# Telegram Monitor: Monitoring Brazilian Political Groups and Channels on
Telegram

Manoel Júnior<EMAIL_ADDRESS>Universidade Federal de Minas GeraisBelo
HorizonteMinas GeraisBrazil ,  Philipe Melo<EMAIL_ADDRESS>Universidade
Federal de Minas GeraisBelo HorizonteMinas GeraisBrazil ,  Daniel Kansaon
<EMAIL_ADDRESS>Universidade Federal de Minas GeraisBelo
HorizonteMinas GeraisBrazil ,  Vitor Mafra<EMAIL_ADDRESS>Universidade Federal de Minas GeraisBelo HorizonteMinas GeraisBrazil ,  Kaio
Sá<EMAIL_ADDRESS>Universidade Federal de Minas GeraisBelo HorizonteMinas
GeraisBrazil and  Fabrício Benevenuto<EMAIL_ADDRESS>Universidade
Federal de Minas GeraisBelo HorizonteMinas GeraisBrazil

(2022)

###### Abstract.

Instant messaging platforms such as Telegram became one of the main means of
communication used by people all over the world. Most of them are home of
several groups and channels that connect thousands of people focused on
political topics. However, they have suffered with misinformation campaigns
with a direct impact on electoral processes around the world. While some
platforms, such as WhatsApp, took restrictive policies and measures to
attenuate the issues arising from the abuse of their systems, others have
emerged as alternatives, presenting little or no restrictions on content
moderation or actions in combating misinformation. Telegram is one of those
systems, which has been attracting more users and gaining popularity. In this
work, we present the “Telegram Monitor”, a web-based system that monitors the
political debate in this environment and enables the analysis of the most
shared content in multiple channels and public groups. Our system aims to
allow journalists, researchers, and fact-checking agencies to identify
trending conspiracy theories, misinformation campaigns, or simply to monitor
the political debate in this space along the 2022 Brazilian elections. We hope
our system can assist the combat of misinformation spreading through Telegram
in Brazil.

Telegram, misinformation, fake news, politics

††copyright: acmcopyright††journalyear: 2022††doi:
XXXXXXX.XXXXXXX††conference: Make sure to enter the correct conference title
from your rights confirmation emai; April 25–29, 2022; Online††price:
15.00††isbn: 978-1-4503-XXXX-X/18/06††ccs: Human-centered computing Empirical
studies in collaborative and social computing

Figure 1. General framework architecture of Telegram Monitor.

Framework of Telegram Monitor.

##  1\. Introduction

Instant messaging platforms have become one of the main ways to communicate
with close friends and also find communities of people who share the same
interest, creating large public groups and channels, gathering thousands of
users around a topic of discussion. Telegram ranks among one of the most
popular with 400
million111https://www.statista.com/statistics/234038/telegram-messenger-mau-
users/ monthly active users around the world and has seen a huge growth in its
user base. Together with their biggest competitor, WhatsApp, these platforms
operate a network that exchanges billions of messages every
day222https://blog.whatsapp.com/connecting-one-billion-users-every-day.
Unfortunately, with the formation of these communities, instant messaging apps
also attracted the attention for the spreading of misinformation campaigns and
fake news during electoral campaigns (Resende et al., 2019). These problems,
along with public pressure to companies such as Facebook, motivated the
establishment of measures to make it difficult to spread content quickly, such
as limiting the forwarding of messages (Melo et al., 2019b) and banning
accounts that violate the platform’s terms of use. In this scenario, other
messaging platforms, including Telegram itself, end up emerging as an
unmoderated alternative in which users are free to discuss whatever they want
without any restrictions.

Although Telegram gained some prominence with a large user base, especially in
Brazil, it has been criticized for spreading terrorist content (Tan, 2017),
and hate speech (Anti-Defamation League, 2019), and little has been
investigated about this service. Unlike other social media, it is a hard task
to have any information about the huge content circulating on this platform.
Due to the closed nature of these systems, not even the most discussed trends
or topics are known to those who intend to analyze this environment.

This work is built on the creation of a Telegram public group monitoring
system – the Telegram Monitor. With this system, we seek to help better
understand the ecosystem of messaging applications in Brazil, creating an
interface that brings out some of the most shared media and messages in
political groups and public channels on the platform, facilitating the
investigation of the content that circulates on this network. The Telegram
Monitor is an online system that displays, in a period of time selected by the
user, the most shared media on these days in the investigated public groups,
similar to other initiatives with the same proposal, but for WhatsApp (Melo et
al., 2019a). This system is updated daily with images, audios, videos, and
messages shared, allowing users to discover the most popular topics that are
present on the platform given a period of time. This can help many researchers
and journalists that have access to the system to observe how these platforms
are being used and easily identify false stories getting viral on its network,
working as a tool to fight against misinformation. The system’s architecture
integrates an extensive collection of content from public political groups in
Brazil, with the storage and processing of all media shared in these
communities. To understand the popularity of each content, the collected media
are grouped by similarity, based on a perceptual hash algorithm that generates
a unique identifier for each content, thus being able to aggregate similar
content and quantify how many times each one was shared by the users. Finally,
this content is processed on a daily basis, ranking each type of media by
popularity and displayed in an online access system through a URL with a link
to the system: http://blackbird.dcc.ufmg.br/monitor-telegram/brazil/. Through
an account with a username and password, it is then possible to access all the
data and browse the days exploring what was shared in the monitored groups and
channels.

##  2\. System Architecture

Telegram is a cloud-based, cross-platform instant messaging application, where
the accounts are validated by a phone number. With this service, users can
send messages to their contacts and participate in chat groups of up to
200,000 members and unlimited size channels. They can also invite others to
their groups/channels by sharing an invitation URL link. Our methodology uses
data collected from these public groups and channels. The system monitors
groups and channels for political discussion in Brazil. Those are operated
either by individuals affiliated with political parties, local community
leaders, or ordinary users with an interest in the topic, and can be freely
accessed by anyone with the invite link.

In this section, we describe all methodology behind the development of the
system of Telegram Monitor. This main architecture is explained through the
flowchart in Figure 1. In this Figure, we have an overview of all the steps
performed to build the online system: (i) A phone with a valid Telegram
account is set. (ii) We find a set of relevant groups and channels on the Web
to monitor. (iii) A script extracts and structures all messages received. (iv)
Media files (image, video and audio) are downloaded. (v) Another step is
responsible for parsing these files and grouping the messages by similarity.
(vi) Merged messages are saved containing the total numbers of times they were
shared. (vii) The online system accesses the database and displays it in the
interface according to the user’s filters. Next, we go through each step
separately giving details on how they were performed.

###  2.1. Data Collection

The first step in gathering data on Telegram is to define which are the groups
and channels of interest to the collection. For this work, we focus on
monitoring Brazilian political public groups and channels. A common practice
for gathering groups in messaging apps is to use keywords along with the group
invitation link pattern to identify potential groups and channels of interest
for collection (Garimella and Tyson, 2018; Melo et al., 2019a; Resende et al.,
2019; Machado et al., 2019; Bursztyn and Birnbaum, 2019). Telegram group
invite links follow the pattern https://t.me/joinchat/<GroupID> or
https://telegram.me/<GroupID>. In this way, we created a wide list of terms
related to the Brazilian political context. The terms in this list were
searched together with the invitation URL pattern on search engines (Google)
and social media (Twitter and Facebook). In addition to this, Telegram has a
group search system within the application itself, where terms were also
searched to find more groups.

With this process, we identified and joined a total of 232 public groups and
channels related to politics with thousands of members. Some of the selected
channels have more than a million members as shown in Figure 2. It is
interesting to note that around 66% of the groups and channels found have more
than 256 members or subscribers (maximum size of a WhatsApp group). That gives
us a hint of how big is the audience that information spreading can reach on
those Telegram groups/channels which is much bigger than WhatsApp groups
maximum capacity.

Figure 2. Cumulative Distribution Function (CDF) of total of members per
groups and channels on Telegram.

###  2.2. Content Processing

The next step in the development of the system is extracting and processing
all the data coming from the groups. The Telegram platform has an official
API333Telegram API available at: https://core.telegram.org/api. in which it is
possible to obtain conversation data, in addition to performing actions on
behalf of the user, such as joining groups and sending messages
programmatically. With that, a retroactive collection of data from the
conversations was carried out. This data includes text messages and media such
as images, videos, audios, documents and any other type of content that the
platform supports in its official clients.

To create a system that allows you to interact with the data and understand
the content shared in these groups, it is still necessary to process and
organize these messages and media in a way that makes sense to the user who
wishes to explore it. For this, we chose to separate the data by day and, for
each day, rank each type of media according to its popularity on the selected
date.

Here, it is worth noting a crucial step for the functioning of the system: the
grouping of similar content. In other social platforms, such as Facebook or
Twitter, the data collection of their posts usually already comes with
metadata about the number of likes or shares that content gained within the
platform. However, with messaging apps like Telegram, each message, image,
video, or audio is posted independently. Therefore, to define popularity, it
is necessary to calculate how many times each media was shared in the period,
as this information is not available. To be able to track and count how many
times the same piece of content has been shared (in other words, measuring
their popularity) we need to do the whole process of tracking, grouping, and
counting the data, by finding all copies of that message (image, audio or
video) among others. To find copies of the same content, we use the pHash –
perceptual hashing algorithm (Monga and Evans, 2006) on images. This process
generates a visual hash that works as a kind of “fingerprint” of the content,
making it possible to compare two images. For audio and video we also
calculate a hash from the checksum (MD5) that generate a unique value for each
different file. Similarly, we also use the Jaccard index to compare text
messages and group them. With those hashes, we can detect content that is
identical or very similar (with minor changes) and group them. As a result, it
is possible to calculate aggregated information about each piece of content
shared, such as determining its popularity, counting in how many groups it
appeared and how many different users sent it as shown in Figure 3.

Figure 3. Process to merge files by visual hashes similarity.

###  2.3. Ethics

Telegram Monitor gathers a considerable amount of data from many Telegram
groups and users. To ensure users’ privacy, we do not share or disclose any
personally identifiable information, such as phone numbers or username. To
prevent misuse, even of aggregated information, we also limit access to our
system to a restricted number of people, through an account with a password.
In addition, they are also informed about the limitations of the data and the
potential bias present in our system. As we only use publicly available
Telegram groups and official tools provided by the system’s API, our data
collection does not violate Telegram’s terms of service.

###  2.4. Online Interface

After the data is properly separated by date and ranked by popularity, this
content is displayed in an online system that fetches the data from the
database and displays it to the end user based on the searched period. The
Telegram Monitor has restricted access through an account with a username and
password on a web page. On this page, the user can select a day (or a period
of time with several days), as shown in Figure 4, and the system displays a
ranking of the most popular content of the selected period. After picking a
date, it is possible to find the most shared content according to the type of
media (images, videos, audios, or text). When choosing the media, the content
will be ranked based on how many shares it had in that period, showing first
those with more shares. An example of the system view can be seen in Figure 5.

For each content displayed, it is also possible to see more details of it with
information about how much it has spread. In details, we have the total
shares, the number of groups in which this content appeared and the number of
different users who sent it. It is also displayed the list of names of the
groups in which it has been shared. Finally, there is a button that searches
for the image on Google to check if that same image appears elsewhere on the
Internet.

Since Telegram has such a closed nature, this possibility of navigating
through days and seeing what has been popular in groups and channels can help
users to better understand that environment, a process that would be hard to
execute without such a system.

Figure 4. Monitor interface for selecting the period in which one wants to
view the content. Figure 5. Snapshot of top images of a day in our monitor.

###  2.5. Relevance of the Monitored Groups

By joining groups on Telegram, we usually have access not only to messages
sent to it after the joining date, but also to the entire history of messages
in the groups since their creation. Because of that, we can observe the volume
of messages exchanged in the groups and channels we monitor since their
creation. In Figure 6 we show the amount of messages sent per week on those
groups and channels. It is possible to notice a relevant increase in the total
number of messages in 2020, reaching 20K messages per week and an even bigger
increase in early 2021 when that number grew up to four times, jumping to
almost 80K messages per week. This result can give us a hint of the importance
Telegram has gained, specially in the last year. With groups up to a thousand
times larger than WhatsApp, this messaging app has a considerable potential to
spread misinformation, although a great part of the content remains distant
and away from fact-checking agencies, researchers and journalists, given the
difficulty of finding it in such a closed environment.

##  3\. Social Impact

Given the upcoming Brazilian elections, scheduled for October 2022, and the
serious concerns about the abuse of Telegram by misinformation campaigns, our
monitoring system has already attracted some attention as a tool to combat
misinformation.

So far, we gave access to our system to a few journalists and fact-checking
agencies, who explicitly mentioned it as a data source for their
investigation444https://bit.ly/3umiFMc. A key challenge for fact-checking
organizations and investigative journalism in this space is to avoid giving
too much attention to conspiracy theories and fake stories that are not
getting viral. The concern here is not only to work on something irrelevant,
but also to boost the attention to some unpopular misbeliefs. Our monitoring
system allows fact-checkers to get a sense if a claim is getting viral at
least in this specific system. In this sense, our university already became a
technical partner of
Comprova555https://projetocomprova.com.br/about/partners/, a project for
collaborative fact-checking in Brazil, recognized by the International Fact-
checking network, with more than forty Brazilian media portals. Other key
fact-checking agencies in Brazil already showed interest in accessing our
system, which we plan to freely provide in the upcoming weeks.

Finally, it is important to mention that Telegram does not have an office or
direct representation in Brazil. It has so far ignored Brazilian
authorities666https://bit.ly/34w1Mnk, which are specially worried about
misinformation campaigns questioning the validity of the 2022 elections and
the electoral process. Thus, the Superior Electoral Court (TSE), which is
responsible for conducting the Brazilian 2022 elections, has also partnered
with our project777https://bit.ly/3oIuRDF. Our system may represent a way for
TSE to get known about possible misinformation campaigns attacking the
democrat process through Telegram and take proper actions if necessary.

Figure 6. Volume of messages sent in monitored groups and channels on Telegram
per week.

