# CFTs with $U(m)\times U(n)$ Global Symmetry in 3D
and the Chiral Phase Transition of QCD

Stefanos R. Kousvosa and Andreas Stergioub

(September 2022)

###### Abstract

Conformal field theories (CFTs) with $U(m)\times U(n)$ global symmetry in
$d=3$ dimensions have been studied for years due to their potential relevance
to the chiral phase transition of quantum chromodynamics (QCD). In this work
such CFTs are analyzed in $d=4-\varepsilon$ and $d=3$. This includes
perturbative computations in the $\varepsilon$ and large-$n$ expansions as
well as non-perturbative ones with the numerical conformal bootstrap. New
perturbative results are presented and a variety of non-perturbative bootstrap
bounds are obtained in $d=3$. Various features of the bounds obtained for
large values of $n$ disappear for low values of $n$ (keeping $m<n$ fixed), a
phenomenon which is attributed to a transition of the corresponding fixed
points to the non-unitary regime. Numerous bootstrap bounds are found that are
saturated by large-$n$ results, even in the absence of any features in the
bounds. A double scaling limit is also observed, for $m$ and $n$ large with
$m/n$ fixed, both in perturbation theory as well as in the numerical
bootstrap. For the case of two-flavor massless QCD existing bootstrap evidence
is reproduced that the chiral phase transition may be second order, albeit
associated to a universality class unrelated to the one usually discussed in
the $\varepsilon$ expansion. Similar evidence is found for the case of three-
flavor massless QCD, where we observe a pronounced kink.

Contents

toc

##  1 Introduction

The Lagrangian of quantum chromodynamics (QCD) with $N_{f}$ massless flavors
of fundamental Dirac fermions (quarks) $q^{i}$, $i=1,\ldots,N_{f}$, has
$U(N_{f})_{\text{L}}\times U(N_{f})_{\text{R}}$ global symmetry. This is
easiest to describe if we decompose $q^{i}$ into left- and right-handed
quarks, $q_{\text{\hskip 0.5ptL}}^{i},q_{\text{\hskip 0.5ptR}}^{i}$, in which
case it acts on $q_{\text{\hskip 0.5ptL}}^{i}$ and $q_{\text{\hskip
0.5ptR}}^{i}$ by independent $N_{f}\times N_{f}$ unitary matrices and is thus
chiral. Two $U(1)$ symmetries are part of this symmetry group: the vector
$U(1)$, commonly denoted by $U(1)_{\text{V}}$, which acts on left- and right-
handed quarks by the same phase, and the axial $U(1)$, commonly denoted by
$U(1)_{\text{A}}$, which acts on left- and right-handed quarks with opposite
phases. With the exception of $U(1)_{\text{A}}$, which is broken by an
anomaly, the global symmetry group of the Lagrangian of QCD persists in the
quantum theory. Starting from the original symmetry group of the classical
theory, henceforth denoted by $G_{\text{LRVA}}=SU(N_{f})_{\text{L}}\times
SU(N_{f})_{\text{R}}\times U(1)_{\text{V}}\times U(1)_{\text{A}}$,111We ignore
factors of $\mathbb{Z}_{N_{f}}$ that result from the isomorphism
$U(N)\simeq[SU(N)\times U(1)]/\mathbb{Z}_{N}$. the remaining symmetry at the
quantum level is then $G_{\text{LRV}}=SU(N_{f})_{\text{L}}\times
SU(N_{f})_{\text{R}}\times U(1)_{\text{V}}$.

If we consider nuclear matter at finite temperature, the global symmetry
$G_{\text{LRV}}$ (or $G_{\text{LRVA}}$ depending on the fate of
$U(1)_{\text{A}}$ at finite temperature) may or may not be broken depending on
the temperature. More specifically, chiral symmetry is spontaneously broken at
low temperatures due to a non-zero vacuum expectation value for a quark
bilinear (quark condensate), namely $\langle\bar{q}_{\text{\hskip
0.5ptL}}^{\hskip 1.0pti}q_{\text{\hskip 0.5ptR}}^{\hskip 1.0ptj}\rangle$,
which breaks $G_{\text{LRV}}$ (or $G_{\text{LRVA}}$) to
$SU(N_{f})_{\text{V}}\times U(1)_{\text{V}}$, where $SU(N_{f})_{\text{V}}$ is
the diagonal subgroup of $G_{\text{LR}}=SU(N_{f})_{\text{L}}\times
SU(N_{f})_{\text{R}}$. At high temperatures, however, the quark condensate is
zero and thus $G_{\text{LRV}}$ (or $G_{\text{LRVA}}$) remains unbroken. The
order of the associated phase transition at the critical temperature $T_{c}$,
with an order parameter given by the quark condensate, has phenomenological
consequences and has been the subject of multiple investigations over the
years, starting with the seminal work of Pisarski and Wilczek [1].

Due to its non-chiral nature, the $U(1)_{\text{V}}$ part of $G_{\text{LRV}}$
is expected to play no role in the symmetry breaking and it is common in the
literature to neglect it and discuss the group $G_{\text{LR}}$ instead. On the
contrary, the $U(1)_{\text{A}}$ part of $G_{\text{LRVA}}$ is of paramount
importance. Despite the fact that $U(1)_{\text{A}}$ is anomalous in QCD, it
may be effectively restored when non-zero temperature effects are considered
[1, 2, 3, 4]. If that is the case, then the chiral symmetry to consider in the
unbroken phase ($T>T_{c}$) is $G_{\text{LRA}}=SU(N_{f})_{\text{L}}\times
SU(N_{f})_{\text{R}}\times U(1)_{\text{A}}$ and not $G_{\text{LR}}$. The two
cases that have been considered in the literature are those of symmetry
$G_{\text{LR}}$ and $G_{\text{LRA}}$.

Quarks are not massless and so consequences obtained in the strict chiral
limit are not expected to hold unaltered. However, quark masses much smaller
than $T_{c}$ can be treated as perturbations of the strict chiral case, and
then the $N_{f}=2$ massless case is of particular interest (with all other
quarks treated as infinitely heavy). This is because the critical temperature
is given by $T_{c}\approx 160\text{ MeV}$, which is two orders of magnitude
larger than the masses of the up and down quarks. $T_{c}$ is slightly larger
than the mass of the strange quark, so one may consider the $N_{f}=3$ massless
case and treat the mass of the strange quark as a perturbation as well,
although this approximation may not be as well justified as in the case of up
and down quarks.

For the case $N_{f}=2$, $G_{\text{LR}}$ becomes $SU(2)_{\text{L}}\times
SU(2)_{\text{R}}\simeq SO(4)$, while $G_{\text{LRA}}$ becomes
$SU(2)_{\text{L}}\times SU(2)_{\text{R}}\times U(1)_{\text{A}}\simeq
SO(4)\times SO(2)$.222Note that we drop factors of $\mathbb{Z}_{2}$ in the
isomorphism $SU(2)\times SU(2)/\mathbb{Z}_{2}\simeq SO(4)$ for convenience of
notation. In the $G_{\text{LRA}}$ case the order of the chiral phase
transition was originally suggested to be first order using the $\varepsilon$
expansion [1], and supporting evidence for this conclusion has also been
reported [5, 6]. However, the conformal bootstrap method applied to this
scenario in [7] (for a review see [8]) produced evidence for the existence of
a potentially relevant $O(4)\times O(2)$ universality class in $d=3$
dimensions. The work [9] has also provided evidence in favor of a second order
phase transition in the case of effective restoration of $U(1)_{\text{A}}$
using renormalization group methods and resummations. For $N_{f}=3$
$G_{\text{LRA}}$ is $SU(3)_{\text{L}}\times SU(3)_{\text{R}}\times
U(1)_{\text{A}}$ and perturbative methods have not found a fixed point that
would open the possibility of a second-order chiral phase transition in three-
flavor massless QCD. A recent Monte Carlo analysis for $O(4)\times O(2)$ can
be found in [10], whereas the three-flavor case has been studied recently in
[11, 12]; see also [13] for an overview.

In this work we study conformal field theories (CFTs) with $mn$ complex scalar
fields and $U(m)\times U(n)$ global symmetry. The scalar fields are assembled
into a complex $m\times n$ matrix $\Phi$ which transforms as a bifundamental
under the action of $U(m)\times U(n)$. Our results for the case $m=n=2,3$ are
of potential relevance to the case of the chiral phase transition of two- and
three-flavor massless QCD, respectively.333We may think of $\Phi$ as the order
parameter $\langle\bar{q}_{L}q_{R}\rangle$ of the phase transition. We note
that $U(m)\times U(n)$ is the symmetry group that naturally arises in the
corresponding Landau–Ginzburg model built with $\Phi$ as a fundamental field
[1, 5, 9, 6], which has been discussed in the context of the chiral phase
transition of QCD for many years (as discussed above).

Before proceeding to outline our methodology and results, let us point out
which results will be relevant if the symmetry of QCD at high temperature is
$G_{\text{LRA}}$, and which will be relevant if it is $G_{\text{LR}}$. Our
non-perturbative numerical results, due to the bootstrap, will apply to both
cases. The logic is very similar to that of [14]. The relevant difference
between the groups $U(n)$ and $SU(n)$ for our purposes is in the existence of
the $n$-index Levi–Civita tensor, which is an invariant of $SU(n)$ but not
$U(n)$. This will affect the case $U(4)\times U(4)$, which will not be
equivalent to $SU(4)\times SU(4)$, in that there will be more sum rules in the
$SU(4)\times SU(4)$ case. However, without further assumptions these
additional sum rules would not yield different results for the bounds obtained
in this work using the $U(4)\times U(4)$ sum rules. On the other hand, our
perturbative results will apply to the case of $G_{\text{LRA}}$ only. That is
because, as discussed in [1] for example, the absence of the $U(1)_{\text{A}}$
symmetry allows one to add the schematic term
$g(\det(\Phi)+\det(\Phi^{\dagger}))$ to the Lagrangian, where $g$ is some
coupling. To find controlled perturbative fixed points in our work we
necessarily take $g=0$, which enhances the symmetry to $G_{\text{LRA}}$.

We analyze the $U(m)\times U(n)$ model in the standard $\varepsilon$ expansion
below $d=4$ up to three loops and also in the large-$n$ expansion at leading
order in $1/n$ using analytic bootstrap methods. Our results include
expressions for the scaling dimensions of scalar operators quadratic
(bilinear) in $\Phi$ that belong to various irreducible representations
(irreps) of the global symmetry group. Dimensions of such non-singlet
operators determine crossover exponents. We also use the non-perturbative
numerical conformal bootstrap [15] (for a review see [16] and [17]; for a
pedagogical introduction see [18]) to obtain upper bounds on various operator
dimensions by considering the constraints of unitarity and crossing symmetry
in the four-point function of $\Phi$.

For the $N_{f}=2$ case our numerical bootstrap bounds coincide with those
obtained in [7]. This includes a kink that indicates the possible existence of
a CFT with $O(4)\times O(2)$ symmetry (see also [19]). For $N_{f}=3$ we also
obtain bounds with kinks. Our results provide possible evidence in favor of a
second order chiral phase transition in the case of QCD with two or three
massless flavors. The associated universality classes, however, do not appear
to be continuations of the ones predicted by the standard $\varepsilon$
expansion for $U(m)\times U(n)$ theories with $n$ sufficiently larger than
$m$.

While not immediately relevant for finite temperature QCD with a small number
of massless flavors, we also probe various parameter limits of $U(m)\times
U(n)$ symmetric CFTs, such as large $n$ and $m/n$ fixed with both $m$ and $n$
large (for a pedagogical discussion around the significance of fixed points
with $m/n$ fixed see [20] and [21]). These are expected to be interesting from
the point of view of field theory, given that we make numerous comparisons
between perturbative and non-perturbative predictions. Notably, we observe
that at large $n$ a lot of our exclusion plots are almost exactly saturated by
the perturbative predictions. In some cases this happens even in the absence
of any feature in the exclusion plot. Typically, in the numerical conformal
bootstrap, kinks are seen as signals of an exclusion bound being saturated by
a CFT. In the present work we see explicit examples where this is not strictly
necessary. We also see kinks due to theories, that at least naively according
to the $\varepsilon$ expansion, should be non-unitary.

This paper is organized as follows. In the next section we review known
results regarding theories with $U(m)\times U(n)$ global symmetry in the
$\varepsilon=4-d$ expansion. In section 3 we work out the group theory
required for our analysis of $U(m)\times U(n)$ CFTs. These results are used in
section 4 to derive $\varepsilon$ expansion results up to three loops
(following the methods developed in [22]), and in section 5 to derive results
in the $1/n$ expansion valid in any $d$. In section 6 we obtain non-
perturbative numerical bootstrap bounds relevant for $U(m)\times U(n)$ CFTs in
$d=3$. We conclude in section 7. In two appendices we include two different
but equivalent ways to derive the crossing equations required in our bootstrap
problem, which are of course also equivalent to the way described in section 3
of the main text.

##  2 Fixed points of theories with $\boldsymbol{U(m)\times U(n)}$ global
symmetry in the $\boldsymbol{\varepsilon}$ expansion

In the theories we consider, the $mn$ complex scalar fields ($2mn$ real scalar
fields) are assembled into an $m\times n$ complex matrix $\Phi_{ar}$,
$a=1\ldots,m$, $r=1,\ldots,n$. The Hermitian conjugate of $\Phi_{ar}$ is
$\Phi^{\dagger}_{ra}$. Using these fields, we may construct the $U(m)\times
U(n)$ invariant Lagrangian [1, 5, 6]

$\mathscr{L}=\partial^{\mu}\Phi^{\dagger}_{ra}\hskip
1.0pt\partial_{\mu}\Phi_{ar}+\tfrac{1}{4}u\hskip
1.0pt(\Phi^{\dagger}_{ra}\Phi_{ar})^{2}+\tfrac{1}{4}v\hskip
1.0pt\Phi^{\dagger}_{ra}\Phi_{as}\Phi^{\dagger}_{sb}\Phi_{br}\,,$ (2.1)

where repeated indices are summed over and we consider up to quartic terms but
neglect the mass term $m^{2}\hskip 1.0pt\Phi^{\dagger}_{ra}\Phi_{ar}$.444We
note that either $U(m)$ or $U(n)$ can realize the $U(1)$ transformation
$\Phi_{ar}\to e^{i\alpha}\Phi_{ar}$. Therefore, strictly speaking, the global
symmetry group of (2.1) is $[U(m)\times U(n)]/U(1)$. With this in mind, we
will continue to refer to the global symmetry of (2.1) as $U(m)\times U(n)$
for brevity. For completeness, let us mention that it is also possible to
construct a fully $U(m)\times U(n)$ symmetric multiscalar Lagrangian using
results of [23]. This would have two distinct mass terms. To our knowledge the
existence of a non-trivial fixed point in such a theory in the $\varepsilon$
expansion has not been studied in the literature. As far as interactions are
concerned, we have the two couplings $u$ and $v$. A theory with $v=0$
preserves $O(2mn)$ symmetry. To examine stability of the quartic potential we
choose $m\leqslant n$ without loss of generality and we find that stability
requires $u+v\geqslant 0$ if $v<0$ and $u+\frac{1}{m}v\geqslant 0$ if
$v\geqslant 0$.555Let $X=\Phi\Phi^{\dagger}$. $X$ is an $m\times m$ Hermitian
matrix and by the Cauchy–Schwarz inequality with inner product $\langle
A,B\rangle=\operatorname*{Tr}(AB)$ for two Hermitian matrices $A,B$, by taking
$A=X$ and $B=\mathds{1}_{m}$ we find $(\operatorname*{Tr}X)^{2}\leqslant
m\operatorname*{Tr}X^{2}$. Additionally, since $X$ is positive-definite, we
have $(\operatorname*{Tr}X)^{2}\geqslant\operatorname*{Tr}X^{2}$.

