# ABJM at Strong Coupling from M-theory, Localization, and Lorentzian
Inversion

###### Abstract

We study the stress tensor multiplet four-point function in the 3d maximally
supersymmetric ABJ(M) theory with Chern-Simons level $k=2$, which in the large
$N$ limit is holographically dual to weakly coupled M-theory on $AdS_{4}\times
S^{7}/\mathbb{Z}_{2}$. We use the Lorentzian inversion to compute the 1-loop
correction to this holographic correlator coming from Witten diagrams with
supergravity $R$ and the first higher derivative correction $R^{4}$ vertices,
up to a finite number of contact terms that contribute to low spins where the
inversion formula does not converge. We find a precise match with the
corresponding terms in the 11d M-theory S-matrix by taking the flat space
limit, which is not sensitive to these contact terms. We then conjecturally
fix these contact terms by analytically continuing the inversion formula below
its expected range of convergence, and verify this conjecture using
supersymmetric localization. Finally, we compare some of the 1-loop CFT data
to non-perturbative in $N$ bounds from the numerical conformal bootstrap,
which we compute at unprecedently high accuracy, and find that the 1-loop
corrections saturate the bounds in the large $N$ regime, which extends the
previously observed match at tree level.

###### Contents

  1. 1 Introduction
  2. 2 Four-point functions at large $c_{T}$
    1. 2.1 Block expansion of $\langle qqpp\rangle$
    2. 2.2 Generalized free field theory at $c_{T}\to\infty$
    3. 2.3 Tree level $\langle 22pp\rangle$
    4. 2.4 Large $c_{T}$ expansion of $\langle 2222\rangle$
  3. 3 $\langle 2222\rangle$ at 1-loop
    1. 3.1 One-loop from tree level
    2. 3.2 Mellin amplitude and comparison to 11d
    3. 3.3 Extracting CFT data
  4. 4 Fixing 1-loop contact terms
    1. 4.1 Analytic continuation of Lorentzian inversion
    2. 4.2 Supersymmetric localization
  5. 5 Numerical bootstrap
    1. 5.1 Setup
    2. 5.2 Bootstrap bound saturation
  6. 6 Conclusion
  7. A Conformal blocks and $\bar{D}_{r_{1},r_{2},r_{3},r_{4}}(U,V)$
  8. B Lorentzian inversion at large $c_{T}$

##  1 Introduction

Holographic correlators are correlation functions of single trace operators in
CFTs with holographic duals, such that the large $N$ limit of the CFT
correlators is dual to scattering of gravitons or KK modes in weakly coupled
AdS gravity. One reason to study these correlators is to learn about quantum
gravity on curved spacetime. Another motivation is that there is a concrete
limit [1, 2, 3, 4, 5, 6] of the AdS correlator that gives the flat space
S-matrix, so we can use CFT to study quantum gravity in flat space. For CFTs
with M-theory duals, which is the subject of this paper, this motivation is
especially compelling, because the M-theory S-matrix cannot be computed from
flat space methods even in principle beyond the lowest few terms in the small
Planck length expansion [7, 8, 9]. Instead, [10] proposed that the M-theory
S-matrix could be defined to all orders by first computing the dual CFT
correlator at large $N$ and then taking the flat space limit.

Holographic correlators were originally computed at large $N$ using Witten
diagrams derived from the explicit AdS supergravity action, see e.g. [11, 12,
13, 14, 15, 16]. Recently, a more powerful analytic bootstrap approach was
introduced in [17] that computes tree level holographic correlators in the
leading supergravity approximation purely based on crossing symmetry,
analyticity, superconformal symmetry, and the flat space limit. This approach
was generalized to higher derivative corrections to tree level supergravity in
[18, 19] following the general discussion in [20], where the coefficients of
these corrections are no longer completely determined by symmetry, but can be
fixed using theory-specific inputs like supersymmetric localization [10, 21,
22, 23, 24, 25, 26, 27] for 3d and 4d CFTs or protected sectors [28] for 6d
CFTs.

The cutting edge of the large $N$ analytic bootstrap is 1-loop, which can be
computed up to a finite number of contact terms by “squaring” the tree level
CFT data of all double-trace operators in the correlator [29]. More precisely,
we apply crossing symmetry to this data to compute the double-discontinuity,
which according to the Lorentzian inversion formula [30] determines the entire
correlator up to a finite number of contact terms discussed in [20] that
contribute to CFT data of low spins.111At finite $N$ the Lorentzian inversion
formula in fact converges for CFT data of all spins [30, 31, 32], but this
convergence gets worse in the large $N$ expansion. This program was carried
out for 1-loop diagrams involving the supergravity $R$ and first higher
derivative correction $R^{4}$ vertices for 4d $SU(N)$ $\mathcal{N}=4$ SYM [33,
34, 35, 36, 37, 38, 39, 40, 41, 42, 43], which is dual to Type IIB string
theory on $AdS_{5}\times S^{5}$, and 6d $(2,0)$ theory [44], which is dual to
$AdS_{7}\times S^{4}$ for the $A_{N-1}$ theories and $AdS_{7}\times
S^{4}/\mathbb{Z}_{2}$ for the $D_{N}$ theories.222For CFTs dual to higher spin
gravity, 1-loop terms were also computed in 3d in [45], see also [46, 47, 48,
49] for tree level results. The $R|R$ diagram has a single contact term
ambiguity that was fixed using localization for $\mathcal{N}=4$ SYM in [27].
However, higher loops diagrams such as $R|R^{4}$ have too many ambiguities to
be fixed using localization, and for 6d $(2,0)$ theory there are no
localization results available. These ambiguities also cannot be fixed by
comparing to the known S-matrix in the flat space limit, which was used to fix
tree level higher derivative terms in [19, 22, 25, 24, 26], since the contact
term ambiguities are purely AdS features that disappear in the flat space
limit.

In this paper we will extend the 1-loop analytic bootstrap to 3d maximally
supersymmetric ABJ(M) theory [50, 51] with Chern-Simons level $k=2$, which is
holographically dual to M-theory on $AdS_{4}\times S^{7}/\mathbb{Z}_{2}$, and
propose how to fix 1-loop contact term ambiguities using an analytic
continuation of the Lorentzian inversion formula. There are two such maximally
supersymmetric CFTs with gauge groups $U(N)_{2}\times U(N)_{-2}$ or
$U(N+1)_{2}\times U(N)_{-2}$, which are called ABJM or ABJ respectively, but
they are identical when expanded at large central charge333The central charge
is defined in (2.10) as the coefficient of the canonically-normalized stress-
tensor two point function, which has been calculated to all orders in $1/N$
through supersymmetric localization [52] using the results of [53] and [54].
$c_{T}\sim N^{\frac{3}{2}}$,444This is because from the M-theory point of
view, the two theories differ by a torsion flux, i.e. a discrete holonomy of
the 3-form field on a torsion 3-cycle of $S^{7}/\mathbb{Z}_{2}$ [51]. This
torsion flux affects the CFT data only through non-perturbative effects. so we
will refer to both theories collectively as ABJ(M). We will study the stress
tensor multiplet correlator $\langle 2222\rangle$, where $\langle pppp\rangle$
denotes the correlator of of the bottom component of the $p$th lowest single
trace half-BPS multiplet, which are dual to the corresponding $p$th lowest
scalar KK mode in the dimensional reduction of M-theory on AdS. We will find
it convenient to work with the Mellin transform $M(s,t;\sigma,\tau)$ of
$\langle 2222\rangle$, where $\sigma,\tau$ parameterize the $R$-symmetry
dependence and $s,t$ are Mellin variables that are related to the 11d
Mandelstam variables in the flat space limit. The large $c_{T}$ expansion of
$M$ is constrained by the analytic bootstrap to take the form

$\begin{split}M(s,t;\sigma,\tau)&=c_{T}^{-1}M^{R}+c_{T}^{-\frac{5}{3}}B^{R^{4}}_{4}M^{4}+c_{T}^{-2}(M^{R|R}+B^{R|R}_{4}M^{4})\\\
&+c_{T}^{-\frac{7}{3}}(B^{D^{6}R^{4}}_{4}M^{4}+B^{D^{6}R^{4}}_{6}M^{6}+B^{D^{6}R^{4}}_{7}M^{7})\\\
&+c_{T}^{-\frac{23}{9}}(B^{D^{8}R^{4}}_{4}M^{4}+B^{D^{8}R^{4}}_{6}M^{6}+B^{D^{8}R^{4}}_{7}M^{7}+B^{D^{8}R^{4}}_{8}M^{8})\\\
&+c_{T}^{-\frac{8}{3}}(M^{R|R^{4}}+B^{R|R^{4}}_{4}M^{4}+B^{R|R^{4}}_{6}M^{6}+B^{R|R^{4}}_{7}M^{7}+B^{R|R^{4}}_{8}M^{8})+\dots\,,\end{split}$
(1.1)

where the $M$’s are functions of $s,t,\sigma,\tau$ with numerical coefficients
$B$ that can depend on $k$. The tree level terms555Since $c_{T}$ is the only
expansion parameter, we can only distinguish between tree and loop terms at
low orders where they have different power of $c_{T}$. at orders $c_{T}^{-1}$,
$c_{T}^{-\frac{5}{3}}$, and $c_{T}^{-\frac{7}{3}}$ were previously computed
for both $k=1,2$ ABJ(M) in [55], [10], and [21], respectively, while this
paper will focus on the 1-loop terms $R|R$ at order $c_{T}^{-2}$ and $R|R^{4}$
at order $c_{T}^{-\frac{8}{3}}$ for $k=2$ ABJ(M).666We will also show some
results for the $R^{4}|R^{4}$ term at order $c_{T}^{-{10}/{3}}$, but we will
not consider the $R|D^{6}R^{4}$ term that contributes at the same order

As in the 4d and 6d cases, double trace long operators are degenerate in the
generalized free field theory (GFFT) at $c_{T}\to\infty$ limit, so their tree
level CFT data at orders $c_{T}^{-1}$ for $R$ and $c_{T}^{-\frac{5}{3}}$ for
$R^{4}$ must be unmixed to get the 1-loop corrections we consider. For $k=2$
ABJ(M), this unmixing requires the average of GFFT OPE coefficients obtained
from $\langle pppp\rangle$ for even $p$, as well as the average of
$c_{T}^{-1}$ and $c_{T}^{-\frac{5}{3}}$ anomalous dimensions obtained from
$\langle 22pp\rangle$ for even $p$. For $k=1$ ABJM, the 1-loop double
discontinuity would also receive contributions from the OPE coefficients of
double trace long operators with odd twists, which generically contribute to
the 1-loop double discontinuity of large $N$ 3d CFTs [45]. These degenerate
contributions must be similarly unmixed using GFFT OPE coefficients in
$\langle pppp\rangle$ for odd $p$ as well as the average of $c_{T}^{-1}$ and
$c_{T}^{-\frac{5}{3}}$ OPE coefficients from $\langle 22pp\rangle$ for odd
$p$. These odd $p$ terms do not contribute to the $k=2$ theory, because they
are projected out by the orbifold.777In the 4d and 6d cases, all double trace
multiplets have even twists, so only anomalous dimensions contribute to the
1-loop double discontinuity. The only difference between the orbifold and the
non-orbifold cases is that we restrict the sum over tree level anomalous
dimensions to even $p$ for the orbifold cases. We computed the average GFFT
OPE coefficients from $\langle pppp\rangle$ for general $p$ by computing the
full superconformal block expansion up to $p=9$ using the superconformal Ward
identities in [56] and then guessing the general $p$ formula, similar to what
we did in [44] for the 6d case. For the tree $\langle 22pp\rangle$ data for
even $p$, we extracted the average anomalous dimension from the $R$ correlator
given in [57] as well as the $R^{4}$ correlator that we compute here using the
known M-theory term in the flat space limit. The odd $p$ case is technically
much harder, because the tree level supergravity $\langle 22pp\rangle$ cannot
be written in terms of a finite number of
$\bar{D}_{r_{1},r_{2},r_{3},r_{4}}(U,V)$ functions, unlike the even $p$ case
in 3d or the general $p$ case in 4d and 6d.888In 4d and 6d, the correlator can
be written in terms of just a few $\bar{D}_{r_{1},r_{2},r_{3},r_{4}}(U,V)$
with integer arguments for any $p$. In 3d and even $p$, the number of
$\bar{D}_{r_{1},r_{2},r_{3},r_{4}}(U,V)$ grows with $p$ and has negative and
half-integer arguments, as we will show in the main text. We will thus only
discuss $k=2$ ABJ(M) in this paper, where we use the even $p$ data to compute
$R|R$, $R|R^{4}$ and $R^{4}|R^{4}$ using the Lorentzian inversion formula up
to the finite number of contact terms discussed above, extract the low-lying
CFT data for spins unaffected by these contact terms, and successfully compare
to the M-theory S-matrix terms computed in [8, 7, 44] in the flat space limit,
which is unaffected by these contact terms.

We then analytically continue the Lorentzian inversion formula to extract CFT
data of spins that are affected by the contact term ambiguities. For $R|R$, we
find that this analytic continuation works for all CFT data, and in particular
allows us to fix the contact term $B^{R|R}_{4}M^{4}$ in (1.1) to zero, where
$M^{R|R}$ is defined so that its CFT data is analytic in spin for all values.
For $R|R^{4}$, we find that the analytic continuation of the inversion formula
works for all CFT data except that affected by the $B^{R|R^{4}}_{4}M^{4}$
contact term, which allows us to fix the other three contact terms in (1.1).
We then apply two localization constraints from [10, 58] and [21, 23] to
further constrain the amplitude. For $R|R$ we find that one of the
localization constraints independently fixes $B^{R|R}_{4}=0$,999We are only
able to write $M^{R|R}$ up to a finite number of polynomial in $s,t$
ambiguities that are in principle fixed by superconformal symmetry, but are
difficult to fix in practice. As such we could not impose the localization
constraint in [21, 23] that requires us to integrate $M^{R|R}$, but we were
still able to impose the localization constraint from [10, 58] that simply
fixes certain protected OPE coefficients, which can be extracted from $R|R$
using the Lorentzian inversion formula without knowing the explicit Mellin
amplitude as we will discuss in the main text. For $M^{R|R^{4}}$, we managed
to compute the complete Mellin amplitude, so we could impose both localization
constriants. which confirms the result from the conjectured analytic
continuation, while for $R|R^{4}$ we use one constraint to fix the remaining
$B^{R|R^{4}}_{4}$ coefficient and the second constraint as a nontrivial check.

Finally, we compare the CFT data extracted from the 1-loop Mellin amplitudes
to the numerical bootstrap. In [58], it was conjectured that the $k=2$ ABJ(M)
theory saturates the numerical bootstrap bounds for $\langle 2222\rangle$,
which was motivated by comparing the bounds to the all orders in $1/c_{T}$
calculation of short operator OPE coefficients computed from supersymmetric
localization [58, 59, 60, 61]. This conjecture was further checked in [62],
where the $c_{T}^{-1}$ correction to all CFT data in $\langle 2222\rangle$ was
found to saturate the bootstrap bounds at sufficiently large $c_{T}$, but this
correction does not depend on $k$. To compare higher order in $1/c_{T}$
corrections that depend on $k$, one must keep in mind that the large $c_{T}$
expansion is asymptotic, so either one must look at very high values of
$c_{T}$, which require extremely high numerical bootstrap accuracy, or focus
on terms in the $1/c_{T}$ expansion where the asymptotic expansion still
converges (i.e. subsequent terms have decreasing coefficients). We focus on
the $c_{T}^{-2}$ terms in the OPE coefficients of semishort operators, which
are the most converged corrections beyond $O(c_{T}^{-1})$, and find that these
corrections noticeably improve the saturation of the bootstrap bounds, which
we compute at much higher accuracy than the previous studies [52, 63, 58, 64].
We do not have enough accuracy yet to extract sufficient CFT data to fix the
$D^{8}R^{4}$ term in (1.1), which would fulfill the goal of [10] of deriving
new terms in the M-theory S-matrix from CFT. Nevertheless, this is the first
successful comparison of analytic $c_{T}^{-2}$ terms to numerical bootstrap,
which improves the many tree level checks performed in various other contexts
in [65, 66, 62, 46, 67], and is a necessary step toward the goal of deriving
$D^{8}R^{4}$.

The rest of this paper is organized as follows. In Section 2 we compute the
explicit superblock decomposition of $\langle qqpp\rangle$ for $q\leq p$ and
$p\leq 9$. We use these superblock expansions to extract the average GFFT OPE
coefficients and tree level anomalous dimensions that we will need to compute
the 1-loop data for $k=2$ ABJM. In Section 3, we use this data to compute the
1-loop corrections to the stress tensor correlator up to contact term
ambiguities. We then match these 1-loop terms to the M-theory S-matrix in the
flat space limit. We also extract CFT data from the 1-loop correlators using
the Lorentzian inversion formula as well as a projection method. In Section 4,
we fix the contact term ambiguities at orders $c_{T}^{-2}$ and
$c_{T}^{-\frac{8}{3}}$ by combining constraints from supersymmetric
localization with an analytic continuation of the Lorentzian inversion
formula. In Section 5 we compare some of these 1-loop analytic results to
bounds from the numerical conformal bootstrap, which we compute at much higher
accuracy than previous studies. Finally, in Section 6 we discuss future
directions. Several technical details are given in various Appendices. We also
include an ancillary Mathematica notebook, which includes many of our more
complicated explicit results.

##  2 Four-point functions at large $c_{T}$

We start by discussing the large $c_{T}\sim N^{3/2}$ expansion of four-point
functions of the dimension $\frac{p}{2}$ scalar bottom component of half-BPS
supermultiplets in $\mathcal{N}=8$ ABJ(M) theory, and derive the data needed
for the 1-loop terms for $k=2$ ABJ(M) in the following sections. First we will
review the constraints of the 3d $\mathcal{N}=8$ superconformal algebra
$\mathfrak{osp}(8|4)\supset\mathfrak{so}(5)\oplus\mathfrak{so}(8)_{R}$ on
$\langle ppqq\rangle$ following [64], and we explicitly perform the superblock
expansion for $p,q\leq 9$. We then discuss the generalized free field theory
(GFFT) that describes the $c_{T}\to\infty$ limit, which we use to compute
average OPE coefficients of double-trace singlet long multiplets in $\langle
qqpp\rangle$. Afterwards, we consider tree level corrections to $\langle
22pp\rangle$ for even $p$, which we use to derive the average anomalous
dimension of singlet long multiplets at orders $1/c_{T}$ and $1/c_{T}^{5/3}$
that correspond to tree level supergravity and $R^{4}$, respectively. Finally,
we discuss higher orders in the large $c_{T}$ expansion of $\langle
2222\rangle$, which will be our main focus in the rest of the paper.