## References

  * (1)
  * Anti-Defamation League (2019) Anti-Defamation League. 2019\.  Telegram: The Latest Safe Haven for White Supremacists.  ADL - Fighting Hate for Good.  https://www.adl.org/blog/telegram-the-latest-safe-haven-for-white-supremacists Acessado em 20 de Julho de 2020.
  * Bursztyn and Birnbaum (2019) Victor S Bursztyn and Larry Birnbaum. 2019.  Thousands of small, constant rallies: A large-scale analysis of partisan WhatsApp groups. In _2019 IEEE/ACM International Conference on Advances in Social Networks Analysis and Mining (ASONAM)_. IEEE, 484–488.
  * Garimella and Tyson (2018) Kiran Garimella and Gareth Tyson. 2018.  Whatapp doc? a first look at whatsapp public group data. In _Twelfth International AAAI Conference on Web and Social Media_.
  * Machado et al. (2019) Caio Machado, Beatriz Kira, Vidya Narayanan, Bence Kollanyi, and Philip Howard. 2019.  A Study of Misinformation in WhatsApp groups with a focus on the Brazilian Presidential Elections.. In _Companion proceedings of the 2019 World Wide Web conference_. 1013–1019.
  * Melo et al. (2019a) Philipe Melo, Johnnatan Messias, Gustavo Resende, Kiran Garimella, Jussara Almeida, and Fabrício Benevenuto. 2019a.  WhatsApp Monitor: A Fact-Checking System for WhatsApp. In _Proceedings of the International AAAI Conference on Web and Social Media_ _(ICWSM ’19, Vol. 13)_. 676–677.
  * Melo et al. (2019b) Philipe Melo, Carolina Coimbra Vieira, Kiran Garimella, Pedro OS de Melo, and Fabrício Benevenuto. 2019b.  Can WhatsApp Counter Misinformation by Limiting Message Forwarding?. In _Proc. of the Int’l Conference on Complex Networks and their Applications (Complex Networks)_.
  * Monga and Evans (2006) Vishal Monga and Brian L. Evans. 2006.  Perceptual image hashing via feature points: performance evaluation and tradeoffs.  _IEEE Transactions on Image Processing_ 15, 11 (2006), 3452–3465.
  * Resende et al. (2019) Gustavo Resende, Philipe Melo, Hugo Sousa, Johnnatan Messias, Marisa Vasconcelos, Jussara Almeida, and Fabrício Benevenuto. 2019\.  (Mis)Information Dissemination in WhatsApp: Gathering, Analyzing and Countermeasures. In _The World Wide Web Conference_ (San Francisco, CA, USA) _(WWW ’19)_. ACM, 818–828.
  * Tan (2017) Rebecca Tan. 2017\.  Terrorists’ love for Telegram, explained.  https://www.vox.com/world/2017/6/30/15886506/terrorism-isis-telegram-social-media-russia-pavel-durov-twitter.


aainstitutetext:  Niels Bohr International Academy and Discovery Center,
University of Copenhagen
The Niels Bohr Institute, Blegdamsvej 17, DK-2100, Copenhagen Ø, Denmark
&bbinstitutetext: Physics Division, National Center for Theoretical Sciences,
National Tsing-Hua University,
Hsinchu, Taiwan 30013, Republic of China.

# Mellin-(Schwinger) representation of One-loop Witten diagrams in AdS

Carlos Cardona<EMAIL_ADDRESS>
###### Abstract

In this paper we consider Witten diagrams at one loop in AdS space for scalar
$\phi^{3}+\phi^{4}$ theory. After using Schwinger parametrization to
trivialize the space-time loop integration, we extract the Mellin-Barnes
representation for the one-loop corrections to the four-particle scattering up
to an integration over the Schwinger parameters corresponding to the
propagators of the internal particles running into the loop. We then discuss
an approach to deal with those integrals.

NCTS-TH/1713

##  1 Introduction

Since the conception of the AdS/CFT correspondence Maldacena:1997re ;
Witten:1998qj , the scattering of particles and strings on AdS spaces has
become an important subject of study, not only because it would improve our
understanding on scattering amplitudes in curved space-times, but also because
it will teach us about the large-$N$ expansion of conformal correlation
functions in theories with holographic interpretation. On those theories, an
expansion of a given correlation function in terms of $1/N$ should corresponds
to an expansion in loops Witten diagrams in AdS gravity. Therefore, the
development of techniques to compute loop Witten diagrams is imperative for
the study of large $N$ conformal field theories. In addition, very recently
Pasterski:2016qvg , has been suggested that the on-shell kinematics of the
flat-space S-matrix can be described by momentum diagrams in AdS, and in
Cardona:2017keg  it was shown that indeed the factorization residues for the
S-matrix in flat-space scalar theories might be described by Witten diagrams
in AdS, implying that the perturbative flat-space S-matrix in $d+2$ dimensions
might be related to correlation functions of some conformal field theory in
$d-$dimension, which presents itself as a new interesting motivation to study
Witten diagrams in AdS.

Right after the formulation of the AdS/CFT conjecture, some tree-level Witten
diagrams were computed by conventional methods Arutyunov:2002fh ;
Freedman:1998bj ; Liu:1998ty ; Muck:1998rr ; DHoker:1999kzh ; Dolan:2000uw ,
but more efficient and illuminating techniques were developed only recently
starting with the introduction of the Mellin representation of correlation
functions for conformal field theories constructed by Mack Mack:2009mi  and
further developed in Penedones:2010ue ; Fitzpatrick:2011ia . Even more
recently, a new formalism called geodesic-Witten diagrams, that emulate the
conformal block expansion of correlation functions from the gravity
perspective, have been developed in Hijano:2015zsa  and extended to the
inclusion of spinning particles in Dyer:2017zef ; Sleight:2017fpc ;
Chen:2017yia ; Castro:2017hpx ; Nishida:2016vds . However, despite those very
interesting technical developments, the computation of loop Witten diagrams
have proven to be very challenging and keeps being very unexplored. With the
exception of few previous considerations Penedones:2010ue ; Fitzpatrick:2011dm
; Fitzpatrick:2011hu ; Cornalba:2007zb  and some recent progress in, not even
the most basic scalar loop integrals have been computed yet.

It is the purpose of this note to initiate a modest study on the computation
of loop Witten diagrams from the gravity side of the duality. Through, to some
extend, brute force approach, we intend to explicitly pin point the basic
obstacles preventing us to make progress in the perturbative computation of
Witten diagrams and identify the building block integrals that we should focus
in order to tackle the computation of loop Witten diagrams, hopefully in a
wider scope that the ones considered here.

The most recent studies of Witten diagrams in AdS are strongly dependent on
the so-called split representation, on which every bulk-to-bulk propagator is,
roughly speaking, replaced by an integration over a product of two bulk-to-
boundary propagators, which at the loop level increase dramatically the number
of integrations over propagators, converting a given loop diagram in an
effective higher order loop diagram. From the experience on flat-space loop
computations, the integration over higher loops is, of course, harder than the
simplest one loop case, and therefore we don’t want to rephrase a loop
integral in terms of higher loops, but on the contrary, try to do the opposite
if possible.

In this note, we propose a different approach, by using directly bulk-to-bulk
propagators in a representation that allow us to treat the bulk coordinates on
AdS in almost the same footing as the boundary coordinates, avoiding us the
introduction of additional integrations over propagators. The advantage of
this procedure is that loop-level diagrams are computed by essentially
following the same route used to compute tree-level diagrams.

For concreteness, we focus on the simplest $\phi^{3}+\phi^{4}$ scalar theory
on AdS. By using Schwinger parametrization to exponentiate the propagators we
manage to integrate the space-time loop and subsequently extract the Mellin
representation for the loop Witten diagrams corresponding to the corrections
to the four particles scattering, expressed as function of Mellin variables
plus a remaining integration over polynomials built out from Schwinger
parameters associated to the internal propagators of the loop. Very recently,
an algorithm to construct those polynomials at tree level and for the
triangular loop, has been developed in Nizami:2016jgt . We identify those
integrals as the hardest nut to crack on this approach and realized that
better techniques should be developed to the evaluation of either, the
particular integrals described in section 7 or the Witten diagram itself. We
however press as hard as we can for the time being and for the case of the
three particles scattering we manage to write the corresponding integral in
terms of a lengthy combination of Hypergeometric functions. As well as, for
the four particles scattering triangle and box, we sketch a proposal for the
evaluation of the integrals by reinterpreting them as contour integrals.

Recently the study of one loop Witten diagrams has been considered from the
dual perspective by computing the $1/N$ expansion of scalar correlators in
conformal field theories with holographic description by means of the analityc
bootstrap. More concretely, in Alday:2016njk  a method was developed to solve
the crossing equations as an expansion in large spin. The given expansion can
be resumed and then extrapolated to lower spin (see also Caron-Huot:2017vep ;
Simmons-Duffin:2016wlq ). This approach has been used in Alday:2017xua ;
Aharony:2016dwx ; Aprile:2017bgs  to compute some one loop corrections to the
four-point function of conformal fields with low conformal dimension,
specifically $\Delta=2$ (for related recent work see also Li:2017lmh ;
Costa:2017twz ). It is our hope that we can reproduce those result in the near
future by improving the approach initiated in this note.

After the completion of this work, an interesting paper Giombi:2017hpr
appeared in the arXiv, where the authors compute explicitly the one loop
correction to 2-point Witten diagrams.

The remainder of this paper is organized as follows. In section 2 we
illustrate the general approach by applying it to a simple tree-level scalar
exchange. In section 3 we realized that the simplest loop correction, namely,
the bubble Witten diagram follows from the scalar tree-level exchange without
much effort. In section 4 we consider the one-loop correction to the three
particle scattering, or in other words, to the structure constant for the
scalar correlator. Section 5 and 6 deal with the most complicate diagrams for
the loop correction to four particles scattering in $\phi^{3}+\phi^{4}$,
namely, the four particle triangle and the box. In section 7 we make a
proposal to treat the leftover integral on loop-related Schwinger parameters.

##  2 Warming up: Tree-level scalar exchange

It is convenient to start by quickly introducing embedding coordinates
Dirac:1936fq ; Penedones:2007ns  since the computations quite simplify on
those. Consider the embedding of euclidean AdSd+1 in $(d+2)-$Minkowski space
$\mathbb{M}^{d+2}$, namely, $X$ lives in AdSd+1 if for $X\in\mathbb{M}^{d+2}$
it satisfies $X^{2}=-1$, where we have set the radius of AdSd+1 space as
$R=1$. The boundary of AdSd+1 is given by null vectors $P\in\mathbb{M}^{2+d}$,
i.e $P^{2}=0$. The usual parametrization of this coordinates is given by,

$X=(X^{+},X^{-},X^{\mu})={1\over z}(1,z^{2}+x^{2},x^{\mu})\,,\quad
P=(P^{+},P^{-},P^{\mu})=(1,x^{2},x^{\mu})\,,$ (1)

where $x^{\mu}\in\mathbb{R}^{d}$.

Let us now consider the simple but still not-trivial Witten diagram shown in
the figure 1.111The simplest not trivial diagram corresponds to a 4-point
contact diagram., corresponding to a single scalar exchange. We will use this
example to illustrate the approach we are going to use in the following over
more complicated examples. This diagram was previously computed long time ago
in DHoker:1999kzh  and later revisited in Penedones:2010ue  by means of the
split-representation which we will review in the appendix A.

Figure 1: Scalar Exchange.

Let us recall some few basic facts about Witten diagrams, as the one
represented in figure 1. The outer circle represents the boundary of $AdS$.
Lines connecting points at the boundary to points in the bulk, represents
bulk-to-boundary propagators which are given by

$G^{\Delta}_{\partial B}(X,P)=\frac{{\cal C}_{\Delta}}{(-2P\cdot
X)^{\Delta}},\quad{\cal
C}_{\Delta}=\frac{\Gamma(\Delta)}{2\pi^{h}\Gamma(\Delta-h+1)}$ (2)

where $\Delta$ denotes the conformal dimension of the conformal field at the
boundary point $P$. Lines connecting bulk points represents bulk-to-bulk
propagators, for which we have choosen the following representation
Penedones:2010ue ,

$\begin{split}G^{\Delta}_{BB}(X_{1},X_{2})=\int_{-i\infty}^{i\infty}{\mathrm{d}c\,\mathrm{d}\gamma\over(2\pi
i)^{2}}{\Gamma(h+c-\gamma)\Gamma(h-c-\gamma)\over\Gamma(c)\Gamma(-c)((\Delta-h)^{2}-c^{2})}{\Gamma(\gamma)\Gamma(h-\gamma)\over\Gamma(2h-2\gamma)\,u^{\gamma}}\,,\end{split}$
(3)

where $u$ corresponds to the geodesic distance between the points $X_{1}$ and
$X_{2}$ and is defined as,

$u\equiv(X_{1}-X_{2})^{2}=-2(1+X_{1}\cdot X_{2})\,.$ (4)

Finally, bulk points have to be integrated over. Therefore, the amplitude
associated to the diagram represented in figure 1 is given by

$\begin{split}&{\cal
A}^{\mathrlap{\Yright}~{}\Yleft}_{4}(P_{1},P_{2},P_{3},P_{4})=\\\
&g^{2}\int_{AdS}\mathrm{d}X_{1}\mathrm{d}X_{2}G^{\Delta_{1}}_{\partial
B}(P_{1},X_{1})G^{\Delta_{2}}_{\partial
B}(P_{2},X_{1})G^{\Delta}_{BB}(X_{1},X_{2})G^{\Delta_{3}}_{B\partial}(X_{2},P_{3})G^{\Delta_{4}}_{B\partial}(X_{2},P_{4})\;.\end{split}$
(5)

where $g$ denotes the coupling of the particles entering the vertex of the
diagram (see figure 1). The crucial difference between the approach we are
going to use in this note and previous computations (particularly the split
representation approach) is that we would like to treat the bulk coordinates
$X$ at the same footing as the boundary coordinates $P$. In order to do that,
we use the following Mellin-Barnes identity,