The number of real fixed points of the Lagrangian (2.1) depends on the values
of $m$ and $n$. There are four regimes:

  1. (I)

For $n>n^{+}(m)$ there are four real fixed points (Gaussian, $O(2mn)$,
$U_{-}$, $U_{+}$). Stable666A fixed point with only one relevant scalar
singlet operator, namely the mass operator $\Phi^{\dagger}_{ra}\Phi_{ar}$, is
called stable. fixed point: $U_{+}$.

  2. (II)

For $n^{-}(m)<n<n^{+}(m)$ there are two real fixed points (Gaussian and
$O(2mn)$). They are both unstable.

  3. (III)

For $n_{H}(m)<n<n^{-}(m)$ there are four real fixed points (Gaussian,
$O(2mn)$, $U_{-}$, $U_{+}$). Stable fixed point: $U_{+}$.

  4. (IV)

For $n<n_{H}(m)$ there are four real fixed points (Gaussian, $O(2mn)$,
$U_{-}$, $U_{+}$). Stable fixed point: $O(2mn)$.

The Gaussian fixed point has $u=v=0$, while the $O(2mn)$ fixed point has
$u>0,v=0$. The fully-interacting fixed points (i.e. the ones besides Gaussian
and $O(2mn)$) both have $uv\neq 0$ and $U(m)\times U(n)$ global symmetry.
These fixed points move around in the $u$-$v$ coupling plane as $m,n$ change.
For every $m$ there is a value of $n$, indicated by $n^{+}(m)$ above, for
which $U_{-}$ and $U_{+}$ collide in the real $u$-$v$ plane and subsequently
move to the complex $u$-$v$ plane as we go below $n^{+}(m)$. For $n>n^{+}(m)$
the $U_{+}$ fixed point is stable and has $v>0$, as does $U_{-}$, but for
$n<n^{+}(m)$ there is no real stable fixed point. However, for some
$n^{-}(m)<n^{+}(m)$ these two fixed points reappear in the $u$-$v$ plane—this
time they have $v<0$ and $U_{+}$ is again stable. Furthermore, there is a
value $n^{H}(m)<n^{-}(m)$ below which the $O(2mn)$ fixed point is stable,
since one of the fully interacting fixed points of the $n^{H}(m)<n<n^{-}(m)$
regime crosses the $v=0$ line and acquires $v>0$, while the other remains with
$v<0$. These four regimes are depicted in Fig. 1.

Fig. 1: Schematic flow diagrams corresponding to the four regimes mentioned
in the text. The location of the fixed points should not be viewed as precise,
but the fixed points are placed at locations consistent with the sign of the
coupling $v$ for which they occur. The region between the hatched lines in
each diagram represents the basin of attraction of the stable fixed point.

The values of $n^{\pm}(m)$ can be estimated in the $\varepsilon$ expansion:

$\begin{split}n^{\pm}(m)&=5m\pm
2\sqrt{6(m-1)(m+1)}-\bigg{(}5m\pm\frac{(5m-4)(5m+4)}{2\sqrt{6(m-1)(m+1)}}\bigg{)}\hskip
1.0pt\varepsilon+\text{O}(\varepsilon^{2})\,.\end{split}$ (2.2)

In a recent paper, these results have been extended to six loops, or order
$\varepsilon^{5}$ [6]. The value of $n^{+}(m)$ is of interest due to
applications to $m$-flavor QCD. In particular, if $n^{+}(m)<m$ in $d=3$, then
there exists a unitary 3D CFT with $U(m)\times U(m)$ global symmetry
(corresponding to $U_{+}$ in regime (I) above). However, the essential
conclusion of the $\varepsilon$ expansion after resummations is that
$n^{+}(m)>m$ [5, 6] in $d=3$. This has been used to argue that there is no
stable unitary fixed point of (2.1) for $m=n$ that could describe the
$m$-flavor chiral phase transition of QCD, which requires a stable fixed point
as well as $v>0$ for the appropriate symmetry-breaking pattern and thus could
not lie in regimes (II), (III) or (IV).

##  3 Group theoretic considerations

In this section we describe the group theoretic ingredients that will allow us
to derive results in the $\varepsilon$ expansion as well as the crossing
equation that we will use for our analytic and numerical bootstrap studies.

For our purposes777Converting to a one-index real-field notation allows us to
readily extract results from existing $\varepsilon$ expansion computations in
the literature. The reader directly interested in numerical bootstrap sum
rules may see Appendix Appendix A. Kronecker tensor structures for complex
fields. it is more economical (index-wise) to consider the replacement [24]

$\Phi_{ar}=T_{ar}{\\!}^{i}\phi_{i}\,,\qquad\Phi^{\dagger}_{ra}=T^{\dagger}{\\!}_{ra}{\\!}^{i}\phi_{i}\,,\qquad
i=1,\ldots,2mn\,,$ (3.1)

where the $2mn$ complex $m\times n$ matrices $T_{ar}{\\!}^{i}$ encode the
complex nature of $\Phi_{ar}$, while the $2mn$ fields $\phi_{i}$ are real.
Essentially, the fields $\Phi$ and $\Phi^{\dagger}$ are repackaged into their
real and complex parts, schematically $(\Phi+\Phi^{\dagger})$ and
$-i(\Phi-\Phi^{\dagger})$.888As an explicit example, consider
$\Phi_{11}=\phi_{1}+i\phi_{2}$; then $T_{11}{\\!}^{1}=1$ and
$T_{11}{\\!}^{2}=i$. Also, all other elements, such as e.g. $T_{12}{\\!}^{1}$
or $T_{44}{\\!}^{1}$, are zero. With this example in mind, the reader may
convince themselves by inspection that (3.2) holds (up to normalization). The
$T$ matrices satisfy

$\begin{gathered}T^{\dagger}{\\!}_{ra}{\\!}^{i}\hskip
1.0ptT_{bs}{\\!}^{i}=\delta_{ab}\delta_{rs}\,,\qquad T_{ar}{\\!}^{i}\hskip
1.0ptT_{bs}{\\!}^{i}=0\qquad T^{\dagger}{\\!}_{ra}{\\!}^{i}\hskip
1.0ptT^{\dagger}{\\!}_{sb}{\\!}^{i}=0\,,\\\
T^{\dagger}{\\!}_{ra}{\\!}^{i}\hskip
1.0ptT_{ar}{\\!}^{j}+T^{\dagger}{\\!}_{ra}{\\!}^{j}\hskip
1.0ptT_{ar}{\\!}^{i}=\operatorname*{Tr}(T^{\dagger\hskip 1.0pti}\hskip
1.0ptT^{j}+T^{\dagger\hskip 1.0ptj}\hskip
1.0ptT^{i})=\delta^{ij}\,.\end{gathered}$ (3.2)

###  3.1 Rank-four invariant tensors

The $T$ matrices allow us to construct the rank-four (in the indices
$i,j,\ldots$) invariant tensors of $U(m)\times U(n)$. These can be used to
derive a variety of results in the $\varepsilon$ expansion up to three loops
as described in [22].999With the recent work of [24] some of these results can
be extended to six loops.

For $m\neq n$ there is one fully symmetric traceless rank-four primitive
invariant tensor, $\zeta_{ijkl}$, three rank-four primitive invariant tensors
$\omega_{u,ijkl}$, $u=1,2,3$, with symmetry properties

$\omega_{u,ijkl}=\omega_{u,jikl},\qquad\omega_{u,klij}=\omega_{u,ijkl}\,,\qquad\omega_{u,i(jkl)}=0\,,\qquad\omega_{u,iikl}=0\,,$
(3.3)

and two fully antisymmetric rank-four primitive invariant tensors,
$\psi_{x,ijkl}$, $x=1,2$. With the addition of the non-primitive rank-four
invariant tensors $\delta_{ij}\delta_{kl}$, $\delta_{ik}\delta_{jl}$ and
$\delta_{il}\delta_{jk}$ we have a total of 12 independent rank-four invariant
tensors.101010There are two inequivalent index permutations and therefore two
independent invariant tensors for each $\omega$. One can check that products
of these 12 tensors with four free indices (such as e.g.
$\omega_{u,ijkl}\hskip 1.0pt\zeta_{klmn}$) close on themselves, i.e. they do
not produce any additional tensors. For $m=n$ we can treat the two $U(n)$’s as
distinguishable or indistinguishable. In the latter case one of the $\omega$
and one of the $\psi$ tensors disappear and we are left with 9 independent
invariant tensors.

With the use of the real scalar fields $\phi_{i}$, the $U(m)\times U(n)$
invariant Lagrangian takes the form111111We consider terms up to quartic in
$\phi$ and neglect the mass term $\tfrac{1}{2}m^{2}\phi^{2}$.

$\mathscr{L}=\tfrac{1}{2}\partial^{\mu}\phi_{i}\partial_{\mu}\phi_{i}+\tfrac{1}{8}\lambda\hskip
0.5pt(\phi^{2})^{2}+\tfrac{1}{24}g\hskip
1.0pt\zeta_{ijkl}\phi_{i}\phi_{j}\phi_{k}\phi_{l}\,,$ (3.4)

where $\phi^{2}=\phi_{i}\phi_{i}$ and

$\zeta_{ijkl}=\tfrac{1}{2}\operatorname*{Tr}\big{[}T^{\dagger}{\\!}_{(i}T_{j}T^{\dagger}{\\!}_{k}T_{l)}\big{]}-\frac{m+n}{mn+1}(\delta_{ij}\delta_{kl}+\delta_{ik}\delta_{jl}+\delta_{il}\delta_{jk})\,,$
(3.5)

where parentheses around indices are used to denote symmetrization of the
enclosed indices.121212Here and hereafter symmetrization and
antisymmetrization of indices is defined without an overall factorial
normalization factor. Due to cyclicity of the trace there are 12 distinct
terms among the 24 produced by the symmetrization of the indices. Choosing,
without loss of generality, $m\leqslant n$, the bound

$\frac{3(m-1)(m+1)}{m(mn+1)}(\phi^{2})^{2}\leqslant\zeta_{ijkl}\phi_{i}\phi_{j}\phi_{k}\phi_{l}\leqslant\frac{3(m-1)(n-1)}{mn+1}(\phi^{2})^{2}$
(3.6)

is satisfied. The couplings $\lambda,g$ of (3.4) are related to the couplings
$u,v$ of (2.1) via

$\lambda=\tfrac{1}{2}\Big{(}u+\frac{m+n}{mn+1}v\Big{)}\,,\qquad
g=\tfrac{1}{2}\hskip 1.0ptv\,.$ (3.7)

The tensor $\zeta_{ijkl}$ satisfies

$\zeta_{ijmn}\hskip 1.0pt\zeta_{mnkl}=\frac{1}{2mn-1}\hskip 1.0pta\hskip
1.0pt\big{(}mn(\delta_{ik}\delta_{jl}+\delta_{il}\delta_{jk})-\delta_{ij}\delta_{kl}\big{)}+e^{u}\hskip
1.0pt\omega_{u,ijkl}+b\,\zeta_{ijkl}\,,$ (3.8)

with

$\begin{split}a&=\frac{6(m-1)(m+1)(n-1)(n+1)}{(mn+1)^{2}}\,,\\\
e^{1}&=\frac{2(m+n)}{3}\,,\qquad e^{2}=m-n\,,\qquad
e^{3}=\frac{2}{3}(2m+2n+3)\,,\\\
b&=\frac{2(m+n)(mn-5)}{3(mn+1)}\,,\end{split}$ (3.9)

where

$\begin{split}\omega_{1,ijkl}&=\operatorname*{Sym}_{ij}\operatorname*{Sym}_{kl}\big{\\{}\tfrac{1}{2}\operatorname*{Tr}\big{(}T^{\dagger}{\\!}_{i}T_{j}T^{\dagger}{\\!}_{k}T_{l}-T^{\dagger}{\\!}_{i}T_{k}T^{\dagger}{\\!}_{j}T_{l}-T^{\dagger}{\\!}_{k}T_{i}T^{\dagger}{\\!}_{l}T_{j}+T^{\dagger}{\\!}_{i}T_{l}T^{\dagger}{\\!}_{k}T_{j}\big{)}\\\
&\qquad\qquad-2\big{[}\operatorname*{Tr}\big{(}T^{\dagger}{\\!}_{i}T_{k}\big{)}\operatorname*{Tr}\big{(}T^{\dagger}{\\!}_{j}T_{l}\big{)}+\operatorname*{Tr}\big{(}T^{\dagger}{\\!}_{k}T_{i}\big{)}\operatorname*{Tr}\big{(}T^{\dagger}{\\!}_{l}T_{j}\big{)}-2\operatorname*{Tr}\big{(}T^{\dagger}{\\!}_{i}T_{j}\big{)}\operatorname*{Tr}\big{(}T^{\dagger}{\\!}_{k}T_{l}\big{)}\big{]}\big{\\}}\\\
&\quad+\frac{4mn+m+n}{2(2mn-1)}(\delta_{ik}\delta_{jl}+\delta_{il}\delta_{jk}-2\hskip
1.0pt\delta_{ij}\delta_{kl})\,,\\\
\omega_{2,ijkl}&=\operatorname*{Sym}_{ij}\operatorname*{Sym}_{kl}\big{[}\operatorname*{Tr}\big{(}T^{\dagger}{\\!}_{i}T_{j}T^{\dagger}{\\!}_{k}T_{l}-T^{\dagger}{\\!}_{i}T_{l}T^{\dagger}{\\!}_{k}T_{j}\big{)}\big{]}+\frac{m-n}{2mn-1}(\delta_{ik}\delta_{jl}+\delta_{il}\delta_{jk}-2\hskip
1.0pt\delta_{ij}\delta_{kl})\,,\\\
\omega_{3,ijkl}&=\operatorname*{Sym}_{ij}\operatorname*{Sym}_{kl}\big{[}\operatorname*{Tr}\big{(}T^{\dagger}{\\!}_{i}T_{k}\big{)}\operatorname*{Tr}\big{(}T^{\dagger}{\\!}_{j}T_{l}\big{)}+\operatorname*{Tr}\big{(}T^{\dagger}{\\!}_{k}T_{i}\big{)}\operatorname*{Tr}\big{(}T^{\dagger}{\\!}_{l}T_{j}\big{)}-2\operatorname*{Tr}\big{(}T^{\dagger}{\\!}_{i}T_{j}\big{)}\operatorname*{Tr}\big{(}T^{\dagger}{\\!}_{k}T_{l}\big{)}\big{]}\\\
&\quad-\frac{mn}{2mn-1}(\delta_{ik}\delta_{jl}+\delta_{il}\delta_{jk}-2\hskip
1.0pt\delta_{ij}\delta_{kl})\,.\end{split}$ (3.10)

The operator $\operatorname*{Sym}_{ij}$ simply symmetrizes the indices $i,j$
of the expression on which it acts.