###  2.1 Block expansion of $\langle qqpp\rangle$

We consider half-BPS superconformal primaries $S_{p}$ in 3d $\mathcal{N}=8$
SCFTs that are scalars with $\Delta=\frac{p}{2}$ and transform in the $[00p0]$
of $\mathfrak{so}(8)_{R}$,101010The convention we use in defining these
multiplets is that the supercharges transform in the ${\bf 8}_{v}=[1000]$
irrep of $\mathfrak{so}(8)_{R}$. where $p=1,2,\dots$. The first such
interacting operator is $S_{2}$, which is the bottom component of the stress
tensor multiplet. We can denote these operators as traceless symmetric tensors
$S_{I_{1}\dots I_{p}}(x)$ of $\mathfrak{so}(8)_{R}$, where $I_{i}=1,\dots 8$.
We can avoid indices by introducing an auxiliary polarization vector $Y^{I}$
that is constrained to be null, $Y_{i}\cdot Y_{i}=0$, and then define

$\begin{split}S_{p}(x,Y)\equiv S_{I_{1}\dots I_{p}}Y^{I_{1}}\cdots
Y^{I_{p}}\,.\end{split}$ (2.1)

Consider the four-point functions $\langle qqpp\rangle$ of four
$S_{p}(x,Y)$’s, where $q\leq p$. Conformal and $\mathfrak{so}(8)_{R}$ symmetry
fixes these correlators to take the form

$\begin{split}\langle
S_{q}(x_{1},Y_{1})S_{q}(x_{2},Y_{2})S_{p}(x_{3},Y_{3})S_{p}(x_{4},Y_{4})\rangle=\frac{(Y_{1}\cdot
Y_{2})^{q}(Y_{3}\cdot
Y_{4})^{p}}{|x_{12}|^{q}|x_{34}|^{p}}\mathcal{G}_{qp}(U,V;\sigma,\tau)\,,\end{split}$
(2.2)

where we define

$\begin{split}U\equiv\frac{{x}_{12}^{2}{x}_{34}^{2}}{{x}_{13}^{2}{x}_{24}^{2}}\,,\qquad
V\equiv\frac{{x}_{14}^{2}{x}_{23}^{2}}{{x}_{13}^{2}{x}_{24}^{2}}\,,\qquad\sigma\equiv\frac{(Y_{1}\cdot
Y_{3})(Y_{2}\cdot Y_{4})}{(Y_{1}\cdot Y_{2})(Y_{3}\cdot
Y_{4})}\,,\qquad\tau\equiv\frac{(Y_{1}\cdot Y_{4})(Y_{2}\cdot
Y_{3})}{(Y_{1}\cdot Y_{2})(Y_{3}\cdot Y_{4})}\,,\end{split}$ (2.3)

with $x_{ij}\equiv x_{i}-x_{j}$. Since (2.2) is a degree $q$ polynomial in
each $Y_{i}$ separately, the quantity $\mathcal{G}_{qp}(U,V;\sigma,\tau)$ is a
degree $q$ polynomial in $\sigma$ and $\tau$. We parametrize these polynomials
in terms of eigenfunctions $Y_{[0\,m\,\,2n-2m\,0]}(\sigma,\tau)$ of the
$\mathfrak{so}(8)_{R}$ quadratic Casimir for irreps $[0\,m\,2n-2m\,0]$ that
appear in the tensor product of $[00q0]\otimes[00q0]$, so that $n=0,1,\dots q$
and $m=0,\dots n$. The polynomials $Y_{[0\,m\,2n-2m\,0]}(\sigma,\tau)$ can be
computed explicitly [68], and we give the explicit values for $q=2,\dots 9$ in
the attached Mathematica file. We then expand
$\mathcal{G}_{qp}(U,V;\sigma,\tau)$ in terms of this basis as

$\begin{split}\mathcal{G}_{qp}(U,V;\sigma,\tau)=\sum^{q}_{n=0}\sum_{m=0}^{n}Y_{[0\,m\,2n-2m\,0]}(\sigma,\tau)\mathcal{G}_{qp}^{[0\,m\,2n-2m\,0]}(U,V)\,,\end{split}$
(2.4)

and furthermore expand $\mathcal{G}_{qp}^{[0\,m\,2n-2m\,0]}(U,V)$ in conformal
blocks $G_{\Delta,\ell}(U,V)$ as

$\begin{split}\mathcal{G}_{qp}^{[0\,m\,2n-2m\,0]}(U,V)=\sum_{\Delta,\ell}\lambda_{qq{\mathcal{O}}_{\Delta,\ell,[0\,m\,2n-2m\,0]}}\lambda_{pp{\mathcal{O}}_{\Delta,\ell,[0\,m\,2n-2m\,0]}}G_{\Delta,\ell}(U,V)\,,\end{split}$
(2.5)

where ${\mathcal{O}}_{\Delta,\ell,[0\,m\,2n-2m\,0]}$ are conformal primaries
with scaling dimension $\Delta$ and spin $\ell$ in irrep $[0\,m\,2n-2m\,0]$
that appear in $S_{q}\times S_{q}$ with OPE coefficient
$\lambda_{qq{\mathcal{O}}_{\Delta,\ell,[0\,m\,2n-2m\,0]}}$. The 3d conformal
blocks were computed in various series expansions in [69, 70, 71], which we
review in our conventions in Appendix A.

The correlator is further constrained by the superconformal Ward identities
[56]:

$\begin{split}\left[z\partial_{z}-\frac{1}{2}\alpha\partial_{\alpha}\right]\mathcal{G}_{qp}(z,\bar{z};\alpha,\bar{\alpha})|_{\alpha=\frac{1}{z}}=\left[\bar{z}\partial_{\bar{z}}-\frac{1}{2}{\bar{\alpha}}\partial_{\bar{\alpha}}\right]\mathcal{G}_{qp}(z,\bar{z};\alpha,\bar{\alpha})|_{\bar{\alpha}=\frac{1}{\bar{z}}}&=0\,,\end{split}$
(2.6)

where $z,\bar{z}$ and $\alpha,\bar{\alpha}$ are written in terms of $U,V$ and
$\sigma,\tau$, respectively, as

$\begin{split}U=z\bar{z}\,,\quad
V=(1-z)(1-\bar{z})\,,\qquad\qquad\sigma=\alpha\bar{\alpha}\,,\quad\tau=(1-\alpha)(1-\bar{\alpha})\,.\end{split}$
(2.7)

We can satisfy these constraints by expanding $\mathcal{G}_{qp}$ in
superconformal blocks as

$\begin{split}\mathcal{G}_{qp}(U,V;\sigma,\tau)=\sum_{\mathcal{M}\in
S_{q}\times
S_{q}}\lambda_{qq\mathcal{M}}\lambda_{pp\mathcal{M}}\mathfrak{G}_{\mathcal{M}}(U,V;\sigma,\tau)\,,\end{split}$
(2.8)

where $\mathfrak{G}_{\mathcal{M}}$ are superblocks for each supermultiplet
$\mathcal{M}$ that appears in $S_{q}\times S_{q}$ (and $S_{p}\times S_{p}$)
with OPE coefficients $\lambda_{qq\mathcal{M}}$ (and
$\lambda_{pp\mathcal{M}}$). The multiplets that can appear in the OPE are [72,
64]:

$\begin{split}&S_{q}\times
S_{q}=\text{Id}+\sum_{n=1}^{q}{(B,+)}_{n,0}^{[0\,0\,2n\,0]}+\sum_{n=1}^{q}\sum_{m=2,4,\dots,n}{(B,2)}_{n,0}^{[0\,m\,2n-2m\,0]}\\\
&+\sum_{\ell=0,2,\dots}\sum_{n=1}^{q-1}{(A,+)}_{n+1+\ell,\ell}^{[0\,0\,2n\,0]}+\sum_{n=1}^{q-1}\Big{[}\sum_{\ell=0,2,\dots}\sum_{m=2,4,\dots,n}{(A,2)}_{n+1+\ell,\ell}^{[0\,m\,2n-2m\,0]}+\sum_{\ell=1,3,\dots}\sum_{m=1,3,\dots,n}{(A,2)}_{n+1+\ell,\ell}^{[0\,m\,2n-2m\,0]}\Big{]}\\\
&+\sum_{n=0}^{q-2}\Big{[}\sum_{\ell=0,2,\dots}\sum_{m=0,2,\dots,n}{(A,0)}_{\Delta>n+1+\ell,\ell}^{[0\,m\,2n-2m\,0]}+\sum_{\ell=1,3,\dots}\sum_{m=1,3,\dots,n}{(A,0)}_{\Delta>n+1+\ell,\ell}^{[0\,m\,2n-2m\,0]}\Big{]}\,,\end{split}$
(2.9)

where we denote superconformal multiplets other than the identity Id by
$X_{\Delta,\ell}^{[a_{1}\,a_{2}\,a_{3}\,a_{4}]}$, with $(\Delta,\ell)$ and
$[a_{1}\,a_{2}\,a_{3}\,a_{4}]$ representing the $\mathfrak{so}(3,2)$ and
$\mathfrak{so}(8)_{R}$ quantum numbers of the superconformal primary, while
$X$ denotes the type of shortening condition. The
$(A,0)^{[0\,m\,2n-2m\,0]}_{\Delta>n+\ell+1,\ell}$ multiplets that appear here
are unprotected. When $\Delta$ saturates the bound we in general get semishort
multiplets $(A,2)^{[0\,m\,2n-2m\,0]}_{n+\ell+1,\ell}$ or
$(A,+)^{[0\,0\,2n\,0]}_{n+\ell+1,\ell}$, which are $\frac{1}{4}$ or
$\frac{1}{8}$ BPS, respectively. The only exception is
$(A,0)^{[0000]}_{\Delta>\ell+1,\ell}$ whose unitarity bound gives conserved
currents that cannot appear in the interacting theories we consider. Finally,
the $(B,2)^{[0\,n-m\,2n\,0]}_{n,0}$ and $(B,+)^{[0\,0\,2n\,0]}_{n,0}$ are
short multiplets, where the former is $\frac{1}{4}$ BPS, while the latter are
the half-BPS multiplets whose bottom component we called $S_{p}$. The lowest
such multiplet is always the stress tensor multiplet $(B,+)_{1,0}^{[0020]}$,
whose OPE coefficient squared is fixed by the conformal Ward identity [73] to
be inversely proportional to the coefficient of the canonically normalized
stress tensor two-point function:

$\begin{split}\langle
T_{\mu\nu}(\vec{x})T_{\rho\sigma}(0)\rangle=\frac{c_{T}}{64}\left(P_{\mu\rho}P_{\nu\sigma}+P_{\nu\rho}P_{\mu\sigma}-P_{\mu\nu}P_{\rho\sigma}\right)\frac{1}{16\pi^{2}\vec{x}^{2}}\,,\qquad
P_{\mu\nu}\equiv\eta_{\mu\nu}\nabla^{2}-\partial_{\mu}\partial_{\nu}\,,\end{split}$
(2.10)

where $c_{T}$ is normalized so that $c_{T}^{\text{free}}=16$ for the free
$\mathcal{N}=8$ theory of eight massless real scalars and Majorana fermions.
In this normalization we get the precise relationship

$\begin{split}\lambda_{pp{(B,+)}_{1,0}^{[0020]}}=\frac{8p}{{c_{T}}}\,.\end{split}$
(2.11)

We will be mostly interested in $\langle 2222\rangle$, whose multiplets are
summarized in Table 1. Note that we introduce simpler notation for these
multiplets, e.g. $(B,+)^{[0040]}_{2,0}\equiv(B,+)$, since not counting the
stress tensor multiplet, only one operator of each type appears.

Type | $(\Delta,\ell)$ |  $\mathfrak{so}(8)_{R}$ irrep | spin $\ell$ | Name
---|---|---|---|---
$(B,+)$ | $(2,0)$ | ${\bf 294}_{c}=[0040]$ | $0$ | $(B,+)$
$(B,2)$ | $(2,0)$ | ${\bf 300}=[0200]$ | $0$ | $(B,2)$
$(B,+)$ | $(1,0)$ | ${\bf 35}_{c}=[0020]$ | $0$ | Stress
$(A,+)$ | $(\ell+2,\ell)$ | ${\bf 35}_{c}=[0020]$ | even | $(A,+)_{\ell}$
$(A,2)$ | $(\ell+2,\ell)$ | ${\bf 28}=[0100]$ | odd | $(A,2)_{\ell}$
$(A,0)$ | $\Delta\geq\ell+1$ | ${\bf 1}=[0000]$ | even | $(A,0)_{\Delta,\ell}$
Id | $(0,0)$ | ${\bf 1}=[0000]$ | even | Id
Table 1: The possible superconformal multiplets in the
${\mathcal{O}}_{\text{Stress}}\times{\mathcal{O}}_{\text{Stress}}$ OPE. The
$\mathfrak{so}(3,2)\oplus\mathfrak{so}(8)_{R}$ quantum numbers are those of
the superconformal primary in each multiplet.

We then compare (2.8) to (2.4) and (2.5) to see that the superblocks are
finite linear combinations of conformal blocks

$\begin{split}\mathfrak{G}_{\mathcal{M}}=\sum^{q}_{n=0}\sum_{m=0}^{n}Y_{[0\,m\,2n-2m\,0]}(\sigma,\tau)\sum_{{\mathcal{O}}\in\mathcal{M}}\frac{\lambda_{qq{\mathcal{O}}_{\Delta,\ell,[0\,m\,2n-2m\,0]}}\lambda_{pp{\mathcal{O}}_{\Delta,\ell,[0\,m\,2n-2m\,0]}}}{\lambda_{qq\mathcal{M}}\lambda_{pp\mathcal{M}}}G_{\Delta,\ell}(U,V)\,,\end{split}$
(2.12)

where ${\mathcal{O}}_{\Delta,\ell,[0\,m\,2n-2m\,0]}$ are conformal primaries
that appear in $\mathcal{M}$, which can be derived using the Racah-Speiser
algorithm in [74]. For $\langle qqpp\rangle$, we fixed all of the pairs of OPE
coefficients
$\lambda_{qq{\mathcal{O}}_{\Delta,\ell,[0\,m\,2n-2m\,0]}}\lambda_{pp{\mathcal{O}}_{\Delta,\ell,[0\,m\,2n-2m\,0]}}$
in terms of the single pair of OPE coefficients
$\lambda_{qq\mathcal{M}}\lambda_{pp\mathcal{M}}$ in (2.8) by applying the Ward
identities to the small $z,\bar{z}$ expansion of the superblocks, where we
used the small $z,\bar{z}$ expansion of the conformal blocks as given in [70]
and reviewed in Appendix A. We give the results for the $s$-channel of
$\langle qqpp\rangle$ for $q=2,\dots,9$ in the attached Mathematica file.

###  2.2 Generalized free field theory at $c_{T}\to\infty$

We now use the superblocks computed in the previous section to perform the
superblock expansion for the GFFT that describes the $c_{T}\to\infty$ limit of
$\langle qqpp\rangle$ and $\langle pppp\rangle$ for $k=1,2$ ABJ(M) theory.
Recall from the Introduction that in fact both $k=1,2$ ABJ(M) have the same
half-BPS correlators at this order, except that all correlators involving
$S_{p}$ for odd $p$ vanish for the $k=2$ theory. In particular, both theories
are described by a GFFT where the operators $S_{p}$ are treated as generalized
free fields with two point functions $\langle
S_{p}(x_{1},Y_{1})S_{q}(x_{2},Y_{2})\rangle=\delta_{pq}\frac{(Y_{1}\cdot
Y_{2})^{p}}{|x_{12}|^{p}}$. We can then compute $\langle qqpp\rangle$ (for
$q\leq p$) using Wick contractions to get

$\begin{split}{\mathcal{G}}_{qp}^{(0)}=1+\delta_{qp}\left(U^{\frac{p}{2}}\sigma^{p}+\frac{U^{\frac{p}{2}}}{V^{\frac{p}{2}}}\tau^{p}\right)\,,\end{split}$
(2.13)

which can be expanded in the superblocks of the previous section to extract
OPE coefficients. If several operators have the same quantum numbers at this
order, then we can only compute the average of their OPE coefficients. Such a
degeneracy occurs for the double trace long multiplet
$(A,0)^{[0000]}_{\Delta,\ell}$ operator
$S_{p}\partial_{\mu_{1}}\dots\partial_{\mu_{\ell}}(\partial^{2})^{n}S_{p}$
with spin $\ell$ and twist $t\equiv\Delta-\ell=p+2n\geq 2$. For $t\geq 2$,
there are $t-1$ such degenerate operators because of the different ways of
adding $p$ and $n$ to get the same twist, which we label using the degeneracy
label $I$. We denote the GFFT OPE coefficient of these operators in the
$S_{p}\times S_{p}$ OPE by $\lambda^{(0)}_{p,t,\ell,I}$, and note that only
twists $t=p,p+2,\dots$ appear in this OPE at leading order. By expanding in
superblocks for $p=2,\dots,9$ we found the general formula

$\begin{split}\langle(\lambda^{(0)}_{p,t,\ell})^{2}\rangle&=\frac{45\sqrt{\pi}(2\ell+1)4^{p-2}(p-1)p\Gamma\left(\frac{t-1}{2}\right)\Gamma\left(\ell+\frac{t}{2}\right)\Gamma(\ell+t+3)\Gamma\left(\frac{p}{2}+\frac{t}{2}+2\right)\Gamma\left(\ell+\frac{p}{2}+\frac{t}{2}+\frac{5}{2}\right)}{\Gamma(p+2)\Gamma(p+4)\Gamma\left(\frac{t+6}{2}\right)\Gamma\left(\ell+t+\frac{5}{2}\right)\Gamma\left(\ell+\frac{t}{2}+\frac{7}{2}\right)\Gamma\left(-\frac{p}{2}+\frac{t}{2}+1\right)\Gamma\left(\ell-\frac{p}{2}+\frac{t}{2}+\frac{3}{2}\right)}\,,\end{split}$
(2.14)

which reproduces the $p=2,3$ values given in [52, 64]. Note that the average
$\langle\lambda^{(0)}_{q,t,\ell}\lambda^{(0)}_{p,t,\ell}\rangle$ trivially
vanishes because ${\mathcal{G}}_{qp}=1$ for $q\neq p$ at GFFT.

###  2.3 Tree level $\langle 22pp\rangle$

We now consider the $1/c_{T}$ and $1/c_{T}^{5/3}$ terms in $\langle
22pp\rangle$, which corresponds to tree level supergravity $R$ and $R^{4}$ in
the bulk description, respectively, and whose CFT data is needed to compute
loops with these vertices in the following section. We expand
${\mathcal{G}}_{2p}$ (which we will denote as ${\mathcal{G}}_{p}$) in (2.2) as
well as the long multiplet CFT data to this order as

$\begin{split}{\mathcal{G}}_{p}(U,V;\sigma,\tau)&={\mathcal{G}}^{(0)}_{p}+c_{T}^{-1}{\mathcal{G}}^{R}_{p}+c_{T}^{-\frac{5}{3}}{\mathcal{G}}^{R^{4}}_{p}+\dots\,\\\
\Delta_{t,\ell,I}&=t+\ell+c_{T}^{-1}\gamma^{R}_{t,\ell,I}+c_{T}^{-\frac{5}{3}}\gamma^{R^{4}}_{t,\ell,I}+\dots\,,\\\
(\lambda_{p,t,\ell,I})^{2}&=(\lambda^{(0)}_{p,t,\ell,I})^{2}+c_{T}^{-1}(\lambda^{R}_{p,t,\ell,I})^{2}+c_{T}^{-\frac{5}{3}}(\lambda^{R^{4}}_{p,t,\ell,I})^{2}+\dots\,.\\\
\end{split}$ (2.15)

A similar expansion exists for the OPE coefficients of the protected
operators, although of course their scaling dimensions are fixed. Using these
expansions, we can write the superblock expansion for ${\mathcal{G}}_{p}$ in
(2.8) at large $c_{T}$ as

$\begin{split}&{\mathcal{G}}_{p}^{R}(U,V)={128p}\mathfrak{G}_{\text{Stress}}(U,V;\sigma,\tau)+\sum_{\mathcal{M}_{\Delta,\ell}\in\\{(B,+),(B,2),(A,2)_{\ell},(A,+)_{\ell}\\}}\lambda^{R}_{22\mathcal{M}}\lambda^{R}_{pp\mathcal{M}}\mathfrak{G}_{\mathcal{M}}(U,V;\sigma,\tau)\\\
&\quad+\sum_{t,\ell,I}\left[\lambda^{R}_{2,t,\ell,I}\lambda^{R}_{p,t,\ell,I}+\lambda^{(0)}_{2,t,\ell,I}\lambda^{(0)}_{p,t,\ell,I}\gamma_{t,\ell,I}^{R}(\partial_{t}^{\text{no-
log}}+\frac{1}{2}\log
U)\right]\mathfrak{G}_{t+\ell,\ell}(U,V;\sigma,\tau)\,.\end{split}$ (2.16)

Here, the first line includes the protected multiplets, and the OPE
coefficient for the stress tensor multiplet was written explicitly using
(2.11). In the second line we denote the singlet long multiplet superblock by
$\mathfrak{G}_{\Delta,\ell}$ and $\partial_{t}^{\text{no-
log}}\mathfrak{G}_{t+\ell,\ell}(U,V;\sigma,\tau)$ denotes that we consider the
term after taking the derivative that does not include a $\log U$, which has
already been written separately. The expansion of ${\mathcal{G}}_{p}^{R^{4}}$
takes the same form with $R\to R^{4}$, except the stress tensor block does not
appear.

