $\frac{O(N)^{4}}{(\mathbb{Z}_{2})^{3}}$
$\frac{O(N)^{4}\rtimes S_{4}}{(\mathbb{Z}_{2})^{3}}$
$\frac{O(N)^{5}}{(\mathbb{Z}_{2})^{4}}$ $\frac{O(N)^{5}\rtimes
S_{5}}{(\mathbb{Z}_{2})^{4}}$ $n_{\text{quartic}}$ 1 2 5 3 14 4 41 5
$n_{\text{fixed points}}$ 2 2 16 4 32 4 67 108 864 16 $n_{\text{non-
isomorphic}}$ 2 2 8 4 10 4 612 032 16

Table 5: Numbers of operators and fixed points in quartic $O(N)^{r}$ theory
and $O(N)^{r}\rtimes S_{r}$ theory

The general answer in the $S_{r}\ltimes O(N)^{r}/(\mathbb{Z}_{2})^{r-1}$ case
is that $n_{\text{quartic}}$ equals $p_{3}(r)$, ie. the number of integer
partitions of $r$ into exactly three parts. This number has the following
generating function:

$\displaystyle\frac{1}{(1-x)(1-x^{2})(1-x^{3})}=1+x+2x^{2}+3x^{3}+4x^{4}+5x^{5}+7x^{6}+8x^{7}+10x^{8}+...\,,$
(91)

and in fact a closed-form expression exists:

$\displaystyle\frac{O(N)^{r}\rtimes S_{r}}{(\mathbb{Z}_{2})^{r-1}}:\hskip
34.1433ptn_{\text{quartic}}(r)=\frac{1}{72}\bigg{(}47+9(-1)^{r}+6\,r(6+r)+16\cos\big{(}\frac{2\pi
r}{3}\big{)}\bigg{)}\,.$ (92)

####  2.5.2 Number of fixed points

Through explicit computation we have seen in Sections 2.1, 2.2, 2.3, and 2.4
that for all the real large $N$ fixed points of quartic $O(N)^{r}$ theory with
$r\leq 5$, the only non-vanishing couplings at the fixed points are those
associated to vectorial operators, by which we mean operators, like the first
four in (90), for which more than half the indices are contracted between the
same pairs of fundamental fields. In other words, operators with melonic or
planar large $N$ limits are absent at the perturbative large $N$ fixed points.
In Appendix B we argue that this same property holds true for all odd values
of $r$, while other types of fixed points may be present for even $r$. By
diagonalizing the vectorial RG flow at large $N$, we also show that the number
of vectorial large fixed points equals two raised to the number of vectorial
operators. In consequence, we find that the number of vectorial large $N$
fixed points for general tensor rank $r$ is given by

$\displaystyle\frac{O(N)^{r}}{(\mathbb{Z}_{r})^{r-1}}:\hskip
56.9055pt\log_{2}n_{\text{fixed
points}}(r)=1+\sum_{m=1}^{\left\lfloor\frac{r-1}{2}\right\rfloor}\binom{r}{m}2^{m-1}\,.$
(93)

The combinatorial explosion witnessed in Table 5 is made further manifest by
this formula. While the fixed points counted by the formula (93) include
isomorphic copies, the number of non-isomorphic fixed points is bounded below
by $n_{\text{vectorial fixed points}}(r)/r!$, and for $r=7$ this lower bound
exceeds Avogadro’s number, while for $r=9$ it exceeds one googol. The
asymptotic behaviour of the number of vectorial fixed points at large rank is
given by

$\displaystyle\frac{O(N)^{r}}{(\mathbb{Z}_{r})^{r-1}}:\hskip
56.9055pt\log_{2}n_{\text{fixed
points}}(r)\rightarrow\begin{cases}\displaystyle\frac{2^{3r/2}}{\sqrt{\pi
r}}\hskip 34.1433pt\text{for }r\text{ odd}\,,\\\ \\\
\displaystyle\frac{2^{3r/2}}{\sqrt{2\pi r}}\hskip 28.45274pt\text{for }r\text{
even}\,,\end{cases}$ (94)

where the limit should be understood to mean that the ratio tends to one (and
not that the difference tends to zero) as $r$ goes to infinity.