There are further identities like (3.8) involving the $\omega_{u}$ tensors in
the left-hand side. These take the form

$\omega_{u,ijmn}\hskip 1.0pt\zeta_{mnkl}=f_{u}{\\!}^{v}\hskip
1.0pt\omega_{v,ijkl}+h_{u}\hskip 1.0pt\zeta_{ijkl}\,,$ (3.11)

with

$\begin{split}f_{1}{\\!}^{1}&=\frac{m^{2}n+mn^{2}+8mn-5m-5n+8}{3(mn+1)}\,,\qquad
f_{1}{\\!}^{2}=\tfrac{1}{2}(m-n)\,,\qquad
f_{1}{\\!}^{3}=\tfrac{2}{3}(m+n+5)\,,\\\
f_{2}{\\!}^{1}&=\tfrac{2}{3}(m-n)\,,\qquad
f_{2}{\\!}^{2}=\frac{(m+n)(mn-1)}{mn+1}\,,\qquad
f_{2}{\\!}^{3}=\tfrac{4}{3}(m-n)\,,\\\ f_{3}{\\!}^{1}&=-\tfrac{4}{3}\,,\qquad
f_{3}{\\!}^{2}=0\,,\qquad f_{3}{\\!}^{3}=-\frac{2(4mn+3m+3n+4)}{3(mn+1)}\,,\\\
h_{1}&=\frac{2(m+n+2)(mn+1)}{3(2mn-1)}\,,\qquad
h_{2}=\frac{4(m-n)(mn+1)}{3(2mn-1)}\,,\qquad
h_{3}=-\frac{2(mn+1)}{3(2mn-1)}\,,\\\ \end{split}$ (3.12)

and

$\begin{split}\omega_{u,ijmn}\hskip
1.0pt\omega_{v,mnkl}=\frac{1}{2mn-1}a^{\prime}_{uv}\big{(}mn(\delta_{ik}\delta_{jl}+\delta_{il}\delta_{jk})-\delta_{ij}\delta_{kl}\big{)}+e^{\prime}_{uv}{\\!}^{w}\hskip
1.0pt\omega_{w,ijkl}+b^{\prime}_{uv}\hskip 1.0pt\zeta_{ijkl}\,,\end{split}$
(3.13)

with the parameters appearing also determined but not quoted here.

Further relations involving the $\omega_{u}$ tensors in the left-hand side
require the $\psi_{x}$ tensors in the right-hand side:

$\begin{split}\omega_{u,imjn}\hskip
1.0pt\omega_{v,kmln}&=\frac{1}{(mn+1)(2mn-1)^{2}}\big{(}\tilde{a}_{uv}\hskip
1.0pt\delta_{ik}\delta_{jl}+\hat{a}_{uv}\hskip
1.0pt\delta_{il}\delta_{jk}+\check{a}_{uv}\hskip
1.0pt\delta_{ij}\delta_{kl}\big{)}\\\
&\quad+\frac{1}{2mn-1}(\tilde{e}_{uv}{\\!}^{w}\hskip
1.0pt\omega_{w,ijkl}+\hat{e}_{uv}{\\!}^{w}\hskip
1.0pt\omega_{w,ikjl})+b^{\prime\prime}_{uv}\hskip
1.0pt\zeta_{ijkl}+d_{uv}{\\!}^{x}\hskip 1.0pt\psi_{x,ijkl}\,,\end{split}$
(3.14)

and there are similar relations for $\omega_{u,imjn}\hskip 1.0pt\psi_{x,mnkl}$
and $\psi_{x,ijmn}\psi_{y,mnkl}$. The tensors $\psi_{x}$ are given by

$\displaystyle\psi_{1,ijkl}=\operatorname*{Tr}\big{(}T^{\dagger}{\\!}_{[i}T_{j}T^{\dagger}{\\!}_{k}T_{l]}\big{)}\,,\qquad\psi_{2,ijkl}=\operatorname*{Tr}\big{(}T^{\dagger}{\\!}_{[i}T_{j}\big{)}\operatorname*{Tr}\big{(}T^{\dagger}{\\!}_{k}T_{l]}\big{)}\,,$

where brackets around indices are used to denote antisymmetrization of the
enclosed indices. There are 12 distinct terms in $\psi_{1}$ and 6 in
$\psi_{2}$.

Finally, there is a relevant identity involving four $\zeta$ tensors,

$\zeta_{ii^{\prime}j^{\prime}k^{\prime}}\hskip
1.0pt\zeta_{ji^{\prime}l^{\prime}m^{\prime}}\hskip
1.0pt\zeta_{kj^{\prime}l^{\prime}n^{\prime}}\hskip
1.0pt\zeta_{lk^{\prime}m^{\prime}n^{\prime}}=\tfrac{1}{4}\hskip
1.0pt\mathcal{A}\hskip
1.0pt\big{(}\delta_{ij}\delta_{kl}+\delta_{ik}\delta_{jl}+\delta_{il}\delta_{jk}\big{)}+c\,\zeta_{ijkl}\,,$
(3.15)

with

$\begin{split}\mathcal{A}&=\frac{1}{N-1}\,a\big{(}(N-2)a+2(N-1)\,b^{2}\big{)}-a\hskip
1.0pte^{u}h_{u}\,,\\\
c&=-\frac{2(m+n)(5m^{3}n^{3}-m^{2}n^{2}+71mn-20m^{3}n-20mn^{3}+4m^{2}+4n^{2}+29)}{(mn+1)^{3}\,,}\,.\end{split}$
(3.16)

When $m=n$ the two $U(n)$ factors in the global symmetry group $U(n)\times
U(n)$ may be treated as distinguishable or indistinguishable. In the former
case one simply needs to take $m=n$ in the various expressions given in this
work. In the latter case the indices carried by $\Phi$ are indistinguishable,
the global symmetry is enhanced to $U(n)^{2}\rtimes\mathbb{Z}_{2}$ and the
tensors $\omega_{2}$ and $\psi_{1}$ vanish. We will comment on this case
separately at various points below.

###  3.2 Rank-four projectors

To derive the projectors we will convert to invariant tensors that are not
traceless. This is not essential and is only done for simplicity of the
expressions for the projectors. We thus define

$\begin{split}\tilde{\zeta}_{ijkl}&=\zeta_{ijkl}+\frac{m+n}{mn+1}(\delta_{ik}\delta_{jl}+\delta_{il}\delta_{jk}+\delta_{ij}\delta_{kl})\,,\\\
\tilde{\omega}_{1,ijkl}&=\omega_{1,ijkl}-\frac{m+n+2}{2(2mn-1)}(\delta_{ik}\delta_{jl}+\delta_{il}\delta_{jk}-2\hskip
1.0pt\delta_{ij}\delta_{kl})\,,\\\
\tilde{\omega}_{2,ijkl}&=\omega_{2,ijkl}-\frac{m-n}{2mn-1}(\delta_{ik}\delta_{jl}+\delta_{il}\delta_{jk}-2\hskip
1.0pt\delta_{ij}\delta_{kl})\,,\\\
\tilde{\omega}_{3,ijkl}&=\omega_{3,ijkl}+\frac{1}{2(2mn-1)}(\delta_{ik}\delta_{jl}+\delta_{il}\delta_{jk}-2\hskip
1.0pt\delta_{ij}\delta_{kl})\,.\end{split}$ (3.17)

The $\tilde{\omega}_{u}$ tensors satisfy all but the last of the properties in
(3.3). Using these tensors one can define the 12 rank-four projectors

$\displaystyle P_{ijkl}^{S_{\text{even}}}$
$\displaystyle=\frac{1}{2mn}\delta_{ij}\delta_{kl}\,,$ $\displaystyle
P_{ijkl}^{S_{\text{odd}}}$
$\displaystyle=\frac{1}{3mn}\big{(}\tilde{\omega}_{3,ijkl}+2\hskip
1.0pt\tilde{\omega}_{3,ikjl}+\psi_{2,ijkl}\big{)}\,,$ $\displaystyle
P_{ijkl}^{RS_{\text{even}}}$
$\displaystyle=-\frac{1}{2mn}\delta_{ij}\delta_{kl}+\frac{1}{6n}(\tilde{\zeta}_{ijkl}+\tilde{\omega}_{1,ijkl}-\tfrac{3}{2}\hskip
1.0pt\tilde{\omega}_{2,ijkl}+2\hskip 1.0pt\tilde{\omega}_{3,ijkl})\,,$
$\displaystyle P_{ijkl}^{RS_{\text{odd}}}$
$\displaystyle=-\frac{1}{6n}(\tilde{\omega}_{1,ijkl}+2\hskip
1.0pt\tilde{\omega}_{1,ikjl}+\tfrac{1}{2}\hskip
1.0pt\tilde{\omega}_{2,ijkl}+\tilde{\omega}_{2,ikjl}+\psi_{1,ijkl})$
$\displaystyle\quad-\frac{1}{3mn}[(m+1)(\tilde{\omega}_{3,ijkl}+2\hskip
1.0pt\tilde{\omega}_{3,ikjl})+\psi_{2,ijkl}]\,,$ $\displaystyle
P_{ijkl}^{SR_{\text{even}}}$
$\displaystyle=-\frac{1}{2mn}\delta_{ij}\delta_{kl}+\frac{1}{6m}(\tilde{\zeta}_{ijkl}+\tilde{\omega}_{1,ijkl}+\tfrac{3}{2}\hskip
1.0pt\tilde{\omega}_{2,ijkl}+2\hskip 1.0pt\tilde{\omega}_{3,ijkl})\,,$
$\displaystyle P_{ijkl}^{SR_{\text{odd}}}$
$\displaystyle=-\frac{1}{6m}(\tilde{\omega}_{1,ijkl}+2\hskip
1.0pt\tilde{\omega}_{1,ikjl}-\tfrac{1}{2}\hskip
1.0pt\tilde{\omega}_{2,ijkl}-\tilde{\omega}_{2,ikjl}+\psi_{1,ijkl})$
$\displaystyle\quad-\frac{1}{3mn}[(n+1)(\tilde{\omega}_{3,ijkl}+2\hskip
1.0pt\tilde{\omega}_{3,ikjl})+\psi_{2,ijkl}]\,,$ $\displaystyle
P_{ijkl}^{RR_{\text{even}}}$
$\displaystyle=\tfrac{1}{4}(\delta_{ik}\delta_{jl}+\delta_{il}\delta_{jk})-\frac{1}{2mn}\delta_{ij}\delta_{kl}-\frac{m+n}{6mn}\tilde{\zeta}_{ijkl}$
$\displaystyle\quad-\frac{1}{6mn}[(m+n)\tilde{\omega}_{1,ijkl}-\tfrac{3}{2}(m-n)\tilde{\omega}_{2,ijkl}+(3mn+2m+2n)\tilde{\omega}_{3,ijkl}]\,,$
$\displaystyle P_{ijkl}^{RR_{\text{odd}}}$
$\displaystyle=-\tfrac{1}{4}(\delta_{ik}\delta_{jl}-\delta_{il}\delta_{jk})+\frac{1}{6mn}[(m+n)(\tilde{\omega}_{1,ijkl}+2\hskip
1.0pt\tilde{\omega}_{1,ikjl})+\tfrac{1}{2}(m-n)(\tilde{\omega}_{2,ijkl}+2\hskip
1.0pt\tilde{\omega}_{2,ikjl})$
$\displaystyle\quad+(mn+2m+2n+2)(\tilde{\omega}_{3,ijkl}+2\hskip
1.0pt\tilde{\omega}_{3,ikjl})]+\frac{m-n}{6mn}\psi_{1,ijkl}-\frac{mn-1}{3mn}\psi_{2,ijkl}\,,$
$\displaystyle P_{ijkl}^{TT_{\text{even}}}$
$\displaystyle=\tfrac{1}{8}(\delta_{ik}\delta_{jl}+\delta_{il}\delta_{jk})+\tfrac{1}{12}\tilde{\zeta}_{ijkl}-\tfrac{1}{6}\tilde{\omega}_{1,ijkl}-\tfrac{1}{12}\tilde{\omega}_{3,ijkl}\,,$
$\displaystyle P_{ijkl}^{TA_{\text{odd}}}$
$\displaystyle=-\tfrac{1}{8}(\delta_{ik}\delta_{jl}-\delta_{il}\delta_{jk})+\tfrac{1}{12}(\tilde{\omega}_{2,ijkl}+2\hskip
1.0pt\tilde{\omega}_{2,ikjl})-\tfrac{1}{12}(\tilde{\omega}_{3,ijkl}+2\hskip
1.0pt\tilde{\omega}_{3,ikjl})-\tfrac{1}{12}\psi_{1,ijkl}+\tfrac{1}{6}\psi_{2,ijkl}\,,$
$\displaystyle P_{ijkl}^{AT_{\text{odd}}}$
$\displaystyle=-\tfrac{1}{8}(\delta_{ik}\delta_{jl}-\delta_{il}\delta_{jk})-\tfrac{1}{12}(\tilde{\omega}_{2,ijkl}+2\hskip
1.0pt\tilde{\omega}_{2,ikjl})-\tfrac{1}{12}(\tilde{\omega}_{3,ijkl}+2\hskip
1.0pt\tilde{\omega}_{3,ikjl})+\tfrac{1}{12}\psi_{1,ijkl}+\tfrac{1}{6}\psi_{2,ijkl}\,,$
$\displaystyle P_{ijkl}^{AA_{\text{even}}}$
$\displaystyle=\tfrac{1}{8}(\delta_{ik}\delta_{jl}+\delta_{il}\delta_{jk})-\tfrac{1}{12}\tilde{\zeta}_{ijkl}+\tfrac{1}{6}\tilde{\omega}_{1,ijkl}+\tfrac{7}{12}\tilde{\omega}_{3,ijkl}\,,$
(3.18)

where the subscripts “even” and “odd” refer to the Lorentz spins with which
the corresponding irreps appear in the $\phi_{i}\times\phi_{j}$ OPE. Note that
$\tilde{\omega}_{u,jikl}+2\hskip
1.0pt\tilde{\omega}_{u,jkil}=-\tilde{\omega}_{u,ijkl}-2\hskip
1.0pt\tilde{\omega}_{u,ikjl}$ so that the “odd” projectors are odd under
$i\leftrightarrow j$ (and under $k\leftrightarrow l$). The projectors satisfy

