* [BBV08] Maria-Florina Balcan, Avrim Blum, and Santosh Vempala.  A discriminative framework for clustering via similarity functions.  In Proceedings of the fortieth annual ACM symposium on Theory of computing, pages 671–680. ACM, 2008.
  * [BDST08] Justin Brickell, Inderjit S Dhillon, Suvrit Sra, and Joel A Tropp.  The metric nearness problem.  SIAM Journal on Matrix Analysis and Applications, 30(1):375–396, 2008.
  * [BEH+10] Georg Baier, Thomas Erlebach, Alexander Hall, Ekkehard Köhler, Petr Kolman, Ondřej Pangrác, Heiko Schilling, and Martin Skutella.  Length-bounded cuts and flows.  ACM Transactions on Algorithms (TALG), 7(1):1–27, 2010.
  * [CC17] Moses Charikar and Vaggos Chatziafratis.  Approximate hierarchical clustering via sparsest cut and spreading metrics.  In Proceedings of the Twenty-Eighth Annual ACM-SIAM Symposium on Discrete Algorithms, pages 841–854. Society for Industrial and Applied Mathematics, 2017.
  * [CCN19] Moses Charikar, Vaggos Chatziafratis, and Rad Niazadeh.  Hierarchical clustering better than average-linkage.  In Proceedings of the Thirtieth Annual ACM-SIAM Symposium on Discrete Algorithms, pages 2291–2304. SIAM, 2019.
  * [CCNY19] Moses Charikar, Vaggos Chatziafratis, Rad Niazadeh, and Grigory Yaroslavtsev.  Hierarchical clustering for euclidean data.  In The 22nd International Conference on Artificial Intelligence and Statistics, AISTATS 2019, 16-18 April 2019, Naha, Okinawa, Japan, pages 2721–2730, 2019.
  * [CDK+21] Vincent Cohen-Addad, Debarati Das, Evangelos Kipouridis, Nikos Parotsidis, and Mikkel Thorup.  Fitting distances by tree metrics minimizing the total error within a constant factor.  In 62nd Annual IEEE Symposium on Foundations of Computer Science (FOCS 2021), 2021.
  * [CdL21] Vincent Cohen-Addad, Rémi de-Joannis-de-Verclos, and Guillaume Lagarde.  Improving ultrametrics embeddings through coresets.  In Proceedings of the 38th International Conference on Machine Learning, ICML 2021, Proceedings of Machine Learning Research. PMLR, 2021\.
  * [CGCR20] Ines Chami, Albert Gu, Vaggos Chatziafratis, and Christopher Ré.  From trees to continuous embeddings and back: Hyperbolic hierarchical clustering.  In Hugo Larochelle, Marc’Aurelio Ranzato, Raia Hadsell, Maria-Florina Balcan, and Hsuan-Tien Lin, editors, Advances in Neural Information Processing Systems 33: Annual Conference on Neural Information Processing Systems 2020, NeurIPS 2020, December 6-12, 2020, virtual, 2020.
  * [CGW05] Moses Charikar, Venkatesan Guruswami, and Anthony Wirth.  Clustering with qualitative information.  Journal of Computer and System Sciences, 71(3):360–383, 2005.
  * [CKK+06] S. Chawla, R. Krauthgamer, R. Kumar, Y. Rabani, and D. Sivakumar.  On the hardness of approximating multicut and sparsest-cut.  Computational Complexity, 15(2):94–114, 2006.
  * [CKL20] Vincent Cohen-Addad, Karthik C. S., and Guillaume Lagarde.  On efficient low distortion ultrametric embedding.  In Proceedings of the 37th International Conference on Machine Learning, ICML 2020, 13-18 July 2020, Virtual Event, volume 119 of Proceedings of Machine Learning Research, pages 2078–2088. PMLR, 2020.
  * [CKM17] Vincent Cohen-Addad, Varun Kanade, and Frederik Mallmann-Trenn.  Hierarchical clustering beyond the worst-case.  In Isabelle Guyon, Ulrike von Luxburg, Samy Bengio, Hanna M. Wallach, Rob Fergus, S. V. N. Vishwanathan, and Roman Garnett, editors, Advances in Neural Information Processing Systems 30: Annual Conference on Neural Information Processing Systems 2017, December 4-9, 2017, Long Beach, CA, USA, pages 6201–6209, 2017.
  * [CKMM19] Vincent Cohen-Addad, Varun Kanade, Frederik Mallmann-Trenn, and Claire Mathieu.  Hierarchical clustering: Objective functions and algorithms.  J. ACM, 66(4):26:1–26:42, 2019.  Announced at SODA 2018.
  * [CLM+21] Vincent Cohen-Addad, Silvio Lattanzi, Slobodan Mitrovic, Ashkan Norouzi-Fard, Nikos Parotsidis, and Jakub Tarnawski.  Correlation clustering in constant many parallel rounds.  In Marina Meila and Tong Zhang, editors, Proceedings of the 38th International Conference on Machine Learning, ICML 2021, 18-24 July 2021, Virtual Event, volume 139 of Proceedings of Machine Learning Research, pages 2069–2078. PMLR, 2021.
  * [CM10] Gunnar E Carlsson and Facundo Mémoli.  Characterization, stability and convergence of hierarchical clustering methods.  J. Mach. Learn. Res., 11(Apr):1425–1470, 2010.
  * [CM15] Michael Cochez and Hao Mou.  Twister tries: Approximate hierarchical agglomerative clustering for average distance in linear time.  In Proceedings of the 2015 ACM SIGMOD international conference on Management of data, pages 505–517. ACM, 2015.
  * [CST01] Pierluigi Crescenzi, Riccardo Silvestri, and Luca Trevisan.  On weighted vs unweighted versions of combinatorial optimization problems.  Information and Computation, 167(1):10–26, 2001.
  * [CYL+20] Vaggos Chatziafratis, Grigory Yaroslavtsev, Euiwoong Lee, Konstantin Makarychev, Sara Ahmadian, Alessandro Epasto, and Mohammad Mahdian.  Bisect and conquer: Hierarchical clustering via max-uncut bisection.  In Silvia Chiappa and Roberto Calandra, editors, The 23rd International Conference on Artificial Intelligence and Statistics, AISTATS 2020, 26-28 August 2020, Online [Palermo, Sicily, Italy], volume 108 of Proceedings of Machine Learning Research, pages 3121–3132. PMLR, 2020.
  * [DPS+13] Geet Duggal, Rob Patro, Emre Sefer, Hao Wang, Darya Filippova, Samir Khuller, and Carl Kingsford.  Resolving spatial inconsistencies in chromosome conformation measurements.  Algorithms for Molecular Biology, 8(1):1–10, 2013.
  * [DS78] M. O. Dayhoff and R. M. Schwartz.  Chapter 22: A model of evolutionary change in proteins.  In in Atlas of Protein Sequence and Structure, 1978.
  * [Elk03] Charles Elkan.  Using the triangle inequality to accelerate k-means.  In Proceedings of the Twentieth International Conference on International Conference on Machine Learning, ICML’03, pages 147–153, 2003.
  * [FGR+20] Chenglin Fan, Anna C Gilbert, Benjamin Raichel, Rishi Sonthalia, and Gregory Van Buskirk.  Generalized metric repair on graphs.  In 17th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2020). Schloss Dagstuhl-Leibniz-Zentrum für Informatik, 2020.
  * [FRVB18] Chenglin Fan, Benjamin Raichel, and Gregory Van Buskirk.  Metric violation distance: Hardness and approximation.  In Proceedings of the Twenty-Ninth Annual ACM-SIAM Symposium on Discrete Algorithms, pages 196–209. SIAM, 2018.
  * [GJ17] Anna C Gilbert and Lalit Jain.  If it ain’t broke, don’t fix it: Sparse metric repair.  In 2017 55th Annual Allerton Conference on Communication, Control, and Computing (Allerton), pages 612–619. IEEE, 2017.
  * [GVY96] N. Garg, V. Vazirani, and M. Yannakakis.  Approximate max-flow min-(multi)cut theorems and their applications.  SIAM J. Comput., 25(2):235–251, 1996.
  * [HKM05] Boulos Harb, Sampath Kannan, and Andrew Mcgregor.  Approximating the best-fit tree under lp norms.  In In Proc. 8th Workshop on Approximation Algorithms for Combinatorial Optimization Problems (APPROX’05, pages 123–133, 2005.
  * [Kho02] Subhash Khot.  On the power of unique 2-prover 1-round games.  In Proceedings of the thiry-fourth annual ACM symposium on Theory of computing, pages 767–775, 2002.
  * [KKMO07] Subhash Khot, Guy Kindler, Elchanan Mossel, and Ryan O’Donnell.  Optimal inapproximability results for max-cut and other 2-variable csps?  SIAM Journal on Computing, 37(1):319–357, 2007.
  * [KLN10] Guy Kortsarz, Michael Langberg, and Zeev Nutov.  Approximating maximum subgraphs without short cycles.  SIAM Journal on Discrete Mathematics, 24(1):255–269, 2010.
  * [Lee15] Euiwoong Lee.  Hardness of graph pricing through generalized max-dicut.  In Proceedings of the forty-seventh annual ACM symposium on Theory of Computing, pages 391–399, 2015.
  * [Lee17] E. Lee.  Improved hardness for cut, interdiction, and firefighter problems.  In 44th International Colloquium on Automata, Languages, and Programming (ICALP), pages 92:1–92:14, 2017.
  * [Mos10] Elchanan Mossel.  Gaussian bounds for noise correlation of functions.  Geometric and Functional Analysis, 19(6):1713–1756, 2010.
  * [MW17] Benjamin Moseley and Joshua Wang.  Approximation bounds for hierarchical clustering: Average linkage, bisecting k-means, and local search.  In Advances in Neural Information Processing Systems, pages 3094–3103, 2017.
  * [O’D14] Ryan O’Donnell.  Analysis of boolean functions.  Cambridge University Press, 2014.
  * [Rag08] Prasad Raghavendra.  Optimal algorithms and inapproximability results for every csp?  In Proceedings of the fortieth annual ACM symposium on Theory of computing, pages 245–254, 2008.
  * [RP16] Aurko Roy and Sebastian Pokutta.  Hierarchical clustering via spreading metrics.  In Advances in Neural Information Processing Systems, pages 2316–2324, 2016.
  * [Sey95] Paul D. Seymour.  Packing directed circuits fractionally.  Combinatorica, 15(2):281–288, 1995.
  * [SG20] Rishi Sonthalia and Anna Gilbert.  Tree! i am no tree! i am a low dimensional hyperbolic embedding.  Advances in Neural Information Processing Systems, 33, 2020.
  * [SWW17] Anastasios Sidiropoulos, Dingkang Wang, and Yusu Wang.  Metric embeddings with outliers.  In Philip N. Klein, editor, Proceedings of the Twenty-Eighth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2017, Barcelona, Spain, Hotel Porta Fira, January 16-19, pages 670–689. SIAM, 2017.
  * [Vaz13] Vijay V Vazirani.  Approximation algorithms.  Springer Science & Business Media, 2013.
  * [VCC+21] Danny Vainstein, Vaggos Chatziafratis, Gui Citovsky, Anand Rajagopalan, Mohammad Mahdian, and Yossi Azar.  Hierarchical clustering via sketches and hierarchical correlation clustering.  In Arindam Banerjee and Kenji Fukumizu, editors, The 24th International Conference on Artificial Intelligence and Statistics, AISTATS 2021, April 13-15, 2021, Virtual Event, volume 130 of Proceedings of Machine Learning Research, pages 559–567. PMLR, 2021.
  * [Wen13] Cenny Wenner.  Circumventing d-to-1 for approximation resistance of satisfiable predicates strictly containing parity of width at least four.  Theory OF Computing, 9(23):703–757, 2013.
  * [WW18] Virginia Vassilevska Williams and R Ryan Williams.  Subcubic equivalences between path, matrix, and triangle problems.  Journal of the ACM (JACM), 65(5):1–38, 2018.

##  Appendix A Performance of Algorithm 2 for Ultrametric Violation Distance

In this section, we give a proof of the following theorem. The proof is a
simplification of the proof for Algorithm 1 for Metric Violation Distance.

######  Theorem A.1.

Algorithm 2 achieves an $O(\min(T,\log n))$-approximation for Ultrametric
Violation Distance, where $T$ is the number of different distance values of
the input.

We recall the high-level intuition of Algorithm 2 as a hierarchical clustering
algorithm. Consider an iteration where $i$ is chosen as the pivot. Let
$x_{1}<\dots<x_{t}$ be the distinct distances of the edges incident on $i$
(i.e. $\\{x_{1},\dots,x_{t}\\}=\\{x(ij):j\in[n]\setminus\\{i\\}\\}$), and for
$r\in[t]$, $V_{r}=\\{j\in[n]\setminus\\{i\\}:x(ij)=x_{r}\\}$. Call each
$V_{r}$ a cluster. Note that the algorithm ensures that, after the for loop,
$x(jk)=\max(x(ij),x(ik))$ if $j$ and $k$ belong to different clusters and
$x(jk)\leq x(ij)=x(ik)$ if $j$ and $k$ belong to the same cluster.

######  Observation A.2.

Any triangle that is not entirely contained in one cluster becomes balanced.
Therefore, in the subsequent iterations, different clusters do not interact at
all (e.g., having a pivot in one cluster only changes the distances within the
cluster).

###### Proof.

Suppose that $i$ is chosen as the pivot in the current iteration. For any
triangle $i^{\prime}j^{\prime}k^{\prime}$ such that $i^{\prime}\in V_{r}$ and
$j^{\prime},k^{\prime}\in V_{p}$ with $p\neq r$,
$x(i^{\prime}j^{\prime})=x(i^{\prime}k^{\prime})=\max(x(ii^{\prime}),x(ij^{\prime}))\geq
x(j^{\prime}k^{\prime})$. ∎

We denote by $\mathcal{T}$ the set of all triangles, and by
$\mathcal{T}^{\prime}$ the set of all unbalanced triangles (i.e., a triangle
such that the largest length is strictly bigger than the other two lengths).
We first prove that Algorithm 2 makes progress, i.e., the set of unbalanced
triangles shrinks as the algorithm progresses.

######  Lemma A.3.

At each execution of a pivoting step in Algorithm 2, no new unbalanced
triangle is created.

###### Proof.

Let $i,j,k,m$ be a $4$-tuple of vertices, and say that we pivot at $i$ during
the algorithm. By Observation A.2, $jkl$ can possibly become unbalanced if all
$j,k,l$ belong to the same cluster $V_{r}$ for some $r$. (I.e.,
$x(ij)=x(ik)=x(il)=x_{r}$ before the modification according to $i$.) In this
case, the algorithm states that each of $x(jk),x(kl),x(lj)$ takes the minimum
of its own value and $x_{r}$. If $jkl$ was balanced before the modification,
it means that there are at two edges that had the same largest distance. The
modification preserves this property. ∎

We are now ready to prove Theorem A.1.

