$\displaystyle\gamma_{\phi^{2}}^{(\text{four-
loop})}=32\gamma_{\phi}^{(\text{four-loop})}\,.$ (121)

The six-loop anomalous dimensions are listed in the supplementary Mathematica
file. Even at large $N$ they are rather lengthy, but we note that
$\gamma_{\phi}^{(\text{six-loop})}$ only depends on the same coupling
constants $\lambda_{i}$ as $\gamma_{\phi}^{(\text{four-loop})}$ does, whereas
$\gamma_{\phi^{2}}^{(\text{six-loop})}$ introduces dependencies on
$\lambda_{4}$, $\lambda_{5}$, $\lambda_{7}$, $\lambda_{8}$, and
$\lambda_{13}$.

By setting to zero the large $N$ beta functions to quadratic order, we can
determine the perturbative fixed points with couplings proportional to
$\epsilon$. This amounts to solving 21 coupled quadratic equations. It can be
directly seen from the large $N$ beta functions that $\lambda_{12}$,
$\lambda_{19}$, and $\lambda_{21}$ must all be zero to leading order in
$\epsilon$ at any fixed point. This leaves 18 coupled equations, which are
still difficult to solve analytically. However, numerical evidence strongly
suggests that $\lambda_{10}$ must also vanish on the fixed points, and once we
enforce the constraint $\lambda_{10}=0$, we are able to solve for the two-loop
fixed points analytically. We find a total of 18 perturbative fixed points,
which we will label with Roman numerals (i) to (xviii), where (i) is the
trivial fixed point. The values of the couplings at each of the fixed points
(ii) to (xviii) are listed to leading order in $\epsilon$ in Appendix C. The
order $\epsilon^{2}$ corrections can be found in the supplementary Mathematica
file. These corrections are required to compute the anomalous dimensions at
the fixed points up to order $\epsilon^{3}$, the results of which computations
we display below. For fixed points (ii) to (x), we find that

$\displaystyle\gamma_{\phi}^{\text{(ii)}}=\gamma_{\phi}^{\text{(iii)}}=\frac{15}{16}\epsilon^{2}-\frac{683-3\pi^{2}}{128}\epsilon^{3}+\mathcal{O}(\epsilon^{4})\,,$
$\displaystyle\gamma_{\phi^{2}}^{\text{(ii)}}=\gamma_{\phi^{2}}^{\text{(iii)}}=30\epsilon^{2}-\frac{1}{4}(1103+51\pi^{2})\epsilon^{3}+\mathcal{O}(\epsilon^{4})\,,$
$\displaystyle\gamma_{\phi}^{\text{(iv)}}=6\epsilon^{2}-(328-6\pi^{2})\epsilon^{3}+\mathcal{O}(\epsilon^{4})\,,$
$\displaystyle\gamma_{\phi^{2}}^{\text{(iv)}}=192\epsilon^{2}-32(349+21\pi^{2})\epsilon^{3}+\mathcal{O}(\epsilon^{4})\,,$
$\displaystyle\gamma_{\phi}^{\text{(v)}}=\frac{1}{9}\epsilon^{2}+\frac{2}{81}(12+\pi^{2})\epsilon^{3}+\mathcal{O}(\epsilon^{4})\,,$
$\displaystyle\gamma_{\phi^{2}}^{\text{(v)}}=\frac{32}{9}\epsilon^{2}+\frac{16}{81}(-15+22\pi^{2})\epsilon^{3}+\mathcal{O}(\epsilon^{4})\,,$
$\displaystyle\gamma_{\phi}^{\text{(vi)}}=\gamma_{\phi}^{\text{(vii)}}=\frac{151}{144}\epsilon^{2}-\frac{7(10797-133\pi^{2})}{10368}\epsilon^{3}+\mathcal{O}(\epsilon^{4})\,,$
$\displaystyle\gamma_{\phi^{2}}^{\text{(vi)}}=\gamma_{\phi^{2}}^{\text{(vii)}}=\frac{302}{9}\epsilon^{2}+\frac{49}{324}(-2319+469\pi^{2})\epsilon^{3}+\mathcal{O}(\epsilon^{4})\,,$
$\displaystyle\gamma_{\phi}^{\text{(viii)}}=\frac{55}{9}\epsilon^{2}-\frac{64}{81}(435-11\pi^{2})\epsilon^{3}+\mathcal{O}(\epsilon^{4})\,,$
$\displaystyle\gamma_{\phi^{2}}^{\text{(viii)}}=\frac{1760}{9}\epsilon^{2}+\frac{80}{81}(-11829+3104\pi^{2})\epsilon^{3}+\mathcal{O}(\epsilon^{4})\,,$
$\displaystyle\gamma_{\phi}^{\text{(ix)}}=\gamma_{\phi}^{\text{(x)}}=\frac{1395}{1331}\epsilon^{2}-\frac{6(261739892-2985147\pi^{2})}{214358881}\epsilon^{3}+\mathcal{O}(\epsilon^{4})\,,$
$\displaystyle\gamma_{\phi^{2}}^{\text{(ix)}}=\gamma_{\phi^{2}}^{\text{(x)}}=\frac{44640}{1331}\epsilon^{2}+\frac{48(-3580720525867+898956393598\pi^{2})}{488523889799}\epsilon^{3}+\mathcal{O}(\epsilon^{4})\,.$

For fixed points (xi) to (xviii) the coefficients of $\epsilon^{2}$ and
$\epsilon^{3}$ in $\gamma_{\phi}$ and $\gamma_{\phi^{2}}$ invole roots of
quartic polynomials, so we will merely display their numerical values, which
we list Table 6. For comparison, we also include the numerical values for the
coefficients at the other fixed points.

fixed point | $\gamma_{\phi}-\mathcal{O}(\epsilon^{4})$ | $\gamma_{\phi^{2}}-\mathcal{O}(\epsilon^{4})$
---|---|---
(i) | 0 | 0
(ii) | $0.9375\epsilon^{2}-5.1046186\epsilon^{3}$ | $30\epsilon^{2}-401.58746\epsilon^{3}$
(iii) | $0.9375\epsilon^{2}-5.1046186\epsilon^{3}$ | $30\epsilon^{2}-401.58746\epsilon^{3}$
(iv) | $6\epsilon^{2}-268.78237\epsilon^{3}$ | $192\epsilon^{2}-17800.374\epsilon^{3}$
(v) | $0.11111111\epsilon^{2}+0.53999023\epsilon^{3}$ | $3.5555556\epsilon^{2}+39.927170\epsilon^{3}$
(vi) | $1.0486111\epsilon^{2}-6.4033949\epsilon^{3}$ | $33.555556\epsilon^{2}+349.32833\epsilon^{3}$
(vii) | $1.0486111\epsilon^{2}-6.4033949\epsilon^{3}$ | $33.555556\epsilon^{2}+349.32833\epsilon^{3}$
(viii) | $6.1111111\epsilon^{2}-257.92344\epsilon^{3}$ | $195.55556\epsilon^{2}+18574.076\epsilon^{3}$
(ix) | $1.0480841\epsilon^{2}-6.5015549\epsilon^{3}$ | $33.538693\epsilon^{2}+519.92939\epsilon^{3}$
(x) | $1.0480841\epsilon^{2}-6.5015549\epsilon^{3}$ | $33.538693\epsilon^{2}+519.92939\epsilon^{3}$
(xi) | $4.8136566\epsilon^{2}-177.89072\epsilon^{3}$ | $154.03701\epsilon^{2}-11073.347\epsilon^{3}$
(xii) | $0.85551344\epsilon^{2}-4.4862499\epsilon^{3}$ | $27.376430\epsilon^{2}-342.60322\epsilon^{3}$
(xiii) | $0.62994776\epsilon^{2}-2.6804986\epsilon^{3}$ | $20.158328\epsilon^{2}-205.70175\epsilon^{3}$
(xiv) | $3.1684748\epsilon^{2}-78.122861\epsilon^{3}$ | $101.39119\epsilon^{2}-4647.5684\epsilon^{3}$
(xv) | $4.9247677\epsilon^{2}-175.81337\epsilon^{3}$ | $157.59257\epsilon^{2}+9810.5193\epsilon^{3}$
(xvi) | $0.96662455\epsilon^{2}-5.7167280\epsilon^{3}$ | $30.931986\epsilon^{2}+262.06325\epsilon^{3}$
(xvii) | $0.74105887\epsilon^{2}-3.5240650\epsilon^{3}$ | $23.713884\epsilon^{2}+131.62751\epsilon^{3}$
(xviii) | $3.2795859\epsilon^{2}-80.724969\epsilon^{3}$ | $104.94675\epsilon^{2}+3331.7552\epsilon^{3}$

Table 6: Table of anomalous dimensions at the large $N$ fixed points of sextic
$O(N)^{4}\rtimes S_{4}$ theory

Based on the sizes of the coefficients in Table 6, it is doubtful that the
fixed points extend all the way down to $d=2$, nor does it bode well that the
$\epsilon^{3}$ coefficients are everywhere larger than the $\epsilon^{2}$
coefficients.

####  3.3.1 Degeneracy of fixed points

The fixed point (iv) stands out from the others. It is special in that a one-
parameter degeneracy in the values of the couplings emerges at order
$\epsilon^{2}$, as we show explicitly in Appendix C. If this one-dimensional
continuum of fixed points in the $\epsilon$ expansion were to be meaningfully
continued to an integer dimension, the degeneracy would be indicative of the
presence of a conformal manifold. The degeneracy may be a perturbative
artefact, but either way higher-order corrections in $\epsilon$ or in $1/N$
will not lift the degeneracy for the simple reason that any linearized
perturbation at a given order in an expansion always produces 21 equations,
$0=\delta\beta_{\lambda_{n}}$, in 21 unknowns, $\delta\lambda_{n}$.

####  3.3.2 Melonic fixed points?

As for the rank-three model, we may ask of the sextic rank-four model if it
possesses fixed points with couplings that scale as $\sqrt{\epsilon}$. In
fact, for the so-called subchromatic sextic theories, ie. the sextic theories
of tensors with rank less than five, Ref. [109] identified a total of three
operators whose contributions to the free energy are melonic and for this
reason can be resummed: the prismatic and wheelic operators $\mathcal{O}_{1}$
and $\mathcal{O}_{2}$ of the rank-three model and also what they referred to
as the octahedral operator, namely $\mathcal{O}_{21}$ of the rank-four model.
To search for a fixed point dominated by the octahedron, we set to zero the
couplings for the non-octahedral operators that also have a large $N$ scaling
exponent of four, ie. we set

$\displaystyle\lambda_{9}=\lambda_{10}=\lambda_{11}=\lambda_{12}=\lambda_{14}=\lambda_{15}=\lambda_{16}=\lambda_{19}=\lambda_{20}=0\,.$
(128)

Imposing the condition (128), the octahedral beta function simplifies to

$\displaystyle\beta_{\lambda_{21}}=-2\epsilon\lambda_{21}+\frac{48}{(6!)^{2}(8\pi)^{4}}\lambda_{21}^{3}+\mathcal{O}(\lambda^{4})\,.$
(129)

Plugging either of the fixed point values $\lambda_{21}=\pm
3840\sqrt{6\epsilon}\pi^{2}$ into the expressions for the six-loop mass and
field anomalous dimensions, while still imposing (128), we recover the melonic
values for a sextic theory:

$\displaystyle\gamma_{\phi}=\frac{\epsilon}{3}+\mathcal{O}(\lambda^{4})\,,\hskip
42.67912pt\gamma_{\phi^{2}}=\frac{32}{3}\epsilon+\mathcal{O}(\lambda^{4})\,.$
(130)

However, in order for the model to posses a genuine fixed point, it is of
course necessary that all beta functions vanish, not just
$\beta_{\lambda_{21}}$. And it can be checked that on imposing (128) and
setting $\lambda_{21}=\pm 3840\sqrt{6\epsilon}\,\pi^{2}$, the beta function
for the 17th coupling equals

$\displaystyle\beta_{\lambda_{17}}=30\,720\pi^{2}\epsilon+\mathcal{O}(\lambda^{4})\,.$
(131)

We conclude that the operators whose couplings exhibit a greater large $N$
suppression than that of the octahedron destroy the stability of the melonic
fixed point, unless we accept the exotic possibility of fixed points with
couplings that scale as $\lambda\sim\epsilon^{1/4}$ or with even smaller
fractional powers of epsilon.

###  3.4 Overview and discussion

In Table 7 we present a summary of the number of perturbative fixed points in
sextic scalar theories after applying the optimal scalings and taking the
large $N$ limit. This tally includes only such fixed points as manifest
themselves already at two-loop level, including the trivial fixed point. In
contrast to the case of quartic interactions, the sextic vector model does not
posses a large $N$ perturbative fixed point under the optimal scaling. At
ranks two and three, a single such fixed point appears, while at rank four,
even after restricting ourselves to the subsector that is symmetric with
respect to permutations of indices, we observe a sudden proliferation, with 18
fixed points present. This number might be significantly higher for the theory
without $S_{4}$ permutation theory, but the computation to check whether this
is so or not is hindered by the fact that $n_{\text{sextic}}=132$ for the
theory with only $O(N)^{4}/(\mathbb{Z}_{2})^{3}$ symmetry. Similarly, in
attempting to extend the above table to higher ranks, one encounters an
obstruction in the fact that $n_{\text{sextic}}=52$ for the theory with
$S_{5}\ltimes O(N)^{5}/(\mathbb{Z}_{2})^{4}$ symmetry, while
$n_{\text{sextic}}=1439$ for the theory with only
$O(N)^{5}/(\mathbb{Z}_{2})^{4}$ symmetry. Unlike the quartic case, we do not
have formulas for the number of operators—let alone the numbers of fixed
points—at arbitrary rank, ultimately for the reason that the set of graphs of
order six is vastly more complicated than that of order-four graphs. Nor are
we able to present general large $N$ solutions as we were for the quartic
theories, owing to the fact that while the quartic theories were exclusively
dominated by vectorial operators, the dominant operators for the sextic
theories also include melonic and matrix-like operators. Purely melonic
theories are solvable, and Ref. [103] successfully obtained a large $N$
solution of the non-trivial rank-three fixed point in Table 7, a fixed point
dominated by the the prismatic operator $\mathcal{O}_{1}$. This is likely to
be the only non-trivial fixed point in the table to admit such an elegant
large $N$ solution. All the other fixed points in the table contain non-zero
matricial couplings among the coupling constants with the smallest suppression
in $N$. This entails that the classes of diagrams that dominate at large $N$
include the set of all planar diagrams built out of those matricial operators,
and the set of planar diagrams is much more unwieldy than its melonic subset.

global symmetry $O(N)$ $\frac{O(N)^{2}\rtimes S_{2}}{\mathbb{Z}_{2}}$
$\frac{O(N)^{3}}{(\mathbb{Z}_{2})^{2}}$ $\frac{O(N)^{3}\rtimes
S_{3}}{(\mathbb{Z}_{2})^{2}}$ $\frac{O(N)^{4}\rtimes
S_{4}}{(\mathbb{Z}_{2})^{3}}$ $n_{\text{sextic}}$ 1 3 16 8 21 $n_{\text{fixed
points}}$ 1 2 2 2 18

Table 7: Numbers of operators and fixed points in sextic $O(N)^{r}\rtimes
S_{r}$ theory

Of all the operators in the sub-chromatic sextic theories we have reviewed,
Ref. [109] identified three for which the sets of diagrams constructed
exclusively out of each of these can be resummed. Besides the prismatic
operator, these are the rank-three operator $\mathcal{O}_{2}$ and the rank-
four operator $\mathcal{O}_{21}$. But as we have seen, the melonic fixed
points associated to these operators, if they are perturbatively accessible,
rely on cancellations between linear and cubic terms in beta functions and
require some coupling constants to scale unusually with $\epsilon$, such as
$\lambda\sim\epsilon^{0}$ or $\lambda\sim\epsilon^{1/4}$, and they are not
among the Wilson-Fisher fixed points listed in Table 7.

