††institutetext: DESY Hamburg, Theory Group, Notkestraße 85, D-22607 Hamburg,
Germany

# Bootstrapping the $(A_{1},A_{2})$ Argyres-Douglas theory

Martina Cornagliotto  Madalena Lemos  Pedro Liendo
<EMAIL_ADDRESS>
###### Abstract

We apply bootstrap techniques in order to constrain the CFT data of the
$(A_{1},A_{2})$ Argyres-Douglas theory, which is arguably the simplest of the
Argyres-Douglas models. We study the four-point function of its single Coulomb
branch chiral ring generator and put numerical bounds on the low-lying
spectrum of the theory. Of particular interest is an infinite family of semi-
short multiplets labeled by the spin $\ell$. Although the conformal dimensions
of these multiplets are protected, their three-point functions are not. Using
the numerical bootstrap we impose rigorous upper and lower bounds on their
values for spins up to $\ell=20$. Through a recently obtained inversion
formula, we also estimate them for sufficiently large $\ell$, and the
comparison of both approaches shows consistent results. We also give a
rigorous numerical range for the OPE coefficient of the next operator in the
chiral ring, and estimates for the dimension of the first R-symmetry neutral
non-protected multiplet for small spin.

###### Keywords:

conformal field theory, supersymmetry, conformal bootstrap

††preprint: DESY 17-175

##  1 Introduction and summary

The revival of the conformal bootstrap program Rattazzi:2008pe , has provided
new tools to study non-perturbative physics. The numerical techniques
introduced in Rattazzi:2008pe  have given a wealth of results, with the most
impressive being the high-precision estimates of the critical exponents of the
$3d$ Ising model ElShowk:2012ht ; El-Showk:2014dwa ; Kos:2014bka ; Simmons-
Duffin:2015qma ; Kos:2016ysd . In a parallel line of development, analytic
approaches to the bootstrap have also been explored, and recent progress has
given access to the spectrum of conformal field theories (CFTs) at large spin
by means of the lightcone limit Fitzpatrick:2012yx ; Komargodski:2012ek .
These two methods were combined in Alday:2015ota ; Simmons-Duffin:2016wlq ,
where knowledge of operator dimensions and operator product expansion (OPE)
coefficients, obtained numerically for the Ising model, was used to derive
analytic approximations for the CFT data at large spin. Remarkably, the
analytic results obtained matched the numerical data down to spin two.

The _superconformal_ bootstrap has also seen substantial progress. Apart from
numerical explorations of crossing symmetry Poland:2010wg ; Poland:2011ey ;
Poland:2015mta ; Li:2017ddj ; Berkooz:2014yda ; Beem:2014zpa ; Beem:2013qxa ;
Alday:2013opa ; Alday:2014qfa ; Chester:2014fya ; Chester:2014mea ;
Lemos:2015awa ; Chester:2015qca ; Bobev:2015vsa ; Bobev:2015jxa ; Bae:2016jpi
; Beem:2015aoa ; Lin:2015wcg ; Lin:2016gcl ; Beem:2016wfs ; Lemos:2016xke ;
Cornagliotto:2017dup ; Chang:2017xmr ; Chang:2017cdx , the bootstrap line of
thinking helped uncover a solvable subsector in four-dimensional
superconformal theories Beem:2013sza .111See also Beem:2014kka  and
Chester:2014mea ; Beem:2016cbd  for similar results in six and three
dimensions. More precisely, the results of Beem:2013sza  imply that any $4d$
${\mathcal{N}}\geqslant 2$ superconformal field theory (SCFT) contains a
closed subsector isomorphic to a $2d$ chiral algebra.

The subsector captures certain protected quantities, and in order to access
non-protected data the numerical bootstrap is still a necessary tool. In some
cases, similarly to the $3d$ Ising model and $O(N)$ models, known
supersymmetric theories appear on special points, “kinks”, of the numerically
produced exclusion curves, and the numerical machinery of Rattazzi:2008pe  can
be applied in order to extract the CFT data. However, kinks seem to be scarce,
in particular, while the numerical bounds for $4d$ ${\mathcal{N}}=2$ SCFTs
obtained in Beem:2014zpa ; Lemos:2015awa  put strong constraints on the
landscape of theories, they did not single out any particular solution to
crossing.

In this work we focus on the “simplest” four-dimensional ${\mathcal{N}}=2$
Argyres-Douglas SCFT: the $(A_{1},A_{2})$ (or $H_{0}$) theory Argyres:1995jj ;
Argyres:1995xn . It has the lowest possible $c$-anomaly coefficient among
interacting SCFTs Liendo:2015ofa , and the lowest $a$-anomaly coefficient
among the known ones. The $(A_{1},A_{2})$ SCFT can be realized by going to a
special point on the Coulomb branch of an ${\mathcal{N}}=2$ supersymmetric
gauge theory, with gauge group $\mathrm{SU}(3)$, where electric and magnetic
particles become simultaneously massless Argyres:1995jj ; Argyres:1995xn . It
is an isolated ${\mathcal{N}}=2$ SCFT, with no exactly marginal deformations,
and thus no weak-coupling description. As such, despite being known for a very
long time, little is known about the spectrum of this theory. Known data
includes the scaling dimension, $\Delta_{\phi}$, of the single generator of
the Coulomb branch chiral ring, whose vev parametrizes the Coulomb branch, and
the $a$\- and $c$-anomaly coefficients Aharony:2016kai :

$\Delta_{\phi}=\frac{6}{5}\,,\qquad c=\frac{11}{30}\,,\qquad
a=\frac{43}{120}\,.$ (1)

The full superconformal index Kinney:2005ej ; Gadde:2011uv ; Rastelli:2014jja
was recently computed using an ${\mathcal{N}}=1$ Lagrangian that flows to the
$(A_{1},A_{2})$ SCFT in the IR Maruyoshi:2016tqk . The chiral algebra of this
theory is conjectured to be the Yang-Lee minimal model rastelli_harvard ;
Beem:2017ooy , which gives access to the spectrum of a particular class of
short operators, dubbed “Schur” operators. However, the chiral algebra is
insensitive to the Coulomb branch data of the theory, and even though the
dimensions of the operators parameterizing the Coulomb branch chiral ring are
known, not much is known about the values of the corresponding three-point
functions.222See Hellerman:2017sur  for a recent computation of the two-point
function (in normalizations where the OPE coefficients are one) of a Coulomb
branch chiral ring operator, for theories with a single chiral ring generator,
in the limit of large $\mathrm{U}(1)_{r}$ charge.

The relatively low values of its central charge and of the dimension of its
Coulomb branch chiral ring generator make the $(A_{1},A_{2})$ Argyres-Douglas
theory amenable to numerical bootstrap techniques. In fact, one could argue
that this is the ${\mathcal{N}}=2$ SCFT with the best chance to be “solved”
numerically. We approach this theory based on the existing Coulomb branch
data, by considering four-point functions of ${\mathcal{N}}=2$ chiral
operators, whose superconformal primaries are identified with the elements of
the Coulomb branch chiral ring.333Another natural operator to consider in the
correlation functions would be the ${\mathcal{N}}=2$ stress-tensor multiplet,
however, the superconformal blocks for this multiplet are not known, and we
leave this for future work. While the values of $c$ and $\Delta_{\phi}$ in (1)
are not selected by the numerical bootstrap, thanks to supersymmetry they are
exactly known and thus we can use them as input in our analysis. We note
however, that nothing is known about the spectrum of non-supersymmetry
preserving relevant deformations of the $(A_{1},A_{2})$ theory, and this type
of information was essential to corner the $3d$ Ising model to a small
“island” Kos:2014bka .

The results we find are encouraging, and provide the first estimates for
unprotected quantities in this theory. We start by obtaining a lower bound on
the central charge valid for any ${\mathcal{N}}=2$ theory with a Coulomb
branch chiral ring operator of dimension $\Delta_{\phi}=\frac{6}{5}$. This
bound appears to be converging to a value _close_ to $c=\tfrac{11}{30}$,
however the numerics are not conclusive enough. If the bound on $c$ converges
to $\tfrac{11}{30}$, then there is a unique solution to the crossing equations
at $\Delta_{\phi}=\frac{6}{5}$ that corresponds to the $(A_{1},A_{2})$ theory.
If the numerical bound falls short of $\tfrac{11}{30}$, we present evidence,
in the form of valid bounds on OPE coefficients and estimates on operator
dimensions, that the various solutions around $c\sim\tfrac{11}{30}$ do not
look so different, as far as certain observables are concerned. While the
results we obtain are not at the level of the precision numerics of the $3d$
Ising model, we are able to provide estimates for the CFT data of this theory.
For example, we constrain the OPE coefficient of the square of the Coulomb
branch chiral ring generator (after unit normalizing its two-point function)
to lie in the interval

$2.1418\leqslant\lambda_{{\mathcal{E}}_{\frac{12}{5}}}^{2}\leqslant 2.1672\,.$
(2)

While this is a true bound, due to slow convergence it is still far from being
optimal, and will improve as more of the constraints of the crossing equations
are taken into account. In section 3.2 we present estimates for the optimal
range, based on conservative extrapolations of the bounds. Similarly, we
constrain the OPE coefficients of a family of semi-short multiplets, appearing
in the self-OPE of ${\mathcal{N}}=2$ chiral operators, to lie in a narrow
range, quoted in (20) for $\ell=2,4$, and in figure 6 for even spins up to
$\ell=20$.

We also provide in (22) the first estimate of the dimension of the lowest-
lying unprotected scalar appearing in the OPE of the ${\mathcal{N}}=2$ chiral
operator with its conjugate. This operator corresponds to a long multiplet
that is a singlet under $\mathrm{SU}(2)_{R}$ symmetry, and neutral under
$U(1)_{r}$, and we find it is relevant. These estimates are obtained from the
extremal functionals ElShowk:2012hu  that gave rise to the aforementioned OPE
coefficient bounds. From these extremal functionals we also obtain rough
estimates for the dimensions of the lowest-twist long operator for higher
values of the spin, shown in figure 7. Surprisingly, for spin greater than
zero these operators are very close to being double-twist operators, i.e.,
$\Delta=2\Delta_{\phi}+\ell$.

Finally, we make use of the inversion formula of Caron-Huot:2017vep  to obtain
large-spin estimates of the CFT data. As our numerical results are much
further away from convergence than Simmons-Duffin:2016wlq , we refrain from
using them as input in the inversion formula. As such the only input we
provide is the identity and stress-tensor supermultiplet exchange (with the
appropriate central charge). Interestingly, we find that this input already
provides a reasonable estimate of the numerically-bounded quantities for small
spin.

A hybrid approach, combining both the numerical bootstrap and the inversion
formula seems to be the most promising way to proceed, perhaps along the lines
of the one suggested in Simmons-Duffin:2016wlq . The results of this paper are
a first step in this direction, and give us hope that a large amount of CFT
data can be bootstrapped for the $(A_{1},A_{2})$ theory.

##  2 The $(A_{1},A_{2})$ Argyres-Douglas theory

Argyres-Douglas theories Argyres:1995jj ; Argyres:1995xn  were first obtained
by going to a special point on the Coulomb branch of an ${\mathcal{N}}=2$
theory in which several BPS particles, with mutually non-local charges, become
massless simultaneously. Among the various Argyres-Douglas models a particular
class appears to be the “simplest”, that is, the $(A_{1},A_{2n})$ theories
obtained in Eguchi:1996vu . They are rank $n$ theories, i.e., their Coulomb
branches have complex dimension $n$, and have trivial Higgs branches
Argyres:2012fu . The chiral algebras associated to the $(A_{1},A_{2n})$
theories have been conjectured to be the non-unitary series of Virasoro
minimal models ${\mathcal{M}}_{2,2n+3}$ rastelli_harvard ; Beem:2017ooy , and
in this sense the theories could be argued to be “simple”.