While the superblock expansion is best expressed in position space, in the
large $c_{T}$ expansion it is also useful to consider the Mellin transform
$M_{p}(s,t;\sigma,\tau)$ of the connected correlator
$\mathcal{G}^{\text{con}}_{p}(U,V;\sigma,\tau)\equiv\mathcal{G}_{p}(U,V;\sigma,\tau)-\mathcal{G}^{(0)}_{p}(U,V;\sigma,\tau)$,
which is defined as [55]:

$\begin{split}\mathcal{G}^{\text{con}}_{p}(U,V;\sigma,\tau)&=\int\frac{ds\,dt}{(4\pi
i)^{2}}U^{\frac{s}{2}}V^{\frac{t}{2}-\frac{p}{4}-\frac{1}{2}}M_{p}(s,t;\sigma,\tau)\\\
&\qquad\qquad\times\Gamma\left[\frac{p}{2}-\frac{s}{2}\right]\Gamma\left[1-\frac{s}{2}\right]\Gamma^{2}\left[\frac{p}{4}+\frac{1}{2}-\frac{t}{2}\right]\Gamma^{2}\left[\frac{p}{4}+\frac{1}{2}-\frac{{u}}{2}\right],\\\
\end{split}$ (2.17)

where $u=p+2-s-t$ and the integration contours here is defined to include all
poles of the Gamma functions on one side of the contour. The Mellin amplitude
is defined such that a bulk contact Witten diagram coming from a vertex with
$2m$ derivatives gives rise to a polynomial in $s,t$ of degree $m$, and
similarly an exchange Witten diagrams corresponds to a Mellin amplitude with
poles for the twists of each exchanged operator. The Mellin amplitude must
also obey the crossing relations

$\begin{split}M_{p}(s,t;\sigma,\tau)=M_{p}(s,u;\tau,\sigma)\,,\qquad
M_{2}(s,t;\sigma,\tau)=\tau^{2}M_{2}(t,s;\sigma/\tau,1/\tau)\,,\end{split}$
(2.18)

which follow from interchanging the first and second operators, and, for
$p=2$, the first and third. Lastly, $M_{p}(s,t;\sigma,\tau)$ must satisfy the
Ward identities (2.6), which can be implemented in Mellin space as shown in
[55]. Using all these constraints, $M_{p}(s,t)$ can be expanded similar to the
position space expression (2.15) to get

$\begin{split}M_{p}(s,t)=c_{T}^{-1}M^{R}_{p}+c_{T}^{-\frac{5}{3}}B^{R^{4}}(p)M^{R^{4}}_{p}+\dots\,,\end{split}$
(2.19)

where $M^{R^{4}}_{p}$ is a complicated degree 4 polynomial in $s,t$ whose
explicit form we give in the attached Mathematica notebook, while the tree
level supergravity amplitude $M^{R}_{p}$ was written in [57] as an infinite
sum of the supergravity multiplet and its descendents:

$\begin{split}M^{R}_{p}=&\sum_{m=0}^{\infty}\Bigg{[}\frac{2^{2m+5}p((p-2t+2)(p+2(s+2)\sigma-2(s+t-1))-2(s+2)\tau(p-2(s+t-1)))}{\pi^{2}(2m+3)(2m-s+1)\Gamma\left(\frac{1}{2}-m\right)\Gamma(2m+2)\Gamma\left(\frac{1}{2}(-2m+p-1)\right)}\\\
&+\frac{16p\tau\Gamma\left(\frac{p}{2}+1\right)((p+2t+2)(p(2\sigma-1)+2(-\sigma
s+s+t-1))+2\tau(p-s)(p-2(s+t-1)))}{\pi\Gamma\left(\frac{1}{2}-m\right)^{2}\Gamma(m+1)\Gamma\left(\frac{p-1}{2}\right)(4m+p-2t)\Gamma\left(m+\frac{p+3}{2}\right)}\\\
&+\frac{2^{2m+5}p((p-2t+2)(p+2(s+2)\sigma-2(s+t-1))-2(s+2)\tau(p-2(s+t-1)))}{\pi^{2}(2m+3)(2m-s+1)\Gamma\left(\frac{1}{2}-m\right)\Gamma(2m+2)\Gamma\left(\frac{1}{2}(-2m+p-1)\right)}\,,\end{split}$
(2.20)

where the overall coefficient was fixed by extracting the stress tensor OPE
coefficient and comparing to (2.11). Note that this expression is independent
of $k$. For even $p$, the sum can be performed to get an expression in terms
of Gamma functions, which can then be written in terms of a finite sum of
$\bar{D}_{r_{1},r_{2},r_{3},r_{4}}(U,V)$ functions using its Mellin space
definition in Appendix A, as was pointed out in a similar case in [46]. For
instance, for $p=2$ first we resum to get

$\begin{split}M^{R}_{2}=&\frac{32\tau\left(\frac{4\Gamma\left(\frac{1}{2}-\frac{t}{2}\right)}{\Gamma\left(1-\frac{t}{2}\right)}-\sqrt{\pi}(t+4)\right)((t+2)(-\sigma
s+s+2\sigma+t-2)+(s-2)\tau(s+t-2))}{\pi^{5/2}t(t+2)}\\\
&+\frac{32\sigma\left(\sqrt{\pi}(s+t-8)+\frac{4\Gamma\left(\frac{1}{2}(s+t-3)\right)}{\Gamma\left(\frac{1}{2}(s+t-2)\right)}\right)((t-2)(-\sigma
s+s+2\sigma+t-6)+(s-2)\tau(s+t-6))}{\pi^{5/2}(s+t-6)(s+t-4)}\\\
&+\frac{32\left(\frac{4\Gamma\left(\frac{1}{2}-\frac{s}{2}\right)}{\Gamma\left(1-\frac{s}{2}\right)}-\sqrt{\pi}(s+4)\right)((t-2)(-(s+2)\sigma+s+t-2)+(s+2)\tau(s+t-2))}{\pi^{5/2}s(s+2)}\,.\end{split}$
(2.21)

Note that the Gamma functions that appear in the denominator exactly cancel
one of the Gamma functions in the Mellin transform (2.17), while the poles
just shift the arguments of the Gamma functions. We can thus use (A.9) to
write this expression in terms of a finite number of $\bar{D}(U,V)$ functions.
For instance, the contribution to the $[0040]$ irrep in (2.4) is

$\begin{split}A^{R}_{[0040]}(U,V)&=-\frac{32U^{3}}{3\pi^{5/2}V^{2}}\Big{[}2\sqrt{\pi}V^{2}\bar{D}_{1,3,-1,1}(U,V)-2V^{2}\bar{D}_{1,\frac{5}{2},-1,\frac{1}{2}}(U,V)\\\
&+\sqrt{\pi}V^{2}\bar{D}_{1,3,0,2}(U,V)+\sqrt{\pi}V^{2}\bar{D}_{2,3,-1,2}(U,V)-2V^{\frac{3}{2}}\bar{D}_{1,\frac{5}{2},\frac{1}{2},-1}(U,V)\\\
&+3\sqrt{\pi}\bar{D}_{3,1,-1,1}(U,V)-\sqrt{\pi}\bar{D}_{4,1,-1,2}(U,V)\Big{]}\,.\end{split}$
(2.22)

While the appearance of half integer and negative arguments might seem
nonstandard, in fact these have standard expansions to all orders in $U$ and
$V$, including a $\log U$ term, as shown in [75] and reviewed in Appendix A.
We can then easily expand this and the other channels in superblocks to get
the average anomalous dimensions
$\langle\lambda^{(0)}_{2,t,\ell}\lambda^{(0)}_{p,t,\ell}\gamma_{t,\ell}\rangle\equiv\sum_{I}\lambda^{(0)}_{2,t,\ell,I}\lambda^{(0)}_{p,t,\ell,I}\gamma_{t,\ell,I}$
weighted by OPE coefficients for $p=2$:

$\begin{split}\langle\lambda^{(0)}_{2,t,\ell}\lambda^{(0)}_{2,t,\ell}\gamma^{R}_{t,\ell}\rangle=&-\frac{64}{\pi^{2}}\frac{\sqrt{\pi}(2\ell+1)\Gamma\left(\frac{t-1}{2}\right)\Gamma\left(\ell+\frac{t}{2}\right)\Gamma(\ell+t+3)}{8\Gamma\left(\frac{t}{2}\right)\Gamma\left(\ell+\frac{t}{2}+\frac{1}{2}\right)\Gamma\left(\ell+t+\frac{5}{2}\right)}\\\
&\times\left(\left(2\ell^{2}+2\ell
t+6\ell+t^{2}+5t+4\right)\left(\psi({\ell+t+3})-\psi({\ell+1})\right)+(-t-2)(2\ell+t+3)\right)\,,\end{split}$
(2.23)

which matches the values originally computed in [62]. We can similarly compute
$\langle\lambda^{(0)}_{2,t,\ell}\lambda^{(0)}_{p,t,\ell}\gamma^{R}_{t,\ell}\rangle$
for higher even $p$, and we show the results for $p\leq 36$ in the attached
Mathematica notebook. Since the $M_{p}^{R}$ did not depend on $k$, these
average anomalous dimensions are also the same for $k=1,2$.

For the $R^{4}$ amplitude, we need to fix the overall coefficient
$B^{R^{4}}(p)$, which will depend nontrivially on $k$ unlike $M_{p}^{R}$. For
$p=2$ this was fixed using localization in [10], and this derivation could in
principle be extended to $p>2$ for $k=1$ using the localization results from
[76]. For $k=2$ and $p>2$ however, which is our primary interest, we can only
fix the coefficient by comparing to the known 11d M-theory S-matrix term in
the flat space limit, as was done for the 6d $(2,0)$ theory in [44]. In
particular, the 11d M-theory S-matrix ${\mathcal{A}}$ can be expanded at small
Planck length $\ell_{11}$ as

$\begin{split}\mathcal{A}(s,t)=\ell_{11}^{9}\mathcal{A}_{R}+\ell^{15}_{11}\mathcal{A}_{R^{4}}+\ell^{18}_{11}\mathcal{A}_{R|R}+\ell^{21}_{11}\mathcal{A}_{D^{6}R^{4}}+\ell^{23}_{11}\mathcal{A}_{D^{8}R^{4}}+\ell_{11}^{24}\mathcal{A}_{R|R^{4}}+\dots\,,\end{split}$
(2.24)

where $s,t,u$ are 11d Mandelstam variables. The lowest few terms
${\mathcal{A}}_{R}$, ${\mathcal{A}}_{R^{4}}$, and ${\mathcal{A}}_{D^{6}R^{4}}$
are protected, and so can be computed from Type IIA string theory by
compactifying on a circle [7, 8, 9]111111${\mathcal{A}}_{D^{4}R^{4}}$ can also
be computed in this way, but it vanishes and so we did not write it. Also, the
1-loop supergravity term ${\mathcal{A}}_{R|R}$ was computed in [8, 7], while
${\mathcal{A}}_{R|R^{4}}$ and ${\mathcal{A}}_{R^{4}|R^{4}}$ were computed in
[44]. to get

$\begin{split}\frac{\mathcal{A}_{R^{4}}}{\mathcal{A}_{R}}=\frac{stu}{3\cdot
2^{7}}\,,\qquad\frac{\mathcal{A}_{D^{6}R^{4}}}{\mathcal{A}_{R}}=\frac{(stu)^{2}}{15\cdot
2^{15}}\,.\end{split}$ (2.25)

The small $\ell_{11}$ expansion in 11d maps to the large $N$ expansion in the
CFT according to the dictionary [50, 51]:

$\begin{split}\frac{L^{6}}{\ell_{11}^{6}}=\left(\frac{3\pi
c_{T}k}{2^{11}}\right)^{\frac{2}{3}}+O(c_{T}^{0})\,.\end{split}$ (2.26)

The flat space limit formula [1, 10] then relates a Mellin amplitude
$M^{a}_{p}(s,t)$ of large $s,t$ degree $a$ to the 11d amplitude defined in
(2.24) as

$\begin{split}c_{T}^{\frac{2(1-a)}{9}}\frac{\pi^{5/2}2^{-a-p-3}\Gamma(p-1)}{\Gamma\left(\frac{1}{2}(2a+p-1)\right)}\lim_{s,t\to\infty}\frac{st(s+t)M^{a}(s,t)}{(t(-\sigma
s+s+t)+s\tau(s+t))^{2}}=\ell_{11}^{2a-2}\frac{{\mathcal{A}_{2a+7}}}{{\mathcal{A}}_{R}}\,,\end{split}$
(2.27)

where $\mathcal{A}_{2a+7}$ is a term in the amplitude with length dimension
$(2a+7)$, and $\ell_{11}$ is the 11d Planck length. For instance,
$\mathcal{A}_{15}\equiv\mathcal{A}_{R^{4}}$ has length dimension 15 in (2.24)
and corresponds to $M_{p}^{4}\equiv M_{p}^{R^{4}}$. The 11d amplitude of
course is the same for all $p$, and the ratio of
$\mathcal{A}_{R^{4}}/\mathcal{A}_{R}$ was given in (2.25). Using this 11d
amplitude and the flat space limit we find

$\begin{split}B^{R^{4}}(p)=\frac{32\cdot{2^{\frac{1}{3}}}(p-1)(p+1)(p+3)(p+5)}{3\
3^{2/3}\pi^{8/3}k^{2/3}\Gamma\left(\frac{p}{2}\right)}\,,\end{split}$ (2.28)

which matches the $p=2$ result computed from localization in [10]. Note that
the simple $k$ dependence comes from the AdS/CFT dictionary (2.26). Since
$M_{p}^{R^{4}}$ is just a polynomial Mellin amplitude for every $p$, we can
convert to a finite number of $\bar{D}(U,V)$ to get

$\begin{split}A^{R^{4}}_{[0040]}(U,V)&=\
\frac{64\cdot{2^{\frac{1}{3}}}\left(p^{2}-1\right)U^{p/2}}{3^{8/3}\pi^{8/3}k^{2/3}\Gamma\left(\frac{p}{2}\right)}\Big{[}6(p-2)p\bar{D}_{\frac{p}{2},\frac{p}{2},1,1}(U,V)+4(p-3)(p-2)(p+2)\bar{D}_{\frac{p}{2},\frac{p}{2},2,2}(U,V)\\\
&+192\bar{D}_{\frac{p}{2},\frac{p}{2},3,3}(U,V)-312\bar{D}_{\frac{p}{2},\frac{p}{2},4,4}(U,V)+p(p-1)((p-1)p-26)\bar{D}_{\frac{p}{2},\frac{p}{2},3,3}(U,V)\\\
&+4p(p(p+2)-29)\bar{D}_{\frac{p}{2},\frac{p}{2},4,4}(U,V)+4p(p+8)\bar{D}_{\frac{p}{2},\frac{p}{2},5,5}(U,V)+60\bar{D}_{\frac{p}{2},\frac{p}{2},5,5}(U,V)\Big{]}\,,\end{split}$
(2.29)

and similarly for the other channels. We can then expand in superblocks to get
the even $p$ average anomalous dimensions

$\begin{split}\langle\lambda^{(0)}_{2,t,\ell}\lambda^{(0)}_{p,t,\ell}\gamma^{R^{4}}_{t,\ell}\rangle=&\delta_{\ell,0}(-1)^{\frac{p}{2}}\frac{(t+1)2^{p+t+\frac{25}{3}}\Gamma\left(\frac{t}{2}+2\right)\Gamma\left(\frac{t}{2}+4\right)\Gamma\left(\frac{1}{2}(p+t+5)\right)}{3^{5/3}\pi^{8/3}k^{2/3}\Gamma(p-1)\Gamma\left(t+\frac{5}{2}\right)\Gamma\left(\frac{1}{2}(-p+t+2)\right)}\,,\end{split}$
(2.30)

which are only nonzero for zero spin, and for $p=2$ match the results in [10].

###  2.4 Large $c_{T}$ expansion of $\langle 2222\rangle$

Finally, we restrict to the stress tensor correlator $\langle 2222\rangle$,
which is our primary interest. For simplicity, we will drop the $p=2$
subscript from all further expressions. The Mellin amplitude $M(s,t)$ is fixed
by the analytic structure, growth at infinity, and crossing symmetry to take
the form (1.1) given in the Introduction, where the coefficient of each
$c_{T}^{-b}$ must include all allowed Mellin amplitudes of large $s,t$ degree
$(9/2b-7/2)$ or less. These can include the polynomial Mellin amplitudes
$M^{a}$ given in [10], which we also include in the attached Mathematica file.
For the amplitudes we consider above, this implies one allowed polynomial
Mellin amplitude of each degree $a$, which correspond to contact Witten
diagrams with $2a$ derivatives. These contact diagrams only contribute to a
finite number of spins that grows with the degree [20]. For the multiplets in
$\langle 2222\rangle$, the contribution from each $M^{a}$ in (1.1) is
summarized in Table 4 of [10], which we repeat here in Table 2. The other
Mellin amplitudes shown in (1.1) include the tree level supergravity term
$M^{R}$ discussed in the previous section, which includes poles for the single
trace supergravity multiplet and its descendents, as well as the 1-loop Mellin
amplitudes $M^{R|R}$ and $M^{{R^{4}}|R}$ of degrees $5.5$ and $8.5$,
respectively, that we will discuss more in the following section.

CFT data: | $M^{4}$ | $M^{6}$ | $M^{7}$ | $M^{8}$
---|---|---|---|---
$\lambda^{2}_{(B,+)}$ | $\frac{256}{35}$ | $-\frac{59392}{693}$ | $-\frac{477184}{429}$ | $\frac{4022091776}{4448925}$
$\lambda^{2}_{(B,2)}$ | $\frac{256}{7}$ | $-\frac{296960}{693}$ | $-\frac{2385920}{429}$ | $\frac{4022091776}{889785}$
$\lambda^{2}_{(A,+)_{0}}$ | 0 | $\frac{16384}{1485}$ | $\frac{950272}{6435}$ | $-\frac{131396796416}{467137125}$
$\lambda^{2}_{(A,+)_{2}}$ | 0 | $0$ | $0$ | $\frac{67108864}{557375}$
$\lambda^{2}_{(A,2)_{1}}$ | 0 | $\frac{131072}{1155}$ | $\frac{21889024}{15015}$ | $-\frac{3848847491072}{1089986625}$
$\lambda^{2}_{(A,2)_{3}}$ | 0 | $0$ | $0$ | $\frac{268435456}{121275}$
$\gamma_{2,0}$ | -192 | $\frac{15360}{11}$ | $\frac{192000}{11}$ | $-\frac{18059264}{1521}$
$\gamma_{2,2}$ | 0 | $-1536$ | $-18432$ | $\frac{509591552}{12675}$
$\gamma_{2,4}$ | 0 | $0$ | $0$ | $-32768$
Table 2: Contributions from large $s,t$ degree $a$ contact Mellin amplitudes
$M^{a}(s,t)$ to the OPE coefficients squared $\lambda^{2}_{22{\mathcal{M}}}$
of some protected multiplets $(B,+)$, $(B,2)$, $(A,+)_{\ell}$ for even $\ell$,
and $(A,2)_{\ell}$ for odd $\ell$, as well as to the anomalous dimensions
$\gamma_{t,\ell}$ for even $\ell$ of the lowest twist $t=2$ unprotected
multiplet $(A,0)_{t+\ell,\ell}$. Adapted from [10] with typos fixed.