Moving on from sub-chromatic sextic theories to sextic rank-five tensor
theory, one encounters the kind of melonic interaction present in the original
Klebanov-Tarnopolsky models proposed in Ref. [28] and further studied in Ref.
[71], which determined an explicit $d$-dependent melonic large $N$ solution.
For their solution, the quadratic operator $\phi^{abcde}\phi^{abcde}$ has a
real dimension only for $d_{\text{crit}}<d\leq 3$ with $d_{\text{crit}}\approx
2.97$. An analysis of the theory through the lense of renormalized
perturbation theory, incorporating the operator mixing of different sextic
operators, was carried out in Ref. [110], which circumvented the issue of the
multitudinous sextic rank-five operators by computing the beta functions only
for the subset of operators relevant to their analysis and only at large $N$.
Through this computation, the authors arrived at the conclusion that there is
no rank-five melonic fixed point in the $\epsilon$ expansion after noticing
the same phenomenon as we observed in the rank-four theory with equation
(131): the fixed point value for the melonic coupling induces a linear term in
another beta function that cannot be cancelled to third order in perturbation
theory. Again the presence of couplings with scalings like
$\lambda\sim\epsilon^{0}$ or $\lambda\sim\epsilon^{1/4}$ provide a possible
resolution to the issue. If scalings $\lambda\sim\epsilon^{0}$ are the true
reason melonic sextic fixed points elude the $\epsilon$ expansion, then it
signifies an intriguing scenario where, as we vary $d$, the melonic theory in
$d_{\text{crit}}<d<3$ interpolates, not to a trivial theory in 3$d$ in the way
that the Ising model interpolates to a trivial theory in $4d$, but to an
interacting yet non-melonic three-dimensional theory, which however cannot be
accessed through perturbation theory. A less interesting possibility is that a
melonic fixed point simply does not exist in this theory.

##  4 Outlook

In this paper we have presented studies of tensorial $\phi^{4}$ and $\phi^{6}$
theories with multiple copies of $O(N)$ symmetry, applying small $\epsilon$
and large $N$ methods. As the behaviour of the quartic and sextic models
differ drastically, each class of models offers its own set of lessons and
further questions.

The sextic theories are by far the least understood. While the strict large
$N$ limit of the sextic $O(N)$ model yields a trivial beta function, the same
no longer applies in the multi-index cases. Instead, the large $N$ dynamics
exhibit an admixture of vectorial, matricial, and melonic operators. Through
our study of sextic $O(N)^{4}$ theory, we have shown that rank-four tensor
theory marks the onset of a sudden increase in the number of perturbative RG
fixed points at large $N$ and in the diverseness among them—with couplings
given in some cases by simple expressions, in others by perplexingly lengthy
ones, and with one perturbative fixed point at subleading order degenerating
into a one-parameter family of continuously connected fixed points. Which of
these fixed points in the $\epsilon$ expansion admit large $N$ solutions exact
in $d$, and for which ranges of dimensions they exist, are questions that
remain to be answered. Higher rank theories are bound to produce even more
fixed points and yet wider variety among them, but the numbers, types, and
ranges of validity of fixed points for general tensor rank elude
classification, as the combinatorial explosion of graphs of order six with
increasing size quickly stymies searches for underlying patterns.

Contrariwise, for quartic orthogonal tensor theories, owing to the relative
simplicity of graphs of order four, we have been able to provide large $N$
solutions of the $O(N)^{r}$ theories with $r\leq 5$, and to chart out a vast
general family of large $N$ fixed points present at any $r$, all of vector-
model type, meaning that the dominant Feynman graphs are bubble diagrams. The
large $N$ and small $\epsilon$ expansions encouragingly indicate that the
fixed points do extend to bona fide $3d$ CFTs, meaning that the critical
$O(N)$ model admits a vast generalization to critical $O(N)^{2}$ models,
critical $O(N)^{3}$ models, etc. Note, however, that our results are all
perturbative, either in $1/N$ or in $\epsilon$, and so more work is needed to
firmly establish these wider classes of critical models. If this can be
achieved, the most intriguing outlook would be a future experimental
realization of these prospective CFTs, but the fact that a large number of
relevant deformations exist for most of the fixed point poses a serious
challenge. On the theoretical side, the $O(N)$ model may just be the single-
most studied of all quantum field theories, and if the $O(N)^{r}$ models
indeed, as our study suggests, furnish separate instances of slightly more
complicated, but still entirely tractable, relatives of the beloved $O(N)$
model, then all of the QFT phenomena that have been investigated with the
$O(N)$ model as a test case can be extended to the larger playfield of the
$O(N)^{r}$ models. Specific avenues for future inquiry include the following:

  * •

It would be desirable to verify or refute the existence of the vectorial fixed
points in three dimensions and at finite $N$ with non-perturbative methods
like Monte Carlo simulations or the numerical bootstrap.

  * •

The critical $O(N)$ model can be studied in $2+\epsilon$ dimensions through a
perturbative analysis of the non-linear sigma model [112, 113, 114, 113, 115,
116]. The status of $O(N)^{r}$ CFTs would be substantially solidified if
$2+\epsilon$ expansions could likewise be developed for these models.

  * •

It remains to be investigated whether the large $N$ fixed points we have
studied are stable at finite temperature. In the case of the $O(N)$, the free
energy at finite temperature was computed to leading order in large $N$ by
Ref. [117], while the subleading correction was computed by Ref. [118] and,
recently and with increased precision, by Ref. [119]. As explained in the
latter reference, the value of the subleading correction hinges on a delicate
renormalization of the fluctuations of the $\sigma$ field, and similar
analyses may be possible for the additional Hubbard-Stratonovich fields
$\chi$, $\tau$, $\rho$, etc. present in the tensorial theories.

  * •

In the $4-\epsilon$ expansion, there exist, in addition to the exhaustive list
of fixed points for four- and five-component scalar theories, several known
infinite families, and it is conceivable that in some cases, for intermediate
values of $N$, the tensorial fixed points we studied in Section 2 can be
identified with members of the known families, such as the Paulos fixed points
[120, 70, 121, 122, 68], which generalize the MN fixed points. A comprehensive
classification of fixed points in the $\epsilon$ expansion requires
determining which, if any, of the $O(N)^{r}$ fixed points that can secretly be
identified with known fixed points.

  * •

Besides the large $N$ fixed points, the perturbative expansions of the quartic
$O(N)^{r}$ theories reveal the presence of numerous fixed points at the
specific value of $N=2$. It seems likely that these $O(2)^{r}$ fixed points
can be classified for general values of $r$.

  * •

The singlet-sector of the critical $O(N)$ model was famously conjectured in
Ref. [123] to be dual to a higher-spin Vasiliev theory, and an abundance of
convincing evidence has since been discovered in favour of the conjecture. How
does this duality generalize to tensorial models?181818Initial investigations
in this direction can be found in Ref. [124].

  * •

It has been known for decades that on sending $N$ to $-N$, the Feynman
diagrams in theories with $O(2N)$ symmetry are transformed into Feynman
diagrams of theories with $Sp(2N)$ symmetry [125, 126]. This relation extends
from vector models to matrix models [127] and to tensor models [86, 128].
Consequently, for all the models studied in this paper, the perturbative beta
functions and anomalous dimensions for the symplectic versions of the theories
are immediately available, so that their fixed points in the $\epsilon$
expansion and their large $N$ limits can be readily analyzed. When the tensor
rank is even, the symplectic theory is a theory of commuting scalars, while
for odd tensor rank the scalars in the symplectic theory anti-commute, as
required for the kinetic term to be non-vanishing. Therefore, the odd rank
theories with $Sp(2N)$ symmetry are non-unitary, but it can nonetheless be
checked that their perturbative RG flows are gradient and monotonic. And as
pointed out in Ref. [129], the CFTs relevant to dS/CFT are Euclidean and not
continued into Lorentzian signature, meaning that non-unitarity in itself does
not despoil symplectic theories from being physically applicable. Higher-
dimensional Liouville field theories are examples of non-unitary CFTs that are
solvable in Euclidean signature [130, 131].

  * •

The large $N$ analysis of the $O(N)$ model indicates the presence of a UV
fixed point in dimensions $4<d<6$. Refs. [3, 132] developed a $6-\epsilon$
expansion for this fixed point in terms of a theory with cubic interactions
$\sigma\phi^{a}\phi^{a}$ and $\sigma^{3}$. Can theories with cubic terms
involving the fields $\chi$, $\tau$, $\rho$, etc. be used to determine similar
$6-\epsilon$ expansions for the $O(N)^{r}$ theories? For theories with
symplectic symmetry, $6-\epsilon$ expansions were developed in Refs. [133],
[134], and tensorial generalizations likely exist for these expansions as
well.

  * •

The $\epsilon$ expansions of the Gross-Neveu model in $2+\epsilon$ dimensions
and of the Gross-Neveu-Yukawa model in $4-\epsilon$ expansion can both be
matched against a large $N$ solution using a Hubbard-Stratonovich
transformation similar to that of the $O(N)$ model [135]. Do these fermionic
theories also admit similar tensorial generalizations?

  * •

A straightforward follow-up to the present paper would be to carry out similar
analyses for tensors with $U(N)^{r}$ rather than $O(N)^{r}$ symmetry. In fact,
because symmetric tensors furnish irreps of the $U(N)$ group without the need
to subtract off traces, the large $N$ field content of $U(N)^{r}$ fixed points
as well as the associated Hubbard-Stratonovich transformations will almost
certainly be simpler than in our case. A study of $U(N)^{r}$ theories would
also offer an avenue to connect with the field of tensor rank decompositions
of Hermitian tensors. Other variations of the present work would be to study
theories where the fundamental fields transform under different
representations than the $r$-fold multi-fundamental representation of the
$O(N)$ group, and attempting to classify e.g. the RG fixed points of quartic
theories of symmetric traceless tensors, antisymmetric tensors, or mixed
representation tensors. Existing results on theories of tensors transforming
under non-fundamental representations can be found in Refs. [136], [137],
[138], [139], [140].

  * •

Among the topics for which the $O(N)$ model has served as an important testing
ground (besides leading to experimental predictions) is the study of line and
surface defects, a study which has seen a recent surge of progress. The
$O(N)^{r}$ models offer the opportunity to broadly extend this testing ground.

  * •

In Appendix B.2, we present an argument that melonic operators are generally
absent at the perturbative large $N$ fixed points of quartic $O(N)^{r}$
models. For odd $r$, the absence of melonic fixed points implies that all
fixed points belong to the vectorial class, whose members we have exhaustively
cataloged. But, in the notation of Appendix B.1, for even $r$ the possibility
remains that matricial and vector-melonic fixed points appear at higher rank.
In the interest of completeness, it would be desirable to disprove such fixed
points or alternatively flesh out counterexamples at higher rank. A
comprehensive understanding of $O(N)^{r}$ theory at any rank would pave the
way for studying the large rank limit of these theories and exploring emergent
phenomena in the spirit of Philip Anderson’s _More is different_ [141]. For
the tensor theories that are related to the SYK model, the study of the limit
of large rank derives motivation from the physical insight that has been
gleaned from the double-scaling limit of the SYK model.

On a closing note, we remark that while some questions can be answered
straightforwardly for quartic tensor models of arbitrarily large rank—such as
how many vectorial large $N$ fixed points there are, or what their anomalous
dimensions are for $\phi$ and $\phi^{2}$—there other basic questions that we
are unable to fully answer even for rank-three tensors. One such questions is
this: what are the single-trace operators of the theory? More precisely, what
is the set of operators that do not factor into separate groups of fundamental
fields with index contractions only among themselves? In the rank-one case,
there is but a single such operator: $\phi^{a}\phi^{a}$. For rank-two tensors,
there is one operator for each even positive integer $n$, constructed by
cyclically contracting $n$ fields together: $\phi^{ab}\phi^{ab}$,
$\phi^{a_{1}b_{1}}\phi^{a_{1}b_{2}}\phi^{a_{2}b_{2}}\phi^{a_{2}b_{1}}$, etc.
In the case of rank-three tensors, the number of such operators proliferates
tremendously with increasing number of fields. Determining the single-trace
operators for a given number of fundamental fields is tantamount to a graph
theoretical problem of a kind so difficult that the mathematical literature
does not attempt to provide exhaustive constructions for the full set of
graphs or to determine general formulas for their numbers but instead contents
itself with searching for efficient algorithms for counting the number of
graphs at a given order.191919To get an idea of the complexity involved in
enumerating graphs, we refer the reader to the growth of cubic connected
graphs in table 2 of Ref. [142]. However, for three-tensor invariants, only
those cubic graphs are relevant which have three-colourable edges. It turns
out that the four colour theorem is equivalent to the statement that the
three-colourable cubic graphs are precisely the subset of cubic graphs that
bridgeless, or 1PI in physics notation. Building on and implementing key
results of Read [143], the authors of [144] were able to write down (quite
complicated) generating functions for the numbers of $O(N)^{r}$ tensor
invariants and to provide software by which to determine these numbers in the
first many cases. Using their results, we can tabulate the number of single
trace operators built from $n$ copies of a tensor fields $\phi$ of rank $r$,
as shown below.

