aainstitutetext: Mathematical Institute, University of Oxford, Andrew Wiles
Building, Radcliffe Observatory Quarter, Woodstock Road, Oxford, OX2 6GG,
U.K.bbinstitutetext: Department of Physics and Astronomy, Uppsala University,
Box 516, SE-751 20 Uppsala, Sweden.ccinstitutetext: Kavli Institute for
Theoretical Sciences, University of Chinese Academy of Sciences, Beijing
100190, China.ddinstitutetext: Princeton Center for Theoretical Science,
Princeton University, Princeton, NJ 08544, USA.

#  One-loop Gluon Amplitudes in AdS

Luis F. Alday b Agnese Bissi c,d Xinan Zhou<EMAIL_ADDRESS>
<EMAIL_ADDRESS>
<EMAIL_ADDRESS>
###### Abstract

We initiate the study of one-loop gluon amplitudes in AdS space. These
amplitudes were recently computed at tree level for a variety of backgrounds
of the form $AdS_{d+1}\times S^{3}$. For concreteness, we compute the one-loop
correction to the massless gluon amplitude on $AdS_{5}\times S^{3}$, which
corresponds to the four-point correlator of the flavor current multiplet in
the dual 4d $\mathcal{N}=2$ SCFT. This requires solving a mixing problem that
involves tree-level amplitudes of arbitrarily massive Kaluza-Klein modes. The
final answer has the same color structure as in flat space but the dependence
on Mandelstam variables is more complicated, with logarithms replaced by
polygamma functions.

††preprint: UUITP-51/21

##  1 Introduction

Via the AdS/CFT correspondence, correlation functions in holographic CFTs are
mapped to on-shell scattering amplitudes in AdS space. These fundamental
observables encode important theoretical data, and play a central role in
testing and exploiting the correspondence. Computing these holographic objects
was once a notoriously difficult task because the complexity of perturbation
theory is amplified by the spacetime curvature. Explicit results were
available only in a handful special examples. In recent years, however, the
application of bootstrap ideas has led to drastic simplifications in the
calculation, prompting a great deal of new developments. A vast amount of
results for amplitudes have now been obtained in a variety of supergravity
theories, both at tree level Rastelli:2016nze ; Rastelli:2017udc ;
Rastelli:2017ymc ; Zhou:2017zaw ; Caron-Huot:2018kta ; Rastelli:2019gtj ;
Giusto:2019pxc ; Goncalves:2019znr ; Giusto:2020neo ; Alday:2020lbp ;
Alday:2020dtb ; Wen:2021lio  and at loop level Alday:2017xua ; Aprile:2017bgs
; Alday:2017vkk ; Aprile:2017qoy ; Alday:2018kkw ; Aprile:2019rep ;
Alday:2019nin ; Alday:2020tgi ; Bissi:2020wtv ; Bissi:2020woe ; Alday:2021ymb
. The results exhibit a remarkable simplicity which is obscured by the
diagrammatic expansion and is reminiscent of the situation in flat space.
Moreover, these AdS supergravity amplitudes display many unexpected
interesting structures, such as hidden conformal symmetries Caron-Huot:2018kta
; Rastelli:2019gtj ; Giusto:2020neo  and Parisi-Sourlas-like dimensional
reduction Behan:2021pzk . By contrast, amplitudes of super Yang-Mills theory
in AdS so far have received much less attention. Tree-level four-point
amplitudes were computed only very recently in Alday:2021odx  for a class of
backgrounds of the form $AdS_{d+1}\times S^{3}$, generalizing the result of an
earlier work Zhou:2018ofp .111For AdS gluon amplitudes in bosonic Yang-Mills
theory with four or more points, recent works include Roehrig:2020kck ;
Armstrong:2020woi ; Albayrak:2020fyp ; Diwakar:2021juk . Many aspects of AdS
supergravity amplitudes have not been similarly explored in the gluon context.
Nevertheless, there is a lot of incentives to further pursue the study of AdS
gluon scattering. On the one hand, gluon amplitudes in general have simpler
structures compared to graviton amplitudes. This makes them easier targets to
study and also more suitable arenas for new ideas. On the other hand, gluon
and graviton amplitudes are known to be intimately connected in flat space by
double copy relations.222See Bern:2019prr  for a recent review. Gluon
amplitudes are in a sense more fundamental, as graviton amplitudes can be
constructed by “squaring” them. It is conceivable that some of these flat
space properties extend to AdS space as well.

In this paper, we continue the investigation of gluon scattering in AdS and
initiate the study of loop-level amplitudes. For concreteness, we focus on the
cases where the hyperbolic space is $AdS_{5}$ and require the system to
preserve eight Poincaré supercharges (i.e., the dual SCFTs have 4d
$\mathcal{N}=2$ superconformal symmetry). Such backgrounds arise in several
top-down constructions. For example, we can consider a stack of D3-branes
probing F-theory singularities Fayyazuddin:1998fb ; Aharony:1998xz , or a
large number of D3-branes with a few probe D7-branes Karch:2002sh . In the
near horizon geometry, both classes of theories contain an $AdS_{5}\times
S^{3}$ locus which carries localized degrees of freedom organized into an
eight dimensional $\mathcal{N}=1$ vector multiplet. This vector multiplet
transforms in the adjoint representation of the flavor group $G_{F}$ which is
a gauge group from the bulk perspective. The Kaluza-Klein reduction of this
multiplet onto $AdS_{5}$ gives rise to an infinite tower of states which have
at most Lorentz spin 1. These are the massless and massive gluons and their
super partners. An important feature of these theories is that the coupling
between the gluons with gravitons is parametrically smaller than the gluon
self-coupling in the large central charge limit. Therefore, it is natural to
decouple gravity and consider a spin-1 gauge theory on $AdS_{5}\times S^{3}$
which describes the leading order dynamics of the gluonic sector. Furthermore,
we can also consider a consistent truncation of the gluon theory to its lowest
Kaluza-Klein level (the massless sector). This gives rise to an $AdS_{5}$ toy
model without the $S^{3}$ internal manifold.

Figure 1: The unitarity method in AdS. The $\langle 2222\rangle$ one-loop
super gluon amplitude in $AdS_{5}\times S^{3}$ can be constructed by “gluing”
together all $\langle 22pp\rangle$ tree-level amplitudes. For the case of
$AdS_{5}$ without internal manifold, there are no higher Kaluza-Klein modes
and $p$ is restricted to 2.

Specifically, we will consider scattering amplitudes of super gluons. They are
scalar super partners of the spin-1 gluons, and are the super primaries of the
superconformal multiplets. The super gluons are labelled by an integer Kaluza-
Klein level $k=2,3,\ldots$, which determines their squared masses in AdS to be
$m^{2}=k(k-4)$. Furthermore, we will use the Mellin representation Mack:2009mi
; Penedones:2010ue  to facilitate the computation. In this formalism AdS
scattering amplitudes become Mellin amplitudes, which share a lot of
similarities with flat space amplitudes. Thanks to supersymmetry, we can
further express the Mellin amplitudes in terms of simpler reduced Mellin
amplitudes. In flat space, this reduction is analogous to extracting a
fermionic delta function from the super amplitudes. One of the main results of
this paper is a closed form expression for the four-point one-loop reduced
Mellin amplitude of $k_{i}=2$ super gluons, for any gauge group $G_{F}$. To
obtain this result, our main technical tool is the AdS unitarity method
developed in Aharony:2016dwx , which is illustrated in Figure 1. Intuitively,
the one-loop amplitude can be obtained by “gluing” together pairs of tree-
level four-point amplitudes and summing over all the modes which run in the
loop. From the figure, we can see that the necessary tree-level input for
$AdS_{5}\times S^{3}$ is the $\langle 22pp\rangle$ amplitudes for arbitrary
values of $p$, which have already been computed in Alday:2021odx . The gluing
procedure will be made precise in the paper, and its realization in Mellin
space turns out to be similar to the supergravity case Alday:2018kkw ;
Alday:2019nin . However, a major difference is that the super gluons transform
in the adjoint representation of the color group. This makes the amplitude
contain multiple independent color structures which are reshuffled under
crossing. These color structures add an extra layer of complication which is
absent in the supergravity case. We will perform gluing in each color channel,
which gives part of the answer. By further using crossing symmetry we will
show that the one-loop reduced amplitude is uniquely fixed. However, this
procedure does not make it clear whether the result should sensitively depend
on the chosen gauge group. To answer this question, we find that the reduced
Mellin amplitude can be rewritten in the following remarkably simple form

$\widetilde{\mathcal{M}}(s,t)\sim\mathtt{d}_{st}\mathcal{B}_{st}(s,t)+\mathtt{d}_{su}\mathcal{B}_{su}(s,t)+\mathtt{d}_{tu}\mathcal{B}_{tu}(s,t)$
(1)

where the three terms are related by crossing symmetry. The function
$\mathcal{B}_{st}(s,t)$ is essentially a box diagram depicted in the LHS of
Figure 1, and has the form of an infinite sum of simultaneous simple poles

$\mathcal{B}_{st}(s,t)=\sum_{m,n=2}^{\infty}\frac{c_{mn}}{(s-2m)(t-2n)}$ (2)

where $c_{mn}$ are constants independent of the Mellin-Mandelstam variables.
Interestingly, similar building block amplitudes also appeared in the
supergravity one-loop reduced amplitudes Alday:2018kkw ; Alday:2019nin . In
this paper we give a closed form expression for $\mathcal{B}_{st}(s,t)$ in
terms of special functions, see (123). On the other hand, the factors
$\mathtt{d}_{st}$, $\mathtt{d}_{su}$, $\mathtt{d}_{tu}$ are “box diagrams” in
color space and are formed by contracting the structure constants of $G_{F}$
in the following way

$\mathtt{d}_{st}=f^{JI_{1}K}f^{KI_{2}L}f^{LI_{3}M}f^{MI_{4}J}\;.$ (3)

Note that the dependence on the gauge group only enters the reduced amplitude
through the structure constants $f^{IJK}$ (and also in the overall constant
suppressed here). Therefore the answer above is in fact valid for any choice
of the gauge group. One might also notice that the structure of the above
result is reminiscent of the gluon one-loop amplitude in flat space. In fact,
we can take the zero-curvature limit of the AdS amplitude and show that
$\widetilde{\mathcal{M}}(s,t)$ reproduces exactly the 8d flat space amplitude.
As another interesting example, we will also compute the super gluon one-loop
amplitude in the $AdS_{5}$ toy model with no internal manifold. In applying
the unitarity method, the Kaluza-Klein level $p$ is now restricted to 2
because the higher Kaluza-Klein modes have been excluded from the spectrum. It
turns out that this modification does not alter the structure of the reduced
Mellin amplitude but changes only the $c_{mn}$ coefficients. Taking the flat
space limit, we find that the amplitude $\mathcal{B}_{st}(s,t)$ becomes the 5d
one-loop box diagram instead.

The rest of the paper is organized as follows. In Section 2 we review the
setup of the problem and the basic kinematics of four-point functions. In
Section 3 we analyze the tree-level super gluon amplitudes and extract data
from them. The one-loop computation was first performed for the simpler
$AdS_{5}$ toy model in Section 4 and then for the full-fledged $AdS_{5}\times
S^{3}$ theory in Section 5. In Section 6, we consider another contribution to
the super gluon four-point function, which arises from coupling to gravity and
is at the same order as the one-loop gluon contribution. We conclude with a
brief discussion of future directions in Section 7. Various technical details
are relegated to the three appendices. In Appendix A and B we collect some
useful results for superconformal blocks and $D$-functions. In Appendix C we
explain how to take the flat space limit of one-loop Mellin amplitudes and how
they reduce to flat space box integrals.

##  2 Preliminaries

###  2.1 Setup

In this paper, we study scattering amplitudes of super gluons in $AdS_{5}$.
Such super gluons can arise in two basic setups. The first is to consider a
stack of $N$ D3-branes probing an F-theory 7-brane singularity
Fayyazuddin:1998fb ; Aharony:1998xz . The near horizon geometry has a metric
identical to that of $AdS_{5}\times S^{5}$, except that one of the angular
coordinates of $S^{5}$ has a changed periodicity. More precisely, we have

$ds^{2}=d\theta^{2}+\sin^{2}\theta d\phi^{2}+\cos^{2}\theta d\Omega_{3}^{2}$
(4)

where $d\Omega_{3}^{2}$ is the metric of $S^{3}$, and
$0\leq\theta\leq\frac{\pi}{2}$. The angular variable $\phi$ has a period of
$2\pi(1-\nu/2)$, where $\nu$ takes values

$\nu=\frac{1}{3}\;,\;\frac{1}{2}\;,\;\frac{2}{3}\;,\;1\;,\;\frac{4}{3}\;,\;\frac{3}{2}\;,\;\frac{5}{3}\;,$
(5)

depending on the type of the 7-brane singularity. The 7-brane is located at
the slice $\theta=0$, which fills the $AdS_{5}$ and occupies an $S^{3}$ in the
compact space. On this singular locus, there is a $7+1$ dimensional
$\mathcal{N}=1$ vector multiplet which transforms in the adjoint
representation of a gauge group $G_{F}$. Corresponding to the values of $\nu$
in (5), $G_{F}$ is given by

$G_{F}=U(1)\;,\;SU(2)\;,\;SU(3)\;,\;SO(8)\;,\;E_{6}\;,\;E_{7}\;,\;E_{8}.$ (6)

The Kaluza-Klein reduction of the $\mathcal{N}=1$ vector multiplet gives rise
to an infinite tower of states which are organized into different
$\frac{1}{2}$-BPS super multiplets of the 4d $\mathcal{N}=2$ superconformal
algebra. We refer to the superprimaries of these $\frac{1}{2}$-BPS super
multiplets as the super gluons. The $SU(2)_{R}$ R-symmetry of the 4d
$\mathcal{N}=2$ superconformal group is identified with one of the $SU(2)$
factors of the $S^{3}$ isometry group $SO(4)\simeq SU(2)_{R}\times SU(2)_{L}$,
while the other $SU(2)_{L}$ factor is a global symmetry. On the other hand,
gravity is not restricted to this subspace. Instead, it propagates in the full
ten dimensional space. The Kaluza-Klein reduction of the ten dimensional
supergravity multiplet gives rise to an infinite tower of super graviton
multiplets in $AdS_{5}$. An interesting feature of such systems is that at
large $N$ the self-interactions of the super gluons are parametrically larger
than their couplings with the super gravitons. More precisely, the cubic
couplings of super gluons scale as $1/\sqrt{N}$, while the couplings involving
two super gluons and one super graviton scale as $1/N$. Therefore, in $1/N$
expansion the super gluon four-point functions have the form

$G_{4\mathrm{-gluon}}=G_{\rm disc}+\frac{1}{N}G_{\text{gluon
tree}}+\frac{1}{N^{2}}(G_{\text{gluon 1-loop}}+G_{\text{graviton
tree}})+\ldots\;,$ (7)

where $G_{\rm disc}$ is the disconnected contribution, $G_{\text{gluon tree}}$
and $G_{\text{graviton tree}}$ respectively come from tree-level gluon and
graviton exchanges, and $G_{\text{gluon 1-loop}}$ is the gluon one-loop
contribution.

The second setup is to consider a stack of $N_{F}$ D7-branes wrapping an
$AdS_{5}\times S^{3}$ subspace in the $AdS_{5}\times S^{5}$ near horizon
geometry of $N$ D3-branes Karch:2002sh . In the limit $N_{F}\ll N$, the probe
D7-branes do not back-react the $AdS_{5}\times S^{5}$ geometry. The theory is
conformally invariant and preserves a 4d $\mathcal{N}=2$ superconformal
symmetry. The low energy degrees of freedom on the D7-branes are again
described by an eight dimensional $\mathcal{N}=1$ vector multiplet which
transforms in the adjoint representation of a gauge group $SU(N_{F})$.
Reducing this multiplet onto $AdS_{5}$ leads to the same spectrum of super
gluons. Just as in the previous setup, there is a separation of scales in the
super gluon self-couplings and their couplings to the super gravitons at large
$N$. These couplings also have the same $1/N$ scaling. Therefore at
$\mathcal{O}(1/N)$, the super gluon four-point functions consist of only tree-
level exchange contributions of super gluons. As was shown in Alday:2021odx ,
the tree-level gluon exchange amplitudes are completely determined by the
spectrum and superconformal symmetry, and have the same form for all choices
of gauge groups. Therefore the super gluon tree-level amplitudes at
$\mathcal{O}(1/N)$ are the same in both setups up to an overall constant.

Finally, we can also consistently truncate the $AdS_{5}\times S^{3}$ super
gluon theory to the lowest Kaluza-Klein mode, and obtain an $AdS_{5}$ theory
without an internal $S^{3}$. The tree-level super gluon exchange amplitude of
the lowest Kaluza-Klein mode remains the same after the truncation. We will
use this truncated theory as a toy model when studying the super gluon
amplitudes at one loop.