###### Proof of Theorem A.1.

We first observe that the solution output by Algorithm 2 indeed forms an
ultrametric. By Observation A.2, at each round the algorithm repairs the
triangles incident to the pivot, and by Lemma A.3, these triangles stay
repaired as the algorithm progresses. Since at the end of the algorithm, every
vertex has been chosen once as a pivot, all the triangles are repaired.

We denote by ALG the output of the algorithm and by OPT the optimal solution
and recall that $\mathcal{T}$ denotes the set of all triples and
$\mathcal{T}^{\prime}$ the set of all unbalanced triangles with respect to the
input distances. For a triangle $ijk\in\binom{n}{3}$, let $A_{ijk}$ be the
indicator of the event that one of them is chosen as a pivot and one of the
edges (i.e., the edge not incident to the pivot) was modified as a result. (By
modification we only consider the event when the value strictly changes.) Note
that in the execution of the algorithm one edge can be modified many times but
this event happens at most once for each triangle since after pivoting at a
vertex its adjacent edges are frozen. Let $p_{ijk}=\mathbb{E}[A_{ijk}]$, where
the expectation is taken over the algorithm’s randomness. Then
$\mathbb{E}[ALG]\leq\sum_{t\in\mathcal{T}}p_{t}$. By Lemma A.3, a triangle
never becomes unbalanced in the course of the algorithm, therefore $p_{t}=0$
for $t\not\in\mathcal{T}^{\prime}$. Thus
$\mathbb{E}[ALG]\leq\sum_{t\in\mathcal{T^{\prime}}}p_{t}$.

To show that the algorithm is an $\alpha$-approximation, we prove that for
every edge $e$, $q_{e}:=\sum_{t\in\mathcal{T}^{\prime},t\ni
e}p_{t}\leq\alpha$. This shows that $\frac{p_{t}}{\alpha}$ is a fractional
packing for the unbalanced triangles, and therefore, we have

$\sum_{t\in\mathcal{T^{\prime}}}\frac{p_{t}}{\alpha}\leq OPT,$

and thus

$\mathbb{E}[ALG]\leq\sum_{t\in\mathcal{T}^{\prime}}p_{t}\leq\alpha OPT.$

Hence, for the rest of the proof, we show that $q_{e}=O(\log n)$ for every
$e$. Fix an edge $e$ that belongs to at least one unbalanced triangle. Without
loss of generality, assume $e=(1,2)$. Note that, for any unbalanced triangle
$t$, when the event indicated by $A_{t}$ happens, one of the three vertices of
$t$ gets chosen uniformly at random, and the opposite edge gets modified.
Therefore, we have that $\sum_{t\in\mathcal{T}^{\prime},t\ni
e}p_{t}/3=\mathbb{E}[\\#\textrm{times that $e$ is modified}]$, and thus it
suffices to bound the latter expectation by $O(\log n)$.

In fact, we count the expected number of bad modifications, where bad
modifications are the ones that still put the two endpoints of $e$ in the same
cluster. Since $e$ cannot modified further if the endpoints belong to
different clusters, the number of bad modifications differs the total number
of modifications by at most $1$. Note that in a bad modification, the distance
value strictly decreases, so $T$ is an upper bound for the number of bad
modifications. It only remains the show another upper bound $O(\log n)$.

Assume without loss of generality that the input distance values are integers
between $1$ and $n^{2}$; by the design of the algorithm, only those values
will ever appear during the algorithm. Bad modifications happen when we choose
$i\geq 3$ as a pivot in the current cluster with $x(1i)=x(2i)<x(12)$. (Let
$b=x(12)$ in the current cluster.) We call such $i$ a bad pivot at distance
$x(1i)$. Note that a bad pivot cannot be created by the design of the
algorithm.

For $a\in\\{1,\dots,b-1\\}$, let $n^{\prime}_{a}$ be the number of $i$ such
that $x(1i)=x(2i)=a$, and $n^{\prime}=\sum_{a=1}^{b-1}n^{\prime}_{a}$. If a
bad pivot $i$ at distance $a$ is chosen, then in the next cluster of
$\\{1,2\\}$, $x(12)$ becomes $a$ and every pairwise distance is at most $a$,
so there cannot be bad pivots at distance $\geq a$.

We bound the expected number of bad modifications by induction on the initial
number of bad pivots $n^{\prime}$. Let $c_{e}(n^{\prime})$ be an upper bound
on the expected number of bad modifications when the number of bad pivots is
$n^{\prime}$. We will prove that $c_{e}(i)\leq 2\ln(i+1)$. For the base case,
$c_{e}(1)\leq 1/3\leq 2\ln 2$. Also, let
$s^{\prime}_{a}=\sum_{i=1}^{a-1}n^{\prime}_{a}$. For $n^{\prime}\geq 2$,

$\displaystyle c_{e}(n^{\prime})$ $\displaystyle=\Pr[\mbox{bad pivot is
chosen}]+\mathbb{E}[c_{e}(n^{\prime\prime})|\mbox{bad pivot is chosen}]$
$\displaystyle(n^{\prime\prime}=\mbox{(number of remaining bad pivots)})$
$\displaystyle\leq
1+\bigg{(}\sum_{a=1}^{b-1}\frac{n^{\prime}_{a}}{n^{\prime}}\cdot
c_{e}(s^{\prime}_{a})\bigg{)}$ $\displaystyle\leq
1+\frac{1}{n^{\prime}}\cdot\bigg{(}\sum_{i=1}^{n^{\prime}}c_{e}(i-1)\bigg{)}$
$\displaystyle\leq 1+\frac{1}{n^{\prime}}\cdot\int_{i=1}^{n^{\prime}}(2\ln
i)di$ $\displaystyle=1+\frac{1}{n^{\prime}}\cdot\bigg{(}2n^{\prime}\ln
n^{\prime}-2n^{\prime}+2\bigg{)}\leq 2\ln(n^{\prime}+1),$

finishing the proof. ∎


# Momentum-space formulae for AdS correlators for diverse theories in diverse
dimensions

Soner
Albayrak${}^{\text{\faEuro}\;,\;\text{\faGbp}\;,\;\text{\faTurkishLira}}$
Savan Kharel${}^{\text{\faDollar}}$ Xinkang Wang${}^{\text{\faDollar}}$
${}^{\text{\faEuro}}$Institute of Physics, University of Amsterdam, Amsterdam,
1098 XH, The Netherlands  ${}^{\text{\faGbp}}$Center for Theoretical Physics,
National Taiwan University, Taipei 10617, Taiwan
${}^{\text{\faTurkishLira}}$Department of Physics, Middle East Technical
University, Ankara 06800, Turkey  ${}^{\text{\faDollar}}$Department of
Physics, University of Chicago, Chicago, IL 60637, USA

(August 28, 2024)

###### Abstract

In this paper, we explore correlators of a series of theories in anti-de
Sitter space: we present comprehensive results for interactions involving
scalars, gluons, and gravitons in multiple dimensions. One aspect of our
investigation is the establishment of an intriguing connection between the
kinematic factors of these theories; indeed, such a connection directly
relates these theories among themselves and with other theories of higher spin
fields. Besides providing several explicit results throughout the paper, we
also highlight the interconnections and relationships between these different
theories, providing valuable insights into their similarities and
distinctions.

###### Contents

  1. I Introduction
  2. II AdS Perturbation Theory in Momentum Space and Classification of Spinning Fields
    1. II.1 Primer: scalar fields in curved background
    2. II.2 Classifications of spinning fields in AdS
    3. II.3 Pure Yang-Mills and Einstein gravity
    4. II.4 Orthogonal decomposition of gauge and gravity propagators
    5. II.5 Interactions among scalars, gluons, and gravitons
  3. III Explicit Computations in AdSd+1
    1. III.1 Scalar-scalar interaction
    2. III.2 Scalar-QED
    3. III.3 Scalar-gravity
    4. III.4 Pure Yang-Mills
  4. IV Conclusion

##  I Introduction

In the last decade, scattering amplitudes have been at the forefront of
theoretical research. These developments have positioned scattering amplitudes
at the vanguard of theoretical research, unlocking profound insights across
various applications. These range from the obvious usage of making
experimental predictions at the Large Hadron Collider (LHC) to innovative
strides in the field of gravitational wave physics [1, 2]. In this period of
prolific research, scientists have elucidated both basic tree-level and
intricate multi-loop amplitude expressions [3]. The hallmark of these
formulations is their remarkable simplicity, contrasting sharply with the
complexity typically encountered in intermediate computations of these
structures. This observation has motivated researchers to produce explicit
results, thereby amassing a substantial repository of “theoretical data” on
scattering amplitudes. This endeavor aims to uncover numerous latent patterns
and structures within scattering amplitudes. Indeed, this approach has proven
exceptionally fruitful, leading to seminal developments, including the
incorporation of positive geometry concepts and the pioneering discovery of
the double copy principle [4, 5].

Another area of significant interest in contemporary theoretical physics is
the holographic duality, particularly epitomized in the Anti-de
Sitter/Conformal Field Theory (AdS/CFT) correspondence [6, 7]. This duality
intriguingly relates the boundary operators in a Conformal Field Theory (CFT)
to bulk fields in Anti-de Sitter (AdS) space, offering a novel perspective to
understand CFT correlation functions as scattering amplitudes in AdS space.
Despite plethora of momentum basis results in the flat space scattering,
momentum basis correlators in AdS remain less explored, partly due to the
complexities of bulk integrals unique to AdS geometries. Nevertheless, the
last decade has witnessed considerable progress in this domain, as evidenced
by a range of studies [8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21,
22, 23, 24, 25, 26, 27, 28, 29].111For some position, Mellin, embedding space
approaches, see for instance: [30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41,
36, 42, 43, 44, 45]

This paper endeavors to make a contribution to the exploration of theories
within the AdS/CFT framework, specifically in momentum space. It offers
comprehensive explicit computations of correlators of several type of scalar
and spinning fields, along with discussions of the interconnections among
these diverse theories. Drawing inspiration from scattering amplitude studies,
our exploration extends into various dimensions (see for instance [46]). We
then conclude the paper with thoughtful speculation on potential avenues for
future research.

##  II AdS Perturbation Theory in Momentum Space and Classification of
Spinning Fields

A Poincaré patch of a maximally symmetric homogeneous curved space can be
described by the metric

$\displaystyle ds^{2}=\frac{\rho^{2}}{z^{2}}\left(\pm
dz^{2}+\eta_{ij}dx^{i}dx^{j}\right)$ (2.1)

where $\rho$ (or $i\rho$) is the curvature radius and $\eta$ is the metric for
the boundary at $z\rightarrow 0$. Depending on the signature of $\eta$ and the
sign of $dz^{2}$, this metric describes Euclidean or Lorentzian (A)dS. In the
rest of the paper, we will mostly focus on AdS (hence $+dz^{2}$) and work with
spacelike momenta, making all our computations agnostic to the signature of a
mostly positive $\eta$. In addition, we will set $\rho=1$ for convenience.

The Poincaré coordinates are particularly advantageous due to the evident
translational invariance in the boundary coordinates, denoted by $x^{i}$. This
invariance simplifies analyses by allowing us to directly employ momentum
space techniques for these coordinates, a method that aligns well with the
established practices in the existing literature. Opting for this coordinate
system offers significant benefits, which we underline as follows: (a) flat
space limit is extremely simple [47], making comparison with flat space data
rather straightforward; (b) cosmological data is conventionally collected and
stored in momentum coordinates, making comparison with dS literature far more
intuitive; (c) diagrammatic rules are quite analogous with (and in some cases
even exactly same to) the flat space diagrammatic rules, making this
coordinate system an appealing choice for perturbation theory.

The steps in the derivation of perturbation theory rules in (A)dS are same
with those of the flat space: we start with a weakly-coupled Lagrangian,
derive the bulk to bulk propagator from the quadratic action, and derive the
vertex factors from the interaction pieces. We will first illustrate these
steps in case of scalars as a brief review, and then move on to the discussion
of the spinning fields.

###  II.1 Primer: scalar fields in curved background

Consider the following action

$\displaystyle S=\int
d^{d+1}x\sqrt{g}\left[\frac{1}{2}\left(g^{\mu\nu}(\partial_{\mu}\phi)(\partial_{\nu}\phi)+\mu^{2}\phi^{2}\right)+\mathcal{L}_{\text{int}}(\phi)\right],$
(2.2)

where one is usually interested in polynomial interaction piece
$\mathcal{L}_{\text{int}}$ for its relevance to the cosmology though we will
keep it generic.222 The parameter $\mu^{2}$ is the _effective mass square_ ,
which is defined as $\mu^{2}\coloneqq m^{2}+\xi R$ for the Ricci scalar $R$
—in AdS, $R=-d(d+1)/\rho^{2}$ [11]. From the quadratic action, one can derive
the equation for the bulk-to-bulk propagator via the Green’s method [48]:333We
follow the same conventions as in [8, 10, 9, 11, 12, 13, 14]: $\mathbf{k}$
denotes the boundary spacelike momenta with $\mathbf{k}^{2}>0$, and $k$ its
norm $\sqrt{\mathbf{k}^{2}}$.

$\displaystyle\left(z^{d+1}\partial_{z}z^{1-d}\partial_{z}-z^{2}\mathbf{k}^{2}-\mu^{2}\right)\mathcal{G}(\mathbf{k};z,z^{\prime})=i\delta(z-z^{\prime})z^{d+1}.$
(2.3)

For the right hand side, we can use the identity

$\displaystyle\int\limits_{0}^{\infty}pdpJ_{\beta}(pz)J_{\beta}(pz^{\prime})=\frac{\delta(z-z^{\prime})}{z},$
(2.4)

where $J_{\beta}$ is the Bessel function of the first kind. In the left hand
side, we use the ansatz $\mathcal{G}(\mathbf{k};z,z^{\prime})=\int
pdp\;c(p)(zz^{\prime})^{\alpha}J_{\beta}(pz)J_{\beta}(pz^{\prime})$. The
differential operator then generates $(d-2\alpha)J_{\beta-1}(pz)$ on the left-
hand side, which we kill by setting $\alpha=d/2$. The rest of the equation
fixes the unknown $\beta$ and the function $c(p)$, leading to the following
expression of bulk-to-bulk propagator

$\displaystyle\mathcal{G}(\mathbf{k};z,z^{\prime})=\int\limits_{0}^{\infty}\frac{-ipdp}{p^{2}+\mathbf{k}^{2}-i\epsilon}(zz^{\prime})^{d/2}J_{\nu}(pz)J_{\nu}(pz^{\prime})$
(2.5)

where we define the parameter $\nu$ as444This parameter is actually related to
the scaling dimension $\Delta$ of the boundary operator that is dual to the
scalar field: $\Delta=\nu+d/2$.