${1\over(A+B)^{\gamma}}={1\over 2\pi
i\Gamma(\gamma)}\int_{-i\infty}^{i\infty}\mathrm{d}f\,\Gamma(\gamma+f)\Gamma(-f){A^{f}\over
B^{\gamma+f}}\,,$ (6)

to represent the bulk-to-bulk propagator as,

$\begin{split}&G^{\Delta}_{BB}(X_{1},X_{2})=\\\
&\int_{-i\infty}^{i\infty}{\mathrm{d}c\,\mathrm{d}\gamma\mathrm{d}f\over(2\pi
i)^{3}}{\Gamma(h+c-\gamma)\Gamma(h-c-\gamma)\over\Gamma(c)\Gamma(-c)((\Delta-h)^{2}-c^{2})}{\Gamma(\gamma+f)\Gamma(h-\gamma)\Gamma(-f)(-2)^{f}\over\Gamma(2h-2\gamma)\,(-2X_{1}\cdot
X_{2})^{\gamma+f}}\\\
&\equiv\int_{-i\infty}^{i\infty}{\mathrm{d}c\,\mathrm{d}\gamma\mathrm{d}f\over(2\pi
i)^{3}}\,\widehat{G}^{\Delta}_{BB}(c,\gamma,f)\,{1\over(-2X_{1}\cdot
X_{2})^{\gamma+f}}\,.\end{split}$ (7)

For the sake of notation we have defined,

$\widehat{G}^{\Delta}_{BB}(c,\gamma,f)={\Gamma(h+c-\gamma)\Gamma(h-c-\gamma)\over\Gamma(c)\Gamma(-c)((\Delta-h)^{2}-c^{2})}{\Gamma(\gamma+f)\Gamma(h-\gamma)\Gamma(-f)(-2)^{f}\over\Gamma(2h-2\gamma)}$
(8)

Putting this definitions back into (5), we get

$\begin{split}&{\cal
A}^{\mathrlap{\Yright}~{}\Yleft}_{4}(P_{i})=g^{2}\prod_{i=1}^{4}{\cal
C}_{\Delta_{i}}\int_{-i\infty}^{i\infty}{\mathrm{d}c\,\mathrm{d}\gamma\mathrm{d}f\over(2\pi
i)^{3}}\widehat{G}^{\Delta}_{BB}(c,\gamma,f)\\\
&\int_{AdS}{\mathrm{d}X_{1}\mathrm{d}X_{2}\over(-2P_{1}\cdot
X_{1})^{\Delta_{1}}(-2P_{2}\cdot X_{1})^{\Delta_{2}}(-2(X_{1}\cdot
X_{2}))^{\gamma+f}(-2P_{3}\cdot X_{2})^{\Delta_{3}}(-2P_{4}\cdot
X_{2})^{\Delta_{4}}}\,.\end{split}$ (9)

The first observation from the representation (LABEL:mybulk) is that it allows
to think the amplitudes in the schematic form

$A\sim\int{\prod\mathrm{d}X_{i}\over(X_{j}\cdot
X_{k})^{\Delta_{ij}}(X_{r}\cdot X_{s})^{\Delta_{rs}}}\,,$ (10)

such us some of the $X_{j}^{\prime}s$ coordinates are projected to the
boundary and the remaining should be integrated. We will see also that this
representation is convenient when computing the integrals by using Schwinger
parametrization. The remaining of the computation proceed similarly as the
computation of loop amplitudes in flat space-time. First, we introduce
Schwinger parameters to exponentiate all the propagators as,

$\begin{split}&{\cal
A}^{\mathrlap{\Yright}~{}\Yleft}_{4}(P_{i})=g^{2}\prod_{i=1}^{4}{\cal
C}_{\Delta_{i}}\int_{-i\infty}^{i\infty}{\mathrm{d}c\,\mathrm{d}\gamma\mathrm{d}f\over(2\pi
i)^{3}}\widehat{G}^{\Delta}_{BB}(c,\gamma,f)\\\
&\int^{\infty}_{0}\prod_{i=1}^{5}{\mathrm{d}t_{i}\over
t_{i}}{t_{i}^{\Delta_{i}}\over\Gamma(\Delta_{i})}\int_{AdS}\mathrm{d}X_{1}\mathrm{d}X_{2}\,{\rm
e}^{2\left(\sum_{i=1}^{2}t_{i}\,P_{i}\cdot X_{1}+t_{5}\,X_{1}\cdot
X_{2}+\sum_{i=3}^{4}t_{i}P_{i}\cdot X_{2}\right)}\,.\end{split}$ (11)

where for convenience of notation we have defined $\Delta_{5}={\gamma+f}$.
Integration over bulk coordinates can be carried out straightforwardly now.
All of them take the following form,

$\int\mathrm{d}X_{i}{\rm e}^{2Q_{i}\cdot X_{i}}$ (12)

which has been computed in Penedones:2010ue  and we review here. The key
observation is that, since $Q$ is a vector in ${\mathbb{M}}^{d+2}$ and the
integral should be Lorenz invariant, we can set $Q=|Q|(1,1,0)$. Then by using
the parametrization (1) we can write,

$\begin{split}\int\mathrm{d}X_{i}{\rm e}^{2Q_{i}\cdot
X_{i}}&=\int_{0}^{\infty}{\mathrm{d}z_{i}\over
z_{i}}z_{i}^{-d}\int\mathrm{d}^{d}x_{i}\,{\rm
e}^{(1+z_{i}^{2}+x_{i}^{2})|Q_{i}|/z}\\\
&=\pi^{h}\int_{0}^{\infty}{\mathrm{d}z_{i}\over z_{i}}(z_{i}|Q_{i}|)^{-h}{\rm
e}^{(1+z_{i}^{2})|Q_{i}|/z_{i}}\\\
&=\pi^{h}\int_{0}^{\infty}{\mathrm{d}z_{i}\over z_{i}}z_{i}^{-h}{\rm
e}^{-z_{i}+Q_{i}^{2}/z_{i}}\,,\end{split}$ (13)

with $h\equiv d/2$. Integration over $X_{1}$ have the form (13) with the
corresponding $Q$ given by,

$Q_{1}=t_{1}P_{1}+t_{2}P_{2}+t_{5}X_{2}\,.$ (14)

In order to factor out $Q_{1}^{2}$ from the $z_{1}$ integration at the last
line of (13), we rescalate
$(t_{1},t_{2},t_{5})\to\sqrt{z_{1}}(t_{1},t_{2},t_{5})$ such as 222Thought as
part of the total integral (11).,

$\begin{split}\int\mathrm{d}X_{1}{\rm e}^{2Q_{1}\cdot X_{1}}&=\pi^{h}{\rm
e}^{Q_{1}^{2}}\int_{0}^{\infty}{\mathrm{d}z_{1}\over z_{1}}{\rm
e}^{-z_{1}}z_{1}^{\Delta_{1}+\Delta_{2}+\gamma+f-2h\over 2}\\\ &={\rm
e}^{Q_{1}^{2}}\Gamma\left({\Delta_{1}+\Delta_{2}+\gamma+f-2h\over
2}\right)\,.\end{split}$ (15)

In the same manner, integration over $X_{2}$ produces an expression as (13)
with

$Q_{2}=t_{1}t_{5}P_{1}+t_{2}t_{5}P_{2}+t_{3}P_{3}+t_{4}P_{4}\,,$ (16)

and again we re-escalate
$(t_{3},t_{4},t_{5})\to\sqrt{z_{2}}(t_{3},t_{4},t_{5})$ to factor out the
$Q_{2}$ dependence from the $z_{2}-$integral such as,

$\begin{split}\int\mathrm{d}X_{2}{\rm e}^{2Q_{2}\cdot X_{2}}&=\pi^{h}{\rm
e}^{Q_{2}^{2}}\int_{0}^{\infty}{\mathrm{d}z_{2}\over z_{2}}{\rm
e}^{-z_{2}(1+t_{5}^{2})}z_{2}^{\Delta_{3}+\Delta_{4}+\gamma+f-2h\over 2}\\\
&={{\rm
e}^{Q_{2}^{2}}\over(1+t_{5}^{2})^{\Delta_{3}+\Delta_{4}+\gamma+f-2h\over
2}}\Gamma\left({\Delta_{3}+\Delta_{4}+\gamma+f-2h\over
2}\right)\,.\end{split}$ (17)

Notice that the term $(1+t_{5}^{2})$ comes from squaring $Q_{2}$, rescaling
$z_{2}$ as above and the fact that $X_{3}^{2}=-1$ (or $X_{i}^{2}=-1$ for
general bulk points). Later on we will see that those factors coming from the
square of the bulk coordinates are the ultimate responsible for the increasing
difficulty in performing the integrals at loop level of Witten diagrams on AdS
by the Schwinger parametrization approach. This is analogous to the situation
in flat space, where theories containing only massless fields $P_{i}^{2}=0$
are much more simpler than massive theories, i.e containing particles such as
$X_{i}^{2}=-1$.

After integration over the bulk coordinates, we end up with

$\begin{split}&{\cal
A}^{\mathrlap{\Yright}~{}\Yleft}_{4}(P_{i})=g^{2}\prod_{i=1}^{4}{\cal
C}_{\Delta_{i}}\int_{-i\infty}^{i\infty}{\mathrm{d}c\,\mathrm{d}\gamma\mathrm{d}f\over(2\pi
i)^{3}}\widehat{G}^{\Delta}_{BB}(c,\gamma,f)\\\
&\int^{\infty}_{0}\prod_{i=1}^{5}{\mathrm{d}t_{i}\over
t_{i}}{t_{i}^{\Delta_{i}}\over\Gamma(\Delta_{i})}{1\over(1+t_{5}^{2})^{\Delta_{3}+\Delta_{4}+\gamma+f-2h\over
2}}\Gamma\left({\Delta_{1}+\Delta_{2}+\gamma+f-2h\over
2}\right)\Gamma\left({\Delta_{3}+\Delta_{4}+\gamma+f-2h\over 2}\right)\\\
&~{}~{}~{}~{}{\rm e}^{2(t_{1}t_{2}t_{5}P_{1}\cdot
P_{2}+t_{1}t_{3}t_{5}P_{1}\cdot P_{3}+t_{2}t_{3}t_{5}P_{2}\cdot
P_{3}+t_{2}t_{4}t_{5}P_{2}\cdot P_{4}+t_{3}t_{4}P_{3}\cdot
P_{4})}\,.\end{split}$ (18)

Now we can use the Symanzik formula Symanzik:1972wj ; Mack:2009mi ,

$\int_{0}^{\infty}\int^{\infty}_{0}\prod_{i=1}^{n}{\mathrm{d}t_{i}\over
t_{i}}t_{i}^{\Delta_{i}}{\rm e}^{\sum_{i<j}^{n}t_{i}t_{j}Q_{ij}}={1\over
2(2\pi i)^{n(n-3)\over
2}}\int_{\Sigma_{n}}\prod_{i<j}^{n}\Gamma(\delta_{ij})(Q_{ij})^{-\delta_{ij}}\,,$
(19)

where $\Sigma_{n}$ is the ${n(n-3)\over 2}-$dimensional Manifold defined by
the solution to the system,

$\delta_{ii}=-\Delta_{i}\,,\quad\sum_{i=1}^{n}\delta_{ij}=0\,.$ (20)

Using this identity on the integration over the Schwinger parameters
associated to the bulk-to-boundary propagators, namely $t_{i}$ with
$i\in\\{1,\cdots,4\\}$, we get,

$\begin{split}&{\cal
A}^{\mathrlap{\Yright}~{}\Yleft}_{4}(P_{i})=g^{2}\prod_{i=1}^{4}{\cal
C}_{\Delta_{i}}\int_{-i\infty}^{i\infty}{\mathrm{d}c\,\mathrm{d}\gamma\mathrm{d}f\over(2\pi
i)^{3}}\widehat{G}^{\Delta}_{BB}(c,\gamma,f){1\over 2(2\pi
i)^{2}}\int_{\Sigma_{4}}\prod_{i<j}^{4}{\Gamma(\delta_{ij})\over\Gamma(\Delta_{i})}(P_{i}\cdot
P_{j})^{-\delta_{ij}}\\\
&~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\Gamma\left({\Delta_{1}+\Delta_{2}+\gamma+f-2h\over
2}\right)\Gamma\left({\Delta_{3}+\Delta_{4}+\gamma+f-2h\over 2}\right)\\\
&~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\int^{\infty}_{0}{\mathrm{d}t_{5}\over
t_{5}}{t_{5}^{\gamma+f-\delta_{13}-\delta_{14}-\delta_{23}-\delta_{24}}\over\Gamma(\gamma+f)}(1+t_{5}^{2})^{-\delta_{12}-{\Delta_{3}+\Delta_{4}+\gamma+f-2h\over
2}}\end{split}$ (21)

