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On Convexity of Charged Operators in CFTs
and the Weak Gravity Conjecture

Ofer Aharony1 and Eran Palti2

_1 Department of Particle Physics and Astrophysics, Weizmann Institute of
Science,
Rehovot 7610001, Israel_

_2 Department of Physics, Ben-Gurion University of the Negev, Beer-Sheva
84105, Israel_

e-mail<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

The Weak Gravity Conjecture is typically stated as a bound on the mass-to-
charge ratio of a particle in the theory. Alternatively, it has been proposed
that its natural formulation is in terms of the existence of a particle which
is self-repulsive under all long-range forces. We propose a closely related,
but distinct, formulation, which is that it should correspond to a particle
with non-negative self-binding energy. This formulation is particularly
interesting in anti-de Sitter space, because it has a simple conformal field
theory (CFT) dual formulation: let $\Delta(q)$ be the dimension of the lowest-
dimension operator with charge $q$ under some global $U(1)$ symmetry, then
$\Delta(q)$ must be a convex function of $q$. This formulation avoids any
reference to holographic dual forces or even to locality in spacetime, and so
we make a wild leap, and conjecture that such convexity of the spectrum of
charges holds for any (unitary) conformal field theory, not just those that
have weakly coupled and weakly curved duals. This Charge Convexity Conjecture,
and its natural generalization to larger global symmetry groups, can be tested
in various examples where anomalous dimensions can be computed, by
perturbation theory, $1/N$ expansions and semi-classical methods. In all
examples that we tested we find that the conjecture holds. We do not yet
understand from the CFT point of view why this is true.

###### Contents

  1. 1 Introduction
  2. 2 The Weak Gravity Conjecture and Binding Energy
  3. 3 A Convex Dimension Conjecture
  4. 4 Application to General CFTs
  5. 5 Tests in Specific Theories
    1. 5.1 The $U(1)$ model in $4-\epsilon$ dimensions
    2. 5.2 The quartic $O(N)$ model in $4-\epsilon$ dimensions
    3. 5.3 The sextic $U(1)$ and $O(N)$ models in $3-\epsilon$ dimensions
    4. 5.4 A simple supersymmetric theory
    5. 5.5 The quartic $O(N)$ model in $d$ dimensions at large $N$
    6. 5.6 $U(N_{c})$ gauge theory in $3$ dimensions with fermions
    7. 5.7 $U(N_{c})$ Chern-Simons theories in 3 dimensions with fermions
    8. 5.8 $U(1)$ Chern-Simons theories in 3 dimensions with scalars
    9. 5.9 Banks-Zaks fixed point in 4 dimensions
  6. 6 Summary

##  1 Introduction

In this note we consider properties of local operators in unitary Conformal
Field Theories (CFTs) with continuous global symmetries (in $d>2$ space-time
dimensions). We propose that such operators should satisfy a certain
convexity-like property:

Abelian Convex Charge Conjecture: Consider any CFT with a $U(1)$ global
symmetry. Denote by $\Delta\left(q\right)$ the dimension of the lowest
dimension operator of charge $q$. Then this must satisfy a convex-like
constraint

$\Delta\left(n_{1}q_{0}+n_{2}q_{0}\right)\geq\Delta\left(n_{1}q_{0}\right)+\Delta\left(n_{2}q_{0}\right)\;,$
(1.1)

for any positive integers $n_{1}$, $n_{2}$, for some $q_{0}$ of order one.

Equivalently, the conjecture states that the operator product expansion (OPE)
of the lowest-dimension operators with positive charges has no singular terms,
whenever the charges are integer multiples of $q_{0}$. The $U(1)$ charge can
always be normalized to be an integer (with the minimal charge equal to one).
As we will discuss below, the non-trivial statement here is that $q_{0}$ is
not parameterically large in any parameter of the CFT. We propose that this is
an exact property of any CFT whose continuous global symmetry is precisely
$U(1)$.

There is a natural generalization of the proposal also to theories with larger
symmetry groups :

Convex Charge Conjecture: Consider any CFT with a continuous global symmetry
group $G$, and consider a simple factor $G_{0}\subset G$. Denote by
$\Delta\left(r\right)$ the dimension of the lowest dimension operator in the
representation $r$ of $G$. Then, there is always some representation $r_{0}$,
which is non-trivial in $G_{0}$ and has weights of order one, such that the
dimensions ${\tilde{\Delta}}(q)\equiv\Delta({\rm Sym}^{q}(r_{0}))$ satisfy a
convex-like constraint

${\tilde{\Delta}}\left(n_{1}+n_{2}\right)\geq{\tilde{\Delta}}\left(n_{1}\right)+{\tilde{\Delta}}\left(n_{2}\right)\;,$
(1.2)

for any positive integers $n_{1}$, $n_{2}$.111By ${\rm Sym}^{q}(r_{0})$ we
mean the symmetric product of $q$ copies of the representation $r_{0}$. In
cases where this symmetric product is reducible, we mean the specific
representation whose highest weight is $q$ times the highest weight of
$r_{0}$.

Our proposal is motivated by, but is much more general than, the Weak Gravity
Conjecture [1] (WGC) in Anti-de Sitter space. Indeed, we propose that the WGC
should be formulated (also in flat space) as a statement about the self-
binding energy of a particle:

Positive Binding Conjecture: For a (weakly coupled) gravitational theory with
a $U(1)$ gauge field, there should exist at least one charged particle in the
theory, with charge of order one, which has a non-negative self-binding
energy.

By a self-binding energy here we mean the difference of energies between the
lowest two-particle state and twice the energy of the one-particle state. This
formulation is closely related to previous formulations, most closely to self-
repulsive statements [2, 3], but has some important differences which become
more pronounced in anti-de Sitter (AdS) space. Specifically, in AdS space the
binding energy receives important contributions from contact terms, and not
only from long-range forces.

The organization of this paper is as follows. In section 2 we discuss the
formulation of the WGC in terms of binding energy, in flat space and in anti-
de-Sitter space. In section 3 we introduce the CFT dual statements, in terms
of convexity of the spectrum, and discuss some related details. In section 4
we discuss how the conjecture fits in with what is known generally about CFTs.
Our discussion in these sections is mostly for the $U(1)$ case, but it can be
generalized in a straightforward manner also to the non-Abelian cases
mentioned above (at least in dimensions where the gauge theory in the bulk is
IR-free). In section 5 we perform preliminary tests of the conjecture in many
different simple examples of CFTs for which the anomalous dimensions are
computable in some expansion, and we find that it is always satisfied. We also
find that it is not satisfied in theories like the $O(N)$ model in $4<d<6$,
which are believed not to be unitary. We end with a summary in section 6.

##  2 The Weak Gravity Conjecture and Binding Energy

The Swampland program aims to understand constraints on effective theories
coming from the requirement of an ultraviolet completion into quantum gravity
[4] (see [5, 6] for reviews). One such conjectured constraint is the Weak
Gravity Conjecture [1], which places a bound on the mass of a charged particle
under a $U(1)$ gauge field – there must be a particle obeying

$m\leq\sqrt{2}gqM_{p}\;.$ (2.1)

Here $m$ is the mass of the particle, $g$ is the gauge coupling, $q$ its
quantised charge, and $M_{p}$ the Planck scale. There are no known counter-
examples to the conjecture coming from string theory to date. Here we wrote
down the equation for the case of a four dimensional space-time, but it can be
generalized to other dimensions as well.

In [2] (see also [3]) it was argued that the more precise formulation of the
WGC should be in terms of the existence of a self-repulsive charged particle.
This formulation differs from (2.1) in the presence of massless scalar fields,
since they mediate an additional self-attractive long-range force.
Specifically, the generalization to this case reads

$2g^{2}q^{2}M_{p}^{2}\geq m^{2}+\mu^{2}M_{p}^{2}\;,$ (2.2)

where $\mu$ is the coupling of the particle to the scalar fields. This has
been tested in string theory, for example see [7, 8, 3].

One motivation for the self-repulsive statement is that it ensures the absence
of a large number $n\sim g^{-1}$ of stable bound states formed from copies of
the particle with the largest charge-to-mass ratio in the theory [1]. It is as
yet unclear why such states would be inconsistent. But we can assume such a
requirement and study its consequences.

The reason that a self-attractive charged particle would form a stable bound
state with itself (or $n$ copies of itself) is because the charge of the bound
state would be twice that of the particle, while the energy of the bound state
would be less than twice the mass of the particle due to the binding energy.
If that particle is the one with the largest charge-to-mass ratio in the
theory, then the bound state cannot decay to it or to any other state in the
theory.

If one is motivated by the absence of such stable bound states, then actually
the most direct and natural formulation of the WGC is as the Positive Binding
Conjecture discussed in the introduction.

In flat space we can apply this statement to a particle placed an infinite
distance from its copy. In that case, the binding energy positivity is mapped
directly to the particle being self-repulsive under the long-range Coulomb
forces222More precisely, this is true for four non-compact space-time
dimensions. In higher dimensions it is possible [3] that the long-distance
force could be attractive but still there will not be any bound state.,
yielding the formulation (2.2). However, global AdS space behaves effectively
like a box, so there is a limit as to how far the particle can be displaced
from its copy. In particular there is no obvious relation between the long-
range force and the binding energy. This makes the binding energy formulation
of the WGC somewhat different from both the statements about unstable
Reissner-Nordstrom black holes (2.1) and about repulsive long-range forces
(2.2). In AdS space there is a contribution from contact interactions to the
binding energy, which cannot be dismissed.