In this paper we focus on the $n=1$ case of the $(A_{1},A_{2n})$ Argyres-
Douglas family, which is of rank one and thus the simplest in this class. In
fact, among all interacting rank one SCFTs obtained through the systematic
classification of Argyres:2015ffa ; Argyres:2015gha ; Argyres:2016xua ;
Argyres:2016xmc ; Argyres:2016yzz , it corresponds to the theory with the
smallest $a$-anomaly coefficient,444This assumes the standard lore that the
Coulomb branch chiral ring is freely generated. While this is true for all
known SCFTs, there is no proof of this fact in a generic SCFT. See
Argyres:2017tmj  for an exploration of theories which could have relations on
the Coulomb branch. which provides a measure of degrees of freedom in CFT
Komargodski:2011vj . This theory was originally obtained on the Coulomb branch
of a pure $\mathrm{SU}(3)$ gauge theory, or alternatively from an
$\mathrm{SU}(2)$ gauge theory with a single hypermultiplet Argyres:1995jj ;
Argyres:1995xn . There is no standard nomenclature for this model, and in this
paper we follow the $(A_{1},A_{2})$ naming convention based on its BPS quiver
Cecotti:2010fi . To emphasize its original construction it was also named
$AD_{N_{f}=0}(\mathrm{SU}(3))$ and $AD_{N_{f}=1}(\mathrm{SU}(2))$ in
Tachikawa:2013kta . Finally, it can also be realized in F-theory, on a single
D3-brane probing a codimension one singularity of type $H_{0}$ where the
dilaton is constant Dasgupta:1996ij ; Aharony:1998xz . For this reason the
theory is often referred to as the $H_{0}$ theory.

The $(A_{1},A_{2})$ theory is an intrinsically interacting isolated fixed
point with no marginal coupling: it does not have a conformal manifold nor a
weak-coupling expansion. Recently, there has been progress in obtaining RG
flows from ${\mathcal{N}}=1$ Lagrangian theories that end on Argyres-Douglas
SCFTs in the IR Maruyoshi:2016aim ; Maruyoshi:2016tqk ; Agarwal:2016pjo ;
Agarwal:2017roi ; Benvenuti:2017bpg , and in particular the $(A_{1},A_{2})$
theory can be obtained starting from a deformation of $\mathrm{SU}(2)$
${\mathcal{N}}=2$ superconformal QCD. This allows for the computation of some
information about the theory, such as the superconformal index. As quoted in
(1) the values of the $a$\- and $c$-anomaly coefficients are known, first
obtained through a holographic computation in Aharony:2016kai , and the
dimension of the single generator of the Coulomb branch chiral ring is also
known, and given in (1). Coulomb branch chiral ring operators can be
associated with the scalar primaries of ${\mathcal{N}}=2$ chiral operators,
${\mathcal{E}}_{r}$ multiplets in the notation of Dolan:2002zh , and this
implies this SCFT must contain an operator with
$r_{0}=\Delta_{\phi}=\frac{6}{5}$, as well as its conjugate.

The chiral algebra of the $(A_{1},A_{2})$ theory is conjectured to be the
${\mathcal{M}}_{2,5}$ minimal model rastelli_harvard ; Beem:2017ooy , also
known as the Yang-Lee edge singularity. The first indication of this
conjecture comes from the central charge. The basic chiral algebra dictionary
states that $4d$ and $2d$ central charges are related by $c_{2d}=-12c_{4d}$;
for the $(A_{1},A_{2})$ theory this gives $c_{2d}=-\frac{22}{5}$, which is
indeed the correct value for the Yang-Lee model. Thanks to the interplay
between $2d$ and $4d$ descriptions one can actually prove that
$c_{4d}\geqslant\frac{11}{30}$ for any interacting ${\mathcal{N}}=2$ SCFTs
Liendo:2015ofa .555Similar bounds can be obtained for ${\mathcal{N}}=3$
Cornagliotto:2017dup  and ${\mathcal{N}}=4$ Beem:2013qxa ; Beem:2016wfs
theories, and also for ${\mathcal{N}}=2$ theories with flavor symmetries
Beem:2013sza ; Lemos:2015orc ; Beem:2017ooy . This bound is saturated by the
$(A_{1},A_{2})$ theory, which in some sense sits at the origin of the
${\mathcal{N}}=2$ theory space, as all other interacting SCFTs must have
higher values of the $c$-central charge. Another entry of the chiral algebra
dictionary states that the Schur limit of the superconformal index
Kinney:2005ej ; Gadde:2011uv ; Rastelli:2014jja  should match the $2d$ vacuum
character. For the Yang-Lee minimal model the vacuum character seems to match
the expression for the Schur index proposed in Cordova:2015nma , while the
character of the non-vacuum module has been matched to the index in the
presence of a surface defect Cordova:2017mhb . Using the Yang-Lee model we can
compute three-point functions of Schur operators, i.e., the operators captured
by the chiral algebra, modulo ambiguities when lifting operators from the $2d$
chiral algebra to representations of the four-dimensional superconformal
algebra. A conjectured prescription on how to lift these ambiguities for the
$(A_{1},A_{2})$ Argyres-Douglas theory has been put forward in Song:2016yfd .
Coulomb branch chiral ring operators, however, are not captured by the chiral
algebra.

The features described above suggest that the $(A_{1},A_{2})$ theory might be
the simplest ${\mathcal{N}}=2$ interacting SCFT. Despite this, apart from the
aforementioned quantities not much is known about the CFT data of this theory.
The fact that the theory has a Coulomb branch operator of relatively low
dimension, $r_{0}=\frac{6}{5}$, and a very low $c$ central charge, makes it
well suited for the bootstrap program. Hence, the goal of this paper is to use
modern bootstrap tools in order to access _non-protected_ dynamical data. The
natural first step is to study the two operators that are guaranteed to be
present: the stress tensor, and the ${\mathcal{N}}=2$ chiral operator that
parametrizes the Coulomb branch. Since the superconformal blocks of the former
remain elusive we focus on the latter. A preliminary analysis of chiral
correlators was already started in Beem:2014zpa ; Lemos:2015awa , however the
main goal of those papers was the exploration of the landscape of
${\mathcal{N}}=2$ SCFTs through their Coulomb branch data. In the following
sections we instead focus exclusively on the $(A_{1},A_{2})$ theory, and
attempt to “zoom in” on it by studying an ${\mathcal{N}}=2$ chiral operator of
fixed dimension $r_{0}=\frac{6}{5}$.666There are other known SCFTs with a
Coulomb branch chiral ring operator of dimension $r_{0}=\frac{6}{5}$, in
particular higher rank theories whose lowest dimensional Coulomb branch
generator has this dimension are obtained in F-theory by probing a singularity
of type $H_{0}$ with $N$ D3-branes Dasgupta:1996ij ; Aharony:1998xz . However,
these theories have larger values of the $c$-anomaly coefficient
Aharony:2016kai , and by fixing the central charge we can focus on the
$(A_{1},A_{2})$ theory.

###  2.1 OPE decomposition and crossing symmetry

As we just discussed, our angle to attack the $(A_{1},A_{2})$ theory is
through its Coulomb branch, and thus we are interested in the
${\mathcal{N}}=2$ chiral and anti-chiral operators, respectively
${\mathcal{E}}_{r}$ and $\bar{{\mathcal{E}}}_{r}$ multiplets, using the naming
conventions of Dolan:2002zh . These are short representations of the
superconformal algebra that are half-BPS, where the superconformal primary is
annihilated by all supercharges of one chirality. We denote the superconformal
primary of the chiral (anti-chiral) multiplets ${\mathcal{E}}_{r}$
($\bar{{\mathcal{E}}}_{r}$) by $\phi_{r}$ ($\bar{\phi}_{-r}$), where $r$ is
the $U(1)_{r}$ charge of the superconformal primary, with unitarity requiring
$r\geqslant 1$ ($-r\geqslant 1$). The dimensions of the superconformal
primaries $\phi_{r}$ ($\bar{\phi}_{-r}$) are fixed in terms of their
$U(1)_{r}$ charges by $\Delta_{\phi}=r$ ($\Delta_{\bar{\phi}}=-r$). We refer
the reader to, e.g., Dolan:2002zh , for more on representation theory of the
${\mathcal{N}}=2$ superconformal algebra.

The numerical bootstrap program applied to chiral correlators was considered
in Beem:2014zpa ; Lemos:2015awa  for the case of two identical operators, and
their conjugates, and in Lemos:2015awa  for two distinct operators, and their
conjugates. Here we briefly review the setup for two identical operators
${\mathcal{E}}_{r}$, and conjugates, and refer the reader to Beem:2014zpa ;
Lemos:2015awa  for a more detailed account. Considering all four-point
functions involving the superconformal primaries of these multiplets, we write
down the OPE selection rules and conformal block decompositions for all of the
channels, and the crossing equations to be studied in sections 3 and 4. In
this work we are only concerned with the $(A_{1},A_{2})$ theory and thus we
fix $r$ to $r_{0}=\frac{6}{5}$, according to (1).

####  2.1.1 Non-chiral channel

The OPE selection rules in the non-chiral channel are Beem:2014zpa

$\phi_{r}\times\bar{\phi}_{-r}\sim\mathbf{1}+\hat{{\mathcal{C}}}_{0(j,j)}+{\mathcal{A}}^{\Delta>2j+2}_{0,0(j,j)}\,.$
(3)

Here the $\hat{{\mathcal{C}}}_{0(j,j)}$ multiplets include conserved currents
of spin $2j+2$, which for $j>0$ are absent in interacting theories
Maldacena:2011jn ; Alba:2013yda  and thus we will set them to zero. The
multiplet $\hat{{\mathcal{C}}}_{0(0,0)}$ corresponds to the superconformal
multiplet that contains the stress tensor. By an abuse of notation we will
often replace the subscript $(j,j)$ by $\ell$, with $\ell=2j$. The
superconformal block decomposition in this channel can be written as

$\langle\phi_{r}(x_{1})\bar{\phi}_{-r}(x_{2})\phi_{r}(x_{3})\bar{\phi}_{-r}(x_{4})\rangle=\frac{1}{x_{12}^{2\Delta_{\phi}}x_{34}^{2\Delta_{\phi}}}\sum_{{\mathcal{O}}_{\Delta,\ell}}|\lambda_{\phi\bar{\phi}{\mathcal{O}}}|^{2}{\mathcal{G}}_{\Delta,\ell}(z,\bar{z})\,,$
(4)

where the superblocks ${\mathcal{G}}_{\Delta,\ell}(z,\bar{z})$, capturing the
supersymmetric multiplets being exchanged in (3), were computed in
Fitzpatrick:2014oza ,

${\mathcal{G}}_{\Delta,\ell}(z,\bar{z})=(z\bar{z})^{-\frac{{\mathcal{N}}}{2}}g_{\Delta+{\mathcal{N}},\ell}^{{\mathcal{N}},{\mathcal{N}}}(z,\bar{z})\,.$
(5)

Here we wrote the blocks for ${\mathcal{N}}=1,2$ chiral operators, since both
cases can be treated almost simultaneously Fitzpatrick:2014oza ; Lemos:2015awa
, but hereafter we focus only on the case ${\mathcal{N}}=2$. The function
$g_{\Delta,\ell}^{\Delta_{12},\Delta_{34}}(z,\bar{z})$ is the standard bosonic
block for the decomposition of a correlation function with four distinct
operators, defined in (39). Although not immediately obvious, the bosonic
block with shifted arguments in (5) can be written as a finite sum of
$g_{\Delta,\ell}^{0,0}(z,\bar{z})$ blocks, as expected from supersymmetry. The
block reduces to $1$ for the identity exchange, i.e., $\Delta=\ell=0$.

The stress-tensor multiplet $\hat{{\mathcal{C}}}_{0(0,0)}$ corresponds to
$\Delta=2$, $\ell=0$ in (5), and its OPE coefficient can be fixed using the
Ward identities (see for example Beem:2014zpa ):

$\left|\lambda_{\phi\bar{\phi}{\mathcal{O}}_{\Delta=2,\ell=0}}\right|^{2}=\frac{\Delta_{\phi}^{2}}{6c}\,,$
(6)

while long multiplets ${\mathcal{A}}^{\Delta>\ell+2}_{0,0,\ell}$ contribute as
(5) with $\Delta>\ell+2$.

