Two-loop integrals of half-BPS six-point functions on a line

Ricardo Rodrigues<EMAIL_ADDRESS>
Centro de F$\acute{\imath}$sica do Porto, Departamento de
F$\acute{\imath}$sica e Astronomia, Faculdade de Ci$\hat{e}$ncias da
Universidade do Porto, Rua do Campo Alegre 687, 4169-007 Porto, Portugal

Abstract

We evaluated all two-loop conformal integrals of scalar half-BPS six-point
functions in $\mathcal{N}=4$ SYM restricted to a configuration where all
points lie on a line. Moreover, we also computed some of these integrals in
the kinematical limit where adjacent points become null-separated. Our results
can serve as cross-checks for future works which obtain these integrals for
general kinematics or by different methods such as integrability.

###### Contents

  1. 1 Introduction
  2. 2 Correlation functions
  3. 3 Integrals
    1. 3.1 Methodology
    2. 3.2 Two-loop half-BPS six-point integrals
    3. 3.3 Line geometry
    4. 3.4 Light-cone limit and asymptotic expansions
  4. 4 Results and analysis
    1. 4.1 Line configuration
    2. 4.2 Light-cone limit
  5. 5 Discussion
  6. A Graph Polynomials $\Psi$ and $\Phi$
  7. B Asymptotic expanded integrals

##  1 Introduction

The applicability of conformal field theories ranges from the study of
critical phenomena to that of quantum gravity. As such, these theories have
been intensively studied over the past few decades. Among these, the
$\mathcal{N}=4$ supersymmetric Yang-Mills theory plays a special role not only
because it is integrable in the planar limit but also because it is one of the
few known examples of an integrable theory in four dimensions. As is the case
with most CFT, correlation functions of local operators are one of the most
interesting observables to study. For this theory in particular it has been
shown to exist certain dualities between some of these correlation functions,
scattering amplitudes and Wilson loops [1, 2, 3, 4].

The $\mathcal{N}=4$ SYM theory has also appeared in the first example of the
AdS/CFT duality, having been shown to be dual to a type IIB string theory in a
$\text{AdS}_{5}\times\text{S}_{5}$ geometry[5]. Since then, this theory has
proved to be of particular usefulness in having a better understanding of
strong coupling phenomena, given that AdS/CFT is a strong/weak coupling
duality. Within its spectrum a special kind of operators called half-BPS
operators are dual to Kaluza-Klein spherical harmonics of the graviton on the
sphere. As such, the analysis of correlators of such operators can offer us
important information about the internal sector of the dual bulk theory and
how its information is encoded in the CFT side of the duality.

The above are some of the reasons why we are concerned with studying
correlation functions in $\mathcal{N}=4$ SYM, in particular of half-BPS
operators. These, however, are not always easy to obtain. In fact, although
some four-point functions of this theory have been studied both for weak [6,
7, 8, 9] and strong [10, 11, 12, 13, 14, 15] coupling, where the computations
have been taken to a relatively high number of loops, the situation for
higher-point functions is not as promising. Five and six-point functions,
however, are somewhat of an exception to this as they have been analyzed
relatively recently. In particular, the integrands of five-point functions are
known up to three loops [16], while the integrals have been explicitly
obtained up to two loops when null limits are taken or when we restrict the
points to a plane [17]. There have also been some bootstrap approaches to
certain five-point functions of this theory at strong coupling in [18, 19]. On
the other hand, the integrands of six-point functions have been obtained up to
two loops [1, 3, 16] but we still have not been able to obtain the integrated
contributions to the corrections.

Here our main concern will be the two loop conformal integrals associated to
six-point functions of scalar single-trace half-BPS operators. In particular,
since the complexity of these integrals for general kinematics is high enough
that we cannot evaluate them explicitly, we will restrict the configuration to
a line geometry which simplifies things. Additionally, we will also evaluate
some of these integrals when null limits are taken.

The structure of this paper is as follows. In section 2 we introduce the
single-trace half-BPS operators, giving special focus to twenty-prime
operators, their correlation functions and respective properties. Section 3
describes the procedure to compute the integrals on the line geometry and
presents the different types of two loop six-point conformal integrals.
Additionally, it tackles some important points about the line geometry and the
null limit cases. In section 4 we comment the results regarding the line
configuration and then move on to present the explicit outcome of evaluating
one of the integrals in the light-cone limit. In section 5 we make a general
assessment of our work and elaborate on possible future directions. Appendix A
contains explicit expressions of quantities concerning the integrand of a
conformal integral considered in section 3.3. In appendix B we briefly explain
how to generalize a result for asymptotic expanded integrals which was
necessary to evaluate the integral considered in section 4.

##  2 Correlation functions

Here, we consider the four dimensional $\mathcal{N}=4$ SYM theory, with gauge
group $SU(N_{c})$, where $N_{c}$ is the number of colors, which among other
fields contains six bosonic scalars denoted by
$\Phi^{I}=(\phi^{1},\dots,\phi^{6})$. Moreover, this theory has an associated
coupling constant $g_{\text{YM}}$ which is related with $N_{c}$ by the so-
called ’t Hooft coupling

$\lambda=g_{\text{YM}}^{2}\,N_{c}\,\,.$ (1)

Within the spectrum of this theory there is a class of important operators
called the scalar single-trace half-BPS operators, which transform in the
symmetric and traceless representation of the $SO(6)_{R}\simeq SU(4)_{R}$
global symmetry group. They are generally written as

$\mathcal{O}_{k}(x,Y)=Y^{i_{1}}\dots
Y^{i_{k}}\,\text{tr}\left(\phi^{i_{1}}(x)\dots\phi^{i_{k}}(x)\right)\,\,,$ (2)

where $k\geq 2$ is an integer, $Y^{i}$ are six-dimensional null polarization
vectors ($Y^{i}\cdot Y^{i}=0$) contracting with the $SO(6)_{R}$ indices, and
the trace is taken over the indices of $SU(N_{c})$, which are omitted for
simplicity. Such operators $\mathcal{O}_{k}$ have conformal dimension
$\Delta_{\mathcal{O}_{k}}=k$.

An interesting property of these operators is that their two- and three-point
functions are protected from quantum corrections by supersymmetry [20], which
means they remain unchanged independently of the theory’s coupling. On the
other hand, while their higher-point functions no longer enjoy this property
they are of extreme importance to us as they encode useful information about
the theory itself and its holographic dual.

In general, any planar $n$-point function of scalar single-trace half-BPS
operators is given by

$\langle\mathcal{O}_{k_{1}}\,\mathcal{O}_{k_{2}}\dots\mathcal{O}_{k_{n}}\rangle=\sum_{\ell=0}^{\infty}\lambda^{2\ell}\,G_{n}^{(\ell)}\,,$
(3)

while the non-planar contributions are subleading in the large $N_{c}$ limit.
In the above expression $G_{n}^{(0)}$ concerns the free correlator, whereas
the terms with $\ell\geq 1$ account for loop corrections, which are given by
specific spacetime integrals. In practice, the integrands of these corrections
can be computed through the Lagrangian insertion method [20, 3, 1, 21, 7], in
which they are given by $(n+\ell)$-point functions with $\ell$ insertions of
the Lagrangian density operator. However, as we start considering more points
and/or higher loops computing the integrands becomes harder as it requires the
evaluation of higher-point correlators. Additionally, once we have the
integrands we still have to compute the integrals which can be quite
challenging.

Inside this special class of operators one of the most important and studied
ones is the lowest length half-BPS operator ($k=2$), which is usually denoted
by $20^{\prime}$ operator

$O_{2}(x,Y)\equiv
O_{20^{\prime}}(x,Y)=Y^{i_{1}}\,Y^{i_{2}}\,\text{tr}\left(\phi^{i_{1}}(x)\,\phi^{i_{2}}(x)\right)\,\,.$
(4)

It belongs to the stress-tensor supermultiplet along with the stress-tensor
itself, the Lagrangian density operator and others. Given the above
definition, we expect the correlation functions of these operators to be given
in terms of polynomials of polarization vectors that have weight two in each
$Y_{i}$.

The two- and three-point functions of the twenty-prime operators are
completely fixed by conformal symmetry and supersymmetry to be given by

$\displaystyle\langle\mathcal{O}_{20^{\prime}}(x_{1},Y_{1})\,\mathcal{O}_{20^{\prime}}(x_{2},Y_{2})\rangle=\left(\frac{Y_{12}}{x_{12}^{2}}\right)^{2}\,,$
(5)
$\displaystyle\langle\mathcal{O}_{20^{\prime}}(x_{1},Y_{1})\,\mathcal{O}_{20^{\prime}}(x_{2},Y_{2})\,\mathcal{O}_{20^{\prime}}(x_{3},Y_{3})\rangle=\frac{C_{20^{\prime}20^{\prime}20^{\prime}}\,Y_{12}Y_{13}Y_{23}}{(x_{12}^{2}x_{13}^{2}x_{23}^{2})}\,\,,$
(6)

where we have used the notation $x_{ij}^{2}\equiv(x_{i}-x_{j})^{2}$,
$Y_{ij}\equiv Y_{i}\cdot Y_{j}$, and where
$C_{20^{\prime}20^{\prime}20^{\prime}}$ is the structure constant of its
three-point function, which is independent of the theory’s coupling. On the
other hand, the higher-point functions, which are not protected by
supersymmetry, are completely non-trivial.