Let us make this more precise for the case of 5-dimensional AdS space,
following the calculation in [9]. We consider a 5-dimensional gravitational
theory with a $U(1)$ gauge field and a scalar field $\varphi$ charged under it
with charge $q$, that has an $AdS_{5}$ solution:

$\displaystyle S$ $\displaystyle=$ $\displaystyle\frac{1}{\kappa^{2}}\int
d^{5}x\sqrt{-g}\left(\frac{1}{2}R+6-\frac{1}{4g^{2}}F_{\mu\nu}^{2}+\left|D_{\mu}\varphi\right|^{2}\right.$
(2.3)
$\displaystyle\left.\qquad\qquad\qquad-m^{2}\left|\varphi\right|^{2}-a\left|\varphi\right|^{4}+b\left|\varphi\right|^{2}\left|D_{\mu}\varphi\right|^{2}\right)\;.$

Here $\kappa$ is the 5-dimensional Planck scale. We measure every dimensionful
quantity in units of the AdS radius $R_{AdS}$, and set this radius to unity
$R_{AdS}=1$. The coefficients $a$ and $b$ are arbitrary constants. The action
(2.3) captures all the relevant contributions to the self-binding energy of
$\varphi$ (up to two-derivative order). Let us denote the self-binding energy
of $\varphi$ as $\gamma_{\varphi^{2}}$. This was calculated at leading order
in perturbation theory in [9], which found

$\gamma_{\varphi^{2}}=\gamma_{\varphi^{2}}^{\mathrm{photon}}+\gamma_{\varphi^{2}}^{\mathrm{graviton}}+\gamma_{\varphi^{2}}^{\mathrm{quartic}}\;,$
(2.4)

with

$\displaystyle\gamma_{\varphi^{2}}^{\mathrm{photon}}$ $\displaystyle=$
$\displaystyle\frac{\pi^{2}N_{\Delta}^{4}g^{2}q^{2}}{2\Delta-1}\;,$
$\displaystyle\gamma_{\varphi^{2}}^{\mathrm{graviton}}$ $\displaystyle=$
$\displaystyle-\frac{2\pi^{2}N_{\Delta}^{4}\Delta^{2}\left(\Delta-2\right)}{3\left(\Delta-1\right)\left(2\Delta-1\right)}\;,$
$\displaystyle\gamma_{\varphi^{2}}^{\mathrm{quartic}}$ $\displaystyle=$
$\displaystyle\frac{\pi^{2}N_{\Delta}^{4}\left(a+b\Delta\left(2-\Delta\right)\right)}{\left(\Delta-1\right)\left(2\Delta-1\right)}\;.$
(2.5)

Here we have introduced $\Delta$ (the dimension of the CFT operator dual to
$\varphi$) and $N_{\Delta}$ as

$m^{2}=\Delta\left(\Delta-4\right)\;,\;\;N_{\Delta}=\sqrt{\frac{\Delta-1}{2\pi^{2}}}\;.$
(2.6)

The Positive Binding Conjecture implies that in all UV-complete theories
$\gamma_{\varphi^{2}}\geq 0$, and there are no counter-examples to this as far
as we know. In the case where the action (2.3) is completed into a
supersymmetric one, and $\varphi$ is taken as a BPS state, we have the
relations

$\mathrm{SUSY}\;:-g^{2}q^{2}=a+b\Delta\left(2-\Delta\right)=-\frac{2}{3}\Delta^{2}\;.$
(2.7)

This leads to the exact relation $\gamma_{\varphi^{2}}=0$. It also manifests
explicitly that the contact contribution to the binding energy becomes
negligible relative to the other contributions in the
$\Delta\rightarrow\infty$ limit, where the particle is very heavy compared to
the AdS scale.

Note that the action (2.3) is not the most general low-energy effective
action, in that it does not include the contribution from additional massless
scalar fields. But the main point is that in AdS the contact interactions are
important, and so the binding energy formulation of the WGC is significantly
different from the one based on black holes or long-range forces. It is also
the one which is precisely saturated with supersymmetry, and so it seems like
the more natural one.

##  3 A Convex Dimension Conjecture

The AdS/CFT correspondence [10, 11, 12] maps gravitational theories on
$(d+1)$-dimensional AdS space to $d$-dimensional CFTs. This correspondence
maps the energy of a state in global AdS space to the dimension of the
corresponding local operator (related to the state by the state/operator
correspondence) in the CFT, and it maps gauge fields in the bulk to global
symmetries of the CFT.333The last statement depends on the boundary
conditions, but it is always true for $d\geq 3$. In any case the inverse
statement is always true; a continuous global symmetry in the CFT is related
to a gauge field in the bulk. Thus, it is natural to try to map the WGC in AdS
space to a statement about CFTs, which should hold at least for CFTs with
weakly coupled and weakly curved gravitational duals, and, as we propose, also
more generally.

The holographic dual of the WGC in AdS was first explored in [13], where it
was formulated as a statement on the ratio of the dimension to the charge of
an operator in the CFT (for 4-dimensional CFTs)

$\frac{\Delta^{2}}{q^{2}}\leq\frac{9}{40}\frac{C_{T}}{C_{V}}\;.$ (3.1)

Here $\Delta$ is the dimension of the operator, $q$ its charge, and $C_{T}$
and $C_{V}$ are the coefficients of the two-point functions of the energy-
momentum tensor and the global symmetry current, respectively (schematically,
these measure the number of degrees of freedom in the CFT, with $C_{T}$
counting all of them and $C_{V}$ only the ones charged under the global
$U(1)$). Subsequently, the holographic dual of the Weak Gravity Conjecture was
further developed in [14, 15, 16, 17, 18, 19, 20, 21, 22]. In [23, 24] the
holographic dual of the distance conjecture was developed.

In this paper we discussed a different formulation of the WGC in terms of
binding energy, and this has a simpler formulation in the dual CFT, as the
Convex Charge Conjecture (1.1) that we presented in the introduction.444Note
that (1.1) is not quite equivalent to convexity of the function
$\Delta\left(q\right)$. More precisely, (1.1) follows if
$\Delta\left(q\right)$ is convex as a function for all real $q\geq 0$. But
since in the CFT $\Delta(q)$ is defined only over the integers (and more
precisely, in the conjecture it appears just for integer multiples of
$q_{0}$), it is possible to have $\Delta\left(q\right)$ which is not convex as
a real function, and still satisfy (1.1). While the motivation for the
conjecture comes from CFTs with weakly coupled and weakly curved gravitational
duals, we will boldly conjecture that it holds for all CFTs, and perform
various tests of this conjecture below555Note that our conjecture is not
directly related to previous conjectures on the convexity of the black hole
spectrum in flat space made by N. Arkani-Hamed [25], where it was conjectured
that $m/q$ is a convex function of the charge $q$ in flat space..

In the case of weakly-coupled theories, we can formulate the conjecture in
terms of specific operators related to the fields appearing in the action. For
example, consider a primary scalar operator $\phi$ charged under a global
symmetry, with charge of order one. Denote by $\phi^{n}$ the first (lowest
dimension) primary operator appearing in the (symmetrised) OPE of $n$
$\phi$’s. Then, the dimension of $\phi^{n}$ should satisfy (1.1) (or its
generalization (1.2)). We can introduce a more condensed notation as

$\gamma_{n_{1},n_{2}}\equiv\Delta\left(\phi^{n_{1}+n_{2}}\right)-\Big{(}\Delta\left(\phi^{n_{1}}\right)+\Delta\left(\phi^{n_{2}}\right)\Big{)}\;,$
(3.2)

and then write (1.1) or (1.2) as

$\gamma_{n_{1},n_{2}}\geq 0\;.$ (3.3)

We will use this notation in the weakly-coupled tests that we will perform
below.

There are various extensions of the Weak Gravity Conjecture that have been
studied in the literature, see the review [5], which will also play a role
here. Specifically, it is possible that when the global symmetry group $G$
contains multiple $U(1)$’s that can mix together, one has to define a more
subtle notion of convexity in the multi-dimensional space of $U(1)$ charges,
perhaps along the lines of the analysis of [16, 26, 27, 3]. A possible simple
restriction which avoids needing to consider this higher dimensional space is
that if there are $n$ $U(1)$’s, then the conjecture should hold when $G_{0}$
is chosen to be one of $n$ specific linearly independent combinations of the
$U(1)$’s. However, we expect that stronger statements are also likely to hold.

We note that all the examples we will discuss below are consistent with a
stronger version of our conjecture, which is that the charge $q_{0}$ (or the
representation $r_{0}$ which appears) is the charge of the charged operator
with the lowest dimension in the CFT. This is related to stronger versions of
the WGC, stating that it should hold for the charged particle with the
smallest mass. Such strong versions of the Weak Gravity Conjecture have been
studied on the gravity side, starting already from the original paper [1], and
we refer to the review [5] for more details. Note that in the case of multiple
$U(1)$’s there is a candidate counter-example to such a statement proposed in
[26]. More specifically, it is proposed as a possible counter-example to the
statement that along any ray in the $U(1)$ charge space the lightest state
satisfies the strong Weak Gravity Conjecture. In the case of a single $U(1)$
there are no proposed counter-examples in string theory to this strong
version.666We note that another possible strong version of the conjecture
would be that $q_{0}$ should be associated to the smallest possible charge,
but the example in section 5.4 contradicts this, as does the example in [26].

Another possible stronger version of the conjecture, which is consistent with
all of our examples, is that the spectrum of all charged scalar operators is
convex. In particular, $q_{0}$ can always be chosen to be the lowest charge of
a charged scalar operator.