Integration over $t_{5}$ can be expressed as a beta function and results in,

$\begin{split}&{\cal
A}^{\mathrlap{\Yright}~{}\Yleft}_{4}(P_{i})=g^{2}\prod_{i=1}^{4}{\cal
C}_{\Delta_{i}}\int_{-i\infty}^{i\infty}{\mathrm{d}c\,\mathrm{d}\gamma\mathrm{d}f\over(2\pi
i)^{3}}\widehat{G}^{\Delta}_{BB}(c,\gamma,f){1\over 2(2\pi
i)^{2}}\int_{\Sigma_{4}}\prod_{i<j}^{4}{\Gamma(\delta_{ij})\over\Gamma(\Delta_{i})}(P_{i}\cdot
P_{j})^{-\delta_{ij}}\\\
&~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\Gamma\left({\Delta_{1}+\Delta_{2}+\gamma+f-2h\over
2}\right)\Gamma\left({\Delta_{3}+\Delta_{4}+\gamma+f-2h\over 2}\right)\\\
&~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\frac{\Gamma\left(\frac{\gamma+f-\delta_{1,3}-\delta_{1,4}-\delta_{2,3}-\delta_{2,4}}{2}\right)\Gamma\left(\frac{-2h+2\delta_{1,2}+\delta_{1,3}+\delta_{1,4}+\delta_{2,3}+\delta_{2,4}+\Delta(3)+\Delta(4)}{2}\right)}{2\Gamma\left(-h+\delta_{1,2}+\frac{\gamma+f+\Delta(3)+\Delta(4)}{2}\right)}\,,\end{split}$
(22)

also the integration over $f$ can be performed straightforwardly leading to

$\begin{split}&{\cal A}^{\mathrlap{\Yright}~{}\Yleft}_{4}(P_{i})={1\over
2(2\pi
i)^{2}}\int_{\Sigma_{4}}\prod_{i<j}^{4}{\Gamma(\delta_{ij})\over\Gamma(\Delta_{i})}(P_{i}\cdot
P_{j})^{-\delta_{ij}}\\\ &\times\,g^{2}\prod_{i=1}^{4}{\cal
C}_{\Delta_{i}}\int_{-i\infty}^{i\infty}{\mathrm{d}c\,\mathrm{d}\gamma\over(2\pi
i)^{2}}{\Gamma(h+c-\gamma)\Gamma(h-c-\gamma)\over\Gamma(c)\Gamma(-c)((\Delta-h)^{2}-c^{2})}{\Gamma(h-\gamma)\over\Gamma(2h-2\gamma)}\\\
&~{}~{}\times\,\frac{\Gamma\left(\frac{\gamma-s}{2}\right)\Gamma\left(\frac{-2h+\gamma+\Delta_{1}+\Delta_{2}}{2}\right)\Gamma\left(\frac{-2h+\gamma+\Delta_{3}+\Delta_{4}}{2}\right)}{\Gamma\left(\frac{-s+\Delta_{1}+\Delta_{2}}{2}\right)\Gamma\left(\frac{-s+\Delta_{3}+\Delta_{4}}{2}\right)}\,.\end{split}$
(23)

Where $s$ and $t$ are defined by the following solutions of the system (20)
for $n=4$,

$\begin{split}\delta_{12}&=\frac{1}{2}\left(\Delta_{1}+\Delta_{2}-s\right)\,,\quad\delta_{13}=\frac{1}{2}\left(\Delta_{1}+\Delta_{3}-t\right)\,,\\\
\delta_{14}&=\frac{1}{2}\left(-\Delta_{2}-\Delta_{3}+s+t\right)\,,\quad\delta_{23}=\frac{1}{2}\left(-\Delta_{1}-\Delta_{4}+s+t\right)\,,\\\
\delta_{24}&=\frac{1}{2}\left(\Delta_{2}+\Delta_{4}-t\right)\,,\quad\delta_{34}=\frac{1}{2}\left(\Delta_{3}+\Delta_{4}-s\right)\,.\end{split}$
(24)

Following Mack Mack:2009mi  and knowing that Witten diagrams correspond to
correlation functions on the boundary (or pieces of the whole correlator), we
expect the following Mellin representation,

$A(P_{i})={{\cal N}\over(2\pi i)^{n(n-3)\over
2}}\int_{\Sigma_{n}}\mathrm{d}\delta_{ij}M(\delta_{ij})\prod_{i<j}^{n}\Gamma(\delta_{ij})(P_{i}\cdot
P_{j})^{-\delta_{ij}}\,,$ (25)

with

${\cal N}={1\over 2}\Gamma\left({\sum_{i=1}^{n}\Delta_{i}-2h\over
2}\right)\prod_{i=1}^{n}{{\cal C}_{\Delta_{i}}\over\Gamma(\Delta_{i})}$ (26)

therefore comparing with (23) we conclude that the Mellin amplitude for the
exchaged scalar diagram can be represented by 333In this case the Mellin
amplitude does not depend on $t$ because we are considering only the
$s-$channel exchange.,

$\begin{split}&M^{\mathrlap{\Yright}~{}\Yleft}=g^{2}\int_{-i\infty}^{i\infty}{\mathrm{d}c\,\mathrm{d}\gamma\over(2\pi
i)^{2}}{\Gamma(h+c-\gamma)\Gamma(h-c-\gamma)\over\Gamma(c)\Gamma(-c)((\Delta-h)^{2}-c^{2})}{\Gamma(h-\gamma)\over\Gamma(2h-2\gamma)}\\\
&~{}~{}\frac{\Gamma\left(\frac{\gamma-s}{2}\right)\Gamma\left(\frac{-2h+\gamma+\Delta_{1}+\Delta_{2}}{2}\right)\Gamma\left(\frac{-2h+\gamma+\Delta_{3}+\Delta_{4}}{2}\right)}{\Gamma\left(\frac{-2h+\Delta_{1}+\Delta_{2}+\Delta_{3}+\Delta_{4}}{2}\right)\Gamma\left(\frac{-s+\Delta_{1}+\Delta_{2}}{2}\right)\Gamma\left(\frac{-s+\Delta_{3}+\Delta_{4}}{2}\right)}\,.\end{split}$
(27)

##  3 Bubble Witten diagram

Figure 2: Bubble Witten diagram.

The bubble Witten diagram corresponds to a similar expression as (5) but with
two bulk-to-bulk propagators connecting the same points $X_{1},\,X_{2}$
instead of one, as shown in figure 2. In practice, we can think of the product
of the two bulk-to-bulk propagators that depend on the same $u$, as a single
bulk-to-bulk propagator of the form (3) but with two $c-$integrations and two
$\gamma-$integrations. That particularity allow us to do a single expansion of
the type (6) on the product, instead of one for each propagator. The amplitude
associated to this diagram is given by

$\begin{split}&{\cal
A}^{\mathrlap{\vee}\wedge}_{4}(P_{1},P_{2},P_{3},P_{4})=\\\
&g^{2}\int_{AdS}\mathrm{d}X_{1}\mathrm{d}X_{2}G^{\Delta_{1}}_{\partial
B}(P_{1},X_{1})G^{\Delta_{2}}_{\partial
B}(P_{2},X_{2})G^{\Delta}_{BB}(X_{1},X_{2})G^{\Delta^{\prime}}_{BB}(X_{1},X_{2})G^{\Delta_{3}}_{B\partial}(X_{2},P_{3})G^{\Delta_{4}}_{B\partial}(X_{2},P_{4})\;.\end{split}$
(28)

Replacing the definitions for the bulk-to-boundary propagators, the
representation (3) and using (6) we get,

$\begin{split}&{\cal
A}^{\mathrlap{\vee}\wedge}_{4}(P_{i})=g^{2}\prod_{i=1}^{4}{\cal
C}_{\Delta_{i}}\prod_{j=1}^{2}\int_{-i\infty}^{i\infty}{\mathrm{d}c_{j}\,\mathrm{d}\gamma_{j}\mathrm{d}f\over(2\pi
i)^{4}}\\\
&{\Gamma(h+c_{j}-\gamma_{j})\Gamma(h-c_{j}-\gamma_{j})\Gamma(\gamma_{j})\Gamma(h-\gamma_{j})\over\Gamma(c_{j})\Gamma(-c_{j})((\Delta-h)^{2}-c_{1}^{2})((\Delta^{\prime}-h)^{2}-c_{2}^{2})\Gamma(2h-2\gamma_{j})}{\Gamma(\gamma_{1}+\gamma_{2}+f)\Gamma(-f)(-2)^{f}\over\Gamma(\gamma_{1}+\gamma_{2})}\\\
&\int_{AdS}{\mathrm{d}X_{1}\mathrm{d}X_{2}\over(-2P_{1}\cdot
X_{1})^{\Delta_{1}}(-2P_{2}\cdot X_{1})^{\Delta_{2}}(-2(X_{1}\cdot
X_{2}))^{\gamma_{j}+f_{j}}(-2P_{3}\cdot X_{2})^{\Delta_{3}}(-2P_{4}\cdot
X_{2})^{\Delta_{4}}}\,.\end{split}$ (29)

we can notice that the computation should follows the same lines as the
exchange diagram in section above with $\Delta_{5}$ in (11) replaced by
$\Delta_{5}=\gamma_{1}+\gamma_{2}+f$, leading us to the following
representation of the corresponding Mellin amplitude,

$\begin{split}&M^{\mathrlap{\vee}\wedge}_{4}=g^{2}\prod_{j=1}^{2}\int_{-i\infty}^{i\infty}{\mathrm{d}c_{j}\,\mathrm{d}\gamma_{j}\over(2\pi
i)^{4}}\\\
&{\Gamma(h+c_{j}-\gamma_{j})\Gamma(h-c_{j}-\gamma_{j})\Gamma(\gamma_{j})\Gamma(h-\gamma_{j})\over\Gamma(c_{j})\Gamma(-c_{j})((\Delta-h)^{2}-c_{1}^{2})((\Delta^{\prime}-h)^{2}-c_{2}^{2})\Gamma(2h-2\gamma_{j})}{1\over\Gamma(\gamma_{1}+\gamma_{2})}\\\
&~{}~{}\frac{\Gamma\left(\frac{\gamma_{1}+\gamma_{2}-s}{2}\right)\Gamma\left(\frac{-2h+\gamma_{1}+\gamma_{2}+\Delta_{1}+\Delta_{2}}{2}\right)\Gamma\left(\frac{-2h+\gamma_{1}+\gamma_{2}+\Delta_{3}+\Delta_{4}}{2}\right)}{\Gamma\left(\frac{-2h+\Delta_{1}+\Delta_{2}+\Delta_{3}+\Delta_{4}}{2}\right)\Gamma\left(\frac{-s+\Delta_{1}+\Delta_{2}}{2}\right)\Gamma\left(\frac{-s+\Delta_{3}+\Delta_{4}}{2}\right)}\,.\end{split}$
(30)

##  4 One-loop vertex correction

Figure 3:  One loop Vertex correction.

Before moving onto the remaining loop corrections to the four point scattering
amplitude, let us consider the one-loop correction to the three point
function, represented in figure 3 and given by,

${\cal A}^{\rhd}_{3}(P_{i})=g^{3}\prod_{i=1}^{3\,{\rm
Mod}3}\int_{AdS}\mathrm{d}X_{i}\,G^{\Delta_{i}}_{\partial
B}(P_{i},X_{i})G^{\Delta_{ii+1}}_{BB}(X_{i},X_{i+1})\,.$ (31)

Introducing Schwinger parameters in the usual way we have,

$\begin{split}&{\cal A}^{\rhd}_{3}(P_{i})=g^{3}\prod_{i=1}^{3\,{\rm
Mod}3}\,\widehat{G}^{\Delta_{ii+1}}_{BB}(c_{i,i+1},\,\gamma_{i,i+1},\,f_{i,i+1})\\\
&~{}~{}~{}~{}\times\prod_{j=1}^{7}\int_{0}^{\infty}{\mathrm{d}t_{j}\over
t_{j}\,\Gamma(\Delta_{j})}t_{j}^{\Delta_{j}}\int_{AdS}\mathrm{d}X_{i}{\rm
e}^{2(t_{1}P_{1}+t_{5}X_{2}+t_{7}X_{3})\cdot X_{1}}{\rm
e}^{2(t_{2}P_{2}+t_{6}X_{3})\cdot X_{2}}{\rm e}^{2(t_{3}P_{3})\cdot
X_{3}}\,,\end{split}$ (32)

where we have redefined the conformal weights of the bulk-to-bulk propagators
in (31) as
$\Delta_{5}\equiv\Delta_{12}=\gamma_{12}+f_{12},\,~{}\Delta_{6}\equiv\Delta_{23}=\gamma_{23}+f_{23},\,~{}\Delta_{7}\equiv\Delta_{13}=\gamma_{31}+f_{31}$.
We then proceed to integrate the bulk coordinates in the exact same manner as
in the previous cases to get

$\begin{split}&{\cal M}^{\rhd}_{3}(P_{i})=\prod_{i=1}^{3\,{\rm
Mod}3}\int_{-i\infty}^{i\infty}{\mathrm{d}c_{i\,i+1}\,\mathrm{d}\gamma_{i\,i+1}\mathrm{d}f_{i\,i+1}\over(2\pi
i)^{4}}{\Gamma(h+c_{i\,i+1}-\gamma_{i\,i+1})\Gamma(h-c_{i\,i+1}-\gamma_{i\,i+1})\over\Gamma(c_{i\,i+1})\Gamma(-c_{i\,i+1})((\Delta-h)^{2}-c_{i\,i+1}^{2})}\\\
&{\Gamma(h-\gamma_{i\,i+1})\Gamma(-f_{i\,i+1})(-2)^{f}_{i\,i+1}\over\Gamma(2h-2\gamma_{i\,i+1})}\\\
&\Gamma\left({\Delta_{1}+\gamma_{12}+f_{12}+\gamma_{31}+f_{31}-2h\over
2}\right)\Gamma\left({\Delta_{2}+\gamma_{12}+f_{12}+\gamma_{23}+f_{23}-2h\over
2}\right)\\\
&\Gamma\left({\Delta_{3}+\gamma_{23}+f_{23}+\gamma_{31}+f_{31}-2h\over
2}\right)\\\ &\prod_{j=5}^{7}\int{\mathrm{d}t_{j}\over
t_{j}}\,t_{j}^{\Delta_{j}}\,{T_{7}^{-\delta_{12}}\,T_{6}^{-\delta_{13}}\,T_{5}^{-\delta_{23}}\,(1+t_{7}^{2}+T_{5}^{2})^{-{\Delta_{3}+\gamma_{23}+f_{23}+\gamma_{31}+f_{31}-2h\over
2}}\over(1+t_{5}^{2})^{{\Delta_{2}+\gamma_{12}+f_{12}+\gamma_{23}+f_{23}-2h\over
2}}}\end{split}$ (33)

where for convenience of notation we have defined
$T_{5}=t_{5}t_{7}+t_{6},\,T_{6}=t_{5}T_{5}+t_{7},\,T_{7}=T_{6}T_{5}$. Notice
that in this case, the Mellin amplitude does not depend on any kinematical
degrees of freedom, or in other words it is just a constant, since the
solution of the system of equation (20) is simply,

$\delta_{12}={\Delta_{1}+\Delta_{2}-\Delta_{3}\over
2},\,~{}\delta_{13}={\Delta_{1}+\Delta_{3}-\Delta_{2}\over
2},\,~{}\delta_{23}={\Delta_{2}+\Delta_{3}-\Delta_{1}\over 2}\,.$ (34)

This is just the reflection of the fact that the three-point function is fixed
by conformal invariance up to a constant, so what we are really computing here
is the loop correction of the three point structure constant.