The number $\log_{2}n_{\text{fixed points}}(r)$ equals the number of vectorial
operators but also equals the distinct possible Hubbard-Stratonovich fields in
$O(N)^{r}$ theory, as we explain in detail in Appendix B. Each of these fields
contains up to $\left\lfloor\frac{r-1}{2}\right\rfloor$ index pairs, each of
which transforms as a traceless symmetric or as an antisymmetric matric under
its own $O(N)$ group, with the number of antisymmetric index pairs being even.
For example, the rank seven theory contains $2^{190}$ large $N$ fixed points,
and the list of possible Hubbard-Stratonovich fields consists of

  * •

one $\sigma$ field,

  * •

seven fields $\chi$ with one traceless symmetric index pair,

  * •

21 doubly antisymmetric fields $\tau$,

  * •

21 doubly traceless symmetric fields $\rho$,

  * •

35 triply traceless symmetric fields, and

  * •

105 fields with two antisymmetric index pairs and one traceless symmetric
index pair.

For all of the vectorial fixed points, the leading large $N$ anomalous
dimensions are given by simple modifications of the $O(N)$ model answers,
obtained by summing over Feynman diagrams (34) and (36) for each of the
associated Hubbard-Stratonovich fields. Many of the
$O(N)^{r}/(\mathbb{Z}_{2})^{r-1}$ fixed points exhibit enhanced symmetry, with
each enhanced symmetry being given by the semi-direct product of a group $G$
acting on
$\mathbb{Z}_{2}\times\prod_{i=1}^{i_{\text{max}}}\frac{O(N^{\ell_{i}})}{\mathbb{Z}_{2}}$,
where $1\leq\ell_{i}\leq r$, and $\sum_{i}\ell_{i}=r$, and $G$ is a subgroup
of $S_{r}$ that permutes $O(N^{\ell_{i}})$ groups with the same value for
$\ell_{i}$. For most vectorial fixed points, non-vectorial operators are
generated at subleading order in $1/N$, but in cases of an enhancement that
includes an $O(N^{m})$ subgroup with $m>r/2$, the symmetry prevents any non-
vectorial operators from being generated, and so for these theories we expect
the $1/N$ expansions developed by the Hubbard-Stratonovich transformation to
remain valid to any order. For a given symmetry enhancement of this kind, the
fixed points with the most Hubbard-Stratonovich fields associated to them are
stable under any quartic deformation that respects the enhanced symmetry. But
among the general space of quartic operators with
$O(N)^{r}/(\mathbb{Z}_{2})^{r-1}$ symmetry, an inordinate amount of tuning is
required to flow to any of the vectorial fixed points since any non-vectorial
operator provides a relevant deformation.

The explosion in number of fixed points is significantly slowed down if we
restrict ourselves to quartic $O(N)^{r}$ tensor theories with additional
$S_{r}$ symmetry under permutations of indices. The formula for the number of
vectorial large $N$ fixed points is in this case given by

$\displaystyle\frac{O(N)^{r}\rtimes S_{r}}{(\mathbb{Z}_{2})^{r-1}}:\hskip
8.53581pt\log_{2}n_{\text{fixed
points}}(r)=\left\lfloor\frac{r+3}{4}\right\rfloor\,\left\lfloor\frac{r+5}{4}\right\rfloor=\begin{cases}l(l+1)\hskip
8.53581pt\text{ for }r=4l-1\,,\\\ l(l+1)\hskip 8.53581pt\text{ for }r=4l\,,\\\
(l+1)^{2}\hskip 8.53581pt\text{ for }r=4l+1\,,\\\ (l+1)^{2}\hskip
8.53581pt\text{ for }r=4l+2\,,\end{cases}$ (95)

all of which fixed points are non-isomorphic.