##  4 Application to General CFTs

Before discussing the conjecture for general CFTs, let us begin by reviewing
the behavior of the spectrum of operators $\Delta(q)$ in general CFTs (see
[28] for a recent review), at very large charge $q$ for a $U(1)$ global
symmetry (where large here means larger than any other large parameter that
the CFT may have). Recall that $\Delta(q)$ is the same as the energy of the
lightest state of charge $q$ in the CFT on $S^{d-1}$. It is believed (though
not proven) that in all CFTs this behavior falls into one of three classes:

  * •

In generic CFTs, one expects the large charge behavior on a sphere to be
similar locally to the behavior at large charge density in flat space. In this
regime, the low-energy theory includes a Nambu-Goldstone boson related to the
spontaneous breaking of the global symmetry. This leads to a universal
behavior [29, 30, 31] of the energy at large $q$,
$\Delta(q)=Aq^{d/(d-1)}+\cdots$, where the constant $A$ depends on the
specific theory (and a specific subleading term in $\Delta(q)$ is universal
and can be computed explicitly [29, 32]).

  * •

In free scalar theories where the scalar $\phi$ is charged under the $U(1)$,
the operators $\phi^{q}$ give $\Delta(q)=q\Delta(1)$. A similar behavior
arises also in supersymmetric theories with a moduli space, whenever there is
some BPS operator ${\cal O}(x)$ carrying the global charge which can obtain an
expectation value on the moduli space, since this implies $\Delta({\cal
O}^{n})=n\Delta({\cal O})$. The operators ${\cal O}^{n}$ are BPS operators in
this case, for all $n$ (namely, they do not obey any relations in the chiral
ring). Note that generally theories with moduli spaces have a continuous
R-symmetry group, and then the charged operators discussed here would always
carry some R-symmetry charge (and perhaps also additional charges).

  * •

In free fermionic theories, where the fermion carries $U(1)$ charge, the
lightest charged operators are not just products of fermions, but rather
because of Pauli’s principle they involve products of fermions and their
derivatives. This leads to states filling a Fermi sea (similar to the
situation in flat space). In this case one also finds
$\Delta(q)=Aq^{d/(d-1)}$, where the constant $A$ depends on the type of spinor
and on $d$, but the low-lying excitations over the Fermi sea look different
than in the generic case mentioned above. It is interesting that in this case
the function $\Delta(q)$ is not analytic even in the large $q$ limit beyond
the leading order [33], but this does not affect our analysis.

Note that in all these cases, the large charge spectrum is convex (marginally
so in the free scalar / SUSY case), so that for a large enough $q_{0}$ (where
the large charge expansion in valid) our conjecture is always satisfied. In
the second and third cases, we can compute the spectrum exactly and the
conjecture is satisfied (for $q_{0}$ that is equal to the charge of the free
scalar/fermion, or to the lowest charge carried by a BPS operator ${\cal O}$
that is non-zero on the moduli space). In the fermion case, if we take $q_{0}$
equal to the number of components of the fermion field, the spectrum is not
even marginally convex [33]. This means that convexity (for that value of
$q_{0}$) is maintained under any small-parameter perturbation of the free
fermion theory. Therefore, non-trivial tests of the conjecture in fermionic
theories require a large number of fermion fields, for example as studied
below in section 5.9.

Thus, the non-trivial question is whether the conjecture is satisfied in the
first case, for some $q_{0}$ of order one. This conjecture is only non-trivial
if the CFT has some large parameter (which can be the inverse of some small
parameter), since in this case the large charge behavior described above only
sets in when $q$ scales as some power of this large parameter. Our conjecture
is that for smaller values of $q$, even though the large charge expansion is
not valid, convexity of the spectrum still holds, with $q_{0}$ of order one
(namely, not scaling with the large parameter).

The original motivation for the conjecture, discussed above, is relevant for
CFTs with a weakly-curved and weakly-coupled gravitational dual (in such
theories the large charge expansion mentioned above is related to large black
holes in AdS space). But since there is no sharp distinction in the space of
CFTs between such CFTs and general CFTs, it is believed that any CFT has in
some sense a quantum gravity AdS dual, and it seems reasonable to assume that
any conjecture applying to “holographic CFTs” could apply to all CFTs. It is
therefore natural to formulate the general version of the conjecture (1.1) (or
(1.2)), which would apply to any CFT.

We do not have any direct arguments for this conjecture, but we can check it
in various examples (see below), corresponding to various types of expansions
where the dimensions $\Delta(q)$ are computable, and it seems to always hold.
In generic perturbative expansions in some parameter $\epsilon$ (which can be
for instance a small coupling, the space-time dimension, or the inverse number
of fields), the leading order correction to $\Delta(q)-q\Delta(1)$ will go as
$\epsilon^{a}P(q)$ for some positive $a$ and for some polynomial $P(q)$ (often
proportional to $q(q-1)$), whose leading term can be positive or negative. In
the first case, convexity holds at the leading order in perturbation theory.
Higher orders in perturbation theory cannot change this within the regime of
validity of perturbation theory, so one has to go beyond perturbation theory
to provide further checks of the conjecture. On the other hand, if the first
correction is negative, then the spectrum is not convex up to some $q_{0}$
going as a negative power of $\epsilon$, contradicting the conjecture (1.1).
Thus, every example where $\Delta(q)$ is amenable to a perturbative expansion
gives us a single test of the conjecture, and so far it has successfully
passed all of these tests.

It would be nice to test our conjecture in the framework of CFTs with a
weakly-curved and weakly-coupled gravitational dual (where perturbative
computations in the bulk are valid). However, the specific examples of such
theories that are under control tend to be supersymmetric, and the charged
spectrum typically contains some BPS operator ${\cal O}(x)$. As mentioned
above, if this operator parameterizes a moduli space, then $\Delta(q)$ behaves
linearly and the conjecture is obvious. However, even if there is no moduli
space, so that ${\cal O}^{n}$ is not a BPS operator for large enough values of
$n$, we still have for all $n$ $\Delta({\cal O}^{n})\geq n\Delta({\cal O})$ by
the BPS bound. While this by itself does not imply the full convexity
conjecture (1.1), it does imply that the first non-BPS operator will obey it
(and obviously all the BPS operators do), and in particular the leading
perturbative correction will be consistent with the conjecture (assuming that
the charge of ${\cal O}$ is not parameterically large). Thus one has to go
beyond perturbation theory to check the full conjecture in these cases.

Note that in theories with such bulk duals, the spectrum naively contains also
small extremal black holes (with a size much smaller than the AdS radius).
There are then two options. In many cases (and in particular in supersymmetric
examples) the extremal black holes are BPS (and saturate the weak gravity
conjecture in the flat space limit); in this case the small extremal black
holes and the multi-particle BPS states all have a linear spectrum $\Delta(q)$
with the same slope. The other option allowed by the weak gravity conjecture
in flat space is that the extremal black holes may have a lower charge-to-mass
ratio than some light charged particle. In this case if we consider the
extremal black hole states, they would not obey the convexity condition (1.1).
However, precisely in this case the extremal black holes are also unstable
towards decay to light particles, so the lowest-energy states with a specific
charge would have energies much smaller than the small extremal black holes,
avoiding a contradiction with the conjecture.

Note also that our conjecture about convexity of the spectrum as a function of
the charge is not related as far as we can see to the convexity of the
spectrum as a function of the spin (at least for large spin), which was
analyzed in [34]. In particular, in the case of spin, the twist spectrum goes
to a constant in the large spin limit, while in the case of charge, in generic
theories $\Delta(q)$ grows faster than linearly at large $q$. It would be
interesting to investigate the spectrum as a function of both the charge and
the spin, but this is beyond the scope of the current paper (the results of
[35] should be useful for this).

We have restricted our conjecture to $d>2$, so that it does not apply to two-
dimensional CFTs. It is possible to find CFTs in two dimensions which have an
integer parameter $N$, such that for large $N$, $q_{0}$ needs to be taken to
scale like $N$.777We thank C. Vafa for bringing these examples to our
attention, and M. Montero and G. Shiu for discussions on relevant work in
progress on similar themes. Note that for two-dimensional CFTs the dual three
dimensional bulk physics is very different from higher dimensions, since
gravity is not dynamical and gauge fields can be massive.

##  5 Tests in Specific Theories

In this section we discuss tests of our conjectures (1.1) and (1.2) in
specific examples. As mentioned above, from the gravitational side the
conjecture does not offer significantly new testing opportunities which go
beyond existing tests of the WGC. The non-trivial new tests that can be
performed are from the CFT side, in theories which do not have weakly-curved
gravitational duals. This also shows the power of the binding energy
formulation, since from the gravity side such theories are typically not
local, and there are no notions of forces and distance. Further, the
gravitational and $U(1)$ gauge fields in the bulk are part of an infinite
tower of higher spin fields (which are all light for weakly coupled CFTs), and
so are not particularly distinguished as mediators of interactions between the
states. The binding energy automatically sums over all interactions, including
higher-spin modes.

Note that when the lightest charged operator is a scalar, the spectrum in the
free theory is linear in the charge, so the first perturbative correction has
to take a specific sign for the perturbative spectrum to be convex. On the
other hand, when the light charged operators are fermions, the spectrum is
convex (growing faster than linearly) already in the free theory, so we do not
have non-trivial tests of our conjecture in such cases (except when the number
of fermion fields is very large, as in large $N$ gauge theories, such that the
faster-than-linear growth of the free fermion theory sets in only at large
charges).