When writing the crossing equations it will be useful to have the block
expansion with a slightly different ordering

$\langle\bar{\phi}_{-r}(x_{1})\phi_{r}(x_{2})\phi_{r}(x_{3})\bar{\phi}_{-r}(x_{4})\rangle=\frac{1}{x_{12}^{2\Delta_{\phi}}x_{34}^{2\Delta_{\phi}}}\sum_{{\mathcal{O}}_{\Delta,\ell}}(-1)^{\ell}|\lambda_{\phi\bar{\phi}{\mathcal{O}}}|^{2}\tilde{{\mathcal{G}}}_{\Delta,\ell}(z,\bar{z})\,,$
(7)

where the function $\tilde{{\mathcal{G}}}_{\Delta,\ell}(z,\bar{z})(z,\bar{z})$
is defined as

$\tilde{{\mathcal{G}}}_{\Delta,\ell}(z,\bar{z})=(z\bar{z})^{-\frac{{\mathcal{N}}}{2}}g_{\Delta+{\mathcal{N}},\ell}^{{\mathcal{N}},-{\mathcal{N}}}(z,\bar{z})\,,$
(8)

and again we are only interested in the case ${\mathcal{N}}=2$.

####  2.1.2 Chiral channel

The OPE selection rules of two identical ${\mathcal{N}}=2$ chiral primary
operators read Beem:2014zpa

$\phi_{r}\times\phi_{r}\sim{\mathcal{E}}_{2r}+{\mathcal{C}}_{0,2r-1(j-1,j)}+{\mathcal{B}}_{1,2r-1(0,0)}+{\mathcal{C}}_{\frac{1}{2},2r-\frac{3}{2}(j-\frac{1}{2},j)}+{\mathcal{A}}_{0,2r-2(j,j)}^{\Delta>2+2r+2j}\,,$
(9)

where we already imposed Bose symmetry, and we assumed $\phi_{r}$ to be above
the unitarity bound, i.e., $r>1$. If one considers different operators, or if
$r=1$, additional multiplets are allowed to appear (see e.g., Lemos:2015awa ).
Chirality of $\phi_{r}$ requires each supermultiplet contributes with a single
conformal family, and therefore the superblock decomposition contains only
bosonic blocks:

$\langle\phi_{r}(x_{1})\phi_{r}(x_{2})\bar{\phi}_{-r}(x_{3})\bar{\phi}_{-r}(x_{4})\rangle=\sum_{\Delta,\ell}|\lambda_{\phi\phi{\mathcal{O}}_{\Delta,\ell}}|^{2}g^{0,0}_{\Delta,\ell}(z,\bar{z})\,.$
(10)

Since we are considering the OPE between two identical $\phi_{r}$ multiplets,
Bose symmetry requires the above sum to include only even $\ell$. The precise
contribution from each of the multiplets appearing in (9) is the following

$\displaystyle{{\mathcal{A}}_{0,2r-2(j,j)}}$ $\displaystyle:$ $\displaystyle
g^{0,0}_{\Delta,\ell}\,,\qquad$
$\displaystyle\Delta>2+2r+\ell\,,\;\ell\;\mathrm{even}\,,$ (11)
$\displaystyle{{\mathcal{C}}_{\tfrac{1}{2},2r-\tfrac{3}{2}(j-\tfrac{1}{2},j)}}$
$\displaystyle:$ $\displaystyle g^{0,0}_{\Delta=2r+\ell+2,\ell}\,,\qquad$
$\displaystyle\ell\geqslant 2\,,\;\ell\;\mathrm{even}\,,$
$\displaystyle{{\mathcal{B}}_{1,2r-1(0,0)}}$ $\displaystyle:$ $\displaystyle
g^{0,0}_{\Delta=2r+2,\ell=0}\,,\qquad$
$\displaystyle{{\mathcal{C}}_{0,2r-1(j-1,j)}}$ $\displaystyle:$ $\displaystyle
g^{0,0}_{\Delta=2r+\ell,\ell}\,,\qquad$ $\displaystyle\ell\geqslant
2\,,\;\ell\;\mathrm{even}\,,$ $\displaystyle{{\mathcal{E}}_{2r}}$
$\displaystyle:$ $\displaystyle g^{0,0}_{\Delta=2r,\ell=0}\,,\qquad$

where $\ell=2j$ is even (see Lemos:2015awa  for the contribution in the case
of different operators). While the short multiplets being exchanged in this
channel have their dimensions fixed by supersymmetry, their OPE coefficients
are not known. In fact, it is not even guaranteed all these multiplets are
present as their physical meaning is not as clear as the short multiplets
exchanged in the non-chiral channel. The ${\mathcal{E}}_{2r}$ multiplet
corresponds to an operator in the Coulomb branch chiral ring and therefore
must be present in the $(A_{1},A_{2})$ theory, although the value of its OPE
coefficient is not known. The scalar primary of the
${\mathcal{B}}_{1,2r-1(0,0)}$ multiplet may be identified with a mixed branch
chiral ring operator Argyres:2015ffa , and since the $(A_{1},A_{2})$ has no
mixed branch, one might expect this multiplet to be absent. We must point out,
however, that the identification of this multiplet with the mixed branch is
conjectural, it could be that the multiplet is present but it does not
correspond to a flat direction Tachikawa:2013kta ; Argyres:2015ffa . Note that
the contribution of the two short operators ${\mathcal{B}}_{1,2r-1(0,0)}$ and
${\mathcal{C}}_{\frac{1}{2},2r-\frac{3}{2}(j-\frac{1}{2},j)}$ is identical to
that of a long multiplet saturating the unitarity bound $\Delta=2+2r+\ell$, as
follows directly from the decomposition of the long multiplet when hitting the
unitarity bound Dolan:2002zh . On the other hand, the contribution of the
short multiplets ${\mathcal{E}}_{2r}$ and ${\mathcal{C}}_{0,2r-1(j-1,j)}$ is
isolated from the continuous spectrum of long operators by a gap; this will be
relevant for the numerical analysis of section 3.

####  2.1.3 Crossing symmetry

We are now ready to quote the crossing equations to be studied in the
subsequent sections, where we recall only even spins are allowed in the
$\phi\phi$ OPE,

$(z{\bar{z}})^{\Delta_{\phi}}\sum\limits_{{\mathcal{O}}\in\phi\phi}|\lambda_{\phi\phi{\mathcal{O}}}|^{2}g^{0,0}_{\Delta_{\mathcal{O}},\ell_{\mathcal{O}}}(1-z,1-{\bar{z}})=((1-z)(1-{\bar{z}}))^{\Delta_{\phi}}\sum\limits_{{\mathcal{O}}\in\phi\bar{\phi}}|\lambda_{\phi\bar{\phi}{\mathcal{O}}}|^{2}(-1)^{\ell}\tilde{{\mathcal{G}}}_{\Delta_{\mathcal{O}},\ell_{\mathcal{O}}}(z,{\bar{z}})\,,$
(12a)
$((1-z)(1-{\bar{z}}))^{\Delta_{\phi}}\sum\limits_{{\mathcal{O}}\in\phi\bar{\phi}}|\lambda_{\phi\bar{\phi}{\mathcal{O}}}|^{2}{\mathcal{G}}_{\Delta_{\mathcal{O}},\ell_{\mathcal{O}}}(z,{\bar{z}})=(z{\bar{z}})^{\Delta_{\phi}}\sum\limits_{{\mathcal{O}}\in\phi\bar{\phi}}|\lambda_{\phi\bar{\phi}{\mathcal{O}}}|^{2}{\mathcal{G}}_{\Delta_{\mathcal{O}},\ell_{\mathcal{O}}}(1-z,1-{\bar{z}})\,.$
(12b)

The full system of equations comprises (12b) together with equation (12a) with
$z\to 1-z$ and ${\bar{z}}\to 1-{\bar{z}}$. These are collected in a form
suitable for the numerical implementation in (41).

####  2.1.4 Numerical bootstrap

In this short section we give details of the numerical implementation that
will be necessary to understand the results of subsequent sections.
Schematically, the final form of the crossing equations given in (41) is

$|\lambda_{{\mathcal{O}}^{\star}}|^{2}\vec{V}_{{\mathcal{O}}^{\star}}+\sum\limits_{{\mathcal{O}}}|\lambda_{{\mathcal{O}}}|^{2}\vec{V}_{{\mathcal{O}}}+\vec{V}_{\mathrm{fixed}}=0\,.$
(13)

Here ${\mathcal{O}}^{\star}$ is a superconformal multiplet whose OPE
coefficient we would like to bound numerically. The term
$\vec{V}_{\mathrm{fixed}}$ encodes the contribution of the identity, or of the
identity and stress tensor if we fix the central charge $c$, and is given in
(43). OPE coefficient bounds are obtained using the SDPB solver of Simmons-
Duffin:2015qma  to solve the following optimization problem

$\begin{split}&\vec{\Psi}\cdot\vec{V}_{{\mathcal{O}}}\geqslant
0\,,\qquad\forall\;{\mathcal{O}}\in\\{\text{trial spectrum}\\}\,,\\\
&\vec{\Psi}\cdot\vec{V}_{{\mathcal{O}}^{\star}}=\pm 1\,,\\\
&\mathrm{Maximize}\left(\vec{\Psi}\cdot\vec{V}_{\mathrm{fixed}}\right)\,,\end{split}$
(14)

where the minus sign in the second line can be consistently imposed at the
same time as the first line, only when the contribution of
${\mathcal{O}}^{\star}$ is isolated from the contribution of the remaining
${\mathcal{O}}\in\\{\text{trial spectrum}\\}$ Poland:2011ey . As is standard
in the bootstrap literature, we truncate the infinite-dimensional functional
as

$\vec{\Psi}=\sum\limits_{m,n}^{m+n\leqslant\Lambda}\vec{\Psi}_{m,n}\partial_{z}^{m}\partial_{{\bar{z}}}^{n}\Big{|}_{z={\bar{z}}=\frac{1}{2}}\,.$
(15)

The result of the extremization problem (14) provides a bound on the OPE
coefficient of ${\mathcal{O}}^{\star}$ as

$\pm|\lambda_{{\mathcal{O}}^{\star}}|^{2}\leqslant-\mathrm{Max}\left(\Psi\cdot\vec{V}_{\mathrm{fixed}}\right)\,.$
(16)

When the bound is saturated, there is a unique solution to the (truncated)
crossing equations Poland:2010wg ; ElShowk:2012hu , with different
extremization problems possibly leading to different solutions.777See,
however, Behan:2017rca  for subtitles that arise when considering systems of
mixed correlators. At finite $\Lambda$, this corresponds to an approximate
solution to the full crossing system, with the spectrum encoded in the
extremal functional ElShowk:2012hu . We refer the reader to, e.g.,
Poland:2011ey ; Simmons-Duffin:2015qma ; Simmons-Duffin:2016gjk  for more
technical details pertaining the numerical bootstrap.