###  2.2 Super gluon four-point functions

The above mentioned super gluons are dual to scalar superprimaries
$\mathcal{O}_{k}^{I;a_{1}\ldots a_{k};\bar{a}_{1}\ldots\bar{a}_{k-2}}$ of the
$\frac{1}{2}$-BPS multiplets with $k=2,3,\ldots$. Here $I=1,\ldots,{\rm
dim}(G_{F})$ is the color index, in the adjoint representation of the flavor
group $G_{F}$. The $SU(2)$ indices $a_{i}=1,2$ and $\bar{a}_{i}=1,2$ are
respectively for $SU(2)_{R}$ and $SU(2)_{L}$. The former is the R-symmetry
group of the 4d $\mathcal{N}=2$ superconformal algebra, while the latter is an
additional global symmetry. These operators have spin $\frac{k}{2}$ under
$SU(2)_{R}$ and spin $\frac{k}{2}-1$ under $SU(2)_{L}$. Because the operators
are $\frac{1}{2}$-BPS, their conformal dimensions are determined by the
$SU(2)_{R}$ spins to be $k$.

It is convenient to keep track of the $SU(2)$ indices by contracting them with
auxiliary two-component spinors $v^{a}$ and $\bar{v}^{\bar{a}}$

$\mathcal{O}_{k}^{I}(x;v,\bar{v})=\mathcal{O}_{k}^{I;a_{1}\ldots
a_{k};\bar{a}_{1}\ldots\bar{a}_{k-2}}v_{a_{1}}\ldots
v_{a_{k}}\bar{v}_{\bar{a}_{1}}\ldots\bar{v}_{\bar{a}_{k-2}}$ (8)

where indices are lowered with $\epsilon$ tensors $v_{a}=v^{b}\epsilon_{ab}$,
$\bar{v}_{\bar{a}}=\bar{v}^{\bar{b}}\epsilon_{\bar{a}\bar{b}}$. We can then
consider four-point correlators of such operators

$G_{k_{1}k_{2}k_{3}k_{4}}^{I_{1}I_{2}I_{3}I_{4}}(x_{i};v_{i},\bar{v}_{i})=\langle\mathcal{O}_{k_{1}}^{I_{1}}(x_{1};v_{1},\bar{v}_{1})\mathcal{O}_{k_{2}}^{I_{2}}(x_{2};v_{2},\bar{v}_{2})\mathcal{O}_{k_{3}}^{I_{3}}(x_{3};v_{3},\bar{v}_{3})\mathcal{O}_{k_{4}}^{I_{4}}(x_{4};v_{4},\bar{v}_{4})\rangle\;,$
(9)

which are functions of both the spacetime and the internal coordinates. For
the purposes of this paper it is enough to consider pairwise equal external
weights $k_{1}=k_{2}=p$ and $k_{3}=k_{4}=q$, and will focus on such
correlators from now on. For a discussion on the kinematics of correlators
with most general weights, see Alday:2021odx . Exploiting conformal symmetry
and the two $SU(2)$ symmetries, we can write the correlators as functions of
four cross ratios

$G_{ppqq}^{I_{1}I_{2}I_{3}I_{4}}=\frac{(v_{1}\cdot v_{2})^{p}(v_{3}\cdot
v_{4})^{q}(\bar{v}_{1}\cdot\bar{v}_{2})^{p-2}(\bar{v}_{3}\cdot\bar{v}_{4})^{q-2}}{x_{12}^{2p}x_{34}^{2q}}\mathcal{G}_{ppqq}^{I_{1}I_{2}I_{3}I_{4}}(U,V;\alpha,\beta)\;.$
(10)

Here $x_{ij}=x_{i}-x_{j}$, $v_{i}\cdot v_{j}=v_{i}^{a}v_{j}^{b}\epsilon_{ab}$,
$\bar{v}_{i}\cdot\bar{v}_{j}=\bar{v}_{i}^{\bar{a}}\bar{v}_{j}^{\bar{b}}\epsilon_{\bar{a}\bar{b}}$,
and the cross ratios are defined as

$U=\frac{x_{13}^{2}x_{24}^{2}}{x_{12}^{2}x_{34}^{2}}\;,\quad
V=\frac{x_{14}^{2}x_{23}^{2}}{x_{12}^{2}x_{34}^{2}}\;,$ (11)
$\alpha=\frac{(v_{1}\cdot v_{3})(v_{2}\cdot v_{4})}{(v_{1}\cdot
v_{2})(v_{3}\cdot
v_{4})}\;,\quad\beta=\frac{(\bar{v}_{1}\cdot\bar{v}_{3})(\bar{v}_{2}\cdot\bar{v}_{4})}{(\bar{v}_{1}\cdot\bar{v}_{2})(\bar{v}_{3}\cdot\bar{v}_{4})}\;.$
(12)

It is easy to see that $\mathcal{G}_{ppqq}^{I_{1}I_{2}I_{3}I_{4}}$ is a
polynomial of degree $\min\\{p,q\\}$ in $\alpha$, and a polynomial of degree
$\min\\{p,q\\}-2$ in $\beta$.

We have so far only exploited the bosonic symmetries. The fermionic generators
in the superconformal group generate extra constraints known as the
superconformal Ward identities Nirschl:2004pa . To write down these
identities, we make a change of variables

$U=z\bar{z}\;,\quad V=(1-z)(1-\bar{z})\;.$ (13)

Then superconformal Ward identities read

$\begin{split}&(z\partial_{z}-\alpha\partial_{\alpha})\mathcal{G}_{ppqq}^{I_{1}I_{2}I_{3}I_{4}}(z,\bar{z};\alpha,\beta)\big{|}_{\alpha=1/z}=0\;,\\\
&(\bar{z}\partial_{\bar{z}}-\alpha\partial_{\alpha})\mathcal{G}_{ppqq}^{I_{1}I_{2}I_{3}I_{4}}(z,\bar{z};\alpha,\beta)\big{|}_{\alpha=1/\bar{z}}=0\;.\end{split}$
(14)

These constraints can be solved and lead to solutions of the form

$\mathcal{G}_{ppqq}^{I_{1}I_{2}I_{3}I_{4}}=\mathcal{G}_{0,ppqq}^{I_{1}I_{2}I_{3}I_{4}}+\mathcal{R}\,\mathcal{H}_{ppqq}^{I_{1}I_{2}I_{3}I_{4}}$
(15)

where $\mathcal{R}$ is a factor determined by superconformal symmetry

$\mathcal{R}=(1-z\alpha)(1-\bar{z}\alpha)\;,$ (16)

and $\mathcal{G}_{0,ppqq}^{I_{1}I_{2}I_{3}I_{4}}$ is a function that contains
coupling-independent information and becomes holomorphic (or anti-holomorphic)
when setting $\alpha=1/\bar{z}$ (or $\alpha=1/z$). The function $\mathcal{H}$
is called the reduced correlator and contains all the dynamical information.
It is always possible to find $\mathcal{G}_{0}$ and $\mathcal{H}$ such that
each of them separately enjoys Bose symmetry333In the case of identical
operators, Bose symmetry is just crossing symmetry.. More precisely, we can
restore the stripped kinematic factor and rewrite (15) as

$G_{ppqq}^{I_{1}I_{2}I_{3}I_{4}}=G_{0,ppqq}^{I_{1}I_{2}I_{3}I_{4}}+{\rm
R}\,H_{ppqq}^{I_{1}I_{2}I_{3}I_{4}}$ (17)

where

${\rm R}=(v_{1}\cdot v_{2})^{2}(v_{3}\cdot
v_{4})^{2}x_{13}^{2}x_{24}^{2}\mathcal{R}$ (18)

is crossing symmetric. The protected part $G_{0}$ has the original conformal
dimensions and $SU(2)_{R}\times SU(2)_{L}$ weights. On the other hand, $H$ can
be viewed as a four-point correlator of operators with shifted conformal
dimensions $k_{i}\to k_{i}+1$ and $SU(2)_{R}$ R-symmetry spins
$\frac{k_{i}}{2}\to\frac{k_{i}}{2}-1$. These shifts are due to the nontrivial
weights carried by the factor ${\rm R}$ under conformal and $SU(2)_{R}$
transformations. Note that in particular the reduced correlator with $k_{i}=2$
is a singlet under both $SU(2)_{R}$ and $SU(2)_{L}$.

A convenient language to discuss holographic correlators is the Mellin
formalism Mack:2009mi ; Penedones:2010ue . In this representation, holographic
correlators have simple analytic structure which closely resembles that of
flat space scattering amplitudes. The Mellin amplitudes are defined by444Here
we have restricted the discussion to $\langle ppqq\rangle$ correlators. For
the most general weight configurations, we refer the reader to Alday:2021odx
for details.

$\mathcal{G}_{ppqq}^{I_{1}I_{2}I_{3}I_{4}}=\int_{-i\infty}^{i\infty}\frac{dsdt}{(4\pi
i)^{2}}U^{\frac{s}{2}}V^{\frac{t-p-q}{2}}\mathcal{M}_{ppqq}\Gamma[\frac{2p-s}{2}]\Gamma[\frac{2q-s}{2}]\Gamma^{2}[\frac{p+q-t}{2}]\Gamma^{2}[\frac{p+q-u}{2}]$
(19)

where $s+t+u=\sum_{i}k_{i}=2(p+q)$. However, we can also define a reduced
Mellin amplitude from the reduced correlator

$\mathcal{H}_{ppqq}^{I_{1}I_{2}I_{3}I_{4}}=\int_{-i\infty}^{i\infty}\frac{dsdt}{(4\pi
i)^{2}}U^{\frac{s}{2}}V^{\frac{t-p-q}{2}}\widetilde{\mathcal{M}}_{ppqq}\Gamma[\frac{2p-s}{2}]\Gamma[\frac{2q-s}{2}]\Gamma^{2}[\frac{p+q-t}{2}]\Gamma^{2}[\frac{p+q-\tilde{u}}{2}]\;.$
(20)

Note that the $u$ variable is replaced by its shifted version $\tilde{u}=u-2$,
and this shift is necessary due to the nontrivial conformal weights of ${\rm
R}$. Bose symmetry then acts on the Mellin amplitude $\mathcal{M}$ by
permuting $s$, $t$, $u$, in addition to permuting the quantum numbers of each
operator. However, acting on the reduced Mellin amplitudes Bose symmetry
permutes instead $s$, $t$, $\tilde{u}$. Thanks to the superconformal Ward
identities, the reduced amplitude determines the full amplitude. The protected
part $\mathcal{G}_{0}$ for holographic correlators turn out to take a form
resembling Wick contractions in mean field theories. As was argued in
Rastelli:2017udc , such functions do not contribute to Mellin amplitudes. By
translating (15) into Mellin space, we find

$\mathcal{M}_{ppqq}=\mathbb{R}\circ\widetilde{\mathcal{M}}_{ppqq}$ (21)

where $\mathbb{R}$ is a difference operator descending from the factor
$\mathcal{R}$. The explicit action of this operator can be found in
Alday:2021odx  but is not needed in this paper. When computing one-loop
correlators, we will focus on the reduced Mellin amplitudes which have already
taken into account superconformal symmetry.

###  2.3 Flavor symmetry

As noted in the previous subsection, the four-point correlators of super
gluons carry flavor indices. This adds an extra layer of complication on top
of the superconformal kinematics discussed above. Below we give a detailed
discussion about how to deal with the flavor structures.

A concrete way to discuss these structures is to decompose the correlator into
different flavor channels that appear in the tensor product of adjoint
representations in the s-channel

$G^{I_{1}I_{2}I_{3}I_{4}}=\sum_{a\in{\rm adj}\otimes{\rm adj}}{\rm
P}^{I_{1}I_{2}|I_{3}I_{4}}_{a}G_{a}\;.$ (22)

Here ${\rm P}^{I_{1}I_{2}|I_{3}I_{4}}_{a}$ are projectors which project the
external indices to the irreducible representation $a$ appearing in ${\rm
adj}\otimes{\rm adj}$, and form a complete basis for flavor structures. For
example, the projectors associated with exchanging the identity and adjoint
representations are given by

${\rm P}^{I_{1}I_{2}|I_{3}I_{4}}_{\bf 1}=\frac{1}{{\rm
dim}(G_{F})}\delta^{I_{1}I_{2}}\delta^{I_{3}I_{4}}\;,\quad{\rm
P}^{I_{1}I_{2}|I_{3}I_{4}}_{\rm
adj}=\frac{1}{\psi^{2}h^{\vee}}f^{I_{1}I_{2}I_{5}}f^{I_{5}I_{3}I_{4}}$ (23)

where $h^{\vee}$ is the dual Coxeter number, $\psi^{2}$ is the length squared
of the longest root, and $f^{IJK}$ are the structure constants of the flavor
group $G_{F}$. The projectors satisfy symmetry properties

${\rm P}^{I_{1}I_{2}|I_{3}I_{4}}_{a}=(-1)^{{\rm R}_{a}}{\rm
P}^{I_{2}I_{1}|I_{3}I_{4}}_{a}\;,\quad{\rm P}^{I_{1}I_{2}|I_{3}I_{4}}_{a}={\rm
P}^{I_{3}I_{4}|I_{1}I_{2}}_{a}$ (24)

where ${\rm R}_{a}$ is the parity of the exchanged representation, i.e., ${\rm
R}_{a}=0$ for symmetric representations and ${\rm R}_{a}=1$ for antisymmetric
representations. They are also idempotent

${\rm P}^{I_{1}I_{2}|I_{3}I_{4}}_{a}{\rm
P}^{I_{4}I_{3}|I_{5}I_{6}}_{b}=\delta_{ab}{\rm
P}^{I_{1}I_{2}|I_{5}I_{6}}_{a}\;.$ (25)

Contracting the external indices gives a delta function for the internal
representations

${\rm P}^{I_{1}I_{2}|I_{3}I_{4}}_{a}{\rm
P}^{I_{1}I_{2}|I_{3}I_{4}}_{b}=\delta_{ab}{\rm dim}({\rm R}_{a})\;.$ (26)

We can use this property to compute the dimension of representation $a$.

We will also encounter the problem of decomposing exchanged flavor
representation in a different channel. This amounts to performing a change of
basis. Consider exchanging the representation $a^{\prime}$ in the t-channel,
which is captured by the t-channel projector ${\rm
P}^{I_{3}I_{2}|I_{1}I_{4}}_{a^{\prime}}$. We can decompose it into various
s-channel representations as

${\rm P}^{I_{3}I_{2}|I_{1}I_{4}}_{a^{\prime}}=\sum_{a}\mu_{a}{\rm
P}^{I_{1}I_{2}|I_{3}I_{4}}_{a}\;,$ (27)

using the completeness of the basis. To get the overlap coefficients, we use
(26) and get

$\mu_{a}=\frac{1}{{\rm dim}({\rm R}_{a})}{\rm
P}^{I_{3}I_{2}|I_{1}I_{4}}_{a^{\prime}}{\rm P}^{I_{1}I_{2}|I_{3}I_{4}}_{a}\;.$
(28)

Let us define the t-channel flavor crossing matrix

$({\rm F}_{t})_{a}{}^{a^{\prime}}\equiv\frac{1}{{\rm dim}({\rm R}_{a})}{\rm
P}^{I_{3}I_{2}|I_{1}I_{4}}_{a}{\rm P}^{I_{1}I_{2}|I_{3}I_{4}}_{a^{\prime}}\;,$
(29)

then we can express the overlap coefficients as

$\mu_{a}=({\rm F}_{t})_{a}{}^{a^{\prime}}\;.$ (30)

Diagrammatically, this is represented by Figure 2.

Figure 2: Decomposition of a t-channel projector into s-channel projectors.
The projectors in the two channels are related by the flavor crossing matrix
${\rm F}_{t}$. Note that here the vertices are oriented, and we read the lines
around a vertex anti-clock-wisely.