##  3 Tensorial $\phi^{6}$ Theory

Following our study of quartic theories in the previous section, we now
advance to sextic theories and study tensorial generalizations of $\phi^{6}$
theory, where each index transforms under its own $O(N)$ group. A high degree
of tuning is associated to these theories since not only must the quadratic
operator be tuned away, the same applies also to each of the quartic
operators, whose number increases exponentially with tensor rank $r$ as shown
in equation (89), although for the subsector of theories with additional
$S_{r}$ permutation theory the increase with $r$ slows down to a quadratic
growth, as attested by equation (92).

We will not be able to uncover the general structure of sextic tensor theories
at large $N$ for the reason that sextic tensor theories do not exhibit the
kind of simple pattern inherent in the large $N$ dynamics of quartic theories,
as appears already on comparing the qualitatively different large $N$ limits
of sextic vector, matrix, and three-tensor theories. Instead, we will review
the theories that have been studied in the literature and then proceed to
analyze the next sextic theory theory in order of increasing rank by
presenting a case study for rank-four tensor theory at four-loop order, in $d$
just below the critical dimension three.

For the rank-one theory, ie. the sextic $O(N)$ model, the perturbative beta
function has been known since work in the 70ies [87, 88], and a series of
subsequent papers demonstrated that at large $N$ all loop corrections vanish
at leading order, while the $1/N$ corrections indicate the existence of an
interacting UV fixed point in three dimensions [89, 90, 91, 92, 93]. More
concretely, these papers found that if we write the action as

$\displaystyle S=\int
d^{3-\epsilon}x\bigg{(}\frac{1}{2}(\partial_{i}\phi^{a})(\partial_{i}\phi^{ab})+\frac{1}{6!}\,g\,(\phi^{a}\phi^{a})^{3}\bigg{)}\,,$
(96)

and we scale the coupling constant according to its optimal scaling,
$g=\frac{\lambda}{N^{2}}$, then the large $N$ beta function to subleading
order in $1/N$ terminates at cubic order in the coupling and is given by

$\displaystyle\beta_{\lambda}=-2\epsilon\lambda+\frac{4\lambda^{2}}{5(8\pi)^{2}N}\Big{(}1-\frac{\lambda}{32\cdot
6!}\Big{)}+\mathcal{O}(\frac{1}{N^{2}})\,.$ (97)

From this equation, we see that at $\lambda=32\cdot 6!$ the theory has a large
$N$ UV fixed point in $3d$.141414Leading large $N$ expressions for several OPE
coefficients and anomalous dimensions of operators $(\phi^{2})^{n}$ for $n\geq
3$ at this 3$d$ UV fixed point were computed a few years ago in Ref. [94],
while Ref. [95] determined the scaling dimension of $\phi^{n}$ to sub-leading
order at large $n$. In principle, the fixed point also exists away from three
dimensions, but it becomes complex for $\epsilon>\frac{36}{\pi^{2}N}$, so as
we take $N$ to infinity, the range of dimensions where the fixed point is real
shrinks to just $d=3$.

The picture of the sextic $O(N)$ model in $3d$ as an asymptotically safe
theory suggested by the above considerations, with the RG flow interpolating
between a free theory in the IR and an interacting one in the UV, was however
cast into doubt by the realization that the effective potential of the theory
becomes unstable already for $\lambda>30(8\pi)^{2}$ [96, 97, 98] (note that
$30(8\pi)^{2}<32\cdot 6!$), and a much more recent study of the theory in the
presence of a planar boundary also points to an instability [99]. But the
alternative picture that arose wherein the scale invariance is spontaneously
broken before the UV fixed point is reached was itself called into question by
the observation that subleading corrections endow the associated dilaton with
an order $1/N$ mass that turns out to be tachyonic [100]. In brief, the fate
of the theory in the UV is not known with certainty, but see Refs. [101, 102]
for recent discussions.