Let us focus now on the integral on the last line of (33),

$\begin{split}&\int{\mathrm{d}t_{5}\over
t_{5}}{\mathrm{d}T_{5}}{\mathrm{d}t_{7}\over
t_{7}}\,t_{5}^{\Delta_{5}}(T_{5}-t_{5}t_{7})^{\Delta_{6}-1}\,t_{7}^{\Delta_{7}}\,\\\
&\times{(t_{5}T_{5}+t_{7})^{-\delta_{12}-\delta_{13}}\,T_{5}^{-\delta_{12}-\delta_{23}}\,(1+t_{7}^{2}+T_{5}^{2})^{-{\Delta_{3}+\gamma_{23}+f_{23}+\gamma_{31}+f_{31}-2h\over
2}}\over(1+t_{5}^{2})^{{\Delta_{2}+\gamma_{12}+f_{12}+\gamma_{23}+f_{23}-2h\over
2}}}\end{split}$ (35)

where we have replaced $t_{6}$ by $T_{5}$. This looks like a difficult
integral to perform, let alone in a clean compact way, so we choose at this
point to perform a brute force approach. In order to do so, we expand the
factor $(1+t_{7}^{2}+T_{5}^{2})$ in the numerator (35) by means of applying
the Mellin-Barnes transform (6) one more time,

$\begin{split}&\int_{-i\infty}^{i\infty}{\mathrm{d}\ell\over 2\pi
i}{\Gamma(a+\ell)\Gamma(-\ell)\over\Gamma(a)}I^{\rhd}_{S}\,,\end{split}$ (36)

where we have defined the new integral,

$I^{\rhd}_{S}\equiv\int{\mathrm{d}t_{5}\over
t_{5}}{\mathrm{d}T_{5}}{\mathrm{d}t_{7}\over
t_{7}}\,t_{5}^{\Delta_{5}}(T_{5}-t_{5}t_{7})^{\Delta_{6}-1}\,t_{7}^{\Delta_{7}}{(t_{5}T_{5}+t_{7})^{-\delta_{12}-\delta_{13}}\,T_{5}^{2\ell-\delta_{12}-\delta_{23}}\over(1+t_{5}^{2})^{b}(1+t_{7}^{2})^{a+\ell}}$
(37)

and $a={{\Delta_{3}+\gamma_{23}+f_{23}+\gamma_{31}+f_{31}-2h\over 2}}$,
$b={\Delta_{2}+\gamma_{12}+f_{12}+\gamma_{23}+f_{23}-2h\over 2}$. With the
help of Mathematica Mathematica  this expansion allow us to represent the
integral over all the remaining Schwinger parameters in terms of a lengthy
combination of Hypergeometric functions, as is shown in Appendix equation
(80). However, we will discuss an approach to deal with a more general family
of integrals in section 7 which includes the one above.

##  5 Triangle Witten diagram

Figure 4:  One loopTriangle.

The 1-loop four-particle triangle Witten diagram is represented by the figure
4 and can be written in the form,

${\cal
A}^{\rhd}(P_{i})=g^{3}\prod_{i=1}^{3}\int_{AdS}\mathrm{d}X_{i}G^{\Delta_{i}}_{\partial
B}(P_{i},X_{i})G^{\Delta_{4}}_{B\partial}(P_{4},X_{3})\prod_{i<j}^{3}G^{\Delta_{ij}}_{BB}(X_{i},X_{j})\,.$
(38)

We use again the representation (LABEL:mybulk) and start by introducing
Schwinger parameters to exponenciate the propagators,

$\begin{split}{\cal A}^{\rhd}(P_{i})&=g^{3}\prod_{i=1}^{4}{\cal
C}_{\Delta_{i}}\prod_{i<j}^{3}\int_{-i\infty}^{i\infty}{\mathrm{d}c\,\mathrm{d}\gamma_{ij}\mathrm{d}f_{ij}\over(2\pi
i)^{3}}\widehat{G}^{\Delta_{ij}}_{BB}(c,\gamma_{ij},f_{ij})\\\
&\int^{\infty}_{0}\prod_{i=1}^{7}{\mathrm{d}t_{i}\over
t_{i}}{t_{i}^{\Delta_{i}}\over\Gamma(\Delta_{i})}\prod_{i=1}^{3}\int_{AdS}\mathrm{d}X_{i}{\rm
e}^{2(t_{1}P_{1}+t_{5}X_{2}+t_{7}X_{3})\cdot X_{1}}{\rm
e}^{2(t_{2}P_{2}+t_{6}X_{3})\cdot X_{2}}{\rm e}^{2(t_{3}P_{3}+t_{4}P_{4})\cdot
X_{3}}\,.\end{split}$ (39)

where for convenience of notation in the last line we have redefined the
conformal weights of the bulk-to-bulk propagators as,

$\Delta_{5}\equiv\Delta_{12}=\gamma_{12}+f_{12},\,~{}~{}\Delta_{6}\equiv\Delta_{23}=\gamma_{23}+f_{23},\,~{}~{}\Delta_{7}\equiv\Delta_{13}=\gamma_{13}+f_{13}\,.$
(40)

By taking $Q_{1}=(t_{1}P_{1}+t_{5}X_{2}+t_{7}X_{3})$, using (13) and scaling
$(t_{1},t_{5},t_{7})\to\sqrt{z_{1}}(t_{1},t_{5},t_{7})$ we can write,

$\begin{split}{\cal A}^{\rhd}(P_{i})&=g^{3}\prod_{i=1}^{4}{\cal
C}_{\Delta_{i}}\prod_{i<j}^{3}\int_{-i\infty}^{i\infty}{\mathrm{d}c_{ij}\,\mathrm{d}\gamma_{ij}\mathrm{d}f_{ij}\over(2\pi
i)^{3}}\widehat{G}^{\Delta_{ij}}_{BB}(c_{ij},\gamma_{ij},f_{ij})\\\
&\int^{\infty}_{0}\prod_{i=1}^{7}{\mathrm{d}t_{i}\over
t_{i}}{t_{i}^{\Delta_{i}}\over\Gamma(\Delta_{i})}{\rm
e}^{Q_{1}^{2}}\prod_{i=2}^{3}\int_{AdS}\mathrm{d}X_{i}{\rm
e}^{2(t_{2}P_{2}+t_{6}X_{3})\cdot X_{2}}{\rm e}^{2(t_{3}P_{3}+t_{4}P_{4})\cdot
X_{3}}\\\
&~{}~{}~{}\times\Gamma\left({\Delta_{1}+\gamma_{12}+f_{12}+\gamma_{13}+f_{13}-2h\over
2}\right)\,.\end{split}$ (41)

Following the same steps for the integration over $X_{2}$ and $X_{3}$ we end
up with the expression,

$\begin{split}&{\cal A}^{\rhd}(P_{i})=g^{3}\prod_{i=1}^{4}{\cal
C}_{\Delta_{i}}\prod_{i<j}^{3}\int_{-i\infty}^{i\infty}{\mathrm{d}c\,\mathrm{d}\gamma_{ij}\mathrm{d}f_{ij}\over(2\pi
i)^{3}}\widehat{G}^{\Delta_{ij}}_{BB}(c_{ij},\gamma_{ij},f_{ij})\,{\rm
e}^{2t_{1}t_{2}P_{1}\cdot P_{2}}\\\
&\int^{\infty}_{0}\prod_{i=1}^{7}{\mathrm{d}t_{i}\over
t_{i}}{t_{i}^{\Delta_{i}}\over\Gamma(\Delta_{i})}\,{\rm
e}^{Q_{3}^{2}}\,\Gamma\left({\Delta_{1}+\gamma_{12}+f_{12}+\gamma_{13}+f_{13}-2h\over
2}\right)\Gamma\left({\Delta_{2}+\gamma_{12}+f_{12}+\gamma_{23}+f_{23}-2h\over
2}\right)\\\ &\int_{0}^{\infty}{\mathrm{d}z_{3}\over
z_{3}}z_{3}^{\Delta_{3}+\Delta_{4}+\gamma_{23}+f_{23}+\gamma_{13}+f_{13}-2h\over
2}{{\rm
e}^{-z_{3}(1+(t_{5}t_{7}+t_{6})^{2})}\over(t_{5}^{2}+z_{3}t_{7}^{2})^{\Delta_{2}+\gamma_{12}+f_{12}+\gamma_{23}+f_{23}-2h\over
2}}\,,\end{split}$ (42)

with
$Q_{3}=(t_{7}+t_{5}(t_{5}t_{7}+t_{6}))t_{1}P_{1}+(t_{5}t_{7}+t_{6})t_{2}P_{2}+t_{3}P_{3}+t_{4}P_{4}$.

As we have pointed out already, we can notice from this expression that the
increasing complexity of the integral over $z_{i}$ comes from the $t_{i}^{2}$
factors left behind by the on-shell condition $X_{i}^{2}=-1$. The integral
over $z_{3}$ can be represented as a combination of Hypergeometric functions,
but since is somewhat lengthy we choose instead to introduce yet another
Mellin parameter to trivialize this integral by Mellin transform the exponent
at the last line of (42),

$\begin{split}&\int_{-i\infty}^{i\infty}{\mathrm{d}g\over(2\pi
i)}\Gamma(-g)\int_{0}^{\infty}{\mathrm{d}z_{3}\over
z_{3}}z_{3}^{\Delta_{3}+\Delta_{4}+\gamma_{23}+f_{23}+\gamma_{13}+f_{13}+g-2h\over
2}{(1+(t_{5}t_{7}+t_{6})^{2})^{g}\over(t_{5}^{2}+z_{3}t_{7}^{2})^{\Delta_{2}+\gamma_{12}+f_{12}+\gamma_{23}+f_{23}-2h\over
2}}\\\ &=\int_{-i\infty}^{i\infty}{\mathrm{d}g\over(2\pi
i)}\Gamma(-g)\frac{\Gamma\left(\frac{-g+f_{12}-f_{13}+\gamma_{12}-\gamma_{13}+\Delta_{2}-\Delta_{3}-\Delta_{4}}{2}\right)\Gamma\left(\frac{g-2h+f_{13}+f_{23}+\gamma_{13}+\gamma_{23}+\Delta_{3}+\Delta_{4}}{2}\right)}{\Gamma\left(\frac{-2h+f_{12}+f_{23}+\gamma_{12}+\gamma_{23}+\Delta_{2}}{2}\right)}\\\
&~{}~{}~{}t_{5}^{g-\gamma_{12}-f_{12}-\gamma_{13}-f_{13}-\Delta_{2}+\Delta_{3}-\Delta_{4}}t_{7}^{-\gamma_{13}-\gamma_{23}-\Delta_{3}-\Delta_{4}-f_{13}-f_{23}-g+2h}(1+(t_{5}t_{7}+t_{6})^{2})^{g}\,.\end{split}$
(43)

Putting it back in (42) and using Symanzik identity (19) we have that the
Mellin amplitude associated to the one-loop triangle can be written as,

$\begin{split}&M^{\rhd}(s,\,t)=g^{3}\prod_{i=1}^{4}{\cal
C}_{\Delta_{i}}\prod_{i<j}^{3}\int_{-i\infty}^{i\infty}{\mathrm{d}c\,\mathrm{d}\gamma_{ij}\mathrm{d}f_{ij}\mathrm{d}g\over(2\pi
i)^{4}}\Gamma(-g)\,\widehat{G}^{\Delta_{ij}}_{BB}(c_{ij},\gamma_{ij},f_{ij})\\\
&\Gamma\left({\Delta_{1}+\gamma_{12}+f_{12}+\gamma_{13}+f_{13}-2h\over
2}\right)\Gamma\left({\Delta_{2}+\gamma_{12}+f_{12}+\gamma_{23}+f_{23}-2h\over
2}\right)\\\
&\frac{\Gamma\left(\frac{-g+f_{12}-f_{13}+\gamma_{12}-\gamma_{13}+\Delta_{2}-\Delta_{3}-\Delta_{4}}{2}\right)\Gamma\left(\frac{g-2h+f_{13}+f_{23}+\gamma_{13}+\gamma_{23}+\Delta_{3}+\Delta_{4}}{2}\right)}{\Gamma\left(\frac{-2h+f_{12}+f_{23}+\gamma_{12}+\gamma_{23}+\Delta_{2}}{2}\right)}\\\
&\int^{\infty}_{0}\prod_{i=5}^{7}{\mathrm{d}t_{i}\over
t_{i}}{1\over\Gamma(\Delta_{i})}\,t_{5}^{g+\gamma_{13}+f_{13}-\Delta_{2}+\Delta_{3}+\Delta_{4}}\,t_{6}^{\gamma_{23}+f_{23}}\,t_{7}^{-\gamma_{23}-f_{23}-\Delta_{3}-\Delta_{4}-g+2h}(1+T_{5}^{2})^{g}\\\
&~{}~{}~{}~{}~{}~{}T_{7}^{-\delta_{12}}T_{6}^{-\delta_{13}-\delta_{14}}T_{5}^{-\delta_{23}-\delta_{24}}\,,\end{split}$
(44)

where we have defined for convenience on the notation,
$T_{5}=(t_{5}t_{7}+t_{6}),\,T_{6}=(t_{7}+t_{5}T_{5}),\,T_{7}=(t_{5}+T_{6}T_{5})$.
Interestingly, the integration over the remaining Schwinger parameters does
not depend on $\gamma_{12}$ and $f_{12}$. Below we will propose a way to treat
the remaining integration over Schwinger parameters on the last line of
(LABEL:trianglemellin), but before doing so, let us discuss first the box
Witten diagram in the next section.

##  6 Box Witten diagram

Figure 5:  One loop box.