###  5.1 The $U(1)$ model in $4-\epsilon$ dimensions

The simplest tests of the conjecture, which we will start from, can be
performed in generalizations of the Wilson-Fisher fixed point, which can be
analyzed perturbatively in an expansion in small $\epsilon$, where
$d=d_{0}-\epsilon$ for some integer $d_{0}$. We will show that in all the
examples which are believed to correspond to unitary CFTs when $\epsilon=1$,
our conjecture holds. However, we should note that strictly speaking the
theories with small $\epsilon$ are not unitary CFTs by themselves [36], and
the extrapolation to $\epsilon=1$ is not under control. So these examples do
not provide rigorous tests of the conjecture, but it seems that if we ignore
the non-unitarity at non-integer dimensions (which is related to specific
operators which vanish as $\epsilon\to 0$), the conjecture does non-trivially
hold in all of these cases.

The simplest non-trivial CFT of this type that we can consider is the
$U(1)=O(2)$ model at the Wilson-Fisher fixed point [37]. We follow here the
analysis of the anomalous dimension in [38]. The Euclidean action is given by

${\cal
L}=\partial^{\mu}\phi^{\dagger}\partial_{\mu}{\phi}+\frac{\lambda}{4}\left(\phi^{\dagger}\phi\right)^{2}\;.$
(5.1)

In $d=4-\epsilon$ dimensions there is a Wilson-Fisher fixed point (when the
mass is fine-tuned to zero) at a coupling

$\frac{\lambda_{\star}}{\left(4\pi\right)^{2}}=\frac{\epsilon}{5}+\frac{3}{25}\epsilon^{2}+{\cal
O}\left(\epsilon^{3}\right)\;.$ (5.2)

The dimension of $\phi^{n}$ in this model is [38]

$\Delta\left(\phi^{n}\right)=\frac{d-2}{2}n+\frac{\epsilon}{10}n\left(n-1\right)-\frac{\epsilon^{2}}{100}n\left(2n^{2}-8n+5\right)+{\cal
O}\left(\epsilon^{3}\right)\;.$ (5.3)

This gives

$\gamma_{n_{1},n_{2}}=\frac{n_{1}n_{2}}{5}\epsilon-\frac{n_{1}n_{2}}{50}\left(3n_{1}+3n_{2}-8\right)\epsilon^{2}+{\cal
O}\left(\epsilon^{3}\right)\;,$ (5.4)

which satisfies (3.3) at leading order in $\epsilon$.888Note that the
expression (5.3) is derived assuming a small expansion parameter $\epsilon
n\ll 1$, and then the first term in (5.4) always dominates. For large
$\epsilon n$ the theory goes over to the large charge regime discussed in the
previous section, which is convex. The spectrum remains convex [39] throughout
the continuous interpolation between these two regimes [38] (see also [40,
41]).

###  5.2 The quartic $O(N)$ model in $4-\epsilon$ dimensions

More generally, we can consider the $O(N)$ model. Here we will follow the
analysis of [42, 43]. The Euclidean Lagrangian is

${\cal
L}=\frac{1}{2}\partial^{\mu}\phi_{i}\partial_{\mu}\phi_{i}+\frac{\lambda}{4!}\left(\phi_{i}\phi_{i}\right)^{2}\;,$
(5.5)

with $i=1,\cdots,N$. In $d=4-\epsilon$ dimensions the fixed point is at

$\frac{\lambda_{\star}}{4\pi^{2}}=\frac{3\epsilon}{N+8}+\frac{9\left(3N+14\right)\epsilon^{2}}{\left(N+8\right)^{3}}+{\cal
O}\left(\epsilon^{3}\right)\;.$ (5.6)

The global group here is non-Abelian, so we need to use the general conjecture
(1.2); in particular $\phi$ itself is in the fundamental representation of
$O(N)$, and we can test the conjecture where $r_{0}$ is taken to be this
representation.

In particular, we can consider the operators $\varphi^{n}$, where
$\varphi\equiv\phi_{1}+i\phi_{2}$; these lie in the $n$’th symmetric product
of the fundamental representation of $O(N)$. Their dimension is given by [43]

$\Delta\left(\varphi^{n}\right)=\frac{d-2}{2}n+\frac{\lambda_{\star}n}{3(4\pi^{2})}\left(n-1\right)-\frac{\left(\lambda_{\star}n\right)^{2}}{18(4\pi^{2})^{2}}\left(N+4n-6\right)+{\cal
O}\left(\left(\lambda_{\star}n\right)^{3}\right)\;.$ (5.7)

We therefore have

$\displaystyle\gamma_{n_{1},n_{2}}=\frac{2n_{1}n_{2}}{N+8}\epsilon$
$\displaystyle-$
$\displaystyle\frac{n_{1}n_{2}}{\left(N+8\right)^{3}}\epsilon^{2}\Big{(}48n_{1}+48n_{2}$
$\displaystyle\qquad\qquad\qquad+6n_{1}N+6n_{2}N-16N+N^{2}-132\Big{)}+{\cal
O}\left(\epsilon^{3}\right)\;,$

and so (3.3) is satisfied at leading order in $\epsilon$.

###  5.3 The sextic $U(1)$ and $O(N)$ models in $3-\epsilon$ dimensions

For the sextic $U(1)$ model in $d=3-\epsilon$ (where both the $|\phi|^{2}$ and
the $|\phi|^{4}$ operators are fine-tuned to zero), we will follow the
analysis of [44, 45]. The Euclidean Lagrangian is

${\cal
L}=\partial^{\mu}\phi\partial_{\mu}\bar{\phi}+\left(\frac{\lambda}{3!}\right)\left(\phi\bar{\phi}\right)^{3}\;.$
(5.9)

The fixed point is at

$\left(\frac{\lambda_{\star}}{8\pi}\right)^{2}=\frac{3}{28}\epsilon+{\cal
O}\left(\epsilon^{2}\right)\;.$ (5.10)

The dimension of $\phi^{n}$ is given by

$\displaystyle\Delta\left(\phi^{n}\right)$ $\displaystyle=$
$\displaystyle\frac{d-2}{2}n+\left(\frac{\lambda_{\star}}{8\pi}\right)^{2}\frac{n\left(n-1\right)\left(n-2\right)}{9}$
$\displaystyle-$
$\displaystyle\left(\frac{\lambda_{\star}}{8\pi}\right)^{4}\frac{n\left(n-1\right)}{72}\Big{[}9\pi^{2}\left(n-2\right)\left(n-3\right)$
$\displaystyle\qquad\qquad\qquad\qquad+8n^{3}-56n^{2}+272n-456\Big{]}+{\cal
O}\left(\lambda_{\star}^{6}\right)\;.$

This gives

$\gamma_{n_{1},n_{2}}=\frac{n_{1}n_{2}}{28}\left(n_{1}+n_{2}-2\right)\epsilon+{\cal
O}\left(\epsilon^{2}\right)\;.$ (5.12)

In this case the leading order perturbative correction takes a more
complicated form, related to the fact that it arises at two-loop order. In
particular we see that for $n_{1},n_{2}>1$ we have a positive anomalous
dimension at leading order, which is the 2-loop level. But for
$n_{1}=n_{2}=1$, the two-loop contribution vanishes, and we have to look at
the 4-loop contribution. Using (5.3) to compute $\Delta(2)$, we find

$\gamma_{1,1}=2\left(\frac{\lambda_{\star}}{8\pi}\right)^{4}+{\cal
O}(\lambda_{\star}^{6})\;,$ (5.13)

consistent with (3.3).

The $O(N)$ sextic model in $3-\epsilon$ dimensions was also analysed in [45].
We can consider the dimension of an operator of total charge $n$ (as in
section 5.2), with $r_{0}$ the fundamental representation of $O(N)$. 999In
[45] certain contributions to the dimension of $\phi^{n}$ at four loops were
not included, specifically self-energy insertions to the $n$ external legs of
the $\phi^{n}$ vertex. However, since this contribution scales as $n$, it does
not modify $\gamma_{n_{1},n_{2}}$. They found

$\displaystyle\gamma_{n_{1},n_{2}}=$
$\displaystyle\left(\frac{\lambda_{\star}}{8\pi}\right)^{2}\frac{n_{1}n_{2}\left(n_{1}+n_{2}-2\right)}{3}$
$\displaystyle-\left(\frac{\lambda_{\star}}{8\pi}\right)^{4}\frac{n_{1}n_{2}}{72}\Big{[}8\left(-182+123n_{1}-32n_{1}^{2}+5n_{1}^{3}\right.$
$\displaystyle\left.+123n_{2}-48n_{1}n_{2}+10n_{1}^{2}n_{2}-32n_{2}^{2}+10n_{1}n_{2}^{2}+5n_{2}^{3}\right)$
$\displaystyle+\left(16+N\right)\left(11-9n_{1}+2n_{1}^{2}-9n_{2}+3n_{1}n_{2}+2n_{2}^{2}\right)\pi^{2}\Big{]}+{\cal
O}(\lambda_{\star}^{6})\;.$

This is consistent with (3.3) for all values of $n_{1}$ and $n_{2}$ (including
$n_{1}=n_{2}=1$).