$\displaystyle\begin{matrix}\text{ \scalebox{0.9}{
\begin{tabular}[]{|l|c|c|c|c|c|c|c|}\hline\cr&$n=2$&$n=4$&$n=6$&$n=8$&$n=10$&$n=12$&$n=14$\\\
\hline\cr$r=1$&1&0&0&0&0&0&0\\\ \hline\cr$r=2$&1&1&1&1&1&1&1\\\
\hline\cr$r=3$&1&4&11&60&318&2806&29\,359\\\
\hline\cr$r=4$&1&13&118&3931&228\,316&24\,499\,085&3\,816\,396\,556\\\
\hline\cr\end{tabular}}}\end{matrix}$ (132)

As intimated by this table, the mathematical discipline of graph theory,
closely tied to the physics of tensorial Feynman diagrams, is one wherein the
border region between trivial and impossible inquiries is particularly narrow.

### Acknowledgements

We are thankful to Ofer Aharony, Yasha Neiman, Fedor Popov, Amit Sever, and
Grigory Tarnopolskiy for useful discussions and especially to Igor Klebanov
for numerous valuable comments offered and insights shared. CBJ expresses his
gratitude to the Simons Center for Geometry and Physics (SCGP), Stony Brook
University, where he was employed during much of this project. YO would like
to thank SCGP for a wonderful hospitality during 2021-2022, when this work was
initiated. This work is supported in part by the ISF center of excellence and
the U.S.-Israel Binational Science Foundation and by Korea Institute for
Advanced Study (KIAS) Grant PG095901.

##  Appendix A Optimal Scaling

The subject of this paper is $d$-dimensional scalar theories of rank-$r$
tensor fields $\phi^{a_{1}a_{2}...a_{r}}$, where $a_{j}\in\\{1,..,N\\}$, with
action given by

$\displaystyle S=\int
d^{d}x\,\bigg{(}\frac{1}{2}(\partial_{i}\phi^{a_{1}a_{2}...a_{r}})(\partial_{i}\phi^{a_{1}a_{2}...a_{r}})+\sum_{n=1}^{n_{\text{max}}}g_{n}\,\mathcal{O}_{n}(\phi)\bigg{)}\
,$ (133)

where the interactions $\mathcal{O}_{n}(\phi)$ are homogeneous polynomials in
$\phi$ that are symmetric under separate rotations of each tensor index and
thereby exhibit $O(N)^{r}$ symmetry (quotiented by $(\mathbb{Z}_{2})^{r-1}$ to
account for overcounting of the $\phi\rightarrow-\phi$ symmetry). For the
theories we studied in the paper, the interactions $\mathcal{O}_{n}$ were all
of degree four or all of degree six, but here we will allow for each
interaction $\mathcal{O}_{n}$ to have its own degree $q_{n}$.

The purpose of this appendix is to describe ways of ascertaining what the
appropriate powers of $N$ are by which to scale the coupling constants. We
will denote these powers by $e_{n}$ and introduce rescaled couplings
$\lambda_{n}$ given, for $n\in\\{1,...,n_{\text{max}}\\}$, by

$\displaystyle\lambda_{n}=\frac{g_{n}}{N^{e_{n}}}\,.$ (134)

We then define the large $N$ limit as

$g_{n}\rightarrow
0,~{}~{}~{}~{}N\rightarrow\infty,~{}~{}~{}~{}\lambda_{n}~{}~{}~{}\text{fixed}\
.$ (135)

In the case of matrix theories, $r=2$, the suitable choice of scaling
exponents was determined in the classic work of ’t Hooft [145], with the
exponent of a given operator being determined by the number of traces of the
operator. For general tensorial operators there is no simple known answer for
what values scaling exponents should assume. We want all observables of the
theory to remain finite as we send $N$ to infinity, which means that we cannot
choose the exponents $e_{n}$ to be too small. We also want all the beta
functions for the couplings $\lambda_{n}$ to remain finite, which means that
we cannot choose the exponents to be too large. These conditions generally
permit ranges of allowed values for the exponents $e_{n}$, and among the
allowed values it is desirable to choose the exponents to be as small as
possible in order not to needlessly suppress any field content and thereby
lose generality.

When doing perturbative calculations in an $\epsilon$ expansion, it is
straightforward to determine the range of allowed scaling exponents at a given
order of perturbation theory. But a scaling allowed at a given order may cause
a blow up at higher order. The issue posed by this fact amounts to a question
of order of limits: the $\epsilon\rightarrow 0$ limit and the
$N\rightarrow\infty$ limits do no commute. Picking a scaling valid only up to
a certain order of perturbation theory amounts to studying the regime where
$\epsilon$ is much smaller than $1/N$ or than some power thereof, which is
usually undesirable since ultimately we would like to set $\epsilon$ equal to
one. One could attempt to guess non-perturbative scalings based on
perturbative answers, but this is a dangerous path to tread, for scalings of
coupling constants are determined by index contractions in Feynman diagrams,
and in the field of graph theory it is not uncommon for patterns present in
families of graphs at low order to unexpectedly disappear at higher
order.202020For example, it was conjecture that the list-chromatic number
$\chi_{\ell}(G)$ of a any plane graph $G$ satisfies the inequality
$\chi_{\ell}(G)\leq\chi(G)+1$, where $\chi(G)$ is the ordinary chromatic
number of $G$. The first known counter-example to this conjecture, pointed out
by Voigt [146], is a graph on 238 vertices, although Voigt and Wirth [147]
subsequently found a counterexample of order 75, and Mirzakhani [148, 149]
discovered a counter-example of order 63. The choice of scaling exponents
should therefore be determined non-perturbatively or at least according to
arguments that apply at all orders of perturbation theory.

In the context of zero-dimensional quantum field theories with $O(N)^{r}$ or
$U(N)^{r}$ symmetry, Ref. [34] introduced the term _optimal scaling_ to denote
the smallest set of scaling exponents compatible with a well-defined $1/N$
expansion. One way to phrase this definition is by identifying, for a given
theory, an optimal scaling as a choice of exponents $\\{e_{n}\\}$ such that
the free energy of the theory is polynomially bounded in $N$ but on decreasing
any one of the exponents $e_{n}$, the free energy ceases to be polynomially
bounded. For tensors of high rank, determining the optimal scaling can be a
non-trivial task. In principle, the optimal scaling need not even be unique.
In fact, we illustrate in Subsection A.5 below that non-unitary theories with
$U(N)^{r}$ symmetry admit multiple inequivalent optimal scalings.

In the following subsection, as a warm-up before we discuss the optimal
scaling of tensor operators, we review the arguments by which ’t Hooft
determined the large $N$ scaling of matrix operators. We then proceed to
explain, for general tensor operators, why no simple known prescription exists
for determining their large $N$ scalings, but we also provide general upper
bounds on scaling exponents.

###  A.1 Matricial optimal scaling

Consider a theory of matrix fields $\phi^{ab}$, $a,b\in\\{1,...,N\\}$, with
some set of interaction terms in the action, each of which is dressed with its
own coupling constant $g_{n}$. We wish to determine scaling exponents $e_{i}$
such that all connected vacuum Feynman diagrams remain finite in the
$N\rightarrow\infty$ limit. To this end, we consider an arbitrary such Feynman
diagram and apply some elementary graph theory. Since there are multiple
distinct ways of representing a Feynman diagram as a graph, we will, for the
sake of clarity, distinguish between three graphical representations of the
Feynman diagram:

  * •

the graph $\mathcal{G}$ which contains one vertex for each interaction term in
the Feynman diagram; this diagram contains no information about the index
contractions of the Feynman diagram,

  * •

the graph $\mathcal{G}^{\prime}$ which contains one vertex for each copy of
the fundamental matrix field in the Feynman diagram and which explicitly shows
the index contractions, and

  * •

the graph $\mathcal{G}^{\prime\prime}$ which contains one vertex for each
trace in the interaction terms in the Feynman diagram.

For all three types of graphs, the edges are determined by the propagators of
the Feynman diagram, but the graph $\mathcal{G}^{\prime}$ uses a double-edge
to indicate the two indices carried by the propagator. See Figure 9 for an
example of the three kinds of graphs associated to a Feynman diagram. Note
that while we require the graph $\mathcal{G}$ to be connected, the graphs
$\mathcal{G}^{\prime}$ and $\mathcal{G}^{\prime\prime}$ can be disconnected.

Figure 9: Different graphs corresponding to the same Feynman diagram built
from a sextic interaction $\text{Tr}[\phi^{2}]\text{Tr}[\phi^{4}]$ ($q_{i}=6$,
$\delta_{i}=2$, $N_{i}=1$) and two different quartic interactions
$\text{Tr}[\phi^{2}]^{2}$ ($q_{i}=4$, $\delta_{i}=2$, $N_{i}=1$) and
$\text{Tr}[\phi^{4}]$ ($q_{i}=4$, $\delta_{i}=1$, $N_{i}=1$) in a matrix
theory with $O(N)^{2}/\mathbb{Z}_{2}$ symmetry. For this example $C=2$, $F=5$,
$E=7$, $V=5$, and $L=6$.

Consider now some arbitrary Feynman diagram with graphs $\mathcal{G}$,
$\mathcal{G}^{\prime}$ and $\mathcal{G}^{\prime\prime}$. We will use the
symbol $C$ to denote the number of disconnected pieces in
$\mathcal{G}^{\prime}$ and $\mathcal{G}^{\prime\prime}$. Furthermore, let $F$
denote the number of faces (including the external face), $V$ the number of
vertices, and $E$ the number of edges in the graph
$\mathcal{G}^{\prime\prime}$. Normalizing the kinetic term such that the
propagator is independent of $N$, the power of $N$ that dresses the Feynman
diagram is equal to the number of index loops, which we will label by the
letter $L$. When the two types of strands of colours in the graph
$\mathcal{G}^{\prime}$ do not intersect, the number of index loops can be
counted from the number of faces plus number of connected pieces minus one.
When the strands do intersect, the number of index loops is lessened.
Consequently,

$\displaystyle L\leq F+C-1\,.$ (136)

Let $\chi$ be the Euler characteristic for the graph
$\mathcal{G}^{\prime\prime}$,

$\displaystyle\chi=V-E+F\,.$ (137)

The above formulas tell us that among the set of Feynman diagrams built from a
given set of interaction terms and a given number of propagators, the dominant
diagrams at large $N$ are those for which the Euler characteristic is the
largest, ie. the Feynman diagrams for which $\mathcal{G}^{\prime\prime}$ is a
planar graph, so that $\chi$ equals $C+1$. Combining this maximal value for
$\chi$ with (136) and (137), we obtain the inequality

$\displaystyle L\leq E-V+2C\,.$ (138)

Suppose now we use an index $n$ to label the different interaction terms of
our theory and that we use the symbol $q_{n}$ to denote the degrees of each of
these interaction terms, and the symbol $\delta_{n}$ to denote the number of
traces in each of them. Letting $N_{n}$ denote the number of times the
interaction term $n$ is present in the Feynman diagram under consideration, we
then have the following two formulas

$\displaystyle V=\sum_{n}N_{n}\,\delta_{n}\,,\hskip
56.9055pt2E=\sum_{n}N_{n}\,q_{n}\,.$ (139)

To determine the general large $N$ scaling for matricial operators, we need
one last formula in addition to the above. For any graph, the number of
disconnected pieces is less than or equal to the number of vertices: $C\leq
V$. But we can obtain a stricter inequality by using the fact that we are
dealing with a connected Feynman diagram. This means that the graph
$\mathcal{G}$ is connected, and since $\mathcal{G}$ has $\sum_{n}N_{n}$
vertices, the implication for the graph $\mathcal{G}^{\prime\prime}$ is that
this latter graph has at minimum $\sum_{n}N_{n}-1$ edges connecting together
different vertices, so that

$\displaystyle C\leq V-(\sum_{n}N_{n}-1)=1+\sum_{n}(\delta_{n}-1)N_{n}\,.$
(140)

Combining (138), (139), and (140), we finally arrive at the inequality

$\displaystyle L\leq
2+\sum_{n}N_{n}\Big{(}\frac{q_{n}}{2}+\delta_{n}-2\Big{)}\,.$ (141)

For a given set of values for $N_{n}$, there always exist Feynman diagrams for
which this inequality is saturated. We therefore conclude that the appropriate
large $N$ scaling of a coupling constant $g_{n}$ is given by

$\displaystyle g_{n}=\frac{\lambda_{n}}{N^{\frac{q_{n}}{2}+\delta_{n}-2}}\,.$
(142)

It is sometimes customary to rescale the fields
$\phi^{ab}\rightarrow\sqrt{N}\phi^{ab}$, so that each propagator carries a
factor of $1/N$ and the scaling exponents become simply $\delta_{n}-2$, but we
will not adopt this convention.