$P^{I}_{ijmn}P^{J}_{nmkl}=P^{I}_{ijkl}\hskip
1.0pt\delta^{IJ}\,,\qquad\sum_{I}P^{I}_{ijkl}=\delta_{il}\delta_{jk}\,,\qquad
P^{I}_{ijkl}\hskip 1.0pt\delta_{il}\delta_{jk}=d_{r}^{I}\,,$ (3.19)

where $d_{r}^{I}$ is the dimension of the irrep indexed by $I$:

$\begin{split}\vec{d}_{r}&=\big{(}[1]_{2},[(m-1)(m+1)]_{2},[(n-1)(n+1)]_{2},[(m-1)(m+1)(n-1)(n+1)]_{2},\\\
&\quad\quad\tfrac{1}{2}m(m+1)n(n+1),\tfrac{1}{2}m(m+1)(n-1)n,\tfrac{1}{2}(m-1)mn(n+1),\tfrac{1}{2}(m-1)m(n-1)n\big{)}\,,\end{split}$
(3.20)

where by $[x]_{2}$ we mean that $x$ appears two consecutive times.

In the case where $m=n$ and we treat the two $U(n)$ factors as
indistinguishable, then, as a consequence of the disappearance of
$\tilde{\omega}_{2}$ and $\psi_{1}$, instead of separate projectors
$P^{RS_{\text{even}}}$ and $P^{SR_{\text{even}}}$ we only have the sum
$P^{RS_{\text{even}}}+P^{SR_{\text{even}}}$,131313in which case we will refer
to the corresponding irrep as $RSSR$ and the same happens for
$P^{RS_{\text{odd}}},P^{SR_{\text{odd}}}$ and
$P^{TA_{\text{odd}}},P^{AT_{\text{odd}}}$. Consequently, we have a total of 9
independent rank-four projectors.

In Appendices Appendix A. Kronecker tensor structures for complex fields and
Appendix B. Kronecker tensor structures for real fields we give projectors
using different ways of parametrizing the scalar fields. Parametrizing the
field with a suitable number of indices, one is able to express the projectors
solely in terms of Kronecker deltas for both real and complex fields. These
have the advantage of being the most straightforward tensors one can write
down.

##  4 Results in the $\boldsymbol{\varepsilon}$ expansion

The results of the previous section suffice to determine beta functions and
anomalous dimensions up to three loops following [22]. With the rescalings
$(\lambda,g)\to 16\pi^{2}(\lambda,g)$ and using $N=2mn$, these are

$\begin{split}\beta_{\lambda}{\\!}^{(1)}&=-\varepsilon\,\lambda+(N+8)\lambda^{2}+a\,g^{2}\,,\\\
\beta_{g}{\\!}^{(1)}&=-\varepsilon\,g+12\,\lambda\,g+3\,b\,g^{2}\,,\end{split}$
(4.1)
$\begin{split}\beta_{\lambda}{\\!}^{(2)}&=-3(3N+14)\,\lambda^{3}-\tfrac{1}{6}(5N+82)\,a\,\lambda
g^{2}\ -2\,ab\,g^{3}\,,\\\
\beta_{g}{\\!}^{(2)}&=-(5N+82)\,\lambda^{2}g+\tfrac{1}{6(N-1)}(N^{2}-17N+34)\,a\,g^{3}-6\,b(6\,\lambda
g^{2}+b\,g^{3})+3\,e^{u}h_{u}\,g^{3}\,,\end{split}$ (4.2)

and

$\displaystyle\beta_{\lambda}{\\!}^{(3)}$
$\displaystyle=\tfrac{1}{8}(33N^{2}+922N+2960)\,\lambda^{4}+12(5N+22)\zeta_{3}\,\lambda^{4}$
$\displaystyle\quad+\tfrac{1}{16}(N^{2}+500N+3492)\,a\,\lambda^{2}g^{2}+12(N+14)\zeta_{3}\,a\,\lambda^{2}g^{2}$
$\displaystyle\quad+\tfrac{1}{8}(27N+470)\,a\,b\,\lambda
g^{3}+48\zeta_{3}\,a\,b\,\lambda g^{3}$
$\displaystyle\quad-\tfrac{1}{16(N-1)}(7\,N^{2}-33\,N+114)\,a^{2}\,g^{4}+\tfrac{13}{2}\,a\,b^{2}\,g^{4}-\tfrac{11}{2}\hskip
1.0pta\hskip 1.0pte^{u}h_{u}\,g^{4}+3\,\mathcal{A}\,\zeta_{3}\,g^{4}\,,$
$\displaystyle\beta_{g}{\\!}^{(3)}$
$\displaystyle=-\tfrac{1}{4}(13N^{2}-368N-3284)\,\lambda^{3}g+48(N+14)\zeta_{3}\,\lambda^{3}g$
$\displaystyle\quad+\tfrac{3}{8}(43N+1334)\,b\,\lambda^{2}g^{2}+432\zeta_{3}\,b\,\lambda^{2}g^{2}$
$\displaystyle\quad+\tfrac{3}{N-1}\,\big{(}3\,N^{2}+33\,N-50\big{)}a\,\lambda
g^{3}+156\,b^{2}\,\lambda g^{3}-42\,e^{u}h_{u}\,\lambda
g^{3}+72\,\mathcal{A}/a\,\zeta_{3}\,\lambda g^{3}$
$\displaystyle\quad-\tfrac{1}{16(N-1)}(11\,N^{2}-289\,N+626)\,a\,b\,g^{4}+\tfrac{39}{2}\,b^{3}g^{4}-\tfrac{3}{4}\big{(}29\,b\,e^{u}h_{u}-8\,e^{u}f_{u}{\\!}^{v}h_{v}\big{)}g^{4}+12\zeta_{3}\,c\,g^{4}\,.$
(4.3)

The $\phi$ anomalous dimension matrix is $\gamma_{\phi}\hskip
1.0pt{\mathds{1}}_{N}$ with

$\gamma_{\phi}^{(2)}=\tfrac{1}{4}(N+2)\big{(}\lambda^{2}+\tfrac{1}{6}\,a\,g^{2}\big{)}$
(4.4)

and

$\gamma_{\phi}^{(3)}=-\tfrac{1}{16}(N+2)(N+8)\,\lambda^{3}-\tfrac{1}{32}(N+2)(6a\,\lambda
g^{2}+a\,b\,g^{3})\,.$ (4.5)

Relevant operators quadratic in $\phi$ can be considered by extending (3.4) by

$\mathscr{L}\to\mathscr{L}+\tfrac{1}{2}\hskip 1.0pt\sigma\hskip
1.0pt\phi^{2}+\tfrac{1}{2}\hskip 1.0pt\rho_{ij}\hskip
1.0pt\phi_{i}\phi_{j}\,,\qquad\rho_{ii}=0\,.$ (4.6)

The corresponding beta functions for $\sigma,\rho$ are then [22]

$\displaystyle\beta_{\sigma}{\\!}^{(1)}={}$
$\displaystyle(N+2)\,\lambda\,\sigma\,,\qquad\beta_{\sigma}{\\!}^{(2)}=-\tfrac{5}{2}(N+2)(\lambda^{2}+\tfrac{1}{6}a\,g^{2})\,\sigma\,,$
$\displaystyle\beta_{\sigma}{\\!}^{(3)}={}$
$\displaystyle\tfrac{1}{16}(N+2)\big{(}12(5N+37)\lambda^{3}+(N+164)\,a\,g^{2}\lambda+27\,ab\,g^{3}\big{)}\,\sigma\,,$
$\displaystyle\beta_{\rho,ij}{\\!}^{(1)}={}$ $\displaystyle
2\lambda\,\rho_{ij}+g\,\zeta_{ijkl}\hskip 1.0pt\rho_{kl}\,,$
$\displaystyle\beta_{\rho,ij}{\\!}^{(2)}={}$
$\displaystyle-\tfrac{1}{2}\big{(}(N+10)\lambda^{2}-\tfrac{N^{2}-5N+10}{6(N-1)}\,a\,g^{2}\big{)}\rho_{ij}-(4\lambda
g+b\,g^{2})\,\zeta_{ijkl}\hskip 1.0pt\rho_{kl}+\tfrac{1}{2}\hskip
1.0pte^{u}\hskip 1.0pt\omega_{u,ijkl}\hskip 1.0pt\rho_{kl}\,,$
$\displaystyle\beta_{\rho,ij}{\\!}^{(3)}={}$
$\displaystyle-\tfrac{1}{4}\big{(}\tfrac{1}{2}(5N^{2}-84N-444)\lambda^{3}-\tfrac{5N^{2}+65N-82}{N-1}\,a\,g^{2}\lambda+\tfrac{N^{2}-35N+54}{4(N-1)}\,ab\,g^{3}\big{)}\rho_{ij}$
$\displaystyle+\tfrac{1}{8}\big{(}3(9N+146)g\,\lambda^{2}+192\,b\,g^{2}\lambda-\tfrac{3N^{2}-25N+66}{2(N-1)}\,a\,g^{3}+32\,b^{2}\,g^{3}-22\,e^{u}h_{u}\,g^{3}\big{)}\,\zeta_{ijkl}\hskip
1.0pt\rho_{kl}$
$\displaystyle-\big{(}3\,g^{2}\lambda+\tfrac{5}{4}\,b\,g^{3}\big{)}e^{u}\hskip
1.0pt\omega_{u,ijkl}\hskip 1.0pt\rho_{kl}+g^{3}\,e^{u}f_{u}{\\!}^{v}\hskip
1.0pt\omega_{v,ijkl}\hskip 1.0pt\rho_{kl}\,.$ (4.7)

In general, $\beta_{\sigma}=\gamma_{\sigma}\hskip 1.0pt\sigma$, but anomalous
dimensions for $\rho$ are determined by the eigenvalue problem

$\zeta_{ijkl}\hskip 1.0ptv_{kl}=\mu\hskip
1.0ptv_{ij}\,,\qquad\omega_{u,ijkl}\hskip 1.0ptv_{kl}=\mu_{u}\hskip
1.0ptv_{ij}\,,\qquad v_{ij}=v_{ji}\,,\ v_{ii}=0\,,$ (4.8)

requiring, using (3.8), (3.11) and (3.13),

$\mu^{2}=e^{u}\mu_{u}+b\,\mu+\tfrac{N}{N-1}\,a\,,\quad\mu\,\mu_{u}=f_{u}{}^{v}\mu_{v}+h_{u}\,\mu\,,\quad\mu_{u}\,\mu_{v}=e^{\prime}{\\!}_{uv}{}^{w}\mu_{w}+b^{\prime}{\\!}_{uv}\,\mu+\tfrac{N}{N-1}\,a^{\prime}{\\!}_{uv}\,.$
(4.9)

The results (4.1)–(4.5) and (4.7), (4.9), for the appropriate
$a,b,c,e^{u},f_{u}{\\!}^{v},h^{u},a^{\prime}_{uv},b^{\prime}_{uv}$ and
$e^{\prime}_{uv}{\\!}^{w}$, apply to any scalar theory with a global symmetry
group that has a unique rank-four symmetric traceless primitive invariant
tensor [22]. In this work we will focus on the two fixed points of (3.4),
labeled $U_{\pm}$, that preserve $U(m)\times U(n)$ symmetry.141414There are
two further fixed points of (3.4): the free theory and the $O(2mn)$ model. At
leading order in the $\varepsilon$ expansion,

$\begin{split}\lambda^{(1)}_{\pm}&=\frac{1}{4(mn+1)D_{mn}}\Big{(}A_{mn}\pm
B_{mn}\sqrt{R_{mn}}\Big{)}\varepsilon\,,\\\ g^{(1)}_{\pm}&=\frac{1}{2\hskip
1.0ptD_{mn}}\Big{(}B_{mn}\mp 3\sqrt{R_{mn}}\Big{)}\varepsilon\,,\end{split}$
(4.10)

where

$\begin{split}A_{mn}&=2m^{2}n^{2}+14mn+m^{3}n+mn^{3}-11m^{2}-11n^{2}+36\,,\qquad
B_{mn}=m^{2}n+mn^{2}-5m-5n\,,\\\
D_{mn}&=2m^{2}n^{2}-16mn+m^{3}n+mn^{3}-8m^{2}-8n^{2}+108\,,\qquad
R_{mn}=m^{2}+n^{2}-10mn+24\,.\end{split}$ (4.11)

The two fixed points coincide when $R_{mn}=0$, in which case the upper bound
of [25] on the quantity $\lambda_{ijkl}\lambda_{ijkl}$ at leading order in the
$\varepsilon$ expansion, namely
$\lambda_{ijkl}\lambda_{ijkl}\leqslant\frac{1}{8}N\varepsilon^{2}$, is
saturated. Using [26] we find that the Diophantine equation $R_{mn}=0$ has an
infinite number of positive integer solutions given by (without loss of
generality we assume $m<n$)

$\begin{split}m_{i+1}&=n_{i}\,,\quad n_{i+1}=-m_{i}+10\hskip
1.0ptn_{i}\,,\quad i=1,2,\ldots,\\\ m_{1}&=1\,,\quad n_{1}=5\,.\end{split}$
(4.12)

The solution with smallest $N$ is $m=5$, $n=49$, $N=490$, since for $m=1$
$g_{\pm}^{(1)}$ is singular at $n=5$. When the $U_{\pm}$ fixed points coincide
they annihilate and move off to the complex $(\lambda,g)$ plane as discussed
in section 2. The solutions (4.12) correspond to $n^{+}$ of (2.2) at
$\varepsilon=0$.