The 1-loop four-particle box Witten diagram is represented by the figure 5 and
can be written in the form,

${\cal A}^{\Box}(P_{i})=g^{3}\prod_{i=1}^{4\,{\rm
Mod}4}\int_{AdS}\mathrm{d}X_{i}\,G^{\Delta_{i}}_{\partial
B}(P_{i},X_{i})G^{\Delta_{ii+1}}_{BB}(X_{i},X_{i+1})\,.$ (45)

Introducing Schwinger parameters in the usual way we have,

$\begin{split}&{\cal A}^{\Box}(P_{i})=g^{4}\prod_{i=1}^{4\,{\rm
Mod}4}\,\widehat{G}^{\Delta_{ii+1}}_{BB}(c_{i,i+1},\,\gamma_{i,i+1},\,f_{i,i+1})\\\
&~{}~{}~{}~{}\times\prod_{j=1}^{8}{\mathrm{d}t_{j}\over
t_{j}\,\Gamma(\Delta_{j})}t_{j}^{\Delta_{j}}\int_{AdS}\mathrm{d}X_{i}{\rm
e}^{2(t_{1}P_{1}+t_{5}X_{2}+t_{8}X_{4})\cdot X_{1}}{\rm
e}^{2(t_{2}P_{2}+t_{6}X_{3})\cdot X_{2}}{\rm e}^{2(t_{3}P_{3}+t_{7}X_{4})\cdot
X_{3}}{\rm e}^{2t_{4}P_{4}\cdot X_{4}}\,.\end{split}$ (46)

where we have defined
$\Delta_{5}=\gamma_{12}+f_{12},\,\Delta_{6}=\gamma_{23}+f_{23},\,\Delta_{7}=\gamma_{34}+f_{34}$
and $\Delta_{8}=\gamma_{41}+f_{41}$. We then proceed to integrate the bulk
coordinates in the exact same manner as in the previous cases to get

$\begin{split}&M^{\Box}(s,t)=\prod_{i=1}^{4\,{\rm
Mod}4}\int_{-i\infty}^{i\infty}{\mathrm{d}c_{i\,i+1}\,\mathrm{d}\gamma_{i\,i+1}\mathrm{d}f_{i\,i+1}\over(2\pi
i)^{4}}{\Gamma(h+c_{i\,i+1}-\gamma_{i\,i+1})\Gamma(h-c_{i\,i+1}-\gamma_{i\,i+1})\over\Gamma(c_{i\,i+1})\Gamma(-c_{i\,i+1})((\Delta-h)^{2}-c_{i\,i+1}^{2})}\\\
&{\Gamma(h-\gamma_{i\,i+1})\Gamma(-f_{i\,i+1})(-2)^{f}\over\Gamma(2h-2\gamma_{i\,i+1})}\\\
&\Gamma\left({\Delta_{1}+\gamma_{12}+f_{12}+\gamma_{41}+f_{41}-2h\over
2}\right)\Gamma\left({\Delta_{2}+\gamma_{12}+f_{12}+\gamma_{23}+f_{23}-2h\over
2}\right)\\\
&\Gamma\left({\Delta_{3}+\gamma_{23}+f_{23}+\gamma_{34}+f_{34}-2h\over
2}\right)\Gamma\left({\Delta_{4}+\gamma_{34}+f_{34}+\gamma_{41}+f_{41}-2h\over
2}\right)\\\ &\prod_{j=5}^{8}{\mathrm{d}t_{j}\over
t_{j}}\,t_{j}^{\Delta_{j}}\,{(1+t_{8}^{2}(1+t_{5}^{2})+T_{5}^{2})^{-{\Delta_{4}+\gamma_{34}+f_{34}+\gamma_{41}+f_{41}-2h\over
2}}\over(1+t_{5}^{2})^{{\Delta_{2}+\gamma_{12}+f_{12}+\gamma_{23}+f_{23}-2h\over
2}}(1+t_{6}^{2})^{{\Delta_{3}+\gamma_{23}+f_{23}+\gamma_{34}+f_{34}-2h\over
2}}}\\\ &~{}~{}~{}~{}~{}\times
T_{10}^{-\delta_{12}}T_{9}^{-\delta_{13}}T_{8}^{-\delta_{23}}\,T_{7}^{-\delta_{14}}\,T_{6}^{-\delta_{24}}\,T_{5}^{-\delta_{34}}\end{split}$
(47)

where in order to simplify the notation we have defined,

$\begin{split}T_{5}&=t_{5}t_{8}t_{6}+t_{7},\,~{}T_{6}=t_{6}T_{5}+t_{5}t_{8},\,~{}T_{7}=t_{5}T_{6}+t_{8},\,~{}T_{8}=t_{6}+T_{5}T_{6},\\\
T_{9}&=t_{5}t_{6}+T_{7}T_{5},\,~{}T_{10}=t_{5}(1+t_{6}^{2})+T_{7}T_{6}\end{split}$
(48)

At this point some comments are in order. In the last two examples we have
been able to write the one loop Mellin representation of the four-point
triangle and box Witten diagrams in terms of multiple contour integrals and
some remaining integrals over Schwinger parameters. Let us assume for a moment
that we are able to evaluate all the integrations over the Schwinger
parameters, then we still will be left with a multiple contour integration of
Mellin-barnes type. Even thought performing this multiple integrals can still
be tricky, is not hard to convince ourselves that those contour integrations
are in principle much more simple than the type of integrals involving
Schwinger parameters, since they are picking up residues only over (a rational
function of) gamma functions, which only have simple poles. Therefore, the
final result from such a multiple contour integration is expected to be, in
general, an infinite sum of all those residues, without any derivatives. In
fact, Hypergeometric functions are represented as contour integrals over
rational functions of gamma functions, similar to the ones we have on the
given examples. 444Actually, the contour integrals we have here are more like
Hypergeometric functions evaluated at $z=1$, namely,
${}_{p}F_{q}(\\{a\\},\\{b\\}|1)$, which are simpler and have simplifying
representations. There is still the possibility of having singularities coming
from poles trapping and pinching the integration contour, but on physical
grounds those are the singularities expected to produce the poles in the
Mellin amplitude.

In order to the above argument to hold, we should hope to be able to write the
leftover integration over Schwinger parameters as a rational function of gamma
functions or even as a similar contour integral over rational functions of
gamma functions. Even though we did not manage to explicitly do it in this
work, in the next section we present a proposal to treat the aforementioned
integrals as contour integrals.

Before proceed to the next section, it is worth to notice that we also can
deal with the remaining Schwinger parameters by iteratively apply formula (6)
to convert those integrals at the last line at (LABEL:trianglemellin) and (47)
as contour integrations over Beta functions. However, on the one hand we would
like to have a representation with the minimal amount of integrations, and
this procedure requires the introduction of at least an additional contour
integral for each polynomial $T_{i}$, since essentially formula (6) would be
used to “binomial’ expand each of them. On the other hand, this procedure
seems quite arbitrary since there is many different ways to choose the factors
playing the role of $A$ and $B$ in (6) and we don’t have a criteria to pick
the more convenient choice. 555We thank to Ellis Ye Yuan for discussions on
this point.

##  7 Proposal on how to treat the integration over the remaining Schwinger
parameters.

In this section we would like to propose a procedure to deal with the
integration of Schwinger parameters associated to the bulk-to-bulk propagators
that were left behind after using the Symanzik formula (19). The type of
integrals we would like to consider have the schematic form,

${\cal I}_{n_{int}}\equiv\int^{\infty}_{0}\prod_{i>n_{\rm ext}}^{n_{\rm
ext}+n_{\rm
int}}{\mathrm{d}t_{i}}\,R(\\{t_{i}\\})\prod_{i<j}^{n_{ext}}\left({\over}T_{ij}(\\{t_{i}\\})\right)^{-\delta_{ij}}\,,$
(49)

where $n_{\rm ext},\,n_{\rm int}$ are the number of external particles
(attached to the boundary of AdS)666In this paper we are only focused on
$n_{\rm ext}=4$, but we prefer to keep calling it $n_{\rm ext}$ since we think
most of the discussion in this section applies equally to the more general
$n-$gon case. and the number of internal particles (or bulk-to-bulk
propagators) respectively, $\\{t_{i}\\}$ denotes the set of integration
variables, $T_{i}(\\{t_{i}\\})$ are polynomials with integer coefficients and
$R(\\{t_{i}\\})$ is in general rational function.

Let us start illustrating the procedure with an over simplified example,
namely a beta-like function with a negative parameter,

${\cal I}_{0}=\int^{\infty}_{0}{\mathrm{d}t\over
t}\,t^{\alpha}\,(1+t)^{\beta}\,.$ (50)

We take the exponents $\alpha$ as negative integers and at the end we analytic
continue them to complex values. We first Wick rotate the integration contour
to the imaginary axes and pick up the residues sitting at the positive
positive quarter of the complex plane. Lets us start performing the
integration by using the Cauchy integration formula picking up the poles at
$t=0$,

${\cal
I}_{0}={\Gamma(\beta+1)\over\Gamma(-\alpha+1)\Gamma(\beta+\alpha+1)}\,.$ (51)

As desired, the integration over this hypothetical Schwinger parameter is
given by a rational function of gamma functions. Notice as well that in this
form $\alpha$ can be analytically continue to generic complex values. For
example, it can be related to the more familiar Beta function by analitically
continue it to $\mathbb{R}(\alpha)>0$ & $\alpha\notin\mathbb{Z}$, by means of
the identity,

$\Gamma(z)\Gamma(1-z)={\pi\over{\rm sin}(\pi z)},\quad z\notin\mathbb{Z}\,,$
(52)

leading us to

${\cal I}_{0}={\rm B}(\alpha,\beta+1){{\rm sin}\pi(-\alpha+1)\over\pi}\,.$
(53)

Maybe a more familiar example is given by writing the Beta function as a
contour integral over the Pochhammer contour,

$(1-{\rm e}^{2\pi\alpha})(1-{\rm e}^{2\pi\beta})B(\alpha,\beta)=\int_{\cal
C}t^{\alpha-1}(1-t)^{\beta-1}\,,$ (54)

which corresponds to the analytic continuation of the Beta function for all
values of $\alpha$ and $\beta$.

###  7.1 Sketching the general approach

In this section we would like to sketch, without getting into details, a
generalization to the procedure illustrated above to deal with the more
general integral (49). As in the previous example we consider the parameters
$\delta_{ij}$ as positive integers and we start by Wick rotate the integration
contour to pick up the residues at the roots of the polynomials in the
denominator of (49), including the ones in the function $R(t_{i})$,
analytically continuing the result to more general values at the end.

Before moving on, is interesting to notice that we can homogenize every
polynomial by the introduction of a delta function, namely,

$1=\int dt\delta(t-1)\,$ (55)

such as that on every monomial composing the polynomial we include as many
factors of $t$ as necessary to make the whole polynomial homogeneous. For
example, the integral over the Schwinger parameters at (LABEL:trianglemellin)
can be rewritten as,

$\begin{split}&\int dt\int^{\infty}_{0}\prod_{i=5}^{7}{\mathrm{d}t_{i}\over
t_{i}}{\delta(t-1)\over\Gamma(\Delta_{i})}\,t_{5}^{g+\gamma_{13}+f_{13}-\Delta_{2}+\Delta_{3}+\Delta_{4}}\,t_{6}^{\gamma_{23}+f_{23}}\,t_{7}^{-\gamma_{23}-f_{23}-\Delta_{3}-\Delta_{4}-g+2h}(1+\hat{T}_{5}^{2})^{g}\\\
&~{}~{}~{}~{}~{}~{}\hat{T}_{7}^{-\delta_{12}}\hat{T}_{6}^{-\delta_{13}-\delta_{14}}\hat{T}_{5}^{-\delta_{23}-\delta_{24}}\,,\end{split}$
(56)

where we have redefined the homogenized polynomials

$\hat{T}_{5}=(t_{5}t_{7}+t_{6}\,t),\,~{}~{}\hat{T}_{6}=(t_{7}\,t^{2}+t_{5}T_{5}),\,~{}~{}\hat{T}_{7}=(t_{5}\,t^{4}+T_{6}T_{5})\,.$
(57)

Even more, the polynomials $(\hat{T}_{i})^{\rho_{i}}$ are still homogeneous
with the same set of zeros as $\hat{T}_{i}$ but with higher multiplicity.
Therefore the problem of finding the zero locus of the set of polynomials can
be rephrased in terms of the problem of looking for the intersections of
algebraic curves in ${\mathbb{P}}^{n}$. This can be an interesting point of
view, linking the problem at hand with similar approaches to the flat space
S-matrix, where the scattering amplitudes elements are written as contour
integrals on the support of the intersection of rational curves
ArkaniHamed:2009dg , as in the present case. It would be interesting to
explore that point of view in future studies.