Let us then start by considering the four-point function, which is the lowest
of the higher-point correlators. From a simple analysis it is possible to
infer that the polynomial in the polarization vectors can contain up to two
different structures. In particular, all the structures we can find are

12434312 Figure 1: Diagrams representing the $Y_{ij}$ structures of the
twenty-prime four-point function.

$Y_{12}^{2}\,Y_{34}^{2}\,\,,\hskip
14.22636ptY_{12}\,Y_{23}\,Y_{34}\,Y_{41}\,\,,$ (7)

which correspond to the diagrams in figure 1, plus the ones obtained from
permuting the positions. As such, we can write the four-point function as

$\langle\mathcal{O}_{20^{\prime}}(x_{1},Y_{1})\,\mathcal{O}_{20^{\prime}}(x_{2},Y_{2})\,\mathcal{O}_{20^{\prime}}(x_{3},Y_{3})\,\mathcal{O}_{20^{\prime}}(x_{4},Y_{4})\rangle=\left(\frac{Y_{12}\,Y_{34}}{x_{12}^{2}\,x_{34}^{2}}\right)^{2}\,\mathcal{G}_{4}(u,v;\sigma,\tau)\,,$
(8)

where the conformal and the R-symmetry111R-symmetry is a symmetry that is
present in supersymmetric theories and which is responsible by transforming
the different supercharges into each other. In this case, it will be
equivalent to the SO(6) symmetry associated with rotations among the six
scalar fields $\Phi^{I}$. cross-ratios are respectively defined as

$\displaystyle
u=\frac{x_{12}^{2}x_{34}^{2}}{x_{13}^{2}x_{24}^{2}}=z\bar{z}\,,\hskip
8.5359ptv=\frac{x_{14}^{2}x_{23}^{2}}{x_{13}^{2}x_{24}^{2}}=(1-z)(1-\bar{z})\,,$
(9)
$\displaystyle\sigma=\frac{Y_{12}Y_{34}}{Y_{13}Y_{24}}=\alpha\bar{\alpha}\,,\hskip
8.5359pt\tau=\frac{Y_{14}Y_{23}}{Y_{13}Y_{24}}=(1-\alpha)(1-\bar{\alpha})\,.$

More can be said about the cross-ratios function $\mathcal{G}$. Indeed,
superconformal Ward identities impose extra non-trivial conditions on it [22,
23, 13] which in due turn enforce it to have a particular structure, namely

$\mathcal{G}_{4}(z,\bar{z};\alpha,\bar{\alpha})=\mathcal{G}_{4}^{(0)}(z,\bar{z};\alpha,\bar{\alpha})+R\,\mathcal{F}_{4}(z,\bar{z})\,,$
(10)

where $\mathcal{G}_{4}^{(0)}(z,\bar{z};\alpha,\bar{\alpha})$ is the free part
of the correlator $G_{4}^{(0)}$ with the position-dependent prefactor
extracted, $R$ is given by

$R=\frac{(1-z\,\alpha)\,(1-\bar{z}\,\alpha)\,(1-z\,\bar{\alpha})\,(1-\bar{z}\,\bar{\alpha})}{(1-z)\,(1-\bar{z})\,\alpha^{2}\,\bar{\alpha}^{2}}\,,$
(11)

and $\mathcal{F}_{4}(z,\bar{z})$ contains all the loop-corrections mentioned
in (3) also without the prefactor. This latter function is important in the
sense that it is contains all the dynamical information.

When it comes to the four-point function of twenty-prime operators, this
comprises one of the most well studied four-point functions in all conformal
field theories. Indeed, the integrands of the corrections for this object are
known in the weak coupling perturbative regime up to ten loops [24, 25]. In
spite of this, the conformal integrals have only been evaluated up to three
loops for fixed cross-ratios [26] and up to five loops in the Euclidean OPE
limit [9]. As for the strong coupling regime, the first few orders of the
$1/N$ corrections have been obtained by means of conformal bootstrap methods
[27, 14, 28, 29].

The five-point function of these same operators is also interesting to look
at. Performing a similar analysis as for the four-point case we can see that
the allowed structures in the null polarization vectors are

$Y_{12}^{2}Y_{34}Y_{45}Y_{53}\,\,,\,\,\,\,Y_{12}Y_{23}Y_{34}Y_{45}Y_{51}\,,$
(12)

corresponding to the diagrams of figure 2,

12345

12345

Figure 2: Diagrams representing the $Y_{ij}$ structures of the twenty-prime
five-point function.

together with the ones obtained from indices permutations. Consequently, we
can write this correlation function as

$\langle\mathcal{O}_{20^{\prime}}(x_{1},Y_{1})\dots\mathcal{O}_{20^{\prime}}(x_{5},Y_{5})\rangle=\frac{x_{13}^{2}Y_{12}^{2}Y_{34}^{2}Y_{15}Y_{35}}{x_{12}^{4}x_{34}^{4}x_{15}^{2}x_{35}^{2}Y_{13}}\,\mathcal{G}_{5}(u_{i};\sigma_{i})\,,$
(13)

which depends on five conformal cross-ratios $u_{1},\dots,u_{5}$ defined as

$u_{1}=\frac{x_{12}^{2}x_{35}^{2}}{x_{13}^{2}x_{25}^{2}}\,\,,\hskip
14.22636ptu_{i+1}=\left.u_{i}\right|_{x_{j}\rightarrow x_{j+1}}\,,$ (14)

where the index $j$ is taken modulo $5$, and on the R-symmetry cross-ratios
$\sigma_{1},\dots,\sigma_{5}$ given as

$\sigma_{1}=\frac{Y_{12}Y_{35}}{Y_{13}Y_{25}}\,\,,\hskip
14.22636pt\sigma_{i+1}=\left.\sigma_{i}\right|_{Y_{j}\rightarrow Y_{j+1}}\,,$
(15)

with $j$ also taken modulo $5$. Similarly, the cross-ratios function
$\mathcal{G}_{5}$ also needs to satisfy superconformal Ward identities like in
the four-point case. However, as far as we know no explicit solution was found
for five points and we do not know if it should have a similar structure to
(10) or not.

Regarding the status of the five-point function corrections, the current
situation is as referred previously. In particular, while the integrands have
been obtained up to three loops [16], the integrals have only been computed up
to $2$ loops for special cases, such as when null limits are taken or when all
the points are set to the plane [17].

Next, we have the twenty-prime six-point function, which is of particular
interest to us since we will concern ourselves with the two-loop six-point
conformal integrals. For this case, the polynomial in $Y_{i}$ which describes
this function should contain up to four different kinds of structures

$Y_{12}^{2}\,Y_{34}^{2}\,Y_{56}^{2}\,\,,\hskip
14.22636ptY_{12}^{2}\,Y_{34}\,Y_{45}\,Y_{56}\,Y_{63}\,\,,\hskip
14.22636ptY_{12}\,Y_{23}\,Y_{31}\,Y_{45}\,Y_{56}\,Y_{64}\,\,,\hskip
14.22636ptY_{12}\,Y_{23}\,Y_{34}\,Y_{45}\,Y_{56}\,Y_{61}\,\,,$ (16)

which can be seen from the diagrams of figure 3,

123456

123456

123456

123456

Figure 3: Diagrams representing the $Y_{ij}$ structures of the twenty-prime
six-point function.

together with all the possible permutations. Therefore, a reasonable choice to
represent this six-point function is

$\langle\mathcal{O}_{20^{\prime}}(x_{1},Y_{1})\,\dots\,\mathcal{O}_{20^{\prime}}(x_{6},Y_{6})\rangle=\left(\frac{Y_{12}\,Y_{34}\,Y_{56}}{x_{12}^{2}\,x_{34}^{2}\,x_{56}^{2}}\right)^{2}\,\mathcal{G}_{6}\left(u_{i},U_{i};\sigma_{i},\tau_{i}\right)\,\,,$
(17)

which now depends on nine cross-ratios $u_{1},\dots,u_{6}$ and
$U_{1},\dots,U_{3}$ defined as

$u_{1}=\frac{x_{12}^{2}x_{35}^{2}}{x_{13}^{2}x_{25}^{2}}\,\,,\hskip
14.22636ptu_{i+1}=\left.u_{i}\right|_{x_{j}\rightarrow x_{j+1}}\,\,,\hskip
19.91684ptU_{1}=\frac{x_{13}^{2}x_{46}^{2}}{x_{14}^{2}x_{36}^{2}}\,\,,\hskip
14.22636ptU_{i+1}=\left.U_{i}\right|_{x_{j}\rightarrow x_{j+1}}\,,$ (18)

where the superscript in $x_{j}$ is taken modulo $6$, and where the respective
R-symmetry cross-ratios $\sigma_{1},\dots,\sigma_{6}$ and
$\tau_{1},\dots,\tau_{3}$ are given by the above expressions with the squared
distances $x_{ij}^{2}$ replaced by the products of the null polarization
vectors $Y_{ij}$.