###  A.2 Three-tensor optimal scaling

In a theory where the fundamental fields are tensors $\phi^{abc}$, we can for
each Feynman diagram introduce a 3-index generalization of the graph type
$\mathcal{G}^{\prime}$, where now each propagator carries three strands of
colours, and the vertices in an interaction term are connected amongst each
other with three coloured edges incident to each vertex, as exemplified by the
following sextic operator:

$\displaystyle\begin{matrix}\text{ \includegraphics[scale={1}]{sexticEx.pdf}
}\end{matrix}$ (143)

For any Feynman diagram we can choose to completely disregard one index,
whereby we are left with a matrix-theory Feynman diagram, to which the
arguments of the previous subsection and the inequality (141) apply. Let
$L^{xy}$ denote the number of index loops in indices $x$ and $y$, and let
$\delta_{n}^{xy}$ denote the number of traces in the interaction term $n$ wrt.
indices $x$ and $y$, ie. when forgetting about the remaining index and viewing
the tensorial interaction as a matrix operator. We then have that

$\displaystyle L=\frac{L^{ab}+L^{ac}+L^{bc}}{2}\leq
3+\sum_{n}N_{n}\Big{(}\frac{3q_{n}}{4}+\frac{\delta^{ab}_{n}+\delta^{ac}_{n}+\delta^{bc}_{n}}{2}-3\Big{)}\,.$
(144)

A Feynman diagram that saturates this inequality possesses the property that
the graph $\mathcal{G}^{\prime}$ remains planar after erasing any one colour.
From the above inequality we recover the large $N$ scaling of Ref. [55]:

$\displaystyle
e_{n}=\frac{\delta_{n}^{ab}+\delta_{n}^{ac}+\delta_{n}^{bc}}{2}-3+\frac{3}{4}q_{n}\,.$
(145)

For example, the operator (143) has the following three two-colour
subdiagrams:

$\displaystyle\begin{matrix}\text{
\includegraphics[scale={1}]{subdiagrams.pdf} }\end{matrix}$ (146)

Consequently we have that

$\displaystyle\delta_{n}^{ab}=1\,,\hskip 42.67912pt\delta_{n}^{ac}=2\,,\hskip
42.67912pt\delta_{n}^{bc}=2\,,$ (147)

and so the large $N$ exponent equals

$\displaystyle e_{n}=\frac{2+1+2}{2}-3+\frac{3}{4}\,6=4\,.$ (148)

###  A.3 Optimal scaling at higher ranks

At rank four and higher there are multiple ways of adding together index loops
from two-colour subdiagrams such as to count the index loops in each colour
once. For example, for the number of index loops $L$ of a Feynman diagram in a
rank-four theory we have that

$\displaystyle
L=L^{ab}+L^{cd}=L^{ac}+L^{bd}=\frac{L^{ab}+L^{ac}+L^{ad}+L^{bc}+L^{bd}+L^{cd}}{3}\,.$
(149)

In general, introducing an ordering $<$ among indices $x$ and $y$ to avoid
double counting, we can choose any non-negative coefficients $C^{xy}$ and
write

$\displaystyle L=\sum_{x<y}C_{xy}\,L^{xy}$ (150)

provided that the coefficients satisfy the conditions

$\displaystyle\forall x:\,\,\sum_{y\neq x}C_{xy}=1\,,\hskip
28.45274pt\text{where}\hskip 28.45274ptC_{xy}=C_{yx}\,.$ (151)

Incidentally, the conditions on the coefficients at rank $r$ are equivalent to
the equations satisfied by Mandelstam invariants for $r$ points scattering or
by Mellin variables in the context of Mellin amplitudes.212121If we introduce
a set of $r$ auxiliary momenta $\vec{k}_{x}$ satisfying momentum conservation,
$\sum_{x}\vec{k}_{x}=0$, and carrying unit mass,
$\vec{k}_{x}\cdot\vec{k}_{x}=-1$, the conditions on the coefficients $C_{xy}$
are automatically satisfied if we set $C_{xy}=\vec{k}_{x}\cdot\vec{k}_{y}$.
This entails that the number of degrees of freedom in the choice of
coefficients equals $\frac{r(r-3)}{2}$, so that the coefficients are uniquely
fixed for $r=3$, while for $r=4$ there are two free parameters in choosing the
coefficients, for $r=5$ there are five free parameters, and so on.

For any valid choice of coefficients $C_{xy}$ we can make repeated use of the
inequality (141) to derive bounds on $L$ and thereby obtain bounds on the
scaling exponents $e_{n}$. Unlike the matrix case, for tensors the bounds we
obtain need not ever be saturated. To obtain the sharpest upper bound, we
should pick among all valid choices of coefficients the set that minimizes the
upper bound on $L$, whereby we obtain the following upper bound on scaling
exponents

$\displaystyle
e_{n}\leq\underset{C_{xy}}{\text{min}}\Big{[}\sum_{x<y}C_{xy}\,\delta_{n}^{xy}-r+\frac{r}{4}q_{n}\Big{]}\,.$
(152)

For the specific quartic and sextic theories we study in this paper, the above
upper bound is saturated and provides us with the optimal scaling. This can be
shown for each operator by constructing explicit classes of Feynman diagrams
for which the above scaling is realized.

In general, however, even the minimal bound (152) need not be saturated, and
it is not always possible to determine the large $N$ scaling of an operator by
separately considering its two-colour subdiagrams. As an example to
demonstrate this fact, consider the octic operator in a theory with $O(N)^{7}$
symmetry shown below.

$\displaystyle\begin{matrix}\text{ \includegraphics[scale={1}]{octicEx.pdf}
}\end{matrix}$ (153)

All the two-colour subgraphs for this operator consist of two disconnected
pieces: $\delta_{n}^{xy}=2$ for any two colours $x$ and $y$. Consequently,
(152) tells us that $e_{n}\leq 14$. To saturate the inequality for any given
choice of coefficients $C_{xy}$, one has to construct Feynman diagrams out of
the octic operator that are planar with respect to any two colours $x$ and $y$
for which $C_{xy}$ is non-zero. But for the particular operator (153) it is
impossible to impose planarity with respect to all such two colours.

The operator (153) actually has a large $N$ exponent of $21/2$ and is of
melonic type, which means that the leading class of Feynman diagrams at large
$N$ are constructed by pairwise grouping together interaction terms and
contracting fields that are isomorphic according to the automorphism group of
the interaction term under permutations $\sigma$ of fields as shown below.

$\displaystyle\begin{matrix}\text{
\includegraphics[scale={1}]{detailedMelonic.pdf} }\end{matrix}$ (154)

This melonic construction of Feynman diagrams is possible for any operator and
provides a universal lower bound on scaling exponents:

$\displaystyle e_{n}\geq\frac{r}{2}\Big{(}\frac{q}{2}-1\Big{)}\,.$ (155)

###  A.4 Other scalings

The optimal scaling we discussed in the previous subsection is rooted in the
idea of picking the smallest exponents such that connected Feynman diagrams
are bounded by fixed powers of $N$. There are however cases where yet smaller
exponents can be considered as well as cases where larger exponents are more
appropriate.

Scaling exponents smaller than those determined by the optimal scaling can be
appropriate if the connected diagrams that would cause a blow up of the theory
for any reason do not contribute to the values of observables. For example,
there may be a cancellation of these diagrams, and in some cases the leading
diagrams at large $N$ that determine the optimal scaling are tadpole diagrams,
which for massless theories in higher dimensions vanish under dimensional
regularization—a subtlety that does not apply to the tensor model literature
studying QFTs in zero dimensions, where tadpole diagrams contribute on equal
footing with any other Feynman diagrams.

Raising scaling exponents to sub-optimal values is relevant for theories where
coupling constants vanish to leading but not subleading order at large $N$. In
the context of RG flows, the values of coupling constants at fixed points need
not—and indeed sometimes do not—scale according to the optimal scaling. In the
presence of multiple fixed points, the leading large $N$ scaling of a given
coupling constant does not have to be the same across the fixed points. And
conversely if a coupling constant grows without bound along all RG flows, then
there is no meaningful scaling exponent to associate to that coupling.

###  A.5 $U(N)^{5}$ theory with multiple optimal scalings

In a tensor model with multiple operators present in the potential, a kind of
synergy may occur wherein combinations of operators together produce Feynman
diagrams with a larger scaling with $N$ than the operators do separately. This
phenomenon engenders one or more degrees of freedom in the choice of how to
scale coupling constants with $N$ while retaining a finite free energy
density. For garden variety unitary tensor models with $O(N)$ or $U(N)$
symmetry, this behaviour does not arise, but it may happen for other classes
of theories and it certainly occurs in non-unitary tensor models, as the
following example illustrates.

Consider a zero-dimensional theory of complex rank-five tensors $\phi^{abcde}$
with $U(N)^{5}$ symmetry,

$\displaystyle
Z(g_{1},g_{2})=\int_{\mathbb{C}\times\mathbb{C}}d\phi\,d\overline{\phi}\exp\bigg{[}-\phi^{abcde}\overline{\phi}^{\,abcde}-g_{1}\,O_{\text{dec}}(\phi,\overline{\phi})-g_{2}\,O^{\ast}_{\text{dec}}(\phi,\overline{\phi})\bigg{]}\,,$
(156)

where the decic interaction terms are given by

$\displaystyle\begin{matrix}\text{ \includegraphics[scale={1}]{UNint.pdf}
}\end{matrix}$ (157)

and by its complex conjugate. We will not assume that $g_{2}=g_{1}^{\ast}$,
and so the action of this theory is generally not real. The free energy
density of the theory has the perturbative expansion

$\displaystyle\frac{\log Z}{N^{5}}=\,$
$\displaystyle\log\pi-N^{5}\Big{(}26+43N+30N^{2}+17N^{3}+4N^{4}\Big{)}\big{(}g_{1}+g_{2}\big{)}$
(158)
$\displaystyle+\frac{1}{2}\Big{(}144+2154N^{2}+32254N^{4}+50668N^{5}+195982N^{6}+343550N^{7}+494568N^{8}$
$\displaystyle\hskip
28.45274pt+648742N^{9}+615478N^{10}+526730N^{11}+355811N^{12}+194258N^{13}$
$\displaystyle\hskip
28.45274pt+96808N^{14}+39768N^{15}+13601N^{16}+3484N^{17}+400N^{18}\Big{)}\big{(}g_{1}^{2}+g_{2}^{2}\big{)}$
$\displaystyle+\Big{(}120+2433N^{2}+32135N^{4}+50996N^{5}+194296N^{6}+342343N^{7}+497778N^{8}$
$\displaystyle\hskip
22.76219pt+650311N^{9}+613461N^{10}+525964N^{11}+356057N^{12}+194292N^{13}+96912N^{14}$
$\displaystyle\hskip
22.76219pt+39789N^{15}+13607N^{16}+3505N^{17}+400N^{18}+N^{20}\Big{)}g_{1}g_{2}+\mathcal{O}(g^{3})\,.$

Focusing just on the leading terms in the large $N$ limit and using the
elipsis “…” to signify terms that are suppressed at order $\frac{1}{N}$ or
higher, we have that

$\displaystyle\frac{\log Z}{N^{5}}=\,$
$\displaystyle\log\pi-4N^{9}(1+...)\big{(}g_{1}+g_{2}\big{)}+200N^{18}(1+...)\big{(}g_{1}^{2}+g_{2}^{2}\big{)}+N^{20}(1+...)g_{1}g_{2}+\mathcal{O}(g^{3})\,.$
(159)

If $g_{1}$ and $g_{2}$ scale as $N^{-e_{1}}$ and $N^{-e_{2}}$ respectively,
then we see that to obtain a finite energy density in the large $N$ limit, we
must require that

$\displaystyle e_{1}\geq 9\,,\hskip 42.67912pte_{2}\geq 9\,,\hskip
42.67912pte_{1}+e_{2}\geq 20\,.$ (160)

These inequalities hold true generally at higher orders in perturbation
theory, with the leading diagrams built of $g_{1}$ and $g_{2}$ being melonic.
In order for the scaling exponents to be as small as possible, we can choose
$e_{1}=9+t$ and $e_{2}=11-t$ with $0\leq t\leq 2$. This produces the following
large $N$ limits:

$\displaystyle\lim_{N\rightarrow\infty}\frac{\log{\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@color@gray@fill{1}\bigg{|}}Z\Big{(}\displaystyle\frac{\lambda_{1}}{N^{9+t}},\frac{\lambda_{2}}{N^{11-t}}\Big{)}}{N^{5}}=\log\pi+\begin{Bmatrix}[l](-4)\lambda_{1}+200\lambda_{1}^{2}&\hskip
5.69054pt\text{for }t=0\\\\[2.0pt] 0&\hskip 5.69054pt\text{for
}0<t<2\\\\[2.0pt] (-4)\lambda_{2}+200\lambda_{2}^{2}&\hskip 5.69054pt\text{for
}t=2\end{Bmatrix}+\lambda_{1}\lambda_{2}+\mathcal{O}(\lambda^{3})\,.$ (161)

In this example we see that there is a line segment of optimal scalings
parametrized by the parameter $t$. We may interpret the RHS in (161) as three
different theories in the large $N$ limit, of which two are isomorphic. If we
impose unitarity, then we single out the theory with $t=1$.

##  Appendix B Quartic Large $N$ $O(N)^{r}$ Fixed Points at Higher Rank

In this appendix we discuss features of quartic $O(N)^{r}$ theory at general
rank. In Subsection B.1, we propose a categorization of quartic singlet
operators into vectorial, matricial, melonic, and vector-melonic types and
introduce notation to be used later in this appendix. In Subsection B.2, we
argue that melonic operators are always absent at large $N$ fixed points.
Subsection B.3 explains why the large $N$ RG flow of vectorial operators
constitutes an invariant subspace of the full RG flow, and Subsection B.4
shows how to diagonalize the RG flow along this subspace. Lastly, Subsection
B.5 provides an enumeration of vectorial large $N$ fixed points through a
description of the large $N$ field content arrived at by Hubbard-Stratonovich
transformations.

###  B.1 Types of operators

A quartic singlet in $O(N)^{r}$ theory is constructed by contracting indices
among four fields $\phi$. Picking any field, the number of times it is
respectively contracted with the other three fields is given by three integers
$n_{1},n_{2},n_{3}\in\mathbb{N}_{0}$ with $n_{1}+n_{2}+n_{3}=r$. We can assume
without loss of generality that $n_{1}\geq n_{2}\geq n_{3}$. For example,

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

The type of large $N$ limit exhibited by a quartic operator and the optimal
large $N$ scaling of its associated coupling constant depend on these three
integers. We can group the operators as follows:

  * •

vectorial operators: $n_{1}>r/2$, large $N$ scaling:
$g=\displaystyle\frac{\lambda}{N^{n_{1}}}\,$,

  * •

matricial operators: $n_{1}=n_{2}=r/2$, large $N$ scaling:
$g=\displaystyle\frac{\lambda}{N^{r/2}}\,$,

  * •

melonic operators: $n_{1}<r/2$, large $N$ scaling:
$g=\displaystyle\frac{\lambda}{N^{r/2}}\,$,

  * •

vector-melonic operators: $n_{1}=r/2$, $n_{2}<n_{1}$, large $N$ scaling:
$g=\displaystyle\frac{\lambda}{N^{r/2}}\,$.

The second and fourth classes of operators are only present when $r$ is even.
The fourth class of operators is characterized by the fact that the dominant
large $N$ Feynman diagrams include both bubble diagrams and melonic diagrams,
but not all planar diagrams. In Section 2 we observed that only vectorial
operators are present at the perturbative large $N$ fixed points for $r\leq
5$. Below we present arguments that there are no melonic fixed points for any
$r$. This implies that all fixed points are vectorial whenever $r$ is odd. In
the case of even $r$, although we observe no large $N$ fixed points with
matricial or vector-melonic operators present at ranks two and four, we cannot
rule out the possibility that such fixed points appear at higher rank.

The integers $(n_{1},n_{2},n_{3})$ introduced above do not uniquely identify a
quartic singlet since they do not distinguish which of the $r$ indices are
contracted where. To fully specify an operator, we can label indices from 1 to
$r$ and introduce a triplet $(S_{1},S_{2},S_{3})$ where $S_{1}$, $S_{2}$, and
$S_{3}$ are disjoint subsets of the indices whose union equals the full set of
indices, ie. $S_{1}\cup S_{2}\cup S_{3}=\\{1,2,...,r\\}$. Without loss of
generality, we can assume that $|S_{1}|\geq|S_{2}|\geq|S_{3}|$.