###  5.4 A simple supersymmetric theory

Next, consider the supersymmetric theory with four supercharges, a single
chiral superfield $\Phi$, and a superpotential

$W=g\Phi^{3}.$ (5.15)

For $d=3$ this theory is believed to flow to a Wilson-Fisher-like fixed point,
while for $d=4$ the theory is IR-free. The only global symmetry of this theory
is a $U(1)_{R}$ symmetry, under which the scalar component $\phi$ of $\Phi$
carries charge $(2/3)$, and its fermionic component $\psi$ carries charge
$-(1/3)$. In our conjecture we assumed that the $U(1)$ charges are quantized
to be integers, so we should consider three times the $U(1)_{R}$ charge. Note
that already in the free limit of this theory (say in $d=4$), we can see that
the lowest-dimension operator with charge $1$, which is $\bar{\psi}$, has
dimension $3/2$, so $\Delta(1)=3/2$, while the lowest-dimension operator with
charge $2$ is $\phi$ which has dimension $1$, so $\Delta(2)=1$. This is a
simple example where in the phrasing of the conjecture (1.1) we cannot take
$q_{0}=1$ but we must take $q_{0}=2$. However, for $q_{0}=2$ the conjecture
trivially holds in the free limit, since $\Delta(2n)=\Delta(\phi^{n})=n$.

For finite coupling (as in the $d=3$ fixed point), the operator $\Phi$ is a
BPS operator so its dimension (and those of $\phi$ and $\psi$) is the same as
in the free theory, but the higher $\Phi^{n}$ are not BPS operators, so they
have positive anomalous dimensions (by the BPS bound). As discussed in the
previous section, the BPS bound means that our conjecture is always satisfied
at leading perturbative order in supersymmetric theories, as long as there is
at least one BPS operator charged under the group $G_{0}$ we are considering
(with a charge that is not parameterically large; note that this operator will
always carry some R-charge). We do not know how to compute the anomalous
dimensions of $\phi^{n}$ for $d=3$ (beyond the knowledge that
$\Delta(\phi^{n})>n\Delta(\phi)$). If these can be computed (perhaps by
numerical bootstrap methods) they could provide nice tests of our convexity
conjecture (e.g. by confirming that $\Delta(\phi^{4})\geq 2\Delta(\phi^{2})$).

We can take the Lagrangian of the $d=4$ theory and consider it also in
$(4-\epsilon)$ dimensions, keeping four-component fermions. The theory is no
longer supersymmetric, but it still has just the $U(1)_{R}$ symmetry, and it
has a weakly coupled fixed point for $g$ that is controllable in the
$\epsilon$-expansion, and we can ask if our conjecture it still satisfied. At
leading order in the $\epsilon$ expansion the computation of anomalous
dimensions (as a function of $g$) is the same as in $d=4$ where the theory is
supersymmetric, so we know that $\phi$ has no anomalous dimension (at order
$g^{2}\propto\epsilon$) while $\phi^{2}$ has a positive anomalous dimension.
On general grounds, as above, the anomalous dimension of $\phi^{n}$ would be
proportional to $\epsilon n(n-1)$ (as in the previous sections where we had
one-loop anomalous dimensions), and we know from the $n=2$ case that the
coefficient is positive. Thus, our conjecture holds for these theories, but
only when we take $q_{0}=2$.

###  5.5 The quartic $O(N)$ model in $d$ dimensions at large $N$

There is an alternative way to compute the anomalous dimensions in the quartic
$O(N)$ model of section 5.2, by performing a $1/N$ expansion for general
dimension $d$, instead of an $\epsilon$ expansion. The two expansions agree in
their overlap. The advantage of this expansion is that it can be used also for
the physical case of $d=3$, and thus it gives us a real test of the conjecture
(in the large $N$ limit, for $r_{0}$ the fundamental representation of
$O(N)$). In this case we should consider a general operator of charge $q$,
with dimension $\Delta\left(q\right)$. Expanding in $\frac{q}{N}$ for $d=3$
one obtains [46, 47]

$\Delta\left(q\right)=N\left[\frac{q}{2N}+\frac{4}{\pi^{2}}\left(\frac{q}{N}\right)^{2}+{\cal
O}\left(\left(\frac{q}{N}\right)^{3}\right)\right]\;.$ (5.16)

This gives $\gamma_{n_{1},n_{2}}>0$, consistent with the conjecture.

It is possible to formally calculate the dimension also in $d=5$ (where the
non-trivial fixed point is a UV fixed point rather than an IR fixed point),
and this gives [47]

$\Delta\left(q\right)=N\left[\frac{3q}{2N}-\frac{32}{3\pi^{2}}\left(\frac{q}{N}\right)^{2}+{\cal
O}\left(\left(\frac{q}{N}\right)^{3}\right)\right]\;.$ (5.17)

This leads to $\gamma_{n_{1},n_{2}}<0$, violating the conjecture. Similarly,
one can calculate in $6-\epsilon$ dimensions for $\sigma|\phi|^{2}$ theories
(which are believed to flow to IR fixed points related to the UV fixed points
mentioned above) in the $\epsilon$ expansion, and this gives [48, 49]

$\gamma_{n_{1},n_{2}}=-264\epsilon\frac{n_{1}n_{2}}{N^{2}}+{\cal
O}\left(\left(\frac{\epsilon n}{N}\right)^{2}\right)\;,$ (5.18)

which is again in contradiction with the conjecture.

However, in $d>4$ (and in particular in $d=5$) it is believed that the CFTs
appearing in the computations above are non-unitary, related to the non-
boundedness of the scalar potential when we flow to these fixed points from
well-defined UV theories (see, for example, [50, 48, 47]). Thus, the examples
of the previous paragraph do not contradict our conjecture, which is only for
unitary theories. It is intriguing that if our conjecture is correct, the
computations in the previous paragraph can be viewed as providing evidence for
the non-unitarity of these CFTs already from perturbative computations, while
usually the non-unitarity only shows up through non-perturbative effects.

###  5.6 $U(N_{c})$ gauge theory in $3$ dimensions with fermions

Gauge theories in 3 dimensions with gauge group $U(N_{c})$ and $N_{f}$
fermions in the fundamental representation are expected to flow to an
interacting CFT in the infrared for sufficiently large $N_{f}$. This is true
both for QED (with $N_{c}=1$) and for the non-Abelian case. Such theories have
a $SU\left(N_{f}\right)\times U(1)_{\mathrm{top}}$ global symmetry, where
$U(1)_{\mathrm{top}}$ is the symmetry whose current is the Hodge dual of the
$U(1)$ field strength. The operators charged under $U(1)_{\mathrm{top}}$ are
monopole operators. The spectrum of the monopoles can be studied in a large
$N_{f}$ expansion, and this was done in particular in [51, 52, 53, 54], whose
results we use101010We thank Shai Chester for suggesting this example..

In perturbation theory, the monopoles are specified by Goddard-Nuyts-Olive
(GNO) charges, associated to the Cartan of $U(N_{c})$, such that the sum of
these charges is the charge under $U(1)_{\mathrm{top}}$. The monopole
dimension does depend in general on the specific GNO charges, but at large
$N_{f}$ this dependence drops out, and the dimension is determined just by the
$U(1)_{\mathrm{top}}$ charge $q$.

The monopoles also transform as non-trivial representations of
$SU\left(N_{f}\right)$, with different dimensional representations for
different $q$’s. This comes from the quantization of fermion zero modes in the
monopole background. To agree with previous sections we rescale the monopole
charge by a factor of $2$ compared to the literature, such that it is an
integer. In this normalization, the monopoles with $q=1$ transform in some
$SU(N_{f})$ representation with $N_{f}/2$ boxes (note that $N_{f}$ must be
even due to the parity anomaly), and all these representations are degenerate
at leading order in $1/N_{f}$. Monopoles with higher $q$’s transform in
$q$’th-products of these representations. We can thus take $r_{0}$ in the
formulation of the conjecture (1.2) to be some representation with charge $1$
under $U(1)_{\mathrm{top}}$, and some representation (for instance the
symmetric one) with $N_{f}/2$ boxes of $SU(N_{f})$. These are the lowest-
dimension operators carrying $U(1)_{\mathrm{top}}$ charge; there are higher-
dimension operators in other $SU(N_{f})$ representations (that one can
construct, say, by products of fermions with the simplest monopoles).

The results for the monopole dimension $\Delta_{q}$ for charge $q$ are given
in table 1 taken from [53] (rescaling the charges by a factor of 2). It is
simple to check that they are indeed convex at large $N_{f}$, and so they
satisfy the conjecture (1.1).

$q$ | $\Delta_{q}$
---|---
$1$ | $0.265\,N_{f}-0.0383+{\cal O}(1/N_{f})$
$2$ | $0.673\,N_{f}-0.194+{\cal O}(1/N_{f})$
$3$ | $1.186\,N_{f}-0.422+{\cal O}(1/N_{f})$
$4$ | $1.786\,N_{f}-0.706+{\cal O}(1/N_{f})$
$5$ | $2.462\,N_{f}-1.04+{\cal O}(1/N_{f})$
Table 1: The first few monopole operator dimensions $\Delta_{q}$ for monopole
charges $q$ in the $U(N_{c})$ gauge theory in $3$ dimensions with $N_{f}$
fermions in the fundamental, taken from [53].

Note that for these monopole operators, as well as for the ones mentioned in
the next two subsections and in [55, 56, 57], the large $q$ expansion works
quite well all the way down to $q=1$ (this was first observed in the context
of the $\epsilon$ expansion of their dimensions in [58]). The convexity of the
spectrum naturally follows from this behavior. However, note that this
statement about the validity of the large $q$ expansion is not true for the
other examples we discuss in this paper. Presumably it is related to the fact
that the dimensions of the lowest-charge monopole operators are
parameterically large (of order $N_{f}$) in the limit where we can compute
them, while the other low-charge operators we discuss have dimensions of order
one, and to the fact that the gap to the massive states in the effective
theory of the Nambu-Goldstone bosons (expanded around the monopole states) is
not small even for $q=1$.