Concerning the corrections of this object, the integrands are only known up to
$2$ loops [1, 3, 16], but the respective integrals have not yet been computed
up to this level. The simplest of these integrals has been studied in [30],
where they have not only been able to obtain its symbol but also infer that
these integrals must be expressed in terms of elliptic functions. These
functions, however, are not as simple as we would like. Fortunately, it has
been observed for higher point functions that under certain kinematical limits
the associated conformal integrals simplify. Therefore, although we might not
be able to evaluate the integrals in full generality we can compute them in
several simpler cases and try to learn something from it. Here we shall
consider two distinct circumstances: a configuration in which the operators
lie in a line, and another where adjacent points will be taken to become null
separated from each other $x_{i,i+1}^{2}\rightarrow 0$, also known as the
light-cone limit.

It is also important to mention that the higher-point functions have
interesting non-trivial properties, namely the Drukker-Plefka twist [31] and
the chiral algebra twist [32], which impose extra constraints on the
correlators. Something nice about the integrals that will be studied here is
that they can be used to verify these properties.

Furthermore, the results we will obtain can not only be used to provide useful
information, such as CFT data, but they can also serve later as cross-checks.
Indeed, if these integrals happen to be computed for general kinematics or by
means of other methods, one can always restrict the results to the same
kinematics as here and see if both results agree. This has already been the
case for four- and five-point correlation functions, where the results from
the evaluation of conformal integrals matched those obtained by means of
integrability methods [33, 34, 35, 36]. In particular, integrability can be
used to obtain the correlators as a power series over the cross-ratios, where
generally one also considers the points to lie on a plane or some other
simpler kinematics. Because the results these methods provide us are already
full integrated functions, simply knowing the integrands of the $\ell$ loop
corrections is not enough. As such, evaluating the conformal integrals for the
six-point function, even if for rather particular kinematics, would already be
useful to compare with results from integrability. Furthermore, given that
from the integrability viewpoint it is hard to make an identification between
a certain integrability contribution and a conformal integral, our results
could also help us make predictions regarding this issue.

##  3 Integrals

Our goal is to compute two-loop six-point conformal integrals of the form

$I=\int\frac{d^{d}x_{7}\,d^{d}x_{8}}{x_{78}^{2}\prod_{i=1}^{6}\prod_{j=7}^{8}(x_{ij}^{2})^{a_{ij}}}\,,$
(19)

where $a_{ij}$ are integers that satisfy $\sum_{i}a_{ij}=d-1$. Below we
explain a possible way of evaluating these integrals which was the one used
here to compute the integrals for the line configuration.

###  3.1 Methodology

The first step of the computation is to use conformal symmetry to simplify the
integrand. In particular, we send one of the external points to infinity.
Although the choice of the point is completely arbitrary it is good practice
to choose one of the points that appears more times in the denominator or one
that appears in the numerator. Additionally, it is also useful to send one
point to lie at the origin and another to a fixed position, usually chosen to
be $(1,0,\dots)$.

In case the remaining integrand has no position-dependent numerator, the next
step is to perform the Schwinger parametrization (see [37] for example), which
makes use of the following identity

$\frac{1}{A^{n}}=\frac{1}{\Gamma(n)}\int_{0}^{\infty}d\alpha\,\alpha^{n-1}e^{-\alpha
A}\,.$ (20)

Upon using this equality, the spacetime integrals transform into Gaussian
integrals, which can easily be done, and we obtain integrals over the
Schwinger parameters [38, 39]

$I(a_{ij})=\Gamma(\omega)\,\left(\prod_{i}\int_{0}^{\infty}\frac{d\alpha_{i}\,\alpha_{i}^{a_{i}-1}}{\Gamma(a_{i})}\right)\,\frac{\Psi^{\omega-\frac{d}{2}}}{\Phi^{\frac{d}{2}}}\,\delta(1-\alpha_{j})\,,$
(21)

where $\Psi$ and $\Phi$ are graph polynomials resulting from the spacetime
integrations, the index $i$ runs over the propagators appearing in the
integral (19), with $a_{i}$ being their powers, and where the superficial
degree of divergence $\omega$ for any number of loops $\ell$ is given by

$\omega=\sum_{i}a_{i}-\frac{d}{2}\cdot\ell\,.$ (22)

The Dirac delta at the end fixes some arbitrary Schwinger parameter
$\alpha_{j}$ to one.

Finally, the integrals over the parameters $\alpha_{i}$ can be evaluated by
means of the computer package HyperInt [39], provided that we consider some
simple geometry or kinematical limits which simplify the integrands. At the
end, we use the function fibrationBasis, belonging to the same package, that
expresses the result in a way that is suitable for expansion around $0$ with
respect to the provided arguments and the specified order.

When it is the case that even after using conformal symmetry we still have a
position-dependent numerator in the integrand the procedure is slightly
different. This has to do with the fact that the identity (20) has problems
for negative $n$, in which case the Gamma function has poles. Nevertheless,
the procedure does not differ significantly from what was just explained,
requiring us to differentiate with respect to some additional Schwinger
parameters and evaluate them at zero. We give a more detailed explanation of
this in an example below but we refer the reader to [40] for a complete
description.

###  3.2 Two-loop half-BPS six-point integrals

The two-loop correction of single-trace half-BPS six-point functions involves
conformal integrals of three different types

$\displaystyle\mathcal{B}_{123,456}\equiv\int\frac{d^{4}x_{7}\,d^{4}x_{8}}{(x_{17}^{2}x_{27}^{2}x_{37}^{2})\,x_{78}^{2}\,(x_{48}^{2}x_{58}^{2}x_{68}^{2})}\,,$
(23)
$\displaystyle\mathcal{P}_{1,23,456}\equiv\int\frac{x_{28}^{2}\,d^{4}x_{7}\,d^{4}x_{8}}{(x_{17}^{2}x_{27}^{2}x_{37}^{2})\,x_{78}^{2}\,(x_{18}^{2}x_{48}^{2}x_{58}^{2}x_{68}^{2})}\,,$
(24)
$\displaystyle\mathcal{T}_{14,23,56}\equiv\int\frac{x_{28}^{2}\,x_{57}^{2}\,d^{4}x_{7}\,d^{4}x_{8}}{(x_{17}^{2}x_{27}^{2}x_{37}^{2}x_{47}^{2})\,x_{78}^{2}\,(x_{18}^{2}x_{48}^{2}x_{58}^{2}x_{68}^{2})}\,.$
(25)

Following the same convention as [16]222The conformal integrals presented here
are the same as the ones from [16] with the difference being that we are using
a different notation and we have ignored the prefactors. we will refer to the
$\mathcal{B}$ integrals as double-box integrals, $\mathcal{P}$ as penta-box
integrals and $\mathcal{T}$ as double-penta integrals. Due to the symmetries
of the integrands, each of these conformal integrals are invariant under
certain permutations of positions. For example, switching $1\leftrightarrow 2$
in the $\mathcal{B}$ integral above amounts to the same integral.
Consequently, the number of independent integrals for each type is: 10
$\mathcal{B}$’s, 60 $\mathcal{P}$’s and 180 $\mathcal{T}$’s. Luckily, we do
not have to compute all of these integrals, which would be quite cumbersome,
but only those with which we can obtain all others from cross-ratios
transformations.

Ideally, we would like to be able to compute these integrals in the most
general configuration of the six-point function. Unfortunately, we have not
yet been able to evaluate integrals of the sort expressed in (19) under
general kinematics. Nonetheless, we can still study them in specific cases for
which the integrands simplify. In principle, this allow us to compute the
integrals and obtain closed form results in terms of well defined functions.

###  3.3 Line geometry

One of the possibilities is to consider a configuration in which all points
lie in the same line. We thus start by using conformal symmetry to send three
points to $0$, $1$ and $\infty$, that we respectively choose to be $x_{1}$,
$x_{2}$ and $x_{6}$, and then impose the remaining points to lie on a line by
fixing $x_{ij}^{2}=(z_{i}-z_{j})(\bar{z}_{i}-\bar{z}_{j})$ with
$z_{i}=\bar{z}_{i}$. In this configuration, the cross-ratios $u_{i}$, $U_{i}$
of the six-point function (18) are related to the $z_{i}$ cross-ratios by

$\displaystyle u_{1}=\frac{z_{2}^{2}\,(1-z_{5})^{2}}{(z_{2}-z_{5})^{2}}\,,$
$\displaystyle\,\,u_{2}=\frac{(z_{2}-1)^{2}}{(z_{2}-z_{4})^{2}}\,,\,\,u_{3}=\frac{(1-z_{4})^{2}\,z_{5}^{2}}{z_{4}^{2}\,(1-z^{5})^{2}}\,,\,\,u_{4}=\frac{(z_{4}-z_{5})^{2}}{(z_{2}-z_{5})^{2}}\,,\,\,u_{5}=\frac{1}{z_{5}^{2}}\,,$
(26) $\displaystyle
U_{1}=\frac{1}{z_{4}^{2}}\,,\,\,U_{2}=\frac{(z_{2}-z_{4})^{2}\,z_{5}^{2}}{z_{4}^{2}\,(z_{2}-z_{5})^{2}}\,,\,\,U_{3}=\frac{(1-z_{5})^{2}}{(z_{2}-z_{5})^{2}}\,.$

Relatedly, the choice of cross-ratios with which we work can be crucial for
the evaluation of the integral. To see this let us consider the one-loop
correction to the four point function of scalars in four dimensions