The incertitude surrounding the sextic $O(N)$ model exemplifies a broader
lesson: once you go beyond the optimal scaling, you open a can of worms, and
the large $N$ and small $\epsilon$ expansions generally cease to commute. But
fortuitously, multi-index sextic theories differ from the vector model in that
they feature loop-corrections not suppressed at large $N$ under the optimal
scaling, entailing that these theories enjoy the benefit of possessing Wilson-
Fisher that do not become complex at values of $\epsilon$ that scale as $1/N$.
In the following subsections, we will inspect the Wilson-Fisher fixed points
for the sextic theories of ranks two, three, and four. Whether these fixed
points present in the $\epsilon$ expansion actually exist for finite values of
$\epsilon$ is a separate issue. In the one case where the answer is known, it
is in the affirmative: the non-trivial large $N$ rank-three fixed point can be
continued along a finite range of dimensions, although not all the way down to
two dimensions [103].

The beta functions and anomalous dimensions of the sextic theories we present
have all been computed by modifying the sextic $O(N)$ theory four-loop beta
function and six-loop anomalous dimensions calculated in Refs. [104, 105] via
suitable index contractions performed on a computer.

###  3.1 The $O(N)^{2}$ model

The matrix theory with $O(N)^{2}/\mathbb{Z}_{2}$ symmetry has three sextic
singlets:

$\displaystyle\begin{matrix}\text{
\includegraphics[scale={1}]{sexticRank2ops.pdf} }\end{matrix}$ (98)

In addition to $O(N)^{2}/\mathbb{Z}_{2}$ symmetry, the model also enjoys an
automatic $S_{2}$ symmetry under permutations of the two indices—unlike for
the higher rank models where permutation theory is not automatic. The action
of the theory can be written as

$\displaystyle S=\int
d^{3-\epsilon}x\bigg{(}\frac{1}{2}(\partial_{i}\phi^{ab})(\partial_{i}\phi^{ab})+\frac{1}{6!}\sum_{n=1}^{3}g_{n}\,\mathcal{O}_{n}\bigg{)}\,.$
(99)

The model was previously studied in Ref. [106], which computed the four-loop
beta functions and studied the perturbative fixed points as functions of $N$.
We will review only the large $N$ fixed points, referring the reader to [106]
for details on the finite $N$ fixed points.151515For the fixed points of the
generalization of this model with $O(N_{1})\times O(N_{2})$ symmetry, see the
recent Ref. [107]. For $N>23.2541-577.350\epsilon+\mathcal{O}(\epsilon^{2})$,
the beta functions have four fixed points including the trivial one. However,
once we impose the optimal large $N$ scaling,

$\displaystyle g_{1}=\frac{\lambda_{1}}{N^{4}}\,,\hskip
56.9055ptg_{2}=\frac{\lambda_{2}}{N^{3}}\,,\hskip
56.9055ptg_{3}=\frac{\lambda_{3}}{N^{2}}\,,$ (100)

we are left with but a single non-trivial fixed point in the large $N$ limit:

$\displaystyle\lambda_{1}=\phantom{-}(8\pi)^{2}\cdot
6!\cdot\Big{(}\frac{295}{108}\epsilon+\frac{4714+6301\pi^{2}}{1944}\epsilon^{2}\Big{)}+\mathcal{O}(\epsilon^{3})\,,$
$\displaystyle\lambda_{2}=-(8\pi)^{2}\cdot
6!\cdot\Big{(}\frac{\epsilon}{2}+\frac{22+7\pi^{2}}{36}\epsilon^{2}\Big{)}+\mathcal{O}(\epsilon^{3})\,,$
(101) $\displaystyle\lambda_{3}=\phantom{-}(8\pi)^{2}\cdot
6!\cdot\Big{(}\frac{1}{36}\epsilon+\frac{17+\pi^{2}}{324}\epsilon^{2}\Big{)}+\mathcal{O}(\epsilon^{3})\,.$

The presence of two perturbative fixed points that exist for arbitrarily large
$N$ but are suppressed away by the optimal scaling represents a new phenomenon
compared with the quartic theories. Unlike the fixed point (101), these other
two fixed points are not reliable in the limit of large $N$ and small
$\epsilon$ unless we also impose that $\epsilon\ll 1/N$.