Therefore, it follows that a t-channel “exchange” can be expressed in the
s-channel as

$\sum_{a^{\prime}\in({\rm adj}\otimes{\rm adj})_{t}}{\rm
P}^{I_{3}I_{2}|I_{1}I_{4}}_{a^{\prime}}\\!\\!\\!\\!\\!\\!\\!\\!\underbrace{H^{(t)}_{a^{\prime}}}_{\text{t-channel
coefficients}}\\!\\!\\!\\!=\sum_{a\in({\rm adj}\otimes{\rm adj})_{s}}{\rm
P}^{I_{1}I_{2}|I_{3}I_{4}}_{a}\underbrace{\sum_{a^{\prime}\in({\rm
adj}\otimes{\rm adj})_{t}}({\rm
F}_{t})_{a}{}^{a^{\prime}}H^{(t)}_{a^{\prime}}}_{\text{s-channel
coefficients}}\;.$ (31)

Similarly, for the u-channel we define the crossing matrix

$({\rm F}_{u})_{a}{}^{a^{\prime}}\equiv\frac{1}{{\rm dim}({\rm R}_{a})}{\rm
P}^{I_{4}I_{2}|I_{3}I_{1}}_{a}{\rm P}^{I_{1}I_{2}|I_{3}I_{4}}_{a^{\prime}}\;.$
(32)

Then the u-channel decomposition can be rewritten in the s-channel as

$\sum_{a^{\prime}\in({\rm adj}\otimes{\rm adj})_{u}}{\rm
P}^{I_{4}I_{2}|I_{3}I_{1}}_{a^{\prime}}\\!\\!\\!\\!\\!\\!\\!\\!\underbrace{H^{(u)}_{a^{\prime}}}_{\text{u-channel
coefficients}}\\!\\!\\!\\!=\sum_{a\in({\rm adj}\otimes{\rm adj})_{s}}{\rm
P}^{I_{1}I_{2}|I_{3}I_{4}}_{a}\underbrace{\sum_{a^{\prime}\in({\rm
adj}\otimes{\rm adj})_{u}}({\rm
F}_{u})_{a}{}^{a^{\prime}}H^{(u)}_{a^{\prime}}}_{\text{s-channel
coefficients}}\;.$ (33)

The elements of these flavor crossing matrices can be expressed in terms of
the $3j$ and $6j$ symbols of the flavor group. We will not go into the details
of computing the matrix elements. Instead we will refer the reader to the book
Cvitanovic:2008zz  for techniques of performing such computations, and Table 6
of Chang:2017cdx  for a list of examples.

Finally, in this paper we will use the group $E_{8}$ as a nontrivial explicit
example to demonstrate various computations – although all our results can
also be phrased abstractly without reference to the explicit crossing
matrices, as we will show. We record here the crossing matrices of $E_{8}$
Chang:2017xmr

${\rm
F}_{t}=\left(\begin{array}[]{ccccc}\frac{1}{248}&\frac{125}{8}&\frac{3375}{31}&1&\frac{245}{2}\\\
\frac{1}{248}&-\frac{3}{8}&\frac{27}{31}&\frac{1}{5}&-\frac{7}{10}\\\
\frac{1}{248}&\frac{1}{8}&\frac{23}{62}&-\frac{1}{30}&-\frac{7}{15}\\\
\frac{1}{248}&\frac{25}{8}&-\frac{225}{62}&\frac{1}{2}&0\\\
\frac{1}{248}&-\frac{5}{56}&-\frac{90}{217}&0&\frac{1}{2}\\\
\end{array}\right)\;,\quad{\rm
F}_{u}=\left(\begin{array}[]{ccccc}\frac{1}{248}&\frac{125}{8}&\frac{3375}{31}&-1&-\frac{245}{2}\\\
\frac{1}{248}&-\frac{3}{8}&\frac{27}{31}&-\frac{1}{5}&\frac{7}{10}\\\
\frac{1}{248}&\frac{1}{8}&\frac{23}{62}&\frac{1}{30}&\frac{7}{15}\\\
-\frac{1}{248}&-\frac{25}{8}&\frac{225}{62}&\frac{1}{2}&0\\\
-\frac{1}{248}&\frac{5}{56}&\frac{90}{217}&0&\frac{1}{2}\\\
\end{array}\right)\;$ (34)

where from top to bottom along each column (or from left to right in each row)
the representations are $a=\bf{1},\bf{3875},\bf{27000},\bf{248}\,({\rm
adj}),\bf{30380}$. Note that the first three representations are symmetric,
while the last two are anti-symmetric.

###  2.4 Superconformal block decomposition

To be able to reconstruct the one-loop correlator, we need to decompose
correlators into superconformal blocks. In order to understand the
contribution of each superconformal multiplet, we study the fusion rules of
the operators $\mathcal{O}$ appearing in our four-point function. Operators in
$\mathcal{N}=2$ SCFTs are organized into different superconformal multiplets
which have been classified in Nirschl:2004pa . Here we will encounter three
types of them: $\hat{\mathcal{B}}_{R}$,
$\hat{\mathcal{C}}_{R,(\frac{\ell}{2},\frac{\ell}{2})}$ and
$\mathcal{A}^{\Delta}_{R,(\frac{\ell}{2},\frac{\ell}{2})}$.555See Appendix A
of Beem:2014zpa  for a summary of the unitary representations and different
notations used in the literature. The $\hat{\mathcal{B}}$ multiplets are
$\frac{1}{2}$-BPS (known as short multiplets), and the conformal dimensions of
the super primaries as fixed by the $SU(2)_{R}$ spin $R$ is $\frac{R}{2}$. The
super gluons belong to this type of multiplets and are the corresponding super
primaries. The $\hat{\mathcal{C}}$ multiplets are also protected, and are
known as semi-short. The super primary has $SU(2)_{R}$ spin $\frac{R}{2}$,
Lorentz spin $\ell$ and dimension $2+2R+\ell$. The multiplets $\mathcal{A}$
are long multiplets, and the conformal dimensions are not protected by
supersymmetry. These are the only three types of multiplets which appear in
the fusion rules of two $\hat{\mathcal{B}}$ Nirschl:2004pa

$\hat{\mathcal{B}}_{\frac{k_{1}}{2}}\times\hat{\mathcal{B}}_{\frac{k_{2}}{2}}\simeq\bigoplus_{p=\frac{k_{2}}{2}-\frac{k_{1}}{2}}^{\frac{k_{1}}{2}+\frac{k_{2}}{2}}\hat{\mathcal{B}}_{p}+\bigoplus_{\ell\geq
0}\left(\bigoplus_{p=\frac{k_{2}}{2}-\frac{k_{1}}{2}}^{\frac{k_{2}}{2}+\frac{k_{1}}{2}-1}\hat{\mathcal{C}}_{p,(\frac{\ell}{2},\frac{\ell}{2})}+\bigoplus_{p=\frac{k_{2}}{2}-\frac{k_{1}}{2}}^{\frac{k_{2}}{2}+\frac{k_{1}}{2}-2}\mathcal{A}^{\Delta}_{p,(\frac{\ell}{2},\frac{\ell}{2})}\right)\;.$
(35)

The exchanged multiplets in the four-point function are those residing in the
overlap of
$\hat{\mathcal{B}}_{\frac{k_{1}}{2}}\times\hat{\mathcal{B}}_{\frac{k_{2}}{2}}$
and
$\hat{\mathcal{B}}_{\frac{k_{3}}{2}}\times\hat{\mathcal{B}}_{\frac{k_{4}}{2}}$.
This is the most general fusion rules, but we will only need a less general
setup to study the four point correlator (10). In particular, it is more
convenient to write the solution to the superconformal Ward identities in a
slightly different form Nirschl:2004pa

$\mathcal{G}(U,V;\alpha)=\frac{(y-\bar{x})f(\bar{x})-(y-x)f(x)}{x-\bar{x}}+(y-x)(y-\bar{x})\mathcal{K}(U,V;y)$
(36)

where we defined the new cross ratios

$y=2\alpha-1\;,\quad x=\frac{2}{z}-1\;,\quad\bar{x}=\frac{2}{\bar{z}}-1\;.$
(37)

We have also temporarily suppressed the flavor indices in (10), as well as the
$SU(2)_{L}$ dependence, to lighten the notation. These extra structures are
independent of the superconformal properties, and therefore can be easily
restored. Given a four-point correlator $\mathcal{G}$, we can first obtain
$f(x)$ by setting $y=\bar{x}$. Then from (36) we can easily determine
$\mathcal{K}$. In contrast to the function $\mathcal{H}$ defined in (15),
$\mathcal{K}$ alone is not invariant under Bose symmetry. Instead, there is an
inhomogeneous term coming from the single-variable function $f$ under
crossing.

The contribution of exchanging a superconformal multiplet to the four-point
function is captured by superconformal blocks, which are linear combinations
of bosonic conformal blocks of the super primary and super descendants of the
superconformal multiplet. They were obtained in Nirschl:2004pa  and are
recorded in Appendix A. In practice, it is most convenient to present them in
terms of contributions to $f$ and $\mathcal{K}$ in (36). Furthermore, the
function $\mathcal{K}$ can be decomposed into different $SU(2)_{R}$ channels.
For computing the one-loop amplitude of $\langle 2222\rangle$, only the
R-symmetry singlet contributions are relevant.

To write down these contributions, let us first introduce the bosonic
conformal blocks

$G_{\Delta,\ell}(z,\bar{z})=\frac{(-\frac{1}{2})^{\ell}z\bar{z}}{z-\bar{z}}\big{(}k_{\frac{\Delta+\ell}{2}}(z)k_{\frac{\Delta-\ell-2}{2}}(\bar{z})-k_{\frac{\Delta+\ell}{2}}(\bar{z})k_{\frac{\Delta-\ell-2}{2}}(z)\big{)}$
(38)

where

$k_{h}(z)=z^{h}{}_{2}F_{1}(h,h;2h;z)\;,$ (39)

and this expression is valid in any four-point function with
$\Delta_{1}=\Delta_{2}$, $\Delta_{3}=\Delta_{4}$. The long super multiplets
$\mathcal{A}^{\Delta}_{R,(\frac{\ell}{2},\frac{\ell}{2})}$ do not contribute
to $f(x)$. Moreover, only the long multiplets with $R=0$ contribute to the
singlet channel of $\mathcal{K}$ by666We introduced a factor of $\frac{1}{4}$
such that when we expand the superconformal block into bosonic conformal
blocks the conformal block of the super primary appears with a unit
coefficient.

$\mathcal{K}^{\mathcal{A}}_{\Delta,R,\ell}\big{|}_{\rm
sing}=\frac{1}{4}\delta_{R,0}G_{\Delta+2,\ell}(z,\bar{z})\;.$ (40)

The semi-short multiplets
$\hat{\mathcal{C}}_{R,(\frac{\ell}{2},\frac{\ell}{2})}$ contribute

$\begin{split}&f^{\mathcal{C}}_{R,\ell}(x)=\frac{(-1)^{\ell-1}2^{2R-\ell}\Gamma\left(R+\frac{1}{2}\right)}{\sqrt{\pi}\Gamma(R+1)}k_{R+2+\ell}(z)\;,\\\
&\mathcal{K}^{\mathcal{C}}_{R,\ell}\big{|}_{\rm
sing}=\frac{(-1)^{R+1}2^{3R-2}\Gamma\left(R+\frac{1}{2}\right)}{\sqrt{\pi}\Gamma(R+1)}G_{R+\ell+4,R+\ell}(z,\bar{z})\;.\end{split}$
(41)

Finally, the short multiplets $\hat{\mathcal{B}}_{R}$ contribute

$f^{\mathcal{B}}_{R}(x)=\frac{4^{R}\Gamma\left(R+\frac{1}{2}\right)}{\sqrt{\pi}\Gamma(R+1)}k_{R}(z)\;,\quad\mathcal{K}^{\mathcal{B}}_{R}\big{|}_{\rm
sing}=\frac{(-1)^{R}2^{3R-4}\Gamma\left(R+\frac{1}{2}\right)}{\sqrt{\pi}\Gamma(R+1)}G_{R+2,R-2}(z,\bar{z})\;.$
(42)

From these expressions, it is easy to see that the contributions to $f$ from
short and semi-short multiplets satisfy the following relations

$f^{\mathcal{C}}_{R,\ell}=\frac{(-1)^{\ell+1}2^{-3\ell-4}\Gamma\left(R+\frac{1}{2}\right)\Gamma(R+\ell+3)}{\Gamma(R+1)\Gamma\left(R+\ell+\frac{5}{2}\right)}f^{\mathcal{B}}_{R+\ell+2}\;,$
(43)
$f^{\mathcal{C}}_{R+a,\ell-a}=\frac{\left(-\frac{1}{8}\right)^{-a}\Gamma(R+1)\Gamma\left(a+R+\frac{1}{2}\right)}{\Gamma\left(R+\frac{1}{2}\right)\Gamma(a+R+1)}f^{\mathcal{C}}_{R,\ell}\;.$
(44)

These relations arise from the fact that short and semi-short multiplets can
recombine and form long multiplets at unitarity bounds Nirschl:2004pa . As a
result, their contributions to $f$ cancel and they only contribute to the
function $\mathcal{K}$.

Generalized free field theory

As a simple application, let us consider the superconformal block
decomposition of generalized free field (GFF) theory correlators. The result
is also needed later for computing the one-loop amplitudes.

The GFF correlators give the disconnected contribution of holographic
correlators, which are leading in the $1/N$-expansion. We are interested in
the correlators of four identical operators with dimension $k$

$\mathcal{G}^{\rm
GFF}_{kkkk}=\delta^{I_{1}I_{2}}\delta^{I_{3}I_{4}}+\delta^{I_{1}I_{3}}\delta^{I_{2}I_{4}}\mathcal{G}^{{\rm
GFF},(u)}_{kkkk}+\delta^{I_{1}I_{4}}\delta^{I_{2}I_{3}}\mathcal{G}^{{\rm
GFF},(t)}_{kkkk}\;.$ (45)

where

$\mathcal{G}^{{\rm
GFF},(t)}_{kkkk}=(\alpha-1)^{k}(\beta-1)^{k-2}\frac{U^{k}}{V^{k}}\;,\quad\mathcal{G}^{{\rm
GFF},(u)}_{kkkk}=\alpha^{k}\beta^{k-2}U^{k}\;.$ (46)

Let us look at the three independent flavor structures
$\delta^{I_{1}I_{2}}\delta^{I_{3}I_{4}}$,
$\delta^{I_{1}I_{3}}\delta^{I_{2}I_{4}}$,
$\delta^{I_{1}I_{4}}\delta^{I_{2}I_{3}}$ separately. The coefficient functions
of these structures can be straightforwardly rewritten in the form of (36).
Note that the first term $\delta^{I_{1}I_{2}}\delta^{I_{3}I_{4}}$ just
corresponds to exchanging the identity operator in the s-channel. For the
other two terms, we denote the $SU(2)_{R}\times SU(2)_{L}$ singlets of
functions $\mathcal{K}$ from $\mathcal{G}^{{\rm GFF},(t,u)}_{kkkk}$ by
$\mathcal{K}^{(t,u)}_{k,{\rm sing}}$. We would like to know the OPE
coefficients of long multiplets in these functions. To this end, we decompose
$\mathcal{K}^{(t,u)}_{k,{\rm sing}}$ into conformal blocks and find

$\displaystyle\mathcal{K}^{(t)}_{k,{\rm sing}}$ $\displaystyle=$
$\displaystyle\sum_{n=-k+1}^{\infty}\sum_{\ell=0}^{\infty}A_{n,\ell}^{(k)}G_{2k+2+2n+\ell,\ell}(z,\bar{z})\;,$
(47) $\displaystyle\mathcal{K}^{(u)}_{k,{\rm sing}}$ $\displaystyle=$
$\displaystyle\sum_{n=-k+1}^{\infty}\sum_{\ell=0}^{\infty}(-1)^{\ell}A_{n,\ell}^{(k)}G_{2k+2+2n+\ell,\ell}(z,\bar{z})\;,$
(48)

where

$\begin{split}A_{n,\ell}^{(k)}=&\frac{\pi(-1)^{\ell}(n+1)_{k-2}(k+n+1)_{k-2}(n+\ell+2)_{k-2}(k+n+\ell+2)_{k-2}}{2^{4(k+n)+\ell}\Gamma(k)^{4}\Gamma\left(k+n-\frac{1}{2}\right)}\\\
&\times\frac{(\ell+1)(2(k+n)+\ell)\Gamma(k+n)\Gamma(k+n+\ell+1)}{\Gamma\left(k+n+\ell+\frac{1}{2}\right)}\;.\end{split}$
(49)

Note that due to the aforementioned multiplet recombination, there is an
ambiguity in interpreting these conformal blocks. On the other hand, we note
that recombination only happens at the unitarity bound. Since we are
restricting ourselves to the $R=0$ sector, the conformal twist of the
recombined long multiplets is $\tau=2$, i.e., the conformal blocks
$G_{4+\ell,\ell}(z,\bar{z})$.777In fact, we observe that the above
$A_{n,\ell}^{(k)}$ coefficients with $n<0$ are all zero except for one value
$n=-k+1$, which corresponds to $\tau=2$. In the holographic limit to be
discussed, which corresponds to turning on interactions for the generalized
free field theory, we expect that all long multiplets are double-trace
operators with minimal twist 4.888For the conformal block
$G_{6+2n+\ell,\ell}(z,\bar{z})$, the associated operators have conformal twist
$\tau=4+2n$ and are linear combinations of the double-trace operators
$:\mathcal{O}_{2}\square^{n}\partial^{\ell}\mathcal{O}_{2}:$,
$\mathcal{O}_{3}\square^{n-1}\partial^{\ell}\mathcal{O}_{3}:$, $\ldots$,
$:\mathcal{O}_{n+2}\partial^{\ell}\mathcal{O}_{n+2}:$. Therefore, their
coefficients are not affected by recombination and are simply given by (49)
with $n\geq 0$.