$\int\frac{d^{4}x_{0}}{x_{01}^{2}\,x_{02}^{2}\,x_{03}^{2}\,x_{04}^{2}}\,\,.$
(27)

Following the procedure mentioned above we start by sending
$x_{4}\rightarrow\infty$, we then introduce the Schwinger parameters and
integrate over $x_{0}$. After doing this we obtain the integral

$\int\frac{d\alpha_{1}d\alpha_{2}}{(1+\alpha_{1}+\alpha_{2})(\alpha_{1}+v\,\alpha_{2}+u\,\alpha_{1}\,\alpha_{2})^{2}}\,,$
(28)

where the third Schwinger parameter has been fixed to $1$ by the Dirac delta.
It is not hard to do one of the two above integrals. If we choose to do the
one in $\alpha_{1}$ we find that this double integral gets reduced to

$\int
d\alpha_{2}\,\frac{1+\left\\{1+u-v\left[1-\log(v\alpha_{2})+\log\big{(}(1+\alpha_{2})(1+u\,\alpha_{2})\big{)}\right]\right\\}\alpha_{2}+u\,\alpha_{2}^{2}}{v\,\alpha_{2}\,\left[1+\alpha_{2}(1-v+u(1+\alpha_{2}))\right]^{2}}\,.$
(29)

The issue with the above expression is that some of the factors in the
denominator are not linear in the integration variable. We want to avoid these
kinds of terms as the method used by HyperInt requires the integrands to
factor linearly. Generally, we can overcome this problem simply by using
different variables. In this particular case if we use the cross-ratios $z$
and $\bar{z}$ instead of $u$ and $v$ the integral over Schwinger parameters is
given by

$\int\frac{d\alpha_{1}d\alpha_{2}}{(1+\alpha_{1}+\alpha_{2})((1-z)(1-\bar{z})\alpha_{2}+\alpha_{1}(1+z\bar{z}\alpha_{2}))^{2}}\,,$
(30)

and the one we obtain after performing one of the integrations is

$\int\\!\\!d\alpha_{2}\,\tfrac{1\,+\,\alpha_{2}\,\big{\\{}z+\bar{z}+(1-z)\,(1-\bar{z})\big{[}\log\left[(1-z)\,(1-\bar{z})\,\alpha_{2}\right]-\log\left[(1+\alpha_{2})\,(1+z\bar{z}\alpha_{2})\right]\big{]}+z\,\bar{z}\,\alpha_{2}\big{\\}}}{(1-z)\,(1-\bar{z})\,\alpha_{2}\,(1+z\alpha_{2})^{2}\,(1+\bar{z}\alpha_{2})^{2}}\,.$
(31)

Now, we no longer have terms in $\alpha_{2}$ as the one found above, thus
making the second integral more tractable.

For higher-point functions the choice of variables such as not to have these
non-linear terms is in general not straightforward. For the six-point
integrals considered here (23,24,25) there might exist some possible choice
that allows us to only have linearized terms in the integrands, which in turn
makes the integrals easier to evaluate. As far as we know such a choice was
not yet found. Therefore, we opted to work with the $z,\bar{z}$ cross-ratios
and further restrict to the line configuration $z=\bar{z}$ in order to avoid
the non-linear terms and be able to compute the integrals.

As mentioned previously, some of the integrals will still have numerator terms
even after using conformal symmetry to fix some of the points. We would now
like to illustrate what the procedure for these cases is when evaluating them
on a line. For this, we will consider one of the double-penta integrals,
namely

$\mathcal{T}_{16,23,45}=\int\frac{x_{28}^{2}\,x_{47}^{2}\,d^{4}x_{7}\,d^{4}x_{8}}{x_{17}^{2}x_{27}^{2}x_{37}^{2}x_{78}^{2}x_{18}^{2}x_{48}^{2}x_{58}^{2}}\,,$
(32)

where we have already used conformal symmetry to send
$x_{6}\rightarrow\infty$. Following the recipe we gave before, we then use
Schwinger parametrization for every term in the integrand and integrate over
the Gaussian space-time integrals. Next, we restrict the points to the plane,
send $x_{1}\rightarrow 0$ and $x_{3}\rightarrow 1$ and further restrict to the
line kinematics $z_{i}=\bar{z}_{i}$. The outcome is an integral over Schwinger
parameters of the form (21), with the graph polynomials $\Psi$ and $\Phi$ as
given in appendix A. Their expressions depend on nine Schwinger parameters,
given that we have nine squared distances terms, but should only depend on
seven since it is the number of terms on the denominator. This has to do with
the aforementioned fact that the Schwinger parametrization as in (20) is
problematic for the numerator terms. In these cases what we must do according
to [40] is to use the more general form of the Schwinger trick (see [37] for
example). This will involve taking two derivatives for each of the two extra
Schwinger parameters $\alpha_{8}$ and $\alpha_{9}$ evaluated at zero. Since
one of the two derivatives for each Schwinger parameter cancels the respective
integration, the integral will then be given by

$\mathcal{T}_{16,23,45}\propto\prod_{i=1}^{7}\int_{0}^{\infty}d\alpha_{i}\,\delta(1-\alpha_{j})\,\frac{\partial}{\partial\alpha_{8}}\left[\frac{\partial}{\partial\alpha_{9}}\left[\frac{\Psi}{\Phi^{2}}\right]_{\alpha_{9}=0}\right]_{\alpha_{8}=0}\,,$
(33)

where we already used that $d=4$ and $\omega=1$ for this case. A small comment
is in order here. In particular, although we can take the two derivatives and
then evaluate the integrals, we found that in some cases the resulting
integrands after the derivatives are complex enough that the integrals take a
very long time to be evaluated with HyperInt. What one must do is to take one
derivative, evaluate a few of the integrals over Schwinger parameters, say
$\alpha_{1}$, $\alpha_{3}$ and $\alpha_{4}$ for example333The order of
integration on the $\alpha$’s is also important to prevent the evaluation of
the integrals from taking too long. To find the best order one uses the
functions irreducibles, cgReduction and suggestIntegrationOrder described in
[39]., take the other derivative and only then compute the remaining
integrals. This is really a trial and error game that has to be played for
each integral to see what works best. At the end, we use fibrationBasis to
write the result in a way that is suitable for being expanded in the limit of
small arguments.

Finally, we would just like to note that because on the line configuration we
will be working with general cross-ratios $z_{i}$, we simply have to evaluate
a conformal integral of each type. All others follow from appropriately
transforming the cross-ratios according to the respective positions
permutations.

###  3.4 Light-cone limit and asymptotic expansions

Another choice that can be taken in order to simplify the conformal integrals
is to consider physically relevant kinematical limits. In particular, limits
in which the leading order terms of these integrals are relatively easier to
compute. One of these cases is the light-cone limit, where we take
$x_{i,i+1}^{2}\rightarrow 0$. When in this kinematics the conformal integrals
can be separated in distinct regions according to whether the integrations
variables are small or not and use certain identities that make the integrals
easier. This procedure is called asymptotic expansion method (see [41, 42, 43]
for example).

Let us then consider we are taking the null limit $x_{12}^{2}\rightarrow 0$.
The idea behind the asymptotic expansions is that we may start by separating
each integration into two regions: one where the integration variable is small
and thus comparable to the small parameter $x_{12}^{2}$, and another where
this variable is big such that dropping the squared distance is a good
approximation. Given that we are dealing with two-loop integrals, we will have
four regions:

$\text{I.}\,\,x_{7}\ll 1,x_{8}\ll 1\,;\,\,\,\,\,\text{II.}\,\,x_{7}\ll
1,x_{8}\gg 1\,;\,\,\,\,\,\text{III.}\,\,x_{7}\gg 1,x_{8}\ll
1\,;\,\,\,\,\,\text{IV.}\,\,x_{7}\gg 1,x_{8}\gg 1\,\,.$ (34)

In general, when we take the light-cone limit between any two points, an
$\ell$-loop integral will give rise to $2^{\ell}$ regions. After separating
the integral in these four regions, it is possible to simplify the integrand
by making use of the identity

$\frac{1}{(x_{ij}^{2})^{b}}=\frac{1}{(x_{1j}^{2})^{b}}\sum_{n=0}^{\infty}\binom{-b}{n}\,\frac{(x_{1i}^{2}-2x_{1i}\cdot
x_{1j})^{n}}{(x_{1j}^{2})^{n}}\,\,,$ (35)

where $x_{i}$ is assumed to be small and $x_{j}$ to be large. Naturally these
expansions have to be done in a suitable manner for each region.