$\displaystyle\nu\coloneqq\frac{\sqrt{d^{2}+4\mu^{2}}}{2}.$ (2.6)

One can now go ahead and derive _bulk-to-boundary_ propagator
$\mathcal{G}(\mathbf{k},z)$ by starting from the bulk-to-bulk propagator and
taking one of the points to the boundary. However, the regularity in the bulk
for spacelike momenta [48] and the compatibility with Ward identities already
bootstraps the expression

$\displaystyle\mathcal{G}(\mathbf{k},z)=\sqrt{\frac{2}{\pi}}z^{d/2}k^{\nu}K_{\nu}(kz)$
(2.7)

where $K_{\nu}$ is the modified Bessel function of the second kind.555 The
overall factor $\sqrt{\frac{2}{\pi}}$ is chosen conventionally. When we move
on to the spinning bulk fields dual to conserved non-scalar boundary
operators, we will work with the generalization of this formula as
$\displaystyle\mathcal{G}_{i_{1}\dots
i_{\ell}}(\mathbf{k},z)=\epsilon_{i_{1}\dots
i_{\ell}}\sqrt{\frac{2}{\pi}}z^{d/2-\ell}k^{\nu}K_{\nu}(kz)$ (2.8) for the
appropriate $\nu$.

With the bulk-to-bulk and bulk-to-boundary propagators given as (2.5) and
(2.7), the only ingredient left to crank the perturbation theory machinery is
the vertex factor: in the case of scalars with polynomial interactions (our
focus in this paper), this is rather trivial —the vertex factor is a simple
constant. However, we will see non-trivial examples in the later sections
where vertex factor will have a tensor part and a monomial $z-$dependence.

With all the ingredients derived, we can now consider the simplest
computation: three-point Witten diagram of three generic scalar fields dual to
scalar operators of arbitrary scaling dimensions $\Delta_{a}$. The amplitude
is simply the product of bulk to boundary propagators integrated over the bulk
point:

$\mathcal{A}_{3}=\int_{0}^{\infty}\frac{dz}{z^{d+1}}\prod_{a=1}^{3}\left[z^{\frac{d}{2}}\sqrt{\frac{2}{\pi}}k_{a}^{\nu_{a}}K_{\nu_{a}}(k_{a}z)\right]$
(2.9)

as seen for instance in [49]. By taking into account the appropriate
normalization factors, one could also derive the three-point correlation
function of the boundary CFT from this amplitude.666 In principle,
$\expectationvalue{\mathcal{O}_{1}\mathcal{O}_{2}\mathcal{O}_{3}}=b_{\mathcal{O}_{1}\mathcal{O}_{2}\mathcal{O}_{3}}\mathcal{A}_{3}$
where $b_{\mathcal{O}_{1}\mathcal{O}_{2}\mathcal{O}_{3}}$ contains the
information of the conformal weights of the boundary operators, see eqn. 4.20
and 4.21 of [50] as an example.

In extending our analysis to tree-level Witten diagrams with a four and higher
number of points, the approach remains largely unchanged. the process still
involves multiplying all the relevant propagator and vertex factors.
Furthermore, an integration over all points in the bulk is necessary,
incorporating the volume factor $z^{-(d+1)}$. However, it is important to note
that these higher-point functions might necessitate renormalization, as
detailed in the work of Bzowski et al. [18]. Moving forward, we encounter
additional complexities when dealing with spinning fields, which will be the
focus of the subsequent section.

###  II.2 Classifications of spinning fields in AdS

In the preceding discussion, we outlined a methodology that can be adapted for
the analysis of spinning fields within the AdS framework. This adaptation
involves a series of systematic steps:

  1. 1.

Derive the propagators from the quadratic action

  2. 2.

Derive vertex factors from the interaction pieces

  3. 3.

To compute any given tree-level Witten diagram, contract propagators and
vertex factors and integrate them over the bulk points

Although the almost-same prescription is well used for decades in the
Minkowski space scattering amplitudes computations, there is actually a
technical complication in AdS amplitudes that is absent in the flat space. We
will discuss this novelty (and its implications) in this section, and turn to
the physically relevant spinning fields in the next section.

In flat space, the unitarity ensures that the internal legs of a Feynamn
diagrams takes the form

$\displaystyle\mathcal{G}(x_{1},x_{2})=\int
d^{d}p\;\frac{\sum\limits_{j}\epsilon_{j}\epsilon_{j}^{\dagger}}{p^{2}+m^{2}+i\epsilon}e^{ip\cdot(x_{1}-x_{2})}$
(2.10)

upto an overall phase, whereas the external legs are

$\displaystyle\mathcal{G}(x)=\int d^{d}p\;\epsilon_{j}e^{ip\cdot x}.$ (2.11)

Here $\epsilon_{j}$ is a polarization vector with $j$ going from 1 to the
number of physical degrees of freedom. Thus, any tree-level Feynman diagram
can be brought to the form

$\displaystyle\textrm{FD}=\int\prod_{i}d^{d}p_{i}\left[\frac{\prod_{i,j}c_{i,j}e^{ip_{i}\cdot
x_{j}}}{\prod_{i}(p_{i}^{2}+m_{i}^{2})}\right]\textrm{contraction}\left(V,\epsilon^{(1)},\epsilon^{(2)},\dots\right)$
(2.12)

for some scalars $c_{i,j}$, whereas $V$ denotes the interaction vertex.

A crucial property of the flat space is that the functional form of the
_kinematic factor_ of the Feynman diagram, the term within square brackets
above, does not explicitly depend on the spin of the fields or the dimension
of the spacetime. However, this property does not extend to AdS: we can
quickly check this by analyzing the form of the bulk-to-boundary propagator
for a spin-$\ell$ gauge field in AdSd+1 –see eqn. (2.8).

This novelty in AdS brings a new challenge that is absent in the flat space
computations: case by case calculation of kinematic factors for the Witten
diagrams. Fortunately, one can actually classify these kinematic factors into
universality classes: not only does this simplify the explicit computations,
but can this also help with making connections between CFT correlators of
various operators in different dimensions.

In what follows, let us restrict to theories with cubic interactions. We will
take the interaction to be rather generic by inducing it via a piece in the
Lagrangian that includes three arbitrary conserved fields of spin
$\ell_{1}\geq\ell_{2}\geq\ell_{3}$ and $n$ derivatives.777In AdS4, $\ell$ is
the standard spin, i.e. the only representation label of $\mathrm{so}(3)$
group in the stabilizer subgroup of the AdS isometries. In higher dimensions,
we refer the length of the first row of the Young tableaux of the
$\mathrm{so}(d)$ representation as the spin. The vertex factor due to such an
interaction should have $\ell_{1}+\ell_{2}+\ell_{3}$ contravariant indices to
contract with three legs (each with $\ell_{1}$, $\ell_{2}$, and $\ell_{3}$
indices) that enter into the vertex. In our position-momentum coordinate
system in the Poincaré patch, $n$ of such indices are induced by the boundary
momenta $\mathbf{k}^{i}$ whereas remaining indices by
$(\ell_{1}+\ell_{2}+\ell_{3}-n)/2$ many inverse boundary metrics
$g^{ij}=z^{2}\eta^{ij}$. But since each momenta $\mathbf{k}^{i}$ needs one
inverse metric as well, we end up with the requirement of
$(\ell_{1}+\ell_{2}+\ell_{3}+n)/2$ many inverse boundary metrics for such an
interaction, which leads to the correspondence

$\boxed{\begin{aligned} &\begin{aligned} \text{Kinematic factor of gluon
cubic}\\\ \text{self-interaction in
AdS${}_{d-1}$}\quad\end{aligned}\quad\Longleftrightarrow\quad\begin{aligned}
&\text{Kinematic factor of graviton cubic}\\\ &\qquad\text{self-interaction in
AdS${}_{d-3}$}\end{aligned}\\\
&\qquad\qquad\qquad\qquad\quad\rotatebox[origin={c}]{135.0}{$\Longleftrightarrow$}\qquad\qquad\qquad\qquad\qquad\quad\rotatebox[origin={c}]{45.0}{$\Longleftrightarrow$}\\\
&\qquad\text{The Witten diagrams for an AdS${}_{d+1}$ scalar with cubic
interactions}\end{aligned}}$

Figure 1:  An example map of connections between the web of theories

$\boxed{\text{cubic interaction with $n$ derivatives between
}\mathcal{O}_{\ell_{1}},\;\mathcal{O}_{\ell_{2}},\;\mathcal{O}_{\ell_{3}}}\\\
\rotatebox[origin={c}]{135.0}{$\Longleftrightarrow$}\\\
\boxed{z^{\ell_{1}+\ell_{2}+\ell_{3}+n}\text{ factor in bulk-point
integration}}$ (2.13)

This argument extends to scalars as well; in fact, we can immediately write
down

$\text{cubic interaction}\sim\int
dzz^{\frac{d-2}{2}+n}E_{\nu_{1}}(k_{1}z)E_{\nu_{2}}(k_{2}z)E_{\nu_{3}}(k_{3}z)\\\
\nu_{i}=\left\\{\footnotesize\begin{aligned} \ell_{i}-2+d/2&\;\text{ conserved
$\ell>0$ operators}\\\ \sqrt{d^{2}+4\mu^{2}}/2&\;\text{
scalars}\end{aligned}\right.$ (2.14)

where $E_{\nu}$ is Bessel $K_{\nu}$ or $J_{\nu}$. This is a generalization of
several case studies in the literature.888For instance, graviton cubic self-
interaction, gluon cubic self-interaction, scalar-scalar-graviton interaction,
and scalar cubic nonderivative-interaction have $z^{\\#}$ consistent with the
general formula above, since these cases have respective
$(n,\ell_{1},\ell_{2},\ell_{3},\\#)$ as $(2,2,2,2,8)$, $(1,1,1,1,4)$,
$(2,2,0,0,4)$, and $(0,0,0,0,0)$. We can summarize this observation as
follows:

The kinematic factors of two theories in AdSd and AdS${}_{d^{\prime}}$ are the
same if their perturbative spectrum and interaction Lagrangian are
appropriately related!

The simplest implication of the result above is the correspondence in Figure
1. Unsurprisingly, this shows that gluon and graviton kinematic factors can be
re-interpreted as scalar Witten diagrams. But perhaps surprisingly, this also
indicates that we can find an equivalence between gauge and graviton kinematic
factors if we focus on appropriate spacetime dimensions.

By analyzing (2.2) we can obtain further correspondences; for instance,

$\boxed{\text{$\ell_{1}-\ell_{2}-\ell_{3}$ interaction with $n-$derivative in
AdS${}_{d+1}$}}\\\ \rotatebox[origin={c}]{135.0}{$\Longleftrightarrow$}\\\
\boxed{\begin{aligned} \text{$(\ell_{1}+1)-(\ell_{2}+1)-(\ell_{3}+1)$
interaction}\\\ \text{with $(n+1)-$derivative in AdS${}_{d-1}$}\end{aligned}}$
(2.15a) and $\boxed{\text{$\ell_{1}-\ell_{2}-\ell_{3}$ interaction with
$n-$derivative in AdS${}_{d+1}$}}\\\
\rotatebox[origin={c}]{135.0}{$\Longleftrightarrow$}\\\ \boxed{\begin{aligned}
\text{all scalars of $\Delta_{i}=d-2+\ell_{i}+n$ with}\\\ \text{polynomial
interaction in AdS${}_{d+2n+1}$}\end{aligned}}$ (2.15b)

Although we present such relations, we will not dwell on their utility in
detail in this paper. Indeed, higher spin theories are of particular
importance in several contexts, most importantly in the Vasiliev theories;
nevertheless, analysis of such theories would take us too far from our main
point so we will stick to spin 0, 1, and 2 fields in the rest of the paper.999
We note that AdS does allow consistent models with higher spin fields as the
flat-space no-go theorems are no longer valid here [51].

###  II.3 Pure Yang-Mills and Einstein gravity

Consider Yang-Mills action in curved background

$\displaystyle S=-\frac{1}{4}\int d^{d+1}x\sqrt{g}F_{\mu\nu}^{a}F^{\mu\nu,a}$
(2.16a) where
$F_{\mu\nu}^{a}=\nabla_{\mu}A_{\nu}^{a}-\nabla_{\nu}A_{\mu}^{a}+ef^{abc}A_{\mu}^{b}A_{\nu}^{c}$
with the coupling constant $e$. Likewise, the usual Einstein-Hilbert bulk
action with a cosmological constant term reads as $\displaystyle S=\int
d^{d+1}x\sqrt{g}(R[g]-2\Lambda).$ (2.16b)

Since we are interested in doing a well-defined perturbation expansion in AdS
background, we will take $e$ small, $\Lambda=-\frac{d(d+1)}{2}$, and
$g=g_{\text{AdS}}+h$ for small $h$: this leads to the following well-known
bulk-to-boundary propagators

$\displaystyle\mathcal{G}_{i}^{a}(\mathbf{k},z)=$
$\displaystyle\sqrt{\frac{2}{\pi}}\epsilon_{i}^{a}(\mathbf{k})z^{d/2-1}k^{d/2-1}K_{d/2-1}(kz)$
(2.17a) $\displaystyle\mathcal{G}_{ij}(\mathbf{k},z)=$
$\displaystyle\sqrt{\frac{2}{\pi}}\epsilon_{ij}(\mathbf{k})z^{d/2-2}k^{d/2}K_{d/2}(kz)$
(2.17b)

which are compatible with the general result in (2.8). We only present the
boundary components of $\mathcal{G}_{\mu}$ and $\mathcal{G}_{\mu\nu}$ as we
are working in the _axial gauge_ , see [48] for further details. This also
means that the _polarization tensors_ $\epsilon_{i}$ and $\epsilon_{ij}$ are
practically identical to their flat space counterparts in $\mathbb{R}^{d}$:
they are transverse, null, and traceless. Whenever there is no room for
confusion, we will drop their explicit momentum dependence —same with the
color indices of $\mathcal{G}_{i}^{a}$.