The coefficient $B^{R^{4}}_{4}\equiv B^{R^{4}}(2)$ was fixed in the previous
section. The three $B^{D^{6}R^{4}}$ coefficients were fixed in [21] from the
two localization constraints and the flat space limit (2.27) to get

$\begin{split}B^{D^{6}R^{4}}_{4}=-\frac{1352960\
6^{2/3}}{9\pi^{10/3}k^{\frac{4}{3}}}\,,\qquad
B^{D^{6}R^{4}}_{6}=-\frac{220528\ 6^{2/3}}{\pi^{10/3}k^{\frac{4}{3}}}\,,\qquad
B^{D^{6}R^{4}}_{7}=\frac{16016\
6^{2/3}}{\pi^{10/3}k^{\frac{4}{3}}}\,.\end{split}$ (2.31)

There are not enough constraints to fully fix the other tree amplitudes, while
the 1-loop amplitudes will be considered in the next section, so for now we
will only extract CFT data up to order $c_{T}^{-\frac{7}{3}}$ and leave the
$R|R$ term unknown for now. The short multiplets OPE coefficients
$\lambda^{2}_{(B,+)}$ and $\lambda^{2}_{(B,+)}$ were in fact computed to all
orders in $1/c_{T}$ using localization in [58], and take the form for $k=2$:

$\begin{split}\lambda^{2}_{(B,2)}&=\frac{32}{3}-\frac{1024\left(4\pi^{2}-15\right)}{9\pi^{2}c_{T}}+\frac{20480\left(\frac{2}{3}\right)^{\frac{2}{3}}}{\pi^{8/3}c_{T}^{5/3}}+\frac{16384\left(2\pi^{2}-25\right)}{9\pi^{4}c_{T}^{2}}-\frac{327680({\frac{2}{3}})^{\frac{1}{3}}}{3\pi^{\frac{10}{3}}c_{T}^{7/3}}+\frac{7536640\left(\frac{2}{3}\right)^{\frac{2}{3}}}{9\pi^{\frac{14}{3}}c_{T}^{8/3}}+O(c_{T}^{-3})\,,\\\
\lambda^{2}_{(B,+)}&=\frac{16}{3}-\frac{1024\left(\pi^{2}+3\right)}{9\pi^{2}c_{T}}+\frac{4096\left(\frac{2}{3}\right)^{2/3}}{\pi^{8/3}c_{T}^{5/3}}+\frac{16384\left(2\pi^{2}-25\right)}{45\pi^{4}c_{T}^{2}}-\frac{65536(\frac{2}{3})^{\frac{1}{3}}}{3\pi^{\frac{10}{3}}c_{T}^{7/3}}+\frac{1507328\left(\frac{2}{3}\right)^{2/3}}{9\pi^{14/3}c_{T}^{8/3}}+O(c_{T}^{-3})\,,\\\
\end{split}$ (2.32)

where note that these OPE coefficients are related due to crossing in the 1d
topological sector as [63]

$\begin{split}\frac{1024}{c_{T}}-5\lambda^{2}_{(B,+)}+\lambda^{2}_{(B,2)}+16=0\,,\end{split}$
(2.33)

so in fact only one is independent. For the other multiplets in $S_{2}\times
S_{2}$ we get for $k=1,2$

$\begin{split}\lambda^{2}_{(A,+)_{\ell}}&=\frac{\pi\Gamma(\ell+3)^{2}}{\Gamma\left(\ell+\frac{5}{2}\right)^{2}}-\frac{64\Gamma(\ell+3)^{2}(-2\ell+2(\ell(\ell+5)+5)\psi^{(1)}(\ell+3)-5)}{\pi
c_{T}\Gamma\left(\ell+\frac{5}{2}\right)^{2}}\\\
&+c_{T}^{-2}(\lambda^{R|R}_{(A,+)_{\ell}})^{2}-\delta_{\ell,0}\frac{1835008\
6^{2/3}}{27\pi^{10/3}c_{T}^{7/3}k^{4/3}}+O(c_{T}^{-\frac{23}{9}})\,,\\\
\lambda^{2}_{(A,2)_{\ell}}&=\frac{\pi\Gamma(\ell+2)\Gamma(\ell+4)}{\Gamma\left(\ell+\frac{3}{2}\right)\Gamma\left(\ell+\frac{7}{2}\right)}+\frac{64\Gamma(\ell+4)^{2}\left((2\ell+3)(\ell(\ell+7)+11)-2(\ell+2)^{2}(\ell+3)^{2}\psi^{(1)}(\ell+2)\right)}{\pi
c_{T}(\ell+2)^{2}(\ell+3)^{2}\Gamma\left(\ell+\frac{3}{2}\right)\Gamma\left(\ell+\frac{7}{2}\right)}\\\
&+c_{T}^{-2}(\lambda^{R|R}_{(A,2)_{\ell}})^{2}-\delta_{\ell,1}\frac{8388608\
6^{2/3}}{5\pi^{10/3}c_{T}^{7/3}k^{4/3}}+O(c_{T}^{-\frac{23}{9}})\,,\\\
\Delta_{2,\ell}&=2+\ell-\frac{256(2\ell+3)(2\ell+5)(2\ell+7)}{\pi^{2}c_{T}(\ell+1)(\ell+2)(\ell+3)(\ell+4)}-\frac{71680\cdot{6}^{\frac{1}{3}}\delta_{0,\ell}}{\pi^{8/3}c_{T}^{5/3}k^{2/3}}\\\
&+c_{T}^{-2}\gamma_{2,\ell}^{R|R}+c_{T}^{-\frac{7}{3}}\left(\delta_{\ell,0}\frac{1433600\
6^{2/3}}{3\pi^{10/3}k^{4/3}}+\delta_{\ell,2}\frac{43524096\
6^{2/3}}{\pi^{10/3}k^{4/3}}\right)+O(c_{T}^{-\frac{23}{9}})\,,\\\ \end{split}$
(2.34)

where $\ell$ is even for $\Delta_{2,\ell}$ and $\lambda^{2}_{(A,+)_{\ell}}$,
odd for $\lambda^{2}_{(A,2)_{\ell}}$, and for $\Delta_{2,\ell}$ we only wrote
the result for the lowest twist because recall that higher twists are
degenerate and so require unmixing beyond leading order. In the following
section, we will determine the 1-loop corrections to some of this non-trivial
CFT data.

##  3 $\langle 2222\rangle$ at 1-loop

We now discuss the 1-loop terms $R|R$ at $c_{T}^{-2}$, $R|R^{4}$ at
$c_{T}^{-\frac{8}{3}}$, and $R^{4}|R^{4}$ at $c_{T}^{-\frac{10}{3}}$ for $k=2$
ABJ(M) theory. For each term we compute the double-discontinuity (DD) from the
tree and GFFT data derived in the previous sections, and then use it as well
as crossing symmetry and the superconformal Ward identity to write the entire
correlator in Mellin space up to contact term ambiguities. We then take the
flat space limit and match these correlators to the relevant 1-loop
corrections to the 11d S-matrix. Finally, we extract low-lying CFT data using
two methods: the Lorentzian inversion integral applied to the DD [77, 30] and
a projection method applied to the entire Mellin amplitude [20, 62]. The
inversion method does not converge for low spins that are affected by contact
term ambiguities, while the projection method can be used to compute that CFT
data in terms of those ambiguities. In the next section, we will discuss how
to use localization and a conjectured analytic continuation of the inversion
method to fix all the contact term ambiguities for $R|R$ and $R|R^{4}$.

###  3.1 One-loop from tree level

We begin by expanding the correlator ${\mathcal{G}}$ for $\langle 2222\rangle$
to 1-loop order at large $c_{T}$ using the block expansion described in
section 2.1. For $R|R$ at order $c_{T}^{-2}$, this takes the form

$\begin{split}{\mathcal{G}}^{R|R}=&\sum_{t=2,4,\dots}\sum_{\ell\in\text{Even}}\Big{[}\frac{1}{8}\langle(\lambda^{(0)}_{t,\ell})^{2}(\gamma^{R}_{t,\ell})^{2}\rangle(\log^{2}U+4\log
U\partial_{t}^{\text{no-log}}+4(\partial_{t}^{\text{no-log}})^{2})\\\
&+\frac{1}{2}\langle(\lambda^{R})^{2}_{t,\ell}\gamma^{R}_{t,\ell}\rangle(\log
U+2\partial_{t}^{\text{no-log}})\\\
&+\frac{1}{2}\langle(\lambda^{(0)}_{t,\ell})^{2}\gamma^{{R}|{R}}_{t,\ell}\rangle(\log
U+2\partial_{t}^{\text{no-
log}})+\langle(\lambda^{{R}|{R}}_{t,\ell})^{2}\rangle\Big{]}\mathfrak{G}_{t+\ell,\ell}(U,V;\sigma,\tau)\\\
&+\sum_{\mathcal{M}_{\Delta,\ell}\in\\{(B,+),(B,2),(A,2)_{\ell},(A,+)_{\ell}\\}}(\lambda^{R|R}_{22\mathcal{M}})^{2}\mathfrak{G}_{\mathcal{M}}(U,V;\sigma,\tau)\,,\end{split}$
(3.1)

where $\partial_{t}^{\text{no-
log}}\mathfrak{G}_{t+\ell,\ell}(U,V;\sigma,\tau)$ was defined in (2.16). The
first three lines describe the double trace singlet long multiplets
$(A,0)^{[0000]}_{t+\ell,\ell}$, where $\langle\rangle$ denotes the average
over the $(t-1)$-fold degenerate operators. The fourth line includes all the
protected multiplets in $\langle 2222\rangle$ except the stress tensor
multiplet, which is $1/c_{T}$ exact. The expression for
${\mathcal{G}}^{R^{4}|R^{4}}$ at order $c_{T}^{-\frac{10}{3}}$ is identical
except we replace $R\to R^{4}$ and the sum for the long multiplets is now
restricted to $\ell=0$, while for ${\mathcal{G}}^{R|R^{4}}$ at order
$c_{T}^{-\frac{8}{3}}$ we furthermore replace the $\frac{1}{8}$ in the first
line by $\frac{1}{4}$, since the vertices are different.

As shown in [29], the entire 1-loop term up to the contact term ambiguities
described in Section 2.4 can in fact be constructed from the $\log^{2}U$ terms
shown above, which are written in terms of GFFT and tree data, since under
$1\leftrightarrow 3$ crossing

$\begin{split}{\mathcal{G}}(U,V;\sigma,\tau)=\frac{U}{V}\tau^{2}{\mathcal{G}}(V,U;\sigma/\tau,1/\tau)\,,\end{split}$
(3.2)

the $\log^{2}U$ terms are related to $\log^{2}V$ terms that are the only
contributions at this order to the DD, which fixes the entire correlator
according to the Lorentzian inversion formula [77, 30]. Note that the average
$\langle(\lambda^{(0)}_{t,\ell})^{2}\gamma^{A}_{t,\ell}\gamma^{B}_{t,\ell}\rangle$
for 1-loop vertices $A,B$ is what appears in the $\log^{2}U$ term, whereas the
different averages
$\langle(\lambda^{(0)}_{t,\ell})^{2}\gamma^{A}_{t,\ell}\rangle$ and
$\langle(\lambda^{(0)}_{t,\ell})^{2}\gamma^{B}_{t,\ell}\rangle$ are what
appear at tree level. As shown in [33, 38, 78, 34] for $\mathcal{N}=4$ SYM and
[44] for 6d $(2,0)$, one can compute
$\langle(\lambda^{(0)}_{t,\ell})^{2}\gamma^{A}_{t,\ell}\gamma^{B}_{t,\ell}\rangle$
from GFFT $\langle ppqq\rangle$ and tree level $\langle 22pp\rangle$ data as

$\begin{split}\langle(\lambda^{(0)}_{t,\ell})^{2}\gamma^{A}_{t,\ell}\gamma^{B}_{t,\ell}\rangle=\sum_{p=2,4,\dots}^{t}\frac{\langle\lambda^{(0)}_{2,t,\ell}\lambda^{(0)}_{p,t,\ell}\gamma^{A}_{t,\ell}\rangle\langle\lambda^{(0)}_{2,t,\ell}\lambda^{(0)}_{p,t,\ell}\gamma^{B}_{t,\ell}\rangle}{{\langle(\lambda^{(0)}_{p,t,\ell})^{2}\rangle}}\,,\end{split}$
(3.3)

where we summed over each $p$ for which a given twist $t$ long multiplet
appears. Unlike the 4d and 6d cases, in 3d the sum only runs over even $p$
regardless of the orbifold.121212As discussed in the introduction, for $k=1$
ABJM the DD would receive additional contributions from the OPE coefficients
of odd twist long multiplets that appear for odd $p$. We computed
${\langle(\lambda^{(0)}_{p,t,\ell})^{2}\rangle}$,
$\langle\lambda^{(0)}_{2,t,\ell}\lambda^{(0)}_{p,t,\ell}\gamma^{R}_{t,\ell}\rangle$,
and
$\langle\lambda^{(0)}_{2,t,\ell}\lambda^{(0)}_{p,t,\ell}\gamma^{R^{4}}_{t,\ell}\rangle$
in (2.14), (2.23), and (2.30), respectively, which is sufficient to compute
$R|R$, $R|R^{4}$, and $R^{4}|R^{4}$ for the $k=2$ ABJ(M) theory. The
$p,t,\ell$ sums for the $\log^{2}U$ term in $R|R$ can be done by expanding at
small $U$ in each $R$-symmetry channel to get:

$\begin{split}\frac{1}{8}\sum_{t=2,4,\dots}&\sum_{\ell\in\text{Even}}\sum_{p=2,4,\dots}^{t}\frac{\langle\lambda^{(0)}_{2,t,\ell}\lambda^{(0)}_{p,t,\ell}\gamma^{R}_{t,\ell}\rangle^{2}}{{\langle(\lambda^{(0)}_{p,t,\ell})^{2}\rangle}}\mathfrak{G}_{t+\ell,\ell}(U,V;\sigma,\tau)=\\\
&Y_{[0000]}(\sigma,\tau)\big{[}Uh_{R|R}^{(1),[0000]}(V)+\cdots\big{]}+Y_{[0100]}(\sigma,\tau)\big{[}Uh_{R|R}^{(1),[0100]}(V)+\cdots\big{]}\\\
&Y_{[0020]}(\sigma,\tau)\big{[}Uh_{R|R}^{(1),[0020]}(V)+\cdots\big{]}+Y_{[0120]}(\sigma,\tau)\big{[}U^{2}h_{R|R}^{(2),[0120]}(V)+\cdots\big{]}\\\
&Y_{[0200]}(\sigma,\tau)\big{[}U^{2}h_{R|R}^{(2),[0200]}(V)+\cdots\big{]}+Y_{[0040]}(\sigma,\tau)\big{[}U^{3}h_{R|R}^{(3),[0040]}(V)+\cdots\big{]}\,,\end{split}$
(3.4)

where the different powers of $U$ in each channel correspond to the lowest
twists that appear in each channel for the $(A,0)_{t+\ell,\ell}^{[0000]}$
superblock as given in Table 8 of [52]. The $U$-slices in each channel take
the form

$\begin{split}h^{(n),[0ab0]}_{R|R}(V)={P^{[0ab0]}_{1,R|R}(V)}\log^{2}V+{P^{[0ab0]}_{2,R|R}(V)}\text{Li}_{2}(1-V)+{P^{[0ab0]}_{3,R|R}(V)}\log
V+{P^{[0ab0]}_{4,R|R}(V)}\,,\end{split}$ (3.5)

where $P^{[0ab0]}_{i,R|R}(V)$ are polynomials in $V$ divided by monomials of
$(1-V)$ whose precise degree varies in each channel. The expressions for
$R|R^{4}$ and $R^{4}|R^{4}$ are also given by (3.3), except the $\ell$ sum is
trivially $\ell=0$ in those cases, and $R|R^{4}$ has an extra factor of 2. The
$U$-slices for $R|R^{4}$ take the simpler form

$\begin{split}h^{(n),[0ab0]}_{R|R^{4}}(V)={P^{[0ab0]}_{1,R|R^{4}}(V)}\log
V+{P^{[0ab0]}_{2,R|R^{4}}(V)}\,,\end{split}$ (3.6)

for similarly defined polynomials divided by monomials, and a similar
expression holds for $R^{4}|R^{4}$. We give the explicit expressions for many
$n$ in the attached Mathematica file.

###  3.2 Mellin amplitude and comparison to 11d

We now show how to complete the position space DD to the entire correlator
using crossing symmetry and the superconformal Ward identity in Mellin space.
For $R|R^{4}$ and $R^{4}|R^{4}$ we will find closed form expressions up to the
expected contact term ambiguities, while for $R|R$ we are able to compute a
closed form up to a certain polynomial in $s,t$ that in principle can be fixed
from the Ward identity, but is difficult to fix in practice. The expressions
we find in all cases are sufficient to check the flat space limit comparison
to 11d, as well as to extract all CFT data except for some low spins.

We can compute the Mellin amplitudes from the resummed DD’s following a
similar but more complicated version of the calculation in the 4d [35] and 6d
[44] cases. In the previous section, we computed the coefficient of
$\log^{2}U$ in the $s$-channel, which gave the DD in the $t$-channel as an
expansion in small $U$. From the definition of the Mellin transform in (2.17),
we can then convert $U^{n}\log^{2}Uh^{(n)}(V)$ to an $s$-pole in $M(s,t)$ as

$\begin{split}U^{n}\log^{2}Uh_{n}(V)\leftrightarrow\frac{res_{n-1}(t)}{s-2(n-1)}\,,\end{split}$
(3.7)

where the residues $res_{n-1}(t)$ follows from the $t$-integral in (2.17). For
$R^{4}|R$ and $R^{4}|R^{4}$, $res_{n-1}(t)$ is analytic because the slices
(3.6) do not contain any $\log^{2}V$ terms. In these cases, we can then use
crossing symmetry (2.18) to fix the other parts of the Mellin amplitude that
are analytic in $s$ to get

$\begin{split}M^{R|R^{4}}(s,t;\sigma,\tau)&=\sum_{m=1}^{\infty}\Bigg{[}\frac{1}{(s-2m)}\left(\frac{\hat{c}(m,s,t;\sigma,\tau)}{m}+\frac{\hat{d}(m,s,t;\sigma,\tau)\Gamma(m)}{\Gamma(m+\frac{1}{2})}\right)\\\
&\qquad\qquad\qquad\qquad\qquad\qquad+\text{crossed}\Bigg{]}+\sum_{i=1}^{50}\hat{k}_{i}{\bf
P}^{(8),i}(s,t;\sigma,\tau)\,,\\\
M^{R^{4}|R^{4}}(s,t;\sigma,\tau)&=\sum_{m=1}^{\infty}\left[\frac{\hat{\hat{d}}(m,s,t;\sigma,\tau)\Gamma(m)}{(s-2m)\Gamma(m+\frac{1}{2})}+\text{crossed}\right]+\sum_{i=1}^{84}\hat{\hat{k}}_{i}{\bf
P}^{(11),i}(s,t;\sigma,\tau)\,,\\\ \end{split}$ (3.8)

where $\hat{c}(m,s,t;\sigma,\tau)$, $\hat{d}(m,s,t;\sigma,\tau)$, and
$\hat{\hat{d}}(m,s,t;\sigma,\tau)$ are quadratic in $\sigma,\tau$ and
polynomials in $m,s,t$, while ${\bf P}^{(8),i}(s,t;\sigma,\tau)$ and ${\bf
P}^{(11),i}(s,t;\sigma,\tau)$ parameterize all crossing symmetric degree 8 and
11 polynomials in $s,t$, respectively, whose coefficients $\hat{k}_{i}$ and
$\hat{\hat{k}}_{i}$ should be fixed by the superconformal Ward identity in
terms of the physical contact term ambiguities in (1.1). Note that one can
swap $s$ for $2m$ in these expressions to get the same residues at the poles,
which only changes the $\hat{k}_{i}$ and $\hat{\hat{k}}_{i}$. The only rule in
performing this swap is that the degree of $M^{R|R^{4}}(s,t)$ and
$M^{R^{4}|R^{4}}(s,t)$ at large $s,t$ does not exceed $8.5$ and $11.5$,
respectively. In practice we can simply set $s=2m$ and similarly for the
crossed terms, which allows us to resum to get

$\begin{split}M^{R|R^{4}}&=\Bigg{[}\frac{\psi\left(1-\frac{s}{2}\right)}{s}{\bf
p}_{1}(s,t;\sigma,\tau)+\frac{\,{}_{3}F_{2}\left(1,1,1-\frac{s}{2};\frac{3}{2},2-\frac{s}{2};1\right)}{2-s}{\bf
p}_{2}(s,t;\sigma,\tau)\\\
&\qquad+\text{crossed}\Bigg{]}+\sum_{i=1}^{50}\hat{k}_{i}{\bf
P}^{(8),i}(s,t;\sigma,\tau)\,,\\\
M^{R^{4}|R^{4}}&=\Big{[}\frac{\,{}_{3}F_{2}\left(1,1,1-\frac{s}{2};\frac{3}{2},2-\frac{s}{2};1\right)}{2-s}{\bf
p}_{3}(s,t;\sigma,\tau)+\text{crossed}\Bigg{]}+\sum_{i=1}^{84}\hat{\hat{k}}_{i}{\bf
P}^{(8),i}(s,t;\sigma,\tau)\,,\\\ \end{split}$ (3.9)

where ${\bf p}_{i}(s,t;\sigma,\tau)$ are various polynomials in
$s,t,\sigma,\tau$ that are given in the attached Mathematica file, along with
the explicit $\hat{k}_{i}$ and $\hat{\hat{k}}_{i}$ that we fix using the
Mellin space Ward identity. At large $s,t$ these amplitudes take the form

$\begin{split}\lim_{s,t\to\infty}M^{R|R^{4}}(s,t;\sigma,\tau)&=\frac{655360\cdot{2}^{\frac{1}{6}}(-s)^{9/2}(t(-\sigma
s+s+t)+s\tau(s+t))^{2}}{3^{5/3}\pi^{19/6}}+\text{crossed}\,,\\\
\lim_{s,t\to\infty}M^{R^{4}|R^{4}}(s,t;\sigma,\tau)&=\frac{63078400\cdot(-s)^{15/2}(t(-\sigma
s+s+t)+s\tau(s+t))^{2}}{2^{\frac{1}{6}}{3}^{\frac{4}{3}}\pi^{23/6}}+\text{crossed}\,,\end{split}$
(3.10)

where we assumed $s,t<0$. We can then use the flat space limit formula (2.27)
for $p=2$ and $a=8.5$ and $a=11.5$ for $M^{R|R^{4}}$ and $M^{R^{4}|R^{4}}$,
respectively, as well as the relation between between $\ell_{11}$ and $c_{T}$
in (2.26) for $k=2$, to get precisely the expected 11d amplitudes as given in
equations 4.28 and 4.29 of [44].