For the sake of exemplification let us then consider one of the simpler six-
point conformal integrals, namely (23). Following the aforementioned
prescription we can write

$\displaystyle\hskip
128.0374pt\mathcal{B}_{123,456}=\mathcal{B}_{1}+\mathcal{B}_{2}+\mathcal{B}_{3}+\mathcal{B}_{4}\,,$
(36)
$\displaystyle\mathcal{B}_{1}=\sum_{n,m,k}\int\frac{d^{d}x_{7}d^{d}x_{8}\,(2x_{13}\cdot
x_{17}-x_{17}^{2})^{n}(2x_{14}\cdot x_{18}-x_{18}^{2})^{m}(2x_{15}\cdot
x_{18}-x_{18}^{2})^{k}}{(x_{13}^{2})^{1+n}\,(x_{14}^{2})^{1+m}\,(x_{15}^{2})^{1+k}\,x_{17}^{2}\,x_{27}^{2}\,x_{78}^{2}}\,,$
$\displaystyle\hskip
56.9055pt\mathcal{B}_{2}=\sum_{n,m}\int\frac{d^{d}x_{7}d^{d}x_{8}\,(2x_{13}\cdot
x_{17}-x_{17}^{2})^{n}(2x_{17}\cdot
x_{18}-x_{17}^{2})^{m}}{(x_{13}^{2})^{1+n}\,(x_{18}^{2})^{1+m}\,x_{17}^{2}\,x_{27}^{2}\,x_{48}^{2}\,x_{58}^{2}}\,,$
$\displaystyle\hskip
99.58464pt\mathcal{B}_{4}=\sum_{n}\int\frac{d^{d}x_{7}d^{d}x_{8}\,(2x_{12}\cdot
x_{17}-x_{12}^{2})^{n}}{(x_{17}^{2})^{2+n}\,x_{37}^{2}\,x_{48}^{2}\,x_{58}^{2}\,x_{78}^{2}}\,,$

where for simplicity we have used conformal symmetry to send
$x_{6}\rightarrow\infty$. The reason why we have not given the explicit
expression for $\mathcal{B}_{3}$ is that it is scaleless [41, 44] and thus
evaluates to zero. Moreover, we point out to the dimensional regularization
$d=4-\epsilon$ performed in the integrals measures. This is a key step of the
asymptotic expansion method which allows to regulate the possible divergences
of each region integral. Importantly, the final result should not depend on
$\epsilon$ since the conformal integrals had no dependence on it to start
with.

Hopefully the above expressions convey the central idea of this method, which
is that we obtain several integrals that are easier to evaluate than the
original one. If needed, however, we could further take the null limits
$x_{34}^{2}\rightarrow 0$ and $x_{56}^{2}\rightarrow 0$ and follow the same
path to obtain even simpler integrals. For the integrals considered in this
work we found no such thing was necessary as one null limit sufficed to
evaluate them.

To compute the integrals in (36) we can start by dropping the terms
$x_{12}^{2}$, $x_{17}^{2}$ and $x_{18}^{2}$ inside the parenthesis terms in
the numerators since they are subleading. Afterwards, we can obtain the final
expressions for these integrals by making use of the following result [45]

$\int\frac{d^{d}x_{7}\,d^{d}x_{8}\,(x_{15}\cdot x_{57})^{k_{1}}\,(x_{25}\cdot
x_{58})^{k_{2}}}{x_{57}^{2}x_{58}^{2}x_{67}^{2}x_{68}^{2}x_{78}^{2}}=c_{k_{1},k_{2}}\,\frac{(x_{15}\cdot
x_{56})^{k_{1}}(x_{25}\cdot x_{56})^{k_{2}}}{(x_{56}^{2})^{d-5}}\,,$ (37)

with

$c_{k_{1},k_{2}}=\binom{k_{1}+k_{2}}{k_{1}}\sum_{j=0}^{k_{2}}(-1)^{j}\binom{k_{2}}{j}\frac{1}{1+j+k_{1}}\left[6\zeta_{3}+S_{1,2}(k_{1}+j)-S_{2,1}(k_{1}+j)\right]\,,$
(38)

where $S_{i,j}$ are harmonic sums defined recursively as

$S_{a,\textbf{b}}(N)=\sum_{n=1}^{N}\frac{(\text{sign}(a))^{n}}{n^{|a|}}\,S_{\textbf{b}}(n)\,,\,\,\,\,\,\text{with}\,\,\,S_{\emptyset}(N)=1\,\,,\,\,\,\,a,b_{i}\in\mathbb{Z}\setminus\\{0\\}\,.$
(39)

Even though our integrals might have more terms in the numerator than what
appears in the equality (37), this does not prevent us from using it, as
explained in appendix B.

In general, when we take kinematical limits it is no longer true that we can
obtain all of the conformal integrals simply by computing one integral of each
kind and appropriately transforming the cross-ratios. This stems from the fact
that in these cases some of these integrals become independent of each other.
Therefore, what must be done is that we have to compute those integrals with
which we can generate all others in these limits. In our case, it is possible
to infer that the conformal integrals we need to assess are

$\mathcal{B}_{123,456}\,\,,\hskip 14.22636pt\mathcal{B}_{135,246}\,\,,$ (40)

for the double-box integrals,

$\mathcal{P}_{2,13,456}\,\,,\,\,\,\,\mathcal{P}_{3,12,456}\,\,,\,\,\,\,\mathcal{P}_{3,14,256}\,\,,\,\,\,\,\mathcal{P}_{3,15,246}\,\,,\,\,\,\,$
(41)

for the penta-box integrals, and

$\mathcal{T}_{46,15,23}\,\,,\,\,\,\,\mathcal{T}_{46,15,32}\,\,,\,\,\,\,\mathcal{T}_{56,14,23}\,\,,\,\,\,\,\mathcal{T}_{56,14,32}\,\,,\,\,\,\,\mathcal{T}_{56,12,34}\,\,,\,\,\,\,\mathcal{T}_{26,15,34}\,\,,\,\,\,\,\mathcal{T}_{24,16,35}\,\,,\,\,\,\,$
(42)

for the double-penta integrals 444Naturally, we are free to permute the
indices and obtain another set of independent integrals, although we gain
nothing new.. As it so happens, some of these integrals are easier to evaluate
than others, namely $\mathcal{T}_{56,14,23}$, $\mathcal{T}_{56,14,32}$ and
$\mathcal{T}_{56,12,34}$. Here we decided to compute only one of these three
integrals, in particular $\mathcal{T}_{56,14,23}$, but the procedure for the
other two is exactly the same. The more difficult ones, however, will be left
to be computed in some future work.

Now that we have given some insights about the kinematics and respective
procedures to deal with the integrals at hand, we will proceed to present the
obtained results and respective comments.

##  4 Results and analysis

###  4.1 Line configuration

We will start by analyzing the results obtained from evaluating the integrals
in the line configuration. We have chosen to compute all ten double-box
integrals, given that the number is fairly reasonable. However, when it comes
to the penta-box integrals and the double-penta integrals, we have only
evaluated the integrals $\mathcal{P}_{1,23,456}$ and $\mathcal{T}_{16,23,45}$,
respectively, as it would be quite time-consuming to compute all others.
Nonetheless, one can still easily obtain the remaining ones by seeing how the
cross-ratios transform under indices permutations, as mentioned in section
3.2. For instance, under the exchange of points $2\leftrightarrow 6$, the
cross-ratios would transform as

$z_{2}\rightarrow
1-z_{2}\,\,,\,\,\,\,z_{4}\rightarrow\frac{z_{4}\,(z_{2}-1)}{z_{2}-z_{4}}\,\,,\,\,\,\,z_{5}\rightarrow\frac{z_{5}\,(z_{2}-1)}{z_{2}-z_{5}}\,\,.$
(43)

Afterwards, we simply have to expand the final expression accordingly to the
kinematical limit we want to analyze.

All the integrals were computed in four different cases, such that the
following kinematical regimes could be taken afterwards if one wanted, namely:
$z_{2},z_{4},z_{5}\rightarrow 0$; $z_{2},z_{5}\rightarrow 0$,
$z_{4}\rightarrow 1$; $z_{2},z_{4}\rightarrow 0$ and $z_{5}\rightarrow\infty$;
$z_{2}\rightarrow 0$, $z_{4}\rightarrow 1$ and $z_{5}\rightarrow\infty$ (with
the remaining points fixed by conformal symmetry as mentioned before). The
results can be found in an auxiliary Mathematica file. We point out that
throughout this file we have introduced two new variables defined as

$y=1-z_{4}\,\,,\hskip 14.22636ptw=\frac{1}{z_{5}}\,\,,$ (44)

which when taken to zero allow for the kinematical limits mentioned above.

Importantly, we found that all of the results were given in terms of rational
functions of the cross-ratios $z_{i}$, zeta functions

$\zeta(s)=\sum_{n=1}^{\infty}\frac{1}{n^{s}}\,\,,$ (45)

with $s\in\mathbb{C}$, and hyperlogarithm functions defined as

$\text{Hlog}(z,[\sigma_{1},\dots,\sigma_{r}])=\int_{0}^{z}\frac{dz^{\prime}}{z^{\prime}-\sigma}\,\text{Hlog}(z^{\prime},[\sigma_{2},\dots,\sigma_{r}])\,,\hskip
14.22636pt\text{with}\,\,\,\text{Hlog}(z)=1\,,$ (46)

where $\sigma_{i}$,$z\in\mathbb{C}$.

Interestingly enough, we observed that in all of the results the simplest type
of integrals $\mathcal{B}$ factorized as a product of a common cross-ratios
prefactor and a linear combination of hyperlogarithms and zeta functions. For
instance, in the case in which the limit $z_{i}\rightarrow 0$ ($i=2,4,5$)
could be taken later, we found one of these integrals to be given by

$\mathcal{B}_{123,456}=\frac{2}{z_{2}(z_{2}-1)(z_{4}-z_{5})}\Big{(}\text{Hlog}(z_{2},[z_{4},0,z_{5}])-\text{Hlog}(z_{2},[z_{4},1,z_{5}])+\dots\Big{)}\,\,.$
(47)

However, in the case of the $\mathcal{P}$ and $\mathcal{T}$ integrals this
property is no longer observed due to the non-existence of a common cross-
ratios prefactor.