###  5.7 $U(N_{c})$ Chern-Simons theories in 3 dimensions with fermions

The theories above can be easily generalized by adding a Chern-Simons level
$k$, as long as $k-N_{f}/2\in\mathbb{Z}$. Let us start from the case of
$N_{c}=1$. Chern-Simons theories with gauge group $U(1)$ in 3-dimensions and
Chern-Simons level $k$, with $N_{f}$ charged fermion fields, flow to an
interacting CFT in the IR for large $N_{f}$ and/or large $k$. The theories
have a $U(1)_{\mathrm{top}}$ with monopoles charged under it. The spectrum of
such monopoles was studied in [54], which we follow. The simplest monopole
operators in this case sit in some $SU(N_{f})$ representations that were
analyzed there, and we can take $r_{0}$ to be any one of these representations
(since they are all degenerate at large $N_{f}$). Note that for $k=N_{f}/2$
the simplest monopole operators are singlets of $SU(N_{f})$, so the
representation $r_{0}$ is trivial in $SU(N_{f})$.

There is a simple expression for the dimension $\Delta_{q}$ for large
$\kappa\gg 1$, where $\kappa=\frac{k}{N_{f}}$ (but not necessarily large $q$),
which is [54]

$\Delta_{q}\simeq\frac{2}{3}\left(q\kappa\right)^{\frac{3}{2}}\;.$ (5.19)

This is convex and satisfies (1.1).

For $\kappa$ between 0 and $\frac{1}{2}$, the spectrum is independent of
$\kappa$, at leading order in $N_{f}$ [54]. Therefore, the leading order
behaviour is as in table 1. For $\kappa>\frac{1}{2}$ which is not very large,
the expression for the dimensions is explicit but complicated; however, the
spectrum remains convex as a function of $q$ [54]. Therefore, in any
calculable regime, the theory has a convex spectrum satisfying (1.1).

Even though the results above were derived for $U(1)$ gauge group, the same
results apply at leading order in $1/N_{f}$ also for $U(N_{c})$.

###  5.8 $U(1)$ Chern-Simons theories in 3 dimensions with scalars

Next, we consider scalar versions of three dimensional QED, involving $U(1)$
gauge theories with $N_{f}$ charged complex scalars, at Chern-Simons level $k$
[59, 54, 60]. There are two types of such theories which flow to CFTs in the
IR (for large enough $N_{f}$), those with quartic terms, which are commonly
known as the $\mathbb{CP}^{N_{f}-1}$ model, and those where the quartic terms
are tuned to zero, which we will refer to as the tri-critical model.

In this case the large $\kappa$ limit again behaves as in (5.19), but with a
different positive prefactor,
$\Delta_{q}\simeq\frac{1}{\sqrt{2}}(q\kappa)^{3/2}$, and so the spectrum is
convex. For small $\kappa$, there is no degeneracy between different values of
$\kappa$ as there was in the fermionic case, and the spectrum becomes more
convex monotonically for increasing $\kappa$ [54].

In this case the simplest monopoles are $SU(N_{f})$ singlets for $\kappa=0$,
while otherwise they sit in specific $SU(N_{f})$ representations that were
analyzed in [54]. For the case of $\kappa=0$ the spectrum for the
$\mathbb{CP}^{N_{f}}$ and tri-critical models at large $N_{f}$ is given in
table 2 as taken from [59, 61]. This is seen to be convex in both cases and so
satisfies (1.1). One can check [61] that the conjecture holds also for non-
zero $\kappa$.

$q$ | $\mathbb{CP}^{N_{f}}$ | tri-critical
---|---|---
$1$ | 0.125 | 0.097
$2$ | 0.311 | 0.226
$3$ | 0.544 | 0.384
$4$ | 0.816 | 0.567
$5$ | 1.121 | 0.771
Table 2: Table showing the coefficient of the leading term in the $1/N_{f}$
expansion, namely the large $N_{f}$ limit of $\Delta(q)/N_{f}$, for the
dimension of the first few monopoles, for the $\mathbb{CP}^{N_{f}}$ model [53]
and the tri-critical model [61].

###  5.9 Banks-Zaks fixed point in 4 dimensions

Four-dimensional $SU(N_{c})$ gauge theories with $N_{f}$ flavors of massless
charged matter have a perturbative fixed point when the number of flavors is
slightly smaller than the value for which the theory becomes IR-free [62, 63,
64]. The coupling constant is parameterically small in the large $N_{c}$ limit
with $N_{f}$ proportional to $N_{c}$. The charged matter can be fermions or
scalars in the fundamental of $SU(N_{c})$.111111In the case of scalars the
mass needs to be fine-tuned to zero at each order in perturbation theory.

When we have scalars, we must also consider $\phi^{4}$ couplings that are
generated by the renormalization group flow, and we need to consider fixed
points for these couplings in addition to the gauge coupling. This question
was analyzed in [65] (see also [66]), where they took $N_{s}$ scalars and a
scalar potential of the form

$V(\phi)={\tilde{h}}{\rm
Tr}(\phi^{\dagger}\phi\phi^{\dagger}\phi)+{\tilde{f}}({\rm
Tr}(\phi^{\dagger}\phi))^{2},$ (5.20)

where $\phi$ is viewed as an $N_{c}\times N_{s}$ matrix. They found that for
small enough values of $N_{s}/N_{c}$, there are perturbative fixed points
which are unitary, with real values for the couplings ${\tilde{h}}$ and
${\tilde{f}}$ and a positive-definite scalar potential. They found $4$ fixed
points of this type, with varying numbers of relevant operators; our analysis
below will apply to all of these fixed points. Since $N_{s}/N_{c}$ needs to be
small, perturbative fixed points of this type require having both fermions and
scalars, where the fermions make up most of the contribution to the beta
function.

In the case of $N_{f}$ fermions the global symmetry is
$SU\left(N_{f}\right)_{L}\times SU\left(N_{f}\right)_{R}\times U(1)_{B}$. We
can consider meson operators which are in the bi-fundamental representation of
$SU(N_{f})_{L}\times SU(N_{f})_{R}$ and neutral under $U(1)_{B}$ and their
symmetric products, so operators of the form

${\cal O}^{\psi}_{n}=\left(\bar{\psi}\psi\right)^{n}\;,$ (5.21)

where we can take all the $\psi$’s and all the $\bar{\psi}$’s to have the same
flavor index (these operators vanish for $n>2N_{c}$, requiring adding
derivatives to obtain higher-charge operators, but in the large $N_{c}$ limit
this is not relevant for our considerations). In the case of $N_{s}$ scalars
the global symmetry is $SU\left(N_{s}\right)\times U(1)_{B}$. We can consider
again meson operators, of type

${\cal O}^{\phi}_{n}=\left(\phi^{*}\phi\right)^{n}\;,$ (5.22)

where now we choose them to transform in symmetric products of the adjoint
representation of $SU(N_{s})$. In particular we can choose, for instance, all
$\phi$’s to have index $1$ and all $\phi^{*}$’s to have index $2$, and we will
implicitly assume this below.

Consider first the scalar case. We are interested in calculating the
difference in dimension between the one-meson and two-meson operators

$\gamma_{1,1}^{\phi}=\Delta\left({\cal O}^{\phi}_{2}\right)-2\Delta\left({\cal
O}^{\phi}_{1}\right)\;.$ (5.23)

At leading order in perturbation theory this is sufficient to determine
convexity of the spectrum.

There are two types of one-loop contributions to the scalar-meson anomalous
dimensions. One contribution comes from gluon exchanges, and it turns out that
this contribution exactly vanishes when we consider the combination (5.23).
The reason for this cancellation between the 1-meson and 2-meson anomalous
dimensions can be simply understood as follows. To calculate the 2-meson
anomalous dimension we consider the correlator121212This correlator is not
gauge-invariant, but we can consider it in a specific gauge if we want, and
this issue is not important at the leading order we discuss here.

$\left<\phi^{*}_{j_{1}}\left(-p_{1}\right)\phi_{i_{1}}\left(q_{1}\right)\phi^{*}_{j_{2}}\left(-p_{2}\right)\phi_{i_{2}}\left(q_{2}\right)(\phi^{*}_{l_{1}}\phi_{k_{1}}\phi^{*}_{l_{2}}\phi_{k_{2}})\left(k\right)\right>\delta^{l_{1}k_{1}}\delta^{l_{2}k_{2}}\;,$
(5.24)

where the first $4$ fields have opposite flavor indices from the ones
appearing in the 2-meson operator (the indices written explicitly in (5.24)
are the color indices). At 1-loop this receives contributions from 4 types of
gluon-exchange diagrams. First, there are diagrams where a gluon connects a
specific external leg to itself. These are wavefunction regularizations, that
cancel when computing differences such as (5.23). Next, there are those
diagrams where the gluon is exchanged between scalars in the same meson, so
say between $\phi^{*}_{l_{1}}$ and $\phi_{k_{1}}$. Those will also cancel with
the 1-meson contributions when we take the difference in dimensions (5.23).
The other two types of diagrams are where the gluon is exchanged between the
mesons, so say between $\phi^{*}_{l_{1}}$ and $\phi^{*}_{l_{2}}$. In one set
of diagrams the gluon is exchanged between two $\phi$’s (or two $\phi^{*}$’s),
and in the other it is exchanged between a $\phi$ and a $\phi^{*}$. The two
possibilities are shown in figure 1.

Figure 1: The two types of diagrams contributing to the anomalous dimension of
the 2-meson operator but not to the 1-meson operator. The colors of the lines
indicate their color-index-contractions at the ${\cal O}_{2}$ operator (red
with red and black with black).