In the supplementary Mathematica file we display the six-loop anomalous
dimensions for $\phi$ and $\phi^{2}$. Imposing the optimal scaling and taking
the large $N$ limit yields the simpler equations

$\displaystyle\gamma_{\phi}=\frac{48\lambda_{3}^{2}}{(6!)^{2}(8\pi)^{4}}-\frac{1152\lambda_{3}^{3}}{(6!)^{3}(8\pi)^{6}}+\mathcal{O}(\lambda^{4})\,.$
(102)
$\displaystyle\gamma_{\phi^{2}}=32\frac{48\lambda_{3}^{2}}{(6!)^{2}(8\pi)^{4}}-\frac{4608\big{(}\pi^{2}\lambda_{2}+(50+6\pi^{2})\lambda_{3}\big{)}\lambda_{3}^{2}}{(6!)^{3}(8\pi)^{6}}+\mathcal{O}(\lambda^{4})\,.$
(103)

At the large $N$ fixed point (101), these anomalous dimension evaluate to

$\displaystyle\gamma_{\phi}\,\,=\frac{1}{27}\epsilon^{2}+\frac{2(14+\pi^{2})}{243}\epsilon^{3}+\mathcal{O}(\epsilon^{4})\hskip
11.38109pt\approx\,0.0370370\epsilon^{2}+0.196458\epsilon^{3}+\mathcal{O}(\epsilon^{4})\,,$
(104)
$\displaystyle\gamma_{\phi^{2}}=\frac{32}{27}\epsilon^{2}+\frac{16(22\pi^{2}-7)}{243}\epsilon^{3}+\mathcal{O}(\epsilon^{4})\approx\,1.18519\epsilon^{2}+13.8358\epsilon^{3}+\mathcal{O}(\epsilon^{4})\,.$
(105)

The sizes of these coefficients do not kindle much hope that this $\epsilon$
expansion yields useful information about a putative 2$d$ CFT.

###  3.2 The $O(N)^{3}$ model

The theory with $O(N)^{3}/(\mathbb{Z}_{2})^{2}$ symmetry contains 16 sextic
singlets. It turns out that under the optimal scaling the two-loop large $N$
beta functions of this model contain but a single large $N$ fixed point and
that this fixed point exhibits an additional $S_{3}$ symmetry under
permutations of tensor indices. To simplify the presentation, and without loss
of large $N$ fixed points, we will therefore restrict our review of rank-three
theory to the subsector that possesses this additional $S_{3}$ symmetry,
whereby the 16-dimensional RG flow reduces to an eight-dimensional dynamical
system. The action is given by

$\displaystyle S=\int
d^{3-\epsilon}x\bigg{(}\frac{1}{2}(\partial_{i}\phi^{abc})(\partial_{i}\phi^{abc})+\frac{1}{6!}\sum_{n=1}^{8}g_{n}\,\mathcal{O}_{n}\bigg{)}\,,$
(106)

iwth the eight sextic operators in the potential shown in Figure 7.

$\displaystyle\begin{matrix}\text{
\includegraphics[scale={0.95}]{sexticRank3ops.pdf} }\end{matrix}$ Figure 7:
Sextic singlet operators in $O(N)^{3}\rtimes S_{3}$ theory

This precise model was studied in detail in Ref. [103]. The authors of this
paper noted the presence of an RG fixed point dominated by the so-called
prismatic operator $\mathcal{O}_{1}$ and demonstrated the separate solvability
of this fixed point at large $N$ and at small $\epsilon$, while also observing
that the large $N$ fixed point becomes complex for $d$ below approximately
2.81. The model was subsequently studied at four-loop level by Igor Klebanov,
Fedor Popov, and one of the present authors in unpublished work, which we
review below. The zero-dimensional version of the model was studied from a
more mathematical perspective in Ref. [108], which analyzed the double scaling
limit of the prismatic model and identified the class of sub-leading diagrams
at large $N$.