Furthermore, we noted that in all the obtained results the integrals only
contained hyperlogarithms with up to three parameters in the arguments list
(see (46)). This implies that the results can be expressed in terms of
polylogarithms, defined as

$\text{Li}_{s}(z)=\sum_{k=1}^{\infty}\frac{z^{k}}{k^{s}}\,\,,\hskip
14.22636pt\,\,\,\text{with}\,\,\,\,\text{Li}_{1}(z)=-\text{log}(1-z)\,\,,$
(48)

for $s\in\mathbb{N}$ and $z\in\mathbb{C}$. This contrasts to what would happen
if the list of parameters had four or more parameters, in which case the
hyperlogarithms are related with the so-called multiple polylogarithms

$\text{Li}_{s_{1},\dots,s_{r}}(z_{1},\dots,z_{r})=\sum_{0<k_{1}<\dots<k_{r}}^{\infty}\frac{z_{1}^{k_{1}}\dots
z_{r}^{k_{r}}}{k_{1}^{k_{1}}\dots k_{r}^{s_{r}}}\,\,,\hskip
14.22636pt\,\,\,s_{1},\dots,s_{r}\in\mathbb{N}\,\,.$ (49)

From looking at expression (47) , in particular its prefactor, one could think
that it diverges in a power-law fashion when the limit $z_{2}\rightarrow 0$ is
taken. If we expand the expression for small cross-ratios $z_{i}\rightarrow 0$
in the order of increasing index, we find that the first few orders are

$\displaystyle\mathcal{B}_{123,456}$ $\displaystyle\underset{z_{i}\rightarrow
0}{\sim}\frac{1}{z_{5}^{2}}\left[2\log(z_{5}^{2})-4\log(z_{2})\log(z_{5})-2\log(z_{4})^{2}+4\log(z_{2})\log(z_{4})\right.$
(50)
$\displaystyle\left.+\,6\log(z_{5})-2\log(z_{4})-4\log(z_{2})+4\right]+\frac{2}{z_{5}}+\dots\,,$

where the dots denote subleading terms in the expansion. From this, it appears
we have power-law divergences due to negative powers in $z_{5}$. However, if
we consider the prefactors that these integrals must have when used for the
loop corrections (see [16] for example) these disappear and only logarithmic
divergences remain. This was also verified in the results for the other
aforementioned limits and types of integrals. Nevertheless, note that even
these logarithmic divergences might disappear when we take the linear
combinations of these integrals to give the full two-loop corrections. This,
however, was not analyzed here.

###  4.2 Light-cone limit

We now discuss the results obtained from evaluating the six-point conformal
integrals in the light-cone limit. In particular, we will only consider the
null limit $x_{56}^{2}\rightarrow 0$ and not others since as previously
explained taking one null limit is sufficient in our case. Moreover, as
mentioned in section 3.4 three of the integrals from (40,41,42) are easier to
evaluate. Here we chose to analyze the double-penta integral
$\mathcal{T}_{56,14,23}$, which is given by

$\mathcal{T}_{56,14,23}=\int\frac{x_{18}^{2}\,x_{27}^{2}\,d^{d}x_{7}\,d^{d}x_{8}}{x_{17}^{2}x_{28}^{2}x_{38}^{2}x_{47}^{2}x_{57}^{2}x_{58}^{2}x_{67}^{2}x_{68}^{2}x_{78}^{2}}\,\,.$
(51)

In order to evaluate it in the light-cone limit we follow the integration by
regions that we explained above. However, for this integral we will only need
to concern with the first region of (34), which will be denoted by
$\mathcal{T}_{I}$, as we will only be interested in the leading order results.
Indeed, from a simple analysis of the leading order behavior in $x_{56}^{2}$,
it is possible to infer that $\mathcal{T}_{I}$ diverges as
$(x_{56}^{2})^{-1-2\epsilon}$ in the light-cone limit, while others are
subleading.

After using the identity (35) accordingly to the assumptions of the first
region, we end up with

$\displaystyle\mathcal{T}_{I}=\sum_{n_{1},n_{2},n_{3},n_{4},n_{5}}\frac{1}{(x_{15}^{2})^{n_{1}+n_{4}}\,(x_{25}^{2})^{n_{2}+n_{5}}\,(x_{35}^{2})^{n_{3}}}\,\times$
(52) $\displaystyle\int\frac{d^{d}x_{7}\,d^{d}x_{8}\,(-2x_{15}\cdot
x_{57})^{n_{1}}(-2x_{25}\cdot x_{58})^{n_{2}}(-2x_{35}\cdot
x_{58})^{n_{3}}(2x_{15}\cdot x_{58})^{n_{4}}(2x_{25}\cdot
x_{57})^{n_{5}}}{x_{57}^{2}x_{58}^{2}x_{67}^{2}x_{68}^{2}x_{78}^{2}}$

where the sums in $n_{4}$ and $n_{5}$ run only over $\\{0,1\\}$555The reason
why these sums run only over $\\{0,1\\}$ is because they come from the
expansion of the numerators. Therefore, the associated binomial coefficients
will be $\binom{1}{n}$, which only make sense for $0\leq n\leq 1$. and where
for simplicity we have sent $x_{4}\rightarrow\infty$. Moreover, we have
dropped the terms $x_{57}^{2}$ and $x_{58}^{2}$ inside the parentheses in the
numerator terms as they are subleading relatively to the ones we have kept.

To evaluate this integral we use (37) while following the reasoning of
appendix B and get to the final result

$\mathcal{T}_{I}=\\!\\!\\!\\!\\!\\!\\!\sum_{n_{1},n_{2},n_{3},n_{4},n_{5}}\\!\\!\\!\\!\\!\\!\frac{(-1)^{n_{1}+n_{2}+n_{3}}\,c_{n_{1}+n_{5},n_{2}+n_{3}+n_{4}}(2x_{15}\cdot
x_{56})^{n_{1}+n_{4}}(2x_{25}\cdot x_{56})^{n_{2}+n_{5}}(2x_{35}\cdot
x_{56})^{n_{3}}}{(x_{15}^{2})^{n_{1}+n_{4}}\,(x_{25}^{2})^{n_{2}+n_{5}}\,(x_{35}^{2})^{n_{3}}\,(x_{56}^{2})^{5-d}}\,.$
(53)

We may take the limit $\epsilon\rightarrow 0$ such that $(5-d)\rightarrow 1$
and explicitly see that this integral indeed diverges as $(x_{56}^{2})^{-1}$
in the light-cone limit. It could also be beneficial to express this result in
terms of the six-point cross-ratios, which could render its analysis simpler.
This will not be done here since it would make the expression longer and thus
not possible to fit it in a single line anymore. Nevertheless, the procedure
would consist on using the identity

$x_{i5}\cdot
x_{56}=-\frac{1}{2}\left(x_{i5}^{2}+x_{56}^{2}-x_{i6}^{2}\right)\,,$ (54)

where we can drop $x_{56}^{2}$ in the light-cone limit, and finally use the
definition of the cross-ratios (18).

##  5 Discussion

Correlation functions of $\mathcal{N}=4$ SYM, especially of half-BPS
operators, have been a subject of interest for some decades as they provide us
information about the CFT itself and also about its dual theory in AdS. The
simpler non-trivial of these are the four-point functions, which have been
studied both in the weak [6, 7, 8, 9] and strong [10, 11, 12, 13, 14, 15]
regimes of the coupling and whose loop corrections are known up to a
relatively high order. When it comes to correlators with even more points,
however, not so much is known. This stems from the contrasting difficulty
between knowing the integrands and actually being able to evaluate the
conformal integrals which give the $\ell$-loop corrections. In the case of
five-point functions, the first few orders of the corrections for some of
these functions have already been obtained under certain kinematical limits
[17]. On the other hand, the integrands of the six-point functions are known
up to $2$-loops [16] and the respective integrals have been analyzed in [30],
even though they were not able to explicitly evaluate them.

Here, we obtained a closed form expression for some of the two loop six-point
integrals on a line configuration, which truly amounts to knowing all of them
since they are related by cross-ratios transformations. For this we have
adopted the Schwinger parameterization and used the HyperInt package [38, 39],
which allows to compute Feynman integrals in the Schwinger parameters
representation. We have found in all the obtained results that the final
expressions were given by rational functions of the cross-ratios $z_{i}$,
hyperlogarithm functions and the zeta functions. Moreover, we took the
kinematical limits mentioned in section 4 for some results and observed that
the results possess divergences. As was commented there, these divergences
might cancel out when we sum the integrals in order to obtain the two-loop
corrections, but this was not verified here.

In addition, we also evaluated one of the double-penta integrals in the
kinematics where two adjacent points become null separated. Although we only
considered one of the three integrals which we claimed were easier to compute
in this kinematical limit the remaining ones can be obtained by following the
exact same path. In order to evaluate this integral we implemented the
asymptotic expansion method. Despite the level of complexity surely rising, it
would be nice to compute the remaining independent integrals (40,41,42) as
they are crucial if we want to obtain the actual two-loop corrections and
extract physical information.