For $R|R$, the residues $res_{n-1}(t)$ in (3.7) now contain poles in $t$
because the slices (3.5) contain $\log^{2}V$ terms. The resulting Mellin
amplitude thus contains both single and double pole terms

$\begin{split}M^{R|R}(s,t;\sigma,\tau)&=\Bigg{[}\sum_{m,n=1}^{\infty}\frac{c(m,n,s,t;\sigma,\tau)}{(s-2m)(t-2n)}\frac{\Gamma(m)\Gamma(n)\Gamma(m+n-\frac{11}{2})}{\Gamma(m+n-1)\Gamma(n-\frac{1}{2})\Gamma(m-\frac{1}{2})}\\\
&+\sum_{m=1}\frac{1}{s-2m}\Bigg{(}\frac{d(m,s,t;\sigma,\tau)\Gamma(m)}{\Gamma(m+\frac{1}{2})(m-4)(m-3)(m-2)(m-1)}\\\
&+\frac{e(m,s,t;\sigma,\tau)}{(m-4)(m-3)(m-2)(m-1)(2m-3)(2m-5)(2m-7)(2m-9)}\Bigg{)}\\\
&+\text{crossed}\Bigg{]}+\hat{\bf P}(s,t;\sigma,\tau)+\sum_{i=1}^{24}k_{i}{\bf
P}^{(5),i}(s,t;\sigma,\tau)\,.\end{split}$ (3.11)

Here, ${c}(m,n,s,t;\sigma,\tau)$, ${d}(m,s,t;\sigma,\tau)$, and
${{e}}(m,s,t;\sigma,\tau)$ are quadratic in $\sigma,\tau$ and polynomials in
$m,s,t$, while ${\bf P}^{(5),i}(s,t;\sigma,\tau)$ are all crossing symmetric
degree 5 polynomial in $s,t$, which in principle should be fixed by the
superconformal Ward identity in terms of just one of the 24 $k_{i}$. For the
double pole residues we can swap $s$ for $2m$ and $t$ for $2n$ to get the same
residue at the poles, but which will change the single pole residues and the
$k_{i}$. When swapping we must be careful that the resulting sums are all
finite, and that the large $s,t$ growth does not exceed $5.5$. In fact, for
all choices of swaps the large $s,t$ degree exceeds $5.5$, which is why we
must also include the polynomial $\hat{\bf P}(s,t;\sigma,\tau)$ that
generically will have degree greater than $5.5$, and is fixed to cancel the
corresponding large $s,t$ terms from the single and double sum terms.

It is difficult to check the large $s,t$ limit of (3.11) with the constraints
just discussed, because we must compute the double sum term to subleading
order in large $s,t$ to take into account the cancellations of the leading
terms by $\hat{\bf P}(s,t;\sigma,\tau)$. Instead, we can more easily compute
the large $s,t$ limit by considering the ansatz (3.11) but with coefficients
with unphysical poles at $s,t,u=0$:

$\begin{split}c_{\text{flat}}(s,t;\sigma,\tau)&\equiv\frac{4mn}{st}c(s/2,t/2,s,t;\sigma,\tau)\,,\\\
d_{\text{flat}}(s,t;\sigma,\tau)&\equiv\frac{1}{t}d(s/2,s,t;\sigma,\tau)\,,\qquad
e_{\text{flat}}(s,t;\sigma,\tau)\equiv\frac{1}{t}e(s/2,s,t;\sigma,\tau)\,,\\\
\end{split}$ (3.12)

as well as replacing the ${\bf P}^{(5),i}(s,t;\sigma,\tau)$ by a higher degree
polynomial multiplied by $\frac{1}{st}$, and without the now unnecessary
$\hat{\bf P}(s,t;\sigma,\tau)$. The resulting explicit Mellin amplitude is
given in the attached Mathematica file. In principle, we can completely fix
the $k_{i}$ using the superconformal Ward identity, which should cancel all
the unphysical poles at $s,t,u=0$. In practice, we checked that all the
residues for both the double and single poles in $s,t$ are satisfied by the
Ward identity, but did not carefully fix the $k_{i}$ since the terms they
multiply are by definition subleading in the large $s,t$ limit and we only use
this formulation to check the flat space limit. In particular, to get the
leading large $s,t$ term that appears in the flat space limit formula (2.27),
we should look at the regime where $m,n,s,t$ all scale equally large, in which
case we can replace the sums over $m,n$ by integrals. For the double pole term
we get

$\begin{split}&\lim_{s,t\to\infty}\sum_{m,n=1}^{\infty}\frac{c_{\text{flat}}(m,n,s,t;\sigma,\tau)}{(s-2m)(t-2n)}\frac{\Gamma(m)\Gamma(n)\Gamma(m+n-\frac{11}{2})}{\Gamma(m+n-1)\Gamma(n-\frac{1}{2})\Gamma(m-\frac{1}{2})}+\text{crossed}\\\
&=\int_{0}^{\infty}dmdn\frac{5120m^{3/2}n^{3/2}st(s+t)(t(-\sigma
s+s+t)+s\tau(s+t))^{2}}{3\pi^{7/2}(m+n)^{9/2}(s-2m)(t-2n)}+\text{crossed}\\\
&=-\frac{1024\sqrt{2}st(t(-\sigma
s+s+t)+s\tau(s+t))^{2}}{63\pi^{5/2}(s+t)^{4}\sqrt{st}}\Bigg{[}6(-t)^{9/2}-41s^{2}(-t)^{5/2}+88s^{3}(-t)^{3/2}\\\
&+88(-s)^{3/2}t^{3}-8\sqrt{-s}t^{4}-45(-s)^{7/2}t-45s(-t)^{7/2}+6(-s)^{9/2}-8s^{4}\sqrt{-t}-41(-s)^{5/2}t^{2}\\\
&+105s^{2}t^{2}\sqrt{-s-t}\log\Big{[}\frac{\left(\sqrt{-s-t}+\sqrt{-s}\right)\left(\sqrt{-s-t}+\sqrt{-t}\right)}{\sqrt{st}}\Big{]}\Bigg{]}+\text{crossed}\,,\\\
\end{split}$ (3.13)

where we assumed that $s,t<0$. For the single pole terms the $d_{\text{flat}}$
term is leading at large $s,t$ and gives

$\begin{split}&\lim_{s,t\to\infty}\sum_{m=1}\frac{1}{s-2m}\frac{d_{\text{flat}}(m,s,t;\sigma,\tau)\Gamma(m)}{\Gamma(m+\frac{1}{2})(m-4)(m-3)(m-2)(m-1)}+\text{crossed}\\\
&\qquad=-\int_{0}^{\infty}dm\frac{18}{7}\sqrt{\pi}m^{3/2}\left(4m^{2}\tau+2mt(-\sigma+\tau+1)+t^{2}\right)^{2}+\text{crossed}\\\
&\qquad=\frac{4096\sqrt{2}}{7\pi^{5/2}}(t(-\sigma
s+s+t)+s\tau(s+t))^{2}\left(it\sqrt{-s-t}+is\sqrt{-s-t}+(-s)^{3/2}+(-t)^{3/2}\right)\,.\\\
\end{split}$ (3.14)

We then plug these large $s,t$ expressions into the flat space limit formula
(2.27) for $p=2$ and $a=5.5$ and use the relation between between $\ell_{11}$
and $c_{T}$ in (2.26) for $k=2$ to precisely get the expected 11d amplitudes
as given in equations 4.19 of [44].

So far, we have a choice of coefficients (3.12) in (3.11) that gives a
putative 1-loop amplitude $M^{R|R}_{\text{flat}}$ that we know has the correct
flat space limit, but which has unphysical poles at $s,t,u=0$ that in
principle could be cancelled by subtracting a polynomial divided by $stu$, but
in practice are hard to fix using the superconformal Ward identity because of
the double sums. We can avoid these unphysical poles by choosing
${c}(m,n,s,t;\sigma,\tau)$, ${d}(m,s,t;\sigma,\tau)$, and
${{e}}(m,s,t;\sigma,\tau)$ in (3.11) that are in fact polynomials in $m,n,s,t$
as originally defined, and then demanding that the resulting expression for
$M^{R|R}$ matches $M^{R|R}_{\text{flat}}$ up to the degree 5 polynomial
ambiguities ${\bf P}^{(5),i}(s,t;\sigma,\tau)$. The resulting expressions for
$c,d,e$ are given in the attached Mathematica result, and are now guaranteed
to have both the correct flat space limit and only physical poles. Finally,
the coefficients $k_{i}$ can in principle be fixed using the superconformal
Ward identity in terms of just a single coefficient, which corresponds to the
single physical degree 4 contact term ambiguity in (1.1). In practice this is
difficult due to the double sums, so instead we fix most of these coefficients
in the next section by demanding a consistent superblock expansion, which is
equivalent to imposing the superconformal Ward identity but easier in
practice.

###  3.3 Extracting CFT data

We now extract all low-lying CFT data from the $R|R$, i.e. $c_{T}^{-2}$, and
$R|R^{4}$, i.e. $c_{T}^{-\frac{8}{3}}$, correlators using two independent
methods. Firstly, we derive an inversion integral formula for each DD in
position space, which lets us to extract all CFT data above a certain spin in
terms of a single integral, as expected from the Lorentzian inversion formula
[30]. Secondly, we expand each entire correlator as written in Mellin space in
superblocks to extract all CFT data for all spins up to the physical contact
term ambiguities that appear in (1.1), as well as some unphysical ambiguities
for $R|R$ that in principle can be fixed by the superconformal Ward identity.
We find that both methods agree in their respective regimes of applicability.
We do not extract CFT data from the $R^{4}|R^{4}$, i.e.
$c_{T}^{-\frac{10}{3}}$, correlator, since we anyway do not know the
$R|D^{6}R^{4}$ term that would contribute at the same order, but it would be
simple to extract the $R^{4}|R^{4}$ data as well from the formulae provided
here.

To extract CFT data, we will look at the superblock expansion in the lightcone
limit of small $U\sim z$, where conformal blocks are expanded as

$\begin{split}G_{\Delta,\ell}(U,V)=\sum_{n=0}^{\infty}U^{\frac{\Delta-\ell}{2}+n}g_{\Delta,\ell}^{[n]}(1-V)\,.\end{split}$
(3.15)

Here, the lowest so-called lightcone block in our normalization is

$\begin{split}g_{\Delta,\ell}^{[0]}(1-V)&=\frac{\Gamma(\ell+1/2)}{4^{\Delta}\sqrt{\pi}\ell!}(1-V)^{\ell}\,{}_{2}F_{1}\left(\frac{\Delta+\ell}{2},\frac{\Delta+\ell}{2},\Delta+\ell,1-V\right)\,,\\\
\end{split}$ (3.16)

and we see that the expansion is naturally organized in terms of twist
$t\equiv\Delta-\ell$. Applying this expansion to the superblocks, we observe
that blocks in different supermultiplets with the same twist can appear in the
same $R$-symmetry channel, so it is convenient to look at channels with the
least mixing between different supermultiplets. For instance, the lowest twist
long multiplet $(A,0)_{\ell+2,\ell}$ contributes at greater than twist two in
the $[0200]$, $[0120]$, and $[0040]$ channels, while all the protected
multiplets contribute at lower twists in these channels, so these channels are
the simplest for extracting OPE coefficients of protected operators. In
particular, if we focus on the lowest twist 2 conformal blocks at $O(U)$, then
from [52] we see that the short superblocks contain the blocks

$\begin{split}&\text{twist =
2:}\qquad\mathfrak{G}_{(B,+)}^{[0040]}=G_{2,0}\,,\qquad\mathfrak{G}_{(B,+)}^{[0120]}=-\frac{4}{3}G_{3,1}\,,\qquad\mathfrak{G}_{(B,+)}^{[0200]}=0\,,\\\
&\qquad\qquad\qquad\;\,\mathfrak{G}_{(B,2)}^{[0040]}=0\,,\qquad\quad\;\,\mathfrak{G}_{(B,2)}^{[0120]}=-\frac{8}{3}G_{3,1}\,,\qquad\mathfrak{G}_{(B,2)}^{[0200]}=G_{2,0}+\frac{64}{45}G_{4,2}\,,\\\
\end{split}$ (3.17)

while the semishort superblocks contain

$\begin{split}&\text{twist =
2:}\quad\mathfrak{G}_{(A,+)_{\ell}}^{[0040]}=\frac{16}{3}G_{\ell+4,\ell+2}\,,\qquad\mathfrak{G}_{(A,+)_{\ell}}^{[0120]}=-4G_{3+\ell,1+\ell}-\frac{64(\ell+3)^{4}G_{5+\ell,3+\ell}}{(2\ell+5)^{2}(2\ell+7)^{2}}\,,\\\
&\quad\qquad\qquad\;\,\mathfrak{G}_{(A,+)_{\ell}}^{[0200]}=\frac{32(\ell+2)(\ell+3)}{3(2\ell+3)(2\ell+7)}G_{4+\ell,2+\ell}\,,\\\
&\quad\qquad\qquad\;\,\mathfrak{G}_{(A,2)_{\ell}}^{[0040]}=0\,,\qquad\qquad\qquad\;\;\mathfrak{G}_{(A,2)_{\ell}}^{[0120]}=-\frac{32(\ell+2)^{2}}{(2\ell+3)(2\ell+5)}G_{\ell+4,\ell+2}\,,\\\
&\quad\qquad\qquad\;\,\mathfrak{G}_{(A,2)_{\ell}}^{[0200]}=4G_{\ell+3,\ell+1}+\frac{64(\ell+2)^{2}(\ell+3)^{2}}{(2\ell+3)(2\ell+5)^{2}(2\ell+7)}G_{\ell+5,\ell+3}\,.\\\
\end{split}$ (3.18)

For the long superblock, we will only extract its lowest twist 2 anomalous
dimension, for which it is convenient to consider the $[0040]$ channel where
only a single block at twist 6 appears:

$\begin{split}\mathfrak{G}_{\ell+2,\ell}^{[0040]}=\frac{16(2\ell+2)(2\ell+4)}{(2\ell+3)(2\ell+5)}G_{\ell+6,\ell}\,.\end{split}$
(3.19)

We can now use this explicit block decomposition to extract CFT data, first
using the Lorentzian inversion formula. In Appendix B we review following the
similar 4d case in [37] how to use the inversion formula to extract CFT data
from the DD of a 3d CFT by comparing to its conformal block expansion in the
large $c_{T}$ limit. We can apply this to the twist 2 block expansion in the
$[0040]$ channel (3.18) for general $\ell$ where only $(A,+)_{\ell}$ appears,
so that we get

$\begin{split}\lambda^{2}_{(A,+),\ell}=\frac{12(2\ell+5)\Gamma(\ell+3)^{4}}{\Gamma\left(\ell+\frac{5}{2}\right)^{2}\Gamma\left(\ell+\frac{7}{2}\right)^{2}}\int_{0}^{1}\frac{d\bar{z}}{\bar{z}}g_{\ell+4,\ell+2}(\bar{z})\text{dDisc}[{\cal
G}^{[0040]}(z\bar{z},1-\bar{z})|_{z}]\,,\end{split}$ (3.20)

where ${\cal G}^{[0040]}(z\bar{z},1-\bar{z})|_{z}$ denotes the leading $z$
term in the basis (2.4), we dropped the superscript from the leading lightcone
block in (3.16), and the overall normalization was fixed using the known GFFT
term in (2.34) as discussed in Appendix B. The $(B,+)$ supermultiplet also
appears at leading twist in this channel, and comparing (3.17) to (3.18) we
see corresponds to the limit131313In other channels negative spins blocks
appear in this limit as discussed in [52], so the comparison is more subtle.

$\begin{split}\mathfrak{G}_{(B,+)}^{[0040]}=\frac{3}{16}\lim_{\ell\to-2}\mathfrak{G}_{(A,+)_{\ell}}^{[0040]}\,,\end{split}$
(3.21)

so can be considered a special case of (3.20). We can similarly analyze the
$[0200]$ and $[0120]$ channels to get

$\begin{split}&\frac{12(\ell+1)^{2}(\ell+2)^{2}}{(2\ell+1)(2\ell+3)^{2}(2\ell+5)}\lambda^{2}_{(A,2)_{\ell-1}}+\frac{2(\ell+2)(\ell+3)}{(2\ell+3)(2\ell+7)}\lambda^{2}_{(A,+)_{\ell}}+\frac{3}{4}\lambda^{2}_{(A,2)_{\ell+1}}=\\\
&\frac{12(2\ell+5)\Gamma(\ell+3)^{4}}{\Gamma\left(\ell+\frac{5}{2}\right)^{2}\Gamma\left(\ell+\frac{7}{2}\right)^{2}}\int_{0}^{1}\frac{d\bar{z}}{\bar{z}}g_{\ell+4,\ell+2}(\bar{z})\text{dDisc}[{\cal
G}^{[0200]}(z\bar{z},1-\bar{z})|_{z}]\,,\end{split}$ (3.22)

and

$\begin{split}&\frac{12(\ell+2)^{4}}{(2\ell+3)^{2}(2\ell+5)^{2}}\lambda^{2}_{(A,+)_{\ell-1}}+\frac{6(\ell+2)^{2}}{(2\ell+3)(2\ell+5)}\lambda^{2}_{(A,2)_{\ell}}+\frac{3}{4}\lambda^{2}_{(A,+)_{\ell+1}}=\\\
&-\frac{12(2\ell+5)\Gamma(\ell+3)^{4}}{\Gamma\left(\ell+\frac{5}{2}\right)^{2}\Gamma\left(\ell+\frac{7}{2}\right)^{2}}\int_{0}^{1}\frac{d\bar{z}}{\bar{z}}g_{\ell+4,\ell+2}(\bar{z})\text{dDisc}[{\cal
G}^{[0120]}(z\bar{z},1-\bar{z})|_{z}]\,,\end{split}$ (3.23)

where the minus sign is expected because the spin is odd. These two equations
give an overconstrained system for $\lambda^{2}_{(A,2)_{\ell}}$ in terms of
$\lambda^{2}_{(A,+)_{\ell-1}}$ and $\lambda^{2}_{(A,+)_{\ell+1}}$, which can
be extracted seperately from (3.20). From comparing (3.17) to (3.18), we see
that $(B,2)$ corresponds to the limit

$\begin{split}\mathfrak{G}_{(B,2)}^{[0200]}=\frac{1}{4}\lim_{\ell\to-1}\mathfrak{G}_{(A,2)_{\ell}}^{[0200]}\,,\end{split}$
(3.24)

and similarly for the $[0040]$ and $[0120]$ channels, so its OPE coefficient
can be extracted from $4\lambda^{2}_{(A,2)_{-1}}$. Finally, we can apply the
inversion analysis in Appendix B for the $R|R$ correction to the anomalous
dimension to the long superblock in the $[0040]$ channel (3.19) to get

$\begin{split}\gamma^{R|R}_{2,\ell}=&\frac{1}{(\lambda^{(0)}_{2,\ell})^{2}}\Big{(}4R^{[0040]}_{1,R|R}(\ell)+\frac{1}{2}\partial_{\ell}\big{[}(\lambda^{(0)}_{2,\ell})^{2}(\gamma^{R}_{2,\ell})^{2}\big{]}-(\lambda^{R}_{2,\ell})^{2}\gamma^{R}_{2,\ell}\Big{)}\,,\end{split}$
(3.25)

where we have the inversion integral

$\begin{split}R^{[0040]}_{1,R|R}(\ell)&=\frac{512(\ell+1)(\ell+2)(2\ell+3)\Gamma(\ell+1)^{4}}{\Gamma\left(\ell+\frac{1}{2}\right)^{2}\Gamma\left(\ell+\frac{5}{2}\right)^{2}}\int_{0}^{1}{d\bar{z}}{\bar{z}}g_{\ell+6,\ell}(\bar{z})\text{dDisc}[\left.{\cal
G}_{R|R}^{[0040]}(z\bar{z},1-\bar{z})\right|_{z^{3}\log z}]\,.\end{split}$
(3.26)

For $R|R^{4}$ we have a similar expression, except without the tree level
terms in (3.25) since for $R^{4}$ they only have support for $\ell=0$.