The derivation of bulk-to-bulk propagators of gluon and graviton is analogous
to our derivation of scalar propagator in § II.1: after some computation, we
end up with

$\displaystyle\mathcal{G}_{ij}(\mathbf{k};z_{1},z_{2})=$
$\displaystyle\int_{0}^{\infty}\frac{-ipdp(z_{1}z_{2})^{\nu}}{p^{2}+\textbf{k}^{2}-i\epsilon}J_{\nu}(pz_{1})J_{\nu}(pz_{2})H_{ij}(\mathbf{k},p)$
(2.18a) $\displaystyle\mathcal{G}_{ij,k\ell}(\mathbf{k};z_{1},z_{2})=$
$\displaystyle\int_{0}^{\infty}\frac{-ipdp(z_{1}z_{2})^{\nu-2}}{p^{2}+\textbf{k}^{2}-i\epsilon}J_{\nu}(pz_{1})J_{\nu}(pz_{2})H_{ij,k\ell}(\mathbf{k},p)$
(2.18b)

where $\nu=d/2-1$ for gluon and $\nu=d/2$ for graviton. Here, the tensor
structures $H$ are defined as

$\displaystyle H_{ij}(\mathbf{k},p)\coloneqq$
$\displaystyle\eta_{ij}+\frac{\textbf{k}_{i}\textbf{k}_{j}}{p^{2}}$ (2.19a)
$\displaystyle H_{ij,k\ell}(\mathbf{k},p)\coloneqq$ $\displaystyle
H_{ik}(\mathbf{k},p)H_{j\ell}(\mathbf{k},p)+H_{i\ell}(\mathbf{k},p)H_{jk}(\mathbf{k},p)$
$\displaystyle-\frac{2}{(d-1)}H_{ij}(\mathbf{k},p)H_{k\ell}(\mathbf{k},p).$
(2.19b)

From the non-quadratic part of the actions in (2.16), one can derive the
vertex factors for the color ordered YM amplitude and the cubic gravitational
interaction:

$\displaystyle
V^{ijk}_{\mathbf{k}_{1},\mathbf{k}_{2},\mathbf{k}_{3}}\coloneqq{}$
$\displaystyle\frac{iz^{4}}{\sqrt{2}}\left(\eta^{ij}(\mathbf{k}_{1}-\mathbf{k}_{2})^{k}+\eta^{jk}(\mathbf{k}_{2}-\mathbf{k}_{3})^{i}+\eta^{ki}(\mathbf{k}_{3}-\mathbf{k}_{1})^{j}\right)$
(2.20a) $\displaystyle V^{ijk\ell}\coloneqq{}$
$\displaystyle{}\frac{iz^{4}}{2}\left(2\eta^{ik}\eta^{j\ell}-\eta^{ij}\eta^{k\ell}-\eta^{i\ell}\eta^{jk}\right)\;,$
(2.20b) $\displaystyle V^{ijk\ell
mn}_{\mathbf{k}_{1},\mathbf{k}_{2},\mathbf{k}_{3}}\coloneqq{}$
$\displaystyle{}\frac{z^{8}}{4}\left[\left(\mathbf{k}_{2}^{i}\mathbf{k}_{3}^{j}\eta^{km}\eta^{\ell
n}-2\mathbf{k}_{2}^{i}\mathbf{k}_{3}^{k}\eta^{jm}\eta^{\ell n}\right)+\text{
permutations}\right]$ (2.20c)

See [12] for further details.

###  II.4 Orthogonal decomposition of gauge and gravity propagators

The tensor structures of gauge and gravity propagators are constructed from
the object $H$, as defined in (2.19a). This object can be elegantly decomposed
into the projectors $\widehat{P}_{ij}^{\perp}$ and
$\widehat{P}_{ij}^{\parallel}$ , adhering to standard projection rules:
$P^{\perp}+P^{\parallel}=\delta$ and $P^{\alpha}\cdot
P^{\beta}=\delta^{\alpha\beta}P^{\alpha}$. Explicitly,

$\displaystyle\widehat{P}_{ij}^{\perp}(\mathbf{k})\coloneqq{}\eta_{ij}-\frac{\textbf{k}_{i}\textbf{k}_{j}}{k^{2}}\;,\quad\widehat{P}_{ij}^{\parallel}(\mathbf{k})\coloneqq{}\frac{\textbf{k}_{i}\textbf{k}_{j}}{k^{2}}.$
(2.21)

Clearly, the tensor $H_{ij}$ of gluon decomposes as

$\displaystyle
H_{ij}(\mathbf{k},p)=\widehat{P}_{ij}^{\perp}(\mathbf{k})+\frac{(k^{2}+p^{2})}{p^{2}}\widehat{P}_{ij}^{\parallel}(\mathbf{k})$
(2.22a) whereas the decomposition of $H_{ij,k\ell}(\mathbf{k},p)$ of graviton
is best written as101010See [9] for further details on this decomposition. A
similar decomposition with a different set of conventions can also be seen in
[52].
$H_{ij,k\ell}(\mathbf{k},p)=\frac{(k^{2}+p^{2})}{p^{2}}\left(\widehat{P}_{ij,\ell
m}^{(\perp,\parallel)}(\mathbf{k})+\widehat{P}_{ij,\ell
m}^{(\parallel,\perp)}(\mathbf{k})\right)\\\ +\widehat{P}_{ij,\ell
m}^{(\perp,\perp)}(\mathbf{k})+\left(\frac{k^{2}+p^{2}}{p^{2}}\right)^{2}\widehat{P}_{ij,\ell
m}^{(\parallel,\parallel)}(\mathbf{k}),$ (2.22b)

where we have defined

$\displaystyle\widehat{P}_{ij,\ell
m}^{(\alpha,\beta)}\coloneqq\widehat{P}_{i\ell}^{(\alpha)}\widehat{P}_{jm}^{(\beta)}+\widehat{P}_{im}^{(\alpha)}\widehat{P}_{j\ell}^{(\beta)}-\frac{2}{d-1}\widehat{P}_{ij}^{(\alpha)}\widehat{P}_{\ell
m}^{(\beta)}.$ (2.23)

One can now decompose the gauge and gravity propagators into their
_transverse_ and _longitudinal_ parts. For brevity, we choose the convention
of defining $\widehat{P}_{ij,\ell m}^{(\perp,\perp)}$ part of the graviton
propagator as its transverse part, and the rest as its longitudinal part.

###  II.5 Interactions among scalars, gluons, and gravitons

In preceding discussions, we methodically derived the propagators pertinent
for gluons, gravitons, and scalars with an arbitrary effective mass; we also
presented the vertex factors for gluon and graviton-only interactions in
(2.20) (recall that the vertex factor for scalar-only interaction is simply a
constant). This section aims to delve into the intricate dynamics of inter-
particle interactions, thereby laying a foundational framework for the ensuing
discourse in this paper.

Consider the simplest self-consistent theory with non-trivial interaction
pieces, i.e. scalar-QCD (and its special case scalar-QED) with the action

$S=\sum\limits_{i=1}^{N_{f}}\int
d^{d+1}x\sqrt{g}\bigg{(}g^{\mu\nu}\left(\mathcal{D}_{\mu}\Phi_{i}\right)^{\dagger}\mathcal{D}_{\nu}\Phi_{i}\\\
+\mu^{2}_{i}\Phi_{i}^{\dagger}\Phi_{i}+\frac{1}{2}\mathrm{tr}\left(F_{\mu\nu}F^{\mu\nu}\right)\bigg{)}$
(2.24)

for the gauge covariant derivative $\mathcal{D}$ and the field stress tensor
$F_{\mu\nu}\coloneqq[\mathcal{D}_{\mu},\mathcal{D}_{\nu}]/(ie)$. The action is
written for $N_{f}$ flavors of scalars with a diagonalized mass matrix —hence
$(\mu^{2})_{ij}\Phi_{i}^{\dagger}\Phi_{j}=\mu^{2}_{i}\Phi_{i}^{\dagger}\Phi_{i}$;
we also suppressed the color indices in $\Phi_{i}$, which can be in any
representation of the color group. Finally, we note that $A^{\dagger}B$ refers
to the singlet in the branching of the product representation
$A^{\dagger}\otimes B$; usually, one considers the fundamental representation
(hence a vector $\Phi$) or the adjoint representation (hence a matrix $\Phi$).

Let us proceed with a scalar field $\Phi_{i}$ in the fundamental
representation of the color group; we can then immediately write down

$\displaystyle(\mathcal{D}_{\mu}\Phi_{i})_{a}=\nabla_{\mu}\Phi_{i,a}+ie(A_{\mu})_{a}^{\;\;b}\Phi_{i,b}$
(2.25)

for the color indices $a$ & $b$, where $A$ is in the adjoint representation as
usual and $\nabla$ is the standard spacetime covariant derivative. By
inserting this in the action above and expanding in the small parameter $e$,
we can extract several leading-order interaction Lagrangians from which the
relevant vertex factors for the perturbation theory can be derived; for
instance, _gluon emission of a scalar_ can be derived from

$\displaystyle S_{\text{int}}=ie\sum\limits_{i=1}^{N_{f}}\int
d^{d+1}x\sqrt{g}\left(\Phi_{i,a}^{*}\overleftrightarrow{\nabla_{\mu}}\Phi_{i,b}\right)(A^{\mu})^{ab}.$
(2.26)

where $A\overleftrightarrow{\partial}B\equiv A\partial B-(\partial A)B$ as
usual. Furthermore one can also derive higher point interactions by inserting
$g=g_{\text{AdS}}+h$ and expanding for small $h$.

The next theory we will analyze is that of a scalar in a _dynamical_
background; as a simple model, consider the action

$S=\int
d^{d+1}x\sqrt{g}\bigg{(}-\frac{1}{2}\left(g^{\mu\nu}(\partial_{\mu}\phi)(\partial_{\nu}\phi)+m^{2}\phi^{2}\right)\\\
+\frac{2-\xi\phi^{2}}{2}R[g]-2\Lambda\bigg{)}$ (2.27)

and expand it around AdS, i.e. $g=g_{\text{AdS}}+h$ for small $h$; as before,
we can get interesting interaction Lagrangians. As an example, _graviton
emission of a scalar_ can be derived from

$\displaystyle S_{\text{int}}=\int
d^{d+1}x\sqrt{g}h^{\mu\nu}\left((\partial_{\mu}\phi)(\partial_{\nu}\phi)-\frac{1}{2}g_{\mu\nu}g^{\alpha\beta}(\partial_{\alpha}\phi)(\partial_{\beta}\phi)\right)$
(2.28)

where the metric $g$ is taken to be the background metric in the leading
order.

In the following sections, we will discuss such interactions in greater
depth.111111We would like to note that the name of this subsection is not
meant to imply a complete discussion of all possible interactions: we only
introduced the cases which we will utilize in the rest of the paper. For
instance, one could add a cubic interaction between two different flavors of
scalars with different $U(1)$ charges in a scalar-QED, i.e.  $\displaystyle
S_{\text{int}}=\lambda\int d^{d+1}x\sqrt{g}\;\Phi\Phi(\Phi^{\prime})^{*}$
(2.29) for $q(\Phi^{\prime})=2q(\Phi)$ where $q$ denotes the $U(1)$ charge.

##  III Explicit Computations in AdSd+1

In this section, our investigation expands into various AdS theories of
physical significance, and compute contributions to the four-point correlation
functions at leading order in the perturbation theory. Our analysis is
distinguished by its breadth, covering purely scalar theories, scalar-QED,
scalar gravity, and pure Yang-Mills gauge theory, and it spans across diverse
dimensions. We leverage the orthogonal decomposition formalism, delineated in
Section II.4, as our computational framework.

###  III.1 Scalar-scalar interaction

Let us start by considering the following two-derivative scalar interaction
action

$\displaystyle
S_{\text{int}}^{\Phi,\Psi}=\int_{\text{AdS}}d^{d+1}x\sqrt{g}(\nabla\Phi)^{2}\Psi$
(3.1)

where $\Psi$ is viewed as a perturbative scalar field (e.g., $\Psi$ absorbs
the weak coupling constant). The contribution to the correlation function of
external $\Phi$’s due to the exchange of an internal $\Psi$, denoted as
$\big{\langle}\Phi(\textbf{x}_{1})\Phi(\textbf{x}_{2})\Phi(\textbf{x}_{3})\Phi(\textbf{x}_{4})\big{\rangle}_{\Psi}$,
is equivalent to the four-point Feynman diagram in leading order:

$\big{\langle}\Phi(\textbf{x}_{1})\Phi(\textbf{x}_{2})\Phi(\textbf{x}_{3})\Phi(\textbf{x}_{4})\big{\rangle}_{\Psi}=\int\frac{dzdz^{\prime}d\textbf{x}^{d}d\textbf{x}^{\prime
d}}{z^{d+1}z^{\prime d+1}}\\\
\times\left(g^{\mu\nu}\partial_{\mu}\mathcal{G}^{\Phi}(z,\mathbf{x};\mathbf{x}_{1})\partial_{\nu}\mathcal{G}^{\Phi}(z,\mathbf{x};\mathbf{x}_{2})\right)\mathcal{G}^{\Psi}(z,\textbf{x};z^{\prime},\textbf{x}^{\prime})\\\
\times\left(g^{\delta\lambda}\partial_{\delta}\mathcal{G}^{\Phi}(z^{\prime},\mathbf{x}^{\prime};\mathbf{x}_{3})\partial_{\lambda}\mathcal{G}^{\Phi}(z^{\prime},\mathbf{x}^{\prime};\mathbf{x}_{4})\right)$
(3.2)

where $g_{\mu\nu}$ is the AdS Poincaré metric and the volume factor
$z^{-(d+1)}$ comes from the metric determinant $g$. Note that
$\mathcal{G}^{\Phi}$ are the bulk-to-boundary propagators of scalar $\Phi$
with points $\textbf{x}_{a}$ on AdSd+1 boundary, while $\mathcal{G}^{\Psi}$ is
the bulk-to-bulk propagator of scalar $\Psi$.

Although it is rather straightforward to generate the leading order correlator
in the position space, we are usually interested in momentum-space
expressions, not least because of their relevance for the observational data.
The translational invariance at the boundary dictates an overall momentum
conservation. Furthermore, note that (3.2) contains two distinct structures

$\displaystyle
g^{zz}(\partial_{z}\mathcal{G}^{\Phi})(\partial_{z}\mathcal{G}^{\Phi})\rightarrow{}$
$\displaystyle{}z^{2}r_{\nu_{\Phi}}(k_{a},k_{b},z)\mathcal{G}^{\Phi}(z,\mathbf{k}_{a})\mathcal{G}^{\Phi}(z,\mathbf{k}_{b})$
(3.3a) $\displaystyle
g^{ij}(\partial_{i}\mathcal{G}^{\Phi})(\partial_{j}\mathcal{G}^{\Phi})\
\rightarrow{}$
$\displaystyle{}-z^{2}\textbf{k}_{a}\cdot\textbf{k}_{b}\mathcal{G}^{\Phi}(z,\mathbf{k}_{a})\mathcal{G}^{\Phi}(z,\mathbf{k}_{b}),$
(3.3b)

where the function $r_{\nu}$ on the first line is

$\displaystyle\bigg{(}\frac{d-2\nu}{2z}-k_{a}\frac{K_{\nu-1}(k_{a}z)}{K_{\nu}(k_{a}z)}\bigg{)}\bigg{(}\frac{d-2\nu}{2z}-k_{b}\frac{K_{\nu-1}(k_{b}z)}{K_{\nu}(k_{b}z)}\bigg{)}.$
(3.4)

Therefore, we have a mixture of terms with different $z$-powers, and the
highest $z$-power term comes from (3.3b) as well as from the ratio of Bessel
$K$ functions within $r_{\nu}$. In the final expression of the correlator, the
highest $z$-power term contributes to the leading order total-energy
structure, which is relevant in the flat-space limit, while the lower
$z$-power terms are correction terms to the subleading total-energy
structures. We demonstrate below the evaluation of the highest $z$-power
contribution in the correlator.