As was mentioned several times throughout this paper, we chose to consider the
line configuration and the light-cone limit in order to simplify the
integrands and be able to evaluate the integrals. This stems from the high
complexity of the same integrals for general configurations, which so far has
prevented us from computing them. It would be nice if we could somehow work
around this problem and obtain the results for a general disposition of the
six-point function. Naturally, this ought to be quite challenging so we could
start off by taking one point out of the line and do the same analysis. We
would then continue to take more points out of the line and gradually move on
to more general but harder configurations and assess these integrals.

Finally, we could try to tackle these issues from an integrability-based
approach, as was done for four- and five-point functions [33, 34, 35, 36], and
compare the outcomes with our results.

## Acknowledgements

We would like to thank Vasco Gonçalves for proposing this project and for many
useful discussions and also Bruno Fernandes for several insights provided
throughout this work and for reviewing the manuscript. We are also grateful to
João Vilas Boas for reviewing the manuscript. Centro de F\́frac{i}{2}sica do
Porto is partially funded by Fundação para a Ciência e a Tecnologia (FCT)
under the grant UID04650-FCUP. The author is supported by Simons Foundation
grant #488637 (Simons collaboration on the non-perturbative bootstrap).

##  Appendix A Graph Polynomials $\Psi$ and $\Phi$

Here we give the explicit expressions for the graph polynomials $\Phi$ and
$\Psi$ that are obtained after writing the double-penta integral
$\mathcal{T}_{16,23,45}$ in the Schwinger parameters representation (21). In
particular, these are given by

$\displaystyle\Psi$
$\displaystyle=\alpha_{1}\alpha_{2}+\alpha_{1}\alpha_{5}+\alpha_{1}\alpha_{6}+\alpha_{1}\alpha_{7}+\alpha_{1}\alpha_{8}+\alpha_{2}\alpha_{3}+\alpha_{2}\alpha_{4}+\alpha_{2}\alpha_{7}+\alpha_{2}\alpha_{9}+\alpha_{3}\alpha_{5}$
(55)
$\displaystyle+\alpha_{3}\alpha_{6}+\alpha_{3}\alpha_{7}+\alpha_{3}\alpha_{8}+\alpha_{4}\alpha_{5}+\alpha_{4}\alpha_{6}+\alpha_{4}\alpha_{7}+\alpha_{4}\alpha_{8}+\alpha_{5}\alpha_{7}+\alpha_{5}\alpha_{9}+\alpha_{6}\alpha_{7}$
$\displaystyle+\alpha_{6}\alpha_{9}+\alpha_{7}\alpha_{8}+\alpha_{7}\alpha_{9}+\alpha_{8}\alpha_{9}\,,$
$\displaystyle\Phi$
$\displaystyle=\alpha_{1}\alpha_{2}\alpha_{3}z_{2}^{2}+\alpha_{2}\alpha_{3}\alpha_{4}z_{2}^{2}+\alpha_{1}\alpha_{3}\alpha_{5}z_{2}^{2}+\alpha_{3}\alpha_{4}\alpha_{5}z_{2}^{2}+\alpha_{1}\alpha_{3}\alpha_{6}z_{2}^{2}+\alpha_{3}\alpha_{4}\alpha_{6}z_{2}^{2}$
(56)
$\displaystyle+\alpha_{1}\alpha_{3}\alpha_{7}z_{2}^{2}+\alpha_{2}\alpha_{3}\alpha_{7}z_{2}^{2}+\alpha_{3}\alpha_{4}\alpha_{7}z_{2}^{2}+\alpha_{3}\alpha_{5}\alpha_{7}z_{2}^{2}+\alpha_{3}\alpha_{6}\alpha_{7}z_{2}^{2}+\alpha_{1}\alpha_{2}\alpha_{8}z_{2}^{2}$
$\displaystyle+\alpha_{1}\alpha_{3}\alpha_{8}z_{2}^{2}+\alpha_{2}\alpha_{3}\alpha_{8}z_{2}^{2}+\alpha_{2}\alpha_{4}\alpha_{8}z_{2}^{2}+\alpha_{3}\alpha_{4}\alpha_{8}z_{2}^{2}+\alpha_{1}\alpha_{5}\alpha_{8}z_{2}^{2}+\alpha_{3}\alpha_{5}\alpha_{8}z_{2}^{2}$
$\displaystyle+\alpha_{4}\alpha_{5}\alpha_{8}z_{2}^{2}+\alpha_{1}\alpha_{6}\alpha_{8}z_{2}^{2}+\alpha_{3}\alpha_{6}\alpha_{8}z_{2}^{2}+\alpha_{4}\alpha_{6}\alpha_{8}z_{2}^{2}+\alpha_{1}\alpha_{7}\alpha_{8}z_{2}^{2}+\alpha_{2}\alpha_{7}\alpha_{8}z_{2}^{2}$
$\displaystyle+\alpha_{4}\alpha_{7}\alpha_{8}z_{2}^{2}+\alpha_{5}\alpha_{7}\alpha_{8}z_{2}^{2}+\alpha_{6}\alpha_{7}\alpha_{8}z_{2}^{2}+\alpha_{2}\alpha_{3}\alpha_{9}z_{2}^{2}+\alpha_{3}\alpha_{5}\alpha_{9}z_{2}^{2}+\alpha_{3}\alpha_{6}\alpha_{9}z_{2}^{2}$
$\displaystyle+\alpha_{3}\alpha_{7}\alpha_{9}z_{2}^{2}+\alpha_{2}\alpha_{8}\alpha_{9}z_{2}^{2}+\alpha_{3}\alpha_{8}\alpha_{9}z_{2}^{2}+\alpha_{5}\alpha_{8}\alpha_{9}z_{2}^{2}+\alpha_{6}\alpha_{8}\alpha_{9}z_{2}^{2}+\alpha_{7}\alpha_{8}\alpha_{9}z_{2}^{2}$
$\displaystyle-2\alpha_{2}\alpha_{3}\alpha_{4}z_{2}-2\alpha_{3}\alpha_{4}\alpha_{5}z_{2}-2\alpha_{3}\alpha_{4}\alpha_{6}z_{2}-2\alpha_{3}\alpha_{4}\alpha_{7}z_{2}-2z_{4}\alpha_{3}\alpha_{5}\alpha_{7}z_{2}$
$\displaystyle-2z_{5}\alpha_{3}\alpha_{6}\alpha_{7}z_{2}-2\alpha_{3}\alpha_{4}\alpha_{8}z_{2}-2z_{4}\alpha_{1}\alpha_{5}\alpha_{8}z_{2}-2z_{4}\alpha_{3}\alpha_{5}\alpha_{8}z_{2}-2z_{4}\alpha_{4}\alpha_{5}\alpha_{8}z_{2}$
$\displaystyle-2z_{5}\alpha_{1}\alpha_{6}\alpha_{8}z_{2}-2z_{5}\alpha_{3}\alpha_{6}\alpha_{8}z_{2}-2z_{5}\alpha_{4}\alpha_{6}\alpha_{8}z_{2}-2\alpha_{4}\alpha_{7}\alpha_{8}z_{2}-2z_{4}\alpha_{5}\alpha_{7}\alpha_{8}z_{2}$
$\displaystyle-2z_{5}\alpha_{6}\alpha_{7}\alpha_{8}z_{2}-2z_{4}\alpha_{2}\alpha_{3}\alpha_{9}z_{2}-2z_{4}\alpha_{3}\alpha_{5}\alpha_{9}z_{2}-2z_{4}\alpha_{3}\alpha_{6}\alpha_{9}z_{2}-2z_{4}\alpha_{3}\alpha_{7}\alpha_{9}z_{2}$
$\displaystyle-2z_{4}\alpha_{3}\alpha_{8}\alpha_{9}z_{2}-2z_{4}\alpha_{5}\alpha_{8}\alpha_{9}z_{2}-2z_{5}\alpha_{6}\alpha_{8}\alpha_{9}z_{2}-2z_{4}\alpha_{7}\alpha_{8}\alpha_{9}z_{2}+\alpha_{1}\alpha_{2}\alpha_{4}$
$\displaystyle+\alpha_{2}\alpha_{3}\alpha_{4}+z_{4}^{2}\alpha_{1}\alpha_{2}\alpha_{5}+z_{4}^{2}\alpha_{2}\alpha_{3}\alpha_{5}+\alpha_{1}\alpha_{4}\alpha_{5}+z_{4}^{2}\alpha_{2}\alpha_{4}\alpha_{5}+\alpha_{3}\alpha_{4}\alpha_{5}+z_{5}^{2}\alpha_{1}\alpha_{2}\alpha_{6}$
$\displaystyle+z_{5}^{2}\alpha_{2}\alpha_{3}\alpha_{6}+\alpha_{1}\alpha_{4}\alpha_{6}+z_{5}^{2}\alpha_{2}\alpha_{4}\alpha_{6}+\alpha_{3}\alpha_{4}\alpha_{6}+z_{4}^{2}\alpha_{1}\alpha_{5}\alpha_{6}+z_{5}^{2}\alpha_{1}\alpha_{5}\alpha_{6}$
$\displaystyle-2z_{4}z_{5}\alpha_{1}\alpha_{5}\alpha_{6}+z_{4}^{2}\alpha_{3}\alpha_{5}\alpha_{6}+z_{5}^{2}\alpha_{3}\alpha_{5}\alpha_{6}-2z_{4}z_{5}\alpha_{3}\alpha_{5}\alpha_{6}+z_{4}^{2}\alpha_{4}\alpha_{5}\alpha_{6}+z_{5}^{2}\alpha_{4}\alpha_{5}\alpha_{6}$
$\displaystyle-2z_{4}z_{5}\alpha_{4}\alpha_{5}\alpha_{6}+\alpha_{1}\alpha_{4}\alpha_{7}+\alpha_{2}\alpha_{4}\alpha_{7}+\alpha_{3}\alpha_{4}\alpha_{7}+z_{4}^{2}\alpha_{1}\alpha_{5}\alpha_{7}+z_{4}^{2}\alpha_{2}\alpha_{5}\alpha_{7}$
$\displaystyle+z_{4}^{2}\alpha_{3}\alpha_{5}\alpha_{7}+z_{4}^{2}\alpha_{4}\alpha_{5}\alpha_{7}-2z_{4}\alpha_{4}\alpha_{5}\alpha_{7}+\alpha_{4}\alpha_{5}\alpha_{7}+z_{5}^{2}\alpha_{1}\alpha_{6}\alpha_{7}+z_{5}^{2}\alpha_{2}\alpha_{6}\alpha_{7}$
$\displaystyle+z_{5}^{2}\alpha_{3}\alpha_{6}\alpha_{7}+z_{5}^{2}\alpha_{4}\alpha_{6}\alpha_{7}-2z_{5}\alpha_{4}\alpha_{6}\alpha_{7}+\alpha_{4}\alpha_{6}\alpha_{7}+z_{4}^{2}\alpha_{5}\alpha_{6}\alpha_{7}+z_{5}^{2}\alpha_{5}\alpha_{6}\alpha_{7}$
$\displaystyle-2z_{4}z_{5}\alpha_{5}\alpha_{6}\alpha_{7}+\alpha_{1}\alpha_{4}\alpha_{8}+\alpha_{3}\alpha_{4}\alpha_{8}+z_{4}^{2}\alpha_{1}\alpha_{5}\alpha_{8}+z_{4}^{2}\alpha_{3}\alpha_{5}\alpha_{8}+z_{4}^{2}\alpha_{4}\alpha_{5}\alpha_{8}$
$\displaystyle+z_{5}^{2}\alpha_{1}\alpha_{6}\alpha_{8}+z_{5}^{2}\alpha_{3}\alpha_{6}\alpha_{8}+z_{5}^{2}\alpha_{4}\alpha_{6}\alpha_{8}+\alpha_{4}\alpha_{7}\alpha_{8}+z_{4}^{2}\alpha_{5}\alpha_{7}\alpha_{8}+z_{5}^{2}\alpha_{6}\alpha_{7}\alpha_{8}$
$\displaystyle+z_{4}^{2}\alpha_{1}\alpha_{2}\alpha_{9}+z_{4}^{2}\alpha_{2}\alpha_{3}\alpha_{9}+z_{4}^{2}\alpha_{2}\alpha_{4}\alpha_{9}-2z_{4}\alpha_{2}\alpha_{4}\alpha_{9}+\alpha_{2}\alpha_{4}\alpha_{9}+z_{4}^{2}\alpha_{1}\alpha_{5}\alpha_{9}$
$\displaystyle+z_{4}^{2}\alpha_{2}\alpha_{5}\alpha_{9}+z_{4}^{2}\alpha_{3}\alpha_{5}\alpha_{9}+z_{4}^{2}\alpha_{4}\alpha_{5}\alpha_{9}-2z_{4}\alpha_{4}\alpha_{5}\alpha_{9}+\alpha_{4}\alpha_{5}\alpha_{9}+z_{4}^{2}\alpha_{1}\alpha_{6}\alpha_{9}$
$\displaystyle+z_{5}^{2}\alpha_{2}\alpha_{6}\alpha_{9}+z_{4}^{2}\alpha_{3}\alpha_{6}\alpha_{9}+z_{4}^{2}\alpha_{4}\alpha_{6}\alpha_{9}-2z_{4}\alpha_{4}\alpha_{6}\alpha_{9}+\alpha_{4}\alpha_{6}\alpha_{9}+z_{4}^{2}\alpha_{5}\alpha_{6}\alpha_{9}$
$\displaystyle+z_{5}^{2}\alpha_{5}\alpha_{6}\alpha_{9}-2z_{4}z_{5}\alpha_{5}\alpha_{6}\alpha_{9}+z_{4}^{2}\alpha_{1}\alpha_{7}\alpha_{9}+z_{4}^{2}\alpha_{2}\alpha_{7}\alpha_{9}+z_{4}^{2}\alpha_{3}\alpha_{7}\alpha_{9}+z_{4}^{2}\alpha_{4}\alpha_{7}\alpha_{9}$
$\displaystyle-2z_{4}\alpha_{4}\alpha_{7}\alpha_{9}+\alpha_{4}\alpha_{7}\alpha_{9}+z_{4}^{2}\alpha_{6}\alpha_{7}\alpha_{9}+z_{5}^{2}\alpha_{6}\alpha_{7}\alpha_{9}-2z_{4}z_{5}\alpha_{6}\alpha_{7}\alpha_{9}+z_{4}^{2}\alpha_{1}\alpha_{8}\alpha_{9}$
$\displaystyle+z_{4}^{2}\alpha_{3}\alpha_{8}\alpha_{9}+z_{4}^{2}\alpha_{4}\alpha_{8}\alpha_{9}-2z_{4}\alpha_{4}\alpha_{8}\alpha_{9}+\alpha_{4}\alpha_{8}\alpha_{9}+z_{4}^{2}\alpha_{5}\alpha_{8}\alpha_{9}+z_{5}^{2}\alpha_{6}\alpha_{8}\alpha_{9}+z_{4}^{2}\alpha_{7}\alpha_{8}\alpha_{9}\,.$