When computing the 2-meson diagrams in figure 1, it is easy to see that there
is a sign difference between (the logarithmic divergence of) the two diagrams
in figure 1 (while their absolute value is equal). This is essentially because
in the diagram on the left the two vertices contribute
$(2q+w)_{\mu}(2p-w)_{\nu}$, while in the diagram on the right the two vertices
contribute $(2q+w)_{\mu}(2p+w)_{\nu}$, while all other factors are the same
(and only the $w^{2}$ term from the vertices contributes to the logarithmic
divergence). So, diagrams where the gluon is exchanged between $\phi^{*}$ and
$\phi$ contribute to the anomalous dimension with opposite sign to those where
the gluon is exchanged between $\phi$ and $\phi$ (or $\phi^{*}$ and
$\phi^{*}$). Therefore, the diagrams cancel between themselves, leading to an
overall vanishing contribution. It is easy to see that the same cancellation
occurs in any computation of $\gamma_{n_{1},n_{2}}$.

Thus, the only one-loop contribution to $\gamma_{n_{1},n_{2}}$ comes from the
$\phi^{4}$ couplings (5.20). Note that like the previous contributions, this
contribution is independent of the number of fermions. It is not difficult to
compute the contribution of the couplings above to (5.23), and we find that it
is proportional to $(\tilde{f}+\tilde{h})$, with a positive coefficient. It
was already noted in [65] that this combination is positive in all the fixed
points, since this is related also to the positivity of the scalar potential.
Thus, this example is consistent with our conjecture.

We can also consider other gauge+scalar fixed points where one can compute
multi-meson dimensions perturbatively, and we will similarly find that the
gluon exchange amplitudes cancel at one-loop order. In particular this applies
also to the $3d$ fixed points for $SU(N_{c})$ (or $U(N_{c})$) theories with
$N_{f}$ scalars in the fundamental representation, where perturbation theory
is valid for large $N_{f}$. In this case (where some anomalous dimension
computations were performed for fermionic theories in [67]) there are no
$\phi^{4}$ couplings (which are fine-tuned to zero), so the one-loop multi-
meson anomalous dimensions cancel, and we need to go to two-loop order to
check our conjecture.

In the case of fermions, the situation is more subtle, since we have several
different meson operators of different spins in the same bi-fundamental flavor
representation (and with the same classical dimension), and even more
operators (including several operators of the same spin) when we look at two-
meson operators. For instance we can consider

$\bar{\psi}\gamma^{\mu}\psi\bar{\psi}\gamma_{\mu}\psi\;,\;\;\bar{\psi}\gamma^{\mu\nu}\psi\bar{\psi}\gamma_{\mu\nu}\psi\;,...$
(5.25)

which will mix with the two-scalar-meson operators. Testing the conjecture
then requires keeping track, at each level $n$, of the lightest operator,
while carefully taking the mixing into account. This is a complicated task,
even at the 1-loop level, and so we leave it for future work. This issue
arises both for the $4d$ Banks-Zaks fixed point, and for $3d$ fixed points
(where the fermion has only two spin indices rather than four, but still there
are several different meson and multi-meson operators with the same classical
dimension and the same flavor representations, complicating the computation).

##  6 Summary

In this paper we pointed out that the natural formulation of the Weak Gravity
Conjecture in AdS space is in terms of the self-binding energy of a charged
state. This has a simple CFT dual, which is that there should exist a charged
operator such that its products with itself have a non-negative anomalous
dimension. This formulation generalizes naturally to all CFTs, even without
weakly-curved gravitational duals, in the form of the conjectures that we
presented in the introduction.

In this paper we performed some simple preliminary tests of the conjecture,
and we found that it is always satisfied. It would be interesting to perform
more tests of the conjecture, in various additional CFTs where operator
dimensions can be computed (for instance, one could look at those fixed points
found in [68] which are unitary). It would also be interesting to study
various generalizations of the conjecture, for instance to non-relativistic
CFTs, or to operators living on boundaries or defects in CFTs.

Assuming that the conjecture is correct, it would be very interesting to try
to derive it. One option is to use conformal bootstrap methods to show that
the spectrum must always be convex. Since we presented an example where
$q_{0}$ needs to be larger than one for the conjecture to hold, it is not
completely clear how to do this; but it is possible that convexity always
holds for scalar charged operators, and this seems like the simplest case to
try to prove by bootstrap methods. Note that just by conformal bootstrap
methods one cannot disprove the conjecture, since generalized free fields
trivially obey the conjecture and are consistent with the bootstrap conditions
(it may be possible to numerically find solutions to the bootstrap equations
where the conjecture would not hold, but one would still need to prove that
these correspond to full-fledged CFTs). Note also that it seems challenging to
derive the conjecture perturbatively, since we have examples above (like the
large $N$ $O(N)$ model in $d=5$) where the conjecture does not hold even
though the CFT is unitary in perturbation theory (and non-unitarity only
appears non-perturbatively in the $1/N$ expansion; in particular, numerical
bootstrap computations at current levels of precision lead to an apparently
unitary solution to the bootstrap equations for the $d=5$ large $N$ $O(N)$
model [69, 70], even though the full CFT is believed to be non-unitary.).

Acknowledgements: We thank N. Arkani-Hamed, S. Chester, G. Cuomo, J. Gracey,
B. Heidenreich, S. Hellerman, I. Jack, Z. Komargodski, A. Kovner, D. Kutasov,
M. Lublinsky, J. Penedones, T. Rudelius, T. Ryttov and M. Watanabe for useful
discussions. The work of OA was supported in part by an Israel Science
Foundation center for excellence grant (grant number 2289/18), by grant no.
2018068 from the United States-Israel Binational Science Foundation (BSF), and
by the Minerva foundation with funding from the Federal German Ministry for
Education and Research. The work of EP was supported by the Israel Science
Foundation (grant No. 741/20). OA is the Samuel Sebba Professorial Chair of
Pure and Applied Physics.