###  2.2 The Superconformal Index

In the non-chiral channel the OPE coefficients of all short multiplets can be
fixed, as they correspond either to the stress-tensor multiplet
$\hat{{\mathcal{C}}}_{0,0}$, which contributes as (6), or to multiplets
containing conserved currents of spin greater than two
$\hat{{\mathcal{C}}}_{0,\ell>0}$ which are absent in interacting theories
Maldacena:2011jn ; Alba:2013yda . In the chiral OPE selection rules (9), there
appear short multiplets whose dimensions are fixed, but whose OPE coefficients
are not known, and could even be absent in the $(A_{1},A_{2})$ theory. To
access the spectrum of short operators, a useful quantity is the
superconformal index Kinney:2005ej ; Gadde:2011uv ; Rastelli:2014jja . The
index is the most general invariant that counts, with signs, short multiplets
up to multiplets that can recombine to form a long multiplet. Because short
multiplets that have the right quantum numbers to recombine in a long
multiplet give zero contribution, the index has some intrinsic ambiguities.
Relevant to us are the following recombination rules

$\begin{split}{\mathcal{A}}^{2R+r+2j_{2}+2}_{R,r(j_{1},j_{2})}&\to{\mathcal{C}}_{R,r(j_{1},j_{2})}\oplus{\mathcal{C}}_{R+\frac{1}{2},r+\frac{1}{2}(j_{1}-\frac{1}{2},j_{2})}\,,\\\
{\mathcal{A}}^{2R+r+2j_{2}+2}_{R,r(0,j_{2})}&\to{\mathcal{C}}_{R,r(0,j_{2})}\oplus{\mathcal{B}}_{R+1,r+\frac{1}{2}(0,j_{2})}\,,\end{split}$
(17)

where the latter can be seen as a special case of the former with the
identification
${\mathcal{C}}_{R,r(-\frac{1}{2},j_{2})}={\mathcal{B}}_{R+\frac{1}{2},r(0,j_{2})}$.
This means that, while the multiplets ${\mathcal{B}}_{1,2r-1(0,0)}$,
${\mathcal{C}}_{0,2r-1(j-1,j)}$, and
${\mathcal{C}}_{\frac{1}{2},2r-\frac{3}{2}(j-\frac{1}{2},j)}$ appearing in (9)
contribute to the superconformal index, we can only see if these multiplets
appear modulo pairs of the type (17). Note that the ${\mathcal{E}}_{r}$,
$\bar{{\mathcal{E}}}_{-r}$ multiplets themselves can never recombine.

The _full_ superconformal index of the $(A_{1},A_{2})$ theory has been
computed in Maruyoshi:2016tqk  using a $4d$ ${\mathcal{N}}=1$ Lagrangian
theory which flows to the $(A_{1},A_{2})$ theory in the IR.888Various limits
of the superconformal index of Argyres-Douglas theories had been obtained
before in Buican:2015ina ; Buican:2015tda ; Cordova:2015nma ; Song:2015wta .
The final expression for the index is given in integral form in equation (16)
of Maruyoshi:2016tqk , which we use to gather information about the spectrum
of short multiplets in the theory. Expanding said expression we find,
unsurprisingly, that the Coulomb branch chiral ring operators
${\mathcal{E}}_{k\frac{6}{5}}$, with integer $k\geqslant 1$ are present (since
we explored the index in an expansion this was only checked for low values of
$k$). In section 3.2 we shall bound, from above and from below, the OPE
coefficient of the ${\mathcal{E}}_{\frac{12}{5}}$ operator appearing in the
${\mathcal{E}}_{\frac{6}{5}}$ self OPE (see (18)). From the index we also find
that the operator ${\mathcal{C}}_{0,\frac{7}{5}(0,1)}$ is present in the
spectrum, which corresponds to the leading-twist contribution of spin two in
(11). We shall recover this result in section 3.2, since we find a non-zero
lower bound for its OPE coefficient (see (20)). Finally, we find no
contributions arising from a ${\mathcal{B}}_{1,\frac{7}{5}(0,0)}$ multiplet
implying that, modulo the recombination ambiguities of (17), it is absent in
the $(A_{1},A_{2})$ theory. This is consistent with the expectation that such
multiplets may be identified with the existence of a mixed branch
Argyres:2015ffa . Although this is not conclusive evidence for the absence of
this multiplet, we take it as an indication that if there are multiple
solutions to the crossing equations for $r_{0}=\frac{6}{5}$ and
$c=\tfrac{11}{30}$, then the one corresponding to the $(A_{1},A_{2})$ theory
likely has the ${\mathcal{B}}_{1,\frac{7}{5}(0,0)}$ multiplet absent.

##  3 Numerical results

In this section we attempt to zoom in on the $(A_{1},A_{2})$ Argyres-Douglas
theory reviewed in the previous section, following up on the numerical
analysis of the Coulomb branch presented in Beem:2014zpa ; Lemos:2015awa ,
where the landscape of theories with one or two Coulomb branch chiral ring
operators was explored.

An interesting question that was left open in Lemos:2015awa  was whether the
$(A_{1},A_{2})$ theory saturates the numerical lower bounds on the central
charge $c$. While it has been established analytically that this theory has
the lowest possible central charge among interacting ${\mathcal{N}}=2$ SCFTs
Liendo:2015ofa , this does not preclude solutions to the crossing equations
(12b) for $r_{0}=\frac{6}{5}$ and values of $c$ smaller than $\frac{11}{30}$.
The central charge bounds from Beem:2014zpa ; Lemos:2015awa  were obtained for
$\Lambda\leqslant 22$, and there was no particularly clear trend that would
allow for an extrapolation to $\Lambda\to\infty$.

In this work we present improved numerical results, with extrapolations
consistent with, but not definite proof of, the saturation of the $c$-bound by
the $(A_{1},A_{2})$ SCFT. Moreover, our results seem to imply that, even if
there is more than one crossing symmetric four-point function for
$r_{0}=\frac{6}{5}$ and $c=\tfrac{11}{30}$, these solutions do not differ by
much as far as some observables are concerned, and can be used as an
approximation to the low-lying spectrum of the $(A_{1},A_{2})$ theory. In
particular, we are able to obtain the first predictions for unprotected OPE
coefficients in the form of _true_ upper and lower bounds for OPE
coefficients, together with conservative extrapolations for
$\Lambda\to\infty$. In addition, we also estimate the value of the lowest-
twist unprotected long multiplets appearing in the non-chiral OPE. In this
section we focus on the lowest spin operators, but numerical results for
larger spins are presented in section 4, where we compare them to estimates
arising from the Lorentzian inversion formula of Caron-Huot:2017vep  adapted
to the supersymmetric case.

###  3.1 Central charge bound

Figure 1: Numerical lower bound (black dots) on the central charge of theories
with an ${\mathcal{N}}=2$ chiral operator of dimension $r_{0}=\frac{6}{5}$ as
a function of the inverse cutoff $\Lambda$. The lines correspond to various
extrapolations to infinitely many derivatives, and the horizontal dashed line
marks $c=\tfrac{11}{30}$ – the central charge of the $(A_{1},A_{2})$ SCFT.

Our first task is to obtain numerical lower bounds on the central charge
$c_{min}(\Lambda)$, for fixed $r_{0}=\frac{6}{5}$, as a function of the cutoff
$\Lambda$. The resulting bound $c_{min}(\Lambda)$ is shown in figure 1,
together with various different extrapolations to $\Lambda\to\infty$. While
the results are consistent with the bound converging to
$c_{min}=\tfrac{11}{30}$ (the dashed line in figure 1), they are still not
conclusive enough. In what follows we will be agnostic about the
$\Lambda\to\infty$ fate of the $c$-bound, and concentrate on a region around
$c\sim\tfrac{11}{30}$ in an attempt to estimate the CFT data of the
$(A_{1},A_{2})$ theory.

###  3.2 OPE coefficient bounds

We now concentrate on OPE-coefficient bounds for the different short
multiplets appearing in the chiral channel, for varying $c\geqslant
c_{min}(\Lambda)$, and external dimension fixed to $r_{0}=\frac{6}{5}$. In
particular, we obtain an upper bound for the OPE coefficient of the
${\mathcal{B}}_{1,\frac{7}{5}(0,0)}$ multiplet, and both _lower and upper_
bounds (see discussion in subsection 2.1.4) for the coefficients of the
${\mathcal{E}}_{\frac{12}{5}}$ and
$\mathcal{C}_{0,\frac{7}{5}\left(\frac{\ell}{2}-1,\frac{\ell}{2}\right)}$
multiplets.

For fixed $\Lambda$, there is a unique solution to the truncated crossing
equations (12b) at $c=c_{min}(\Lambda)$ Poland:2010wg ; ElShowk:2012hu , and
indeed we will see below that upper and lower bounds (when available)
coincide. As already discussed, it is plausible that
$c_{min}(\Lambda)\to\tfrac{11}{30}$ as $\Lambda\to\infty$, and so in this
limit the meeting point of upper and lower bounds would be at
$c=\frac{11}{30}\simeq 0.367$.

An important subtlety in all the plots that follow is that we cannot fix the
central charge exactly: each time we quote a value of $c$, the corresponding
plot captures values less or equal than the given number. This follows from
the fact that we allow for a continuum of long multiplets with dimensions
consistent with the unitarity bounds; for the non-chiral channel this means
long multiplets with $\Delta>\ell+2$ (3). However, as is clear from the
superconformal blocks (5), the contribution of a long multiplet at the
unitarity bound mimics the contribution of a conserved current
$\hat{{\mathcal{C}}}_{0,\ell}$. This has two important consequences. First, we
cannot restrict ourselves to interacting theories, because it is not possible
to set to zero the OPE coefficient of the conserved currents of spin greater
than two ($\hat{{\mathcal{C}}}_{0,\ell\geqslant 1}$), without imposing a gap
on the spectrum of all long multiplets. Second, even if we fix the central
charge according to (6), a long multiplet at the unitarity bound with an
arbitrary (positive) coefficient, will increase the value of the
$\lambda_{\phi\bar{\phi}{\mathcal{O}}_{\Delta=2,\ell=0}}$ coefficient, which
means that we are really allowing for all central charges smaller than the
fixed value. This implies that a given bound can only get weaker as $c$ is
increased, and explains the flatness of some of the bounds presented below.

#### OPE coefficient bound for ${\mathcal{B}}_{1,\frac{7}{5}(0,0)}$

Figure 2: Numerical upper bound on the OPE coefficient squared of the operator
${\mathcal{B}}_{1,\frac{7}{5}(0,0)}$ appearing in the chiral channel for
$\Lambda=26,\ldots,40$, and external dimension $r_{0}=\frac{6}{5}$. Left:
Upper bound on the OPE coefficient for different values of the central charge,
with the strongest bound corresponding to $\Lambda=40$; the dashed lines mark
the minimum central charge as extracted from figure 1 for each cutoff
$\Lambda$, and the solid line marks $c=\tfrac{11}{30}$. Right: Bound on the
OPE coefficient for $c=\tfrac{11}{30}$ as a function of the inverse cutoff
$\Lambda$, together with various extrapolations to infinitely many
derivatives.

Let us first consider the OPE coefficient squared of
${\mathcal{B}}_{1,\frac{7}{5}(0,0)}$. A numerical upper bound, as a function
of the central charge, and for fixed external dimension $r_{0}=\frac{6}{5}$,
is shown on the left-hand side of figure 2 for various values of the cutoff
$\Lambda$. For each value of $\Lambda$, the upper bound vanishes for
$c=c_{min}(\Lambda)$ (marked by the dashed vertical lines in the figure), and
becomes negative for $c<c_{min}(\Lambda)$, implying there is no unitary
solution to the crossing equations. This is consistent with what was found for
$\Lambda=12$ in Lemos:2015awa , and suggests this operator is responsible for
the existence of the central charge bound. Since such a multiplet is
associated with the mixed branch, and the $(A_{1},A_{2})$ theory has no mixed
branch, it would be natural to expect its absence to be a feature of the four-
point function of the $(A_{1},A_{2})$ theory. However, as can be seen on the
right-hand side of figure 2, the numerical results appear to leave room for
solutions to crossing with a small value of this OPE coefficient, as it is not
clear if the upper bound will converge to zero as $\Lambda\to\infty$. If there
is more than one solution, it is plausible that the one corresponding to the
$(A_{1},A_{2})$ theory is one in which ${\mathcal{B}}_{1,\frac{7}{5}(0,0)}$
has zero OPE coefficient. We should point out though, that the absence of a
mixed branch does not guarantee that the aforementioned multiplet is absent,
as it is possible that such a multiplet is present, but one cannot give it a
vev and thus no mixed branch exists Tachikawa:2013kta ; Argyres:2015ffa .

#### OPE coefficient bound for ${\mathcal{E}}_{\frac{12}{5}}$

Turning to the OPE coefficient of the Coulomb branch chiral ring operator
${\mathcal{E}}_{\frac{12}{5}}$, we can now place upper and lower bounds as a
function of the central charge. We present the results on the left-hand side
of figure 3 for several values of $\Lambda$.