##  3 Data from tree-level gluon amplitudes

Let us proceed to tree level. The four-point super gluon Mellin amplitudes
were computed in Alday:2021odx  for arbitrary external weights. To compute the
$\langle 2222\rangle$ correlator at one loop, we will only need the $\langle
22kk\rangle$ tree-level correlators.

###  3.1 Correlators in position space

To extract the CFT data, it is more convenient to rewrite the Mellin space
result in position space. Let us first give the result written in the form of
(17), before outlining how the translation was performed. The protected parts
for correlators $\langle 22kk\rangle$ with $k>2$ are related to the $k=2$ case
by

$G_{0,22kk}=\frac{(v_{3}\cdot
v_{4})^{k-2}(\bar{v}_{3}\cdot\bar{v}_{4})^{k-2}}{x_{34}^{2(k-2)}}G_{0,2222}\;,$
(50)

and the dynamical parts have the form

$H_{22kk}=(C_{2,2,2})^{2}\big{(}\mathtt{c}_{s}H_{22kk}^{(s)}+\mathtt{c}_{t}H_{22kk}^{(t)}+\mathtt{c}_{u}H_{22kk}^{(u)}\big{)}\;.$
(51)

Here $\mathtt{c}_{s,t,u}$ are color structures

$\mathtt{c}_{s}=f^{I_{1}I_{2}J}f^{JI_{3}I_{4}}\;,\quad\quad\mathtt{c}_{t}=f^{I_{1}I_{4}J}f^{JI_{2}I_{3}}\;,\quad\quad\mathtt{c}_{u}=f^{I_{1}I_{3}J}f^{JI_{4}I_{2}}\;,$
(52)

which are related to the projectors defined in (2.3) by

$\mathtt{c}_{s}=\psi^{2}h^{\vee}{\rm P}^{I_{1}I_{2}|I_{3}I_{4}}_{\rm
adj}\;,\quad\mathtt{c}_{t}=-\psi^{2}h^{\vee}{\rm
P}^{I_{3}I_{2}|I_{1}I_{4}}_{\rm
adj}\;,\quad\mathtt{c}_{u}=-\psi^{2}h^{\vee}{\rm
P}^{I_{4}I_{2}|I_{3}I_{1}}_{\rm adj}\;.$ (53)

Thanks to the Jacobi identity, they satisfy

$\mathtt{c}_{s}+\mathtt{c}_{t}+\mathtt{c}_{u}=0\;.$ (54)

The function $G_{0,2222}$ has the form of Wick contractions

$\begin{split}G_{0,2222}=&\frac{(v_{1}\cdot v_{2})^{2}(v_{3}\cdot
v_{4})^{2}}{3x_{12}^{2}x_{34}^{2}}(C_{2,2,2})^{2}\bigg{(}\mathtt{c}_{s}\frac{(\alpha-1)x_{13}^{2}x_{24}^{2}+2\alpha
x_{14}^{2}x_{23}^{2}+\alpha(1-\alpha)x_{12}^{2}x_{34}^{2}}{x_{13}^{2}x_{14}^{2}x_{23}^{2}x_{24}^{2}}\\\
&\quad\quad\quad\quad\quad\quad\quad\quad\quad+\mathtt{c}_{t}\frac{(1-\alpha)x_{13}^{2}x_{24}^{2}+\alpha
x_{14}^{2}x_{23}^{2}+2\alpha(1-\alpha)x_{12}^{2}x_{34}^{2}}{x_{13}^{2}x_{14}^{2}x_{23}^{2}x_{24}^{2}}\bigg{)}\;,\end{split}$
(55)

while $H_{22kk}^{(s,t,u)}$ can be expressed in terms of $D$-functions (see
Appendix B for their definition and properties)

$\displaystyle H_{22kk}^{(s)}$ $\displaystyle=$
$\displaystyle\frac{2(k-1)k}{3\pi^{2}}\bigg{(}\frac{D_{1,2,k+1,k}}{x_{12}^{2}x_{14}^{2}}-\frac{D_{1,2,k,k+1}}{x_{12}^{2}x_{13}^{2}}\bigg{)}(v_{3}\cdot
v_{4})^{k-2}(\bar{v}_{3}\cdot\bar{v}_{4})^{k-2}\;,$ $\displaystyle
H_{22kk}^{(t)}$ $\displaystyle=$
$\displaystyle\frac{4(k-1)}{3\pi^{2}}\bigg{(}\frac{D_{1,3,k,k}}{x_{13}^{2}x_{14}^{2}}-\frac{k}{2}\frac{D_{1,2,k+1,k}}{x_{12}^{2}x_{14}^{2}}\bigg{)}(v_{3}\cdot
v_{4})^{k-2}(\bar{v}_{3}\cdot\bar{v}_{4})^{k-2}\;,$ (56) $\displaystyle
H_{22kk}^{(u)}$ $\displaystyle=$
$\displaystyle\frac{4(k-1)}{3\pi^{2}}\bigg{(}\frac{k}{2}\frac{D_{1,2,k,k+1}}{x_{12}^{2}x_{13}^{2}}-\frac{D_{1,3,k,k}}{x_{13}^{2}x_{14}^{2}}\bigg{)}(v_{3}\cdot
v_{4})^{k-2}(\bar{v}_{3}\cdot\bar{v}_{4})^{k-2}\;.$

Note that $H_{22kk}^{(s,t,u)}$ satisfy the relation

$H_{22kk}^{(s)}+H_{22kk}^{(t)}+H_{22kk}^{(u)}=0\;,$ (57)

which parallels the Jacobi relation (54). Finally, $C_{2,2,2}$ is related to
the cubic coupling of the $AdS_{5}\times S^{3}$ SYM, and appears in the three-
point function of $\mathcal{O}_{2}$ as

$\langle\mathcal{O}_{2}(x_{1};v_{1})\mathcal{O}_{2}(x_{2};v_{2})\mathcal{O}_{2}(x_{3};v_{3})\rangle=C_{2,2,2}\frac{(v_{1}\cdot
v_{2})(v_{2}\cdot v_{3})(v_{3}\cdot
v_{1})}{x_{12}^{2}x_{13}^{2}x_{23}^{2}}\;.$ (58)

It can be expressed in terms of flavor central charge as999Here the flavor
current two-point functions are
$\langle\mathcal{J}_{\mu}^{I}(x)\mathcal{J}_{\nu}^{J}(0)\rangle=\frac{C_{\mathcal{J}}}{2\pi^{2}}\frac{\delta^{IJ}(\delta_{\mu\nu}-2\frac{x^{\mu}x^{\nu}}{x^{2}})}{x^{6}}\;.$
For the $\mathcal{N}=2$ SCFTs arising from D3-branes probing F-theory
singularities, $C_{\mathcal{J}}$ is given by
$C_{\mathcal{J}}=\frac{12}{2-\nu}N$, where $N$ is the number of D3-branes, and
$\nu$ is determined by the type of the singularity according to (5).

$(C_{2,2,2})^{2}=\frac{6}{C_{\mathcal{J}}}\;.$ (59)

The translation from Mellin space to position space was done in two steps. One
writes down an ansatz for $G$ in terms of exchange and contact Witten diagrams
which are expressed in terms of $D$-functions using the method of
DHoker:1999mqo . The coefficients of these diagrams are found by matching the
Mellin amplitudes. Similarly, one finds a position space expression for the
reduced correlator $H$ in terms of $D$-functions. Then the protected piece
$G_{0}$ can be solved from (17), and can be simplified into the form (55)
after using a few $D$-function identities which are summarized in Appendix B.

Moreover, the $D$-functions in (3.1) satisfy a set of differential recursion
relations which shifts their weights. These relations are collected in
Appendix B, and allow us to relate $H_{22kk}^{(s,t,u)}$ with $k>2$ to
$H_{2222}^{(s,t,u)}$ by differential operators. Let us define

$H_{22kk}^{(s,t,u)}(x_{i},v_{i},\bar{v}_{i})=\frac{(v_{3}\cdot
v_{4})^{k-2}(\bar{v}_{3}\cdot\bar{v}_{4})^{k-2}}{x_{12}^{6}x_{34}^{2+2k}}U\mathcal{H}_{22kk}^{(s,t,u)}(U,V)\;.$
(60)

Then using the formulae in Appendix B, one can show

$U\mathcal{H}_{22kk}^{(s,t,u)}=\frac{1}{(k-2)!}\prod_{i=1}^{k-2}(i+2-U\partial_{U})\big{(}U\mathcal{H}_{2222}^{(s,t,u)}\big{)}\;.$
(61)

###  3.2 Extracting tree-level data

We are interested in the tree-level anomalous dimensions which are encoded in
the $\log U$ coefficients of the reduced correlators in the small $U$
expansion. Let us first focus on the case with $k=2$. Using the differential
recursion relations, we find that the reduced correlator can be compactly
written as

$\displaystyle\mathcal{H}_{2222}^{(s)}=$
$\displaystyle\frac{U^{2}}{3}\big{(}2\partial_{U}+(1+V)\partial_{U}\partial_{V}+U\partial_{U}^{2}\big{)}\Phi(z,\bar{z})\;,$
$\displaystyle\mathcal{H}_{2222}^{(t)}=$
$\displaystyle-\frac{U^{2}}{3}\big{(}2\partial_{V}+V\partial_{V}^{2}+(1+U)\partial_{U}\partial_{V}\big{)}\Phi(z,\bar{z})\;,$
(62) $\displaystyle\mathcal{H}_{2222}^{(u)}=$
$\displaystyle\frac{U^{2}}{3}\big{(}2\partial_{V}+V\partial_{V}^{2}-2\partial_{U}+(U-V)\partial_{U}\partial_{V}-U\partial_{U}^{2}\big{)}\Phi(z,\bar{z})$

where $\Phi(z,\bar{z})=\bar{D}_{1,1,1,1}$ is the well known scalar one-loop
box diagram. The $\log U$ coefficient of $\Phi(z,\bar{z})$ is given by

$\Psi(z,\bar{z})\equiv\Phi(z,\bar{z})\big{|}_{\log
U}=\frac{\log(1-z)-\log(1-\bar{z})}{z-\bar{z}}\;.$ (63)

It is then straightforward to obtain the $\log U$ coefficients of
$\mathcal{H}_{2222}^{(s,t,u)}$, simply by replacing $\Phi$ with $\Psi$ in
(3.2), as differential operators acting on $\log U$ would lead to rational
functions. Similarly, the $\log U$ coefficients of
$\mathcal{H}_{22kk}^{(s,t,u)}$ are obtained by further acting on
$\mathcal{H}_{2222}^{(s)}\big{|}_{\log U}$ with the differential operators in
(61). It is not difficult to decompose these $\log U$ coefficients into
conformal blocks, and we find101010We remind the reader the difference between
$\mathcal{K}$ and $\mathcal{H}$. The factor $(x-y)(\bar{x}-y)$ in (36) is
$4U^{-1}\mathcal{R}$ where $\mathcal{R}$ was defined in (16). Therefore a long
multiplet of which the super primary has dimension $\Delta$ and spin $\ell$
contributes to $\mathcal{H}$ by $U^{-1}G_{\Delta+2,\ell}$.

$\displaystyle\mathcal{H}_{22kk}^{(s)}\big{|}_{\log U}$ $\displaystyle=$
$\displaystyle\sum_{n,\ell}\left(\frac{1}{2}-\frac{(-1)^{\ell}}{2}\right)\omega_{n,\ell}^{(k)}U^{-1}G_{2k+2+2n+\ell,\ell}(z,\bar{z})\;,$
(64) $\displaystyle\mathcal{H}_{22kk}^{(t)}\big{|}_{\log U}$ $\displaystyle=$
$\displaystyle\sum_{n,\ell}\left(\frac{1}{2}+(-1)^{\ell}\right)\omega_{n,\ell}^{(k)}U^{-1}G_{2k+2+2n+\ell,\ell}(z,\bar{z})\;,$
(65) $\displaystyle\mathcal{H}_{22kk}^{(u)}\big{|}_{\log U}$ $\displaystyle=$
$\displaystyle-\sum_{n,\ell}\left(1+\frac{(-1)^{\ell}}{2}\right)\omega_{n,\ell}^{(k)}U^{-1}G_{2k+2+2n+\ell,\ell}(z,\bar{z})\;,$
(66)

where

$\omega_{n,\ell}^{(k)}=\frac{\pi(-1)^{k+1}(n+1)_{k-1}2^{-4k-4n-\ell+3}\Gamma(2k+n-1)\Gamma(k+n+\ell+1)}{3\Gamma(k-1)\Gamma(k)\Gamma\left(k+n-\frac{1}{2}\right)\Gamma\left(k+n+\ell+\frac{1}{2}\right)}\;.$
(67)

The coefficients $\omega_{n,\ell}^{(k)}$ are proportional to the schematic
averaged quantity $\langle a^{(0)}_{n,\ell}\gamma^{(1)}_{n,\ell}\rangle_{a}$
in each color channel

$\frac{1}{2}\sum_{n,\ell}\langle
a^{(0)}_{n,\ell}\gamma^{(1)}_{n,\ell}\rangle_{a}U^{-1}G_{2k+2+2n+\ell,\ell}(z,\bar{z})={\rm
P}_{a}^{I_{1}I_{2}|I_{3}I_{4}}\mathcal{H}_{22kk}\big{|}_{\log U}$ (68)

where $a^{(0)}_{n,\ell}$ are generalized free field theory OPE coefficients
and $\gamma^{(1)}_{n,\ell}$ are the tree-level anomalous dimensions of the
double-trace operators.

##  4 Toy model: AdS5 without internal S3

In this section, we first study one-loop amplitudes in a toy model where the
$AdS_{5}$ super gluon theory does not have an internal $S^{3}$ manifold. This
amounts to setting all $k>2$ fields to zero in the $AdS_{5}\times S^{3}$
theories, which consistently truncates the theory to the $k=2$ sector. As a
result, there is no operator mixing in this model. All the double-trace
operators are of the schematic form
$:\mathcal{O}_{2}\square^{n}\partial^{\ell}\mathcal{O}_{2}:$, and there is a
one-to-one correspondence between the conformal twist and $n$.111111By
contrast, the $AdS_{5}\times S^{3}$ theories allow, e.g., operators of the
form $:\mathcal{O}_{3}\square^{n-1}\partial^{\ell}\mathcal{O}_{3}:$ which have
the same conformal twist when $n\geq 1$ and lead to a degeneracy. We will
discuss more about operator mixing in Section 5. We will use this simplified
model to explore the analytic structure of super gluon one-loop amplitudes in
AdS, and to see what color structures can arise. We will first perform
calculations in the $E_{8}$ theory where we use the explicit crossing matrices
of $E_{8}$ and apply the AdS unitarity method in each independent flavor
channel. The AdS unitarity method was first developed in Aharony:2016dwx .
Here we will use the Mellin version of this method, originally presented in
Aharony:2016dwx  and further developed in Alday:2018kkw ; Alday:2019nin , to
manifest the simple analytic structure of the one-loop amplitude. After
solving the $E_{8}$ example, we show the result can be cast in a form agnostic
of the color group choice. We then prove that this form of the answer holds
for general gauge groups.

###  4.1 One-loop amplitude with $E_{8}$ color group

As was pointed out in Aharony:2016dwx , one-loop correlators are determined by
the $\log^{2}U$ coefficients in the small $U$ expansion. Our first task is to
compute these coefficients in each independent color channel.