To present results in compact form we will assume, without loss of generality,
that $m<n$ and present anomalous dimensions in a large-$n$ expansion up to
three loops but at leading order in $1/n$. The full unexpanded in $n$ results
are straightforward to compute with our methods and they are included in an
ancillary file. The anomalous dimension of $\phi$ is

$\gamma_{\phi,+}=\frac{m}{8n}\big{(}\varepsilon^{2}-\tfrac{1}{4}\hskip
1.0pt\varepsilon^{3}\big{)}\,,\qquad\gamma_{\phi,-}=\frac{(m-1)(m+1)}{8mn}\big{(}\varepsilon^{2}-\tfrac{1}{4}\hskip
1.0pt\varepsilon^{3}\big{)}\,.$ (4.13)

The dimension of $\phi$ at the fixed points $U_{\pm}$ is equal to
$\Delta_{\pm}=1-\frac{1}{2}\hskip 1.0pt\varepsilon+\gamma_{\pm}$.

For the $\phi^{2}$ operator (leading scalar in the irrep $S_{\text{even}}$
above) we find

$\gamma_{\sigma,+}=\varepsilon-\frac{m}{n}\big{(}3\hskip
1.0pt\varepsilon-\tfrac{13}{4}\hskip 1.0pt\varepsilon^{2}+\tfrac{3}{16}\hskip
1.0pt\varepsilon^{3}\big{)}\,,\qquad\gamma_{\sigma,-}=\frac{(m-1)(m+1)}{mn}\big{(}3\hskip
1.0pt\varepsilon-\tfrac{13}{4}\hskip 1.0pt\varepsilon^{2}+\tfrac{3}{16}\hskip
1.0pt\varepsilon^{3}\big{)}\,.$ (4.14)

Finally, for the $\rho_{ij}\phi_{i}\phi_{j}$ operators we find a decomposition
into five distinct cases, with

$\displaystyle\gamma_{\rho_{1},+}$
$\displaystyle=\varepsilon-\frac{m}{n}\big{(}\varepsilon-\tfrac{5}{4}\hskip
1.0pt\varepsilon^{2}+\tfrac{3}{16}\hskip
1.0pt\varepsilon^{3}\big{)}\,,\quad\gamma_{\rho_{1},-}=\varepsilon-\frac{1}{mn}\big{[}(m^{2}-5)\varepsilon-\tfrac{5}{4}(m^{2}-5)\varepsilon^{2}+\tfrac{1}{16}(3m^{2}-11)\varepsilon^{3}\big{]}\,,$
$\displaystyle\gamma_{\rho_{2},+}$
$\displaystyle=\frac{m}{n}\big{(}\varepsilon-\tfrac{1}{4}\hskip
1.0pt\varepsilon^{2}-\tfrac{5}{16}\hskip
1.0pt\varepsilon^{3}\big{)}\,,\quad\gamma_{\rho_{2},-}=\frac{(m-1)(m+1)}{mn}\big{(}\varepsilon-\tfrac{1}{4}\hskip
1.0pt\varepsilon^{2}-\tfrac{5}{16}\hskip 1.0pt\varepsilon^{3}\big{)}\,,$
$\displaystyle\gamma_{\rho_{3},+}$
$\displaystyle=\frac{m}{4n}\big{(}\varepsilon^{2}-\tfrac{1}{4}\hskip
1.0pt\varepsilon^{3}\big{)}\,,\quad\gamma_{\rho_{3},-}=-\frac{1}{mn}\big{[}\varepsilon-\tfrac{1}{4}(m^{2}+1)\varepsilon^{2}+\tfrac{1}{16}(m^{2}-5)\varepsilon^{3}\big{]}\,,$
$\displaystyle\gamma_{\rho_{4},+}$
$\displaystyle=\frac{1}{n}\big{[}\varepsilon+\tfrac{1}{4}(m-2)\varepsilon^{2}-\tfrac{1}{16}(m+4)\varepsilon^{3}\big{]}\,,\quad\gamma_{\rho_{4},-}=\frac{m-1}{mn}\big{[}\varepsilon+\tfrac{1}{4}(m-1)\varepsilon^{2}-\tfrac{1}{16}(m+5)\varepsilon^{3}\big{]}\,,$
$\displaystyle\gamma_{\rho_{5},+}$
$\displaystyle=-\frac{1}{n}\big{[}\varepsilon-\tfrac{1}{4}(m+2)\varepsilon^{2}-\tfrac{1}{16}(m-4)\varepsilon^{3}\big{]}\,,\quad\gamma_{\rho_{5},-}=-\frac{m+1}{mn}\big{[}\varepsilon-\tfrac{1}{4}(m+1)\varepsilon^{2}+\tfrac{1}{16}(m-5)\varepsilon^{3}\big{]}\,.$
(4.15)

These correspond to the leading scalar operators in the irreps
$RS_{\text{even}}$, $SR_{\text{even}}$, $RR_{\text{even}}$,
$TT_{\text{even}}$, $AA_{\text{even}}$ above, respectively. The dimensions of
the quadratic in $\phi$ operators at the fixed points $U_{\pm}$ are equal to
$\Delta_{\sigma,\rho,\pm}=2-\varepsilon+\gamma_{\sigma,\rho,\pm}$. Since
$\gamma_{\sigma,+}$ and $\gamma_{\rho_{1},\pm}$ are equal to $\varepsilon$ at
$n\to\infty$, there should exist a large $n$ expansion (independent of the
$\varepsilon$ expansion studied in this section) in which the scaling
dimensions of these operators at the corresponding fixed points go to 2 in the
infinite-$n$ limit. In the next section, using the analytic bootstrap, we will
compute the $\frac{1}{n}$ corrections for $d$ arbitrary. We will see that
indeed the $\varepsilon$ and large-$n$ expansions agree in their region of
overlapping validity. Our non-perturbative numerical bootstrap results in
$d=3$ below are also consistent with the existence of the large-$n$ limit.

When $m=n$ from (4.11) we have $R_{nn}=8(3-n^{2})$ and thus the $\varepsilon$
expansion gives a unitary fixed point for $n^{2}\leqslant 3$.151515This holds
for $\varepsilon$ infinitesimal. When $\varepsilon$ is finite, this value is
expected to change. For positive integer $n$ this is only satisfied for the
uninteresting case $n=1$.

When $m=n>1$ and we treat the two $U(n)$ factors as indistinguishable, then
the indices $u,v,w$ in (4.9) take only two values (as opposed to three in the
$m\neq n$ case). As a result, the $\rho_{ij}\phi_{i}\phi_{j}$ operators
decompose into four distinct cases (as opposed to the five in (4)), due to the
fact that $RS_{\text{even}}$ and $SR_{\text{even}}$ can no longer be
distinguished. The anomalous dimensions of the corresponding operators are
real when $n^{2}\leqslant 3$.

If we take $m,n$ large with $m/n$ held fixed, then we observe that
$\Delta_{\phi,-}=\Delta_{\phi,+}$, $\Delta_{\sigma,-}+\Delta_{\sigma,+}=d$ and
$\Delta_{\rho_{i},-}=\Delta_{\rho_{i},+}$ for $i=1,\ldots,5$. Assuming $m,n>0$
and $n=\alpha\hskip 1.0ptm$, then $R_{mn}$ in (4.10), (4.11) is positive when
$\alpha<5-2\sqrt{6}\sqrt{1-1/m^{2}}$ or $\alpha>5+2\sqrt{6}\sqrt{1-1/m^{2}}$.
If $m$ is assumed large and $n>m$, then we may focus in the region
$\alpha\gtrsim\alpha_{c}=5+2\sqrt{6}$, where the fixed points $U_{\pm}$ are
unitary. The value of $\alpha_{c}$ has an $\varepsilon$ expansion that follows
from (2.2). As we will see below, the numerical conformal bootstrap provides
evidence that for large $m,n$ the fixed points $U_{\pm}$ either remain unitary
down to $\alpha_{c}=1$ in $d=3$, or their non-unitarities are small enough to
still allow the bootstrap to produce a kink. Lastly, let us mention that the
results in (4) have the same strict double scaling limit, $\alpha=m/n$ fixed
with $m$ and $n$ large, with the corresponding results in [19], differing only
in subleading $1/n$ corrections. We will see this reflected in one of our
plots later (see the discussion around Fig. 8).

##  5 Results in the large-$\boldsymbol{n}$ expansion

In this section we use analytic bootstrap methods as outlined in [19, Sec. 3]
to determine scaling dimensions of operators at leading order in $1/n$ as a
function of the spacetime dimension $d$. Our basic assumption is that there
exist auxiliary Hubbard–Stratonovich fields as leading scalar operators in
some irreps. This assumption can be verified a posteriori by means of a
comparison with the $\varepsilon$ expansion results of the previous section.

The essential ingredient needed for our application of the analytic bootstrap
method is the crossing equation. The four-point function of $\phi$ is written
in the form

$\langle\phi_{i}(x_{1})\phi_{j}(x_{2})\phi_{k}(x_{3})\phi_{l}(x_{4})\rangle=\frac{1}{(x_{12}^{2}x_{34}^{2})^{\Delta_{\phi}}}\sum_{I}P^{I}_{ijkl}\,\mathcal{G}_{I}(u,v)\,,$
(5.1)

where $P^{I}_{ijkl}$ are the projectors (3.18), $u,v$ are the usual cross-
ratios defined by

$u=\frac{x_{12}^{2}x_{34}^{2}}{x_{13}^{2}x_{24}^{2}}\,,\qquad
v=\frac{x_{14}^{2}x_{23}^{2}}{x_{13}^{2}x_{24}^{2}}\,,\qquad
x_{ij}=x_{i}-x_{j}\,,$ (5.2)

and

$\mathcal{G}_{I}(u,v)=\sum_{\mathcal{O}_{I}}c_{\phi\phi\mathcal{O}_{I}}^{2}G_{\Delta_{\mathcal{O}_{I}},\ell_{\mathcal{O}_{I}}}(u,v)\,,$
(5.3)

with $G_{\Delta,\ell}(u,v)$ the usual conformal block [27, 28, 29].161616Our
conventions for the normalization of the conformal block are those of [30].
The crossing equation follows from exchanging operators at $x_{2}$ and $x_{4}$
and can we written as

$\mathcal{G}_{I}(u,v)=M_{IJ}\left(\frac{u}{v}\right)^{\Delta_{\phi}}\mathcal{G}_{J}(v,u)\,,$
(5.4)

where the explicit form of the $12\times 12$ matrix $M_{IJ}$ is easy to work
out and is included in an ancillary file.

Imposing that the leading spin-two operator in the irrep $S_{\text{even}}$ is
the stress-energy tensor with dimension $d$, and that the leading spin-one
operators in the irreps $RS_{\text{odd}}$ and $SR_{\text{odd}}$ are conserved
currents with dimensions $d-1$, we may determine the dimensions of operators
at leading order in $1/n$ in each of the $U_{\pm}$ fixed points. Here we
report only the leading scalar operators. We have included operators of higher
spin in an ancillary file. First let us define $\mu=d/2$ and

$\eta_{1}=\frac{(\mu-2)\Gamma(2\mu-1)\sin(\pi\mu)}{\pi\Gamma(\mu)\Gamma(\mu+1)}\,.$
(5.5)

At $U_{+}$ we find, at leading order in $1/n$,

$\displaystyle\Delta_{\phi,+}$
$\displaystyle=\mu-1+\frac{m}{2}\frac{\eta_{1}}{n}$
$\displaystyle\overset{\mathrm{3d}}{=}\frac{1}{2}+\frac{2m}{3\pi^{2}n}\,,$
$\displaystyle\Delta_{S,+}$
$\displaystyle=2-\frac{2(\mu-1)(2\mu-1)m}{2-\mu}\frac{\eta_{1}}{n}$
$\displaystyle\overset{\mathrm{3d}}{=}2-\frac{16m}{3\pi^{2}n}\,,$
$\displaystyle\Delta_{RS,+}$
$\displaystyle=2-\frac{2(\mu-1)^{2}m}{2-\mu}\frac{\eta_{1}}{n}$
$\displaystyle\overset{\mathrm{3d}}{=}2-\frac{4m}{3\pi^{2}n}\,,$
$\displaystyle\Delta_{SR,+}$ $\displaystyle=2\Delta_{\phi+}+\frac{\mu
m}{2-\mu}\frac{\eta_{1}}{n}$
$\displaystyle\overset{\mathrm{3d}}{=}1+\frac{16m}{3\pi^{2}n}\,,$
$\displaystyle\Delta_{RR,+}$ $\displaystyle=2\Delta_{\phi+}$
$\displaystyle\overset{\mathrm{3d}}{=}1+\frac{4m}{3\pi^{2}n}\,,$
$\displaystyle\Delta_{TT,+}$
$\displaystyle=2\Delta_{\phi+}+\frac{\mu}{2-\mu}\frac{\eta_{1}}{n}$
$\displaystyle\overset{\mathrm{3d}}{=}1+\frac{4(m+3)}{3\pi^{2}n}\,,$
$\displaystyle\Delta_{AA,+}$
$\displaystyle=2\Delta_{\phi+}-\frac{\mu}{2-\mu}\frac{\eta_{1}}{n}$
$\displaystyle\overset{\mathrm{3d}}{=}1+\frac{4(m-3)}{3\pi^{2}n}\,.$ (5.6)