To apply these inversion integrals, we need to compute the leading $z$ term of
the DD in various channels for $R|R$ and $R|R^{4}$, which is given by the
coefficient of $\log^{2}(1-\bar{z})$ according to

$\begin{split}{\rm
dDisc}\,[f(z,\bar{z})\log^{2}(1-\bar{z})]=4\pi^{2}f(z,\bar{z})\,,\end{split}$
(3.27)

for arbitrary $f(z,\bar{z})$ analytic at $\bar{z}=1$. We compute the
$\log^{2}(1-\bar{z})$ term by taking the $U$-slices $h^{(n),[0ab0]}(V)$ in
Section 3.1 that multiply $\log^{2}U$, applying the $1\leftrightarrow 3$
crossing (3.2) to get expressions that multiply
$\log^{2}V\sim\log^{2}(1-\bar{z})$, resumming the slices, and reexpanding in
the $Y_{[0ab,0]}(\sigma,\tau)$ to get the DD in each irrep. For $R|R^{4}$, we
were able to find closed form expressions, while for $R|R$ the result is
written in terms of an integral over an auxiliary variable, see the attached
Mathematica notebook for the explicit expressions. For $R|R$ we find that the
inversion integrals with these explicit DDs converges for $\ell>-\frac{1}{2}$
for (3.20), for $\ell>\frac{1}{2}$ for (3.23), and for $\ell>\frac{3}{2}$ for
(3.26), which allows us to compute the CFT data:

$\begin{split}(\lambda^{R|R}_{(A,+)_{0}})^{2}&=285.32043668331375685394087\,,\\\
(\lambda^{R|R}_{(A,+)_{2}})^{2}&=77.098186992813023177613926\,,\\\
(\lambda^{R|R}_{(A,+)_{4}})^{2}&=48.536178605208049991881361\,,\\\
(\lambda^{R|R}_{(A,2)_{1}})^{2}&=2239.9009500059848334084088\,,\\\
(\lambda^{R|R}_{(A,2)_{3}})^{2}&=540.71435539680002180491475\,,\\\
(\lambda^{R|R}_{(A,2)_{5}})^{2}&=328.90127928121821108070743\,,\\\
\gamma_{2,2}^{R|R}&=\frac{1645242368}{1125\pi^{4}}-\frac{207785984}{663\pi^{2}}\,,\\\
\gamma_{2,4}^{R|R}&=\frac{80811812224}{25725\pi^{4}}-\frac{2170015744}{6783\pi^{2}}\,,\\\
\gamma_{2,6}^{R|R}&=\frac{14459024425792}{3678675\pi^{4}}-\frac{45500125184}{115115\pi^{2}}\,,\\\
\end{split}$ (3.28)

where we could also compute higher spin data if desired.141414From computing
some higher spins values, we observed that the anomalous dimensions are not
monotonic in spin until $\ell=6$, unlike the the case of 4d $\mathcal{N}=4$
SYM [34] and 6d $(2,0)$ [44] that was monotonic in spin in general. Of course,
monotonicity in spin is only required at sufficiently high spin [79]. Note
that the CFT data that we cannot compute, namely $\gamma_{2,0}^{R|R}$,
$(\lambda^{R|R}_{(B,+)_{0}})^{2}$, and $(\lambda^{R|R}_{(B,2)_{0}})^{2}$, is
what is affected by the degree 4 contact term $B_{4}^{R|R}M^{4}$ in (1.1), as
shown in Table 2, which is analogous to the 4d [33] and 6d [44] cases. For
$R|R^{4}$ we find that the inversion integrals with the explicit $R|R^{4}$ DDs
converges for $\ell>\frac{7}{2}$ for (3.20), for $\ell>\frac{9}{2}$ for
(3.23), and for $\ell>\frac{11}{2}$ for (3.26), which allows us to compute the
CFT data:

$\begin{split}(\lambda^{R|R^{4}}_{(A,+)_{4}})^{2}&=\frac{22291954008064\left(\frac{2}{3}\right)^{2/3}}{4357815\pi^{14/3}}+\frac{1561306511441920\left(\frac{2}{3}\right)^{2/3}}{6298655363\pi^{8/3}}\,,\\\
(\lambda^{R|R^{4}}_{(A,2)_{5}})^{2}&=\frac{254814018760343552\left(\frac{2}{3}\right)^{2/3}}{3277699425\pi^{14/3}}+\frac{281474976710656\left(\frac{2}{3}\right)^{2/3}}{72177105\pi^{8/3}}\,,\\\
\gamma_{2,6}^{R|R^{4}}&=-\frac{512640462848\left(\frac{2}{3}\right)^{2/3}}{693\pi^{14/3}}-\frac{110655386419200\left(\frac{2}{3}\right)^{2/3}}{2956811\pi^{8/3}}\,,\\\
\end{split}$ (3.29)

where we could also compute higher spin data if desired. The CFT data we
cannot compute is what is affected by the degree 8 and smaller contact terms
in (1.1), as shown in Table 2, which is analogous to the 4d [34] and 6d [44]
cases.

Finally, we can also extract CFT data from the entire correlator as written in
Mellin space in terms of the contact term ambiguities described before. We
extract this data in the lightcone expansion following [62], by taking the
relevant $s$-pole, doing the $t$-integral, projecting against a block of the
corresponding spin using the projectors introduced in [20], and comparing
against the lightcone expansion of the superblocks in (3.17), (3.18), and
(3.19). For $R|R$, we first demand that $\log U$ terms, which correspond to
anomalous dimensions, should only show up in the $[0200]$, $[0120]$, and
$[0040]$ channels starting with the appropriate twists, which fixes 12 of the
24 coefficients $k_{i}$ of the polynomial ambiguity in the Mellin amplitude
(3.11). After fixing these, we find that $\gamma^{R|R}_{2,\ell}$ for $\ell\geq
4$, $(\lambda_{(A,+)_{\ell}})^{2}$ for $\ell\geq 2$, and
$(\lambda_{(A,2)_{\ell}})^{2}$ for $\ell\geq 3$ are unaffected by the
remaining $k_{i}$, so we could extract this data by computing the double sums
numerically and confirm the inversion results in (3.28). For $R|R^{4}$, we
already completely fixed the Mellin amplitude up to the physical contact term
ambiguities in (1.1), so we can compute all CFT data, which confirms the
inversion results in (3.29), and also gives the complete result at order
$c_{T}^{-\frac{8}{3}}$ for the low spin data:

$\begin{split}(\lambda^{R|R^{4}}_{(B,+)})^{2}&=-\frac{59609927581958144\left(\frac{2}{3}\right)^{2/3}}{14189175\pi^{14/3}}+\frac{256}{35}B_{4}^{R|R^{4}}\,,\\\
(\lambda^{R|R^{4}}_{(B,2)})^{2}&=-\frac{59609927581958144\left(\frac{2}{3}\right)^{2/3}}{2837835\pi^{14/3}}+\frac{256}{7}B_{4}^{R|R^{4}}\,,\\\
(\lambda^{R|R^{4}}_{(A,+)_{0}})^{2}&=-\frac{7798563930112\left(\frac{2}{3}\right)^{2/3}}{1216215\pi^{14/3}}-\frac{134217728\left(\frac{2}{3}\right)^{2/3}}{429\pi^{8/3}}\\\
&\quad+\frac{16384}{1485}B_{6}^{RR|R^{4}}+\frac{950272}{6435}B_{7}^{RR|R^{4}}-\frac{131396796416}{467137125}B_{8}^{RR|R^{4}}\,,\\\
(\lambda^{R|R^{4}}_{(A,+)_{2}})^{2}&=-\frac{148820650360832\left(\frac{2}{3}\right)^{2/3}}{2786875\pi^{14/3}}-\frac{229076375699456\left(\frac{2}{3}\right)^{2/3}}{56581525\pi^{8/3}}+\frac{67108864}{557375}B_{8}^{R|R^{4}}\,,\\\
(\lambda^{R|R^{4}}_{(A,2)_{1}})^{2}&=\frac{3402914332672\left(\frac{2}{3}\right)^{2/3}}{218295\pi^{14/3}}-\frac{794568949760\left(\frac{2}{3}\right)^{2/3}}{29393\pi^{8/3}}\\\
&\quad+\frac{131072}{1155}B_{6}^{R|R^{4}}+\frac{21889024}{15015}B_{7}^{R|R^{4}}-\frac{3848847491072}{1089986625}B_{8}^{R|R^{4}}\,,\\\
(\lambda^{R|R^{4}}_{(A,2)_{3}})^{2}&=-\frac{3672876448219136\left(\frac{2}{3}\right)^{2/3}}{2546775\pi^{14/3}}-\frac{614077244112896\left(\frac{2}{3}\right)^{2/3}}{6292363\pi^{8/3}}+\frac{268435456}{121275}B_{8}^{R|R^{4}}\,,\\\
\gamma_{2,0}^{R|R^{4}}&=\frac{5112797289066496\left(\frac{2}{3}\right)^{2/3}}{45045\pi^{14/3}}+\frac{359760199680\left(\frac{2}{3}\right)^{2/3}}{46189\pi^{8/3}}\\\
&\quad-192B_{4}^{R|R^{4}}+\frac{15360}{11}B_{6}^{R|R^{4}}+\frac{192000}{11}B_{7}^{R|R^{4}}-\frac{18059264}{1521}B_{8}^{R|R^{4}}\,,\\\
\gamma_{2,2}^{R|R^{4}}&=\frac{12875118112768\left(\frac{2}{3}\right)^{2/3}}{1365\pi^{14/3}}+\frac{162643071467520\left(\frac{2}{3}\right)^{2/3}}{96577\pi^{8/3}}\\\
&\quad-1536B_{6}^{R|R^{4}}-18432B7^{R|R^{4}}+\frac{509591552}{12675}B_{8}^{R|R^{4}}\,,\\\
\gamma_{2,4}^{R|R^{4}}&=\frac{1806876913664\left(\frac{2}{3}\right)^{2/3}}{63\pi^{14/3}}+\frac{709927895040\left(\frac{2}{3}\right)^{2/3}}{391\pi^{8/3}}-32768B_{8}^{R|R^{4}}\,,\\\
\end{split}$ (3.30)

where note that
$(\lambda^{R|R^{4}}_{(B,2)})^{2}=5(\lambda^{R|R^{4}}_{(B,+)})^{2}$ as expected
from (2.33).

##  4 Fixing 1-loop contact terms

So far we computed the 1-loop Mellin amplitudes $M^{R|R}(s,t)$,
$M^{R|R^{4}}(s,t)$, and $M^{R^{4}|R^{4}}(s,t)$, but we did not fix the
polynomial in $s,t$ contact terms ambiguities that appear in (1.1) at orders
$c_{T}^{-2}$, $c_{T}^{-\frac{8}{3}}$, and $c_{T}^{-\frac{10}{3}}$,
respectively. It is necessary to fix these contact terms if we want to extract
low spin CFT data at these orders that are affected by these ambiguities, as
summarized by Table 2. We will fix the contact term ambiguities for
$M^{R|R}(s,t)$ and $M^{R|R^{4}}(s,t)$ using two methods. First, we will
propose a unique analytic continuation of Lorentzian inversion below the spins
where it was shown to converge in the previous section, which will allow us to
fix all CFT data at order $c_{T}^{-2}$, and all but the CFT data affected by
the $B^{R|R^{4}}_{4}M^{4}$ contact term at order $c_{T}^{-\frac{8}{3}}$. We
will then use the two localization constraints from [10, 21] to confirm these
results at order $c_{T}^{-2}$, as well as fix $B^{R|R^{4}}_{4}$ and give an
additional nontrivial consistency check. Note that at order $c_{T}^{-2}$, we
will be able to extract all CFT data even though we will not write down an
explicit Mellin amplitude, because we have not yet fixed all the coefficients
$k_{i}$ in (3.11) that are in principle fixed by the superconformal Ward
identity. Also, we cannot similarly analyze the order $c_{T}^{-\frac{10}{3}}$
term yet, because it receives contributions not only from
$M^{R^{4}|R^{4}}(s,t)$ but also from $M^{R|D^{6}R^{4}}(s,t)$, which has not
yet been computed.

###  4.1 Analytic continuation of Lorentzian inversion

Let us begin by discussing the inversion integral (3.20) for the $[0040]$
channel, which we use to compute $\lambda^{2}_{(A,+)_{\ell}}$ for even $\ell$.
For $R|R$, we can see from the explicit expression for the DD in the attached
Mathematica file that it has the small $\bar{z}$ expansion

$\begin{split}\text{dDisc}[{\cal
G}_{R|R}^{[0040]}(z\bar{z},1-\bar{z})|_{z}]=\frac{3072}{\pi{\bar{z}}^{3/2}}-\frac{61952}{27\pi\sqrt{\bar{z}}}+\dots\,.\end{split}$
(4.1)

Since the measure in (3.20) scales as ${\bar{z}}^{\ell+1}$, we see that this
integral converges for $\ell>-\frac{1}{2}$, which allowed us to compute all
the $(\lambda^{R|R}_{(A,+)_{\ell}})^{2}$ for even $\ell\geq 0$ in the previous
section, but did not allow us to compute
$(\lambda^{R|R}_{(B,+)})^{2}=\frac{16}{3}(\lambda^{R|R}_{(A,+)_{-2}})^{2}$ as
given by (3.21). We can uniquely analytically continue (3.20) to $\ell=-2$ by
writing it as

$\begin{split}\lambda^{2}_{(A,+),\ell}&=\frac{12(2\ell+5)\Gamma(\ell+3)^{4}}{\Gamma\left(\ell+\frac{5}{2}\right)^{2}\Gamma\left(\ell+\frac{7}{2}\right)^{2}}\Bigg{[}\int_{0}^{1}\frac{d\bar{z}}{\bar{z}}g_{\ell+4,\ell+2}(\bar{z})\Big{[}\text{dDisc}[{\cal
G}^{[0040]}(z\bar{z},1-\bar{z})|_{z}]\\\
&\quad-\frac{3072}{\pi}\left(\frac{1-\bar{z}}{\bar{z}}\right)^{3/2}-\frac{62464}{27\pi}\sqrt{\frac{1-z}{z}}\Big{]}+\frac{3072}{\pi}{\bf
f}(\ell,3/2)+\frac{62464}{27\pi}{\bf f}(\ell,1/2)\Bigg{]}\,,\end{split}$ (4.2)

where we define the analytically continued integral

$\begin{split}\int_{0}^{1}\frac{d\bar{z}}{\bar{z}}g_{\ell+4,\ell+2}(\bar{z})\left(\frac{1-\bar{z}}{\bar{z}}\right)^{p}=\frac{\Gamma(2(\ell+3))\Gamma(p+1)^{2}\Gamma(\ell-p+2)}{\Gamma(\ell+3)^{2}\Gamma(\ell+p+4)}\,,\end{split}$
(4.3)

which can be computed using the integral expression for the hypergeometric
function in the lightcone block (3.16). The explicit $\bar{z}$ integral in
(4.2) as well as $\bf{f}(\ell,3/2)$ and $\bf{f}(\ell,1/2)$ are now convergent
for $\ell<-\frac{1}{2}$, which we can compute at high precision for $\ell=-2$
to get

$\begin{split}(\lambda^{R|R}_{(B,+)})^{2}=\frac{16}{3}(\lambda^{R|R}_{(A,+)_{-2}})^{2}=\frac{32768}{45\pi^{2}}-\frac{81920}{9\pi^{4}}\,.\end{split}$
(4.4)

We can similarly analytically continue the inversion integrals in the $[0120]$
(3.23) and $[0200]$ (3.22) channels to compute
$(\lambda^{R|R}_{(B,2)})^{2}=4(\lambda^{R|R}_{(A,2)_{-1}})^{2}$, which is
related to $(\lambda^{R|R}_{(B,+)})^{2}$ by a factor of 5 as expected from
(2.33), which is evidence that our analytic continuation respects
superconformal symmetry. The analytic continuation of the inversion integral
(3.26) for the anomalous dimension then gives

$\begin{split}\gamma_{2,0}^{R|R}=\frac{46224640}{9\pi^{4}}-\frac{117698560}{429\pi^{2}}\,,\end{split}$
(4.5)

which is the only other CFT data that was affected by the
$B_{4}^{R|R}M^{4}(s,t)$ contact term, and so could not be computed in the
previous section.

We can then analytically continue the inversion integrals in the same way for
$R|R^{4}$. In this case we find that the small $\bar{z}$ expansion of the DD
includes a term ${\bar{z}}^{-1}$, which gives a ${\bf{f}}(\ell,1)$ term after
analytic continuation. From (4.3), we see that this ${\bf{f}}(\ell,1)$ has a
logarithmic divergence at $\ell=-2$, so we can only analytically continue for
$\ell>-2$, which allows us to compute all $(\lambda^{R|R}_{(A,+)_{\ell}})^{2}$
for even $\ell\geq 0$, but not
$(\lambda^{R|R^{4}}_{(B,+)})^{2}=\frac{16}{3}(\lambda^{R|R^{4}}_{(A,+)_{-2}})^{2}$.
We see a similar pattern in the other inversion integrals, where we can
compute all CFT except $(\lambda^{R|R^{4}}_{(B,2)})^{2}$ and all
$\gamma^{R|R^{4}}_{2,0}$, which would be affected by the
$B_{4}^{R|R^{4}}M^{4}(s,t)$ contact term. The results for the other CFT data
are151515Curiously, the results for the anomalous dimensions for $R|R^{4}$
become monotonic in spin at $\ell=6$, which was the same value for $R|R$ as
discussed above.

$\begin{split}(\lambda^{R|R^{4}}_{(A,+)_{0}})^{2}&=-\frac{269877248\left(\frac{2}{3}\right)^{2/3}}{45\pi^{14/3}}-\frac{134217728\left(\frac{2}{3}\right)^{2/3}}{429\pi^{8/3}}\,,\\\
(\lambda^{R|R^{4}}_{(A,+)_{2}})^{2}&=-\frac{7322684358656\left(\frac{2}{3}\right)^{2/3}}{91875\pi^{14/3}}-\frac{229076375699456\left(\frac{2}{3}\right)^{2/3}}{56581525\pi^{8/3}}\,,\\\
(\lambda^{R|R^{4}}_{(A,2)_{1}})^{2}&=-\frac{167914766336\left(\frac{2}{3}\right)^{2/3}}{315\pi^{14/3}}-\frac{794568949760\left(\frac{2}{3}\right)^{2/3}}{29393\pi^{8/3}}\,,\\\
(\lambda^{R|R^{4}}_{(A,2)_{3}})^{2}&=-\frac{1634776490442752\left(\frac{2}{3}\right)^{2/3}}{848925\pi^{14/3}}-\frac{614077244112896\left(\frac{2}{3}\right)^{2/3}}{6292363\pi^{8/3}}\,,\\\
\gamma_{2,2}^{R|R^{4}}&=\frac{166203518976\left(\frac{2}{3}\right)^{2/3}}{5\pi^{14/3}}+\frac{162643071467520\left(\frac{2}{3}\right)^{2/3}}{96577\pi^{8/3}}\,,\\\
\gamma_{2,4}^{R|R^{4}}&=\frac{2257848479744\left(\frac{2}{3}\right)^{2/3}}{63\pi^{14/3}}+\frac{709927895040\left(\frac{2}{3}\right)^{2/3}}{391\pi^{8/3}}\,.\\\
\end{split}$ (4.6)

We can compare this against the CFT data in (3.30) that was extracted from the
explicit Mellin amplitude, which fixes the coefficients

$\begin{split}B_{6}^{R|R^{4}}=-\frac{128720195584\left(\frac{2}{3}\right)^{2/3}}{819\pi^{14/3}}\,,\qquad
B_{7}^{R|R^{4}}=\frac{775420813312\left(\frac{2}{3}\right)^{2/3}}{68445\pi^{14/3}}\,,\qquad
B_{8}^{R|R^{4}}=-\frac{655360\left(\frac{2}{3}\right)^{2/3}}{3\pi^{14/3}}\,.\end{split}$
(4.7)

We cannot yet fix the $B_{4}^{R|R^{4}}$ coefficient, because the analytic
continuation of inversion did not converge for the low spin data affected by
$M^{4}(s,t)$.