###  B.2 Absence of melonic fixed points

For any melonic operator with large $N$ coupling $\lambda_{\text{mel}}$, the
leading beta function only receives a correction from the melonic contribution
to the anomalous dimension and so is given by

$\displaystyle\beta_{\lambda_{\text{mel}}}=-\epsilon\lambda_{\text{mel}}+\frac{\lambda_{\text{mel}}}{18(4\pi)^{4}}\sum_{m}\lambda_{m}^{2}\,,$
(162)

where the sum over $m$ runs over all melonic and vector-melonic operators in
the theory. In order to have a fixed point with a non-zero value of
$\lambda_{\text{mel}}$, we must therefore have that

$\displaystyle\sum_{m}\lambda_{m}^{2}=18(4\pi)^{4}\epsilon\,,$ (163)

which means that there must be at least one melonic or vector-melonic coupling
$\lambda_{m^{\prime}}$ that scales as $\sqrt{\epsilon}$.

Consider first the case when $\lambda_{m^{\prime}}$ is a vector-melonic
coupling. In this case, $\lambda_{m^{\prime}}$ feeds into a matricial beta
function. We can denote the triplet for the corresponding matrix operator by
$(S_{1},S_{2},0)$, where $|S_{1}|=|S_{2}|=r/2$. The large $N$ beta function of
a matricial operator is given by

$\displaystyle\beta_{\lambda_{\text{mat}}}=\,$
$\displaystyle-\epsilon\lambda_{\text{mat}}+\frac{2\lambda_{\text{mat}}^{2}+\sum_{w}\lambda_{w}^{2}}{48\pi^{2}}+\mathcal{O}(\lambda^{3})\,,$
(164)

where the index $w$ runs over vector-melonic couplings with triplets
$(S_{1}^{\prime},S_{2}^{\prime},S_{3}^{\prime})$ such that
$S_{1}^{\prime}=S_{1}$ or $S_{1}^{\prime}=S_{2}$. Now if one of these
couplings $\lambda_{w}$ scales as $\sqrt{\epsilon}$, then the beta function
picks up a term given by a positive constant times $\epsilon$. But there is no
way to cancel this term at a real perturbative fixed point. We could cancel
the term against the linear term $-\epsilon\lambda_{\text{mat}}$ by letting
$\lambda_{\text{mat}}$ scale as $\epsilon^{0}$, but then we would not be
situated at a perturbative fixed point. And we could cancel the term against
the $\lambda_{\text{mat}}^{2}$ term or against other other $\lambda_{w}^{2}$
terms, but then we would not be situated at a real fixed point.

The other possibility is that $\lambda_{m^{\prime}}$ is a melonic operator so
that it feeds into vectorial beta functions. Let one of the corresponding
vectorial operators be associated to the triplet
$(\mathcal{S}_{1},\mathcal{S}_{2},0)$, where
$|\mathcal{S}_{1}|>|\mathcal{S}_{2}|>0$. The beta function for the vectorial
coupling $\lambda_{\text{vec}}$ is given by

$\displaystyle\beta_{\lambda_{\text{vec}}}=\,$
$\displaystyle-\epsilon\beta_{\lambda_{\text{vec}}}+\frac{\lambda_{\text{vec}}^{2}+\sum_{v}\lambda_{v}^{2}}{48\pi^{2}}+\text{terms
from other vectorial ops.}$ (165) $\displaystyle+\text{terms from matrical
ops.}+\text{terms from vector-melonic ops.}+\mathcal{O}(\lambda^{3})\,,$

where the index $v$ runs over melonic or vector-melonic couplings with
triplets
$(\mathcal{S}^{\prime}_{1},\mathcal{S}^{\prime}_{2},\mathcal{S}^{\prime}_{3})$
such that
$\mathcal{S}_{1}=\mathcal{S}^{\prime}_{1}\cup\mathcal{S}^{\prime}_{2}$ or
$\mathcal{S}_{1}=\mathcal{S}^{\prime}_{1}\cup\mathcal{S}^{\prime}_{3}$ or
$\mathcal{S}_{1}=\mathcal{S}^{\prime}_{2}\cup\mathcal{S}^{\prime}_{3}$. If one
of these couplings $\lambda_{v}$ scales as $\sqrt{\epsilon}$, then the beta
function again picks up a term given by a positive constant times $\epsilon$,
which cannot be cancelled at a real perturbative fixed point, although it is
not quite as easy to show this time. We already ruled out matricial and
vector-melonic couplings that scale as $\sqrt{\epsilon}$, so we can disregard
those terms. The issue is that the quadratic terms from other vectorial
operators contain cross-terms, so that they are not manifestly positive semi-
definite. However, on diagonalizing the vectorial beta functions according to
the manner we describe in generality below in Subsection B.4, and which we
carried out for the theories of ranks three, four, and five in equations (54),
(74), and (LABEL:rank5h) respectively, the quadratic terms do become
manifestly positive semi-definite, and it can be checked that a term with a
positive constant times $\epsilon$ remains present after the diagonalization
and so cannot be cancelled to produce a real perturbative fixed point.

###  B.3 Invariance of the vectorial RG flow

For any single vectorial operator, the dominant Feynman diagrams at large $N$
are in one-to-one correspondence with the dominant bubble diagrams of a vector
operator at large $N$. For example, there is a correspondence between the
large $N$ diagrams constructed from the quartic operators shown
below.222222The operator on the left represents a quartic singlet operator in
$O(N)^{6}$ theory, the two colours are not meant to indicate that the
contracted indices transform under only two distinct $O(N)$ groups, in this
case the colours are merely used for emphasis.

$\displaystyle\begin{matrix}\text{ \includegraphics[scale={1}]{1to1.pdf}
}\end{matrix}$ (166)

The teal indices of the left diagram act in concert like a single dominant
vector index determining which diagrams contribute in the large $N$ limit,
while the brown indices merely tag along for the ride. When multiple vectorial
operators are present, there is instead a many-to-one correspondence. At the
level of loop-integrals, the dominant Feynman diagrams remain the same, but
for any dominant Feynman diagram in the vector model, there are multiple
dominant diagrams in the tensor model, obtained by replacing each vector
operator by any vectorial operator. This means that in the tensor case, the
dominant operators are built out of multiple distinct vectorial operators. It
is, however, possible to diagonalize the theory, such that the dominant
diagrams are each built out of only one type of operator, as we will see
presently in B.4.

The perturbative beta functions are computed from Feynman diagrams built via
index contractions among quartic operators. The first quantum contribution to
the beta functions is determined by the operation of taking two contractions
across a pair of quartic operators. This operation takes as input two quartic
operators and outputs a linear combination of quartic operators. In the case
when we have $n$ quartic singlets $\mathcal{O}_{i}$ in a theory of rank $r$
tensors, we may write the operation as

$\displaystyle\delta_{AC}\,\delta_{BD}\,\frac{\partial\mathcal{O}_{i}}{\partial\phi^{A}\partial\phi^{B}}\,\frac{\partial\mathcal{O}_{j}}{\partial\phi^{C}\partial\phi^{D}}=\sum_{\ell=1}^{n}C_{ij}^{\ell}(N)\,\mathcal{O}_{\ell}\,,\hskip
42.67912pt\begin{matrix}\text{{\footnotesize$A=a_{1}a_{2}...a_{r}$}}\,,\\\
\text{{\footnotesize$B=b_{1}b_{2}...b_{r}$}}\,,\\\
\text{{\footnotesize$C=c_{1}c_{2}...c_{r}$}}\,,\\\
\text{{\footnotesize$D=d_{1}d_{2}...d_{r}$}}\,.\end{matrix}$ (167)

Below we display an example of one term that contributes to this operation.

$\displaystyle\begin{matrix}\text{
\includegraphics[scale={1}]{contraction.pdf} }\end{matrix}$ (168)

In the above example, pairs of vertices connected by teal edges are contracted
together, and only this kind of contraction contributes to the large $N$ beta
functions. Furthermore, performing this kind of contraction among two
vectorial operators necessarily generates a vectorial operator. If the two
operators are associated to triplets $(S_{1},S_{2},S_{3})$ and
$(S^{\prime}_{1},S^{\prime}_{2},S^{\prime}_{3})$, then they generate a triplet
$(S^{\prime\prime}_{1},S^{\prime\prime}_{2},S^{\prime\prime}_{3})$ with
$S_{1}^{\prime\prime}=S_{1}\cup S_{1}^{\prime}$. On adopting the optimal
scaling, it is easy to verify that the powers of $N$ work out correctly. For
the original operators carry couplings $\frac{\lambda}{N^{|S_{1}|}}$ and
$\frac{\lambda^{\prime}}{N^{|S^{\prime}_{1}|}}$, while the contraction (168)
induces $N^{|S_{1}\cap S_{2}|}$ index loops, and the generated operator
carries an optimal scaling

$\displaystyle\frac{\lambda^{\prime\prime}}{N^{|S_{1}^{\prime\prime}|}}=\frac{\lambda^{\prime\prime}}{N^{|S_{1}\cup
S^{\prime}_{1}|}}=\frac{\lambda^{\prime\prime}}{N^{|S_{1}|+|S^{\prime}_{1}|-|S_{1}\cap
S_{1}^{\prime}|}}\,.$ (169)

In the language of Refs. [150, 70, 151], the operation (167) furnishes an
algebra on the set of operators $\mathcal{O}_{i}$. What we are observing now
is that at large $N$, the set of vectorial operators close on themselves.
Physically, this implies that the RG flow among the couplings of vectorial
operators constitutes an invariant subspace of the large $N$ RG flow of the
full quartic tensor theory.

At higher perturbative order, new kinds of index contractions contribute to
the beta functions. But the only Feynman diagrams built of vectorial operators
that survive in the large $N$ limit are tadpoles and bubble diagrams whose
index contractions can be evaluated by iterating the decomposition (167). As
in the vector model, once the first bubble is renormalized, all the higher
bubbles are rendered finite too, and so the large $N$ RG flow of the vectorial
operators terminates at quadratic order.

###  B.4 Diagonalizing the vectorial RG flow

The operator basis that diagonalizes the vectorial RG flow at large $N$ is the
set of operators of the form
$\mathcal{O}=\big{(}\Phi^{i_{1}j_{1}\,i_{2}j_{2}\,...\,i_{m}j_{m}}\big{)}^{2}$,
with $m<r/2$, where $\Phi^{i_{1}j_{1}\,i_{2}j_{2}\,...\,i_{m}j_{m}}$ is a
homogeneous polynomial in the fundamental fields of degree two that is
traceless symmetric or antisymmetric in each index pair $i_{p}\,j_{p}$, with
an even number of index pairs being antisymmetric. To be more explicit, the
$\Phi$ tensors is constructed by taking the quadratic operator
$\phi^{k_{1}\,...\,k_{r}}\phi^{l_{1}\,...\,l_{r}}$ and multiplying it an even
number of times $N_{A}$ (possibly zero) for different $p\in\\{1,...,r\\}$ by
an antisymmetric tensor

$\displaystyle
A^{i_{p}\,j_{p}}_{k_{p}\,l_{p}}=\delta^{i_{p}}_{k_{p}}\delta^{j_{p}}_{l_{p}}-\delta^{i_{p}}_{l_{p}}\delta^{j_{p}}_{k_{p}}\,,$
(170)

and multiplying it $N_{S}=m-N_{A}$ times for yet other different $p$ by a
symmetric traceless tensor

$\displaystyle
S^{i_{p}\,j_{p}}_{k_{p}\,l_{p}}=\delta^{i_{p}}_{k_{p}}\delta^{j_{p}}_{l_{p}}+\delta^{i_{p}}_{l_{p}}\delta^{j_{p}}_{k_{p}}-\frac{2}{N}\delta^{i_{p}\,j_{p}}\delta^{k_{p}\,l_{p}}\,,$
(171)

and finally multiplying it with $\delta_{k_{p}\,l_{p}}$ a total of $r-m$ times
to contract the remaining $\phi$ indices. Schematically,

$\displaystyle\Phi=(A)^{N_{A}}\,(S)^{m-N_{A}}\,(\delta)^{r-m}\,\phi\,\phi\,.$
(172)

The condition that $N_{A}$ be even is needed in order for $\Phi$ to be non-
vanishing. That operators built according to the above construction
diagonalize the operation (167) can be seen by applying the operation to
operators given, schematically, by $\mathcal{O}=(\Phi)^{2}$ and
$\mathcal{O}^{\prime}=(\Phi^{\prime})^{2}$, and first noting that, at large
$N$, the leading contributions to (167) come from the instances where the two
$\frac{\partial}{\partial\phi}$ derivatives act on two fields $\phi$ in the
same copy of $\Phi$. Roughly speaking, we have that

$\displaystyle\frac{\partial\mathcal{O}}{\partial\phi\,\partial\phi}\cdot\frac{\partial\mathcal{O}^{\prime}}{\partial\phi\,\partial\phi}\sim
4\,\Phi\,(A)^{N_{A}}\,(S)^{m-N_{A}}\,(\delta)^{r-m}\,\times\,\Phi^{\prime}\,(A)^{N_{A}^{\prime}}\,(S)^{m^{\prime}-N_{A}^{\prime}}\,(\delta)^{r-m^{\prime}}\,.$
(173)

Now unless $\mathcal{O}=\mathcal{O}^{\prime}$ so that $\Phi=\Phi^{\prime}$,
one of the following will necessarily happen:

  * •

an $A$ tensor from $\Phi$ is contracted with an $S$ tensor or with a $\delta$
function from $\Phi^{\prime}$,

  * •

an $S$ tensor from $\Phi$ is contracted with an $S$ tensor or with a $\delta$
function from $\Phi^{\prime}$,

  * •

a $\delta$ function from $\Phi$ is contracted with an $A$ tensor or an $S$
tensor from $\Phi^{\prime}$.

In any of the above cases the result will be zero on account of the
antisymmery and tracelessness of the $A$ and $S$ tensors. The product (173) is
non-zero only when $\mathcal{O}=\mathcal{O}^{\prime}$, in which case the
product is proportional to $\mathcal{O}$. In other words, if $\mathcal{O}_{i}$
and $\mathcal{O}_{j}$ in equation (167) are vectorial operators, then all
other coefficients $C_{ij}^{\ell}(N)$ are subdominant at large $N$ compared to
$C_{jj}^{j}(N)$ and $C_{ii}^{i}(N)$.

We have seen now that if we set to zero the couplings for the non-vectorial
operator at any scale, they will remain zero along the large $N$ RG flow,
while the beta functions for the vectorial operators on this subspace can be
explicitly decoupled from each other. For positive $\epsilon$, these decoupled
beta functions all have a negative linear coefficient and a positive quadratic
coefficient, which entails that each of the decoupled beta functions has its
own Wilson-Fisher fixed point, which in turn implies that the number of
vectorial fixed points is two raised to the number of vectorial operators.