At $U_{-}$ and again at leading order in $1/n$ we find

$\displaystyle\Delta_{\phi,-}$
$\displaystyle=\mu-1+\frac{(m-1)(m+1)}{2m}\frac{\eta_{1}}{n}$
$\displaystyle\overset{\mathrm{3d}}{=}\frac{1}{2}+\frac{2(m-1)(m+1)}{3\pi^{2}mn}\,,$
$\displaystyle\Delta_{S,-}$
$\displaystyle=2\Delta_{\phi-}+\frac{\mu(4\mu-5)(m-1)(m+1)}{(2-\mu)m}\frac{\eta_{1}}{n}$
$\displaystyle\overset{\mathrm{3d}}{=}1+\frac{16(m-1)(m+1)}{3\pi^{2}mn}\,,$
$\displaystyle\Delta_{RS,-}$
$\displaystyle=2-\frac{2}{2-\mu}\Big{[}(\mu-1)^{2}m-\frac{4\mu^{2}-6\mu+1}{m}\Big{]}\frac{\eta_{1}}{n}$
$\displaystyle\overset{\mathrm{3d}}{=}2-\frac{4(m-2)(m+2)}{3\pi^{2}mn}\,,$
$\displaystyle\Delta_{SR,-}$
$\displaystyle=2\Delta_{\phi-}+\frac{\mu(m-1)(m+1)}{(2-\mu)m}\frac{\eta_{1}}{n}$
$\displaystyle\overset{\mathrm{3d}}{=}1+\frac{16(m-1)(m+1)}{3\pi^{2}mn}\,,$
$\displaystyle\Delta_{RR,-}$
$\displaystyle=2\Delta_{\phi-}-\frac{\mu}{(2-\mu)m}\frac{\eta_{1}}{n}$
$\displaystyle\overset{\mathrm{3d}}{=}1+\frac{4(m-2)(m+2)}{3\pi^{2}mn}\,,$
$\displaystyle\Delta_{TT,-}$
$\displaystyle=2\Delta_{\phi-}+\frac{\mu(m-1)}{(2-\mu)m}\frac{\eta_{1}}{n}$
$\displaystyle\overset{\mathrm{3d}}{=}1+\frac{4(m-1)(m+4)}{3\pi^{2}mn}\,,$
$\displaystyle\Delta_{AA,-}$
$\displaystyle=2\Delta_{\phi-}-\frac{\mu(m+1)}{(2-\mu)m}\frac{\eta_{1}}{n}$
$\displaystyle\overset{\mathrm{3d}}{=}1+\frac{4(m-4)(m+1)}{3\pi^{2}mn}\,.$
(5.7)

We have checked that the $\mu$-dependent results are consistent with (4.13),
(4.14) and (4) when expanded in $\varepsilon$ with $\mu=2-\varepsilon/2$. To
our knowledge the large-$n$ results presented here are new.

##  6 Numerical bootstrap results

We start this section by noting that in the various plots we will label bounds
for $U(m)\times U(n)$ theories by $U_{m,n}$ for brevity. We emphasize here
that, since our bootstrap bounds are obtained with the four-point function of
$\phi$ only, they apply to theories with $U(m)\times U(n)$, $[U(m)\times
U(n)]/U(1)$, and $SU(m)\times SU(n)$ global symmetry.171717$[U(m)\times
U(n)]/U(1)$ bounds are necessarily weaker than corresponding $U(m)\times U(n)$
bounds, but the two may also coincide. For the case $m=n$ we may treat the two
$U(n)$ factors as distinguishable or indistinguishable and we will explicitly
mention our choice in context. With the latter choice those are bounds for
theories with $U(n)^{2}\rtimes\mathbb{Z}_{2}$ global symmetry, and we will
label these by $\widehat{U}_{n,n}$ in the corresponding
plots.181818$U(n)\times U(n)$ bounds are necessarily weaker than corresponding
$U(n)^{2}\rtimes\mathbb{Z}_{2}$ bounds, but the two may also coincide.
Whenever squares and circles appear in the plots, these correspond to the
location of the $U_{+}$ and $U_{-}$ fixed points of the corresponding
Lagrangian theories, respectively, according to the large-$n$ results (5) and
(5). Straight lines connecting squares and circles are added to illustrate
that the connected fixed points correspond to the same $m,n$. It is emphasized
that the fixed points $U_{\pm}$ are added in plots without examining the issue
of their existence as unitary fixed points. The parameters used in the
numerics are discussed in Appendix Appendix C. Numerical parameters. The
crossing equations are included in an ancillary file.

In Fig. 2 we present bounds for the dimension of the first scalar operator in
the $RS$ irrep as a function of the dimension of $\phi$. We work with
$U(2)\times U(n)$ theories, but similar behavior is seen in bounds at higher
$m$. These bounds display sharp kinks at large $n$. Using our analytic
large-$n$ results of the previous section, we see that the $U_{-}$ fixed
points are responsible for these kinks.191919That being said, we observed that
extracting the spectrum at e.g. the $U(2)\times U(20)$ kink, there was no sign
of the “$\phi^{4}$”-type singlet with dimension $\Delta_{S}=2+O(1/n)$ expected
from the large-$n$ description. A similar observation was made for the bound
corresponding to the $O(2)\times O(10)$ anti-chiral fixed point in [19]. Let
us also mention that operators have been known to be missing from the
extracted spectrum even in theories which are under very good control in the
numerical bootstrap, such as the Ising model. For example, in Figure 11 of
[31], while the second and fourth $\mathbb{Z}_{2}$-odd spin-$0$ operators are
captured by the numerics and agree with perturbative estimates, the third
operator is not seen.

Fig. 2: Upper bound on the dimension of the first scalar $RS$ operator in the
$\phi\times\phi$ OPE as a function of the dimension of $\phi$. Areas above the
curves are excluded in the corresponding theories. The locations of the fixed
points as predicted by the large-$n$ results (5) and (5) for $n=5,10,20$ are
also given as squares and circles for the $U_{+}$ and $U_{-}$ fixed points,
respectively. (The lines between squares and circles are added to illustrate
that the corresponding fixed points have the same symmetry.)

An interesting question is whether there exists a kink in the $U(2)\times
U(2)$ theory. The corresponding $\Delta_{RS}$ bound is seen in Fig. 3. There
we see that the $\Delta_{RS}$ bound for the $U(2)\times U(2)$ theory is much
stronger than the $\Delta_{RS}$ bound for the $U(2)\times U(3)$ theory. The
difference is much more significant than that between the $U(3)\times U(3)$
and $U(3)\times U(4)$ theories, which are also shown in Fig. 3 for comparison.
This indicates that the $U(2)\times U(3)$ theory is sensitive to a potential
fixed point which has no extension to the $U(2)\times U(2)$ theory. Indeed, a
kink is clearly forming in the $U(2)\times U(3)$ bound, while no kink at all
is present in the $U(2)\times U(2)$ bound.202020We have obtained the
$U(2)\times U(2)$ $\Delta_{RS}$ bound up to $\Delta_{\phi}=0.7$ and no kink is
seen. Comparing with Fig. 2, it seems plausible that the kink in the
$U(2)\times U(3)$ bound in Fig. 3 is due to the corresponding $U_{-}$ fixed
point. Note that [6] estimated $n^{+}(2)=4.373(18)$ in $d=3$, which if correct
would imply that both $U(2)\times U(4)$ and $U(2)\times U(3)$ kinks in Fig. 2
correspond to non-unitary fixed points.

Fig. 3: Upper bound on the dimension of the first scalar $RS$ operator in the
$\phi\times\phi$ OPE as a function of the dimension of $\phi$. Areas above the
curves are excluded in the corresponding theories. The $U(2)\times U(2)$ and
$U(3)\times U(3)$ bounds are obtained assuming the two factors of the global
symmetry group are distinguishable. The location of the fixed points $U_{+}$
(square) and $U_{-}$ (circle) as predicted by the large-$n$ results (5) and
(5) for the $U(2)\times U(3)$ theory are also shown. (The line in between is
added to illustrate that the corresponding fixed points have the same
symmetry.)

Our conclusion from the $\Delta_{RS}$ bounds is that the $U(2)\times U(2)$
theory does not have a unitary $U_{-}$ fixed point. This is consistent with
expectations from perturbative methods [5, 6]. We do stress, however, that
this does not in principle exclude some other fixed point of a different type
for this symmetry (e.g. a fixed point inaccessible through standard
perturbation theory); see [9], [7] and also our discussion below pertaining to
Fig. 6.

In Fig. 4 we plot bounds for the dimension of the first scalar operator in the
$SR$ irrep as a function of the dimension of $\phi$, again for $U(2)\times
U(n)$ theories for various values of $n$. Here we do not see kinks as sharp as
those of Fig. 2, but at large $n$ we do observe changes in slope that are
saturated by the $U_{+}$ fixed point. This is more clear for the $U(2)\times
U(20)$ theory in Fig. 5, where we plot bounds on the dimensions of the leading
operators in all five scalar non-singlet irreps that appear in the
$\phi\times\phi$ OPE. The blue circle in each plot in Fig. 5 corresponds to
the $O(80)$ model, which saturates the $\Delta_{RR}$, $\Delta_{TT}$ and
$\Delta_{AA}$ bounds. Note that the $\Delta_{TT}$ bound in Fig. 5 is saturated
both by the $O(80)$ model and $U_{-}$ for different values of $\Delta_{\phi}$,
without sharp kinks in either case.

In Fig. 6 we plot bounds on the leading scalar non-singlet operators in the
case of $U(2)\times U(2)$ symmetry with the $O(8)$ fixed point marked in blue.
As we have already mentioned, the $\Delta_{RS}$ bound does not display a kink,
which is interpreted as the absence of a unitary $U_{+}$ fixed point for
$m=n=2$. However, the $\Delta_{AA}$ bound displays a kink around
$\Delta_{\phi}=0.53$. This kink was first observed in the $O(2)\times O(4)$
studies of [7, 19] (see [7, Fig. 3] and [19, Fig. 3]).212121The bounds in Fig.
6 are valid for 3D CFTs with either $U(2)\times U(2)$ or $[U(2)\times
U(2)]/U(1)\simeq SU(2)\times SU(2)\times U(1)\simeq SO(4)\times SO(2)$ global
symmetry. The fact that the $\Delta_{AA}$ bound in Fig. 6 coincides with a
bound obtained for 3D CFTs with $O(4)\times O(2)$ global symmetry means that
3D CFTs with $U(2)\times U(2)$ symmetry, should any exist, lie in the allowed
region of the $\Delta_{AA}$ bound.

Fig. 4: Upper bound on the dimension of the first scalar $SR$ operator in the
$\phi\times\phi$ OPE as a function of the dimension of $\phi$. Areas above the
curves are excluded in the corresponding theories. The locations of the fixed
points as predicted by the large-$n$ results (5) and (5) for $n=5,10,20$ are
also given as squares and circles for the $U_{+}$ and $U_{-}$ fixed points,
respectively. (The lines between squares and circles are added to illustrate
that the corresponding fixed points have the same symmetry.)

In Fig. 7 we plot bounds on the leading scalar non-singlet operators for 3D
CFTs with $U(3)\times U(3)$ symmetry. The $O(18)$ fixed point is marked in
blue. Here we observe no kinks at all, indicating that the $U_{\pm}$ fixed
points of the $\varepsilon$ expansion do not appear to survive as unitary
fixed points for $m=n=3$.

Fig. 5: Upper bounds on the dimensions of the first scalar operators in the
various irreps that appear in the $\phi\times\phi$ OPE as functions of the
dimension of $\phi$ for $U(2)\times U(20)$ global symmetry. The blue circle
marks the location of the $O(80)$ fixed point, which for all plots in the
vertical axis is given by the dimension of the leading scalar two-index
traceless symmetric operator in that theory. The $U_{\pm}$ fixed points are
also shown as green circles ($U_{-}$) and squares ($U_{+}$), connected by a
thin line to indicate that they have the same global symmetry. Areas above the
curves are excluded. Fig. 6: Upper bounds on the dimensions of the first
scalar operators in the various irreps that appear in the $\phi\times\phi$ OPE
as functions of the dimension of $\phi$ for $U(2)\times U(2)$ global symmetry,
where the two $U(2)$ factors are considered distinguishable. The blue circle
marks the location of the $O(8)$ fixed point, which for all plots in the
vertical axis is given by the dimension of the leading scalar two-index
traceless symmetric operator in that theory. Areas above the curves are
excluded. The $\Delta_{SR}$ bound is identical to the $\Delta_{RS}$ one. Fig.
7: Upper bounds on the dimensions of the first scalar operators in the various
irreps that appear in the $\phi\times\phi$ OPE as functions of the dimension
of $\phi$ for $U(3)\times U(3)$ global symmetry, where the two $U(3)$ factors
are considered distinguishable. The blue circle marks the location of the
$O(18)$ fixed point, which for all plots in the vertical axis is given by the
dimension of the leading scalar two-index traceless symmetric operator in that
theory. Areas above the curves are excluded. The $\Delta_{SR}$ bound is
identical to the $\Delta_{RS}$ one.

From our results so far it appears that the $U_{\pm}$ fixed points of the
$\varepsilon$ expansion are not of importance for the chiral phase transition
of two- and three-flavor massless QCD. Using our setup we can extend this
question to QCD with (parametrically) many massless flavors. In Fig. 8 we
present bounds on the dimension of the leading scalar operator in the $RSSR$
irrep, where we assume $m=n$ and that the two $U(n)$ factors are
indistinguishable. We have also obtained $\Delta_{RS}$ and $\Delta_{SR}$
bounds assuming that the two $U(n)$ factors are distinguishable, which are
identical between themselves and differ slightly from the ones shown in Fig. 8
only for $n=2,3$. For large $n$ the $\Delta_{RS}$, $\Delta_{SR}$ bounds
coincide with the corresponding $\Delta_{RSSR}$ bound of Fig. 8. As we
observe, despite the absence of a kink for low $n$ a kink is clearly seen at
large $n$.222222A similar situation has been encountered in the bootstrap of
four-point functions of scalar adjoint operators in 3D CFTs with $SU(N)$
global symmetry [32, Fig. 2]. From Fig. 9 we expect this kink to be due to the
$U_{-}$ fixed point.