Our starting point is the disconnected correlator. We use (23) to express
$\delta^{I_{1}I_{4}}\delta^{I_{2}I_{3}}$ and
$\delta^{I_{1}I_{3}}\delta^{I_{2}I_{4}}$ as t- and u-channel projectors, and
then the crossing matrices (31) and (33) to decompose them in the s-channel.
Using the results in Section 2.4, we find that the contributions of long
operators are the same in channels of the same parity

$\displaystyle\mathcal{H}^{(0)}_{2222,{\rm long},{\bf
1}}=\mathcal{H}^{(0)}_{2222,{\rm long},{\bf
3875}}=\mathcal{H}^{(0)}_{2222,{\rm long},{\bf
27000}}=\sum_{n,\ell=0}^{\infty}\left(\tfrac{1+(-1)^{\ell}}{2}\right)(8A_{n,\ell}^{(2)})U^{-1}G_{6+2n+\ell,\ell}(z,\bar{z})\;,$
$\displaystyle\mathcal{H}^{(0)}_{2222,{\rm long},{\bf
248}}=\mathcal{H}^{(0)}_{2222,{\rm long},{\bf
30380}}=\sum_{n,\ell=0}^{\infty}\left(\tfrac{1-(-1)^{\ell}}{2}\right)(8A_{n,\ell}^{(2)})U^{-1}G_{6+2n+\ell,\ell}(z,\bar{z})\;.$
(69)

Similarly, at tree level we use (23), (53), (31) and (33) to find the
following s-channel decompositions of the $\log U$ coefficients

$\left(\begin{array}[]{l}\mathcal{H}^{(1)}_{2222,{\bf 1}}\big{|}_{\log U}\\\
\mathcal{H}^{(1)}_{2222,{\bf 3875}}\big{|}_{\log U}\\\
\mathcal{H}^{(1)}_{2222,{\bf 27000}}\big{|}_{\log
U}\end{array}\right)=(C_{2,2,2})^{2}\psi^{2}h^{\vee}\left(\begin{array}[]{c}-3\\\
-\frac{3}{5}\\\
\frac{1}{10}\end{array}\right)\times\sum_{n,\ell=0}^{\infty}\left(\tfrac{1+(-1)^{\ell}}{2}\right)\omega_{n,\ell}^{(2)}U^{-1}G_{6+2n+\ell,\ell}(z,\bar{z})\;,$
(70) $\left(\begin{array}[]{l}\mathcal{H}^{(1)}_{2222,{\bf 248}}\big{|}_{\log
U}\\\ \mathcal{H}^{(1)}_{2222,{\bf 30380}}\big{|}_{\log
U}\end{array}\right)=(C_{2,2,2})^{2}\psi^{2}h^{\vee}\left(\begin{array}[]{c}\frac{3}{2}\\\
0\end{array}\right)\times\sum_{n,\ell=0}^{\infty}\left(\tfrac{1-(-1)^{\ell}}{2}\right)\omega_{n,\ell}^{(2)}U^{-1}G_{6+2n+\ell,\ell}(z,\bar{z})\;.$
(71)

Note that in the adjoint (${\bf 248}$) channel, there are two types of
contributions. The $\log U$ coefficient receives contributions not only from
the two crossed channels, but also from the s-channel.

Because there is no operator mixing, the $\log^{2}U$ coefficients at one loop
are simply given by squaring the tree-level coefficients and then dividing by
the disconnected coefficients. We find that they can be written as

$\left(\begin{array}[]{l}\mathcal{H}^{(2),{\rm 5d}}_{2222,{\bf
1}}\big{|}_{\log^{2}U}\\\ \mathcal{H}^{(2),{\rm 5d}}_{2222,{\bf
3875}}\big{|}_{\log^{2}U}\\\ \mathcal{H}^{(2),{\rm 5d}}_{2222,{\bf
27000}}\big{|}_{\log^{2}U}\end{array}\right)=(C_{2,2,2})^{4}(\psi^{2}h^{\vee})^{2}\left(\begin{array}[]{c}9\\\
\frac{9}{25}\\\ \frac{1}{100}\end{array}\right)\times\mathcal{F}_{\rm
even}(z,\bar{z})\;,$ (72) $\left(\begin{array}[]{l}\mathcal{H}^{(2),{\rm
5d}}_{2222,{\bf 248}}\big{|}_{\log^{2}U}\\\ \mathcal{H}^{(2),{\rm
5d}}_{2222,{\bf
30380}}\big{|}_{\log^{2}U}\end{array}\right)=(C_{2,2,2})^{4}(\psi^{2}h^{\vee})^{2}\left(\begin{array}[]{c}\frac{9}{4}\\\
0\end{array}\right)\times\mathcal{F}_{\rm odd}(z,\bar{z})\;,$ (73)

where the two basic functions of cross ratios are

$\begin{split}\mathcal{F}_{\rm
even}(z,\bar{z})=&\sum_{n,\ell=0}^{\infty}\frac{1}{2}\frac{(\omega_{n,\ell}^{(2)})^{2}}{8A^{(2)}_{n,\ell}}\left(\tfrac{1+(-1)^{\ell}}{2}\right)U^{-1}G_{6+2n+\ell,\ell}(z,\bar{z})\;,\\\
\mathcal{F}_{\rm
odd}(z,\bar{z})=&\sum_{n,\ell=0}^{\infty}\frac{1}{2}\frac{(\omega_{n,\ell}^{(2)})^{2}}{8A^{(2)}_{n,\ell}}\left(\tfrac{1-(-1)^{\ell}}{2}\right)U^{-1}G_{6+2n+\ell,\ell}(z,\bar{z})\;.\end{split}$
(74)

We now follow the strategy of Alday:2018kkw ; Alday:2019nin  and look for a
Mellin amplitude that reproduces these $\log^{2}U$ coefficients. To achieve
this, we look at the small $V$ expansion of $\mathcal{F}_{\rm even}$ and
$\mathcal{F}_{\rm odd}$. We find that they contain at most $\log^{2}V$
singularities in the $V\to 0$ limit. Such $\log^{2}U\log^{2}V$ singularities
of the four-point correlator require the reduced Mellin amplitudes to have
simultaneous poles of the form121212Recall that the Gamma function factor
already have double poles at these locations. The inverse Mellin integrand
then has in total triple poles, which give rise to $\log^{2}U\log^{2}V$
singularities upon taking residues.

$\frac{1}{(s-2m)(t-2n)}\;,\quad m,n=2,3,4,\ldots\;.$ (75)

We then make the following simple ansatz which is consistent with the symmetry
in the even and odd channels

$\displaystyle\widetilde{\mathcal{M}}_{\rm even}$ $\displaystyle=$
$\displaystyle\sum_{m,n=2}^{\infty}c^{\rm
even}_{mn}\left(\frac{1}{(s-2m)(t-2n)}+\frac{1}{(s-2m)(\tilde{u}-2n)}\right)\;,$
$\displaystyle\widetilde{\mathcal{M}}_{\rm odd}$ $\displaystyle=$
$\displaystyle\sum_{m,n=2}^{\infty}c^{\rm
odd}_{mn}\left(\frac{1}{(s-2m)(t-2n)}-\frac{1}{(s-2m)(\tilde{u}-2n)}\right)\;.$
(76)

Here we have assumed that $c^{\rm even}_{mn}$ and $c^{\rm odd}_{mn}$ are
numbers independent of Mandelstam variables. Taking the residues in the
inverse Mellin transformation, we can extract the $\log^{2}U\log^{2}V$
coefficients as a power expansion in $U$ and $V$. By matching them with the
$\log^{2}V$ coefficients $\mathcal{F}_{\rm even,odd}$ in the small $U$, $V$
expansion, we can obtain the coefficients $c^{\rm even,odd}_{mn}$. We find
they are the same function up to a sign

$c^{\rm odd}_{mn}=-c^{\rm even}_{mn}=-c^{\rm 5d}_{mn}\;,$ (77)

and

$\begin{split}c_{mn}^{\rm
5d}=&\frac{\sqrt{\pi}\Gamma(m-1)\Gamma(n-1)\Gamma\left(m+n-\frac{5}{2}\right)}{18\Gamma\left(m+\frac{1}{2}\right)\Gamma\left(n+\frac{1}{2}\right)\Gamma(m+n-1)}\\\
&\times\left(3m^{2}n-2m^{2}+3mn^{2}-11mn+6m-2n^{2}+6n-3\right)\end{split}$
(78)

is symmetric under $n\leftrightarrow m$. Note that so far we have only used
and matched the $\log^{2}V$ coefficients of $\mathcal{F}_{\rm even,odd}$.
However, it turns out that the amplitudes (4.1) with these coefficients
reproduce the full functions $\mathcal{F}_{\rm even,odd}$, which include $\log
V$ singularities and regular terms as well. This implies that there are no
additional single poles in $s$ in the reduced Mellin amplitude, confirming the
assumption that $c^{\rm even}_{mn}$ and $c^{\rm odd}_{mn}$ are independent of
the Mandelstam variables.

Let us denote the above building block amplitudes as

$\displaystyle\mathcal{B}^{\rm 5d}_{st}=\sum_{m,n=2}^{\infty}\frac{c_{mn}^{\rm
5d}}{(s-2m)(t-2n)}\;,$ $\displaystyle\mathcal{B}^{\rm
5d}_{su}=\sum_{m,n=2}^{\infty}\frac{c_{mn}^{\rm 5d}}{(s-2m)(\tilde{u}-2n)}\;,$
(79) $\displaystyle\mathcal{B}^{\rm
5d}_{tu}=\sum_{m,n=2}^{\infty}\frac{c_{mn}^{\rm 5d}}{(t-2m)(\tilde{u}-2n)}\;,$

which are the $AdS_{5}$ box diagrams. This interpretation of the amplitudes
becomes clear when we examine their behavior in the flat space limit. The flat
space limit corresponds to the high energy regime of the Mellin amplitude
where $s$, $t$ are taken to be large Penedones:2010ue . As we show in Appendix
C, any Mellin amplitude which has the form of $\mathcal{B}^{\rm 5d}_{st}$ and
coefficients with the following scaling behavior

$c^{\rm 5d}_{mn}=\frac{(mn)^{\frac{D}{2}-3}}{(m+n)^{\frac{D}{2}-2}}+\cdots$
(80)

in the large $m$, $n$ limit, becomes the $D$-dimensional scalar one-loop box
diagram in flat space. In the current case the coefficients (78) scale as
$1/(mn(m+n))^{\frac{1}{2}}$ and therefore (4.1) become the 5d box diagrams in
the flat space limit.

Continuing the above discussion, the one-loop reduced amplitude of $\langle
2222\rangle$ should be

$\widetilde{\mathcal{M}}_{2222}^{AdS_{5}}=(C_{2,2,2})^{4}(\psi^{2}h^{\vee})^{2}\left(\begin{array}[]{c}9(\mathcal{B}^{\rm
5d}_{st}+\mathcal{B}^{\rm 5d}_{su}+\lambda_{1}\mathcal{B}^{\rm 5d}_{tu})\\\
\frac{9}{25}(\mathcal{B}^{\rm 5d}_{st}+\mathcal{B}^{\rm
5d}_{su}+\lambda_{2}\mathcal{B}^{\rm 5d}_{tu})\\\
\frac{1}{100}(\mathcal{B}^{\rm 5d}_{st}+\mathcal{B}^{\rm
5d}_{su}+\lambda_{3}\mathcal{B}^{\rm 5d}_{tu})\\\ \frac{9}{4}(\mathcal{B}^{\rm
5d}_{su}-\mathcal{B}^{\rm 5d}_{st})\\\ 0\end{array}\right)\;,$ (81)

in order to be consistent with the $\log^{2}U$ coefficients. Note that in the
symmetric channels, we have also included the box diagram $\mathcal{B}^{\rm
5d}_{tu}$ which is compatible with the symmetry of these channels. This box
diagram does not contribute the $\log^{2}U$ coefficients, and therefore the
parameters $\lambda_{i}$ are not fixed by them. However, the one-loop reduced
amplitude should also be crossing symmetric

$\displaystyle\mathcal{M}_{2222,a}^{AdS_{5}}(s,t)$ $\displaystyle=$
$\displaystyle\sum_{a^{\prime}}({\rm
F}_{t})_{a}{}^{a^{\prime}}\mathcal{M}_{2222,a^{\prime}}^{AdS_{5}}(t,s)\;,$
(82) $\displaystyle\mathcal{M}_{2222,a}^{AdS_{5}}(s,t)$ $\displaystyle=$
$\displaystyle\sum_{a^{\prime}}({\rm
F}_{u})_{a}{}^{a^{\prime}}\mathcal{M}_{2222,a^{\prime}}^{AdS_{5}}(u,t)\;.$
(83)

Imposing these conditions uniquely solves the unknown coefficients, and gives

$\lambda_{1}=\frac{1}{2}\;,\quad\lambda_{2}=-\frac{3}{2}\;,\quad\lambda_{3}=16\;.$
(84)

Let us also make a comment regarding the completeness of the ansatz. As
pointed out earlier, the fact that the reduced Mellin amplitude reproduces the
complete $\log^{2}U$ coefficient rules out single poles in the $s$ variable.
Via the full crossing symmetry, this also rules out single poles in $t$ and
$\tilde{u}$. Therefore, the only ambiguities are regular contact terms.

Written as (81), the color structure of the one-loop amplitude is quite
obscure and the answer appears to sensitively depend on the specific choice of
the gauge group. However, we claim that the amplitude can be rewritten in a
more illuminating form as

$\widetilde{\mathcal{M}}_{2222}^{AdS_{5}}=9(C_{2,2,2})^{4}\big{(}\mathtt{d}_{st}\mathcal{B}_{st}^{\rm
5d}+\mathtt{d}_{su}\mathcal{B}_{su}^{\rm
5d}+\mathtt{d}_{tu}\mathcal{B}_{tu}^{\rm 5d}\big{)}$ (85)

where

$\begin{split}\mathtt{d}_{st}=&f^{JI_{1}K}f^{KI_{2}L}f^{LI_{3}M}f^{MI_{4}J}\;,\\\
\mathtt{d}_{su}=&f^{JI_{1}K}f^{KI_{2}L}f^{LI_{4}M}f^{MI_{3}J}\;,\\\
\mathtt{d}_{tu}=&f^{JI_{1}K}f^{KI_{3}L}f^{LI_{2}M}f^{MI_{4}J}\;,\end{split}$
(86)

are box diagrams for the color part (see Figure 3). Clearly, this form of the
answer can be generalized to any color group.

Figure 3: Box diagrams for colors. Each vertex represents a structure constant
and the lines are in the adjoint representation.

To show that the two representations (81) and (85) are equivalent, we
decompose the color structures $\mathtt{d}_{st}$, $\mathtt{d}_{st}$,
$\mathtt{d}_{tu}$ into s-channel projectors and compare the coefficients in
each channel. This can be done by following the procedure outlined in Figure
4. We first cut open the box diagram vertically, and view it as the
contraction of two t-channel projectors associated with the adjoint
representation. We then use the crossing matrix to express each t-channel
projector as a linear combination of the s-channel projectors. Finally, using
the idempotence of projectors (25) we reduce the color box to be a linear
combination of s-channel projectors, and find

$\mathtt{d}_{st}=(\psi^{2}h^{\vee})^{2}\sum_{a}\big{(}({\rm F}_{t})_{a}{}^{\rm
adj}\big{)}^{2}{\rm P}^{I_{1}I_{2}|I_{3}I_{4}}_{a}\;.$ (87)

The other two color boxes can be obtained by substituting
$I_{3}\leftrightarrow I_{4}$ and $I_{2}\leftrightarrow I_{3}$ respectively and
then applying crossing relations of projectors. Using these relations and the
explicit form of the $E_{8}$ crossing matrices, it is straightforward to
verify that (81) and (85) agree.

Figure 4: Writing the color box function $\mathcal{B}_{st}$ as s-channel
projectors. The box diagram can be obtained as the product of two t-channel
projectors in which the adjoint representations are exchanged. We then use the
crossing matrix to expand each t-channel projector in the s-channel, and
idempotence of projectors to write the function as a linear combination of
s-channel projectors.

###  4.2 One-loop amplitude with general color group

In the previous subsection we worked out the explicit example of $E_{8}$ gauge
group. The answer was then rewritten in a different form (85), which depends
only on the structure constants of the group and indicates a straightforward
generalization to any color group. In this subsection, we prove that the one-
loop reduced amplitude for general color groups is indeed given by (85).

Let us first start with some simple facts about projectors. From (25), (26)
and the definition (23), it follows that

$({\rm F}_{t})_{a}{}^{\bf 1}=\frac{1}{{\rm dim}(G_{F})}$ (88)

for any representation $a$ in the tensor product. Similarly, we also have

$({\rm F}_{u})_{a}{}^{\bf 1}=\frac{(-1)^{{\rm R}_{a}}}{{\rm dim}(G_{F})}\;.$
(89)

Therefore, the contribution of the disconnected correlator is always

$\mathcal{H}^{(0)}_{2222,{\rm long},{\rm
even}}=\sum_{n,\ell=0}^{\infty}\left(\tfrac{1+(-1)^{\ell}}{2}\right)(8A_{n,\ell}^{(2)})U^{-1}G_{6+2n+\ell,\ell}(z,\bar{z})$
(90)

for the even channels, and

$\mathcal{H}^{(0)}_{2222,{\rm long},{\rm
odd}}=\sum_{n,\ell=0}^{\infty}\left(\tfrac{1-(-1)^{\ell}}{2}\right)(8A_{n,\ell}^{(2)})U^{-1}G_{6+2n+\ell,\ell}(z,\bar{z})$
(91)

for the odd channels, regardless of the gauge group $G_{F}$. Note that the
$1/{\rm dim}(G_{F})$ factor is cancelled out due to the appearance of the same
factor in the definition (23).