## References

  * [1] N. Arkani-Hamed, L. Motl, A. Nicolis, and C. Vafa, “The String landscape, black holes and gravity as the weakest force,” JHEP 06 (2007) 060, hep-th/0601001.
  * [2] E. Palti, “The Weak Gravity Conjecture and Scalar Fields,” JHEP 08 (2017) 034, 1705.04328.
  * [3] B. Heidenreich, M. Reece, and T. Rudelius, “Repulsive Forces and the Weak Gravity Conjecture,” JHEP 10 (2019) 055, 1906.02206.
  * [4] C. Vafa, “The String landscape and the swampland,” hep-th/0509212.
  * [5] E. Palti, “The Swampland: Introduction and Review,” Fortsch. Phys. 67 (2019), no. 6 1900037, 1903.06239.
  * [6] M. van Beest, J. Calderón-Infante, D. Mirfendereski, and I. Valenzuela, “Lectures on the Swampland Program in String Compactifications,” 2102.01111.
  * [7] S.-J. Lee, W. Lerche, and T. Weigand, “Tensionless Strings and the Weak Gravity Conjecture,” JHEP 10 (2018) 164, 1808.05958.
  * [8] S.-J. Lee, W. Lerche, and T. Weigand, “A Stringy Test of the Scalar Weak Gravity Conjecture,” Nucl. Phys. B 938 (2019) 321–350, 1810.05169.
  * [9] A. L. Fitzpatrick and D. Shih, “Anomalous Dimensions of Non-Chiral Operators from AdS/CFT,” JHEP 10 (2011) 113, 1104.5013.
  * [10] J. M. Maldacena, “The Large N limit of superconformal field theories and supergravity,” Adv. Theor. Math. Phys. 2 (1998) 231–252, hep-th/9711200.
  * [11] E. Witten, “Anti-de Sitter space and holography,” Adv. Theor. Math. Phys. 2 (1998) 253–291, hep-th/9802150.
  * [12] S. S. Gubser, I. R. Klebanov, and A. M. Polyakov, “Gauge theory correlators from noncritical string theory,” Phys. Lett. B 428 (1998) 105–114, hep-th/9802109.
  * [13] Y. Nakayama and Y. Nomura, “Weak gravity conjecture in the AdS/CFT correspondence,” Phys. Rev. D 92 (2015), no. 12 126006, 1509.01647.
  * [14] S. Giombi and E. Perlmutter, “Double-Trace Flows and the Swampland,” JHEP 03 (2018) 026, 1709.09159.
  * [15] N. Benjamin, E. Dyer, A. L. Fitzpatrick, and S. Kachru, “Universal Bounds on Charged States in 2d CFT and 3d Gravity,” JHEP 08 (2016) 041, 1603.09745.
  * [16] M. Montero, G. Shiu, and P. Soler, “The Weak Gravity Conjecture in three dimensions,” JHEP 10 (2016) 159, 1606.08438.
  * [17] J.-B. Bae, S. Lee, and J. Song, “Modular Constraints on Superconformal Field Theories,” JHEP 01 (2019) 209, 1811.00976.
  * [18] M. Montero, “Are tiny gauge couplings out of the Swampland?,” JHEP 10 (2017) 208, 1708.02249.
  * [19] A. Urbano, “Towards a proof of the Weak Gravity Conjecture,” 1810.05621.
  * [20] M. Montero, “A Holographic Derivation of the Weak Gravity Conjecture,” JHEP 03 (2019) 157, 1812.03978.
  * [21] P. Agarwal and J. Song, “Large N Gauge Theories with a Dense Spectrum and the Weak Gravity Conjecture,” JHEP 05 (2021) 124, 1912.12881.
  * [22] L. F. Alday and E. Perlmutter, “Growing Extra Dimensions in AdS/CFT,” JHEP 08 (2019) 084, 1906.01477.
  * [23] F. Baume and J. Calderón Infante, “Tackling the SDC in AdS with CFTs,” 2011.03583.
  * [24] E. Perlmutter, L. Rastelli, C. Vafa, and I. Valenzuela, “A CFT Distance Conjecture,” 2011.10040.
  * [25] N. Arkani-Hamed, “unpublished,”.
  * [26] B. Heidenreich, M. Reece, and T. Rudelius, “Evidence for a sublattice weak gravity conjecture,” JHEP 08 (2017) 025, 1606.08437.
  * [27] S. Andriolo, D. Junghans, T. Noumi, and G. Shiu, “A Tower Weak Gravity Conjecture from Infrared Consistency,” Fortsch. Phys. 66 (2018), no. 5 1800020, 1802.04287.
  * [28] L. A. Gaumé, D. Orlando, and S. Reffert, “Selected Topics in the Large Quantum Number Expansion,” 2008.03308.
  * [29] S. Hellerman, D. Orlando, S. Reffert, and M. Watanabe, “On the CFT Operator Spectrum at Large Global Charge,” JHEP 12 (2015) 071, 1505.01537.
  * [30] A. Monin, D. Pirtskhalava, R. Rattazzi, and F. K. Seibold, “Semiclassics, Goldstone Bosons and CFT data,” JHEP 06 (2017) 011, 1611.02912.
  * [31] L. Alvarez-Gaume, O. Loukas, D. Orlando, and S. Reffert, “Compensating strong coupling with large charge,” JHEP 04 (2017) 059, 1610.04495.
  * [32] G. Cuomo, “A note on the large charge expansion in 4d CFT,” Phys. Lett. B 812 (2021) 136014, 2010.00407.
  * [33] Z. Komargodski, M. Mezei, S. Pal, and A. Raviv-Moshe, “Spontaneously Broken Boosts in CFTs,” 2102.12583.
  * [34] Z. Komargodski and A. Zhiboedov, “Convexity and Liberation at Large Spin,” JHEP 11 (2013) 140, 1212.4103.
  * [35] D. Li, D. Meltzer, and D. Poland, “Non-Abelian Binding Energies from the Lightcone Bootstrap,” JHEP 02 (2016) 149, 1510.07044.
  * [36] M. Hogervorst, S. Rychkov, and B. C. van Rees, “Unitarity violation at the Wilson-Fisher fixed point in 4-$\epsilon$ dimensions,” Phys. Rev. D 93 (2016), no. 12 125025, 1512.00013.
  * [37] K. G. Wilson and M. E. Fisher, “Critical exponents in 3.99 dimensions,” Phys. Rev. Lett. 28 (1972) 240–243.
  * [38] G. Badel, G. Cuomo, A. Monin, and R. Rattazzi, “The Epsilon Expansion Meets Semiclassics,” JHEP 11 (2019) 110, 1909.01269.
  * [39] G. Cuomo and Z. Komargodski, “Private Communication,”.
  * [40] G. Arias-Tamargo, D. Rodriguez-Gomez, and J. G. Russo, “The large charge limit of scalar field theories and the Wilson-Fisher fixed point at $\epsilon=0$,” JHEP 10 (2019) 201, 1908.11347.
  * [41] M. Watanabe, “Accessing large global charge via the $\epsilon$-expansion,” JHEP 04 (2021) 264, 1909.01337.
  * [42] O. Antipin, J. Bersini, F. Sannino, Z.-W. Wang, and C. Zhang, “Charging the $O(N)$ model,” Phys. Rev. D 102 (2020), no. 4 045011, 2003.13121.
  * [43] I. Jack and D. R. T. Jones, “Anomalous dimensions at large charge in d=4 O(N) theory,” Phys. Rev. D 103 (2021), no. 8 085013, 2101.09820.
  * [44] G. Badel, G. Cuomo, A. Monin, and R. Rattazzi, “Feynman diagrams and the large charge expansion in $3-\varepsilon$ dimensions,” Phys. Lett. B 802 (2020) 135202, 1911.08505.
  * [45] I. Jack and D. R. T. Jones, “Anomalous dimensions for $\phi^{n}$ in scale invariant $d=3$ theory,” Phys. Rev. D 102 (2020), no. 8 085012, 2007.07190.
  * [46] L. Alvarez-Gaume, D. Orlando, and S. Reffert, “Large charge at large N,” JHEP 12 (2019) 142, 1909.02571.
  * [47] S. Giombi and J. Hyman, “On the Large Charge Sector in the Critical $O(N)$ Model at Large $N$,” 2011.11622.
  * [48] G. Arias-Tamargo, D. Rodriguez-Gomez, and J. G. Russo, “On the UV completion of the $O(N)$ model in $6-\epsilon$ dimensions: a stable large-charge sector,” JHEP 09 (2020) 064, 2003.13772.
  * [49] O. Antipin, J. Bersini, F. Sannino, Z.-W. Wang, and C. Zhang, “More on the cubic versus quartic interaction equivalence in the $O(N)$ model,” 2107.02528.
  * [50] S. Giombi, R. Huang, I. R. Klebanov, S. S. Pufu, and G. Tarnopolsky, “The $O(N)$ Model in ${4<d<6}$ : Instantons and complex CFTs,” Phys. Rev. D 101 (2020), no. 4 045013, 1910.02462.
  * [51] V. Borokhov, A. Kapustin, and X.-k. Wu, “Topological disorder operators in three-dimensional conformal field theory,” JHEP 11 (2002) 049, hep-th/0206054.
  * [52] S. S. Pufu, “Anomalous dimensions of monopole operators in three-dimensional quantum electrodynamics,” Phys. Rev. D 89 (2014), no. 6 065016, 1303.6125.
  * [53] E. Dyer, M. Mezei, and S. S. Pufu, “Monopole Taxonomy in Three-Dimensional Conformal Field Theories,” 1309.1160.
  * [54] S. M. Chester, L. V. Iliesiu, M. Mezei, and S. S. Pufu, “Monopole Operators in $U(1)$ Chern-Simons-Matter Theories,” JHEP 05 (2018) 157, 1710.00654.
  * [55] E. Dupuis, M. B. Paranjape, and W. Witczak-Krempa, “Transition from a Dirac spin liquid to an antiferromagnet: Monopoles in a QED3-Gross-Neveu theory,” Phys. Rev. B 100 (2019), no. 9 094443, 1905.02750.
  * [56] E. Dupuis, M. B. Paranjape, and W. Witczak-Krempa, “Monopole operators and their symmetries in QED3-Gross-Neveu models,” in 11th International Symposium on Quantum Theory and Symmetries, 11, 2019.  1911.05802.
  * [57] E. Dupuis and W. Witczak-Krempa, “Monopole hierarchy in transitions out of a Dirac spin liquid,” 2102.04885.
  * [58] S. M. Chester, M. Mezei, S. S. Pufu, and I. Yaakov, “Monopole operators from the $4-\epsilon$ expansion,” JHEP 12 (2016) 015, 1511.07108.
  * [59] E. Dyer, M. Mezei, S. S. Pufu, and S. Sachdev, “Scaling dimensions of monopole operators in the $\mathbb{C}{\mathrm{\mathbb{P}}}^{N_{b}-1}$ theory in 2 $+$ 1 dimensions,” JHEP 06 (2015) 037, 1504.00368. [Erratum: JHEP 03, 111 (2016)].
  * [60] S. M. Chester, “Anomalous dimensions of monopole operators in scalar QED3 with Chern-Simons term,” JHEP 07 (2021) 034, 2102.07377.
  * [61] S. Chester, “Private Communication,”.
  * [62] W. E. Caswell, “Asymptotic Behavior of Nonabelian Gauge Theories to Two Loop Order,” Phys. Rev. Lett. 33 (1974) 244.
  * [63] A. A. Belavin and A. A. Migdal, “Calculation of anomalous dimensions in non-abelian gauge field theories,” Pisma Zh. Eksp. Teor. Fiz. 19 (1974) 317–320.
  * [64] T. Banks and A. Zaks, “On the Phase Structure of Vector-Like Gauge Theories with Massless Fermions,” Nucl. Phys. B 196 (1982) 189–204.
  * [65] F. Benini, C. Iossa, and M. Serone, “Conformality Loss, Walking, and 4D Complex Conformal Field Theories at Weak Coupling,” Phys. Rev. Lett. 124 (2020), no. 5 051602, 1908.04325.
  * [66] F. F. Hansen, T. Janowski, K. Langæble, R. B. Mann, F. Sannino, T. G. Steele, and Z.-W. Wang, “Phase structure of complete asymptotically free SU($N_{c}$) theories with quarks and scalar quarks,” Phys. Rev. D 97 (2018), no. 6 065014, 1706.06402.
  * [67] S. M. Chester and S. S. Pufu, “Anomalous dimensions of scalar operators in QED3,” JHEP 08 (2016) 069, 1603.05582.
  * [68] O. Antipin, J. Bersini, F. Sannino, Z.-W. Wang, and C. Zhang, “Untangling scaling dimensions of fixed charge operators in Higgs theories,” Phys. Rev. D 103 (2021), no. 12 125024, 2102.04390.
  * [69] S. M. Chester, S. S. Pufu, and R. Yacoby, “Bootstrapping $O(N)$ vector models in 4 $<d<$ 6,” Phys. Rev. D 91 (2015), no. 8 086014, 1412.7746.
  * [70] Z. Li and N. Su, “Bootstrapping Mixed Correlators in the Five Dimensional Critical O(N) Models,” JHEP 04 (2017) 098, 1607.07077.