###  B.5 Large $N$ field content

The $O(N)$ model admits a familiar large $N$ description via a Hubbard-
Stratonovich transformation in terms of an $O(N)$ scalar field $\sigma$. We
have seen in Subsections 2.2, 2.3, and 2.4 that in quartic $O(N)^{r}$ theory
with $r\geq 3$, the large $N$ field content additionally includes $r$
traceless symmetric fields $\chi_{p}^{i_{p}\,j_{p}}$. And when $r\geq 5$, we
must also introduce $\binom{r}{2}$ doubly antisymmetric fields
$\tau_{ij}^{i_{1}j_{2}\,j_{1}j_{2}}$ as well as the same number of doubly
traceless symmetric fields $\rho_{ij}^{i_{1}j_{2}\,j_{1}j_{2}}$. The
diagonalization we performed above in B.4 in terms of $\Phi$ tensors
immediately answers the question of which further fields appear at arbitrary
higher tensor rank. For the Hubbard-Stratonovich fields by which we integrate
out vectorial operators precisely couple to the $\Phi$ tensors and therefore
possess the same index structure. We conclude that the Hubbard-Stratonovich
fields present at a given rank $r$ are tensor fields with anywhere from 0 to
$\left\lfloor\frac{r-1}{2}\right\rfloor$ pairs of incides, whereof an even
number transform as antisymmetric matrices under separate $O(N)$ groups, while
the remaining index pairs transform as traceless symmetric matrices under yet
other separate $O(N)$ groups. Another way to say this, using multinomial
coefficients $\binom{n_{1}+n_{2}+n_{3}}{n_{1},\,n_{2},\,n_{3}}$ to count
fields, is by stating that on increasing the rank $r$ of the theory, new types
of fields with an extra pair of indices appear at each odd rank, and that for
$r=4l+1$ the fields that appear, and which carry $2l$ pairs of indices, are

  * •

$\binom{r}{2l,\,0,\,r-2l}$ fields with $2l$ pairs of traceless symmetric
indices,

  * •

$\binom{r}{2l-2,\,2,\,r-2l}$ fields with $2l-2$ pairs of traceless symmetric
indices and two antisymmetric index pairs,

  * •

$\binom{r}{2l-4,\,4,\,r-2l}$ fields with $2l-4$ pairs of traceless symmetric
indices and four antisymmetric index pairs,

$\displaystyle\vdots$

  * •

$\binom{r}{0,\,2l,\,r-2l}$ fields with $2l$ antisymmetric index pairs,

while for $r=4l+3$ the new fields appearing, which carry $2l+1$ pairs of
indices, are

  * •

$\binom{r}{2l+1,\,0,\,r-2l-1}$ fields with $2l+1$ pairs of traceless symmetric
indices,

  * •

$\binom{r}{2l-1,\,2,\,r-2l-1}$ fields with $2l-1$ pairs of traceless symmetric
indices and two antisymmetric index pairs,

  * •

$\binom{r}{2l-3,\,4,\,r-2l-1}$ fields with $2l-3$ pairs of traceless symmetric
indices and four antisymmetric index pairs,

$\displaystyle\vdots$

  * •

$\binom{r}{1,\,2l,\,r-2l-1}$ fields with one pair of traceless symmetric
indices and $2l$ antisymmetric index pairs.

##  Appendix C Locations of Fixed Points in Sextic $O(N)^{4}\rtimes S_{4}$
theory

The sextic rank-four tensor theory with action (118) has 18 perturbative fixed
points, which we enumerate with Roman numerals (i) to (xviii). The first is
simply the trivial fixed point with all couplings zero. To leading order in
$\epsilon$, fixed points (ii) to (viii) are situated at the following loci in
the space of coupling constants :

$\displaystyle\vec{\lambda}^{\,\text{(ii)}}=(8\pi)^{2}\epsilon\,\Big{\\{}$
$\displaystyle\frac{135}{4}(47+4\sqrt{3})\,,\,\frac{4995}{4}-540\sqrt{3}\,,\,\frac{2295}{4}\,,\,270\,,\,270(2+\sqrt{3})\,,\,\frac{8235}{4}+540\sqrt{3}\,,$
$\displaystyle
270\,,\,-1080\,,\,0\,,\,0\,,\,270\,,\,0\,,\,-540\sqrt{3}\,,\,0\,,\,0\,,\,0\,,\,0\,,\,0\,,\,0\,,\,135\sqrt{3}\,,\,0\Big{\\}}\,,$
(174)
$\displaystyle\vec{\lambda}^{\,\text{(iii)}}=(8\pi)^{2}\epsilon\,\Big{\\{}$
$\displaystyle\frac{6345}{4}-135\sqrt{3}\,,\,\frac{4995}{4}+540\sqrt{3}\,,\,\frac{2295}{4}\,,\,270\,,\,270(2-\sqrt{3})\,,\frac{8235}{4}-540\sqrt{3}\,,$
$\displaystyle
270\,,\,-1080\,,\,0\,,\,0\,,\,270\,,\,0\,,\,540\sqrt{3}\,,\,0\,,\,0\,,\,0\,,\,0\,,\,0\,,\,0\,,\,-135\sqrt{3},0\Big{\\}}\,,$
(175)
$\displaystyle\vec{\lambda}^{\,\text{(iv)}}=(8\pi)^{2}\epsilon\,\Big{\\{}$
$\displaystyle
69120\,,\,157680\,,\,17280\,,\,1080\,,\,2160\,,\,66960\,,\,4320\,,\,-4320\,,\,0\,,\,0\,,\,1080\,,$
$\displaystyle 0\,,\,\
0\,,\,0\,,\,0\,,\,0\,,\,0\,,\,0\,,\,0\,,\,0\,,\,0\Big{\\}}\,,$ (176)
$\displaystyle\vec{\lambda}^{\,\text{(v)}}=(8\pi)^{2}\epsilon\,\Big{\\{}$
$\displaystyle
5900\,,\,0\,,\,0\,,\,-1080\,,\,0\,,\,0\,,\,0\,,\,0\,,\,60\,,\,0\,,\,0\,,\,0\,,\,0\,,\,0\,,\,0\,,\,0\,,\,0\,,\,0\,,\,0\,,\,0\,,\,0\Big{\\}}\,,$
(177)
$\displaystyle\vec{\lambda}^{\,\text{(vi)}}=(8\pi)^{2}\epsilon\,\Big{\\{}$
$\displaystyle\frac{601625}{4}+135\sqrt{3}\,,\,-\frac{45}{4}(7225+48\sqrt{3})\,,\,\frac{12015}{4}\,,\,-5670\,,\,270(2+\sqrt{3})\,,\,$
$\displaystyle\frac{45}{4}(935+48\sqrt{3})\,,\,1890\,,\,-1080\,,\,60\,,\,0\,,\,270\,,\,0\,,\,-540\sqrt{3}\,,\,0\,,\,0\,,\,0\,,\,0\,,\,0\,,\,0\,,\,$
$\displaystyle 135\sqrt{3}\,,\,0\Big{\\}}\,,$ (178)
$\displaystyle\vec{\lambda}^{\,\text{(vii)}}=(8\pi)^{2}\epsilon\,\Big{\\{}$
$\displaystyle\frac{601625}{4}-135\sqrt{3}\,,\,\frac{45}{4}(-7225+48\sqrt{3})\,,\,\frac{12015}{4}\,,\,-5670\,,\,270(2-\sqrt{3})\,,\,$
$\displaystyle\frac{42075}{4}-540\sqrt{3}\,,\,1890\,,\,-1080\,,\,60\,,\,0\,,\,270\,,\,0\,,\,540\sqrt{3}\,,\,0\,,\,0\,,\,0\,,\,0\,,\,0\,,\,0\,,\,$
$\displaystyle-135\sqrt{3}\,,\,0\Big{\\}}\,,$ (179)
$\displaystyle\vec{\lambda}^{\,\text{(viii)}}=(8\pi)^{2}\epsilon\,\Big{\\{}$
$\displaystyle
6543500\,,\,-4212720\,,\,172800\,,\,-38880\,,\,2160\,,\,560880\,,\,17280\,,\,-4320\,,\,60\,,\,$
$\displaystyle
0\,,\,1080\,,\,0\,,\,0\,,\,0\,,\,0\,,\,0\,,\,0\,,\,0\,,\,0\,,\,0\,,\,0\Big{\\}}\,.$
(180)

Like the above fixed points, the leading values of the couplings at the next
two fixed points, (ix) and (x), are given simply in terms of ratio of integers
and square roots of integers, up to an overall factor of $\pi^{2}$. But in
cases (ix) and (x) the integers involved are particularly lengthy. Although
their precise form is probably of little interest to the reader, we
nonetheless display these solutions explicitly below, simply to give an idea
of the complexity involved in any tentative large $N$ solution that matches
these fixed points in the $\epsilon$ expansion:

$\frac{\vec{\lambda}^{\,\text{(ix)}}}{(8\pi)^{2}\epsilon}$={
$\frac{1620}{156420504648617}(19607370583144755-1815344122912\sqrt{23699})\,,\,\frac{8640}{2951330276389}(-28078989400213+17602913268\sqrt{23699})\,,\,$
$\frac{4320}{1295011091}(793909403-2211396\sqrt{23699})\,,\,-\frac{1080}{567236263}(3067843601+436084\sqrt{23699})\,,\,\frac{164160}{567236263}(3379651-70585\sqrt{23699})\,,\,$
$\frac{2160}{55685476913}(253780228967-486150604\sqrt{23699})\,,\,\frac{4320}{10702571}(3019649+4114\sqrt{23699})\,,\,\frac{17280}{10702571}(-650012+4569\sqrt{23699})\,,\,$
$\frac{7020}{121}\,,\,0\,,\,\frac{32400}{121}\,,\,0\,,\,\frac{51840}{10702571}(-187187+1523\sqrt{23699})\,,\,\frac{28080}{5203}\,,\,-\frac{4320}{5203}\sqrt{23699}\,,\,-\frac{56160}{5203}\,,\,\frac{112320}{27071209}(3126-\sqrt{23699})\,,\,$
$\frac{4320}{27071209}(48243-5134\sqrt{23699})\,,\,0\,,\,\frac{28080}{121}\,,\,0\bigg{\\}}\,,$
(181) $\frac{\vec{\lambda}^{\,\text{(x)}}}{(8\pi)^{2}\epsilon}=\Big{\\{}$
$\frac{1620}{156420504648617}(19607370583144755+1815344122912\sqrt{23699})\,,\,-\frac{8640}{2951330276389}(28078989400213+17602913268\sqrt{23699})\,,\,$
$\frac{4320}{1295011091}(793909403+2211396\sqrt{23699})\,,\,\frac{1080}{567236263}(-3067843601+436084\sqrt{23699})\,,\,\frac{164160}{567236263}(3379651+70585\sqrt{23699})\,,\,$
$\frac{2160}{55685476913}(253780228967+486150604\sqrt{23699})\,,\,\frac{4320}{10702571}(3019649-4114\sqrt{23699})\,,\,-\frac{17280}{10702571}(650012+4569\sqrt{23699})\,,\,$
$\frac{7020}{121}\,,\,0\,,\,\frac{32400}{121}\,,\,0\,,\,-\frac{51840}{10702571}(187187+1523\sqrt{23699})\,,\,\frac{28080}{5203}\,,\,\frac{4320}{5203}\sqrt{23699}\,,\,-\frac{56160}{5203}\,,\,\frac{112320}{27071209}(3126+\sqrt{23699})\,,\,$
$\frac{4320}{27071209}(48243+5134\sqrt{23699})\,,\,0\,,\,\frac{28080}{121}\,,0\Big{\\}}\,.$
(182)

The locations of the last eight fixed points are even more complicated. These
have couplings given in terms of roots of quartic equations, meaning that
closed-form expressions exist but are very lengthy. For this reason, we will
merely display their numerical values:

$\displaystyle\vec{\lambda}^{\,\text{(xi)}}=(8\pi)^{2}\epsilon\,\Big{\\{}$
$\displaystyle
39771.238\,,\,89713.610\,,\,8849.7258\,,\,874.73176\,,\,7.3847425\,,\,38900.990\,,\,$
$\displaystyle
3133.3241\,,\,-1890.5304\,,\,0\,,\,0\,,\,874.73176\,,\,0\,,\,1875.7610\,,\,0\,,\,0\,,\,-804.19829\,,\,$
$\displaystyle-1003.2853\,,\,149.70715\,,\,0\,,\,-66.841092\,,\,0\Big{\\}}\,,$
(183)
$\displaystyle\vec{\lambda}^{\,\text{(xii)}}=(8\pi)^{2}\epsilon\,\Big{\\{}$
$\displaystyle
1140.6966\,,\,1406.3408\,,\,73.935497\,,\,228.84964\,,\,132.13957\,,\,1615.4471\,,\,$
$\displaystyle
255.25587\,,\,-187.72991\,,\,0\,,\,0\,,\,228.84964\,,\,0\,,\,-76.549228\,,\,0\,,\,0\,,\,$
$\displaystyle-363.83433\,,\,-92.906483\,,\,30.642458\,,\,0\,,\,201.05447\,,\,0\Big{\\}}\,,$
(184)
$\displaystyle\vec{\lambda}^{\,\text{(xiii)}}=(8\pi)^{2}\epsilon\,\Big{\\{}$
$\displaystyle
947.47330\,,\,-1228.6423\,,\,716.93412\,,\,146.05887\,,\,1584.1817\,,\,2129.2941\,,\,$
$\displaystyle
200.95056\,,\,-1610.6616\,,\,0\,,\,0\,,\,146.05887\,,\,0\,,\,-1557.7019\,,\,0\,,\,0\,,\,513.21307\,,\,$
$\displaystyle 260.75229\,,\,60.969364\,,\,0\,,\,132.81893\,,\,0\Big{\\}}\,,$
(185)
$\displaystyle\vec{\lambda}^{\,\text{(xiv)}}=(8\pi)^{2}\epsilon\,\Big{\\{}$
$\displaystyle
14455.314\,,\,33686.191\,,\,4452.0897\,,\,610.35973\,,\,2716.2939\,,\,12960.658\,,\,$
$\displaystyle
1750.4695\,,\,-4231.0780\,,\,0\,,\,0\,,\,610.35973\,,\,0\,,\,-1201.5099\,,\,0\,,\,0\,,\,894.81955\,,\,$
$\displaystyle 1382.1062\,,\,185.34769\,,\,0\,,\,-147.03231\,,\,0\Big{\\}}\,,$
(186)
$\displaystyle\vec{\lambda}^{\,\text{(xv)}}=(8\pi)^{2}\epsilon\,\Big{\\{}$
$\displaystyle
3789647.1\,,\,-2416308.7\,,\,97152.004\,,\,-29501.994\,,\,7.3847425\,,\,320793.13\,,\,$
$\displaystyle
12898.899\,,\,-1890.5304\,,\,60\,,\,0\,,\,874.73176\,,\,0\,,\,1875.7610\,,\,0\,,\,0\,,\,$
$\displaystyle-804.19829\,,\,-1003.2853\,,\,149.70715\,,\,0\,,\,-66.841092\,,\,0\Big{\\}}\,,$
(187)
$\displaystyle\vec{\lambda}^{\,\text{(xvi)}}=(8\pi)^{2}\epsilon\,\Big{\\{}$
$\displaystyle
122723.36\,,\,-64076.212\,,\,1956.5468\,,\,-5128.8813\,,\,132.13957\,,\,8172.7084\,,\,$
$\displaystyle
1681.1662\,,\,-187.72991\,,\,60\,,\,0\,,\,228.84964\,,\,0\,,\,-76.549228\,,\,0\,,\,0\,,\,$
$\displaystyle-363.83433\,,\,-92.906483\,,\,30.642458\,,\,0\,,\,201.05447\,,\,0\Big{\\}}\,,$
(188)
$\displaystyle\vec{\lambda}^{\,\text{(xvii)}}=(8\pi)^{2}\epsilon\,\Big{\\{}$
$\displaystyle
71023.715\,,\,-35338.456\,,\,1617.3650\,,\,-3892.3509\,,\,1584.1817\,,\,5292.0617\,,\,$
$\displaystyle
1187.0872\,,\,-1610.6616\,,\,60\,,\,0\,,\,146.05887\,,\,0\,,\,-1557.7019\,,\,0\,,\,0\,,\,$
$\displaystyle
513.21307\,,\,260.75229\,,\,60.969364\,,\,0\,,\,132.81893\,,\,0\Big{\\}}\,,$
(189)
$\displaystyle\vec{\lambda}^{\,\text{(xviii)}}=(8\pi)^{2}\epsilon\,\Big{\\{}$
$\displaystyle
1475365.0\,,\,-917652.49\,,\,37148.251\,,\,-18296.774\,,\,2716.2939\,,\,$
$\displaystyle
118579.60\,,\,7692.8473\,,\,-4231.0780\,,\,60\,,\,0\,,\,610.35973\,,\,0\,,\,-1201.5099\,,\,0\,,$
$\displaystyle\,0\,,\,894.81955\,,\,1382.1062\,,\,185.34769\,,\,0\,,\,-147.03231\,,\,0\Big{\\}}\,.$
(190)