At tree level, the t- and u-channel exchanges
$\mathcal{H}_{22kk}^{(t)}\big{|}_{\log U}$ and
$\mathcal{H}_{22kk}^{(u)}\big{|}_{\log U}$ together contribute

$(C_{2,2,2})^{2}\psi^{2}h^{\vee}\times(-3({\rm F}_{t})_{a}{}^{\rm
adj})\sum_{n,\ell=0}^{\infty}\left(\tfrac{1+(-1)^{\ell}}{2}\right)\omega_{n,\ell}^{(2)}U^{-1}G_{6+2n+\ell,\ell}(z,\bar{z})\;,$
(92)

in the even channels, and

$(C_{2,2,2})^{2}\psi^{2}h^{\vee}\times({\rm F}_{t})_{a}{}^{\rm
adj}\sum_{n,\ell=0}^{\infty}\left(\tfrac{1-(-1)^{\ell}}{2}\right)\omega_{n,\ell}^{(2)}U^{-1}G_{6+2n+\ell,\ell}(z,\bar{z})\;,$
(93)

in the odd channels. To proceed, we need to compute the $({\rm
F}_{t})_{a}{}^{\rm adj}$ matrix elements for $a$ odd. We look at the Jacobi
identity which is diagrammatically represented by Figure 5. We contract both
sides with the s-channel projector ${\rm P}^{I_{1}I_{2}|I_{3}I_{4}}_{a}$. The
second and third terms respectively give $-({\rm F}_{t})_{a}{}^{\rm adj}$ and
$-({\rm F}_{u})_{a}{}^{\rm adj}$. Because $a$ is odd, the last two terms
contribute equally. On the other hand, due to the delta function in (25) the
first term is only nonzero and equals to one when $a$ is the adjoint
representation. We then find

$({\rm F}_{t})_{\rm adj}{}^{\rm adj}=({\rm F}_{u})_{\rm adj}{}^{\rm
adj}=\frac{1}{2}\;,$ (94)

and

$({\rm F}_{t})_{\rm adj}{}^{\rm a}=({\rm F}_{u})_{\rm adj}{}^{\rm a}=0\;,\quad
a\in{\rm odd}\;,\;\;a\neq{\rm adj}\;.$ (95)

On the other hand, we note that in the adjoint channel there is also another
contribution from the s-channel tree-level exchange
$\mathcal{H}_{22kk}^{(s)}\big{|}_{\log U}$. This allows us to write the total
contribution in the odd channels as

$(C_{2,2,2})^{2}\psi^{2}h^{\vee}\times(3({\rm F}_{t})_{a}{}^{\rm
adj})\sum_{n,\ell=0}^{\infty}\left(\tfrac{1-(-1)^{\ell}}{2}\right)\omega_{n,\ell}^{(2)}U^{-1}G_{6+2n+\ell,\ell}(z,\bar{z})\;,$
(96)

which have the same form as the even channels (92) up to an overall minus
sign.

Figure 5: Jacobi identity in the diagrammatic form. The second term has a
minus sign because vertices are oriented.

Let us now move on to the one-loop level. It follows from the above discussion
that the $\log^{2}U$ coefficients are

$\mathcal{H}^{(2),{\rm
5d}}_{2222,a}\big{|}_{\log^{2}U}=9(C_{2,2,2})^{4}(\psi^{2}h^{\vee})^{2}(({\rm
F}_{t})_{a}{}^{\rm adj})^{2}\times\left\\{\begin{array}[]{c}\mathcal{F}_{\rm
even}\,,\;\;a\in{\rm even}\\\ \mathcal{F}_{\rm odd}\,,\;\;a\in{\rm
odd}\end{array}\right.$ (97)

where $\mathcal{F}_{\rm even,odd}$ were defined in (74). Then from (87) and
the discussion of the Mellin amplitudes of $\mathcal{F}_{\rm even,odd}$, we
know that $\widetilde{\mathcal{M}}_{2222}^{AdS_{5}}$ must include
$9(C_{2,2,2})^{4}(\mathtt{d}_{st}\mathcal{B}_{st}^{\rm
5d}+\mathtt{d}_{su}\mathcal{B}_{su}^{\rm 5d})$ as part of the answer. By
requiring that the one-loop reduced amplitude is crossing symmetric, we arrive
at (85). Note that in the above derivation, we have not used any special
properties that are specific to certain groups. Therefore, (85) is the one-
loop reduced Mellin amplitude for any gauge group.

##  5 One-loop gluon amplitude on AdS5$\times$S3

Let us now compute the super gluon one-loop amplitudes in the theories of
D3-branes probing F-theory singularities. Compared with the $AdS_{5}$ toy
model studied in the previous section, here the space is $AdS_{5}\times
S^{3}$. The $S^{3}$ factor gives rise to infinitely many additional massive
super gluons from Kaluza-Klein reduction, which must be summed over in the
one-loop amplitude. The existence of these massive modes leads to a degeneracy
in double-trace operators, and presents additional complexity in using the AdS
unitarity method. We will first review how to solve this mixing problem, and
then compute the one-loop reduced Mellin amplitude for an arbitrary gauge
group $G_{F}$. A remarkable feature of our result is that it admits a closed
form expression, unlike the case without $S^{3}$. We will also show that the
answer reproduces the 8d box diagrams in the flat space limit, which agrees
with the fact that our background is now eight dimensional.

###  5.1 Mixing problem

For a fixed engineering conformal twist $\tau=4+2n$ and Lorentz spin $\ell$,
the space of $SU(2)_{R}\times SU(2)_{L}$ invariant double-trace operators in
each flavor channel is $(n+1)$-dimensional. Since operators with different
flavor representations do not mix, we will suppress the flavor indices in this
subsection to simplify the notation. Schematically, these operators can be
written as

$:\mathcal{O}_{2}\square^{n}\partial^{\ell}\mathcal{O}_{2}:\;,\;\;:\mathcal{O}_{3}\square^{n-1}\partial^{\ell}\mathcal{O}_{3}:\;,\;\;\ldots\;\;:\mathcal{O}_{n+2}\partial^{\ell}\mathcal{O}_{n+2}:$
(98)

where we have also suppressed the $SU(2)_{R}\times SU(2)_{L}$ indices. For
computing the $\langle 2222\rangle$ one-loop amplitude, only the singlets are
relevant. Turning on interactions lifts the degeneracies, and we will denote
the eigenstates $\mathbb{O}_{n,\ell,i}$.

At the disconnected level, these double-trace operators contribute to the
reduced correlator $\langle kkkk\rangle$ as

$\mathcal{H}^{(0)}_{kkkk,{\rm
long}}=\sum_{n=0}^{\infty}\sum_{\ell}a_{n,\ell}^{(k)}U^{-1}G_{2k+2n+2+\ell,\ell}(z,\bar{z})$
(99)

where $a_{n,\ell}^{(k)}$ are related to three-point function coefficients by

$a_{n,\ell}^{(k)}=\sum_{i=1}^{k+n-1}C^{(0)}_{kk\mathbb{O}_{k+n-2,\ell,i}}C^{(0)}_{kk\mathbb{O}_{k+n-2,\ell,i}}\;.$
(100)

We remind the reader that there is a shift in dimension by 2 in the conformal
blocks according to (40) because we are looking at the reduced correlators. At
tree level, we will focus on the $\log U$ coefficients of the reduced
correlator of $\langle 22kk\rangle$. They can be decomposed into long operator
conformal blocks as

$\mathcal{H}^{(1)}_{22kk}\big{|}_{\log
U}=\sum_{n=0}^{\infty}\sum_{\ell}\omega^{(k)}_{n,\ell}U^{-1}G_{2k+2n+2+\ell,\ell}(z,\bar{z})$
(101)

where $\omega^{(k)}_{n,\ell}$ is given by

$\omega^{(k)}_{n,\ell}=\sum_{i=1}^{k+n-1}C^{(0)}_{22\mathbb{O}_{k+n-2,\ell,i}}\gamma^{(1)}_{k+n-2,\ell,i}C^{(0)}_{kk\mathbb{O}_{k+n-2,\ell,i}}$
(102)

with $\gamma^{(1)}_{k+n-2,\ell,i}$ being the tree-level anomalous dimension of
$\mathbb{O}_{k+n-2,\ell,i}$.

Let us now look at the $\langle 2222\rangle$ correlator at one loop. The key
object is the $\log^{2}U$ coefficient of its reduced correlator, which is
computed by

$\mathcal{H}^{(2)}_{2222}\big{|}_{\log^{2}U}=\sum_{n=0}^{\infty}\sum_{\ell}F_{n,\ell}U^{-1}G_{6+2n+\ell,\ell}(z,\bar{z})\;,$
(103)

with

$F_{n,\ell}=\frac{1}{2}\sum_{i=1}^{n+1}C^{(0)}_{22\mathbb{O}_{n,\ell,i}}\big{(}\gamma^{(1)}_{n,\ell,i}\big{)}^{2}C^{(0)}_{22\mathbb{O}_{n,\ell,i}}\;.$
(104)

Due to operator mixing, we cannot compute $F_{n,\ell}$ using only the $\langle
2222\rangle$ correlator at disconnected and tree levels. However, this
quantity can still be expressed in terms of the lower-order data extracted
from $\mathcal{H}^{(0)}_{kkkk,{\rm long}}$ and
$\mathcal{H}^{(1)}_{22kk}\big{|}_{\log U}$ if we consider arbitrary $k$. To
see this, let us rewrite the above results in matrix notation. We first define
a $(q+1)$ by $(q+1)$ matrix

$\mathbf{C}_{q,\ell}=\left(\begin{array}[]{c}C^{(0)}_{22\mathbb{O}_{q,\ell,i}}\\\
\ldots\\\ C^{(0)}_{q+2,q+2\mathbb{O}_{q,\ell,i}}\end{array}\right)\;,$ (105)

which involves the double-trace operators with conformal twist $2q+4$. Then
$\mathbf{C}\mathbf{C}^{T}$ gives the long multiplet coefficients of the
following matrix of disconnected correlators

$\mathbf{C}_{q,\ell}\big{(}\mathbf{C}_{q,\ell}\big{)}^{T}=\left(\begin{array}[]{ccc}\langle
2222\rangle^{(0)}&\ldots&\langle 2,2,q+2,q+2\rangle^{(0)}\\\
\ldots&\ldots&\ldots\\\ \langle q+2,q+2,2,2\rangle^{(0)}&\ldots&\langle
q+2,q+2,q+2,q+2\rangle^{(0)}\end{array}\right)_{\text{long coef.}}$ (106)

Note that due to large $N$ factorization, $\langle rrss\rangle^{(0)}$ contains
no long operators unless $r=s$. Therefore, the matrix
$\mathbf{C}\mathbf{C}^{T}$ is diagonal, and we will denote it as

$\mathbf{C}_{q,\ell}\big{(}\mathbf{C}_{q,\ell}\big{)}^{T}=\mathbf{N}_{q,\ell}\;.$
(107)

Moreover, we can collect the $q+1$ eigenvalues of tree-level anomalous
dimensions into a diagonal matrix

$\mathbf{\Gamma}_{q,\ell}={\rm
diag}(\gamma^{(1)}_{q,\ell,1},\ldots\gamma^{(1)}_{q,\ell,q+1})\;.$ (108)

We can then consider the matrix

$\mathbf{C}_{q,\ell}\mathbf{\Gamma}_{q,\ell}\big{(}\mathbf{C}_{q,\ell}\big{)}^{T}=\left(\begin{array}[]{ccc}\langle
2222\rangle^{(1)}&\ldots&\langle 2,2,q+2,q+2\rangle^{(1)}\\\
\ldots&\ldots&\ldots\\\ \langle q+2,q+2,2,2\rangle^{(1)}&\ldots&\langle
q+2,q+2,q+2,q+2\rangle^{(1)}\end{array}\right)_{\log U\text{ coef.}}$ (109)

The first row (or the first column) are $\omega^{(k)}_{q-k+2,\ell}$ with
$k=2,3,\ldots q+2$, which appear in the tree-level $\langle 22kk\rangle$
correlators. Furthermore, it is easy to see that quantity (104) corresponds to
the matrix elements

$F_{n,\ell}=\frac{1}{2}\bigg{(}\mathbf{C}_{n,\ell}\mathbf{\Gamma}_{n,\ell}\big{(}\mathbf{C}_{n,\ell}\big{)}^{T}\bigg{)}\mathbf{N}_{n,\ell}^{-1}\bigg{(}\mathbf{C}_{n,\ell}\mathbf{\Gamma}_{n,\ell}\big{(}\mathbf{C}_{n,\ell}\big{)}^{T}\bigg{)}\;.$
(110)

Since $\mathbf{N}$ is diagonal, it follows that this matrix element can be
expressed as

$F_{n,\ell}=\frac{1}{2}\sum_{k=2}^{n+2}\frac{\big{(}\omega^{(k)}_{n-k+2,\ell}\big{)}^{2}}{a^{(k)}_{n-k+2,\ell}}\;.$
(111)

This gives the correct answer for the average of the squared tree-level
anomalous dimensions, with operator mixing properly taken into account.

###  5.2 One-loop amplitude

Because the color structure is independent of mixing, the discussion in
Section 4.2 directly applies here. The $\log^{2}U$ coefficients of
$AdS_{5}\times S^{3}$ four-point correlator take the same form as in the
$AdS_{5}$ case (97)

$\mathcal{H}^{(2),{\rm
8d}}_{2222,a}\big{|}_{\log^{2}U}=9(C_{2,2,2})^{2}(\psi^{2}h^{\vee})^{2}(({\rm
F}_{t})_{a}{}^{\rm adj})^{2}\times\left\\{\begin{array}[]{c}\mathcal{F}_{\rm
even}^{\rm 8d}\,,\;\;a\in{\rm even}\\\ \mathcal{F}_{\rm odd}^{\rm
8d}\,,\;\;a\in{\rm odd}\end{array}\right.\;.$ (112)

The only difference is that $\mathcal{F}_{\rm even,odd}$ are now replaced by

$\begin{split}\mathcal{F}^{\rm 8d}_{\rm
even}(z,\bar{z})=&\sum_{n,\ell=0}^{\infty}F_{n,\ell}\left(\tfrac{1+(-1)^{\ell}}{2}\right)U^{-1}G_{6+2n+\ell,\ell}(z,\bar{z})\;,\\\
\mathcal{F}^{\rm 8d}_{\rm
odd}(z,\bar{z})=&\sum_{n,\ell=0}^{\infty}F_{n,\ell}\left(\tfrac{1-(-1)^{\ell}}{2}\right)U^{-1}G_{6+2n+\ell,\ell}(z,\bar{z})\;.\end{split}$
(113)

where $F_{n,\ell}$ is given by (111) as

$F_{n,\ell}=\sum_{k=2}^{n+2}E_{k,n,\ell}\;,$ (114)

with

$\begin{split}E_{k,n,\ell}=\frac{1}{2}\times\frac{\left(\omega^{(k)}_{n+2-k,\ell}\right)^{2}}{8A^{(k)}_{n+2-k,\ell}}=&\frac{\pi(k-1)^{2}(n+1)2^{-4n-\ell-6}(n+\ell+2)}{9(\ell+1)(2n+\ell+4)\Gamma\left(n+\frac{3}{2}\right)}\\\
\times&\frac{\Gamma(n+3)\Gamma(k+n+1)\Gamma(n+\ell+4)\Gamma(-k+n+\ell+4)}{\Gamma(-k+n+3)\Gamma\left(n+\ell+\frac{5}{2}\right)\Gamma(k+n+\ell+2)}\;.\end{split}$
(115)

Following the same procedure, we find that the $\log^{2}V$ coefficients of
$\mathcal{F}^{\rm 8d}_{\rm even}$ and $\mathcal{F}^{\rm 8d}_{\rm odd}$ can be
respectively matched by the reduced Mellin amplitudes

$\widetilde{\mathcal{M}}^{\rm 8d}_{\rm even}=\mathcal{B}^{\rm
8d}_{st}+\mathcal{B}^{\rm 8d}_{su}\;,\quad\quad\widetilde{\mathcal{M}}^{\rm
8d}_{\rm odd}=\mathcal{B}^{\rm 8d}_{su}-\mathcal{B}^{\rm 8d}_{st}\;.$ (116)