Fig. 8: Upper bound on the dimension of the first scalar $RSSR$ operator in
the $\phi\times\phi$ OPE as a function of the dimension of $\phi$. The global
symmetry group for the various bounds here is $U(n)^{2}\rtimes\mathbb{Z}_{2}$.
Areas above the curves are excluded in the corresponding theories. Fig. 9:
Upper bound on the dimension of the first scalar $RS$ operator in the
$\phi\times\phi$ OPE as a function of the dimension of $\phi$. The
$U_{1000,1000}$ bound has been obtained treating the two $U(1000)$ factors as
distinguishable; however, it is identical to the $\widehat{U}_{1000,1000}$
bound in Fig. 8. Areas above the curves are excluded in the corresponding
theories. The location of the $U(100)\times U(1000)$ $U_{-}$ fixed point as
predicted by the large-$n$ results (5) theory is also shown. The $U(100)\times
U(1000)$ $U_{+}$ fixed point is at the same location. Fig. 10: Upper bounds
on the dimensions of the first scalar operators in the various irreps that
appear in the $\phi\times\phi$ OPE as functions of the dimension of $\phi$ for
$U(1000)\times U(1000)$ global symmetry, where the two $U(1000)$ factors are
considered distinguishable. Areas above the curves are excluded. The
$\Delta_{SR}$ bound is identical to the $\Delta_{RS}$ one.

We note that the $m=n=100$ bound in Fig. 8 is essentially identical to the one
in [33, Fig. 12]. This explains the origin of the kink seen in that bound.
Note that the operator $Z^{ab}_{ij}$ in that work is equivalent (when
$a,b=1,2$) to a bifundamental operator $\phi_{ij}$ of $O(n)\times O(n)$ in the
case where the $O(n)$ factors are indistinguishable. Then, following our
discussion at the end of section 4, it becomes clear why these bounds can
coincide at large $n$.

The $\widehat{U}_{1000,1000}$ kink in Fig. 9 (which is identical to the
$\Delta_{RS}$ kink in Fig. 10) is also relevant for the possible existence of
a $U(m)\times U(n)$ 3D CFT at $m,n$ large with $m/n$ fixed and close to 1. As
we discussed at the bottom of section 4, when $m,n$ are both large but $m/n$
is sufficiently small, the $U_{\pm}$ fixed points are unitary. Since the kink
in Fig. 9 survives as we increase the ratio $m/n$ towards 1, we may conclude
that either the $U_{\pm}$ fixed points survive as unitary fixed points in that
case, or the possible non-unitarities are too small to stop the kink from
forming. Therefore, QCD232323Or, more appropriately, Yang–Mills theory with a
sufficiently large number of colors if one wants it to be confining. with 1000
massless flavors may undergo a (near) second order chiral phase transition due
to the presence of the $U_{+}$ fixed point. It would be interesting to compute
the value of $n^{+}(1000)$ in the $\varepsilon$ expansion using the results of
[6] and see which of the two pictures it corroborates. If the fixed point is
indeed non-unitary this could give us a quantitative measure242424Since one
may tune the size of the non-unitarity by tuning $1/n$. of the sensitivity to
non-unitarities in the bootstrap.

Let us comment on Fig. 10. We have already discussed the kink in the
$\Delta_{RS}$ bound. We additionally observe that the
$\Delta_{RR},\Delta_{TT},\Delta_{AA}$ bounds essentially coincide. This
coincidence is also seen in the large-$n$ results (5) and (5) when we take $m$
large and equal to $n$, although taking $m$ large in those results is not
justified. We have also compared with the large-$m,n$ $O(m)\times O(n)$
results of [19] and we find the same set of scaling dimensions.

In Fig. 11 we show the $RS$ exclusion bound for $m=3$ fixed and increasing
$n$. At large $n$ these bounds have kinks that are saturated by the
corresponding $U_{+}$ fixed points. For $n=3,4$ no kinks are found up to
$\Delta_{\phi}=0.54$, and so we expect $n^{+}(3)>4$. Nevertheless, the
$U(3)\times U(3)$ bound does have a kink as we see in Fig. 12. This kink
appears to be unrelated to the $U_{+}$ fixed point of the $\varepsilon$
expansion. As we see from Fig. 13 this kink disappears as we increase $n$ with
$m=3$ fixed. Its relevance for the nature of the chiral phase transition of
QCD remains to be seen, but its existence opens the possibility that it may be
second order. We stress, however, that this fixed point need not necessarily
be due to a multi-scalar theory, but may also be due to a gauge theory or a
Gross–Neveu–Yukawa (GNY) theory.252525As it is located at values of
$\Delta_{\phi}$ larger than the typical ones expected for multi-scalar
theories ($0.5+\text{corrections}$). We remind the reader that, for example in
GNY theories, one obtains a correction to the anomalous dimension of $\phi$ an
order earlier in perturbation theory (at one loop instead of two loops as in a
pure scalar field theory).

Fig. 11: Upper bound on the dimension of the first scalar $RS$ operator in
the $\phi\times\phi$ OPE as a function of the dimension of $\phi$. Areas above
the curves are excluded in the corresponding theories. The locations of the
fixed points as predicted by the large-$n$ results (5) and (5) for
$n=3,4,10,20$ are also given as squares and circles for the $U_{+}$ and
$U_{-}$ fixed points, respectively. The lines between squares and circles (for
$n=10,20$) are added to illustrate that the corresponding fixed points have
the same symmetry. These plots were run with parameters “ _D_ ” in Appendix
Appendix C. Numerical parameters. Fig. 12: Upper bound on the dimension of
the first scalar $RS$ operator in the $\phi\times\phi$ OPE as a function of
the dimension of $\phi$. The area above the curves is excluded. Fig. 13:
Upper bound on the dimension of the first scalar $RS$ operator in the
$\phi\times\phi$ OPE as a function of the dimension of $\phi$. Areas above the
curves are excluded in the corresponding theories.

Before concluding, let us discuss a few more plots that present features which
may be of interest to the bootstrap in general. In Fig. 14 we present bounds
on operators in various representations of the global symmetry for the
$U(3)\times U(20)$ CFTs. There are a couple of features that stand out.
Firstly, the exclusion bound for the $l=1$ $RR$ operator is almost exactly
saturated by the $U_{-}$ fixed point, even though the plot is essentially a
straight line, absent of even the mildest feature. Secondly, the bound on the
$l=0$ $TT$ operator is also saturated very well by the $U_{+}$ fixed point,
albeit in this case it does have a very minor feature (a very minor change of
slope). In fact, throughout this work, we found that the $TT$ bound was always
saturated very well by the analytic predictions despite only having a very
weak feature. These examples show that a lot of mundane looking bootstrap
plots may actually be much richer than initially thought. To reiterate, we saw
explicitly that the $TT$ bound is saturated by not just one, but two fixed
points. Lastly, in Fig. 15 we plot the $TA$ exclusion bound as a function of
increasing spin. For $l=1$ we see that the bound is saturated by the $U_{-}$
fixed point. Then, at $l=3$ the bound is almost saturated by three distinct
fixed points, again despite having no feature. As the spin is further
increased the agreement becomes progressively worse, which may be due to loss
of constraining power.

Fig. 14: Upper bound on the dimension of various operators in the
$\phi\times\phi$ OPE as a function of the dimension of $\phi$ for $m=3$ and
$n=20$. The spin of each operator is labeled by a superscript. Areas above the
curves are excluded in the corresponding theories. The locations of the fixed
points as predicted by the large-$n$ results (5) and (5) are also given as
squares and circles for the $U_{+}$ and $U_{-}$ fixed points, respectively.
The lines between squares and circles are added to illustrate that the
corresponding fixed points have the same symmetry. The blue circle marks the
location of the $O(120)$ fixed point. The first two plots were run with
parameters “ _B_ ” and the rest with parameters “ _C_ ” in Appendix Appendix
C. Numerical parameters. Fig. 15: Upper bound on the dimension of $AT$
operators of various spins in the $\phi\times\phi$ OPE as a function of the
dimension of $\phi$ for $m=3$ and $n=20$. The spin of each operator is labeled
by a superscript. Areas above the curves are excluded in the corresponding
theories. The locations of the fixed points as predicted by the large-$n$
results (5) and (5) are also given as squares and circles for the $U_{+}$ and
$U_{-}$ fixed points, respectively. The lines between squares and circles are
added to illustrate that the corresponding fixed points have the same
symmetry. The blue circle marks the location of the $O(120)$ fixed point.
These plots were run with parameters “ _B_ ” in Appendix Appendix C.
Numerical parameters.

##  7 Discussion and future directions

In the present work we performed a comprehensive study of $U(m)\times U(n)$
symmetric CFTs. This included perturbative computations in the
$\varepsilon=4-d$ and large-$n$ expansions, and non-perturbative computations
with the numerical conformal bootstrap in $d=3$. When $m=n$ we analyzed the
cases where the two $U(n)$ factors in $U(n)\times U(n)$ are considered
distinguishable or indistinguishable. Our study was motivated both by
phenomenological applications to the chiral phase transition of massless QCD,
as well as purely field theoretical considerations.

For the phenomenologically interesting values of $m=n=2$ for two-flavor
massless QCD, we found that as $n$ is lowered from $n=3$ to $n=2$ with $m=2$
fixed (see Fig. 3), there is a large change in the corresponding bootstrap
bound. More specifically, the bound becomes much stronger (i.e. it excludes
much more of parameter space) and lacks a kink that existed for larger values
of $n$ (see Fig. 2). This could be explained by the disappearance of a unitary
fixed point as we lower the value of $n$, such that the bootstrap may then
exclude the region in parameter space originally occupied by that fixed point.
If so, whatever fixed point was responsible for the feature in the $U(2)\times
U(n)$ exclusion bounds for large $n$, cannot exist for $U(2)\times U(2)$. We
stress, though, that this does not exclude the possibility of some other
$U(2)\times U(2)$ fixed point. For example, novel fixed points with
$O(n)\times O(2)$ symmetry were reported in [34] (remember that $O(4)\times
O(2)\sim SU(2)\times SU(2)\times U(1)$). Indeed a kink is observed in the
$\Delta_{AA}$ bound of Fig. 6, which may be attributed to a unitary CFT
unrelated to the fixed points found in the $\varepsilon$ expansion. The
$\Delta_{AA}$ bound of Fig. 6 is identical to a bound obtained for 3D CFTs
with $O(4)\times O(2)$ global symmetry [7, 8, 19]. We note that recently [10]
found the $O(4)\times O(2)$ transition to be first order.

For $m=n=3$, a case relevant for the chiral phase transition of three-flavor
massless QCD, we also observe a pronounced kink in our bootstrap bound; see
Fig. 12. Therefore, our work produces evidence that this transition may be
second order.

On the field theoretical side, we observed that computations in the large-$n$
limit provided very accurate predictions for the scaling dimensions of
numerous operators; see e.g. Fig. 5. Additionally, we found that large-$n$
results saturated bootstrap bounds, even in the complete absence of kinks.
Another interesting observation is that in Fig. 9 the unitary $m=100,n=1000$
fixed point seems to evolve into the $m=n=1000$ fixed point as $m$ is
increased. One interpretation of this is that for large $m,n$ a unitary
$U(m)\times U(n)$ CFT exists even when $m/n\to 1$. An alternative
interpretation is that in the limit $m/n\to 1$ with $n\rightarrow\infty$ the
non-unitarities become suppressed enough for the bootstrap bound to display a
kink. Note that the double scaling limit reported in this work also exists in
the results of [19].

Given our results, it would be interesting to extend the existing perturbative
data available for these theories. More precise perturbative predictions could
allow us to follow theories from infinitesimal values of the control parameter
to the physically interesting values (e.g. $\varepsilon=1$ or $m=n=2,3$). For
example, in [31] in the case of the Ising model within the context of the
$\varepsilon$ expansion, the perturbative data was found, a posteriori, to be
a very accurate description of the full non-perturbative theory (at least in
the absence of operator mixing). The extension of results for $U(m)\times
U(n)$ theories in the $\varepsilon$ expansion to higher order in perturbation
theory and more operators is possible with the results of [24]. Extension of
our large $n$ results to higher orders would also be desirable, especially
seeing the usefulness of even leading order results, when used in conjunction
with the numerical bootstrap. On the numerical side, we would like to be able
to precisely pinpoint the values of $m$ and $n$ which separate the different
regimes of fixed points (which we discussed in section 2).

Acknowledgments

We thank R. Pisarski for insightful correspondence, reading through the
manuscript and pointing out relevant literature. We also thank J. Henriksson
for helpful conversations on the analytic bootstrap. Additionally, we are
grateful to three anonymous referees whose reports helped improve this
manuscript.

AS is funded by the Royal Society under grant URF\R1\211417 “Advancing the
Conformal Bootstrap Program in Three and Four Dimensions.” The research work
of SRK received funding from the European Research Council (ERC) under the
European Union’s Horizon 2020 research and innovation programme (grant
agreement no. 758903). The numerical computations in this work have used
King’s College London’s Rosalind and CREATE [35] computing resources and the
INFN Pisa HPC cluster Theocluster Zefiro. Some computations in this paper have
been performed with the help of _Mathematica_ with the packages xAct [36] and
xTras [37].

## Appendix A. Kronecker tensor structures for complex fields

In this appendix we demonstrate how a product of two operators can be
decomposed onto irreps of $U(m)\times U(n)$ in the picture where we work with
complex fields. For simplicity we will start with just $U(n)$. The
generalization to $U(m)\times U(n)$ is trivial. The main utility of working
with complex fields is that the projectors take their simplest form possible,
namely as combinations of Kronecker deltas with the least number of indices
possible. The real field picture can also have its projectors expressed in
terms of Kronecker deltas, albeit at the cost of more indices. We hope that
presenting our projectors in three different pictures will make our work more
intuitive. In order to capture all irreps that can appear in the real field
notation, and hence not miss any information in the bootstrap algorithm, we
need to consider two OPEs, namely $\Phi_{i}^{\dagger}\times\Phi_{j}$ and
$\Phi_{i}\times\Phi_{j}$. Below we show how they may be decomposed onto
irreps:

$\begin{split}\Phi_{i}^{\dagger}\times\Phi_{j}&\sim\Big{(}\Phi_{i}^{\dagger}\Phi_{j}-\frac{1}{n}\delta_{ij}\Phi_{k}^{\dagger}\Phi_{k}\Big{)}+\frac{1}{n}\delta_{ij}\Phi_{k}^{\dagger}\Phi_{k}\,,\\\
\Phi_{i}\times\Phi_{j}&\sim(\Phi_{i}\Phi_{j}+\Phi_{j}\Phi_{i})+(\Phi_{i}\Phi_{j}-\Phi_{j}\Phi_{i})\,.\end{split}$
(A.1)

The first line in (A.1) shows the decomposition into the adjoint ($R$) and
singlet ($S$) representations, where as the second line shows the
decomposition into the symmetric ($T$) and antisymmetric representations
($A$). To read off the projectors from (A.1) it is useful to remember

$\displaystyle O^{X}_{ij}\sim P^{X}_{ijkl}\Phi_{k}\Phi_{l}\,,\qquad
O^{X}_{ij}\sim P^{X}_{ijkl}\Phi^{\dagger}_{k}\Phi_{l}\,,$ (A.2)

where $O^{X}$ is the exchanged field in some irrep, e.g.
$O^{T}_{12}\sim\Phi_{1}\Phi_{2}+\Phi_{2}\Phi_{1}$. The first relation in (A.2)
can be used to read off the $T$ and $A$ projectors, whereas the second can be
used for $S$ and $R$. Notice that we have implicitly assumed that fields are
inserted at different positions in order for antisymmetric irreps to not
vanish identically. The projectors can be read off as262626Note that we take
the correlator to be
$\langle\Phi_{i}^{\dagger}\Phi_{j}\Phi_{k}\Phi_{l}^{\dagger}\rangle$ which is
why the projector of the adjoint representation is equal to
$P^{R}_{ijkl}=\delta_{ik}\delta_{jl}-\frac{1}{n}\delta_{ij}\delta_{kl}$
instead of
$P^{R}_{ijkl}=\delta_{il}\delta_{jk}-\frac{1}{n}\delta_{ij}\delta_{kl}$.