Let us first consider the contribution of (3.3b) in (3.2). We can conveniently
extract a pair of momentum inner product $(\mathbf{k}_{a}\cdot\mathbf{k}_{b})$
appropriate for the channel we are interested in: this leads to the following
$s$-channel expression

$(2\pi)^{d}\delta^{d}\bigg{(}\sum_{a=1}^{4}\textbf{k}_{a}\bigg{)}\times(\textbf{k}_{1}\cdot\textbf{k}_{2})(\textbf{k}_{3}\cdot\textbf{k}_{4})\mathcal{M}_{\perp;\
2}^{\nu_{\Phi},\nu_{\Phi},\nu_{\Phi},\nu_{\Phi};\nu_{\Psi}}$ (3.5)

for the “seed” amplitudes

$\displaystyle\mathcal{M}_{\perp;\
n}^{\nu_{1},\nu_{2},\nu_{3},\nu_{4};i}\coloneqq$
$\displaystyle\int_{0}^{\infty}\frac{pdp}{s^{2}+p^{2}}\mathcal{K}\mathcal{K}\mathcal{J}^{(n)}_{\nu_{1},\nu_{2},i}(k_{1},k_{2},p)\mathcal{K}\mathcal{K}\mathcal{J}^{(n)}_{\nu_{3},\nu_{4},i}(k_{3},k_{4},p)$
(3.6a) $\displaystyle\mathcal{M}_{\parallel;\
n}^{\nu_{1},\nu_{2},\nu_{3},\nu_{4};i}\coloneqq$
$\displaystyle\int_{0}^{\infty}\frac{dp}{p}\mathcal{K}\mathcal{K}\mathcal{J}^{(n)}_{\nu_{1},\nu_{2},i}(k_{1},k_{2},p)\mathcal{K}\mathcal{K}\mathcal{J}^{(n)}_{\nu_{3},\nu_{4},i}(k_{3},k_{4},p),$
(3.6b)

where $s=\absolutevalue{\mathbf{k}_{1}+\mathbf{k}_{2}}$ and the function
$\mathcal{K}\mathcal{K}\mathcal{J}$ is defined in accordance with (2.14) and
(2.8):

$\displaystyle\mathcal{K}\mathcal{K}\mathcal{J}^{(n)}_{\nu_{a},\nu_{b},i}\coloneqq\frac{2k_{a}^{\nu_{a}}k_{b}^{\nu_{b}}}{\pi}\int_{0}^{\infty}z^{\frac{d-2}{2}+n}dzK_{\nu_{a}}(k_{a}z)K_{\nu_{b}}(k_{b}z)J_{i}(pz)\
.$ (3.7)

Note that the number of derivatives is $n=2$ in the action (3.1). Although the
second line (3.6b) does not appear in the scalar four-point function (3.5), it
is shown here for completeness, and as we will show later that it appears in
the longitudinal components of four-point functions in scalar-QED and other
theories (as the subscript already indicates, the first line (3.6a) would
appear in the corresponding transverse components).

On the other hand, the contribution of (3.3a) in (3.2) depends on the specific
value of $\nu_{\Phi}$. Let us thus examine a simple scalar interaction theory
in which the two scalars $\sigma$ and $\phi$, with their own quadratic actions
given as (2.2), have effective mass squares $\mu_{\sigma}^{2}=1-d$ and
$\mu_{\phi}^{2}=0$, respectively. The interaction of the form
$\int(\nabla\sigma)^{2}\phi$ then yields the following four-point correlator
in $s$-channel121212Other channels can be obtained by a simple permutation of
the momenta.

$\big{\langle}\sigma(\textbf{k}_{1})\sigma(\textbf{k}_{2})\sigma(\textbf{k}_{3})\sigma(\textbf{k}_{4})\big{\rangle}_{\phi}^{(s)}=(2\pi)^{d}\delta^{d}\bigg{(}\sum_{a=1}^{4}\textbf{k}_{a}\bigg{)}\\\
\times(k_{1}k_{2}-\textbf{k}_{1}\cdot\textbf{k}_{2})(k_{3}k_{4}-\textbf{k}_{3}\cdot\textbf{k}_{4})\mathcal{M}_{\perp;\
2}^{\nu_{\sigma},\nu_{\sigma},\nu_{\sigma},\nu_{\sigma};\nu_{\phi}}+c.t.$
(3.8)

where the correction terms due to lower $z$-powers are denoted as “$c.t.$".
Comparing with (3.5), it is evident that the highest $z$-power contribution
from (3.3a) generates momentum scalar products $(k_{a}k_{b})$.

Let us look at two specific dimensions as examples. At $d=3$, the interaction
of the form $\int(\nabla\sigma)^{2}\phi$ would then lead to

$\displaystyle\mathcal{K}\mathcal{K}\mathcal{J}_{\nu_{\sigma},\nu_{\sigma},\nu_{\phi}}^{(2)}(k_{1},k_{2},p)=\sqrt{\frac{2}{\pi}}\frac{2p^{\frac{3}{2}}}{(k_{12}^{2}+p^{2})^{2}},$
(3.9)

where we have used a shorthand notation $k_{ab}=k_{a}+k_{b}$. At this
dimension, $\sigma$ and $\phi$ are commonly referred to as conformally coupled
scalar and massless scalar in literature, respectively, with
$\Delta_{\sigma}=2$ and $\Delta_{\phi}=3$. Using the above
$\mathcal{K}\mathcal{K}\mathcal{J}$ expression, we then obtain the following
evaluation for correlator
$\langle\sigma\sigma\sigma\sigma\rangle_{\phi}^{(s)}$ in AdS4

$(2\pi)^{3}\delta^{3}\bigg{(}\sum_{a=1}^{4}\textbf{k}_{a}\bigg{)}\frac{2(E_{L}E_{R}+sE)}{E_{L}^{2}E_{R}^{2}E^{3}}\\\
\times(k_{1}k_{2}-\textbf{k}_{1}\cdot\textbf{k}_{2})(k_{3}k_{4}-\textbf{k}_{3}\cdot\textbf{k}_{4})+c.t.$
(3.10)

where $E_{L,R}$ are left/right partial energy sums and $E$ is the total
energy.131313 In terms of momenta, these variables read as  $\displaystyle
E=k_{1}+k_{2}+k_{3}+k_{4}\;,\quad E_{L}=s+k_{1}+k_{2}\;,\quad
E_{R}=s+k_{3}+k_{4}.$ (3.11) We can straightforwardly check that we have the
correct flat-space limit as well, which amounts to obtaining the coefficient
of the leading total-energy pole when $E$ approaches zero [53, 47], with which
we arrive at $S$ for four-dimensional Mandelstam variable
$S\coloneqq(k_{1}+k_{2})^{2}-s^{2}$.141414The other two Mandelstam variables
are  $\displaystyle T\coloneqq(k_{1}+k_{4})^{2}-t^{2},\quad
U\coloneqq(k_{1}+k_{3})^{2}-u^{2}$ (3.12) where
$t=|\textbf{k}_{1}+\textbf{k}_{4}|$ and $u=|\textbf{k}_{1}+\textbf{k}_{3}|$
(and $s=|\mathbf{k}_{1}+\mathbf{k}_{2}|$) are the exchange momenta in
respective channels. Therefore, the total four-point amplitude
$\mathcal{A}=S+T+U$ is a constant.  For the same pair of interactive scalars,
we can repeat the process to obtain the correlator expression in AdS6
151515The $R_{n}$ functions in the seed amplitude are defined as
$\displaystyle R_{5}=$ $\displaystyle
12k_{1}k_{2}k_{3}(k_{1}+k_{2}+k_{3})(k_{12}-s)^{2}E_{L}^{2}E_{R}^{3}$ (3.13a)
$\displaystyle R_{4}=$ $\displaystyle
3E_{L}E_{R}^{3}(k_{12}-s)\big{[}k_{1}k_{2}k_{12}(k_{12}^{2}-s^{2})$
$\displaystyle+k_{3}\big{(}k_{12}^{4}-(k_{1}-k_{2})^{2}s^{2}\big{)}+k_{12}k_{3}^{2}(k_{12}^{2}+4k_{1}k_{2}-s^{2})\big{]}$
(3.13b) $\displaystyle R_{3}=$
$\displaystyle\big{[}k_{12}^{6}-s^{2}k_{12}^{4}+s^{2}(k_{1}-k_{2})^{2}(s^{2}-k_{12}^{2})$
$\displaystyle+\big{(}2s^{2}k_{12}k_{3}+k_{3}^{2}(3k_{12}^{2}-s^{2})\big{)}(k_{12}^{2}+4k_{1}k_{2}-s^{2})\big{]}E_{R}^{3}$
(3.13c) $\displaystyle R_{2}=$
$\displaystyle(k_{12}^{2}+4k_{1}k_{2}-s^{2})\big{[}s^{2}(k_{1}+k_{2}+k_{3})+3k_{12}k_{3}^{2}\big{]}E_{R}^{3}$
(3.13d) $\displaystyle R_{1}=$
$\displaystyle(k_{12}^{2}+4k_{1}k_{2}-s^{2})(s^{2}+3k_{3}^{2})E_{R}^{3}$
(3.13e) $\displaystyle R_{0}=$
$\displaystyle(k_{12}^{2}+4k_{1}k_{2}-s^{2})\big{[}-2s^{2}k_{3}(s+k_{3})-s(s^{2}+sk_{3}+3k_{3}^{2})E_{R}$
$\displaystyle-(s^{2}+3k_{3}^{2})E_{R}^{2}\big{]}.$ (3.13f)

$(2\pi)^{5}\delta^{5}\bigg{(}\sum_{a=1}^{4}\textbf{k}_{a}\bigg{)}\frac{2}{E_{L}^{3}E_{R}^{3}(k_{12}-s)^{3}}\bigg{(}\sum_{n=0}^{5}\frac{R_{n}}{E^{n}}\bigg{)}\\\
\times(k_{1}k_{2}-\textbf{k}_{1}\cdot\textbf{k}_{2})(k_{3}k_{4}-\textbf{k}_{3}\cdot\textbf{k}_{4})+c.t.$
(3.14)

which then expectedly leads to the same flat-space amplitude as the AdS4
correlator (up to a pre-factor of $k_{1}k_{2}k_{3}k_{4}$).

###  III.2 Scalar-QED

The procedure of obtaining the contributions to the correlation functions at
the leading order in perturbation theory remains the same when we move from
scalar-only interaction to the scalar-QED. We start with the relevant
interaction action, which in this case is

$\displaystyle
S_{\text{int}}^{\text{sQED}}=e\int_{\text{AdS}}d^{d+1}x\sqrt{g}\big{(}{\widetilde{\Psi}}\nabla_{\mu}\Psi-\Psi\nabla_{\mu}{\widetilde{\Psi}})A^{\mu},$
(3.15)

where $\Psi$ and $\widetilde{\Psi}$ are real and imaginary parts of the
complex scalar field respectively.161616 Explicitly, we define
$\displaystyle\Psi=\frac{\big{(}\Phi+\Phi^{*}\big{)}}{\sqrt{2}},\quad\widetilde{\Psi}=\frac{\big{(}\Phi-\Phi^{*}\big{)}}{i\sqrt{2}}$
(3.16) where $\Phi$ and $\Phi^{*}$ are complex scalar fields.  We will now
proceed with the actual computation of the Feynman diagrams; however, we would
like to specialize in two physically relevant cases in what follows, namely

conformally coupled scalar:
$\displaystyle\nu_{\varphi}=\frac{1}{2}\qquad\rightarrow\;\Delta_{\varphi}=\frac{d+1}{2}$
(3.17a) vectorlike scalar:
$\displaystyle\nu_{\sigma}=\frac{d-2}{2}\;\rightarrow\;\Delta_{\sigma}=d-1$
(3.17b)

where $\nu$ is the index that appears in the propagators, and $\Delta$ is the
scaling dimension of the boundary operator that is dual to the relevant bulk
field. Although these two cases collapses into one at $d=3$, the most commonly
analyzed dimension in the literature, they physically describe different
scalars in general dimensions: $\nu=1/2$ (along with the additional
masslessness condition) makes the theory (2.2) perturbatively invariant under
conformal transformations, whereas $\nu=(d-2)/2$ makes the scalar mimic the
scaling behavior of the spin$-1$ bulk fields as we already discussed within
scalar-scalar interactions.

Conformally coupled scalars have received extensive attention in the
literature, owing to their relevance and inherent simplicity. Indeed, their
straightforward nature not only facilitates deeper understanding but also
allows them to serve as a foundational basis for computing more complex
theories, as discussed in [52, 11]; in fact, their computations are almost
identical to ordinary field theoretical computations in half flat
spacetime.171717This follows from the nice behavior of the conformally coupled
scalar under a Weyl transformation that relates the metrics
$ds^{2}=\frac{dz^{2}+dx_{i}dx^{i}}{z^{2}}$ and $ds^{2}=dz^{2}+dx_{i}dx^{i}$.
On the contrary, vectorlike scalars with $\Delta=d-1$ which are not
conformally invariant are not analyzed as much;181818The exception is $d=3$ at
which vectorlike scalars coincide with the conformally coupled scalars and
hence have been extensively studied. nevertheless, due to the common scaling
dimension, it becomes rather interesting to study the interaction of
vectorlike scalars and spin-1 gauge fields, an interesting special case of
scalar QED.