For the fixed point (iv), there is a one-parameter degeneracy in the order
$\epsilon^{2}$ corrections to the values of the couplings. Parameterizing the
degeneracy with the number $u$, the explicit values of the couplings are given
up to $\mathcal{O}(\epsilon^{3})$ contributions by the following:

$\displaystyle\frac{\lambda^{\text{(iv)}}_{1}}{6!(8\pi)^{2}}=96\epsilon+u\epsilon^{2}\,,$
$\displaystyle\frac{\lambda^{\text{(iv)}}_{2}}{6!(8\pi)^{2}}=219\epsilon+(-14454+286\pi^{2}+u)\epsilon^{2}\,,$
$\displaystyle\frac{\lambda^{\text{(iv)}}_{3}}{6!(8\pi)^{2}}=24\epsilon+\frac{1}{2}(381420-19123\pi^{2}-4u)\epsilon^{2}\,,$
$\displaystyle\frac{\lambda^{\text{(iv)}}_{4}}{6!(8\pi)^{2}}=\frac{3}{2}\epsilon+\frac{1}{4}(97281-4954\pi^{2}-u)\epsilon^{2}\,,$
$\displaystyle\frac{\lambda^{\text{(iv)}}_{5}}{6!(8\pi)^{2}}=3\epsilon-\frac{3}{2}(-94713+4713\pi^{2}+u)\epsilon^{2}\,,$
$\displaystyle\frac{\lambda^{\text{(iv)}}_{6}}{6!(8\pi)^{2}}=93\epsilon-\frac{3}{2}(-96781+5002\pi^{2}+u)\epsilon^{2}\,,$
$\displaystyle\frac{\lambda^{\text{(iv)}}_{7}}{6!(8\pi)^{2}}=6\epsilon+\frac{1}{2}(-94815+4669\pi^{2}+u)\epsilon^{2}\,,$
$\displaystyle\frac{\lambda^{\text{(iv)}}_{8}}{6!(8\pi)^{2}}=-6\epsilon+(-189372+9425\pi^{2}+2u)\epsilon^{2}\,,$
$\displaystyle\frac{\lambda^{\text{(iv)}}_{9}}{6!(8\pi)^{2}}=-\frac{1}{4}\pi^{2}\epsilon^{2}\,,$
$\displaystyle\frac{\lambda^{\text{(iv)}}_{10}}{6!(8\pi)^{2}}=0\,,$
$\displaystyle\frac{\lambda^{\text{(iv)}}_{11}}{6!(8\pi)^{2}}=\frac{3}{2}\epsilon+\frac{3}{4}(-54+\pi^{2})\epsilon^{2}\,,$
$\displaystyle\frac{\lambda^{\text{(iv)}}_{12}}{6!(8\pi)^{2}}=0\,,$ (191)
$\displaystyle\frac{\lambda^{\text{(iv)}}_{13}}{6!(8\pi)^{2}}=2(-94767+4714\pi^{2}+u)\epsilon^{2}\,,$
$\displaystyle\frac{\lambda^{\text{(iv)}}_{14}}{6!(8\pi)^{2}}=0\,,$
$\displaystyle\frac{\lambda^{\text{(iv)}}_{15}}{6!(8\pi)^{2}}=\frac{1}{4}(94767-4714\pi^{2}-u)\epsilon^{2}\,,$
$\displaystyle\frac{\lambda^{\text{(iv)}}_{16}}{6!(8\pi)^{2}}=0\,,$
$\displaystyle\frac{\lambda^{\text{(iv)}}_{17}}{6!(8\pi)^{2}}=18\epsilon^{2}\,,$
$\displaystyle\frac{\lambda^{\text{(iv)}}_{18}}{6!(8\pi)^{2}}=(94767-4714\pi^{2}-u)\epsilon^{2}\,,$
$\displaystyle\frac{\lambda^{\text{(iv)}}_{19}}{6!(8\pi)^{2}}=0\,,$
$\displaystyle\frac{\lambda^{\text{(iv)}}_{20}}{6!(8\pi)^{2}}=0\,,$
$\displaystyle\frac{\lambda^{\text{(iv)}}_{21}}{6!(8\pi)^{2}}=0\,.$

##  Appendix D Gradient Flow and Linear Relations

The RG flows of standard perturbative QFTs are gradient flows.232323See Refs.
[152, 153] for foundational work pertaining to this fact and Refs. [154, 155,
156, 157, 158] for modern discussions and more recent results. What this means
concretely for a given QFT with $n$ coupling constants $g_{i}$ is that there
exits a function $F$ and an $n\times n$ matrix $G_{ij}$, both of which depend
on the coupling constants, such that

$\displaystyle\frac{\partial F}{\partial g_{i}}=G_{ij}\,\beta_{g_{j}}\,.$
(192)

If we define $t\equiv\log\mu$ where $\mu$ is the RG scale, such that
$\beta_{g_{i}}=\frac{d}{dt}g_{i}$, then the gradient condition (192) implies
that

$\displaystyle\frac{dF}{dt}=\frac{\partial F}{\partial
g_{i}}\beta_{g_{i}}=\beta_{g_{i}}G_{ij}\,\beta_{g_{j}}\,,$ (193)

from which we see that the RG flow is monotonic if the matrix $G_{ij}$ is
positive definite or negative definite. Since the partial derivatives of $F$
with respect to couplings $g_{i}$ and $g_{k}$ commute, (192) tells us that

$\displaystyle\forall i,k:\hskip 142.26378pt\frac{\partial
G_{ij}\beta_{g_{j}}}{\partial g_{k}}-\frac{\partial
G_{kj}\beta_{g_{j}}}{\partial g_{i}}=0\,.$ (194)

The above equation provides a powerful way of checking that perturbative beta
functions have been computed correctly. To carry out the check one must first
determine $G_{ij}$, which can be computed by taking second derivatives of the
free energy of the given theory. To the perturbative order we are working at,
the free energy, which equals the sum of connected vacuum Feynman diagrams,
also equals the anomalous dimension for $\phi$, up to rescalings of the
coupling constants that, like the overall normalization of $G_{ij}$, carry no
significance in this context. For the theories studied in this paper, we
computed $\gamma_{\phi}$ to third order in the couplings, which means we can
determine $G_{ij}$ to linear order:

$\displaystyle G_{ij}^{(1)}=\frac{\partial^{2}\gamma_{\phi}^{(3)}}{\partial
g_{i}\partial g_{j}}\,.$ (195)

Consequently, we are able to verify the curl condition (194) to second order:

$\displaystyle\forall i,j:\hskip 142.26378pt\frac{\partial
G^{(1)}_{ij}\beta^{(3)}_{g_{j}}}{\partial g_{k}}-\frac{\partial
G^{(1)}_{kj}\beta^{(3)}_{g_{j}}}{\partial g_{i}}=\mathcal{O}(g^{3})\,.$ (196)

Other checks one can perform that beta functions are computed correctly
include comparing the double trace beta function $\beta_{g_{1}}$ against the
well-known beta function of the $O(N)$ model and checking that the beta
functions respect the linear relations that exists among operators at finite
values of $N$. The linear relations can be determined by direct computation
but can also be extracted from $G_{ij}$, which depends not only on the
coupling constants but also on $N$. In fact, the linear relations can be
conveniently read off already from the piece $G_{ij}^{(0)}$ that is constant
with respect to the coupling constants. When evaluated at values of $N$ for
which linear relations exist among the operator, $G_{ij}^{(0)}$ will contain
zero eigenvalues, and the corresponding eigenvectors reveal which linear
combinations of operators vanish. Below we list all the linear relations that
exist in the theories studied in this paper, except the obvious relation that
when $N=1$, all quartic operators equal one another and ditto for the sextic
operators. In the quartic theories, the operatos are all linearly independent
for all $N\geq 2$, but linear relationships exist at $N=-2$, ie. in the
corresponding $Sp(2)^{r}$ theories. In the sextic theories linear relations
exist also at $N=2$ and $N=-4$.

  * •

quartic $O(N)^{2}$ theory:

$\displaystyle N=-2:\hskip 28.45274pt\mathcal{O}_{1}=-2\,\mathcal{O}_{2}\,,$

  * •

quartic $O(N)^{3}$ theory:

$\displaystyle N=-2:\hskip
28.45274pt\mathcal{O}_{1}=-2\,\mathcal{O}_{2}=-2\,\mathcal{O}_{3}=-2\,\mathcal{O}_{4}=\mathcal{O}_{5}\,,$

  * •

quartic $O(N)^{4}$ theory:

$\displaystyle N=-2:\hskip
28.45274pt\mathcal{O}_{1}=-2\mathcal{O}_{2}=-2\mathcal{O}_{3}=-2\mathcal{O}_{4}=-2\mathcal{O}_{5}=2\,\mathcal{O}_{8}-2\,\mathcal{O}_{13}-2\,\mathcal{O}_{14}\,,$
$\displaystyle\hskip
79.66771pt\mathcal{O}_{6}=\mathcal{O}_{8}-\mathcal{O}_{13}-2\,\mathcal{O}_{14}\,,$
$\displaystyle\hskip
79.66771pt\mathcal{O}_{7}=\mathcal{O}_{8}-2\,\mathcal{O}_{13}-\mathcal{O}_{14}\,,$
$\displaystyle\hskip 79.66771pt\mathcal{O}_{9}=\mathcal{O}_{14}\,,$
$\displaystyle\hskip 79.66771pt\mathcal{O}_{10}=\mathcal{O}_{13}\,,$
$\displaystyle\hskip
79.66771pt\mathcal{O}_{11}=\mathcal{O}_{12}=-\mathcal{O}_{13}-\mathcal{O}_{14}\,,$

  * •

quartic $O(N)^{5}$ theory:

$\displaystyle N=-2:\hskip
28.45274pt\mathcal{O}_{1}=-2\,\mathcal{O}_{2,a}=-2\,\mathcal{O}_{2,b}=-2\,\mathcal{O}_{2,c}=-2\,\mathcal{O}_{2,d}=-2\,\mathcal{O}_{2,e}$
$\displaystyle\hskip
95.31673pt=2\,\mathcal{O}_{3,ab}+2\,\mathcal{O}_{4,ab}=2\,\mathcal{O}_{3,ac}+2\,\mathcal{O}_{4,ac}=2\,\mathcal{O}_{3,ad}+2\,\mathcal{O}_{4,ad}$
$\displaystyle\hskip
95.31673pt=2\,\mathcal{O}_{3,ae}+2\,\mathcal{O}_{4,ae}=2\,\mathcal{O}_{3,bc}+2\,\mathcal{O}_{4,bc}=2\,\mathcal{O}_{3,bd}+2\,\mathcal{O}_{4,bd}$
$\displaystyle\hskip
95.31673pt=2\,\mathcal{O}_{3,be}+2\,\mathcal{O}_{4,be}=2\,\mathcal{O}_{3,cd}+2\,\mathcal{O}_{4,cd}=2\,\mathcal{O}_{3,ce}+2\,\mathcal{O}_{4,ce}$
$\displaystyle\hskip
95.31673pt=2\,\mathcal{O}_{3,de}+2\,\mathcal{O}_{4,de}=-\mathcal{O}_{5,a,bc}-\mathcal{O}_{5,b,ac}-\mathcal{O}_{5,c,ab}-\mathcal{O}_{5,d,ab}-\mathcal{O}_{5,e,ab}\,,$
$\displaystyle\hskip
79.66771pt\mathcal{O}_{5,a,bc}=\mathcal{O}_{5,a,bd}=\mathcal{O}_{5,a,be}\,,$
$\displaystyle\hskip
79.66771pt\mathcal{O}_{5,b,ac}=\mathcal{O}_{5,b,ad}=\mathcal{O}_{5,b,ae}\,,$
$\displaystyle\hskip
79.66771pt\mathcal{O}_{5,c,ab}=\mathcal{O}_{5,c,ad}=\mathcal{O}_{5,c,ae}\,,$
$\displaystyle\hskip
79.66771pt\mathcal{O}_{5,d,ab}=\mathcal{O}_{5,d,ac}=\mathcal{O}_{5,d,ae}\,,$
$\displaystyle\hskip
79.66771pt\mathcal{O}_{5,e,ab}=\mathcal{O}_{5,e,ac}=\mathcal{O}_{5,e,ad}\,,$
$\displaystyle\hskip
79.66771pt\mathcal{O}_{4,ab}=-\mathcal{O}_{5,a,bc}-\mathcal{O}_{5,b,ac}\,,$
$\displaystyle\hskip
79.66771pt\mathcal{O}_{4,ac}=-\mathcal{O}_{5,a,bc}-\mathcal{O}_{5,c,ab}\,,$
$\displaystyle\hskip
79.66771pt\mathcal{O}_{4,ad}=-\mathcal{O}_{5,a,bc}-\mathcal{O}_{5,d,ab}\,,$
$\displaystyle\hskip
79.66771pt\mathcal{O}_{4,ae}=-\mathcal{O}_{5,a,bc}-\mathcal{O}_{5,e,ab}\,,$
$\displaystyle\hskip
79.66771pt\mathcal{O}_{4,bc}=-\mathcal{O}_{5,b,ac}-\mathcal{O}_{5,c,ab}\,,$
$\displaystyle\hskip
79.66771pt\mathcal{O}_{4,bd}=-\mathcal{O}_{5,b,ac}-\mathcal{O}_{5,d,ab}\,,$
$\displaystyle\hskip
79.66771pt\mathcal{O}_{4,be}=-\mathcal{O}_{5,b,ac}-\mathcal{O}_{5,e,ab}\,,$
$\displaystyle\hskip
79.66771pt\mathcal{O}_{4,cd}=-\mathcal{O}_{5,c,ab}-\mathcal{O}_{5,d,ab}\,,$
$\displaystyle\hskip
79.66771pt\mathcal{O}_{4,ce}=-\mathcal{O}_{5,c,ab}-\mathcal{O}_{5,e,ab}\,,$
$\displaystyle\hskip
79.66771pt\mathcal{O}_{4,de}=-\mathcal{O}_{5,d,ab}-\mathcal{O}_{5,e,ab}\,,$