Here we have defined the eight dimensional $AdS^{5}\times S^{3}$ box diagrams
as

$\displaystyle\mathcal{B}^{\rm 8d}_{st}=\sum_{m,n=2}^{\infty}\frac{c_{mn}^{\rm
8d}}{(s-2m)(t-2n)}\;,$ $\displaystyle\mathcal{B}^{\rm
8d}_{su}=\sum_{m,n=2}^{\infty}\frac{c_{mn}^{\rm 8d}}{(s-2m)(\tilde{u}-2n)}\;,$
(117) $\displaystyle\mathcal{B}^{\rm
8d}_{tu}=\sum_{m,n=2}^{\infty}\frac{c_{mn}^{\rm 8d}}{(t-2m)(\tilde{u}-2n)}\;,$

with the coefficients given by

$c_{mn}^{\rm
8d}=\frac{4\left(3m^{2}n-4m^{2}+3mn^{2}-16mn+15m-4n^{2}+15n-12\right)}{27(m+n-4)(m+n-3)(m+n-2)}\;.$
(118)

Using this solution we can check that the functions $\mathcal{F}^{\rm 8d}_{\rm
even}$ and $\mathcal{F}^{\rm 8d}_{\rm odd}$ are completely reproduced by the
amplitudes (116). This implies that the simultaneous poles are again
sufficient. It follows from the discussions in Section 4.2 that the one-loop
reduced Mellin amplitudes for $AdS_{5}\times S^{3}$ are

$\widetilde{\mathcal{M}}_{2222}^{AdS_{5}\times
S^{3}}=9(C_{2,2,2})^{4}\big{(}\mathtt{d}_{st}\mathcal{B}_{st}^{\rm
8d}+\mathtt{d}_{su}\mathcal{B}_{su}^{\rm
8d}+\mathtt{d}_{tu}\mathcal{B}_{tu}^{\rm 8d}\big{)}\;.$ (119)

We also note that the coefficients $c_{mn}^{\rm 8d}$ scale as

$c_{mn}^{\rm 8d}\sim\frac{mn}{(m+n)^{2}}+\ldots$ (120)

in the large $m$, $n$ limit. According to the results in Appendix C, the
Mellin amplitudes (5.2) become eight-dimensional box integrals in the flat
space limit. There is actually a more explicit way to see this. It turns out
that the $AdS_{5}\times S^{3}$ box diagrams can be computed in a closed form.
The basic building block is the following double sum

$\Theta(s,t)=\sum_{m,n=1}\frac{1}{(m+n)(s-2m)(t-2n)}$ (121)

which can be performed in terms of polygamma functions

$\displaystyle\Theta(s,t)=$
$\displaystyle\frac{1}{4(s+t)}\bigg{\\{}\psi^{(1)}\left(-\frac{s}{2}\right)+\psi^{(1)}\left(-\frac{t}{2}\right)-\left[\psi^{(0)}\left(-\frac{s}{2}\right)\right]^{2}$
(122)
$\displaystyle\quad\quad\quad\quad\quad-\left[\psi^{(0)}\left(-\frac{t}{2}\right)\right]^{2}+2\psi^{(0)}\left(-\frac{s}{2}\right)\psi^{(0)}\left(-\frac{t}{2}\right)\bigg{\\}}$
$\displaystyle+\frac{2}{st(s+t)}-\frac{\pi^{2}}{4(s+t)}-\frac{\psi^{(0)}\left(-\frac{s}{2}\right)+\gamma}{t(s+t)}-\frac{\psi^{(0)}\left(-\frac{t}{2}\right)+\gamma}{s(s+t)}\;.$

Here $\gamma$ is the Euler-Mascheroni constant. Mellin amplitudes in various
examples, including the ones in this paper, can be written in terms of this
building block, upon shifting the $s$, $t$ variables, plus edge terms
involving a single sum. For the case at hand the double sum is divergent, but
it can be regularized in different ways. The difference between different
regularizations is simply a constant. This can be seen as follows. Both
$\partial_{s}\mathcal{B}^{\rm 8d}_{st}$ and $\partial_{t}\mathcal{B}^{\rm
8d}_{st}$ are given by convergent sums, and they lead to $\mathcal{B}^{\rm
8d}_{st}$ up to a constant of integration which depends on a single variable.
This together with symmetry under $s\leftrightarrow t$ implies the constant of
integration is actually independent of both $s$ and $t$. The final result is
given by

$\displaystyle\mathcal{B}^{\rm 8d}_{st}=$ $\displaystyle
R_{0}(s,t)\left(\psi^{(1)}\left(2-\frac{s}{2}\right)+\psi^{(1)}\left(2-\frac{t}{2}\right)-\left(\psi^{(0)}\left(2-\frac{s}{2}\right)-\psi^{(0)}\left(2-\frac{t}{2}\right)\right)^{2}\right)$
(123)
$\displaystyle+R_{1}(s,t)\psi^{(0)}\left(2-\frac{s}{2}\right)+R_{1}(t,s)\psi^{(0)}\left(2-\frac{t}{2}\right)+\pi^{2}R_{2}(s,t)-\frac{32}{27(s+t-8)}+a$

where different choices of regularization would lead to different values of
$a$ and the rational functions are given by

$\displaystyle R_{0}(s,t)$ $\displaystyle=$
$\displaystyle-\frac{4\left(3s^{2}t-8s^{2}+3st^{2}-32st+60s-8t^{2}+60t-96\right)}{27(s+t-8)(s+t-6)(s+t-4)}\;,$
(124) $\displaystyle R_{1}(s,t)$ $\displaystyle=$
$\displaystyle\frac{8\left(3s^{2}+3st-26s-10t+48\right)}{27(s+t-8)(s+t-4)}\;,$
(125) $\displaystyle R_{2}(s,t)$ $\displaystyle=$
$\displaystyle\frac{4\left(3s^{2}t-8s^{2}+3st^{2}-32st+60s-8t^{2}+60t-96\right)}{27(s+t-8)(s+t-6)(s+t-4)}\;.$
(126)

Given this explicit expression it is straightforward to compute the flat space
limit where $-s$, $-t$ are large. We obtain

$\mathcal{B}^{\rm
8d}_{st}\sim\frac{4st\log^{2}\left(\frac{-s}{-t}\right)}{9(s+t)^{2}}+\frac{8(s\log(-s)+t\log(-t))}{9(s+t)}+\frac{4\pi^{2}st}{9(s+t)^{2}}+\beta\;,$
(127)

for some constant $\beta$ which depends on $a$. This precisely agrees with the
eight dimensional box function in flat space! Note that also in this case the
computation involves the introduction of a regulator, and we have a
corresponding ambiguity. The method presented here can also be used to resum
the $AdS_{5}\times S^{5}$ one-loop reduced Mellin amplitudes obtained in
Alday:2018kkw ; Alday:2019nin .131313Due to the different form of the $c_{mn}$
coefficients, this strategy does not immediately apply to the 5d amplitude
(85) with just the $AdS_{5}$ factor. However, we expect similar ambiguity
structure which contains contact terms affecting only the CFT data of double-
trace operators with low-lying spins.

##  6 Coupling to gravity

In Section 5, we considered the gluon one-loop contribution to the four-point
function $\langle 2222\rangle$ which is of order $\mathcal{O}(N^{-2})$.
However, at the same order there is another independent contribution arising
from the tree-level exchange of the massless supergravity multiplet. Such
contributions to super gluon four-point functions were considered for
$AdS_{6}$ and $AdS_{7}$ theories in Zhou:2018ofp ,141414However, in these
theories the super graviton exchange contributions are not of the same order
as the super gluon one-loop contributions. and the correlators were shown to
be fixed by the superconformal Ward identities up to an overall normalization
determined by the stress tensor central charge. Here we similarly compute the
super graviton exchange amplitude for $AdS_{5}$.

The exchanged supergravity multiplet includes an $SU(2)_{R}$ singlet scalar
super primary field of dimension 2, a vector field of dimension 3 and
$SU(2)_{R}$ spin 1, and an $SU(2)_{R}$ neutral graviton field of dimension
4.151515Note that supergravity multiplets of higher Kaluza-Klein levels are
not exchanged in this correlator. We can compute the amplitude due to super
graviton multiplet exchange using the position space method developed in
Rastelli:2016nze ; Rastelli:2017udc . We write down an ansatz for the four-
point function as the linear combination of all exchange diagrams and contact
diagrams with no more than two derivatives

$G_{2222,{\rm
gravity}}^{I_{1}I_{2}I_{3}I_{4}}=\delta^{I_{1}I_{2}}\delta^{I_{3}I_{4}}G_{s}+\delta^{I_{1}I_{4}}\delta^{I_{2}I_{3}}G_{t}+\delta^{I_{1}I_{3}}\delta^{I_{2}I_{4}}G_{u}+G_{con}^{I_{1}I_{2}I_{3}I_{4}}\;.$
(128)

In this ansatz, $G_{s}$ is the s-channel exchange contribution

$G_{s}=\frac{(v_{1}\cdot v_{2})^{2}(v_{3}\cdot
v_{4})^{2}}{x_{12}^{4}x_{34}^{4}}\left(\lambda_{s}\mathcal{W}_{2,0}+\lambda_{v}(\alpha-\frac{1}{2})\mathcal{W}_{3,1}+\lambda_{g}\mathcal{W}_{4,2}\right)\;,$
(129)

where $(\alpha-\frac{1}{2})$ is an $SU(2)_{R}$ polynomial capturing the
exchange contribution of the spin-1 contribution, and $\lambda_{s}$,
$\lambda_{v}$, $\lambda_{g}$ are unknown parameters. The exchange Witten
diagrams $\mathcal{W}_{\Delta,\ell}$ can be expressed as a finite sum of the
$D$-functions, and formulae for computing such diagrams can be found in
Appendix A of Rastelli:2017udc . $G_{t}$ and $G_{u}$ are related to $G_{s}$ by
crossing symmetry. We have assumed the contact part
$G_{con}^{I_{1}I_{2}I_{3}I_{4}}$ contains all R-symmetry and color structures,
i.e., it is a degree-2 polynomial in $\alpha$ after extracting the factor
$(v_{1}\cdot v_{2})^{2}(v_{3}\cdot v_{4})^{2}$ and contributes to all color
channels. Furthermore, we assume that the contact term contains at most two
derivatives, so that the correlator has the correct behavior in the flat space
limit. This means we have only $D_{2,2,2,2}$, and $x_{12}^{2}D_{3,3,2,2}$ with
its crossing images. Imposing the superconformal Ward identities (14), we can
fix all coefficients in the ansatz up to an overall
constant.161616Equivalently, one could also write down the ansatz in Mellin
space, and then solve the ansatz by imposing the Mellin space superconformal
Ward identities Zhou:2017zaw ; Alday:2020dtb . The answer is given by

$G_{2222,{\rm gravity}}^{I_{1}I_{2}I_{3}I_{4}}={\rm R}\,H_{2222,{\rm
gravity}}^{I_{1}I_{2}I_{3}I_{4}}$ (130)

where

$H_{2222,{\rm
gravity}}^{I_{1}I_{2}I_{3}I_{4}}=\frac{24(C_{2,2,g})^{2}}{\pi^{2}}\bigg{(}\delta^{I_{1}I_{2}}\delta^{I_{3}I_{4}}\frac{D_{2,2,3,3}}{x_{12}^{2}}+\delta^{I_{1}I_{4}}\delta^{I_{2}I_{3}}\frac{D_{2,3,3,2}}{x_{14}^{2}}+\delta^{I_{1}I_{3}}\delta^{I_{2}I_{4}}\frac{D_{2,3,2,3}}{x_{13}^{2}}\bigg{)}\;.$
(131)

The coefficient $C_{2,2,g}$ appears in the three-point function two
$\mathcal{O}_{2}$ and the super primary of the massless supergravity multiplet
as

$\langle\mathcal{O}_{2}(x_{1},v_{1})\mathcal{O}_{2}(x_{2},v_{2})\mathcal{O}_{g}(x_{3})\rangle=C_{2,2,g}\frac{(v_{1}\cdot
v_{2})^{2}}{x_{12}^{2}x_{13}^{2}x_{23}^{2}}\;,$ (132)

and is related to the stress tensor central charge by

$(C_{2,2,g})^{2}=\frac{80}{3C_{\mathcal{T}}}\;.$ (133)

Here the central charge appears in the stress tensor two-point function as

$\langle\mathcal{T}_{\mu\nu}(x)\mathcal{T}_{\rho\sigma}(0)\rangle=\frac{C_{\mathcal{T}}}{4\pi^{2}x^{8}}\left(I_{\mu\sigma}I_{\nu\rho}+I_{\mu\rho}I_{\nu\sigma}-\frac{1}{2}\delta_{\mu\nu}\delta_{\sigma\rho}\right)$
(134)

where $I_{\mu\nu}=\delta_{\mu\nu}-2\frac{x_{\mu}x_{\nu}}{x^{2}}$. In the
theories of D3 probing F-theory singularities, the central charge is
Aharony:2007dj

$C_{\mathcal{T}}=\frac{35N^{2}}{2\pi^{2}(2-\nu)}+\mathcal{O}(N)$ (135)

where $\nu$ characterizes the singularity type (see foonote 9). Note an
interesting feature of (130) is that it does not contain a protected part.
Under the twist $\alpha=1/z$ or $\alpha=1/\bar{z}$, the four-point correlator
vanishes.

Let us also express this correlator in Mellin space. The Mellin amplitude of
the four-point function is

$\mathcal{M}_{2222,{\rm
gravity}}^{I_{1}I_{2}I_{3}I_{4}}=\delta^{I_{1}I_{2}}\delta^{I_{3}I_{4}}\mathcal{M}_{s}+\delta^{I_{1}I_{4}}\delta^{I_{2}I_{3}}\mathcal{M}_{t}+\delta^{I_{1}I_{3}}\delta^{I_{2}I_{3}}\mathcal{M}_{u}$
(136)

where

$\mathcal{M}_{s}=-3(C_{2,2,g})^{2}(v_{1}\cdot v_{2})^{2}(v_{3}\cdot
v_{4})^{2}\bigg{(}\frac{(u-4)^{2}+2s(u-1)\alpha-8\alpha}{s-2}+(s-4)\alpha^{2}\bigg{)}\;,$
(137)

and $\mathcal{M}_{t}$, $\mathcal{M}_{u}$ can be obtained by using crossing
symmetry. This Mellin amplitude can be further expressed in terms of a compact
reduced Mellin amplitude defined from $H$

$\widetilde{\mathcal{M}}_{2222,{\rm
gravity}}^{I_{1}I_{2}I_{3}I_{4}}=-12(C_{2,2,g})^{2}\bigg{(}\frac{\delta^{I_{1}I_{2}}\delta^{I_{3}I_{4}}}{s-2}+\frac{\delta^{I_{1}I_{4}}\delta^{I_{2}I_{3}}}{t-2}+\frac{\delta^{I_{1}I_{3}}\delta^{I_{2}I_{4}}}{\tilde{u}-2}\bigg{)}$
(138)

where we recall $\tilde{u}=u-2=6-s-t$.

##  7 Discussion and outlook

In this paper, we initiated the study of AdS gluon amplitudes at loop levels.
Our main results are the simple Mellin space formulae (85) and (119) for four-
point one-loop amplitudes of SYM on $AdS_{5}$ and $AdS_{5}\times S^{3}$. For
the case of $AdS_{5}\times S^{3}$, we performed the sums in the amplitude and
obtained a closed form expression, see (123). These results are reminiscent of
the flat space amplitudes, with logarithms replaced by polygamma functions,
and reproduce the latter in the large AdS radius limit. Our work leads to a
number of natural research avenues for the future. We list some of them below.

  * •

In this paper, we focused on the simplest one-loop amplitude with the lowest
Kaluza-Klein level $k_{i}=2$. To further explore the loop-level dynamics, we
should also study more general amplitudes with higher Kaluza-Klein levels.
Computing these amplitudes requires a thorough analysis of the mixing problem
at the tree level which is so far missing in the literature.

  * •

Relatedly, the tree-level super gluon amplitudes on $AdS_{5}\times S^{3}$ was
shown to display a hidden eight dimensional conformal symmetry Alday:2021odx .
It would be interesting to explore how this structure can help to organize the
loop-level amplitudes.

  * •

Recently, a double-copy-like relation was found in Zhou:2021gnu  which relates
all tree-level four-point amplitudes of $AdS_{5}\times S^{5}$ IIB
supergravity, $AdS_{5}\times S^{3}$ SYM, and bi-adjoint scalars on
$AdS_{5}\times S^{1}$. It would be very interesting to see if the tree-level
relation can be extended to the loop level as well.

  * •

For super gluon four-point functions at $\mathcal{O}(N^{-2})$ the only loop
contribution is the one which we computed in this paper with gluons running in
the loop. However, at $\mathcal{O}(N^{-4})$ we can also have one-loop box
diagrams where two internal legs are gluons and the other two legs are
gravitons. Since the gluons are restricted to the eight dimensional subspace
while the gravitons can propagate in the full ten dimensional space, it will
be particularly interesting to examine the flat space limit of this amplitude.
Presumably, it should match a ten dimensional flat space one-loop box integral
where the two of the four internal propagators are confined to a codimension-2
subspace.