For both conformally coupled and vectorlike scalars, we can immediately write
down the perturbative expression for the correlation function in the leading
order; explicitly,

$\expectationvalue{\Psi(\mathbf{x}_{1})\tilde{\Psi}(\mathbf{x}_{2})\Psi(\mathbf{x}_{3})\tilde{\Psi}(\mathbf{x}_{4})}_{A}=\int\frac{dzdz^{\prime}d\textbf{x}^{d}d\textbf{x}^{\prime
d}}{z^{d+1}z^{\prime
d+1}}\mathcal{G}_{i\ell}(z,\textbf{x};z^{\prime},\textbf{x}^{\prime})\\\
\times\left(g^{ij}\mathcal{G}^{\widetilde{\Psi}}(z,\mathbf{x};\mathbf{x}_{1})\partial_{j}\mathcal{G}^{\Psi}(z,\mathbf{x};\mathbf{x}_{2})-(\Psi\leftrightarrow\widetilde{\Psi})\right)\\\
\times\left(g^{\ell
m}\mathcal{G}^{\widetilde{\Psi}}(z^{\prime},\mathbf{x}^{\prime};\mathbf{x}_{3})\partial_{m}\mathcal{G}^{\Psi}(z^{\prime},\mathbf{x}^{\prime};\mathbf{x}_{4})-(\Psi\leftrightarrow\widetilde{\Psi})\right)$
(3.18)

which is the contribution to the four-point correlator due to the photon
exchange.191919We remind the reader that we choose the axial gauge throughout
the paper, hence $A^{z}=A^{0}=0$ and there is no $g^{zz}$ contribution.  We
will be explicitly working in the $s$-channel, but the $t$\- and $u$\-
channels can be obtained via an appropriate permutation of the indices. We can
now go ahead and decompose this correlator into transverse and longitudinal
parts. In momentum space, the correlator expression for conformally coupled
scalars reads as

$\big{\langle}\varphi(\textbf{k}_{1})\tilde{\varphi}(\textbf{k}_{2})\varphi(\textbf{k}_{3})\tilde{\varphi}(\textbf{k}_{4})\big{\rangle}_{A,\alpha}^{(s)}=(2\pi)^{d}\delta^{d}\bigg{(}\sum_{a=1}^{4}\textbf{k}_{a}\bigg{)}\\\
\times(\textbf{k}_{1}-\textbf{k}_{2})^{i}(\textbf{k}_{3}-\textbf{k}_{4})^{j}\widehat{P}_{ij}^{\alpha}(\textbf{k}_{1}+\textbf{k}_{2})\mathcal{M}_{\alpha;\
1}^{\nu_{\varphi},\nu_{\varphi},\nu_{\varphi},\nu_{\varphi};\nu_{A}}$ (3.19)

where $\alpha=\perp,\parallel$ and the projectors $\widehat{P}_{ij}^{\alpha}$
are defined in eqn. (2.21), while the seed amplitudes $\mathcal{M}_{\alpha}$
are defined in (3.6). Note that one can obtain the same set of expressions for
vectorlike scalars by simply switching the index $\nu_{\varphi}$ to
$\nu_{\sigma}$ in the above expressions.

We now present the explicit evaluation for $d=3$ and $d=5$ below. Even though
one can also write down implicit results in general dimensions, explicit
results are harder to obtain as the expression in generic dimensions is an
integral of products of hypergeometric functions. After a little bit algebra,
we can show that the full correlator in AdS4 reads

$\displaystyle\langle\varphi\tilde{\varphi}\varphi\tilde{\varphi}\rangle_{A,\perp}^{(s)}\evaluated{}_{d=3}$
$\displaystyle=\frac{(\textbf{k}_{1}-\textbf{k}_{2})\cdot(\textbf{k}_{3}-\textbf{k}_{4})+s^{-2}(k_{1}^{2}-k_{2}^{2})(k_{3}^{2}-k_{4}^{2})}{E_{L}E_{R}E}$
(3.20)
$\displaystyle\langle\varphi\tilde{\varphi}\varphi\tilde{\varphi}\rangle_{A,\parallel}^{(s)}\evaluated{}_{d=3}$
$\displaystyle=-\frac{(k_{1}-k_{2})(k_{3}-k_{4})}{s^{2}E},$

upto the prefactor
$(2\pi)^{3}\delta^{3}\big{(}\sum_{a=1}^{4}\textbf{k}_{a}\big{)}$.202020 We
remind the reader that these results are applicable both for conformally
coupled and vectorlike scalars as they coincide in AdS4.  It is
straightforward to check that the full correlator expression above is
consistent with the corresponding flat-space amplitude. Indeed, in the flat-
space limit $E\rightarrow 0$, the coefficient of the total-energy pole in the
correlator
$\langle\varphi\tilde{\varphi}\varphi\tilde{\varphi}\rangle_{A}^{(s)}$ becomes
the corresponding flat-space scattering amplitude

$\displaystyle\mathcal{A}_{\varphi\tilde{\varphi}\varphi\tilde{\varphi}}^{(s)}\evaluated{}_{d=3}$
$\displaystyle=\lim_{E\rightarrow
0}\frac{(\textbf{k}_{1}-\textbf{k}_{2})\cdot(\textbf{k}_{3}-\textbf{k}_{4})-(k_{1}-k_{2})(k_{3}-k_{4})}{E_{L}E_{R}},$
(3.21)

which is simply $2S^{-1}(U-T)$, where we have imposed conservation of momentum
and $S,T,U$ are the four-dimensional flat-space Mandelstam variables defined
earlier. The same set of momentum-space expressions can be derived similarly
in AdS6. For the conformally coupled scalar, we have

$\displaystyle\langle\varphi\tilde{\varphi}\varphi\tilde{\varphi}\rangle_{A,\perp}^{(s)}\evaluated{}_{d=5}=$
$\displaystyle\frac{(\textbf{k}_{1}-\textbf{k}_{2})\cdot(\textbf{k}_{3}-\textbf{k}_{4})+s^{-2}(k_{1}^{2}-k_{2}^{2})(k_{3}^{2}-k_{4}^{2})}{E_{L}^{2}E_{R}^{2}}$
$\displaystyle\times\frac{2\big{(}E_{L}E_{R}+sE\big{)}}{E^{3}}$ (3.22a)
$\displaystyle\langle\varphi\tilde{\varphi}\varphi\tilde{\varphi}\rangle_{A,\parallel}^{(s)}\evaluated{}_{d=5}=$
$\displaystyle-\frac{2(k_{1}-k_{2})(k_{3}-k_{4})}{s^{2}E^{3}}$ (3.22b)

with the prefactor
$(2\pi)^{5}\delta^{5}\big{(}\sum_{a=1}^{4}\textbf{k}_{a}\big{)}$; and for the
vectorlike scalar we have212121 The functions $B_{n}$ in the transverse
expression are defined as  $\displaystyle
B_{3}=+2k_{1}k_{2}k_{3}(k_{1}+k_{2}+k_{3})(k_{12}-s)E_{L}E_{R}^{2}$ (3.23)
$\displaystyle
B_{2}=+\big{[}2k_{1}k_{2}k_{3}(s^{2}+k_{3}k_{12})-(k_{12}-s)E_{L}k_{12}k_{13}k_{23}\big{]}E_{R}^{2}$
(3.24) $\displaystyle
B_{1}=+\big{[}k_{12}^{4}-s^{2}(k_{1}^{2}+k_{2}^{2}+k_{3}^{2})+(k_{12}^{2}+2k_{1}k_{2})k_{3}^{2}\big{]}E_{R}^{2}$
(3.25) $\displaystyle
B_{0}=-\big{[}2k_{1}k_{2}+E_{L}(k_{12}-s)\big{]}\big{[}(s+k_{3})sk_{3}+(s^{2}+k_{3}^{2})E_{R}\big{]}.$
(3.26)

$\displaystyle\langle\sigma\tilde{\sigma}\sigma\tilde{\sigma}\rangle_{A,\perp}^{(s)}\evaluated{}_{d=5}=$
$\displaystyle\frac{(\textbf{k}_{1}-\textbf{k}_{2})\cdot(\textbf{k}_{3}-\textbf{k}_{4})+s^{-2}(k_{1}^{2}-k_{2}^{2})(k_{3}^{2}-k_{4}^{2})}{(k_{12}-s)^{2}E_{L}^{2}E_{R}^{2}}$
$\displaystyle\times\bigg{(}\sum_{n=0}^{3}\frac{B_{n}}{E^{n}}\bigg{)}$ (3.27a)
$\displaystyle\langle\sigma\tilde{\sigma}\sigma\tilde{\sigma}\rangle_{A,\parallel}^{(s)}\evaluated{}_{d=5}=$
$\displaystyle-\frac{2k_{1}k_{2}k_{3}k_{4}+(k_{12}k_{3}k_{4}+k_{1}k_{2}k_{34})E+k_{12}k_{34}E^{2}}{s^{2}}$
$\displaystyle\times\frac{(k_{1}-k_{2})(k_{3}-k_{4})}{E^{3}}$ (3.27b)

with the prefactor
$(2\pi)^{5}\delta^{5}\big{(}\sum_{a=1}^{4}\textbf{k}_{a}\big{)}$. As expected,
the flat-space limit of both correlators give the same form of amplitude as in
(3.21), expressable in terms of six-dimensional Mandelstam variables (up to a
pre-factor of $k_{1}k_{2}k_{3}k_{4}$ for $\sigma$).

###  III.3 Scalar-gravity

In the previous subsection, we analyzed _the conformally coupled_ and the so-
defined _vectorlike_ scalars, which are dual to boundary CFT operators with
scaling dimensions $\frac{d+1}{2}$ and $d-1$, respectively. In this section,
however, we will instead consider a new type of scalar that we will dub as
“tensorlike”, i.e.

$\displaystyle\text{tensorlike scalar:
}\nu=\frac{d}{2}\;\rightarrow\;\Delta=d.$ (3.28)

This class of scalars has trivial effective mass $\mu^{2}=0$, in which the
minimally coupled scalar with $m=0$ is often referred to as the massless
scalar in literature. Besides the obvious observation that tensorlike scalars
share the scaling dimension of the graviton with which they mix in the scalar-
gravity interaction (making the resultant theory rather interesting), such
scalars are also relevant as they may be numerically indistinguishable from
the contributions of conserved vector fields in the boundary CFT.222222More
specifically, traditional numerical conformal bootstrap techniques cannot
distinguish a “fake” scalar of scaling dimension of $d$ from a conserved
vector with the scaling dimension $d-1$, as they induce indistinguishable
contributions [54].

For the four-point correlation function of the external tensorlike scalars
with graviton exchange, the leading-order perturbative contribution can be
written down using the scalar-gravity interaction in (2.28). In the position
space, the correlator reads as

$\expectationvalue{\phi(\textbf{x}_{1})\phi(\textbf{x}_{2})\phi(\textbf{x}_{3})\phi(\textbf{x}_{4})}_{G}=\int\frac{dzdz^{\prime}d\textbf{x}^{d}d\textbf{x}^{\prime
d}}{z^{d+1}z^{\prime d+1}}\\\
\times\left(\bar{g}^{i_{1}i_{2}}\bar{g}^{j_{1}j_{2}}\mathcal{T}_{i_{1}j_{1}}(z,\textbf{x};\textbf{x}_{1},\textbf{x}_{2})\right)\mathcal{G}_{i_{2}j_{2},\ell_{2}m_{2}}(z,\textbf{x};z^{\prime},\textbf{x}^{\prime})\\\
\times\left(\bar{g}^{\ell_{1}\ell_{2}}\bar{g}^{m_{1}m_{2}}\mathcal{T}_{\ell_{1}m_{1}}(z^{\prime},\textbf{x}^{\prime};\textbf{x}_{3},\textbf{x}_{4})\right)$
(3.29)

where we define

$\mathcal{T}_{\mu\nu}(z,\textbf{x};\textbf{x}_{a},\textbf{x}_{b})\coloneqq\partial_{\mu}\mathcal{G}^{\phi}(z,\textbf{x};\textbf{x}_{a})\partial_{\nu}\mathcal{G}^{\phi}(z,\textbf{x};\textbf{x}_{b})\\\
-\frac{1}{2}\bar{g}_{\mu\nu}\bar{g}^{\alpha\beta}\partial_{\alpha}\mathcal{G}^{\phi}(z,\textbf{x};\textbf{x}_{a})\partial_{\beta}\mathcal{G}^{\phi}(z,\textbf{x};\textbf{x}_{b})$
(3.30)

for brevity.232323 We remind the reader that the overall metric decomposes as
$g_{\mu\nu}=g_{AdS}+h_{\mu\nu}$ with perturbative $h_{\mu\nu}$, and for
clarity we use $\bar{g}=g_{AdS}$ in the expressions. Note also that we work in
the axial gauge for the metric perturbation ($h_{zz}=0,h_{zi}=0$), and thus
there is no $\bar{g}^{zz}$ contribution.

We proceed to decompose the correlator into its transverse and orthogonal
components, as in § II.4. This analysis is conducted in momentum space,
paralleling our prior approach. Our primary focus is on the $s$-channel, with
the understanding that the $t$\- and $u$-channels can be derived through
suitable permutations. A modest amount of algebra yields the transverse and
longitudinal parts of the four-point graviton-exchanging function as