Figure 3: Numerical upper and lower bounds on the OPE coefficient squared of
the chiral operator ${\mathcal{E}}_{\frac{12}{5}}$ for increasing number of
derivatives and external dimension $r_{0}=\frac{6}{5}$. Left: Bounds on the
OPE coefficient for different values of the central charge, with cutoffs
$\Lambda=26,\ldots,40$, the vertical line marks $c=\tfrac{11}{30}$. Right:
Various extrapolations of the lower and upper bounds at $c=\tfrac{11}{30}$ for
infinite $\Lambda$.

As already discussed, the plots in this section allow for all central charges
$c\geqslant c_{\mathrm{fixed}}$, since a gap in the spectrum of spin zero long
multiplets is not imposed. This explains the flatness of the upper bound:
solutions to crossing saturating it can effectively have central charges equal
to $c_{min}(\Lambda)$.999A natural solution would be to impose small gaps in
the spectrum of long multiplets, this removes the conserved currents of spin
greater than two and fixes the central charge. However, we have no intuition
on the size of these gaps, not even for spin zero, as there is no
understanding of the number of non-supersymmetry preserving relevant
deformations. We experimented imposing that the spectrum of long multiplets
obeys $\Delta\geqslant 2+\epsilon+\ell$ for various small values of
$\epsilon$, and although the upper bound gets stronger than that of figure 3,
it varies smoothly with $\epsilon$ and thus there is no justification to pick
any specific value. The lower bound, on the other hand, shows a much smaller
dependence on $\epsilon$. The lower bound, however, must be saturated by
theories with central charge equal to the fixed value. At $c_{min}(\Lambda)$
the upper and lower bounds coincide, fixing a unique value of the OPE
coefficient, and as $c$ is increased a wider range of values, and distinct
solutions to crossing, are allowed. We show the allowed range, for
$c=\tfrac{11}{30}$, as a function of $1/\Lambda$ on the right-hand side of
figure 3. The lines correspond to different extrapolations through (subsets)
of the data points, and the shaded region aims to give an idea of where the
bounds are converging to. If $c_{min}(\Lambda\to\infty)=\tfrac{11}{30}$, then
the upper and lower bound should converge to the same value, which is not
ruled out by the extrapolations. In any case, our results indicate that the
OPE coefficient of ${\mathcal{E}}_{\frac{12}{5}}$ is constrained to a narrow
range.

We have thus obtained the following _rigorous_ bounds for the value of this
OPE coefficient in the $(A_{1},A_{2})$ theory:

$2.1418\leqslant\lambda_{{\mathcal{E}}_{\frac{12}{5}}}^{2}\leqslant
2.1672\,,\qquad\mathrm{for}\;\Lambda=40\,.$ (18)

Furthermore, the most conservative of the extrapolations presented in figure 3
gives

$2.146\lesssim\lambda_{{\mathcal{E}}_{\frac{12}{5}}}^{2}\lesssim
2.159\,,\qquad\text{extrapolated for}\;\Lambda\to\infty\,.$ (19)

#### OPE coefficient bounds for ${\mathcal{C}}_{0,\frac{7}{5}(0,1)}$ and
${\mathcal{C}}_{0,\frac{7}{5}(1,2)}$

(a) ${\mathcal{C}}_{0,\frac{7}{5}(0,1)}$

(b) ${\mathcal{C}}_{0,\frac{7}{5}(1,2)}$

Figure 4: Numerical upper and lower bounds on the OPE coefficient squared of
the chiral channel multiplet
$\mathcal{C}_{0,\frac{7}{5}\left(\frac{\ell}{2}-1,\frac{\ell}{2}\right)}$, for
$\ell=2,4$, and external dimension $r_{0}=\frac{6}{5}$. The bounds were
obtained for cutoffs $\Lambda=26,\ldots,34$ and the vertical line marks
$c=\tfrac{11}{30}$. The dashed line corresponds to the value obtained from the
Lorentzian inversion formula of Caron-Huot:2017vep  applied to the chiral
channel and using as input only the exchange of the identity and stress-tensor
superblocks in the non-chiral channel, and thus valid for sufficiently large
$\ell$ (see section 4.1 for more details).

Let us now focus on the
$\mathcal{C}_{0,\frac{7}{5}\left(\frac{\ell}{2}-1,\frac{\ell}{2}\right)}$
family of multiplets. Like in the ${\mathcal{E}}_{\frac{12}{5}}$ case, upper
and lower bounds are possible thanks to the gap that separates these
${\mathcal{C}}$-type multiplets from the continuum of long operators. The
bounds for $\ell=2,4$, as a function of $c$, are shown in figures 4(a) and
4(b) respectively, while bounds for higher values of $\ell$ can be found in
figure 6, for fixed $c=\tfrac{11}{30}$. The dashed lines in figure 4 are
estimates of the OPE coefficient valid for sufficiently large $\ell$, that
will be discussed in detail in section 4.1. Similarly to the
${\mathcal{E}}_{\frac{12}{5}}$ multiplet, the OPE coefficients of these
multiplets in the $(A_{1},A_{2})$ theory are constrained to lie in a narrow
range:

$0.46831\leqslant\lambda_{{\mathcal{C}}_{0,\frac{7}{5},(0,1)}}^{2}\leqslant
0.46901\,,\qquad
0.048919\leqslant\lambda_{{\mathcal{C}}_{0,\frac{7}{5},(1,2)}}^{2}\leqslant
0.048945\,,\qquad\text{for}\;\Lambda=34\,.$ (20)

The upper bounds in figure 4 now show a mild dependence on the central charge,
and so we can compare the extrapolations of the upper and lower bounds at
$c=\tfrac{11}{30}$ with the extrapolation of the value of the OPE coefficient
for the unique solution at $c_{min}(\Lambda)$. Like before, the extrapolations
(not shown) do not rule out that $c_{min}\to\tfrac{11}{30}$ as
$\Lambda\to\infty$. As visible in figure 4, the value of the OPE coefficient
at $c_{min}(\Lambda)$ (the meeting point) shows a very mild dependence on
$\Lambda$, unlike the OPE coefficient of ${\mathcal{E}}_{\frac{12}{5}}$, we
can therefore obtain the following estimates

$\begin{split}0.4687&\lesssim\lambda_{{\mathcal{C}}_{0,\frac{7}{5}(0,1)}}^{2}(c=c_{min}(\Lambda))\lesssim
0.4688\,,\\\
0.04892&\lesssim\lambda_{{\mathcal{C}}_{0,\frac{7}{5}(1,2)}}^{2}(c=c_{min}(\Lambda))\lesssim
0.04894\,,\\\ \end{split}\qquad\text{extrapolated for }\;\Lambda\to\infty\,.$
(21)

###  3.3 Dimensions of unprotected operators

(a) Dimension of first long multiplet in the non-chiral channel arising from
different bounds.

(b) Dimension of first long multiplet in the chiral channel arising from
different bounds.

Figure 5: Numerical estimates for the first scalar long operator in the non-
chiral (a) and chiral (b) channels obtained from the functionals of figures 1,
2, 3, and 4. The data points are color-coded according to the bound the
extremal functional was extracted from, and in the cases where the bounds are
plotted as a function of $c$, the functional for $c=\tfrac{11}{30}$ was used.
The lines give an estimate of the extrapolation to infinitely many
derivatives.

Finally, we estimate dimensions of unprotected long operators. In Beem:2014zpa
; Lemos:2015awa  numerical upper bounds on the dimensions of the first long in
the non-chiral and chiral channels were obtained, for various values of $c$
and $r_{0}$. The best bound obtained in Beem:2014zpa  for the dimension of the
first scalar long operator in the non-chiral channel reads
$\Delta_{\phi\bar{\phi}}\leqslant 2.68$, for $\Lambda=18$, $r_{0}=\frac{6}{5}$
and $c\leqslant\tfrac{11}{30}$. On the other hand, the bound obtained for the
first scalar long operator in the chiral channel was very weak and converged
too slowly without further assumptions (figure 2 of Lemos:2015awa ). Removing
the ${\mathcal{B}}_{1,\frac{7}{5}(0,0)}$ multiplet this bound improved to
$\Delta_{\phi\phi}\leqslant 4.93$ for $\Lambda=20$, $r_{0}=\frac{6}{5}$, and
did not appear to depend on $c$ Lemos:2015awa .

Here, instead, we extract the dimensions of the first long
${\mathcal{A}}_{0,0,\ell}^{\Delta_{\phi\bar{\phi}}}$ and
${\mathcal{A}}_{0,\frac{2}{5},\ell}^{\Delta_{\phi\phi}}$ multiplets in the
approximate solutions to crossing saturating the various bounds presented
above. The results for $\ell=0$, and various values of the cutoff $\Lambda$,
are given in figure 5(a) for $\Delta_{\phi\bar{\phi}}$, and in figure 5(b) for
$\Delta_{\phi\phi}$. The dimensions were extracted from the extremal
functionals ElShowk:2012hu  of figures 2–4 for $c=\tfrac{11}{30}$, and from
figure 1, and are color coded according to the bound they came from. Note that
these are the dimensions of the first long operator present in each of the
extremal solutions, and are not rigorous upper bounds, which would require a
different extremization problem. The lines in the figures show various
extrapolations of the dimensions to $\Lambda\to\infty$.

The estimates of the dimensions of the lowest-lying long multiplet in the non-
chiral channel appear all consistent with each other, even at finite
$\Lambda$. This implies that, even if the various extremization problems are
solved by different solutions to the crossing equations, these solutions do
not differ by much as far as $\Delta_{\phi\bar{\phi}}$ is concerned, and we
can take the spread of the values as an estimate for the uncertainty in the
value of this dimension. Then, conservative extrapolations for
$\Lambda\to\infty$ of the values coming from the various functionals give

$2.56\lesssim\Delta_{\phi\bar{\phi}}\lesssim 2.68\,,\qquad\text{from the
extrapolations as }\Lambda\to\infty\,,$ (22)

for $\Delta_{\phi\bar{\phi}}$ in the $(A_{1},A_{2})$ theory. Similarly, the
dimensions of the leading twist non-chiral operators with spin $\ell>0$,
obtained from the various extremal functionals of figures 1–4, are shown in
figure 7 for $\Lambda=34$. We will comment on these results in section 4.2.

Less coherent are the results for $\Delta_{\phi\phi}$, the values extracted
from the extremal functionals of figures 1–4 look very different for finite
$\Lambda$, and the extrapolations are not conclusive. This is shown in figure
5(b). Since the dimensions obtained are so disparate, it is not clear we can
get any meaningful estimate for this operator in the $(A_{1},A_{2})$ theory.

##  4 Inverting the OPEs

In recent years, starting from Fitzpatrick:2012yx ; Komargodski:2012ek , there
has been much progress in understanding the large spin spectrum of CFTs by
studying analytically the crossing equations in a Lorentzian limit. In $d>2$,
by studying the four-point function
$\langle\phi_{1}(x_{1})\phi_{1}(x_{2})\phi_{2}(x_{3})\phi_{2}(x_{4})\rangle$
in the lightcone limit, the authors of Fitzpatrick:2012yx ; Komargodski:2012ek
found that in the $t-$channel there must be exchanged, in a distributional
sense, double-twist operators, i.e., operators whose dimensions approach
$\Delta_{1}+\Delta_{2}+\ell$ for $\ell\to\infty$, and whose OPE coefficients
tend to the values of generalized free field theory. Corrections to these
dimensions and OPE coefficients, or rather weighted averages of these
quantities, in a large spin expansion can be obtained in terms of the leading
twist operators exchanged in the $s-$channel.101010Some of the steps taken in
the derivations Fitzpatrick:2012yx ; Komargodski:2012ek  rely on intuitive
assumptions, some of which have started to be put on a firm footing in
Qiao:2017xif . Assuming the existence of individual operators close to the
average values, this procedure was set up to systematically compute the OPE
coefficients and dimensions of the double-twist operators in an asymptotic
expansion in the inverse spin Alday:2015ewa . The lightcone limit of the
crossing equations has been used to constrain the large spin spectrum of
various CFTs with different global symmetries and supersymmetries
Fitzpatrick:2012yx ; Komargodski:2012ek ; Alday:2013cwa ; Fitzpatrick:2014vua
; Vos:2014pqa ; Fitzpatrick:2015qma ; Kaviraj:2015cxa ; Alday:2015eya ;
Kaviraj:2015xsa ; Alday:2015ota ; Beem:2015aoa ; 1612.03891 ; 1603.05150 ;
1611.01500 ; 1612.00696 ; 1510.08091 ; 1602.04928 ; 1511.08025 . Remarkably,
this has resulted in predictions for OPE coefficients and anomalous dimensions
of operators that match the numerical results down to spin two Simmons-
Duffin:2016wlq ; Alday:2015ota .