  * •

sextic $O(N)^{2}$ theory:

$\displaystyle N=2:\hskip
36.98857pt\mathcal{O}_{1}=3\,\mathcal{O}_{2}-2\,\mathcal{O}_{3}\,,$
$\displaystyle N=-2:\hskip
28.45274pt\mathcal{O}_{1}=-2\,\mathcal{O}_{2}=4\,\mathcal{O}_{3}\,,$
$\displaystyle N=-4:\hskip
28.45274pt\mathcal{O}_{1}=-6\,\mathcal{O}_{2}-8\,\mathcal{O}_{3}\,,$

  * •

sextic $O(N)^{3}\rtimes S_{3}$ theory:

$\displaystyle N=2:\hskip
36.98857pt\mathcal{O}_{1}=2\,\mathcal{O}_{5}-\frac{1}{2}\,\mathcal{O}_{6}-\mathcal{O}_{7}+\frac{1}{2}\,\mathcal{O}_{8}\,,$
$\displaystyle\hskip
79.66771pt\mathcal{O}_{2}=3\,\mathcal{O}_{5}-\frac{3}{2}\,\mathcal{O}_{6}-\frac{3}{2}\,\mathcal{O}_{7}+\mathcal{O}_{8}\,,$
$\displaystyle\hskip
79.66771pt\mathcal{O}_{3}=\mathcal{O}_{5}-\frac{1}{2}\,\mathcal{O}_{6}+\frac{1}{2}\,\mathcal{O}_{7}\,,$
$\displaystyle\hskip
79.66771pt\mathcal{O}_{4}=\frac{3}{2}\,\mathcal{O}_{7}-\frac{1}{2}\,\mathcal{O}_{8}\,,$
$\displaystyle N=-2:\hskip
28.45274pt\mathcal{O}_{1}=-\frac{4}{5}\,\mathcal{O}_{2}=4\,\mathcal{O}_{3}=4\,\mathcal{O}_{4}=-2\,\mathcal{O}_{5}=\mathcal{O}_{6}=-2\,\mathcal{O}_{7}=\mathcal{O}_{8}\,,$
$\displaystyle N=-4:\hskip
28.45274pt\mathcal{O}_{1}=-\frac{1}{3}\,\mathcal{O}_{2}-\frac{3}{2}\,\mathcal{O}_{5}+\frac{1}{8}\,\mathcal{O}_{6}+\frac{3}{4}\,\mathcal{O}_{7}+\frac{1}{12}\,\mathcal{O}_{8}\,,$
$\displaystyle\hskip
79.66771pt\mathcal{O}_{3}=-\frac{1}{2}\,\mathcal{O}_{5}-\frac{1}{8}\,\mathcal{O}_{6}-\frac{1}{4}\,\mathcal{O}_{7}\,,$
$\displaystyle\hskip
79.66771pt\mathcal{O}_{4}=-\frac{3}{4}\,\mathcal{O}_{7}-\frac{1}{8}\,\mathcal{O}_{8}\,,$

  * •

sextic $O(N)^{4}\rtimes S_{4}$ theory:

$\displaystyle N=2:\hskip
36.98857pt\mathcal{O}_{1}=3\,\mathcal{O}_{5}-30\,\mathcal{O}_{16}+4\,\mathcal{O}_{18}+18\,\mathcal{O}_{19}+12\,\mathcal{O}_{20}-6\,\mathcal{O}_{21}\,,$
$\displaystyle\hskip
79.66771pt\mathcal{O}_{2}=\mathcal{O}_{5}+2\,\mathcal{O}_{13}-8\,\mathcal{O}_{16}+2\,\mathcal{O}_{18}+2\,\mathcal{O}_{19}+2\,\mathcal{O}_{20}\,,$
$\displaystyle\hskip
79.66771pt\mathcal{O}_{3}=3\,\mathcal{O}_{13}+3\,\mathcal{O}_{16}+\,\mathcal{O}_{18}-6\,\mathcal{O}_{19}-3\,\mathcal{O}_{20}+3\,\mathcal{O}_{21}\,,$
$\displaystyle\hskip
79.66771pt\mathcal{O}_{4}=\mathcal{O}_{5}+4\,\mathcal{O}_{15}-8\,\mathcal{O}_{16}+2\,\mathcal{O}_{18}-4\,\mathcal{O}_{19}+2\,\mathcal{O}_{20}+4\,\mathcal{O}_{21}\,,$
$\displaystyle\hskip
79.66771pt\mathcal{O}_{6}=2\,\mathcal{O}_{13}-2\,\mathcal{O}_{16}+\mathcal{O}_{18}\,,$
$\displaystyle\hskip
79.66771pt\mathcal{O}_{7}=\mathcal{O}_{13}+2\,\mathcal{O}_{15}-2\,\mathcal{O}_{16}+\mathcal{O}_{18}-3\,\mathcal{O}_{19}+2\,\mathcal{O}_{21}\,,$
$\displaystyle\hskip
79.66771pt\mathcal{O}_{8}=\mathcal{O}_{13}+2\,\mathcal{O}_{16}-\,\mathcal{O}_{19}-\,\mathcal{O}_{20}\,,$
$\displaystyle\hskip
79.66771pt\mathcal{O}_{9}=6\,\mathcal{O}_{15}-9\,\mathcal{O}_{16}+\mathcal{O}_{18}-3\,\mathcal{O}_{19}+3\,\mathcal{O}_{20}+3\,\mathcal{O}_{21}\,,$
$\displaystyle\hskip
79.66771pt\mathcal{O}_{10}=2\,\mathcal{O}_{15}-2\,\mathcal{O}_{16}+\,\mathcal{O}_{20}\,,$
$\displaystyle\hskip
79.66771pt\mathcal{O}_{11}=\mathcal{O}_{16}+\mathcal{O}_{19}-\mathcal{O}_{21}\,,$
$\displaystyle\hskip 79.66771pt\mathcal{O}_{12}=\mathcal{O}_{18}\,,$
$\displaystyle\hskip
79.66771pt\mathcal{O}_{14}=2\,\mathcal{O}_{15}-\mathcal{O}_{16}+\mathcal{O}_{18}-3\,\mathcal{O}_{19}+2\,\mathcal{O}_{21}\,,$
$\displaystyle\hskip
79.66771pt\mathcal{O}_{17}=3\,\mathcal{O}_{19}-2\,\mathcal{O}_{21}\,,$
$\displaystyle N=-2:\hskip
28.45274pt\mathcal{O}_{1}=-2\,\mathcal{O}_{2}=4\,\mathcal{O}_{3}=2\,\mathcal{O}_{4}=4\,\mathcal{O}_{6}=-4\,\mathcal{O}_{7}=-4\,\mathcal{O}_{18}+8\,\mathcal{O}_{21}\,,$
$\displaystyle\hskip
79.66771pt\mathcal{O}_{5}=\mathcal{O}_{8}=\mathcal{O}_{13}=\mathcal{O}_{19}=0\,,$
$\displaystyle\hskip
79.66771pt\mathcal{O}_{9}=-\mathcal{O}_{18}+4\,\mathcal{O}_{21}\,,$
$\displaystyle\hskip
79.66771pt\mathcal{O}_{10}=-\mathcal{O}_{11}=-2\,\mathcal{O}_{21}\,,$
$\displaystyle\hskip 79.66771pt\mathcal{O}_{12}=-\mathcal{O}_{18}\,,$
$\displaystyle\hskip
79.66771pt\mathcal{O}_{14}=-\mathcal{O}_{18}+\mathcal{O}_{21}\,,$
$\displaystyle\hskip
79.66771pt\mathcal{O}_{15}=-\mathcal{O}_{16}=-\frac{1}{2}\,\mathcal{O}_{17}=\frac{1}{2}\,\mathcal{O}_{20}=\mathcal{O}_{21}\,,$
$\displaystyle N=-4:\hskip
28.45274pt\mathcal{O}_{3}=-\frac{1}{8}\,\mathcal{O}_{1}-\frac{3}{4}\,\mathcal{O}_{2}\,,$
$\displaystyle\hskip
79.66771pt\mathcal{O}_{8}=-\frac{1}{4}\,\mathcal{O}_{2}-\frac{1}{4}\,\mathcal{O}_{4}-\frac{1}{4}\,\mathcal{O}_{5}-\mathcal{O}_{6}-\mathcal{O}_{7}\,,$
$\displaystyle\hskip
79.66771pt\mathcal{O}_{12}=\frac{1}{8}\,\mathcal{O}_{1}+\frac{3}{2}\,\mathcal{O}_{2}+\frac{3}{4}\,\mathcal{O}_{4}+\frac{3}{4}\,\mathcal{O}_{5}+3\,\mathcal{O}_{6}-\mathcal{O}_{9}-3\,\mathcal{O}_{10}-3\,\mathcal{O}_{11}\,,$
$\displaystyle\hskip
79.66771pt\mathcal{O}_{13}=-\frac{1}{4}\,\mathcal{O}_{5}\,,$
$\displaystyle\hskip
79.66771pt\mathcal{O}_{18}=\frac{1}{2}\,\mathcal{O}_{5}-\mathcal{O}_{6}-2\,\mathcal{O}_{7}-2\,\mathcal{O}_{14}-4\,\mathcal{O}_{15}-2\,\mathcal{O}_{16}-\mathcal{O}_{17}\,,$
$\displaystyle\hskip
79.66771pt\mathcal{O}_{19}=-\frac{1}{24}\,\mathcal{O}_{1}-\frac{1}{2}\,\mathcal{O}_{2}-\frac{1}{4}\,\mathcal{O}_{4}-\frac{5}{12}\,\mathcal{O}_{5}-\frac{2}{3}\,\mathcal{O}_{6}+\frac{2}{3}\,\mathcal{O}_{7}+\frac{1}{3}\,\mathcal{O}_{9}+\mathcal{O}_{10}$
$\displaystyle\hskip
110.96574pt+\mathcal{O}_{11}+\frac{2}{3}\,\mathcal{O}_{14}+\frac{4}{3}\,\mathcal{O}_{15}+\frac{2}{3}\,\mathcal{O}_{16}+\frac{1}{3}\,\mathcal{O}_{17}\,,$
$\displaystyle\hskip
79.66771pt\mathcal{O}_{20}=\frac{1}{12}\,\mathcal{O}_{1}+\frac{3}{2}\,\mathcal{O}_{2}+\mathcal{O}_{4}+\frac{19}{12}\,\mathcal{O}_{5}+\frac{10}{3}\,\mathcal{O}_{6}+\frac{2}{3}\,\mathcal{O}_{7}-\frac{2}{3}\,\mathcal{O}_{9}-3\,\mathcal{O}_{10}$
$\displaystyle\hskip
110.96574pt-2\,\mathcal{O}_{11}-\frac{4}{3}\,\mathcal{O}_{14}-\frac{20}{3}\,\mathcal{O}_{15}-\frac{16}{3}\,\mathcal{O}_{16}-\frac{2}{3}\,\mathcal{O}_{17}\,,$
$\displaystyle\hskip
79.66771pt\mathcal{O}_{21}=\frac{1}{16}\,\mathcal{O}_{1}+\frac{3}{4}\,\mathcal{O}_{2}+\frac{3}{8}\,\mathcal{O}_{4}+\frac{5}{8}\,\mathcal{O}_{5}+\mathcal{O}_{6}-\mathcal{O}_{7}-\frac{1}{2}\,\mathcal{O}_{9}-\frac{3}{2}\,\mathcal{O}_{10}$
$\displaystyle\hskip
110.96574pt-3\,\mathcal{O}_{11}-\mathcal{O}_{14}-2\,\mathcal{O}_{15}-4\,\mathcal{O}_{16}-\mathcal{O}_{17}\,.$

For each of the $O(N)^{r}$ models, it can be checked explicitly that the
matrix $G_{ij}$ is positive definite, implying that the RG flow is indeed
monotonic, as required by unitarity. When we send $N$ to $-N$, it can be
checked in each case that $G_{ij}$ either remains positive definite or becomes
negative definite, entailing that the RG flows of the corresponding
$Sp(N)^{r}$ theories too are monotonic. When $r$ is odd, however, the
fundamental fields of the $Sp(N)^{r}$ theories obtained by sending
$N\rightarrow-N$ are anti-commuting scalars, so that these theories are non-
unitary. But in Ref. [159] it was shown that the Hamiltonian of the “wrong
statistics” $Sp(N)$ theory possesses the property of being pseudo-Hermitian.
In light of the monotonic RG flows we observe, we are therefore naturally led
to pose the question, does pseudo-unitarity imply monotonicity?

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