###  4.2 Supersymmetric localization

We can also fix the Mellin amplitudes using the two localization constraints
in [10, 21]. The first constraint is simply the value of the short OPE
coefficients $\lambda^{2}_{(B,+)}$ and $\lambda^{2}_{(B,2)}$, which are shown
to all orders in $1/c_{T}$ in (2.32), and impose just one independent
constraint on the 4-point function due to the relation (2.33). For $R|R$, the
localization values exactly match the prediction in (4.4) from the
analytically continued Lorentzian inversion formula, which independently fixes
the CFT data affected by $M^{4}(s,t)$ without needing to assume the
conjectured analytic continuation, and so gives a nontrivial check on that
conjecture. For $R|R^{4}$ we use this constraint to fix the last contact term
ambiguity

$\begin{split}B_{4}^{R|R^{4}}=\frac{65229926487808\left(\frac{2}{3}\right)^{2/3}}{135135\pi^{14/3}}\,,\end{split}$
(4.8)

which we can then use to compute the last remaining unfixed CFT datum

$\begin{split}\gamma^{R|R^{4}}_{2,2}=\frac{25509449728\left(\frac{2}{3}\right)^{2/3}}{15\pi^{14/3}}+\frac{359760199680\left(\frac{2}{3}\right)^{2/3}}{46189\pi^{8/3}}\,.\end{split}$
(4.9)

The second localization constraint involves a nontrivial integral [23]:

$\begin{split}&\frac{\partial\log Z}{\partial m_{+}^{2}\partial
m_{-}^{2}}\Big{|}_{m_{\pm}=0}=\frac{\pi^{2}c_{T}^{2}}{2^{11}}I_{+-}[{\mathcal{S}}^{i}]\,,\\\
&\qquad\quad\;\;\,I_{+-}[{\cal S}^{i}]\equiv\int\frac{ds\ dt}{(4\pi
i)^{2}}\frac{2\sqrt{\pi}}{(2-t)(s+t-2)}\mathfrak{M}_{1}(s,t)\\\
&\qquad\qquad\quad\times\Gamma\left[1-\frac{s}{2}\right]\Gamma\left[\frac{s+1}{2}\right]\Gamma\left[1-\frac{t}{2}\right]\Gamma\left[\frac{t-1}{2}\right]\Gamma\left[\frac{s+t-2}{2}\right]\Gamma\left[\frac{3-s-t}{2}\right]\,,\end{split}$
(4.10)

where $\mathfrak{M}_{1}(s,t)$ is the first element of the Mellin amplitude
basis

$\begin{split}M(s,t;\sigma,\tau)=\mathfrak{M}_{1}+\sigma^{2}\mathfrak{M}_{2}+\tau^{2}\mathfrak{M}_{3}+\sigma\tau\mathfrak{M}_{4}+\tau\mathfrak{M}_{5}+\sigma\mathfrak{M}_{6}\,,\end{split}$
(4.11)

and the mass derivatives of the partition function was computed to all orders
in $1/c_{T}$ in [21] for $k=2$ ABJ(M):

$\begin{split}\frac{\partial\log Z}{\partial m_{+}^{2}\partial
m_{-}^{2}}\Big{|}_{m_{\pm}=0}=-\frac{\pi^{2}}{64c_{T}}+\frac{5\pi^{4/3}\left(\frac{2}{3}\right)^{\frac{2}{3}}}{16c_{T}^{5/3}}-\frac{5}{12c_{T}^{2}}-\frac{4\left(\frac{2\pi^{2}}{3}\right)^{\frac{1}{3}}}{3c_{T}^{7/3}}+\frac{91\left(\frac{2}{3\pi}\right)^{2/3}}{9c_{T}^{8/3}}+O(c_{T}^{-3})\,.\end{split}$
(4.12)

For the Mellin amplitudes that appear at order $c_{T}^{-\frac{8}{3}}$ in (1.1)
we compute the integrals in (4.10) to get

$\begin{split}I_{+-}[M^{4}]&=\frac{8\pi^{2}}{7}\,,\qquad
I_{+-}[M^{6}]=-\frac{448\pi^{2}}{33}\,,\qquad
I_{+-}[M^{7}]=-\frac{529472\pi^{2}}{3003}\,,\\\
I_{+-}[M^{8}]&=\frac{49716397568\pi^{2}}{342567225}\,,\qquad
I_{+-}[M^{R|R^{4}}]=-\frac{1861955980828672\left(\frac{2}{3}\right)^{2/3}}{2837835\pi^{8/3}}\,,\end{split}$
(4.13)

where for the polynomial Mellin amplitudes we used the fact that they are all
proportional to $(2-t)(s+t-2)$ as well as the Barnes Lemma

$\begin{split}\int_{-i\infty}^{i\infty}\frac{ds}{2\pi
i}\Gamma(a+s)\Gamma(b+s)\Gamma(c-s)\Gamma(d-s)=\frac{\Gamma(a+c)\Gamma(b+d)\Gamma(b+c)\Gamma(a+d)}{\Gamma(a+b+c+d)}\,,\end{split}$
(4.14)

while for $M^{R|R^{4}}(s,t)$ we instead numerically computed the integral of
the closed form expression in the Mathematica file to high precision. Plugging
(4.13) and (4.12) into (4.10), we find that $M^{R|R^{4}}$ with the values of
$B_{i}^{R|R^{4}}$ fixed in (4.8) and (4.7) precisely satisfies the constraint,
which is a nontrivial check that the analytically continued Lorentzian
inversion formula gives the correct CFT data and thus fixes the contact term
ambiguities at 1-loop.

Finally, we can use both localization constraints and the explicit integrals
in (4.13) to fix two of the four coefficients $B_{i}^{D^{8}R^{4}}$ in the tree
level $D^{8}R^{4}$ term in (1.1) to get

$\begin{split}B^{D^{8}R^{4}}_{4}=-\frac{3200B^{D^{8}R^{4}}_{7}}{429}-\frac{238578176B^{D^{8}R^{4}}_{8}}{1957527}\,,\quad
B^{D^{8}R^{4}}_{6}=-\frac{177B^{D^{8}R^{4}}_{7}}{13}+\frac{10356296B^{D^{8}R^{4}}_{8}}{494325}\,,\end{split}$
(4.15)

which will be useful in future attempts to fix this amplitude by independently
computing its CFT data.

##  5 Numerical bootstrap

In the previous sections we studied $\langle 2222\rangle$ for the $k=2$ ABJ(M)
theory in the large $c_{T}\sim N^{\frac{3}{2}}$ limit to several orders. In
this section, we will study this correlator non-perturbatively using the
numerical conformal bootstrap, and compute bounds on CFT data as a function of
$c_{T}$ as was done in previous work [52, 63, 58, 64], but now to much higher
numerical accuracy using subsequent technical improvements to the bootstrap
software [80]. Previously, the tree level supergravity correction [55, 62] was
found to saturate the lower bounds [63, 58] for all CFT. These lower bounds
were conjectured to correspond to $k=2$ ABJ(M) theory in [58], because they
were found to be approximately saturated by the values of the short $(B,2)$
and $(B,+)$ OPE coefficients as computed all orders in $1/N$ in [58]. We now
find that the $R|R$ correction continues to saturate these lower bounds for
those semishort $(A,2)_{\ell}$ and $(A,+)_{\ell}$ OPE coefficients where the
asymptotic large $c_{T}$ expansion is well converged.

###  5.1 Setup

We start by briefly reviewing how the numerical bootstrap can be applied to
the stress-tensor multiplet four-point function in $\mathcal{N}=8$ theories,
for further details see [52]. Invariance of the four-point function (2.2) as
expanded in superblocks (2.8) under $1\leftrightarrow 3$ crossing (3.2)
implies crossing equations of the form

$\begin{split}\sum_{{\cal
M}\,\in\,\\{\text{Id},\,\text{Stress},\,(B,+),\,(B,2),\,(A,+)_{\ell},\,(A,2)_{\ell},\,(A,0)_{\Delta,\ell}\\}}\lambda_{\cal
M}^{2}\,\vec{V}_{{\cal M}}=0\,,\end{split}$ (5.1)

where ${\cal M}$ ranges over all the superconformal multiplets listed in Table
1, $\vec{V}_{{\cal M}}$ are functions of superconformal blocks, and
$\lambda_{\cal M}^{2}$ are squares of OPE coefficients that must be positive
by unitarity. As in [52], we normalize the OPE coefficient of the identity
multiplet to $\lambda_{\text{Id}}=1$, and parameterize our theories by the
value of $\lambda_{\text{Stress}}$, which is related to $c_{T}$ through (2.11)
for $p=2$.

To find upper/lower bounds on a given OPE coefficient of a protected multiplet
${\cal M}^{\prime}$ that appears in the ${\cal O}_{\text{Stress}}\times{\cal
O}_{\text{Stress}}$ OPE, we consider linear functionals $\alpha$ satisfying

$\begin{split}&\alpha(\vec{V}_{\cal M^{\prime}})=s\,,\;\quad\qquad\text{$s=1$
for upper bounds, $s=-1$ for lower bounds}\,,\\\ &\alpha(\vec{V}_{\cal M})\geq
0\,,\;\;\quad\qquad\text{for all short and semi-short ${\cal
M}\notin\\{\text{Id},\text{Stress},\cal M^{\prime}\\}$}\,,\\\
&\alpha(\vec{V}_{(A,0)_{\Delta,\ell}})\geq 0\,,\qquad\text{for all $\ell$ with
$\Delta\geq\ell+1$}\,.\\\ \end{split}$ (5.2)

If such a functional $\alpha$ exists, then this $\alpha$ applied to (5.1)
along with the positivity of all $\lambda_{\cal M}^{2}$ except, possibly, for
that of $\lambda_{{\cal M}^{\prime}}^{2}$ implies that

$\begin{split}&\text{if $s=1$, then}\qquad\lambda_{{\cal
M}^{\prime}}^{2}\leq-\alpha(\vec{V}_{\text{Id}})-\frac{256}{c_{T}}\alpha(\vec{V}_{\text{Stress}})\,,\\\
&\text{if $s=-1$, then}\qquad\lambda_{{\cal
M}^{\prime}}^{2}\geq\alpha(\vec{V}_{\text{Id}})+\frac{256}{c_{T}}\alpha(\vec{V}_{\text{Stress}})\,.\\\
\end{split}$ (5.3)

Note that we can get both upper/lower bounds because the protected multiplets
are isolated from the continuum of operators, unlike the long multiplets
$(A,0)_{\Delta,\ell}$ for which we could only compute upper bounds on their
OPE coefficients. To obtain the most stringent upper/lower bound on
$\lambda_{{\cal M}^{\prime}}^{2}$, one should then minimize/maximize the RHS
of (5.3) under the constraints (5.2). In the above algorithms, we fixed the
SCFT by inputting the value of $c_{T}$, which was computed to all orders in
$1/N$ for ABJ(M) in [58]. We can further fix the theory by also putting in the
values of all the short OPE coefficients $\lambda^{2}_{(B,2)}$ and
$\lambda^{2}_{(B,+)}$, which were also computed to all orders in $1/N$. We
should then remove these operators from the second line of (5.2) and put them
on the RHS of (5.3) with their explicit OPE coefficients, just like the stress
tensor multiplet.

The numerical implementation of the minimization/maximization problem
described depends on five parameters: the number of derivatives parameter
$\Lambda$ used to construct $\alpha$, the range of spins of multiplets up to
$\ell_{\text{max}}$ that we consider, the order $r_{\text{max}}$ to which we
expand blocks, the parameter $\kappa$ that parametrizes how many poles we keep
when approximating blocks, and the precision of the solver SDPB [81]. We used
$\Lambda=83$,161616This is equivalent to $n_{\text{max}}=42$, which must be
some kind of record for 3d bootstrap! $\ell_{\text{max}}=90$,
$r_{\text{max}}=140$, $\kappa=70$, and $1116$ binary digits of precision. The
most important parameter is $\Lambda$, which should be compared to the
previously highest value $\Lambda=43$ used in [64].

###  5.2 Bootstrap bound saturation

| $\frac{\lambda_{\text{Stress}}^{2}}{16}=\frac{16}{c_{T}}$ | $\lambda_{(B,+)}^{2}$ | $\lambda_{(B,2)}^{2}$
---|---|---|---
large $N$ | $0.20952$ | $7.36854$ | $7.43343$
exact | $0.20944$ | $7.37115$ | $7.45176$
Table 3: Comparison of the large $N$ formulae to the exact values from [64]
for the OPE coefficients of short operators that appear in $\langle
2222\rangle$ for $U(3)_{2}\times U(3)_{-2}$ ABJM.

The large $c_{T}$ expansion of CFT data is asymptotic, which means that after
a few orders the expansion will actually get worse, unless we look at very
large values of $c_{T}$. The larger the value of $c_{T}$, the more precise our
numerics must be to make a meaningful comparison, so we should focus on CFT
data for which the asymptotic expansion is still pretty good for the lowest
few orders. In general, we observe that the convergence of the asymptotic
expansion is better for more protected operators. For instance, the short
operators $(B,2)$ and $(B,+)$ are the most protected, and their expansion at
large $c_{T}$ as shown to all orders in [58] takes the form

$\begin{split}\lambda^{2}_{(B,2)}=10.6667-17.6369\frac{16}{c_{T}}+7.26668\Big{[}\frac{16}{c_{T}}\Big{]}^{\frac{5}{3}}-0.384051\Big{[}\frac{16}{c_{T}}\Big{]}^{2}-3.25726\Big{[}\frac{16}{c_{T}}\Big{]}^{\frac{7}{3}}+1.88158\Big{[}\frac{16}{c_{T}}\Big{]}^{\frac{8}{3}}+\dots\,,\end{split}$
(5.4)

and similarly for $\lambda^{2}_{(B,+)}$. We expand in $16/c_{T}$ because the
free $\mathcal{N}=8$ theory has $c_{T}=16$, which makes this a natural
quantity. Note that coefficient of each subsequent order is in general getting
smaller, which implies that this asymptotic expansion is expected to be pretty
good even to many orders. This expectation is supported by the fact that the
all orders in $1/N$ expansion matches the finite $N$ values to the sub-percent
level for the $U(3)_{2}\times U(3)_{-2}$ ABJM theory, as computed in
[64]171717Actually, in [64] the values were computed directly for the
interacting sector of $U(4)_{1}\times U(4)_{-1}$ ABJM theory, but this theory
is dual to $U(3)_{2}\times U(3)_{-2}$ theory since in the UV they are both
described by $SU(4)\cong SO(6)$ SYM [82, 83]. and reviewed in Table 3. Another
piece of evidence is that the all orders expression is close to saturating the
lower bound from the numerical bootstrap, as first observed in [58] with
$\Lambda=43$ accuracy, and now further confirmed with $\Lambda=83$ accuracy in
Figure 1. Note that there is still a discrepancy between the all orders
expression in solid red and the lower bound in solid gray, even though the
numerics seem well converged as can be seen from comparing to the old
$\Lambda=43$ value in dashed gray. This suggests that either the lower bound
is not actually saturated by the $k=2$ ABJ(M) theory, even though it is very
close to the curve for a large range of $c_{T}$ as observed in [58], or that
the non-perturbative in $c_{T}$ corrections to the all orders expressions for
$\lambda^{2}_{(B,2)}$ might account for this small discrepancy. Since we
cannot rule out either possibility at this stage, for subsequent plots we will
show bounds where we inputed the values of $\lambda^{2}_{(B,2)}$ and
$\lambda^{2}_{(B,+)}$ as well as bounds with no assumptions, and as expected
these bounds differ by a small amount.

Figure 1: Upper and lower bounds on the $\lambda_{(B,2)}^{2}$ OPE coefficient
with $\Lambda=83$ (solid gray) and $\Lambda=43$ (dashed gray) in terms of the
stress-tensor coefficient $c_{T}$ in the large $c_{T}$ regime. The red dotted
line denotes the large $c_{T}$ expansion to order tree level supergravity
$O(c_{T}^{-1})$, which is independent of $k$, while the red dot-dashed line
also includes the tree level $R^{4}$ correction at order $O(c_{T}^{-5/3})$ and
the red dashed line furthermore includes the 1-loop $R|R$ correction at order
$O(c_{T}^{-2})$, both of which depend on the value $k=2$. The solid red line
includes the all orders in $1/c_{T}$ expression, which only misses non-
perturbative in $c_{T}$ corrections. The blue and brown vertical lines denote
the values of $c_{T}$ for various known $k=2$ ABJM and ABJ theories,
respectively, which are summarized in Table 4. Since there is still a small
discrepancy in this very zoomed in plot (relative to [58], which looked at
$0\leq 16/c_{T}\leq 1$) between the all orders result and the numerical bounds
even at very high $\Lambda$, we have imposed the value of
$\lambda^{2}_{(B,2)}$ in subsequent plots, which should further constrain the
numerical bounds to match $k=2$ ABJM.

The next most protected operators are $(A,+)_{\ell}$ for even $\ell$ and
$(A,2)_{\ell}$ for odd $\ell$. In the previous sections we computed the large
$c_{T}$ expansion of their OPE coefficients to many orders, which we summarize
showing explicit numerical values for each term:

$\begin{split}\lambda^{2}_{(A,+)_{0}}&=7.11111+3.02803\frac{16}{c_{T}}+1.11453\Big{[}\frac{16}{c_{T}}\Big{]}^{2}-3.04011\Big{[}\frac{16}{c_{T}}\Big{]}^{\frac{7}{3}}\\\
&+\Big{[}\frac{9199616}{6435}B_{7}^{D^{8}R^{4}}-\frac{120501022588928}{5138508375}B_{8}^{D^{8}R^{4}}\Big{]}{c_{T}^{-\frac{23}{9}}}-20.4134\Big{[}\frac{16}{c_{T}}\Big{]}^{\frac{8}{3}}+\dots\,,\\\
\lambda^{2}_{(A,+)_{2}}&=13.3747+3.19665\frac{16}{c_{T}}+0.301165\Big{[}\frac{16}{c_{T}}\Big{]}^{2}+\frac{67108864}{557375}B_{8}^{D^{8}R^{4}}{c_{T}^{-\frac{23}{9}}}-268.868\Big{[}\frac{16}{c_{T}}\Big{]}^{\frac{8}{3}}+\dots\,,\\\
\lambda^{2}_{(A,+)_{4}}&=19.6506+3.25967\frac{16}{c_{T}}+0.189594\Big{[}\frac{16}{c_{T}}\Big{]}^{2}+16.9909\Big{[}\frac{16}{c_{T}}\Big{]}^{\frac{8}{3}}+\dots\,,\\\
\end{split}$ (5.5)

and

$\begin{split}\lambda^{2}_{(A,2)_{1}}&=9.75238-6.17281\frac{16}{c_{T}}+8.74961\Big{[}\frac{16}{c_{T}}\Big{]}^{2}-75.0472\Big{[}\frac{16}{c_{T}}\Big{]}^{\frac{7}{3}}\\\
&-\Big{[}\frac{262144}{3003}B_{7}^{D^{8}R^{4}}+\frac{2766298677248}{2397970575}B_{8}^{D^{8}R^{4}}\Big{]}{c_{T}^{-\frac{23}{9}}}-1797.23\Big{[}\frac{16}{c_{T}}\Big{]}^{\frac{8}{3}}+\dots\,,\\\
\lambda^{2}_{(A,2)_{3}}&=16.2118-6.43488\frac{16}{c_{T}}+2.11217\Big{[}\frac{16}{c_{T}}\Big{]}^{2}+\frac{268435456}{121275}B_{8}^{D^{8}R^{4}}{c_{T}^{-\frac{23}{9}}}-6491.08\Big{[}\frac{16}{c_{T}}\Big{]}^{\frac{8}{3}}+\dots\,,\\\
\lambda^{2}_{(A,2)_{5}}&=22.573-6.54041\frac{16}{c_{T}}+1.28477\Big{[}\frac{16}{c_{T}}\Big{]}^{2}+261.161\Big{[}\frac{16}{c_{T}}\Big{]}^{\frac{8}{3}}+\dots\,,\\\
\end{split}$ (5.6)

where we included the $D^{8}R^{4}$ at order $c_{T}^{-\frac{23}{9}}$ term that
we only know up to two as yet unfixed coefficients. As the spin increases, we
observe that the $R|R$ term at order $c_{T}^{-2}$ becomes increasingly smaller
compared to the tree level supergravity term, which means that we can trust it
more for a larger range of $c_{T}$. On the other hand, the $R|R^{4}$ term at
order $c_{T}^{-\frac{8}{3}}$ is much bigger than the previous terms, which
means that we can only trust it at very large $c_{T}$. There is also a
$D^{6}R^{4}$ term at order $c_{T}^{-\frac{7}{3}}$ that only affects the lowest
spin for each multiplet, and also is roughly the same size as the supergravity
term.

Lastly, the least protected multiplet is the long multiplet
$(A,0)_{\Delta,\ell}$. In the previous sections we computed the large $c_{T}$
expansion to the scaling dimension of the lowest twist operator for the lowest
few spins, which we summarize showing explicit numerical values for each term:

$\begin{split}\Delta_{2,0}&=2-7.09248\frac{16}{c_{T}}-38.1501\Big{[}\frac{16}{c_{T}}\Big{]}^{\frac{5}{3}}+97.378\Big{[}\frac{16}{c_{T}}\Big{]}^{2}+21.3758\Big{[}\frac{16}{c_{T}}\Big{]}^{\frac{7}{3}}\\\
&+\Big{[}-\frac{222720}{143}B_{7}^{D^{8}R^{4}}+\frac{18902167552}{1087515}B_{8}^{D^{8}R^{4}}\Big{]}{c_{T}^{-\frac{23}{9}}}+3993.9\Big{[}\frac{16}{c_{T}}\Big{]}^{\frac{8}{3}}+\dots\,,\\\
\Delta_{4,2}&=4-3.12069\frac{16}{c_{T}}-65.3944\Big{[}\frac{16}{c_{T}}\Big{]}^{2}\\\
&+\Big{[}\frac{32256}{13}B_{7}^{D^{8}R^{4}}+\frac{1322266624}{164775}B_{8}^{D^{8}R^{4}}\Big{]}{c_{T}^{-\frac{23}{9}}}+112035.\Big{[}\frac{16}{c_{T}}\Big{]}^{\frac{8}{3}}+\dots\,,\\\
\Delta_{6,4}&=6-2.06985\frac{16}{c_{T}}-0.645987\Big{[}\frac{16}{c_{T}}\Big{]}^{2}-32768B_{8}^{D^{8}R^{4}}{c_{T}^{-\frac{23}{9}}}+120791.\Big{[}\frac{16}{c_{T}}\Big{]}^{\frac{8}{3}}+\dots\,.\\\
\end{split}$ (5.7)

The asymptotic expansion for this quantity seems very poor, as the coefficient
of each subsequent term is growing rapidly.

${\cal N}=8$ SCFT | $c_{T}$ | $\frac{\lambda_{\text{Stress}}^{2}}{16}=\frac{16}{c_{T}}$
---|---|---
$\;\;U(4)_{2}\times U(4)_{-2}$ ABJM | $126.492$ | $0.138133$
$\;\;U(5)_{2}\times U(5)_{-2}$ ABJM | $172.058$ | $0.0998481$
$\;\;U(6)_{2}\times U(6)_{-2}$ ABJM | $221.97$ | $0.0765165$
$\;\;U(7)_{2}\times U(7)_{-2}$ ABJM | $275.879$ | $0.0610587$
$\;\;U(4)_{2}\times U(5)_{-2}$ ABJ | $115.831$ | $0.12649$
$\;\;U(5)_{2}\times U(6)_{-2}$ ABJ | $160.243$ | $0.0929919$
$\;\;U(6)_{2}\times U(7)_{-2}$ ABJ | $209.105$ | $0.0720818$
$\;\;U(7)_{2}\times U(8)_{-2}$ ABJ | $262.043$ | $0.0579965$
Table 4: Several values of $c_{T}$ and $\lambda_{\text{Stress}}^{2}/16$ for
$k=2$ ABJM or ABJ theories, as computed from the all orders in $1/N$ formulae
in [58].