Recently, the work of Caron-Huot:2017vep  has explained this agreement, by
showing that the spectrum organizes in analytic families.111111We thank Marco
Meineri for many discussions on Caron-Huot:2017vep . There, a “Lorentzian”
inversion formula for the $s-$channel OPE of a given correlator was obtained,
with the crucial feature that the result of the inversion is a function that
is analytic in spin (a function valid only for spin greater than one). This
established, for sufficiently large spin, the existence of each individual
double-twist operator. The inversion formula explained the organization of the
spectrum, and allows one to compute individual OPE coefficients and anomalous
dimensions, avoiding the asymptotic expansions, and obtaining the coefficients
instead of averages.

Motivated by the success of Simmons-Duffin:2016wlq , we take the first steps
towards a systematic analysis of the $(A_{1},A_{2})$ theory for large spin,
considering both crossing equations (12b). We apply the inversion formula
obtained in Caron-Huot:2017vep  to invert the chiral (10) and non-chiral (4)
OPEs. The block decomposition of the former happens to be simply a
decomposition in bosonic blocks (11), and thus the inversion formula directly
applies. The latter has a decomposition in superblocks, but as we shall see,
the formula can still be applied, although we must work as if we had a
correlator of unequal external operators. The only required modifications will
be on the spin down to which the formula holds, and on what the crossed-
channel decompositions are. Since the numerical results in supersymmetric
theories are not yet at the level of accuracy of the $3d$ Ising model, we
refrain from using numerical data as input to the analysis, and instead
compare the large-spin estimates coming from the inversion formula with the
numerical results. The only input we give is the exchange of the identity and
stress-tensor supermultiplet in the non-chiral OPE, and thus find results that
are good estimates for sufficiently large spin. We find a reasonable agreement
between the OPE coefficients of the leading-twist short operators exchanged in
the chiral channel, and the analytical estimate, already for low spin, see
figure 6. Our analysis also shows that the anomalous dimensions of the double-
twist operators in the non-chiral channel, arising from the stress tensor
exchange, are small, and we confirm this by matching to numerical estimates of
these dimensions obtained from the bounds of section 3 (see figure 7).

We start with a brief summary of the results of Caron-Huot:2017vep  relevant
for our purposes and refer the reader to that reference for further details.
Starting from the four-point function of unequal scalar operators,

$\langle{\mathcal{O}}_{1}(x_{1}){\mathcal{O}}_{2}(x_{2}){\mathcal{O}}_{3}(x_{3}){\mathcal{O}}_{4}(x_{4})\rangle=\frac{1}{|x_{12}|^{\Delta_{1}+\Delta_{2}}|x_{34}|^{\Delta_{3}+\Delta_{4}}}\left|\frac{x_{14}}{x_{24}}\right|^{\Delta_{21}}\left|\frac{x_{14}}{x_{13}}\right|^{\Delta_{34}}{\mathcal{G}}(z,{\bar{z}})\,,$
(23)

the main result is a “Lorentzian” inversion formula for the $s-$channel
decomposition of ${\mathcal{G}}(z,{\bar{z}})$ in conformal blocks,

${\mathcal{G}}(z,{\bar{z}})=\sum\limits_{\Delta,\ell}\lambda_{\Delta,\ell}^{2}g^{\Delta_{12},\Delta_{34}}_{\Delta,\ell}(z,{\bar{z}})\,.$
(24)

The OPE coefficients $\lambda_{\Delta,\ell}^{2}$ (for $\ell>1$) of the above
decomposition (24) are then encoded in the residues of a function
$c(\ell,\Delta)$ that is analytic in spin, contrasting with the one that can
be obtained from a Euclidean inversion of the OPE. The condition $\ell>1$
arises during the contour manipulations needed to go from the Euclidean
inversion of the OPE, valid only for integer $\ell$, to the “Lorentzian”
formula of Caron-Huot:2017vep . This condition requires looking at the $t-$
and $u-$ channels to bound the growth of ${\mathcal{G}}(z,{\bar{z}})$ in a
particular region, and it is valid for any unitary CFT. The function
$c(\ell,\Delta)$ receives contributions from the $t-$ and $u-$channels, with
the even and odd spin operators defining two independent trajectories, as

$c(\ell,\Delta)=c^{t}(\ell,\Delta)+(-1)^{\ell}c^{u}(\ell,\Delta)\,,$ (25)

where $c^{t}$ and $c^{u}$ are defined in (3.20) of Caron-Huot:2017vep .

The poles of $c(\ell,\Delta)$ in $\Delta$, at fixed $\ell$, encode the
dimensions of the operators in the theory, with the residues giving the OPE
coefficients.121212In some cases the residues need to be corrected as
discussed in (3.9) of Caron-Huot:2017vep , but for the computations carried
out in this section this correction is not needed. As described in section 3.2
of Caron-Huot:2017vep , if one is interested only in getting the poles and
residues, the inversion formula can be written as

$\begin{split}c^{t}(\ell,\Delta)\big{|}_{\mathrm{poles}}&=\int\limits_{0}^{1}\frac{\mathrm{d}z}{2z}z^{\frac{\ell-\Delta}{2}}\left(\sum\limits_{m=0}^{\infty}z^{m}\sum\limits_{k=-m}^{m}B_{\ell,\Delta}^{(m,k)}C^{t}(z,\ell+\Delta+2k)\right)\,,\\\
C^{t}(z,\beta)&=\int\limits_{z}^{1}\mathrm{d}{\bar{z}}\frac{(1-{\bar{z}})^{\frac{\Delta_{21}+\Delta_{34}}{2}}}{{\bar{z}}^{2}}\kappa_{\beta}k_{\beta}^{\Delta_{12},\Delta_{34}}({\bar{z}})\mathrm{dDisc}\left[{\mathcal{G}}(z,{\bar{z}})\right]\,,\end{split}$
(26)

and similarly for $c^{u}(\ell,\Delta)$. Here $\mathrm{dDisc}$ denotes the
double-discontinuity of the function,
$k_{\beta}^{\Delta_{12},\Delta_{34}}({\bar{z}})$ is defined in equation (39),
and

$\kappa_{\beta}=\frac{\Gamma\left(\frac{\beta-\Delta_{21}}{2}\right)\Gamma\left(\frac{\Delta_{21}+\beta}{2}\right)\Gamma\left(\frac{\beta-\Delta_{34}}{2}\right)\Gamma\left(\frac{\Delta_{34}+\beta}{2}\right)}{2\pi^{2}\Gamma(\beta-1)\Gamma(\beta)}\,.$
(27)

The $z\to 0$ limit of the block in (26) gave the collinear block
$k_{\beta}^{\Delta_{12},\Delta_{34}}({\bar{z}})$ which does not take into
account all descendants, these are instead taken into account by the functions
$B_{\ell,\Delta}^{(m,k)}$, as discussed in Caron-Huot:2017vep . Since we shall
focus only on leading twist operators we do not need to subtract descendants
and thus do not need these functions, apart from $B_{\ell,\Delta}^{(0,0)}=1$.
A term $z^{\frac{\tau(\Delta+\ell)}{2}}$ in the bracketed term in (26) implies
there exists a pole at $\Delta-\ell=\tau(\Delta+\ell)$, with its residue,
taken at fixed $\ell$, providing the OPE coefficient; see Caron-Huot:2017vep
for more details.

###  4.1 Inverting the chiral OPE

The Lorentzian inversion formula obtained in Caron-Huot:2017vep  can be
directly applied to invert the $s-$channel OPE of the correlator (10),

$\langle\phi(x_{1})\phi(x_{2})\bar{\phi}(x_{3})\bar{\phi}(x_{4})\rangle=\frac{1}{x_{12}^{2\Delta_{\phi}}x_{34}^{2\Delta_{\phi}}}\sum\limits_{{\mathcal{O}}_{\Delta,\ell}}|\lambda_{\phi\phi{\mathcal{O}}_{\Delta,\ell}}|^{2}g_{\Delta,\ell}^{0,0}(u,v)\,,$
(28)

as it is exactly of the form (23), with ${\mathcal{G}}(z,{\bar{z}})$ admitting
a decomposition in bosonic blocks, with $\Delta_{12}=\Delta_{34}=0$. We can
thus apply (25) directly, with the $t-$ and $u-$channel decompositions as
dictated by crossing symmetry of (28). Since the operators at point one and
two are identical, the $u-$ and $t-$ channels give identical contributions,
and thus only even spins appear in the $s-$channel OPE of (28), precisely in
agreement with Bose symmetry.

We now want to make use of the generating functional (26), to obtain the
dimensions and OPE coefficients of the $s-$channel operators (at least for
large enough spin) by providing information about the $t-$channel
decomposition. For large spin (that is large $\beta$) the leading
contributions in (26) come from the ${\bar{z}}\to 1$ limit of the integrand,
with the leading contribution corresponding to the lowest twist operators
exchanged in the $t-$channel. The $t-$channel decompositions are given by the
non-chiral OPE, as follows from (12a). From (3) we see that, after the
identity, the leading contribution comes from the superconformal multiplet of
the stress tensor $\hat{{\mathcal{C}}}_{0,0}$, and, since we are interested on
interacting theories, there is no other contribution with the same twist. The
next contributions will arise from long multiplets, for which we only
currently have the numerical estimates for their dimensions obtained in
section 3.3. At large spin the contributions of one of these operators of
twist $\tau$ behaves as $\ell^{-\tau}$ Fitzpatrick:2012yx ; Caron-Huot:2017vep
.131313Here we are using the bosonic results of Fitzpatrick:2012yx ; Caron-
Huot:2017vep , while we have a superblock contribution at twist $\tau$.
However, decomposing the superblock in bosonic blocks we find a finite number
of bosonic blocks with twist $\tau$ together with a finite number of higher
twist, and so the presence of the superblock will only modify the coefficient
of the leading behavior for large $\ell$, which is unimportant for our point
here. The leading twist operators have been estimated numerically from the
various extremal functionals obtained in section 3. The leading spin zero
operator could have twist as low as $\tau\sim 2.5$, while the higher spin
operators (figure 7) all appear to have twists close to $2r_{0}=2.4$, with
small corrections depending on the spin. This is to be compared with the
contribution of the stress tensor with exactly $\tau=2$. For sufficiently
large spin the contributions of long multiplets are subleading, so in what
follows we shall consider only the stress-tensor and identity exchanges in the
$t-$ and $u-$channels.

The identity and stress tensor contribute to (26) according to the crossing
equation (12a), with the identity contributing as
$|\lambda_{\phi\bar{\phi}1}|^{2}\tilde{{\mathcal{G}}}_{0,0}(u,v)=1$. The
stress-tensor superblock is given by (8) with $\Delta=2$, $\ell=0$, and with
OPE coefficient given by (6), and we find

$C^{t}(z,\beta)\supset\int\limits_{0}^{1}\frac{\mathrm{d}{\bar{z}}}{{\bar{z}}^{2}}\kappa_{\beta}k_{\beta}^{\Delta_{12},\Delta_{34}}({\bar{z}})\mathrm{dDisc}\left[\frac{(z{\bar{z}})^{r_{0}}}{((1-z)(1-{\bar{z}}))^{r_{0}}}\left(1+|\lambda_{\phi\bar{\phi}\hat{{\mathcal{C}}}_{0,0}}|^{2}\tilde{{\mathcal{G}}}_{2,0}(1-z,1-{\bar{z}})\right)\right]\,.$
(29)

From the identity contribution, which is the leading one for large spin, we
recover the existence of double-twist operators $[\phi\phi]_{m,\ell}$ (see for
example section 4.2 of Caron-Huot:2017vep ), namely operators with dimensions
approaching

$\Delta_{[\phi\phi]_{m,\ell}}\underset{\ell\gg
1}{\longrightarrow}2r_{0}+2m+\ell\,,\qquad\ell\;\mathrm{even}\,,$ (30)

and with OPE coefficients approaching those of generalized free field theory,

$\lambda_{\mathrm{gfft}}^{2}=\frac{\left((-1)^{\ell}+1\right)\left(\left(r_{0}-1\right)_{m}\right){}^{2}\left(\left(r_{0}\right)_{m+\ell}\right){}^{2}}{m!\ell!\left(m+2r_{0}-3\right)_{m}(\ell+2)_{m}\left(m+\ell+2r_{0}-2\right)_{m}\left(2m+\ell+2r_{0}-1\right)_{\ell}}\,,$
(31)

at large spin. In (31) $(a)_{b}$ denotes the Pochhammer symbol.