  * •

It would also be interesting to extend the one-loop analysis to other
backgrounds which have AdS factors other than $AdS_{5}$, and to explore the
structure of one-loop amplitudes across different spacetime dimensions.
Similar supergravity one-loop four-point amplitudes have been computed for
eleven dimensional supergravity on $AdS_{7}\times S^{4}$ Alday:2020tgi  and
$AdS_{4}\times S^{7}$ Alday:2021ymb .

  * •

We can also apply the AdS unitarity method to compute all-loop contributions
that correspond to the iterated s-cuts in flat space, as has been done in the
supergravity case Bissi:2020wtv ; Bissi:2020woe . However, to fully determine
these higher-loop correlators knowing just tree-level four-point functions is
no longer sufficient. We would also need the information of multi-trace
operators which are encoded in higher-point correlators.

###### Acknowledgements.

The work of L.F.A. is supported by funding from the European Research Council
(ERC) under the European Union’s Horizon 2020 research and innovation
programme (grant agreement No 787185), and in part by the STFC grant
ST/T000864/1. The work of A.B is supported by Knut and Alice Wallenberg
Foundation under grant KAW 2016.0129 and by VR grant 2018-04438. The work of
X.Z. in Princeton was supported in part by Simons Foundation Grant No. 488653.
The work of X.Z. in China is supported by funds from University of Chinese
Academy of Sciences (UCAS) and from the Kavli Institute for Theoretical
Sciences (KITS).

##  Appendix A $\mathcal{N}=2$ superconformal blocks

For reader’s convenience, we record below the expressions for superconformal
blocks which were obtained in Nirschl:2004pa .

Long multiplet $\mathcal{A}^{\Delta}_{R,(\frac{\ell}{2},\frac{\ell}{2})}$:

$S^{\mathcal{A}}_{\Delta,R,\ell}=(y-x)(y-\bar{x})P_{R}(y)G_{\Delta+2,\ell}$
(139)

where $P_{R}(y)$ is the Legendre polynomial.

Semi-short multiplet $\hat{\mathcal{C}}_{R,(\frac{\ell}{2},\frac{\ell}{2})}$:

$\displaystyle S^{\mathcal{C}}_{R,\ell}=$ $\displaystyle
P_{R}(y)\big{(}G_{2R+\ell+2,\ell}+\tfrac{1}{4}a_{R}G_{2R+\ell+4,\ell}+a_{R+\ell+2}G_{2R+\ell+4,\ell+2}\big{)}$
(140)
$\displaystyle+P_{R-1}(y)\frac{R}{2R+1}\big{(}G_{2R+\ell+3,\ell+1}+\tfrac{1}{4}G_{2R+\ell+3,\ell-1}+\tfrac{1}{4}a_{R+\ell+2}G_{2R+\ell+5,\ell+1}\big{)}$
$\displaystyle+P_{R-2}(y)\frac{(R-1)R}{4(2R-1)(2R+1)}G_{2R+\ell+4,\ell}+P_{R+1}(y)\frac{R+1}{2R+1}G_{2R+\ell+3,\ell+1}$

where

$a_{R}=\frac{R^{2}}{(2R-1)(2R+1)}\;.$ (141)

Short multiplet $\hat{\mathcal{B}}_{R}$:

$S^{\mathcal{B}}_{R}=P_{R}(y)G_{2R,0}+\frac{R}{2R+1}P_{R-1}(y)G_{2R+1,1}+\frac{(R-1)R}{4(2R-1)(2R+1)}P_{R-2}(y)G_{2R+2,0}\;.$
(142)

Using these expressions, it is straightforward to extract the single variable
function $f$ and the two variable function $\mathcal{K}$ in the representation
(36). In the singlet channel, we find (40), (41) and (42).

##  Appendix B Properties of $D$-functions

In this appendix, we collect a few useful formulae of the $D$-functions. The
$D$-functions are defined as a four-point contact Witten diagram in
$AdS_{d+1}$ with no derivatives

$D_{\Delta_{1},\Delta_{2},\Delta_{3},\Delta_{4}}(x_{i})=\int\frac{d^{d}zdz_{0}}{z_{0}^{d+1}}\prod_{i=1}^{4}G^{\Delta_{i}}_{B\partial}(z,x_{i})\;,\quad
G^{\Delta_{i}}_{B\partial}(z,x_{i})=\left(\frac{z_{0}}{z_{0}^{2}+(\vec{z}-\vec{x}_{i})^{2}}\right)^{\Delta_{i}}$
(143)

Contact diagrams with derivatives can also be expressed as $D$-functions with
shifted weights by using the identity

$\nabla^{\mu}G^{\Delta_{1}}_{B\partial}\nabla_{\mu}G^{\Delta_{2}}_{B\partial}=\Delta_{1}\Delta_{2}(G^{\Delta_{1}}_{B\partial}G^{\Delta_{2}}_{B\partial}-2x_{12}^{2}G^{\Delta_{1}+1}_{B\partial}G^{\Delta_{2}+1}_{B\partial})\;.$
(144)

It is convenient to write the $D$-functions as functions of cross ratios by
extracting a kinematic factor

$\frac{\prod_{i=1}^{4}\Gamma(\Delta_{i})}{\Gamma(\frac{1}{2}\Sigma_{\Delta}-\frac{1}{2}d)}\frac{2}{\pi^{\frac{d}{2}}}D_{\Delta_{1},\Delta_{2},\Delta_{3},\Delta_{4}}(x_{i})=\frac{(x_{14}^{2})^{\frac{1}{2}\Sigma_{\Delta}-\Delta_{1}-\Delta_{4}}(x^{2}_{34})^{\frac{1}{2}\Sigma_{\Delta}-\Delta_{3}-\Delta_{4}}}{(x^{2}_{13})^{\frac{1}{2}\Sigma_{\Delta}-\Delta_{4}}(x^{2}_{24})^{\Delta_{2}}}\bar{D}_{\Delta_{1},\Delta_{2},\Delta_{3},\Delta_{4}}(U,V)\,,$
(145)

where $\Sigma_{\Delta}=\sum_{i=1}^{4}\Delta_{i}$. The $\bar{D}$-functions are
independent of the spacetime dimension.

The $D$-functions satisfy the derivative relations such as

$D_{\Delta_{1},\Delta_{2}+1\Delta_{3}+1,\Delta_{4}}(x_{i})=\frac{d-\Sigma_{\Delta}}{2\Delta_{2}\Delta_{3}}\frac{\partial}{\partial
x_{23}^{2}}D_{\Delta_{1},\Delta_{2},\Delta_{3},\Delta_{4}}(x_{i})\;.$ (146)

We can rewrite the relations in terms of $\bar{D}$-functions and get

$\begin{split}\bar{D}_{\Delta_{1}+1,\Delta_{2}+1,\Delta_{3},\Delta_{4}}={}&-\partial_{U}\bar{D}_{\Delta_{1},\Delta_{2},\Delta_{3},\Delta_{4}}\;,\\\
\bar{D}_{\Delta_{1},\Delta_{2},\Delta_{3}+1,\Delta_{4}+1}={}&(\Delta_{3}+\Delta_{4}-\tfrac{1}{2}\Sigma_{\Delta}-U\partial_{U})\bar{D}_{\Delta_{1},\Delta_{2},\Delta_{3},\Delta_{4}}\;,\\\
\bar{D}_{\Delta_{1},\Delta_{2}+1,\Delta_{3}+1,\Delta_{4}}={}&-\partial_{V}\bar{D}_{\Delta_{1},\Delta_{2},\Delta_{3},\Delta_{4}}\;,\\\
\bar{D}_{\Delta_{1}+1,\Delta_{2},\Delta_{3},\Delta_{4}+1}={}&(\Delta_{1}+\Delta_{4}-\tfrac{1}{2}\Sigma_{\Delta}-V\partial_{V})\bar{D}_{\Delta_{1},\Delta_{2},\Delta_{3},\Delta_{4}}\;,\\\
\bar{D}_{\Delta_{1},\Delta_{2}+1,\Delta_{3},\Delta_{4}+1}={}&(\Delta_{2}+U\partial_{U}+V\partial_{V})\bar{D}_{\Delta_{1},\Delta_{2},\Delta_{3},\Delta_{4}}\;,\\\
\bar{D}_{\Delta_{1}+1,\Delta_{2},\Delta_{3}+1,\Delta_{4}}={}&(\tfrac{1}{2}\Sigma_{\Delta}-\Delta_{4}+U\partial_{U}+V\partial_{V})\bar{D}_{\Delta_{1},\Delta_{2},\Delta_{3},\Delta_{4}}\;.\end{split}$
(147)

These identities allow us to shift the weights of $\bar{D}$-functions by
taking derivatives. Another set of useful identities arise from the invariance
of (143) under permutation of operators. It is straightforward to find

$\begin{split}\bar{D}_{\Delta_{1},\Delta_{2},\Delta_{3},\Delta_{4}}(U,V)=&V^{-\Delta_{2}}\bar{D}_{\Delta_{1},\Delta_{2},\Delta_{4},\Delta_{3}}(U/V,1/V)\\\
=&V^{\Delta_{4}-\frac{1}{2}\Sigma_{\Delta}}\bar{D}_{\Delta_{2},\Delta_{1},\Delta_{3},\Delta_{4}}(U/V,1/V)\\\
=&\bar{D}_{\Delta_{3},\Delta_{2},\Delta_{1},\Delta_{4}}(V,U)\\\
=&V^{\Delta_{1}+\Delta_{4}-\frac{1}{2}\Sigma_{\Delta}}\bar{D}_{\Delta_{2},\Delta_{1},\Delta_{4},\Delta_{3}}(U,V)\\\
=&U^{\Delta_{3}+\Delta_{4}-\frac{1}{2}\Sigma_{\Delta}}\bar{D}_{\Delta_{4},\Delta_{3},\Delta_{2},\Delta_{1}}(U,V)\;.\end{split}$
(148)

Let us now focus on the special $D$-function with $\Delta_{i}=1$. The
associated $\bar{D}$-function is the well known scalar one-loop box integral
in four dimensions, and evaluates to Usyukina:1992jd

$\bar{D}_{1,1,1,1}\equiv\Phi(z,\bar{z})=\frac{1}{z-\bar{z}}\left(2{\rm
Li}_{2}(z)-2{\rm
Li}_{2}(\bar{z})+\log(z\bar{z})\log\big{(}\frac{1-z}{1-\bar{z}}\big{)}\right)\;.$
(149)

Using this explicit formula, it is straightforward to check that in the small
$U$ expansion (i.e., small $z$ and fixed $\bar{z}$), the $\log U$ coefficient
of $\Phi(z,\bar{z})$ is given by

$\Phi(z,\bar{z})\big{|}_{\log
U}=\frac{\log(1-z)-\log(1-\bar{z})}{z-\bar{z}}\;.$ (150)

Finally, from this expression we can verify the following differential
recursion relations

$\begin{split}\partial_{z}\Phi(z,\bar{z})=&-\frac{\Phi(z,\bar{z})}{z-\bar{z}}+\frac{\log
U}{(z-1)(z-\bar{z})}-\frac{\log V}{z(z-\bar{z})}\;,\\\
\partial_{\bar{z}}\Phi(z,\bar{z})=&\frac{\Phi(z,\bar{z})}{z-\bar{z}}-\frac{\log
U}{(\bar{z}-1)(z-\bar{z})}+\frac{\log V}{\bar{z}(z-\bar{z})}\;.\end{split}$
(151)

These relations imply that any $\bar{D}$-function obtained from
$\bar{D}_{1,1,1,1}$ by using the “weight-shifting” operators in (147) can be
written as a linear combination of the basis functions $\Phi(z,\bar{z})$,
$\log U$, $\log V$ and $1$, with rational coefficients in $z$ and $\bar{z}$.
This property was useful for manipulating the tree-level correlators in
position space and extracting the protected part of correlators.

##  Appendix C Flat space limit of Mellin amplitudes

Let us consider a holographic CFT in $1/c$ expansion. Its four-point
correlators at one loop can be reconstructed following Aharony:2016dwx  and in
a wide class of examples Alday:2018kkw ; Alday:2019nin  they have a remarkable
structure in Mellin space, namely, the Mellin amplitudes have only
simultaneous simple poles. We consider a (partial)-Mellin amplitude of such a
form

$\mathcal{M}(s,t)=\sum_{m,n}\frac{c_{mn}}{(s-2m)(t-2n)}$ (152)

where $c_{mn}$ has the following large $m,n$ behavior

$c_{mn}=\frac{(mn)^{\frac{D}{2}-3}}{(m+n)^{\frac{D}{2}-2}}+\cdots\;.$ (153)

We would like to obtain the flat space limit of $\mathcal{M}(s,t)$, defined as
the limit where $s$, $t$ become large. Naively, we would expect
$\mathcal{M}(s,t)\sim 1/(st)$. However, this behavior is enhanced provided

$\sum_{m,n}\frac{(mn)^{\frac{D}{2}-3}}{(m+n)^{\frac{D}{2}-2}}=\text{divergent},$
(154)

which happens for $D>4$. In this case, the leading contribution to the Mellin
amplitude at large $s$, $t$ arises from the region with large $m,n\sim s,t$,
and the Mellin amplitude in the flat space limit is well approximated by

$\mathcal{M}_{\rm
flat}(s,t)=\int^{\infty}_{0}dmdn\frac{(mn)^{\frac{D}{2}-3}}{(m+n)^{\frac{D}{2}-2}}\frac{1}{(s-2m)(t-2n)}\;.$
(155)

Note that this integral is real and finite for $s,t<0$ and $4<D<6$. We can
compute $\mathcal{M}_{\rm flat}(s,t)$ in this region, and then analytically
continue it to other regions. This integral can be performed as follows.
First, we perform the integral over $m$ to obtain

$\mathcal{M}_{\rm
flat}(s,t)=-\int^{\infty}_{0}dn\frac{2n^{\frac{D}{2}-3}\,_{2}F_{1}\left(1,\frac{D-4}{2};\frac{D-2}{2};\frac{2n}{t}+1\right)}{(D-4)t(2n-s)}.$
(156)

The integral over $n$ can be performed by first using the integral
representation for the hypergeometric function

$\,{}_{2}F_{1}(a,b;c;z)=\frac{\Gamma(c)}{\Gamma(b)\Gamma(c-b)}\int_{0}^{1}d\zeta\zeta^{b-1}(1-\zeta
z)^{-a}(1-\zeta)^{c-b-1}$ (157)

and then integrating over $n$ and over $\zeta$, in that order. The result
appears to be quite complicated, but using standard identities for
hypergeometric functions it can be written in the following compact form

$M_{\rm
flat}(s,t)=\frac{\pi\left(s^{2}(-t)^{D/2}\,_{2}F_{1}\left(1,\frac{D}{2}-2;\frac{D}{2}-1;\frac{s+t}{s}\right)+t^{2}(-s)^{D/2}\,_{2}F_{1}\left(1,\frac{D}{2}-2;\frac{D}{2}-1;\frac{s+t}{t}\right)\right)}{2^{\frac{D}{2}-3}(D-4)\sin\left(\frac{\pi
D}{2}\right)s^{3}t^{3}}$ (158)

where the analytic continuation $s\to s+i\epsilon,t\to t+i\epsilon$ is
understood. We see that $M_{\rm flat}(s,t)$ is a function of $s/t$ up an
overall factor

$\mathcal{M}_{\rm flat}(s,t)\sim\frac{s^{D/2-3}}{t}f(s/t)$ (159)

which indeed displays an enhanced behavior with respect to $1/(st)$ for $D>4$.
Furthermore, up to an overall coefficient, $\mathcal{M}_{\rm flat}(s,t)$
exactly agrees with the flat space massless scalar box integral in $D$
dimensions

$\mathcal{M}_{\rm
flat}(s,t)=\frac{2^{\frac{D}{2}-3}\Gamma\left(\frac{D-3}{2}\right)}{\sqrt{\pi}}I^{(D)}_{\rm
box}(s,t)$ (160)

where

$I^{(D)}_{\rm
box}(s,t)=\Gamma\left(4-\frac{D}{2}\right)\int_{0}^{1}\prod_{i=1}^{4}da_{i}\delta(\sum_{i=1}^{4}a_{i}-1)\left(-sa_{1}a_{3}-ta_{2}a_{4}\right)^{\frac{D}{2}-4}\;.$
(161)

In summary, the partial Mellin amplitude (152) with the behavior (153) reduces
to the $D$-dimensional massless scalar box integral in the flat space limit.

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