$\langle\phi(\textbf{k}_{1})\phi(\textbf{k}_{2})\phi(\textbf{k}_{3})\phi(\textbf{k}_{4})\rangle_{G,\perp}^{(s)}\
=\textbf{k}_{1}^{i}\textbf{k}_{2}^{j}\textbf{k}_{3}^{\ell}\textbf{k}_{4}^{m}\\\
\times\widehat{\mathcal{P}}_{ij,\ell
m}^{(\perp,\perp)}(\mathbf{k}_{1}+\mathbf{k}_{2})\mathcal{M}_{\perp;\
2}^{\nu_{\phi},\nu_{\phi},\nu_{\phi},\nu_{\phi};\nu_{G}}$ (3.31a) and
$\displaystyle\langle\phi(\textbf{k}_{1})\phi(\textbf{k}_{2})\phi(\textbf{k}_{3})\phi(\textbf{k}_{4})\rangle_{G,\parallel}^{(s)}\
=\sum_{b=1}^{2}\bar{\mathcal{M}}_{\parallel,b;\
2}^{\nu_{\phi},\nu_{\phi},\nu_{\phi},\nu_{\phi};\nu_{G}}$ (3.31b) upto an
overall factor
$(2\pi)^{d}\delta^{d}\left(\sum_{a=1}^{4}\textbf{k}_{a}\right)$.242424
Remember that the projectors and the transverse seed amplitude are defined in
(2.23) and (3.6a) respectively.  The _modified_ longitudinal seed amplitude
above is defined as  $\bar{\mathcal{M}}_{\parallel,b;\
2}^{\nu_{\phi},\nu_{\phi},\nu_{\phi},\nu_{\phi};\nu_{G}}=2\textbf{k}_{1}^{i_{1}}\textbf{k}_{2}^{j_{1}}\textbf{k}_{3}^{\ell_{1}}\textbf{k}_{4}^{m_{1}}\widehat{\mathcal{P}}_{i_{1}j_{1},\ell_{1}m_{1}}^{b}\int_{0}^{\infty}dp\mathcal{K}\mathcal{K}\mathcal{J}_{\nu_{\phi},\nu_{\phi},\nu_{G}}^{(2)}(k_{1},k_{2},p)\\\
\times
f^{b}\mathcal{K}\mathcal{K}\mathcal{J}_{\nu_{\phi},\nu_{\phi},\nu_{G}}^{(2)}(k_{3},k_{4},p)\\\
+\textbf{k}_{1}^{i}\textbf{k}_{2}^{j}\widehat{\mathcal{P}}_{ij,\ell\ell}^{b}\int_{0}^{\infty}dp\mathcal{K}\mathcal{K}\mathcal{J}_{\nu_{\phi},\nu_{\phi},\nu_{G}}^{(2)}(k_{1},k_{2},p)\\\
\times
f^{b}\overline{\mathcal{K}\mathcal{K}\mathcal{J}}_{\nu_{\phi},\nu_{\phi},\nu_{G}}^{(2)}(k_{3},k_{4},p)\\\
+\textbf{k}_{3}^{\ell}\textbf{k}_{4}^{m}\widehat{\mathcal{P}}_{ii,\ell
m}^{b}\int_{0}^{\infty}dp\overline{\mathcal{K}\mathcal{K}\mathcal{J}}_{\nu_{\phi},\nu_{\phi},\nu_{G}}^{(2)}(k_{1},k_{2},p)\\\
\times
f^{b}\mathcal{K}\mathcal{K}\mathcal{J}_{\nu_{\phi},\nu_{\phi},\nu_{G}}^{(2)}(k_{3},k_{4},p)\\\
+\frac{1}{2}\widehat{\mathcal{P}}_{ii,\ell\ell}^{b}\int_{0}^{\infty}dp\overline{\mathcal{K}\mathcal{K}\mathcal{J}}_{\nu_{\phi},\nu_{\phi},\nu_{G}}^{(2)}(k_{1},k_{2},p)\\\
\times
f^{b}\overline{\mathcal{K}\mathcal{K}\mathcal{J}}_{\nu_{\phi},\nu_{\phi},\nu_{G}}^{(2)}(k_{3},k_{4},p),$
(3.32)

where the $\overline{\mathcal{K}\mathcal{K}\mathcal{J}}$ functions contain
z-derivatives of the Bessel $K$ functions,252525We define
$\overline{\mathcal{K}\mathcal{K}\mathcal{J}}_{\nu_{\phi},\nu_{\phi},\nu_{G}}^{(n)}(\textbf{k}_{a},\textbf{k}_{b},p)$
as
$\displaystyle\overline{\mathcal{K}\mathcal{K}\mathcal{J}}_{\nu_{\phi},\nu_{\phi},\nu_{G}}^{(n)}(\textbf{k}_{a},\textbf{k}_{b},p)\coloneqq-\textbf{k}_{a}\cdot\textbf{k}_{b}\mathcal{K}\mathcal{K}\mathcal{J}_{\nu_{\phi},\nu_{\phi},\nu_{G}}^{(n)}(k_{a},k_{b},p)$
(3.33)
$\displaystyle+\int_{0}^{\infty}\frac{dz}{z^{d-3}}\partial_{z}\bigg{[}\sqrt{\frac{2}{\pi}}(k_{a}z)^{\frac{d}{2}}K_{\frac{d}{2}}(k_{a}z)\bigg{]}\partial_{z}\bigg{[}\sqrt{\frac{2}{\pi}}(k_{b}z)^{\frac{d}{2}}K_{\frac{d}{2}}(k_{b}z)\bigg{]}\bigg{[}z^{\frac{d}{2}-2}J_{\frac{d}{2}}(pz)\bigg{]}$
for $\nu_{\phi}=\nu_{G}=\frac{d}{2}$.  and where $\widehat{\mathcal{P}}$ and
$f^{b}$ are abbreviations for brevity.262626The projector sums here are
defined as $\widehat{\mathcal{P}}_{ij,\ell
m}^{b=1}=\widehat{\mathcal{P}}_{ij,\ell
m}^{(\perp,\parallel)}+\widehat{\mathcal{P}}_{ij,\ell
m}^{(\parallel,\perp)}+\widehat{\mathcal{P}}_{ij,\ell
m}^{(\parallel,\parallel)}$ and $\widehat{\mathcal{P}}_{ij,\ell
m}^{b=2}=\widehat{\mathcal{P}}_{ij,\ell m}^{(\parallel,\parallel)}$; using
definition (2.23), we can compute the sums whose indices are partially or
fully contracted
$\displaystyle\widehat{\mathcal{P}}_{ij,\ell\ell}^{b=1}=-\frac{2}{d-1}\eta_{ij},\quad\widehat{\mathcal{P}}_{ii,\ell
m}^{b=1}=-\frac{2}{d-1}\eta_{\ell m}$ (3.34)
$\displaystyle\widehat{\mathcal{P}}_{ij,\ell\ell}^{b=2}(\textbf{k})=+\frac{2(d-2)}{(d-1)}\frac{\textbf{k}_{i}\textbf{k}_{j}}{k^{2}},\quad\widehat{\mathcal{P}}_{ii,\ell
m}^{b=2}(\textbf{k})=+\frac{2(d-2)}{(d-1)}\frac{\textbf{k}_{\ell}\textbf{k}_{m}}{k^{2}}$
(3.35)
$\displaystyle\widehat{\mathcal{P}}_{ii,\ell\ell}^{b=1}=-\frac{2d}{(d-1)},\quad\widehat{\mathcal{P}}_{ii,\ell\ell}^{b=2}=+\frac{2(d-2)}{(d-1)}.$
(3.36) In addition, the $f^{b}$ functions are defined as $f^{1}=\frac{1}{p}$
and $f^{2}=\frac{k^{2}}{p^{3}}$.

This direct perturbative approach of the computation of this four-point
correlation function can be compared to many other existing techniques in
literature. In particular, similar decomposition for $d=3$ is considered in
[55], which follows from the decomposition of the $3d$ action into the
transverse and longitudinal parts [56]. Our final numerical evaluations of the
correlator agree for $d=3$ and also share the same formal expression of the
transverse part, and it would be interesting to explore if their decomposition
also generalizes to arbitrary $d$ in the same way our decomposition does.
Other techniques include: Arnowitt-Deser-Misner (ADM) and in-in formalism [57,
58], bootstrap formalism with weight-shifting operators [59], lifting
formalism from flat space [60, 61], bootless bootstrap formalism [60]272727 At
$d=3$, we identify our transverse and longitudinal parts with the following
structures in the bootstrap computation  $\displaystyle
12\big{(}\textbf{k}_{1}^{i}\textbf{k}_{2}^{j}\textbf{k}_{3}^{\ell}\textbf{k}_{4}^{m}\big{)}\widehat{\mathcal{P}}_{ij,\ell
m}^{(\perp,\perp)}\mathcal{M}_{\perp;2}^{\nu_{\phi},\nu_{\phi},\nu_{\phi},\nu_{\phi};\nu_{G}}=s^{4}\Pi_{2,2}^{(s)}f_{2,2}^{(s)}$
(3.37) $\displaystyle 12\sum_{b=1}^{2}\bar{\mathcal{M}}_{\parallel,b;\
2}^{\nu_{\phi},\nu_{\phi},\nu_{\phi},\nu_{\phi};\nu_{G}}=\big{[}s^{2}\Pi_{2,1}^{(s)}f_{(2,1)}^{(s)}+(E_{L}E_{R}-sE)\Pi_{2,0}^{(s)}f_{(2,0)}^{(s)}+f_{c}\big{]}$
(3.38) where the polarization tensors $\Pi_{2,i}$ as well as functions
$f_{(2,i)}$ and $f_{c}$ are defined in [60]. In particular, we find
$f_{2,2}^{(s)}=\mathcal{M}_{\perp;2}^{\nu_{\phi},\nu_{\phi},\nu_{\phi},\nu_{\phi};\nu_{G}}$.,
and double-copy formalim with correction terms [26]. The final evaluation of
the correlator $\langle\phi\phi\phi\phi\rangle_{G}$ from all these techniques,
including our version and the previous version of orthogonal decomposition,
have been numerically cross-checked and have been found to match for AdS4.

Unlike AdS4, there are not so many formalisms within which the scalar gravity
correlation functions have been computed for AdS6. To complement this, we
provide the explicit evaluation of $\langle\phi\phi\phi\phi\rangle_{G}$ in
AdS6 (together with the evaluation in AdS4) in a Mathematica file attached to
this paper. We have explicitly checked that our correlators at $d=3,5$ satisfy
the manifestly local test (MLT) [62]; interestingly, our transverse and
longitudinal components pass the MLT separately.

###  III.4 Pure Yang-Mills

As an important example of using perturbation theory to compute external
spinning correlators in momentum space, let us examine the tree-level four-
point gluon correlator $\langle JJJJ\rangle$ in AdSd+1. The full tree-level
four-point gluon correlator $\langle JJJJ\rangle$ in AdSd+1, associated with
action (2.16), consists of three pieces, the $s$-channel expression, the
$t$-channel expression and a contact-diagram expression282828Note that in a
color-ordered correlator, there is no $u$-channel.

$\displaystyle\langle JJJJ\rangle=\langle JJJJ\rangle^{(s)}+\langle
JJJJ\rangle^{(t)}+\langle JJJJ\rangle^{(c)}\ ,$ (3.39)

where the channels are exchanging a gluon in the bulk. In particular, the
formal expression of the gluon-exchanging contribution here directly
generalizes the scalar-gluon version (3.18), with the external bulk-to-
boundary scalar propagators replaced by their gluon counterparts. As usual, we
use the Poincaré coordinates and choose the axial gauge, and we will be
suppressing the color indices.

The perturbative expression of correlator in the leading order can be
expressed with the gluon bulk-to-boundary propagator (2.17a) and the gluon
bulk-to-bulk propagator (2.18a). After some algebra, we find the following
$s$-channel contribution in momentum space

$\langle
JJJJ\rangle^{(s)}=\int_{0}^{\infty}dp\mathcal{K}\mathcal{K}\mathcal{J}_{\nu_{A},\nu_{A},\nu_{A}}^{(1)}(k_{1},k_{2},p)\epsilon_{i}(\mathbf{k}_{1})\epsilon_{j}(\mathbf{k}_{2})\mathcal{V}^{ijm}_{\mathbf{k}_{1},\mathbf{k}_{2},-\mathbf{k}}\\\
\times\bigg{(}\frac{-ipH_{mn}(\textbf{k},p)}{s^{2}+p^{2}}\bigg{)}\\\
\times\epsilon_{k}(\mathbf{k}_{3})\epsilon_{\ell}(\mathbf{k}_{4})\mathcal{V}^{k\ell
n}_{\mathbf{k}_{3},\mathbf{k}_{4},\mathbf{k}}\mathcal{K}\mathcal{K}\mathcal{J}_{\nu_{A},\nu_{A},\nu_{A}}^{(1)}(k_{3},k_{4},p),$
(3.40)

where the $\mathcal{K}\mathcal{K}\mathcal{J}$ function is defined in (3.7),
$\nu_{A}$ is $\frac{d}{2}-1$, and the momentum conservation gives
$\mathbf{k}=\mathbf{k}_{1}+\mathbf{k}_{2}=-\mathbf{k}_{3}-\mathbf{k}_{4}$. In
addition, the vertex operator here
$\mathcal{V}_{\mathbf{k}_{1},\mathbf{k}_{2},\mathbf{k}}^{ijm}=z_{1}^{-4}V_{\mathbf{k}_{1},\mathbf{k}_{2},\mathbf{k}}^{ijm}$
is stripped off of z-powers (see full vertex definition in (2.20)). As before,
we express the correlator as a sum of transverse and longitudinal components

$\langle
JJJJ\rangle^{(s)}=(2\pi)^{d}\delta^{d}\bigg{(}\sum_{a=1}^{4}\textbf{k}_{a}\bigg{)}\epsilon_{i}(\mathbf{k}_{1})\epsilon_{j}(\mathbf{k}_{2})\epsilon_{k}(\mathbf{k}_{3})\epsilon_{\ell}(\mathbf{k}_{4})\\\
\mathcal{V}^{ijm}_{\mathbf{k}_{1},\mathbf{k}_{2},-\mathbf{k}_{1}-\mathbf{k}_{2}}\mathcal{V}^{k\ell
n}_{\mathbf{k}_{3},\mathbf{k}_{4},\mathbf{k}_{1}+\mathbf{k}_{2}}\bigg{(}\widehat{P}_{mn}^{\perp}\mathcal{M}_{\perp;\
1}^{\nu_{A},\nu_{A},\nu_{A},\nu_{A};\nu_{A}}+\widehat{P}_{mn}^{\parallel}\mathcal{M}_{\parallel;\
1}^{\nu_{A},\nu_{A},\nu_{A},\nu_{A};\nu_{A}}\bigg{)}$ (3.41)

where the projectors are defined in (2.21), and the transverse as well as the
longitudinal seed amplitudes are defined in (3.6a) and (3.6b), respectively.
The $t$-channel expression can be extracted from the above result via the
usual permutation

$\displaystyle\langle JJJJ\rangle^{(t)}=\langle
JJJJ\rangle^{(s)}\evaluated{}_{1\to 2\to 3\to 4\to 1}.$ (3.42)

On the other hand, the contact diagram is simply given by

$\displaystyle\langle
JJJJ\rangle^{(c)}=\int_{0}^{\infty}\frac{dz_{1}}{z_{1}^{d+1}}z_{1}^{4}\mathcal{V}^{ijk\ell}\mathcal{G}_{i}\mathcal{G}_{j}\mathcal{G}_{k}\mathcal{G}_{\ell},$
(3.43)

where $\mathcal{G}_{i}$ in this expression are the gluon bulk-to-boundary
propagators (2.17a), and $(z_{1}^{4}\mathcal{V}^{ijk\ell})=V^{ijk\ell}$ is the
relevant vertex factor defined in (2.20).

We show two examples of evaluation as usual. For AdS4, the transverse and the
longitudinal seed amplitudes in the $s$-channel are

$\displaystyle\mathcal{M}_{\perp;\
1}^{\nu_{A},\nu_{A},\nu_{A},\nu_{A};\nu_{A}}\bigg{|}_{d=3}=\frac{1}{E_{L}E_{R}E},\quad\mathcal{M}_{\parallel;\
1}^{\nu_{A},\nu_{A},\nu_{A},\nu_{A};\nu_{A}}\bigg{|}_{d=3}=\frac{1}{k_{12}k_{34}E};$
(3.44)

and the contact-diagram expression becomes

$\displaystyle\langle JJJJ\rangle^{(c)}\bigg{|}_{d=3}$
$\displaystyle=\frac{1}{E}.$ (3.45)

In fact, the above seed amplitudes are identical to those in (3.20), which are
the seed amplitudes of
$\langle\sigma\tilde{\sigma}\sigma\tilde{\sigma}\rangle_{A}^{(s)}=\langle\varphi\tilde{\varphi}\varphi\tilde{\varphi}\rangle_{A}^{(s)}$,
because the vectorlike scalar has the same scaling dimension as the spin-1
gauge field.