Let us return to our goal in this section by rewrite (49) as,

${\cal I}_{n_{int}}\equiv\int_{\cal C}\prod_{i=1}^{\cal
I}{\mathrm{d}t_{i}}\,{G(\\{t_{i}\\})\over\prod_{j=1}^{m}P_{j}(\\{t_{i}\\})}\,,$
(58)

where we have explicitly displayed all the polynomial dependence on the
denominator, such as that the number $m$ of polynomials in the denominator is
larger than the number of index couples $\\{ij\\}$, i.e, $m>(n_{\rm
ext}(n_{\rm ext}-1)/2)\,~{}$, and $G(\\{t_{i}\\})$ is assume to be an
holomorphic function inside the contour ${\cal C}$, given by the cycle
$C_{\cal C}(\\{t_{i}\\})=\\{\\{t_{i}\\}:P_{j}(\\{t_{i}\\})=\epsilon\\}$ with
$\epsilon$ small. ${\cal I}$ is the number of integration variables. Notice
that along the way we have defined some of the polynomials to be,

$P_{j}(\\{t_{i}\\})=\left({\over}T_{j}(\\{t_{i}\\})\right)^{\delta_{j}},\,~{}~{}~{}~{}~{}\text{for}~{}~{}~{}~{}~{}j=1,\cdots,{n_{\rm
ext}(n_{\rm ext}-1)\over 2}\,.$ (59)

In this form, we can define local residues associated to the common roots
$\tilde{t}_{i}$ of a subset of polynomials $\\{P_{q_{1}},\cdots P_{q_{\cal
I}}\\}\in\\{P_{1},\cdots P_{m}\\}$ in the following way Griffiths ;
1994alg.geom:4011C . Take,

$H(\\{\tilde{t}_{i}\\})_{q_{1},\cdots,q_{\cal
I}}={G(\\{\tilde{t}_{i}\\})\over\prod_{j\neq{q_{1},\cdots,q_{\cal
I}}}P_{j}(\\{\tilde{t}_{i}\\})}$ (60)

then if the Jacobian $J(\\{\tilde{t}_{i}\\})$,

$J(\\{\tilde{t}_{i}\\})={\rm Det}\left({\partial(P_{q_{1}},\cdots P_{q_{\cal
I}})\over\partial(\\{t_{i}\\})}\right)(\\{\tilde{t}_{i}\\})$ (61)

is not degenerate, i.e, if $J(\\{\tilde{t}_{i}\\})\neq 0$, then a local
residue can be defined as,

${\rm Res}_{\\{P_{q_{1}},\cdots P_{q_{\cal
I}}\\}}\left({G(\\{t_{i}\\})\over\prod_{j=1}^{m}P_{j}(\\{t_{i}\\})}\right)|_{t_{i}=\tilde{t}_{i}}={H(\\{\tilde{t}_{i}\\})_{q_{1},\cdots,q_{\cal
I}}\over J(\\{\tilde{t}_{i}\\})}$ (62)

otherwise, a local residue can be defined for the modified integrand,

${\rm Res}_{\\{P_{q_{1}},\cdots P_{q_{\cal
I}}\\}}\left({G(\\{t_{i}\\})J(\\{\tilde{t}_{i}\\})\over\prod_{j=1}^{m}P_{j}(\\{t_{i}\\})}\right)|_{t_{i}=\tilde{t}_{i}}=\mu_{{\\{P_{q_{1}},\cdots
P_{q_{\cal
I}}\\}}}(\\{\tilde{t}_{i}\\})\,H(\\{\tilde{t}_{i}\\})_{q_{1},\cdots,q_{\cal
I}}\,$ (63)

where $\mu_{{\\{P_{q_{1}},\cdots P_{q_{\cal I}}\\}}}(\\{\tilde{t}_{i}\\})$
denotes the intersection multiplicity of $\\{P_{q_{1}},\cdots P_{q_{\cal
I}}\\}$ at $\\{\tilde{t}_{i}\\}$.

By Bézout’s theorem, the maximum number of common zeros $n_{0}$ for a system
of polynomials $\\{P_{q_{1}},\cdots P_{q_{\cal I}}\\}$ is given by the product
of their degrees $n_{0}=\prod_{j=q_{1}}^{q_{\cal I}}{\rm deg}(P_{j})$, where
${\rm deg}(P_{j})$ denotes the degree of the given polynomial $P_{j}$. In
order to find them, we can use the method recently developed in
Cardona:2015ouc ; Dolan:2015iln ; Cardona:2015eba  to find solutions to the
scattering equations. This method is based in both, the theory of resultants
(see for example Dickenstein ; Gelfand ) and the theory of elimination (see
for example Sturmfels:2002 ).

In theory, the process described in this section is adequate to solve the
integrals of the type (49), but in practice, the application of this procedure
to the actual integrals we need, namely for example (LABEL:trianglemellin) and
(47), can still be quite challenging. For example, the exponents $\delta_{ij}$
correspond to the Mellin variables, or in other words, to the kinematics
variables of the Mellin amplitude, and therefore take arbitrary values, so in
principle we must be able to apply it to arbitrary values of the exponents.
However, for the particular integrals we need to consider, some
particularities might simplify or facilitate the procedure, as we shown in the
appendix C, when applied to the triangle four points loop the integral
correspond to a simpler, but still involved problem.

##  8 Conclusions and outlook

In this note we have considered one loop Witten diagrams for scattering of
scalars in AdS. We start by considering a contour representation for the bulk-
to-bulk propagator that allow us to consider bulk and boundary coordinates in
almost the same footing from the point of view of the integrated loop. By
using Schwinger parametrization to exponentiate the propagators, the
integration over AdS “radial” coordinate becomes Gaussian and can be
straightforwardly integrated. With the help of a Mellin-Barnes transformation
the remaining dependence on AdS coordinates can be also treated very easily.
We then Mellin transform the integral over the Schwinger parameters associated
to the bulk-to-boundary propagators by means of the Symanzik formula, to
extract the Mellin representation of the Witten diagram under consideration.
However, a leftover integration over the Schwinger parameters associated to
the bulk-to-bulk propagators remains, and we need to look for further
techniques to integrate them. We start a modest study of this leftover
integrals by reinterpreting them in terms of integrals over contours defined
by the zero locus of a system of polynomial equations, which boils down to the
computation of residues on a multidimensional space.

Along the way, we realized that the reinterpretation of those Schwinger
integrals in terms of contour resembles modern formulas for the computation of
scattering amplitudes in flat space in terms of contours defined by the
intersection of algebraic curves in projective space, such as the
Amplituheadron program ArkaniHamed:2012nw  or the most recent CHY formulation
for arbitrary dimensions Cachazo:2013hca . It would be very interesting to
explore those similarities further and maybe apply this well understood
techniques for scattering amplitudes in flat space to the scattering of
particles in AdS spaces.

###### Acknowledgements.

It is my pleasure to thank to Thiago Araujo for collaboration during very
early stages of this work, as well as to Yu-tin Huang, Simon Caron-Huot, Ellis
Ye Yuan and Yoji Koyama for useful and enlightening discussions. The work is
supported in part by the Danish National Research Foundation (DNRF91) and the
National Center for Theoretical Science (NCTS), Taiwan, Republic of China.

##  Appendix A Exchanged scalar by the split representation

In this section we want to compute again the exchange tree-level scalar Witten
diagram as an example to illustrate how the split representation works as well
as to point out why this method becomes harder than the one used in this paper
when it comes to the computation of loops. The split representation have been
introduced in Costa:2014kfa  and states essentially the fact that the bulk-to-
bulk propagator can be rewritten in terms of bulk-to-boundary propagators as
an integration over bulk-to-boundary propagators,

$G^{\Delta}_{BB}(X,Y)=\int_{-i\infty}^{+i\infty}\frac{\mathrm{d}c}{2\pi
i}\frac{\Gamma(h+c)\Gamma(h-c)f^{\Delta}(c)}{{\cal C}_{h+c}{\cal
C}_{h-c}}\int_{\partial AdS}\mathrm{d}KG^{(h+c)}_{\partial
B}(X,K)G^{(h-c)}_{\partial B}(K,Y)$ (64)

The function $f(c)$ is given by,

$f^{\Delta}(c)=\frac{1}{2\pi^{2h}\Gamma(c)\Gamma(-c)}\frac{1}{[(\Delta-h)^{2}-c^{2}]}\;.$
(65)

Plugging in the above definitions back to (5), we get

$\begin{split}&{\cal
A}_{4}(P_{1},P_{2},P_{3},P_{4})=g^{2}\int_{-i\infty}^{+i\infty}\frac{\mathrm{d}c}{2\pi
i}\frac{\Gamma(h+c)\Gamma(h-c)f(c)}{{\cal C}_{h+c}{\cal
C}_{h-c}}\int_{\partial AdS}\mathrm{d}K\times\\\
&\times\int_{AdS}\mathrm{d}XG_{B\partial}(P_{1},X)G_{B\partial}(P_{2},X)G_{B\partial}^{(h+c)}(X,K)\int_{AdS}\mathrm{d}YG^{(h-c)}_{\partial
B}(K,Y)G_{B\partial}(Y,P_{3})G_{\partial B}(Y,P_{4})\,.\end{split}$ (66)

The 4pt-amplitude is decomposed into a product of two 3pt-Witten diagrams as
in Figures (6(a)) and (6(b)) integrated over the new inserted boundary point
$K$,

${\cal A}_{4}(P_{i})=\int_{-i\infty}^{+i\infty}\frac{\mathrm{d}c}{2\pi
i}\frac{\Gamma(h+c)\Gamma(h-c)f(c)}{{\cal C}_{h+c}{\cal
C}_{h-c}}\int_{\partial AdS}\mathrm{d}K{\cal
A}^{(h+c)}_{3}(P_{1},P_{2},K){\cal A}^{(h-c)}_{3}(K,P_{3},P_{4})\;.$ (67)

At this point is already easy to see that the splitting have trade a tree-
level diagram by loop-like diagram since we have to integrate now over space-
time coordinates.

(a) Scalar Exchange.

(b) Product of two 3pt functions.

Figure 6: Decomposition of the scalar exchange diagram.

The contact three-point function of Figure (6(b))

${\cal
A}_{3}(P_{i},P_{j},K)=g\int_{AdS}\mathrm{d}XG_{B\partial}(X,P_{i})G_{B\partial}(X,P_{j})G_{B\partial}(X,K)\;,$
(68)

is easy to computed and is given by Freedman:1998tz

${\cal A}_{3}(P_{i},P_{j},P_{k})=g\frac{\pi^{h}{\cal C}_{\Delta_{i}}{\cal
C}_{\Delta_{j}}{\cal
C}_{\Delta_{k}}}{\Gamma(\Delta_{i})\Gamma(\Delta_{j})\Gamma(\Delta_{k})}\frac{\Gamma(\Delta_{ijk}-h)\Gamma(\Delta_{ij})\Gamma(\Delta_{ik})\Gamma(\Delta_{jk})}{P_{ij}^{\Delta_{ij}}P_{ik}^{\Delta_{ik}}P_{jk}^{\Delta_{jk}}}$
(69)

where we adopt the notation

$\Delta_{ij}=\frac{\Delta_{i}+\Delta_{j}-\Delta_{k}}{2}\;,\qquad\Delta_{ijk}=\frac{\Delta_{i}+\Delta_{j}+\Delta_{k}}{2}\;.\\\
$ (70)

where $\Delta_{c}=h+c$ and $\Delta_{\bar{c}}=h-c$. Inserting these expressions
into (67), we find

$\begin{split}{\cal
A}_{4}(P_{i})&=\frac{g^{2}\pi^{2h}}{4}\int_{-i\infty}^{+i\infty}\frac{\mathrm{d}c}{2\pi
i}f(c)\prod_{i=1}^{4}\frac{{\cal
C}_{\Delta_{i}}}{\Gamma(\Delta_{i})}\frac{1}{P_{12}^{\Delta_{12}}P_{34}^{\Delta_{34}}}\times\\\
&\times\Gamma(\Delta_{12c}-h)\Gamma(\Delta_{34\bar{c}}-h)\Gamma(\Delta_{12})\Gamma(\Delta_{1c})\Gamma(\Delta_{2c})\Gamma(\Delta_{34})\Gamma(\Delta_{3\bar{c}})\Gamma(\Delta_{4\bar{c}})\\\
&\times\int_{\partial AdS}\mathrm{d}K\frac{1}{(-2P_{1}\cdot
K)^{\Delta_{1c}}}\frac{1}{(-2P_{2}\cdot
K)^{\Delta_{2c}}}\frac{1}{(-2P_{3}\cdot
K)^{\Delta_{3\bar{c}}}}\frac{1}{(-2P_{4}\cdot
K)^{\Delta_{4\bar{c}}}}\;.\end{split}$ (71)

computing explicitly the boundary integral we arrive to

$\begin{split}{\cal A}_{4}&(P_{i})=\frac{g^{2}\pi^{3h}}{8(2\pi
i)^{2}}\prod_{i=1}^{4}\frac{{\cal
C}_{\Delta_{i}}}{\Gamma(\Delta_{i})}\int_{-i\infty}^{+i\infty}\frac{\mathrm{d}c}{2\pi
i}f(c)\times\\\
&\times\Gamma\left(\frac{\Delta_{1}+\Delta_{2}+c-h}{2}\right)\Gamma\left(\frac{\Delta_{3}+\Delta_{4}+c-h}{2}\right)\Gamma\left(\frac{\Delta_{1}+\Delta_{2}-c-h}{2}\right)\Gamma\left(\frac{\Delta_{3}+\Delta_{4}-c-h}{2}\right)\times\\\
&\times\int_{\Sigma^{4}}\mathrm{d}\delta_{ij}\frac{\Gamma(\delta_{12}-\Delta_{12})\Gamma(\delta_{34}-\Delta_{34})}{\Gamma(\delta_{12})\Gamma(\delta_{34})}\prod_{i<j=1}^{4}\Gamma(\delta_{ij})(P_{ij})^{-\delta_{ij}}\;.\end{split}$
(72)

where,

$\begin{split}&\delta_{12}=\frac{\Delta_{1}+\Delta_{2}-s_{12}}{2}\;,\qquad\ \
\ \ \delta_{34}=\frac{\Delta_{3}+\Delta_{4}-s_{12}}{2}\\\
&\delta_{12}-\Delta_{12}=\frac{h+c-s_{12}}{2}\;,\qquad\delta_{34}-\Delta_{34}=\frac{h+c-s_{12}}{2}\;.\end{split}$
(73)

Therefore,

$\begin{split}{\cal A}_{4}&(P_{i})=\frac{g^{2}\pi^{3h}}{8(2\pi
i)^{2}}\prod_{i=1}^{4}\frac{{\cal
C}_{\Delta_{i}}}{\Gamma(\Delta_{i})}\int_{\Sigma^{4}}\mathrm{d}\delta_{ij}\prod_{i<j=1}^{4}\Gamma(\delta_{ij})(P_{ij})^{-\delta_{ij}}\times\\\
&\times\left[\Gamma\left(\frac{\Delta_{1}+\Delta_{2}-s_{12}}{2}\right)\Gamma\left(\frac{\Delta_{3}+\Delta_{4}-s_{12}}{2}\right)\right]^{-1}\times\\\
&\times\int_{-i\infty}^{+i\infty}\frac{\mathrm{d}c}{2\pi
i}f(c)\Gamma\left(\frac{h+c-s_{12}}{2}\right)\Gamma\left(\frac{\Delta_{1}+\Delta_{2}+c-h}{2}\right)\Gamma\left(\frac{\Delta_{3}+\Delta_{4}+c-h}{2}\right)\times\\\
&\times\Gamma\left(\frac{h-c-
s_{12}}{2}\right)\Gamma\left(\frac{\Delta_{1}+\Delta_{2}-c-h}{2}\right)\Gamma\left(\frac{\Delta_{3}+\Delta_{4}-c-h}{2}\right)\end{split}$
(74)