To compute the leading correction to these dimensions and OPE coefficients at
large spin we take into account the contribution of the stress-tensor
multiplet to the OPE. To do so we take the $z\to 0$ limit of (29); as pointed
in Caron-Huot:2017vep , the correct procedure should be to subtract a known
sum, such that the limit $z\to 0$ commutes with the infinite sum over
$t-$channel primaries. However, when anomalous dimensions are small this
procedure gives small corrections to the naive one of taking a series
expansion in $z$ and extracting anomalous dimensions from the terms
proportional to $\mathrm{log}z$ (the generating function should have
$z^{\gamma/2}\approx 1+\tfrac{1}{2}\gamma\,\mathrm{log}z+\ldots$) and
corrections to OPE coefficients from the terms without $\mathrm{log}z$. For
the case considered below the situation is even better as the anomalous
dimensions of the operators we are interested in vanish. Taking the small $z$
limit, the first observation is that anomalous dimensions, i.e., log-terms,
only come with a power of $z^{\Delta_{\phi}+2}$, and thus only the operators
$[\phi\phi]_{m\geqslant 2,\ell}$ acquire an anomalous dimension. This is
consistent with the fact that from the block decomposition (11) we identify
the double-twist operators with $m=0,1$ as short multiplets,
${\mathcal{C}}_{0,2r_{0}-1(\frac{\ell}{2}-1,\frac{\ell}{2})}$ and
${\mathcal{C}}_{\frac{1}{2},2r_{0}-\frac{3}{2}(\frac{\ell}{2}-\frac{1}{2},\frac{\ell}{2})}$
respectively, whose dimensions are protected.141414We assume $\ell\geqslant 2$
here since the inversion formula is not guaranteed to converge for $\ell=0$.

Figure 6: Comparison between the numerical bounds on the OPE coefficient
squared of the leading twist operators in the chiral channel and the results
from the inversion formula (29), for $c=\tfrac{11}{30}$ and external dimension
$r_{0}=\frac{6}{5}$. The black boxes mark the numerically allowed range for
the squared OPE coefficients of the
$\mathcal{C}_{0,\frac{7}{5},\left(\frac{\ell}{2}-1,\frac{\ell}{2}\right)}$
operators for different values of $\ell$ (the $\ell=0$ operator should be
interpreted as ${\mathcal{E}}_{\frac{12}{5}}$) and with $\Lambda=36$. The
dashed line shows the result of equation (29), where we considered only the
contribution of the identity and stress-tensor operators in the non-chiral
channel, and thus is an approximate result for sufficiently large spin. The
formula (29) is not guaranteed to be valid for $\ell=0$, and the results here
are just shown as an illustration.

We can now compute corrections to the OPE coefficient of the
$\mathcal{C}_{0,2r_{0}-1,\left(\frac{\ell}{2}-1,\frac{\ell}{2}\right)}$
operators, for $r_{0}=\frac{6}{5}$ and $c=\tfrac{11}{30}$, from (29). The
result is plotted in figure 6, where we performed the integral in (29)
numerically (after taking the leading $z$ term), together with the numerical
upper and lower bonds on the OPE coefficients, obtained in section
3.2.151515Note that by the usual lightcone methods Fitzpatrick:2012yx ;
Komargodski:2012ek ; Alday:2015ewa , we could obtain an asymptotic expansion
in $\tfrac{1}{\ell}$ of the correction to the generalized free field theory
OPE coefficients (31) arising from the stress tensor exchange. By considering
the contributions of the stress tensor superblock to (26) we are effectively
re-summing the lightcone expansion to all orders. The results for $\ell=0$
(where the multiplet becomes an ${\mathcal{E}}_{2r_{0}}$) are also shown even
though the formula is only guaranteed to be valid for $\ell\geqslant
2$.161616The formula could only be valid for $\ell=0$ if, for some reason, the
growth of the four-point function of the $(A_{1},A_{2})$ theory, in the limit
relevant for the dropping of arcs of integration along the derivation of the
inversion formula, was better than the generic growth expected in any CFT and
derived in Caron-Huot:2017vep .

We point out that the only input was the leading $t-$ and $u-$channels
contributions and thus the resulting OPE coefficients are an approximation for
sufficient large spin. Indeed, the neglected contribution of the long
multiplets should behave like $\ell^{-\tau}$ for large spin, with $\tau\sim
2.4$, while the stress tensor contributes as $\tau=2$. Nevertheless, we see in
figure 6 that starting from $\ell=4$ the analytical result is already inside
the numerically allowed range for the OPE coefficient. This is shown clearly
in figure 4(b) where the result of (29) for $\ell=4$ is shown as a dashed blue
line, together with the numerically allowed range. For $\ell=2$, however, the
result of (29) (blue dashed line in figure 4(a)) is clearly insufficient, as
it is outside the numerically allowed region. Note that the numerical results
are not optimal yet, i.e., while they provide true bounds they have not yet
converged, and the optimal bounds will be more restrictive. Thus, the fact
that the $\ell=4$ estimate was inside the numerical bounds should not be taken
to mean the subleading contributions are negligible for such a low spin. What
is in fact surprising is that the estimates from (29) are so close to the
numerically obtained ranges for such low values of the spin. These results
leave us optimistic that better estimates can be obtained by providing a few
of the subleading contributions, as was done in Simmons-Duffin:2016wlq  for
the $3d$ Ising model. The computation used to obtain figure 6 could be easily
extended to obtain estimates for the OPE coefficients of the
${\mathcal{C}}_{\frac{1}{2},2r_{0}-\frac{3}{2}(\frac{\ell}{2}-\frac{1}{2},\frac{\ell}{2})}$
multiplets, and also the dimensions and OPE coefficients of the remaining
operators in (9). One particularly interesting multiplet would be
${\mathcal{B}}_{1,\frac{7}{5}(0,0)}$ since, as discussed before, we expect it
to be absent in the $(A_{1},A_{2})$ theory. However, this corresponds to a
spin zero contribution and thus convergence of the inversion formula is not
guaranteed.

###  4.2 Inverting the non-chiral OPE

Next we turn to the non-chiral channel, where we have a decomposition in
superconformal blocks, and so we must obtain a supersymmetric version of the
inversion formula of Caron-Huot:2017vep . We consider the inversion of the
$s-$channel OPE of (4), with the superblocks given by (5),

$\langle\phi(x_{1})\bar{\phi}(x_{2})\phi(x_{3})\bar{\phi}(x_{4})\rangle=\frac{(z{\bar{z}})^{-\frac{{\mathcal{N}}}{2}}}{x_{12}^{2\Delta_{\phi}}x_{34}^{2\Delta_{\phi}}}\left(\sum\limits_{{\mathcal{O}}_{\Delta,\ell}}|\lambda_{\phi\bar{\phi}{\mathcal{O}}_{\Delta,\ell}}|^{2}g_{\Delta+{\mathcal{N}},\ell}^{{\mathcal{N}},{\mathcal{N}}}(z,{\bar{z}})\right)\,,$
(32)

where we are interested in taking ${\mathcal{N}}=2$, but the same equation is
also valid for ${\mathcal{N}}=1$, and so all that follows generalizes easily
to that case. Fortunately, the fact that, up to the overall prefactor
$(z{\bar{z}})^{-{\mathcal{N}}/2}$ in (32), the blocks relevant for the
$s-$channel decomposition are identical to bosonic blocks of operators with
unequal dimensions makes the task of obtaining an inversion formula very easy.
We can use the results of Caron-Huot:2017vep  with small modifications: The
Lorentzian inversion formula applies to the term between brackets in (32), and
the fact that the pre-factor is not the correct one for operators of unequal
dimension plays a small role in the derivation of Caron-Huot:2017vep . The
only time the prefactor is considered is when bounding the growth of the
correlator, needed to show the inversion formula is valid for spin greater
than one. The modified prefactor here seems to ameliorate the growth: we are
inverting $(z{\bar{z}})^{\tfrac{{\mathcal{N}}}{2}}$ times a CFT correlator
whose growth is bounded as discussed in Caron-Huot:2017vep . The condition
$\ell>1$ on the inversion formula (26) came from the need to have $\ell$ large
such that one could drop the arcs at infinity during the derivation of Caron-
Huot:2017vep . The prefactor’s behavior in this limit means the inversion
formula will be valid for all $\ell>1-{\mathcal{N}}$, and the results we
obtain for ${\mathcal{N}}=2$ should be valid for all spins. Apart from this,
the prefactor will only play a role when representing the correlator by its
$t-$ and $u-$channel OPEs. As such we apply (26) with

${\mathcal{G}}(z,{\bar{z}})=\sum\limits_{{\mathcal{O}}_{\Delta,\ell}}|\lambda_{\phi\bar{\phi}{\mathcal{O}}_{\Delta,\ell}}|^{2}g_{\Delta+{\mathcal{N}},\ell}^{{\mathcal{N}},{\mathcal{N}}}(z,{\bar{z}})\,.$
(33)

The $t-$ and $u-$channels of the correlator (32) are given by a non-chiral and
chiral OPE respectively. Using the crossing equation (12b) we see that the
$t-$channel expansion of ${\mathcal{G}}(z,{\bar{z}})$ is

${\mathcal{G}}(z,{\bar{z}})=(z{\bar{z}})^{\frac{{\mathcal{N}}}{2}}\left(\frac{z{\bar{z}}}{(1-z)(1-{\bar{z}})}\right)^{r_{0}}\sum\limits_{\Delta,\ell}|\lambda_{\phi\bar{\phi}{\mathcal{O}}_{\Delta,\ell}}|^{2}{\mathcal{G}}_{\Delta,\ell}(1-z,1-{\bar{z}})\,,$
(34)

with the superblock given by (5). While the $u-$channel is given by

${\mathcal{G}}(z,{\bar{z}})=(z{\bar{z}})^{r_{0}+\frac{{\mathcal{N}}}{2}}\sum\limits_{\Delta,\ell}|\lambda_{\phi\phi{\mathcal{O}}}|^{2}g_{\Delta,\ell}\left(\frac{1}{z},\frac{1}{{\bar{z}}}\right)\,.$
(35)

Once again, the leading contributions to the $s-$channel spectrum at large
spin, i.e., the leading contributions for ${\bar{z}}\to 1$ in (34), are from
the $t-$channel identity and stress-tensor multiplet. The subleading
contributions in the $t-$channel come from long multiplets with
$\Delta>\ell+2$. On the other hand, the leading twist contribution in the $u-$
channel arises from the ${\mathcal{E}}_{2r_{0}}$ and
$\mathcal{C}_{0,2r_{0}-1,\left(\frac{\ell}{2}-1,\frac{\ell}{2}\right)}$
multiplets, whose twists are all exactly $2r_{0}$, and so one should consider
the infinite sum over $\ell$. From a lightcone computation, e.g., Li:2015rfa ,
we expect an individual chiral operator of twist $\tau_{c}$ to contribute to
the anomalous dimensions of the non-chiral operators at large $\ell$ as
$\frac{(-1)^{\ell}}{\ell^{\tau_{c}}}$. Similarly, a non-chiral operator of
twist $\tau$ contributes to the same anomalous dimension at large $\ell$ as
$\frac{1}{\ell^{\tau}}$. In the case at hand, $\tau=2$ for the stress-tensor
multiplet and $\tau_{c}=2.4$ for each of the infinite number of leading
operators in the chiral channel. The contribution of an individual chiral
operator in the $u-$channel is thus subleading for sufficiently large spin.
This is similar to what happened in section 4.1, and while in this case the
dimensions of the operators are protected, their OPE coefficients are not.
Indeed, the value of these OPE coefficients remains elusive, and the best
estimate we have to go on comes from the numerically obtained bounds for the
operators with $\ell\leqslant 20$ presented in figure 6. An interesting
possibility would be to attempt to combine the numerical ranges for low spin
with the estimate for the large spin OPE coefficients obtained from (29). The
numerical bounds on the OPE coefficients would turn into an estimate, in the
form of an interval, for the anomalous dimension; we leave this exploration
for future work. Here we apply the inversion formula (26) only to the exchange
of the identity and stress-tensor multiplets

$C^{t}(z,\beta)\supset\int\limits_{0}^{1}\frac{\mathrm{d}{\bar{z}}}{{\bar{z}}^{2}}\kappa_{\beta}k_{\beta}^{1,1}({\bar{z}})\mathrm{dDisc}\left[\frac{(z{\bar{z}})^{r_{0}+1}}{((1-z)(1-{\bar{z}}))^{r_{0}}}\left(1+|\lambda_{\phi\bar{\phi}\hat{{\mathcal{C}}}_{0,0}}|^{2}{\mathcal{G}}_{2,0}(1-z,1-{\bar{z}})\right)\right]\,,$
(36)

where one should recall that
$\Delta_{12}=\Delta_{34}=\frac{{\mathcal{N}}}{2}=1$ when taking the double-
discontinuity.