These observations motivate us to focus on comparing to numerical bootstrap
for $\lambda^{2}_{(A,2)_{\ell}}$ for $\ell>1$ and $\lambda^{2}_{(A,+)_{\ell}}$
for $\ell>0$ up to $O(c_{T}^{-2})$, where we can be reasonably confident that
the asymptotic expansion is well converged for a moderately large range of
$c_{T}$. In Figures 2 and 3 we compare the large $c_{T}$ expansion of this CFT
data to non-perturbative lower bounds from the numerical bootstrap in the
large $c_{T}$ regime, which includes many physical examples of $k=2$ ABJ(M)
theories as summarized in Table 4. As discussed above, we show bounds where we
inputed the values of $\lambda^{2}_{(B,2)}$ and $\lambda^{2}_{(B,+)}$ using
their all orders in $1/c_{T}$ expressions, as well as bounds with no
assumptions. The discrepancy between each type of bound is small, and we
observe that both are well approximated by the $O(c_{T}^{-2})$ analytic
expressions for the entire range of $c_{T}$ that we looked at, and that the
discrepancy between the two kinds of bounds is smaller than the improvement of
the $c_{T}^{-2}$ term relative to the $O(c_{T}^{-1})$ approximation. For the
more protected $\frac{1}{4}$-BPS multiplet $(A,+)_{\ell}$, the correction to
tree level supergravity is quite small, so it is somewhat harder to see the
1-loop correction, but for the less protected $\frac{1}{8}$-BPS multiplet
$(A,2)_{\ell}$ we can very clearly see the improvement from the 1-loop
correction to tree level supergravity. We also show upper bounds, which after
imposing $\lambda^{2}_{(B,2)}$ and $\lambda^{2}_{(B,+)}$ become very close the
lower bounds, which suggests that the theory is almost completely fixed.

Figure 2: Upper and lower bounds on the $\lambda_{(A,+)_{\ell}}^{2}$ OPE
coefficient for $\ell=2$ (top) and $\ell=4$ (bottom) in terms of the stress-
tensor coefficient $c_{T}$ in the large $c_{T}$ regime. The black solid lines
are with the all orders in $1/c_{T}$ values of $\lambda_{(B,+)}^{2}$ and
$\lambda_{(B,2)}^{2}$ for $k=2$ ABJ(M) inputed into the bootstrap, while the
gray solid lines are without any assumptions, and note that the allowed region
for the black bounds is much smaller than the gray bounds. The red dotted line
denotes the large $c_{T}$ expansion to order tree level supergravity
$O(c_{T}^{-1})$, which is independent of $k$. The red dashed line also
includes the 1-loop $R|R$ correction at order $O(c_{T}^{-2})$, which depends
on the value $k=2$, and improves the saturation of the lower bound relative to
$O(c_{T}^{-1})$. The blue and brown vertical lines denote the values of
$c_{T}$ for various known $k=2$ ABJM and ABJ theories, respectively, which are
summarized in Table 4. These plots were made with $\Lambda=83$.

Figure 3: Upper and lower bounds on the $\lambda_{(A,2)_{\ell}}^{2}$ OPE
coefficient for $\ell=3$ (top) and $\ell=5$ (bottom) in terms of the stress-
tensor coefficient $c_{T}$ in the large $c_{T}$ regime. The black solid lines
are with the all orders in $1/c_{T}$ values of $\lambda_{(B,+)}^{2}$ and
$\lambda_{(B,2)}^{2}$ for $k=2$ ABJ(M) inputed into the bootstrap, while the
gray solid lines are without any assumptions, and note that the allowed region
for the black bounds is much smaller than the gray bounds. The red dotted line
denotes the large $c_{T}$ expansion to order tree level supergravity
$O(c_{T}^{-1})$, which is independent of $k$. The red dashed line also
includes the 1-loop $R|R$ correction at order $O(c_{T}^{-2})$, which depends
on the value $k=2$, and improves the saturation of the lower bound relative to
$O(c_{T}^{-1})$. The blue and brown vertical lines denote the values of
$c_{T}$ for various known $k=2$ ABJM and ABJ theories, respectively, which are
summarized in Table 4. These plots were made with $\Lambda=83$.

##  6 Conclusion

There are three main results of this work. Firstly, we computed the 1-loop
terms $R|R$, $R|R^{4}$, and $R^{4}|R^{4}$ for $k=2$ ABJ(M) theory up to
contact term ambiguities, and checked that they match the relevant terms in
the 11d M-theory S-matrix in the flat space limit. Secondly, we fixed the
contact terms for $R|R$ and $R|R^{4}$ by combining two constraints from
supersymmetric localization with a conjectured analytic continuation of the
Lorentzian inversion formula, where localization confirms the inversion
results for $R|R$ and provides a nontrivial check for $R|R^{4}$. Finally, we
found that the $R|R$, i.e. $c_{T}^{-2}$, correction to semishort CFT data
saturates the numerical bootstrap bounds for $k=2$ ABJ(M) in the large $c_{T}$
regime.

One could try to perform the same analytic continuation of the Lorentzian
inversion formula for the other maximally supersymmetric holographic CFTs that
have been studied at 1-loop: 4d $SU(N)$ $\mathcal{N}=4$ SYM dual to Type IIB
string theory on $AdS_{5}\times S^{5}$, and 6d $(2,0)$ theory dual to M-theory
on $AdS_{7}\times S^{4}$ for the $A_{N-1}$ theories and $AdS_{7}\times
S^{4}/\mathbb{Z}_{2}$ for the $D_{N}$ theories. In the 4d case, general spin
formulae were found for the $R|R$ correction in [38] and the $R|R^{4}$
correction in [34]. The $R|R$ formula can be analytically continued to spin
zero, which is the spin that is affected by the 4d analogue of the
$B_{4}^{R|R}M^{4}$ contact term in (1.1), but it was shown using
supersymmetric localization in [27] that $B_{4}$ is nonzero, unlike what we
found here in 3d. The $R|R^{4}$ formula has explicit poles for spins $0,2,4$,
which are the spins affected by the 4d analogue of the
$O(c_{T}^{-\frac{8}{3}})$ contact terms in (1.1), so we cannot analytically
continue as we did in here in 3d. In 6d, one can check the Lorentzian
inversion formula results in [34] can be analytically continued to all CFT
data for both $R|R$ and $R|R^{4}$, which is even better than what we observed
in 3d where $R|R^{4}$ could not be continued to CFT data affected by the
$B_{4}^{R|R}M^{4}$ contact term. Unlike 3d, however, in 6d we have no
localization results to check if this conjectured analytic continuation is
correct.

The similarity of ABJ(M) and 6d $(2,0)$ in contrast to $\mathcal{N}=4$ SYM
suggests that the analytically continued Lorentzian inversion formula can be
applied to holographic CFTs dual to 11d M-theory, but not those dual to 10d
string theory. One could try to justify this by observing that contact terms
must always correspond to even powers of Planck length, so they can affect
1-loop terms for 10d duals that are also even in Planck length, but they
cannot effect 1-loop terms for 11d duals that are odd in Plank length. On the
other hand, recall that we did not compute $M^{R|R}$ and $M^{R|R^{4}}$ using
explicit Witten diagrams, so the difference between a 1-loop diagram and a
contact diagram at the same order in $c_{T}$ is not entirely clear from our
approach. One could tentatively define the 1-loop diagram $M^{R|R}$ as
whatever gives a formula for CFT data that is analytic in spin for all spins,
as we did in practice, and then define $B_{4}^{R|R}M^{4}$ as a putative
contact term, which we found was zero. Unfortunately, this description would
not make sense for $M^{R|R^{4}}$, where there is no general spin formula that
converges for the spins that contribute to $B_{4}^{R|R^{4}}M^{4}$, so it is
hard to distinguish between 1-loop diagrams and contact diagrams at the same
order $c_{T}^{-\frac{8}{3}}$.

It would be nice to check if the analytically continued inversion formula at
order $c_{T}^{-\frac{10}{3}}$ can be used to fix all the contact term
ambiguities, perhaps when combined with supersymmetric localization, and if
the conjectured results could be checked with localization. In this work we
already computed the $R^{4}|R^{4}$ term that contributes at order
$c_{T}^{-\frac{10}{3}}$, but we did not yet compute the $R|D^{6}R^{4}$ term
that also contributes at this order. To compute this term we would need to
know $\langle 22pp\rangle$ for all even $p$ at tree level $D^{6}R^{4}$, i.e.
order $c_{T}^{-\frac{7}{3}}$. Currently we only know this for $p=2$, where we
could fix the three coefficients in (1.1) using the two localization
constraints from [10, 58] and [21, 23] as well as comparison to the known term
(2.25) in the M-theory S-matrix in the flat space limit. For $p>2$, we still
have the flat space limit constraint, and it is possible that we could
generalize to $p>2$ the localization constraint in [10, 58], which is just the
value of short operator OPE coefficients that could in principle be computed
from a $k=2$ generalization of the 1d theory [60], which is currently only
known for $k=1$ ABJM. On the other hand, the second localization constraint
[21, 23] is specific to $p=2$, and also the $p>2$ correlator has more
coefficients that need fixing than $p=2$ because crossing constraints are
weaker for mixed correlators. Perhaps the $p>2$ term could be computed from
the known $p=2$ case, as both in principle arise from dimensional reduction of
the same $D^{6}R^{4}$ term in the flat space 11d action effective action for
M-theory.

It would be also be very interesting if more terms in the analytic large
$c_{T}$ expansion of CFT data could be found to saturate the numerical
bootstrap bounds, beyond the $c_{T}^{-2}$ terms for semishort OPE coefficients
that we considered in this work. In particular, if we could successfully match
the $c_{T}^{-\frac{8}{3}}$ terms computed in this work, then this would allow
us to read off the $D^{8}R^{4}$ term at the lower order
$c_{T}^{-\frac{23}{9}}$, which could then be used to fix the corresponding
unknown term in the 11d M-theory S-matrix as outlined in [10]. The
coefficients in the asymptotic large $c_{T}$ expansion start to grow
drastically starting at $c_{T}^{-\frac{8}{3}}$ even for the best behaved case
of semishort OPE coefficients, so to accurately compare them to the numerical
bootstrap we must look at very large $c_{T}$, which requires a very accurate
numerical bootstrap. In this work we pushed the current numerical bootstrap to
very high accuracy, as parameterized by $\Lambda=83$, which is twice the value
used in the previous study [64]. We found that the current numerical bootstrap
has already started to converge, so pushing to higher $\Lambda$ will likely
not improve matters. On the other hand, we observed in this work that
inputting the short OPE coefficients noticeably improved the bounds, such that
the lower bounds actually become very close to the upper bounds for the regime
of $c_{T}$ that we studied, so it is likely that imposing other exact
quantities such as the integrated constraint in [21, 23] will further improve
the accuracy of the bounds, and maybe even fix the theory by having upper and
lower bounds approximately coincide. It is also likely that a third
localization constraint can be computed by considering derivatives of the
squashed sphere free energy, which was computed to all orders in $1/c_{T}$ in
[84] using the localization results in [85, 86]. This additional constraint
could both improve the numerical bootstrap, as well as allow us to
analytically fix another of the coefficients in (1.1) for $D^{8}R^{4}$, so
there would only be a single remaining unfixed coefficient to be fixed from
numerics.

In this paper we focused on $k=2$ ABJ(M) theory, because it is technically
more difficult to compute the $\langle 22pp\rangle$ data for odd $p$ that is
needed to compute $k=1$ ABJM at 1-loop. In particular, while the $\langle
22pp\rangle$ tree level supergravity for even $p$ could be written in terms of
a finite number of $\bar{D}_{r_{1},r_{2},r_{3},r_{4}}(U,V)$, for odd $p$ we
require an infinite number. If we could compute $k=1$ ABJM at 1-loop, then we
could perform another nontrivial check of our conjectured analytic
continuation of the Lorentzian inversion formula, and also try to compare to
numerical bootstrap bounds. The numerical bootstrap for $k=1$ could be more
accurate than the $k=2$ case in this work, because for $k=1$ we could use the
mixed correlator setup in [64] that only applies to the $k=1$ theory, and
includes inputs from more short OPE coefficients that can be computed to all
orders in $1/c_{T}$ using the localization results of [76]. We look forward to
reporting on the $k=1$ case in the future.

Finally, it would be interesting to generalize our 1-loop derivation to
$\mathcal{N}=6$ ABJ(M) with $k>2$, which at small $k$ is dual to M-theory on
$AdS_{4}\times S^{7}/\mathbb{Z}_{k}$ and at large $k$ is dual to Type IIA
string theory on $AdS_{4}\times\mathbb{CP}^{3}$. The single trace operators in
this case can be either $\frac{1}{2}$ or a $\frac{1}{3}$ BPS [87], so the
unmixing problem would be more complicated. In particular, one would need to
derive the superblock expansion of correlators of the various single operator
operators to extract the GFFT and tree level data needed to compute 1-loop. So
far, the superblock expansion was only computed for the stress tensor
multiplet correlator [67], and it seems difficult to generalize that
derivation to arbitrary correlators of other $\frac{1}{3}$-BPS operators.
Instead, it might be easier to compute the relevant CFT data by directly
imposing the superconformal Ward identity on 3-point functions, rather than
expanding 4-point correlators in superblocks. If one could compute the 1-loop
term for general $k$, then in the flat space limit one could interpolate
between an 11d box diagram at small $k$ to a 10d box diagram at large $k$,
just as the tree level $R^{4}$ term computed in [23] was found to interpolate
between M-theory and string theory.

## Acknowledgments

We thank Ofer Aharony, Silvu Pufu, Walter Landry, and Xinan Zhou for useful
conversations, Silviu Pufu and Xinan Zhou for collaboration at an early stage
of the collaboration, and Ofer Aharony for reading through the manuscript. The
work of LFA is supported by the European Research Council (ERC) under the
European Union’s Horizon 2020 research and innovation programme (grant
agreement No 787185). SMC is supported by the Zuckerman STEM Leadership
Fellowship. HR acknowledges the support from the PBC postdoctoral fellowship
program as well as the Israel Science Foundation center for excellence grant
(grant number 1989/14) and by the Minerva foundation with funding from the
Federal German Ministry for Education and Research. The authors would like to
acknowledge the use of the University of Oxford Advanced Research Computing
(ARC) facility in carrying out this
work.(http://dx.doi.org/10.5281/zenodo.22558)

##  Appendix A Conformal blocks and $\bar{D}_{r_{1},r_{2},r_{3},r_{4}}(U,V)$

The $SO(3,2)$ conformal blocks $G_{\Delta,\ell}(U,V)$ which appear in the
decomposition of four-point function of identical scalar operators are
eigenfunctions of the quadratic Casimir of $SO(3,2)$. These eigenfunctions are
not known in closed form but there exists various expansions. In what follows
we present two such expansions which we used heavily for computations.

The first is the lightcone expansion which is a power series expansion around
$U=0$

$G_{\Delta,\ell}(U,V)=\sum_{n=0}^{\infty}U^{(\Delta-\ell)/2+n}g^{[n]}_{\Delta,\ell}(1-V)$
(A.1)

where $g^{[n]}_{\Delta,\ell}(1-V)$ are the lightcone blocks that depend on the
dimension $\Delta$ and spin $\ell$ of the exchanged operator. In our
normalization, these lightcone blocks take the following form [88]

$\displaystyle
g^{[n]}_{\Delta,\ell}(1-V)=\frac{1}{4^{\Delta}}\frac{\left(\frac{1}{2}\right)_{\ell}}{\ell!}\sum_{m=0,2,4,...2n}A_{n,m}f_{n,m}(1-V)$
(A.2)

where the basis functions $f_{n,m}$ are ${}_{2}F_{1}$ hypergeometric functions
given by

$f_{n,m}(1-V)=(1-V)^{\ell-m}{}_{2}F_{1}\left[\begin{array}[]{c}(\Delta+\ell)/2+n-m,(\Delta+\ell)/2+n-m\\\
2((\Delta+\ell)/2+n-m)\end{array};1-V\right]$ (A.3)

and the expansion coefficients $A_{n,m}$ are

$\displaystyle A_{n,m}=$
$\displaystyle\sum_{m_{1},m_{2}=0}^{n_{1}}(-1)^{m+m_{1}+1}4^{m_{1}+m_{2}}\frac{(-\ell)_{m}\left(-n_{1}\right)_{m_{1}+m_{2}}\left(n-n_{1}+1/2\right)_{m_{1}}}{m!m_{1}!m_{2}!\left(n-m+m_{1}\right)!}$
(A.4) $\displaystyle\times\frac{(\Delta-1)_{2n-m}(3/2-\Delta)_{m-n-
m_{1}-m_{2}}(2-\Delta+\ell)_{2\left(n_{1}-m_{2}\right)-m}}{((\Delta-\ell)/2+2\ell-m-1)_{2n-m}((\Delta+\ell)/2+2)_{2\left(n+m_{1}-n_{1}\right)-m}}$
$\displaystyle\times\frac{\left(-1/2-\ell-n+m-m_{1}+m_{2}\right)\left(n_{2}-\ell\right)_{m_{2}}}{(1/2-\ell)_{-n+m+m_{2}}\left(1/2+\ell-
m_{2}\right)_{n-m+m_{1}+m_{2}+1}}$
$\displaystyle\times\left[\left(\frac{\Delta+\ell}{2}\right)_{n-m+m_{1}}\left(\frac{\Delta-\ell-1}{2}\right)_{m_{2}}\right]^{4}$

in which $n_{1},n_{2}$ are defined as

$n_{1}=\lfloor m/2\rfloor,\quad n_{2}=\lfloor(m+1)/2\rfloor~{}.$ (A.5)

Setting $n=0$ in the above formulas gives us the leading lightcone block in
(3.16). This form of the conformal blocks was particularly useful for the
extraction of the tree-level CFT data and in computing the $U^{n}\log^{2}U$
slices in section 3.1.

The second convenient representation of the conformal blocks is in terms of
expansion in Jack polynomials. In [70], it was shown that in three dimensions
the conformal blocks can be written as (in a normalization identical to those
of the lightcone blocks)

$\displaystyle
G_{\Delta,\ell}(z,\bar{z})=\frac{1}{4^{\Delta}}\frac{\left(\frac{1}{2}\right)_{\ell}}{\ell!}\sum_{m,n}r_{m,n}(\Delta,\ell)J_{m,n}^{(\Delta,\ell)}(z,\bar{z})$
(A.6)

where the Jack polynomials $J_{m,n}$ in three dimensions are given in terms of
Legendre polynomials as follows

$J_{m,n}^{(\Delta,\ell)}(z,\bar{z})=(z\bar{z})^{\frac{1}{2}(m+n+\Delta)}P_{m-n+\ell}\left(\frac{z+\bar{z}}{\sqrt{4z\bar{z}}}\right)~{}.$
(A.7)

The expansion coefficients $r_{m,n}(\Delta,\ell)$ are given by

$\displaystyle r_{m,n}(\Delta,\ell)=-\pi
4^{\Delta-2}\Gamma\left(\Delta-\frac{1}{2}\right)\frac{\Gamma(\ell+1)}{\Gamma(\Delta-1)}(2m-2n+2\ell+1)\csc(\pi(m+\ell))$
$\displaystyle\times\frac{\Gamma\left(\frac{\Delta-\ell}{2}\right)\Gamma\left(\frac{1}{2}(\ell+\Delta+1)\right)\Gamma\left(\frac{1}{2}(2m+\ell+\Delta)\right)^{2}\Gamma\left(\frac{1}{2}(2n-\ell+\Delta-1)\right)^{2}}{m!n!\Gamma\left(\ell+\frac{1}{2}\right)\Gamma\left(m+\ell+\frac{3}{2}\right)\Gamma\left(\frac{1}{2}(-\ell+\Delta-1)\right)\Gamma\left(\frac{\ell+\Delta}{2}\right)\Gamma(m+\ell+\Delta)}$
$\displaystyle\times\,_{4}F_{3}\left(\frac{1}{2},-m+n-\ell,-\ell,\Delta-1;1,-m-\ell,n+\Delta-\ell-\frac{1}{2};1\right)$
(A.8)

The poles from the $\csc(\pi(m+\ell))$ cancels with the corresponding zeros of
${}_{4}F_{3}$ and therefore the above coefficients are well defined in the
allowed range of $m$ and $\ell$. This representation of the 3d conformal