$\displaystyle P^{S}_{ijkl}$
$\displaystyle=\frac{1}{n}\delta_{ij}\delta_{kl}\,,\qquad
P^{R}_{ijkl}=\delta_{ik}\delta_{jl}-\frac{1}{n}\delta_{ij}\delta_{kl}\,,$
(A.3) $\displaystyle P^{T}_{ijkl}$
$\displaystyle=\frac{1}{2}(\delta_{ik}\delta_{jl}+\delta_{il}\delta_{jk})\,,\qquad
P^{A}_{ijkl}=\frac{1}{2}(\delta_{ik}\delta_{jl}-\delta_{il}\delta_{jk})\,.$
(A.4)

The dimensions of the corresponding irreps are
$(1,(n-1)(n+1),\frac{1}{2}n(n+1),\frac{1}{2}n(n-1))$. As the reader may have
observed from the main text, or the next appendix, when going from the complex
field picture to the real field picture the above dimensions get multiplied by
a factor of $2$. For example $\frac{n(n+1)}{2}$ becomes $n(n+1)$. This is
because each of the initial $\frac{n(n+1)}{2}$ complex elements contains two
real elements. The last step now is to write down the projectors for
$U(m)\times U(n)$. This is trivial, in the sense that they are just products
of $U(m)$ with $U(n)$ projectors. We have

$\displaystyle P^{S}_{ijklmnop}$
$\displaystyle=P^{S}_{ijkl}P^{S}_{mnop}\,,\quad
P^{RS}_{ijklmnop}=P^{R}_{ijkl}P^{S}_{mnop}\,,\quad
P^{SR}_{ijklmnop}=P^{S}_{ijkl}P^{R}_{mnop}\,,$ (A.5) $\displaystyle
P^{RR}_{ijklmnop}$ $\displaystyle=P^{R}_{ijkl}P^{R}_{mnop}\,,\quad
P^{TT}_{ijklmnop}=P^{T}_{ijkl}P^{T}_{mnop}\,,\quad
P^{TA}_{ijklmnop}=P^{T}_{ijkl}P^{A}_{mnop}\,,$ $\displaystyle
P^{AT}_{ijklmnop}$ $\displaystyle=P^{A}_{ijkl}P^{T}_{mnop}\,,\quad
P^{AA}_{ijklmnop}=P^{A}_{ijkl}P^{A}_{mnop}\,.$

The sum rules that can be derived with the above projectors can be checked to
be completely equivalent to those derived from the projectors of real fields
outlined in the main text. Another observation is that, compared to real
fields, we do not need separate projectors for even and odd spins.

## Appendix B. Kronecker tensor structures for real fields

The projectors that correspond to a four-point function of real fields can be
intuitively presented in terms of Kronecker deltas if we add an additional
index. This form is useful since one may directly extract the form of
exchanged operators, as we will show. The form of exchanged operators is
useful to know since it can guide us with respect to assumptions we may
impose. Also, we expect it to be easier to work with in a mixed correlator
system. We start by labeling the real and complex parts of an operator
$\Phi_{i}$ with an index (we start with $U(n)$ for simplicity)

$\Phi_{i}=\phi^{1}_{i}+i\hskip 1.0pt\phi^{2}_{i}\,,$ (B.1)

where the upper case $\Phi$ denotes the complex operator and the lower case
$\phi$ denote real fields. We must now simply plug in (B.1) to the expressions
for the representations of the previous appendix. For simplicity we will do
this for the singlet representation, and then quote the results for rest of
the representations. Note that implicitly we consider the two external fields
of the OPE at different positions, for otherwise the antisymmetric
combinations would vanish identically. We have

$\Phi_{i}^{\dagger}\Phi_{i}=(\phi^{1}_{i}\phi^{1}_{i}+\phi^{2}_{i}\phi^{2}_{i})+i(\phi^{1}_{i}\phi^{2}_{i}-\phi^{2}_{i}\phi^{1}_{i})\,,$
(B.2)

where the first parenthesis corresponds to what was called $S_{\text{even}}$
in the main text, and the second parenthesis corresponds to what was called
$S_{\text{odd}}$. As expected $S_{\text{odd}}$ vanishes identically if we
don’t insert powers of derivatives between the operators. The projectors are
now very straightforward to write down by recalling the relation

$O^{X}_{ij;ab}=P^{X}_{ijkl;abcd}\hskip 1.0pt\phi^{a}_{i}\phi^{b}_{j}\,,$ (B.3)

where $X$ stands for some specific irrep and indices from the beginning of the
latin alphabet take the values $1,2$. Notice that (B.3) is simply the
statement that projectors must project products of operators onto irreps. We
have

$\begin{split}P^{S_{\text{even}}}_{ijkl;abcd}&=\frac{1}{2n}\delta_{ab}\delta_{cd}\delta_{ij}\delta_{kl}\,,\\\
P^{S_{\text{odd}}}_{ijkl;abcd}&=\frac{1}{2n}(\delta_{ac}\delta_{bd}-\delta_{ad}\delta_{bc})\delta_{ij}\delta_{kl}\,.\end{split}$
(B.4)

Indeed, one may confirm that, for example,

$O^{S_{\text{even}}}_{11;11}\sim(\phi^{1}_{i}\phi^{1}_{i}+\phi^{2}_{i}\phi^{2}_{i})\sim
P^{S_{\text{even}}}_{11kl;11cd}\hskip 1.0pt\phi^{c}_{k}\phi^{d}_{l}\,.$ (B.5)

This procedure can be repeated for the rest of the irreps. The resulting
projectors are

$\begin{split}P^{S_{\text{even}}}_{ijkl;abcd}&=\frac{1}{2n}\delta_{ab}\delta_{cd}\delta_{ij}\delta_{kl}\,,\\\
P^{S_{\text{odd}}}_{ijkl;abcd}&=\frac{1}{2n}(\delta_{ac}\delta_{bd}-\delta_{ad}\delta_{bc})\delta_{ij}\delta_{kl}\,,\\\
P^{R_{\text{even}}}_{ijkl;abcd}&=\tfrac{1}{2}\delta_{ab}\delta_{cd}\Big{(}\delta_{ik}\delta_{jl}-\frac{1}{n}\delta_{ij}\delta_{kl}\Big{)}\,,\\\
P^{R_{\text{odd}}}_{ijkl;abcd}&=\tfrac{1}{2}(\delta_{ac}\delta_{bd}-\delta_{ad}\delta_{bc})\Big{(}\delta_{ik}\delta_{jl}-\frac{1}{n}\delta_{ij}\delta_{kl}\Big{)}\,,\\\
P^{T_{\text{even}}}_{ijkl;abcd}&=\tfrac{1}{4}(\delta_{ac}\delta_{bd}+\delta_{ad}\delta_{bc}-\delta_{ab}\delta_{cd})(\delta_{ik}\delta_{jl}+\delta_{il}\delta_{jl})\,,\\\
P^{A_{\text{odd}}}_{ijkl;abcd}&=\tfrac{1}{4}(\delta_{ac}\delta_{bd}+\delta_{ad}\delta_{bc}-\delta_{ab}\delta_{cd})(\delta_{ik}\delta_{jl}-\delta_{il}\delta_{jl})\,.\end{split}$
(B.6)

The dimensions of the corresponding irreps are
$(1,1,(n-1)(n+1),(n-1)(n+1),n(n+1),n(n-1))$.

Using the above expressions, it is now trivial to write down the $U(m)\times
U(n)$ projectors:

$\displaystyle P^{S_{\text{even}}}_{ijklmnop;abcdefgh}$
$\displaystyle=P^{S_{\text{even}}}_{ijkl;abcd}\hskip
1.0ptP^{S_{\text{even}}}_{mnop;efgh}+P^{S_{\text{odd}}}_{ijkl;abcd}\hskip
1.0ptP^{S_{\text{odd}}}_{mnop;efgh}\,,$ $\displaystyle
P^{S_{\text{odd}}}_{ijklmnop;abcdefgh}$
$\displaystyle=P^{S_{\text{even}}}_{ijkl;abcd}\hskip
1.0ptP^{S_{\text{odd}}}_{mnop;efgh}+P^{S_{\text{odd}}}_{ijkl;abcd}\hskip
1.0ptP^{S_{\text{even}}}_{mnop;efgh}\,,$ $\displaystyle
P^{RS_{\text{even}}}_{ijklmnop;abcdefgh}$
$\displaystyle=P^{R_{\text{even}}}_{ijkl;abcd}P^{S_{\text{even}}}_{mnop;efgh}+P^{R_{\text{odd}}}_{ijkl;abcd}P^{S_{\text{odd}}}_{mnop;efgh}\,,$
$\displaystyle P^{RS_{\text{odd}}}_{ijklmnop;abcdefgh}$
$\displaystyle=P^{R_{\text{even}}}_{ijkl;abcd}P^{S_{\text{odd}}}_{mnop;efgh}+P^{R_{\text{odd}}}_{ijkl;abcd}P^{S_{\text{even}}}_{mnop;efgh}\,,$
$\displaystyle P^{SR_{\text{even}}}_{ijklmnop;abcdefgh}$
$\displaystyle=P^{S_{\text{even}}}_{ijkl;abcd}P^{R_{\text{even}}}_{mnop;efgh}+P^{S_{\text{odd}}}_{ijkl;abcd}P^{R_{\text{odd}}}_{mnop;efgh}\,,$
$\displaystyle P^{SR_{\text{odd}}}_{ijklmnop;abcdefgh}$
$\displaystyle=P^{S_{\text{even}}}_{ijkl;abcd}P^{R_{\text{odd}}}_{mnop;efgh}+P^{S_{\text{odd}}}_{ijkl;abcd}P^{R_{\text{even}}}_{mnop;efgh}\,,$
$\displaystyle P^{RR_{\text{even}}}_{ijklmnop;abcdefgh}$
$\displaystyle=P^{R_{\text{even}}}_{ijkl;abcd}P^{R_{\text{even}}}_{mnop;efgh}+P^{R_{\text{odd}}}_{ijkl;abcd}P^{R_{\text{odd}}}_{mnop;efgh}\,,$
$\displaystyle P^{RR_{\text{odd}}}_{ijklmnop;abcdefgh}$
$\displaystyle=P^{R_{\text{even}}}_{ijkl;abcd}P^{R_{\text{odd}}}_{mnop;efgh}+P^{R_{\text{odd}}}_{ijkl;abcd}P^{R_{\text{even}}}_{mnop;efgh}\,,$
$\displaystyle P^{TT_{\text{even}}}_{ijklmnop;abcdefgh}$
$\displaystyle=P^{T_{\text{even}}}_{ijkl;abcd}P^{T_{\text{even}}}_{mnop;efgh}\,,$
$\displaystyle P^{TA_{\text{odd}}}_{ijklmnop;abcdefgh}$
$\displaystyle=P^{T_{\text{even}}}_{ijkl;abcd}P^{A_{\text{odd}}}_{mnop;efgh}\,,$
$\displaystyle P^{AT_{\text{odd}}}_{ijklmnop;abcdefgh}$
$\displaystyle=P^{A_{\text{odd}}}_{ijkl;abcd}P^{T_{\text{even}}}_{mnop;efgh}\,,$
$\displaystyle P^{AA_{\text{even}}}_{ijklmnop;abcdefgh}$
$\displaystyle=P^{A_{\text{odd}}}_{ijkl;abcd}P^{A_{\text{odd}}}_{mnop;efgh}\,.$
(B.7)

From these expressions we can also see explicitly that when $m=n$, if we
choose to consider the two $U(n)$ symmetries as indistinguishable (which we
remind the reader is not strictly necessary), the $RS$ irreps become the same
as the $SR$ irreps. The same also happens for $TA$ and $AT$.

## Appendix C. Numerical parameters

For most of our plots, the bounds are obtained with the use of PyCFTBoot [30]
and SDPB [38]. We use the numerical parameters
$\texttt{m\\_max}=6,\texttt{n\\_max}=9,\texttt{k\\_max}=36$ in PyCFTBoot, and
we include spins up to $\texttt{l\\_max}=26$. The binary precision for the
produced xml files is 896 digits. SDPB is run with the options
\--precision=896, \--detectPrimalFeasibleJump, \--detectDualFeasibleJump and
default values for other parameters. We refer to this set of parameters as “
_A_ ”. Unless otherwise stated, our plots are run with parameters “ _A_ ”.

For some of the plots we used
$\texttt{m\\_max}=5,\texttt{n\\_max}=7,\texttt{l\\_max}=36,\texttt{k\\_max}=42$
and
$\texttt{m\\_max}=6,\texttt{n\\_max}=9,\texttt{l\\_max}=36,\texttt{k\\_max}=42$,
referred to as “ _B_ ” and “ _C_ ” respectively. Lastly, we also used qboot
[39], with $\Lambda=15$, $\texttt{n\\_max}=500$, $\nu\texttt{\\_max}=25$ and
$l=\\{0\text{--}49,55,56,59,60,64,65,69,70,74,75,\\\ 79,80,84,85,89,90\\}$
referred to as “ _D_ ”.

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