Like before, the exchange of the identity in (36) gives the existence of
double-twist operators $\left[\phi\bar{\phi}\right]_{m,\ell}$, with dimensions

$\Delta_{\left[\phi\bar{\phi}\right]_{m,\ell}}\underset{\ell\gg
1}{\longrightarrow}2r_{0}+2m+\ell\,.$ (37)

Computing the OPE coefficients from the identity exchange we find, for the
leading twist operators,

$|\lambda_{\phi\bar{\phi}\left[\phi\bar{\phi}\right]_{0,\ell}}|^{2}\underset{\ell\gg
1}{\longrightarrow}\frac{4^{2-\ell}r_{0}(r_{0})_{\ell-2}(2r_{0}+1)_{\ell-2}}{(1)_{\ell-2}(r_{0}+\ell-2)\left(r_{0}+\frac{1}{2}\right)_{\ell-2}}\,,$
(38)

which are precisely the OPE coefficients of generalized free field theory, now
decomposed in superblocks instead of bosonic blocks.

Figure 7: Anomalous dimension
($\gamma_{\ell}=\Delta_{\ell}-(2\Delta_{\phi}+\ell)$) of the first spin $\ell$
long multiplet in the non-chiral channel. The colored dots are the dimension
estimates extracted from the extremal functionals of the various bounds
(figures 1 and 2-4 for $c=\tfrac{11}{30}$) as indicated by their colors, with
$\Lambda=34$. The dashed line corresponds to the result from the inversion
formula (36), for $c=\tfrac{11}{30}$ and external dimension
$r_{0}=\frac{6}{5}$, taking into account only the exchange of the identity and
stress tensor in the $t-$channel, and is thus an approximate result for
sufficiently large spin.

The stress-tensor exchange provides corrections to these dimensions and OPE
coefficients. As an illustration we computed its contribution to the anomalous
dimensions of the leading twist operators
$\left[\phi\bar{\phi}\right]_{0,\ell}$,
$\gamma_{\ell}=\Delta_{\ell}-(2\Delta_{\phi}+\ell)$. From the numerical
estimates (see figure 7) we see the anomalous dimensions starting at spin one
are rather small, and so we simply take the zeroth order of the procedure
outlined in Caron-Huot:2017vep  to commute the $z\to 0$ limit with the sum
over primaries in (36). These results are also shown in figure 7 for
$\ell\geqslant 1$ as a dashed blue line, together with estimates for these
values arising from the various extremal functionals of section 3, color coded
according to which bound they came from.171717We omitted two spin seven
dimensions, as we could not accurately estimate them from the functionals. The
two points that appear to be outlying in spin $6$ and $7$ correspond to cases
where there were two zeros of the functional very close to one another, and we
extracted the dimension of the first. We expect that higher derivative orders
would fix both situations. We are plotting the results starting from spin
$\ell=1$. The leading $\ell=0$ operator is the stress tensor itself, which was
not present in the generalized free field theory solution. As such the
dimension of $\left[\phi\bar{\phi}\right]_{0,0}$ must come down from
$2r_{0}=2.4$ to exactly $2$. The value of the anomalous dimensions coming from
(36) is still insufficient for this to happen, as clear from figure 7. For
$\ell\geqslant 1$, however, the numerical estimates of leading twist
operators’ dimensions are very close the values of double-twist operators
(37). Indeed, the maximum anomalous dimension in figure 7, ignoring the two
out-lying points, is of the order of $\gamma_{1}\sim 0.04$, in a dimension
that is close to $2r_{0}+1=3.4$. The anomalous dimensions obtained from (36)
(dashed blue line in (7)) are close to the numerically obtained values
starting from $\ell=2$, despite the fact that our results are only valid for
sufficiently large spin, as we have only considered the identity and stress
tensor contributions in the $t-$channel, and completely disregarded any
$u-$channel contribution. In particular, for spin $\ell\gtrsim 8$ the
numerical estimates arising from the different extremization problems of
section 3 are all cluttered, approaching the value (37), and close to the
values coming from (36).

All in all, we have seen that both in the chiral and non-chiral channels the
estimates coming from applying the inversion formula, and providing only the
leading twist operators (identity plus stress-tensor supermultiplet), come
very close to the numerically obtained bounds/estimates. This leaves us
optimistic that the spectrum of the $(A_{1},A_{2})$ can be bootstrapped,
similarly to the $3d$ Ising model. The numerical results for ${\mathcal{N}}=2$
theories suffer from slow convergence and thus the estimates for OPE
coefficients and anomalous dimensions we obtain are not yet with the precision
of those of the $3d$ Ising model. By using this data as input to the inversion
formulas they would in turn produce ranges for the various quantities
appearing in the chiral and non-chiral OPEs. Finally, another direction
corresponds to using the output of each inversion formula as input for the
other to obtain better estimates. We leave these two directions for future
work.

###### Acknowledgements.

We have greatly benefited from discussions with Philip Argyres, Lorenzo
Bianchi, Mario Martone, Marco Meineri, and Volker Schomerus. The authors thank
the organizers of the Pollica Summer Workshop 2017 for hospitality, and were
partly supported by the ERC STG grant 306260 during the Pollica Summer
Workshop. M.L. and P.L. thank ICTP-SAIFR in São Paulo for hospitality during
the Bootstrap 2017 workshop, and the Simons Collaboration on the Non-
perturbative Bootstrap for providing many stimulating workshops and
conferences during which this work was carried out. We also thank the
participants of the Simons Collaboration’s kick off, “Exact Operator Algebras
in Superconformal Field Theories” and “Numerical bootstrap” meetings for
discussions. M.L. was partially supported by the German Research Foundation
(DFG) via the Emmy Noether program “Exact results in Gauge theories”.

##  Appendix A Blocks and Crossing

We write the bosonic blocks for the exchange of a conformal primary of
dimension $\Delta$ and spin $\ell$, in the four-point function of unequal
scalar operators of dimensions $\Delta_{i=1,\dots,4}$, as Dolan:2000ut

$\displaystyle g_{\Delta,\ell}^{\Delta_{12},\Delta_{34}}(z,{\bar{z}})$
$\displaystyle=$
$\displaystyle\frac{z\bar{z}}{z-\bar{z}}\left(k^{\Delta_{12},\Delta_{34}}_{\Delta+\ell}(z)k^{\Delta_{12},\Delta_{34}}_{\Delta-\ell-2}(\bar{z})-z\leftrightarrow\bar{z}\right)\,,$
$\displaystyle k^{\Delta_{12},\Delta_{34}}_{\beta}(z)$ $\displaystyle=$
$\displaystyle
z^{\frac{\beta}{2}}{{}_{2}F_{1}}\left(\frac{\beta-\Delta_{12}}{2},\frac{\beta+\Delta_{34}}{2};\beta;z\right)\,,$
(39)

where $\Delta_{ij}=\Delta_{i}-\Delta_{j}$, and $z$ and ${\bar{z}}$ are
obtained from the standard conformally invariant cross-ratios

$z{\bar{z}}=\frac{x_{12}^{2}x_{34}^{2}}{x_{13}^{2}x_{24}^{2}}\,,\qquad\qquad(1-z)(1-{\bar{z}})=\frac{x_{14}^{2}x_{23}^{2}}{x_{13}^{2}x_{24}^{2}}\,.$
(40)

The crossing equations (12b) (see Beem:2014zpa ; Lemos:2015awa  for a
derivation) are written here in a form suitable for the numerical analysis of
section 3

$\sum_{{\mathcal{O}}\in\phi\bar{\phi}}|\lambda_{\phi\bar{\phi}{\mathcal{O}}}|^{2}\begin{bmatrix}(-1)^{\ell}\tilde{{\mathcal{F}}}_{\pm,\Delta,\ell}(z,{\bar{z}})\\\
{\mathcal{F}}_{-,\Delta,\ell}(z,{\bar{z}})\end{bmatrix}+\sum_{{\mathcal{O}}\in\phi\phi}|\lambda_{\phi\phi{\mathcal{O}}}|^{2}\begin{bmatrix}\mp(-1)^{\ell}F_{\pm,\Delta,\ell}(z,{\bar{z}})\\\
0\end{bmatrix}=0\,,$ (41)

where the first line encodes two separate crossing equations, differing by the
signs indicated, and where we defined (recall that
$\Delta_{\phi}=\Delta_{\bar{\phi}}=r_{0}$)

$\begin{split}F_{\pm,\Delta,\ell}(z,{\bar{z}})&\equiv\left((1-z)(1-{\bar{z}})\right)^{r_{0}}g^{0,0}_{\Delta,\ell}(z,{\bar{z}})\pm(z{\bar{z}})^{r_{0}}g^{0,0}_{\Delta,\ell}(1-z,1-{\bar{z}})\,,\\\
{\mathcal{F}}_{\pm,\Delta,\ell}(z,{\bar{z}})&\equiv\left((1-z)(1-{\bar{z}})\right)^{r_{0}}{\mathcal{G}}_{\Delta,\ell}(z,{\bar{z}})\pm(z{\bar{z}})^{r_{0}}{\mathcal{G}}_{\Delta,\ell}(1-z,1-{\bar{z}})\,,\\\
\tilde{{\mathcal{F}}}_{\pm,\Delta,\ell}(z,{\bar{z}})&\equiv\left((1-z)(1-{\bar{z}})\right)^{r_{0}}\tilde{{\mathcal{G}}}_{\Delta,\ell}(z,{\bar{z}})\pm(z{\bar{z}})^{r_{0}}\tilde{{\mathcal{G}}}_{\Delta,\ell}(1-z,1-{\bar{z}})\,,\end{split}$
(42)

with the superblocks ${\mathcal{G}}_{\Delta,\ell}$ and
$\tilde{{\mathcal{G}}}_{\Delta,\ell}$ given in (5) and (8). In (41) the stress
tensor and the identity contribute as

$\overrightarrow{V}_{\mathrm{fixed}}=\begin{bmatrix}\tilde{\mathcal{F}}_{\pm,\Delta=0,\ell=0}(z,{\bar{z}})\\\
{\mathcal{F}}_{-,\Delta=0,\ell=0}(z,{\bar{z}})\end{bmatrix}+\frac{r_{0}^{2}}{6c}\begin{bmatrix}\tilde{\mathcal{F}}_{\pm,\Delta=2,\ell=0}(z,{\bar{z}})\\\
{\mathcal{F}}_{-,\Delta=2,\ell=0}(z,{\bar{z}})\end{bmatrix}\,.$ (43)

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