The AdS6 seed amplitudes of $\langle JJJJ\rangle$ in the $s$-channel are

$\displaystyle\mathcal{M}_{\perp;\
1}^{\nu_{A},\nu_{A},\nu_{A},\nu_{A};\nu_{A}}\bigg{|}_{d=5}$
$\displaystyle=\frac{1}{(k_{12}-s)^{2}E_{L}^{2}E_{R}^{2}}\bigg{(}\sum_{n=0}^{3}\frac{A_{n}}{E^{n}}\bigg{)},$
(3.46) $\displaystyle\mathcal{M}_{\parallel;\
1}^{\nu_{A},\nu_{A},\nu_{A},\nu_{A};\nu_{A}}\bigg{|}_{d=5}$
$\displaystyle=\frac{2k_{1}k_{2}k_{3}k_{4}+(k_{12}k_{3}k_{4}+k_{1}k_{2}k_{34})E+k_{12}k_{34}E^{2}}{k_{12}k_{34}E^{3}}$
(3.47)

which are identical to the seed amplitudes of
$\langle\sigma\tilde{\sigma}\sigma\tilde{\sigma}\rangle_{A}^{(s)}$ in (3.27);
in comparison, the contact-diagram expression reads292929The $g_{n}$ functions
here are defined as  $\displaystyle g_{3}=$
$\displaystyle-2k_{1}k_{2}k_{3}(k_{1}+k_{2}+k_{3})$ (3.48a) $\displaystyle
g_{2}=$
$\displaystyle-(k_{1}^{2}+k_{2}k_{3})k_{23}+k_{1}(k_{2}^{2}+k_{3}^{2})$
(3.48b) $\displaystyle g_{1}=$ $\displaystyle-(k_{1}^{2}+k_{2}^{2}+k_{3}^{2})$
(3.48c) $\displaystyle g_{0}=$ $\displaystyle-k_{4}.$ (3.48d)

$\displaystyle\langle
JJJJ\rangle^{(c)}\bigg{|}_{d=5}=\sum_{n=0}^{3}\frac{g_{n}}{E^{n}}.$ (3.49)

The equivalence of the seed amplitudes between $\langle JJJJ\rangle$ and
$\langle\sigma\tilde{\sigma}\sigma\tilde{\sigma}\rangle_{A}$ reflects the
value of studying vectorlike scalars when it comes to computing correlation
functions for spin-1 gauge fields. A reasonable extension of such observation
is the equivalence of the seed amplitudes between $\langle TTTT\rangle$, the
graviton four-point function, and $\langle\phi\phi\phi\phi\rangle_{G}$ which
we computed in (3.31). It is then of considerable interest to directly
generate spinning correlators from the associated scalar functions. Recent
progress in this direction include [60, 63, 64], which rely on the bootstrap
techniques and/or the application of weight-shifting operators. From another
perspective, one can also take advantage of the dimensional map we introduced
in section II.2 and observe that the two-derivative seed amplitudes of
$\langle\phi\phi\phi\phi\rangle_{\phi}$ in AdSd-1 are equivalent to those of
$\langle JJJJ\rangle$ in AdSd+1 and $\langle TTTT\rangle$ in AdSd-1.303030The
scalar function here is the highest $z$-power contribution from (3.3a) and
(3.3b) associated with bulk Lagrangian of the form $\int(\nabla\phi)^{2}\phi$.

##  IV Conclusion

In this study, we have concentrated on momentum space AdS correlators across
multiple dimensions, with a specific focus on QED, Yang-Mills, and exchange
graviton correlators in (Anti-)de Sitter space. We have provided explicit
examples of correlators for scalar QED in both three and five boundary
dimensions. Furthermore, our venture into five-dimensional analysis reveals
concrete cases of exchange graviton correlators, which are meticulously
detailed in the accompanying Mathematica files. Despite their complexity,
these expressions align seamlessly with the expected flat space limits, thus
underscoring the robustness of our approach. Additionally, our observations on
the interrelation of scalar factors across different dimensions potentially
offer robust computational and organizational tools for future studies in the
field.

In terms of the interrelation of theories, our explicit computations across
various theories in different dimensions are set to also expand the
application scope of the double copy concept within curved spacetime [13, 65,
27, 26, 66, 67, 68, 36, 69, 70, 63]. The double copy methodology proves to be
instrumental in determining amplitudes in complex theories by utilising
simpler theories as foundational inputs, and we hope that our results can be
leveraged as valuable input for this research.

The extension and application of loop-level AdS computations represent a
natural progression, as underscored by recent advances in the field [71, 12,
72, 73, 74, 75]. Furthermore, there is a compelling prospect in broadening our
classification to include a more diverse range of theories. This expansion is
particularly pertinent to the cosmological bootstrap initiative and the
computation of de Sitter space correlators, as referenced in several recent
studies [59, 52, 76, 77, 78, 79]. Efforts in this direction are currently in
progress.

###### Acknowledgements.

We thank Jinwei Chu and Hayden Lee for conversations.

## References

  * [1] H. Elvang and Y.-t. Huang, arXiv:1308.1697 [hep-th].
  * [2] A. Buonanno, M. Khalil, D. O’Connell, R. Roiban, M. P. Solon, and M. Zeng,, “Snowmass White Paper: Gravitational Waves and Scattering Amplitudes,” in Snowmass 2021.  4, 2022.  arXiv:2204.05194 [hep-th].
  * [3] N. Arkani-Hamed, J. L. Bourjaily, F. Cachazo, S. Caron-Huot, and J. Trnka, JHEP 01 (2011) 041.
  * [4] Z. Bern, J. J. M. Carrasco, and H. Johansson, Phys. Rev. Lett. 105 (2010) 061602.
  * [5] N. Arkani-Hamed, Y. Bai, and T. Lam, JHEP 11 (2017) 039.
  * [6] J. M. Maldacena, Adv. Theor. Math. Phys. 2 (1998) 231–252.
  * [7] E. Witten, Adv. Theor. Math. Phys. 2 (1998) 253–291.
  * [8] S. Albayrak and S. Kharel, JHEP 02 (2019) 040.
  * [9] S. Albayrak and S. Kharel, JHEP 12 (2019) 135.
  * [10] S. Albayrak, C. Chowdhury, and S. Kharel, JHEP 10 (2019) 274.
  * [11] S. Albayrak, C. Chowdhury, and S. Kharel, Phys. Rev. D 101 no. 12, (2020) 124043.
  * [12] S. Albayrak and S. Kharel, Phys. Rev. D 103 no. 2, (2021) 026004.
  * [13] S. Albayrak, S. Kharel, and D. Meltzer, JHEP 03 (2021) 249.
  * [14] S. Albayrak and S. Kharel, JHEP 05 (2023) 151.
  * [15] A. Bzowski, P. McFadden, and K. Skenderis, Phys. Rev. Lett. 124 no. 13, (2020) 131602.
  * [16] A. Bzowski, P. McFadden, and K. Skenderis, JHEP 01 (2021) 192.
  * [17] A. Bzowski, P. McFadden, and K. Skenderis, JHEP 03 (2014) 111.
  * [18] A. Bzowski, P. McFadden, and K. Skenderis, JHEP 12 (2022) 039.
  * [19] R. Marotta, K. Skenderis, and M. Verma, JHEP 03 (2023) 196.
  * [20] H. Isono, T. Noumi, and G. Shiu, JHEP 07 (2018) 136.
  * [21] H. Isono, T. Noumi, and G. Shiu, JHEP 10 (2019) 183.
  * [22] C. Corianò, M. M. Maglio, and D. Theofilopoulos, Eur. Phys. J. C 80 no. 6, (2020) 540.
  * [23] N. Anand, Z. U. Khandker, and M. T. Walters, JHEP 10 (2020) 095.
  * [24] S. Jain, R. R. John, A. Mehta, A. A. Nizami, and A. Suresh, JHEP 08 (2021) 089.
  * [25] S. Jain, R. R. John, A. Mehta, A. A. Nizami, and A. Suresh, JHEP 09 (2021) 041.
  * [26] C. Armstrong, H. Goodhew, A. Lipstein, and J. Mei, JHEP 08 (2023) 206.
  * [27] J. A. Farrow, A. E. Lipstein, and P. McFadden, JHEP 02 (2019) 130.
  * [28] C. Corianò, M. M. Maglio, and D. Theofilopoulos, Eur. Phys. J. C 81 no. 8, (2021) 740.
  * [29] C. Corianò and M. M. Maglio, Phys. Rept. 952 (2022) 1–95.
  * [30] G. Mack, arXiv:0907.2407 [hep-th].
  * [31] J. Penedones, JHEP 03 (2011) 025.
  * [32] L. Rastelli and X. Zhou, Phys. Rev. Lett. 118 no. 9, (2017) 091602.
  * [33] L. Rastelli and X. Zhou, JHEP 04 (2018) 014.
  * [34] L. F. Alday and X. Zhou, Phys. Rev. Lett. 125 no. 13, (2020) 131604.
  * [35] L. F. Alday and X. Zhou, Phys. Rev. X 11 no. 1, (2021) 011056.
  * [36] L. F. Alday, V. Gonçalves, and X. Zhou, Phys. Rev. Lett. 128 no. 16, (2022) 161601.
  * [37] L. F. Alday, C. Behan, P. Ferrero, and X. Zhou, JHEP 06 (2021) 020.
  * [38] S. Kharel and G. Siopsis, JHEP 11 (2013) 159.
  * [39] M. S. Costa, V. Gonçalves, and J. a. Penedones, JHEP 09 (2014) 064.
  * [40] C. Sleight and M. Taronna, JHEP 06 (2017) 100.
  * [41] V. Gonçalves, R. Pereira, and X. Zhou, JHEP 10 (2019) 247.
  * [42] A. Bissi, A. Sinha, and X. Zhou, Phys. Rept. 991 (2022) 1–89.
  * [43] Y.-Z. Li and J. Mei, JHEP 07 (2023) 156.
  * [44] L. F. Alday, V. Gonçalves, M. Nocchi, and X. Zhou, arXiv:2307.06884 [hep-th].
  * [45] J. Chu and S. Kharel, arXiv:2311.06342 [hep-th].
  * [46] C. Cheung, C.-H. Shen, and C. Wen, JHEP 02 (2018) 095.
  * [47] S. Raju, Phys. Rev. D 85 (2012) 126009.
  * [48] S. Raju, Phys. Rev. D 83 (2011) 126002.
  * [49] A. Bzowski, P. McFadden, and K. Skenderis, JHEP 03 (2016) 066.
  * [50] J. Liu, E. Perlmutter, V. Rosenhaus, and D. Simmons-Duffin, JHEP 03 (2019) 052.
  * [51] S. Giombi,, “Higher Spin — CFT Duality,” in Theoretical Advanced Study Institute in Elementary Particle Physics: New Frontiers in Fields and Strings, pp. 137–214.  2017\.  arXiv:1607.02967 [hep-th].
  * [52] D. Baumann, C. Duaso Pueyo, A. Joyce, H. Lee, and G. L. Pimentel, JHEP 12 (2020) 204.
  * [53] J. M. Maldacena and G. L. Pimentel, JHEP 09 (2011) 045.
  * [54] D. Karateev, P. Kravchuk, M. Serone, and A. Vichi, JHEP 06 (2019) 088.
  * [55] A. Ghosh, N. Kundu, S. Raju, and S. P. Trivedi, JHEP 07 (2014) 011.
  * [56] H. Liu and A. A. Tseytlin, Phys. Rev. D 59 (1999) 086002.
  * [57] D. Seery, J. E. Lidsey, and M. S. Sloth, JCAP 01 (2007) 027.
  * [58] D. Seery, M. S. Sloth, and F. Vernizzi, JCAP 03 (2009) 018.
  * [59] N. Arkani-Hamed, D. Baumann, H. Lee, and G. L. Pimentel, JHEP 04 (2020) 105.
  * [60] J. Bonifacio, H. Goodhew, A. Joyce, E. Pajer, and D. Stefanyszyn, JHEP 06 (2023) 212.
  * [61] D. Baumann, W.-M. Chen, C. Duaso Pueyo, A. Joyce, H. Lee, and G. L. Pimentel, JHEP 09 (2022) 010.
  * [62] S. Jazayeri, E. Pajer, and D. Stefanyszyn, JHEP 10 (2021) 065.
  * [63] H. Lee and X. Wang, Phys. Rev. D 108 no. 6, (2023) L061702.
  * [64] Y.-Z. Li, Phys. Rev. D 107 no. 12, (2023) 125018.
  * [65] C. Armstrong, A. E. Lipstein, and J. Mei, JHEP 02 (2021) 194.
  * [66] P. Diwakar, A. Herderschee, R. Roiban, and F. Teng, JHEP 10 (2021) 141.
  * [67] X. Zhou, Phys. Rev. Lett. 127 no. 14, (2021) 141601.
  * [68] S. Jain, R. R. John, A. Mehta, A. A. Nizami, and A. Suresh, JHEP 07 (2021) 033.
  * [69] A. Herderschee, R. Roiban, and F. Teng, JHEP 05 (2022) 026.
  * [70] C. Cheung, J. Parra-Martinez, and A. Sivaramakrishnan, JHEP 05 (2022) 027.
  * [71] S. Giombi, C. Sleight, and M. Taronna, JHEP 06 (2018) 030.
  * [72] L. F. Alday, A. Bissi, and X. Zhou, JHEP 02 (2022) 105.
  * [73] T. Heckelbacher, I. Sachs, E. Skvortsov, and P. Vanhove, JHEP 08 (2022) 052.
  * [74] A. Herderschee, arXiv:2112.08226 [hep-th].
  * [75] S. F. Stawinski, arXiv:2309.15059 [hep-th].
  * [76] D. Baumann, C. Duaso Pueyo, A. Joyce, H. Lee, and G. L. Pimentel, SciPost Phys. 11 (2021) 071.
  * [77] C. Sleight and M. Taronna, JHEP 02 (2020) 098.
  * [78] C. Sleight and M. Taronna, JHEP 12 (2021) 074.
  * [79] S. Albayrak, P. Benincasa, and C. Duaso Pueyo, arXiv:2305.19686 [hep-th].