##  Appendix B Asymptotic expanded integrals

When evaluating the integral (52) in the light-cone limit
$x_{56}^{2}\rightarrow 0$ we claimed that we could make use of the equality
(37), even though our integral has more terms than what appeared in this
identity. Here we will briefly explain the basic idea of why such result can
be used for our case, as it merely involves making a generalization.

Let us then consider just the terms which involve $x_{57}$ and forget about
the ones with $x_{58}$, where the argument is the same. Suppose then that we
have more than one term of the kind appearing in our integral. For simplicity
let us consider we have two of these terms

$(k_{1}\cdot x_{57})^{a_{1}}\,(k_{2}\cdot x_{57})^{a_{2}}\,,$ (57)

where $k_{1}$, $k_{2}$ and $a_{1}$, $a_{2}$ are arbitrary vectors and
integers, respectively. Now, we introduce some parameter $q$ which allows us
to keep track of the coefficients, just as it happens when working with
generating functions. Moreover, we shall consider the sum over the exponents
just as we have in our case but with a small caveat, we shall introduce by
force specific coefficients which will allow us to use the following equality

$\sum_{a_{1},a_{2}=0}^{\infty}\binom{a_{1}+a_{2}}{a_{2}}(k_{1}\cdot
x_{57})^{a_{1}}(k_{2}\cdot
x_{57})^{a_{2}}q^{a_{2}}=\sum_{n=0}^{\infty}\big{(}(k_{1}+q\,k_{2})\cdot
x_{57}\big{)}^{n}\,.$ (58)

Next, we can consider the integral of this equation with the remaining
factors, such as the denominator and the terms with $x_{58}$, which will be
omitted for simplicity. However, since we have reduced the two terms with
$x_{57}$ to just one we can use the identity (37) on the right-hand side.
Schematically, this would give

$\sum_{n=0}^{\infty}c_{n,b}\,\big{(}(k_{1}+q\,k_{2})\cdot
x_{56}\big{)}^{n}\,\dots\,,$ (59)

where the dots refer to the other terms we are omitting and $b$ denotes the
power of the term with $x_{58}$, which was also integrated. In order to
compare with the initial expression, i.e. the left-hand side of (57), we use
the binomial expansion

$\sum_{n=0}^{\infty}\sum_{m=0}^{n}c_{n,b}\,\binom{n}{m}(k_{1}\cdot
x_{56})^{n-m}\,(k_{2}\cdot x_{56})^{m}\,q^{m}\,\dots\,,$ (60)

and we reorganize the sum by making the transformations $n\rightarrow
a_{1}+a_{2}$ and $m\rightarrow a_{2}$, obtaining therefore

$\sum_{a_{1},a_{2}=0}^{\infty}c_{a_{1}+a_{2},b}\,\binom{a_{1}+a_{2}}{a_{2}}(k_{1}\cdot
x_{56})^{a_{1}}(k_{2}\cdot x_{56})^{a_{2}}q^{a_{2}}\,\dots\,.$ (61)

This result implies that when we have more than one term of the kind $(k\cdot
x_{56})^{a}$, the outcome of the integration is similar to that of (37) but
where in the labels of the coefficients $c_{k_{1},k_{2}}$ (38) appear the sums
of the powers of these terms. Finally, we would just like to point out that
this reasoning generalizes to any number of these terms as well as for the
ones with $x_{58}$.

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