Moreover, using that

$\prod_{i=1}^{4}\frac{{\cal C}_{\Delta_{i}}}{\Gamma(\Delta_{i})}=\frac{2{\cal
N}}{\pi^{h}\Gamma\left(\frac{1}{2}\sum_{i=1}^{4}\Delta_{i}-h\right)}\;,$ (75)

and equation (65), we find

$\begin{split}{\cal A}_{4}&(P_{i})=\frac{g^{2}{\cal N}}{2(2\pi
i)^{2}}\int_{\Sigma^{4}}\mathrm{d}\delta_{ij}\prod_{i<j=1}^{4}\Gamma(\delta_{ij})(P_{ij})^{-\delta_{ij}}\times\\\
&\times\left[\Gamma\left(\frac{1}{2}\sum_{i=1}^{4}\Delta_{i}-h\right)\Gamma\left(\frac{\Delta_{1}+\Delta_{2}-s_{12}}{2}\right)\Gamma\left(\frac{\Delta_{3}+\Delta_{4}-s_{12}}{2}\right)\right]^{-1}\times\\\
&\times\int_{-i\infty}^{+i\infty}\frac{\mathrm{d}c}{2\pi
i}\frac{1}{(\Delta-h)^{2}-c^{2}}\times\\\
&\times\frac{\Gamma\left(\frac{h+c-s_{12}}{2}\right)\Gamma\left(\frac{\Delta_{1}+\Delta_{2}+c-h}{2}\right)\Gamma\left(\frac{\Delta_{3}+\Delta_{4}+c-h}{2}\right)}{2\Gamma(c)}\frac{\Gamma\left(\frac{h-c-
s_{12}}{2}\right)\Gamma\left(\frac{\Delta_{1}+\Delta_{2}-c-h}{2}\right)\Gamma\left(\frac{\Delta_{3}+\Delta_{4}-c-h}{2}\right)}{2\Gamma(-c)}\;.\end{split}$
(76)

Comparing this result with the following Mellin representation

${\cal A}_{4}(P_{i})=\frac{{\cal N}}{(2\pi
i)^{2}}\int_{\Sigma^{4}}\mathrm{d}\delta_{ij}M(\delta_{ij})\prod_{i<j=1}^{4}\Gamma(\delta_{ij})(P_{ij})^{-\delta_{ij}}\;,$
(77)

we can see that the Mellin amplitude for the scalar exchange diagram is given
by

$M(\delta_{ij})=\frac{g^{2}}{\Gamma\left(\frac{1}{2}\sum_{i=1}^{4}\Delta_{i}-h\right)\Gamma\left(\frac{\Delta_{1}+\Delta_{2}-s_{12}}{2}\right)\Gamma\left(\frac{\Delta_{3}+\Delta_{4}-s_{12}}{2}\right)}\int_{-i\infty}^{+i\infty}\frac{\mathrm{d}c}{2\pi
i}\frac{\ell(c)\ell(-c)}{(\Delta-h)^{2}-c^{2}}$ (78)

and

$\ell(c)\equiv\frac{\Gamma\left(\frac{h+c-s_{12}}{2}\right)\Gamma\left(\frac{\Delta_{1}+\Delta_{2}+c-h}{2}\right)\Gamma\left(\frac{\Delta_{3}+\Delta_{4}+c-h}{2}\right)}{2\Gamma(c)}\;.$
(79)

which coincides with the previous result by Penedones Penedones:2010ue .

##  Appendix B $I^{\rhd}_{S}$ integral in terms of Hypergeometric functions.

In this appendix we display explicitly the integral (36) in terms of
Hypergeometric functions,

$\begin{split}&I^{\rhd}_{S}=\frac{\pi^{2}(-1)^{\gamma_{23}+f_{23}-2l}\Gamma\left(\frac{1}{2}\left(-2h+\Delta_{1}+\Delta_{2}+\Delta_{3}-1\right)\right)}{4\Gamma\left(\Delta_{1}\right)\Gamma\left(-f_{23}-\gamma_{23}\right)\Gamma\left(\frac{1}{2}\left(-2h+f_{12}+f_{23}+\gamma_{12}+\gamma_{23}+\Delta_{2}\right)\right)}\\\
&\frac{\csc\left(\pi\left(\gamma_{23}-\Delta_{2}+f_{23}+2l\right)\right)\Gamma\left(\frac{1}{2}\left(2l+f_{13}+f_{23}+\gamma_{13}+\gamma_{23}-\Delta_{1}-\Delta_{2}+1\right)\right)}{\Gamma\left(\frac{1}{2}\left(-2h+2l+f_{23}+f_{31}+\gamma_{23}+\gamma_{31}+\Delta_{3}\right)\right)}\\\
&\left(\frac{2e^{i\pi(3h+l)}\Gamma\left(-h+l+\frac{1}{2}\right)\left(e^{2i\pi\left(\gamma_{23}+f_{23}+2l\right)}-e^{2i\pi\Delta_{2}}\right)}{\left(e^{2i\pi\Delta_{2}}+e^{2i\pi(h+l)}\right)\left(e^{2i\pi\left(\gamma_{23}+f_{23}+l\right)}+e^{2i\pi
h}\right)}\Gamma\left(-h-l+\Delta_{1}+\Delta_{2}-\frac{1}{2}\right)\right.\\\
&\left.\Gamma\left(\frac{1}{2}\left(-2h+f_{12}+f_{23}+\gamma_{12}+\gamma_{23}+\Delta_{2}\right)\right)\,_{3}\tilde{F}_{2}\left(a_{1},b_{1},c_{1};d_{1},e_{1};1\right)\right.\\\
&\left.-\Gamma\left(-f_{23}-\gamma_{23}\right)e^{i\pi\left(\gamma_{23}+f_{23}+2l\right)}\sec\left(\pi\left(-\gamma_{23}-f_{23}+h-l\right)\right)\right.\\\
&\left.\Gamma\left(\frac{1}{2}\left(-2l+f_{12}-f_{23}+\gamma_{12}-\gamma_{23}+\Delta_{2}-1\right)\right)\right.\\\
&\left.\Gamma\left(-2l-f_{23}-\gamma_{23}+\Delta_{1}+\Delta_{2}-1\right)\,_{3}\tilde{F}_{2}\left(a_{2},b_{2},c_{2};d_{2},e_{2};1\right)\right.\\\
&\left.+e^{i\pi\Delta_{2}}\Gamma\left(\Delta_{1}\right)\Gamma\left(2l-\Delta_{2}+1\right)\sec\left(\pi\left(-\Delta_{2}+h+l\right)\right)\right.\\\
&\left.\Gamma\left(\frac{1}{2}\left(2l+f_{12}+f_{23}+\gamma_{12}+\gamma_{23}-\Delta_{2}+1\right)\right)\,_{3}\tilde{F}_{2}\left(a_{3},b_{3},c_{3};d_{3},e_{3};1\right)\right)\end{split}$
(80)

where the tilde on the Hypergeometric functions means they are actually
Hypergeometric regularized and their arguments are given by,

$\begin{split}a_{1}&=-h+l+\frac{1}{2},\,~{}b_{1}=\frac{1}{2}\left(-2h+f_{12}+f_{23}+\gamma_{12}+\gamma_{23}+\Delta_{2}\right),\\\
c_{1}&=-h-l+\Delta_{1}+\Delta_{2}-\frac{1}{2},\,~{}d_{1}=-h+l+f_{23}+\gamma_{23}+\frac{3}{2},\,~{}e_{1}=-h-l+\Delta_{2}+\frac{1}{2}\end{split}$
(81)
$\begin{split}a_{2}&=-f_{23}-\gamma_{23},\,~{}b_{2}=\frac{1}{2}\left(-2l+f_{12}-f_{23}+\gamma_{12}-\gamma_{23}+\Delta_{2}-1\right),\\\
c_{2}&=-2l-f_{23}-\gamma_{23}+\Delta_{1}+\Delta_{2}-1,\,~{}d_{2}=h-l-
f_{23}-\gamma_{23}+\frac{1}{2},\\\
e_{2}&=-2l-f_{23}-\gamma_{23}+\Delta_{2}\end{split}$ (82)
$\begin{split}a_{3}&=\Delta_{1},\,~{}b_{3}=2l-\Delta_{2}+1,\,~{}c_{3}=\frac{1}{2}\left(2l+f_{12}+f_{23}+\gamma_{12}+\gamma_{23}-\Delta_{2}+1\right),\\\
d_{3}&=h+l-\Delta_{2}+\frac{3}{2},\,~{}e_{3}=2l+f_{23}+\gamma_{23}-\Delta_{2}+2\,.\end{split}$
(83)

##  Appendix C Few observations for the one loop triangle four-point
scattering

Now we would like to highlight some particularities of the triangle four
points scattering.

Let us consider the integral at the last line of equation
(LABEL:trianglemellin) which we copy here for quick reference,

$\begin{split}&\int^{\infty}_{0}\prod_{i=5}^{7}{\mathrm{d}t_{i}\over
t_{i}}{1\over\Gamma(\Delta_{i})}\,t_{5}^{g+\gamma_{13}+f_{13}-\Delta_{2}+\Delta_{3}+\Delta_{4}}\,t_{6}^{\gamma_{23}+f_{23}}\,t_{7}^{-\gamma_{23}-f_{23}-\Delta_{3}-\Delta_{4}-g+2h}(1+(t_{5}t_{7}+t_{6})^{2})^{g}\\\
&~{}~{}~{}~{}~{}~{}\left({\over}T_{3}(\\{t_{i}\\})\right)^{-\delta_{12}}\left({\over}T_{2}(\\{t_{i}\\})\right)^{-\delta_{13}-\delta_{14}}\left({\over}T_{1}(\\{t_{i}\\})\right)^{-\delta_{23}-\delta_{24}}\,,\end{split}$
(84)

where as before,

$\begin{split}T_{1}(\\{t_{i}\\})&=t_{5}t_{7}+t_{6},\,~{}~{}~{}T_{2}(\\{t_{i}\\})=t_{7}+t_{5}(t_{5}t_{7}+t_{6}),\\\
T_{3}(\\{t_{i}\\})&=t_{5}+(t_{7}+t_{5}(t_{5}t_{7}+t_{6}))(t_{5}t_{7}+t_{6})\end{split}$
(85)

Since in this case the number of polynomials $T_{j}$ equals the number of
integration variables, we pick them to compute the Jacobian which quite
interestingly is not only non-degenerate but even more, equals to one!. That
allow us to perform a change of variables $x_{i}=T_{i}$, to rewrite the
integral as,

$\begin{split}&\int^{\infty}_{0}\prod_{i=5}^{7}{\mathrm{d}x_{i}\over\Gamma(\Delta_{i})}(x_{5}^{2}+1)^{g-1}(x_{5}^{3}x_{6}^{2}-2x_{5}^{2}x_{6}x_{7}+x_{5}(x_{6}^{2}+x_{7}^{2}+1)-x_{6}x_{7})^{\gamma_{23}+f_{23}-1}\\\
&~{}~{}~{}~{}~{}~{}~{}~{}~{}(x_{7}-x_{5}x_{6})^{\gamma_{13}-\Delta_{2}+\Delta_{3}+\Delta_{4}+f_{13}+g-1}\\\
&~{}~{}~{}~{}~{}~{}{1\over
x_{7}^{\delta_{12}}\,x_{6}^{\delta_{13}+\delta_{14}}x_{5}^{\delta_{23}+\delta_{24}}\,(x_{5}(x_{5}x_{6}-x_{7})+x_{6})^{\gamma_{23}+\Delta_{3}+\Delta_{4}+f_{23}+g-2h+1}}\,,\end{split}$
(86)

from here is clear that the only poles comes from the simpler equations
$x_{i}=0,\,~{}i=5,6,7.$ Deforming the contour such as it enclosed the poles
around $x_{i}=0,\,~{}i=5,6,7,$ we have,

$\int_{{\cal
C}}\,\prod_{i=5}^{7}{\mathrm{d}x_{i}\over\Gamma(\Delta_{i})}{G(x_{5},x_{6},x_{7})\over
x_{7}^{\delta_{12}}\,x_{6}^{\delta_{13}+\delta_{14}}x_{5}^{\delta_{23}+\delta_{24}}\,(x_{5}(x_{5}x_{6}-x_{7})+x_{6})^{\gamma_{23}+\Delta_{3}+\Delta_{4}+f_{23}+g-2h+1}}\,,$
(87)

where we have defined

$\begin{split}G(x_{5},x_{6},x_{7})&=(x_{5}^{2}+1)^{g-1}(x_{5}^{2}x_{6}(x_{5}x_{6}-2x_{7})+x_{5}(x_{6}^{2}+x_{7}^{2}+1)-x_{6}x_{7})^{\gamma_{23}+f_{23}-1}\\\
&~{}~{}~{}~{}(x_{7}-x_{5}x_{6})^{\gamma_{13}-\Delta_{2}+\Delta_{3}+\Delta_{4}+f_{13}+g-1}\,.\end{split}$
(88)

Even thought the integration (87) is simpler that (LABEL:triangle3), on the
surfaces satisfying

$J(0)={\rm
Det}\left({\partial(P_{q_{1}},\,P_{q_{2}},P_{q_{3}})\over\partial(\\{x_{i}\\})}\right)(0)=0\,,$
(89)

where

$(P_{q_{1}},\,P_{q_{2}},P_{q_{3}})\in\\{x_{7}^{\delta_{12}},\,x_{6}^{\delta_{13}+\delta_{14}},\,x_{5}^{\delta_{23}+\delta_{24}},\,(x_{5}(x_{5}x_{6}-x_{7})+x_{6})^{\gamma_{23}+\Delta_{3}+\Delta_{4}+f_{23}+g-2h+1}\\}\,,$
(90)

i.e, on the surfaces where the Jacobians degenerates, we can not write a
simple formula as (62). However, we can still used iteratively the Cauchy
integral formula on each $x_{i}$, which for a given value of $\delta_{ij}$
will give us a sum of rational functions of gamma functions.

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