The optimal scaling of the model (106) is given as follows:161616For some of
the couplings in the theory, Ref. [103] adopted large $N$ scaling exponents
bigger than those in equation (LABEL:sexticRank3scaling). Their scaling enjoys
the benefit of flushing out the order $\epsilon$ terms in the fixed point
(LABEL:rank4fixedPT) that are suppressed in $1/N$ but causes a blow-up of the
four-loop beta functions at large $N$ unless we take $\epsilon$ to be
suppressed in $1/N$.

$\displaystyle g_{1}=\frac{\lambda_{1}}{N^{3}}\,,\hskip
42.67912ptg_{2}=\frac{\lambda_{2}}{N^{3}}\,,\hskip
42.67912ptg_{3}=\frac{\lambda_{3}}{N^{4}}\,,\hskip
42.67912ptg_{4}=\frac{\lambda_{4}}{N^{4}}\,,$ $\displaystyle
g_{5}=\frac{\lambda_{5}}{N^{7/2}}\,,\hskip
34.1433ptg_{6}=\frac{\lambda_{6}}{N^{9/2}}\,,\hskip
34.1433ptg_{7}=\frac{\lambda_{7}}{N^{5}}\,,\hskip
42.67912ptg_{8}=\frac{\lambda_{8}}{N^{6}}\,.$

Adopting this scaling, the two-loop beta functions have a unique large $N$
fixed point. Including four-loop corrections, this fixed point is situated at

$\displaystyle\lambda_{1}=11\,520\pi^{2}\epsilon-34\,560\pi^{2}\epsilon^{2}+\mathcal{O}(\epsilon^{3})\,,$
$\displaystyle\lambda_{2}=\mathcal{O}(\epsilon^{3})\,,$
$\displaystyle\lambda_{3}=69\,120\pi^{2}\epsilon+138\,240\pi^{2}\epsilon^{2}+\mathcal{O}(\epsilon^{3})\,,$
$\displaystyle\lambda_{4}=-5760\pi^{4}\epsilon^{2}+\mathcal{O}(\epsilon^{3})\,,$
$\displaystyle\lambda_{5}=\mathcal{O}(\epsilon^{3})\,,$
$\displaystyle\lambda_{6}=\mathcal{O}(\epsilon^{3})\,,$
$\displaystyle\lambda_{7}=34\,560\pi^{2}\epsilon+34\,560(83\pi^{2}-\pi^{4})\epsilon^{2}+\mathcal{O}(\epsilon^{3})\,,$
$\displaystyle\lambda_{8}=11\,520(126\pi^{2}-\pi^{4})\epsilon^{2}+\mathcal{O}(\epsilon^{3})\,.$

Under the optimal scaling, the large $N$ anomalous dimensions of $\phi$ and
$\phi^{2}$ are given by

$\displaystyle\gamma_{\phi}=\frac{16}{(6!)^{2}(8\pi)^{4}}(\lambda_{1}^{2}+3\lambda_{2}^{2})-\frac{128}{3(6!)^{3}(8\pi)^{6}}\lambda_{1}^{3}+\mathcal{O}(\lambda^{4})\,,$
$\displaystyle\gamma_{\phi^{2}}=\frac{512}{(6!)^{2}(8\pi)^{4}}(\lambda_{1}^{2}+3\lambda_{2}^{2})-\frac{25\,600}{3(6!)^{3}(8\pi)^{6}}\lambda_{1}^{3}+\mathcal{O}(\lambda^{4})\,.$

At the large $N$ fixed point (LABEL:rank4fixedPT), these anomalous dimensions
assume the following values:

$\displaystyle\gamma_{\phi}=\epsilon^{2}-\frac{20}{3}\epsilon^{3}+\mathcal{O}(\epsilon^{4})\,,$
(110)
$\displaystyle\gamma_{\phi^{2}}=32\epsilon^{2}-\frac{976}{3}\epsilon^{3}+\mathcal{O}(\epsilon^{4})\,.$
(111)

These values match the large $N$ solution that Ref. [103] obtained by
expressing the exponential of the prismatic operator as an integral over an
auxiliary field and then writing down a pair of coupled Schwinger-Dyson
equations for the fundamental and auxiliary fields. Specifically, the authors
of [103] found that

$\displaystyle\Delta_{\phi}=\frac{1}{2}-\frac{\epsilon}{2}+\epsilon^{2}-\frac{20\epsilon^{3}}{3}+\Big{(}\frac{472}{9}+\frac{\pi^{2}}{3}\Big{)}\epsilon^{4}+\Big{(}7\zeta(3)-\frac{12\,692}{27}-\frac{56\pi^{2}}{9}\Big{)}\epsilon^{5}+\mathcal{O}(\epsilon^{6})\,,$
(112)
$\displaystyle\Delta_{\phi^{2}}=1-\epsilon+32\epsilon^{2}-\frac{976}{3}\epsilon^{3}+\Big{(}\frac{30\,320}{9}+\frac{32\pi^{2}}{3}\Big{)}\epsilon^{4}+\mathcal{O}(\epsilon^{5})\,.$
(113)

The presence of both $\lambda_{1}$ and $\lambda_{2}$ in equations (110) and
(111) suggests that both the prismatic operator $\mathcal{O}_{1}$ and the
wheelic operator $\mathcal{O}_{2}$ can dominate at large $N$. And indeed Ref.
[109] established that for precisely these two rank-three operators, the
Feynman diagrams that contribute to the free energy at large $N$ are melonic
and can be resummed. We may ask then, does the model (106) posses a whelic
fixed point? In this paper we have so far listed only standard type Wilson-
Fisher fixed points, which rely on a cancellation between linear and quadratic
terms in beta functions, so that the leading coupling constants scale as
$\epsilon$, with higher order terms in the beta functions giving rise to
$\mathcal{O}(\epsilon^{2})$ corrections. But a wheelic fixed point should
match the $q=6$ melonic theory described in Ref [71], whose equations (4.3)
and (5.1) tell us that

$\displaystyle\Delta_{\phi}=\frac{d}{6}\,,$ (114)
$\displaystyle\Delta_{\phi^{2}}=1+\frac{29}{3}\epsilon+\frac{400}{9}\epsilon^{2}+\frac{160}{27}(237+2\pi^{2})\epsilon^{3}+\mathcal{O}(\epsilon^{4})\,,$
(115)

corresponding to anomalous dimensions given by

$\displaystyle\gamma_{\phi}=\Delta_{\phi}-\frac{d-2}{2}=\frac{\epsilon}{3}\,,$
$\displaystyle\gamma_{\phi^{2}}=\Delta_{\phi^{2}}-(d-2)=\frac{32}{3}\epsilon+\mathcal{O}(\epsilon^{2})\,.$

Since the perturbative anomalous dimensions in a sextic theory scale as
$\lambda^{2}$, a wheelic fixed point should manifest itself in pertubation
theory as having couplings with leading scaling $\sqrt{\epsilon}$, meaning
that such a fixed point arises from a cancellation between linear and cubic
terms in the four-loop beta functions. And in fact, the model (106) does
posses a pair of such fixed points:

$\displaystyle\lambda_{1}=0\,,$ $\displaystyle\lambda_{2}=\pm
3840\pi^{2}\sqrt{\epsilon}\,,$ $\displaystyle\lambda_{3}=-17\,280\pi^{2}(1\pm
2\sqrt{\epsilon})\,,$ $\displaystyle\lambda_{4}=0\,,$ (117)
$\displaystyle\lambda_{5}=0\,,$ $\displaystyle\lambda_{6}=0\,,$
$\displaystyle\lambda_{7}=\frac{34\,560}{7}\pi^{2}(5\pm 7\sqrt{\epsilon})\,,$
$\displaystyle\lambda_{8}=-\frac{640}{7}\pi^{2}(109\pm 126\sqrt{\epsilon})\,.$

However, since some of the couplings contain terms of order $O(\epsilon^{0})$,
we cannot trust the perturbative analysis that revealed the above pair of
fixed points, irrespectively of how small a value we assign to $\epsilon$.
Higher-order terms are liable to change even leading values of the above
couplings. But it is compelling to think that the values for $\lambda_{1}$ and
$\lambda_{2}$ in (117) are robust and that the non-robust couplings do not
enter into the large $N$ anomalous dimensions. In any event, plugging the
values (117) into (LABEL:sexticRank3anomalous), one does find agreement with
(LABEL:melonicGamma).171717Melonic fixed points with order $O(\epsilon^{0})$
terms were also observed for the sextic theory with $U(N)^{3}$ symmetry
studied at two loops in Ref. [110] and at four loops in Ref. [111].

###  3.3 The $O(N)^{4}\rtimes S_{4}$ model

We now turn to sextic scalar theory of rank-four tensors. The full theory with
$O(N)^{4}/(\mathbb{Z}_{2})^{3}$ symmetry contains 132 sextic singlet
operators. Computing the two-loop beta functions for this system is
manageable, but determining the number of fixed points for such a large set of
coupled equations is challenging. We therefore impose a global $S_{4}$
symmetry under permutations of indices, whereby the 132-dimensional RG flow
reduces to a more manageable 21-dimensional invariant subspace associated to
the operators show in Figure 8.

$\displaystyle\begin{matrix}\text{
\includegraphics[scale={0.95}]{sexticRank4ops.pdf} }\end{matrix}$ Figure 8:
Sextic singlet operators in $O(N)^{4}\rtimes S_{4}$ theory

The reader may observe that the operators $\mathcal{O}_{12}$ and
$\mathcal{O}_{18}$ differ only by their colouring. Mathematically speaking,
the four-regular graph of order six that is associated to these operators
admits two distinct edge colourings not related by permutations of colours or
vertices. Physically, this means these operators would not be distinct in a
theory of symmetric four-tensors.

The action of the theory we now consider is given explicitly by

$\displaystyle S=\int
d^{3-\epsilon}x\bigg{(}\frac{1}{2}(\partial_{i}\phi^{abcd})(\partial_{i}\phi^{abcd})+\frac{1}{6!}\sum_{n=1}^{21}g_{n}\,\mathcal{O}_{n}\bigg{)}\,.$
(118)

The four-loop beta functions for this theory are listed in the supplementary
Mathematica file. They are quite lengthy but simplify somewhat in the large
$N$ limit. The optimal scalings associated to the operators are given by

$\displaystyle g_{i}=\frac{\lambda_{i}}{N^{e_{i}}}\,,$ (119)

where the exponents take on the following values:

$\displaystyle e=\\{8,7,6,6,6,6,5,5,4,4,4,4,5,4,4,4,5,5,4,4,4\\}\,.$ (120)

Under this scaling, the four-loop anomalous dimensions for $\phi$ and
$\phi^{2}$ are given at large $N$ by

$\displaystyle\gamma_{\phi}^{(\text{four-
loop})}=2\frac{24\lambda_{9}^{2}+4\lambda_{10}^{2}+4\lambda_{11}^{2}+12\lambda_{12}^{2}+4\lambda_{14}^{2}+\lambda_{15}^{2}+\lambda_{16}^{2}+2\lambda_{19}^{2}+8\lambda_{20}^{2}+12\lambda_{21}^{2}}{3(6!)^{2}(8\pi)^{4}}\,,$