DESY-24-004

Two loop five point integrals: light, heavy and large spin correlators

Carlos Bercini${}^{\text{\tiny 1}}$, Bruno Fernandes${}^{\text{\tiny
2}}$,Vasco Gonçalves${}^{\text{\tiny 2}}$

${}^{\text{\tiny 1}}$ Deutsches Elektronen-Synchrotron DESY, Notkestr. 85,
22607 Hamburg, Germany
${}^{\text{\tiny 2}}$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 appearing in five point
correlation functions of protected operators of $\mathcal{N}=4$ Super Yang-
Mills in several kinematical regimes. Starting from the correlation function
of the lightest operators of the theory, we were able to extract structure
constants of up to two spinning operators for small and large values of
polarizations and spin. We conjectured an universal all loop behaviour for the
large spin small polarization structure constants and comment on the
subtleties of analytically continuing it from finite to large spin. We also
consider correlation functions of heavier operators that get factorized in the
more fundamental object called decagon. We fixed this object at two loops in
general kinematics and studied its physical properties under OPE and null
limits.

###### Contents

  1. 1 Introduction
  2. I Integrals
    1. 2 Conformal Integrals
      1. 2.1 Parametric integration
      2. 2.2 Asymptotic expansions
      3. 2.3 Logarithm terms
  3. II Correlators
    1. 3 Lightest correlator
    2. 4 Large weight correlator
    3. 5 Discussion
    4. A Master integral and form of the differential equation
    5. B Perturbative data
    6. C Examples of conformal integrals

###  1 Introduction

Correlation functions of local operators are one of the most natural
observables to consider in a conformal field theory. One of the most famous
and interesting conformal field theories is the four dimensional
$\mathcal{N}=4$ super Yang Mills that is dual, through the AdS/CFT
correspondence, to type IIB super string theory in $AdS_{5}\times S_{5}$.

An important class of such operators in this theory are the scalar single-
trace half-BPS operators. Even though supersymmetry prevents their two and
three point functions to receive quantum corrections [1], their higher point
correlation functions encode a plethora of information about interesting
physical observables of the theory via a web of dualities between correlation
functions, null polygon Wilson loops and gluon scattering amplitudes [2, 3, 4,
5, 6, 7]. Their correlation functions can be written as

$\displaystyle\langle\mathcal{O}_{k_{1}}\mathcal{O}_{k_{2}}\dots\mathcal{O}_{k_{n}}\rangle=\sum_{\ell=0}^{\infty}\lambda^{\ell}\mathcal{G}_{n}^{(\ell)}\,,\quad\quad{\rm{with}}\quad\quad\lambda=\frac{g_{YM}^{2}N_{c}}{4\pi^{2}}\,.$
(1)

where the half-BPS operators
$\mathcal{O}_{k_{i}}=\text{Tr}(Y_{i}\cdot\Phi(x_{i}))^{k_{i}}$ are defined
with the help of an auxiliary six-dimensional null vector $Y_{i}$ and $N_{c}$
is the number of colors of the SU$(N_{c})$ gauge group, which will always
taken to be large.

The first non-trivial higher point correlators are the four point functions,
which have been studied since the early days of AdS/CFT correspondence. Their
integrands have been computed perturbatively in the planar limit up to ten
loops in the t’Hooft coupling $\lambda$, see [8, 9, 10, 11, 12]. Meanwhile,
for higher point functions, beyond the one loop result of [13], very few cases
have been worked out [14, 15, 16, 17].

The central object we will consider in this paper is one of these exceptions:
the integrand of the five point correlation function of the lowest weight
protected operator, which has been recently computed up to three loops in
[18]. At two loops the five point correlator can be expressed in terms of
several new conformal integrals

$\displaystyle\mathcal{L}_{{\color[rgb]{0.92578125,0,0.04296875}1};{\color[rgb]{0,0.640625,0.95703125}2}{\color[rgb]{0.66796875,0.015625,0.83203125}3};{\color[rgb]{1,0.6015625,0}4}{\color[rgb]{0.17578125,0.87890625,0}5}}=\int\frac{d^{4}x_{7}d^{4}x_{8}}{x_{{\color[rgb]{0.92578125,0,0.04296875}1}7}^{2}x_{{\color[rgb]{0.92578125,0,0.04296875}1}8}^{2}x_{{\color[rgb]{0,0.640625,0.95703125}2}7}^{2}x_{{\color[rgb]{0.66796875,0.015625,0.83203125}3}7}^{2}x_{{\color[rgb]{1,0.6015625,0}4}8}^{2}x_{{\color[rgb]{0.17578125,0.87890625,0}5}8}^{2}x_{78}^{2}}$
(2)
$\displaystyle\mathcal{S}_{{\color[rgb]{0.92578125,0,0.04296875}1};{\color[rgb]{0,0.640625,0.95703125}2}{\color[rgb]{0.66796875,0.015625,0.83203125}3};{\color[rgb]{1,0.6015625,0}4}{\color[rgb]{0.17578125,0.87890625,0}5}}=\int\frac{d^{4}x_{7}\,d^{4}x_{8}x_{{\color[rgb]{0.92578125,0,0.04296875}1}7}^{2}}{x_{{\color[rgb]{0.92578125,0,0.04296875}1}8}^{2}x_{{\color[rgb]{0,0.640625,0.95703125}2}7}^{2}x_{{\color[rgb]{0.66796875,0.015625,0.83203125}3}7}^{2}x_{{\color[rgb]{0.66796875,0.015625,0.83203125}3}8}^{2}x_{{\color[rgb]{1,0.6015625,0}4}7}^{2}x_{{\color[rgb]{0.17578125,0.87890625,0}5}8}^{2}x_{{\color[rgb]{0.17578125,0.87890625,0}5}7}^{2}x_{78}^{2}}$
(3) $\displaystyle
D_{{\color[rgb]{0,0.640625,0.95703125}2}{\color[rgb]{1,0.6015625,0}4},{\color[rgb]{0.92578125,0,0.04296875}1}{\color[rgb]{0.66796875,0.015625,0.83203125}3}{\color[rgb]{0.17578125,0.87890625,0}5}}=\int\frac{d^{4}x_{7}d^{4}x_{8}x_{{\color[rgb]{0,0.640625,0.95703125}2}7}^{2}x_{{\color[rgb]{1,0.6015625,0}4}8}^{2}}{x_{{\color[rgb]{0.92578125,0,0.04296875}1}7}^{2}x_{{\color[rgb]{0.92578125,0,0.04296875}1}8}^{2}x_{{\color[rgb]{0,0.640625,0.95703125}2}8}^{2}x_{{\color[rgb]{0.66796875,0.015625,0.83203125}3}7}^{2}x_{{\color[rgb]{0.66796875,0.015625,0.83203125}3}8}^{2}x_{{\color[rgb]{1,0.6015625,0}4}7}^{2}x_{{\color[rgb]{0.17578125,0.87890625,0}5}7}^{2}x_{{\color[rgb]{0.17578125,0.87890625,0}5}8}^{2}x_{78}^{2}}\,.$
(4)

Contrary to their four point two loop counterparts, these integrals are not
known in closed form. In this paper we managed to compute these integrals for
particular kinematic configurations: the plane kinematics (all points in
plane), the light-cone limit (points becoming null separated) and the OPE
limit (points approaching each other). This allowed us to evaluate the five
point correlator and study its physical properties under these several
regimes.

During the last couple of years there has been increasing interest in
analysing the constraints coming from the conformal bootstrap equations for
more than four operators. These are equations that involve kinematical
functions known as conformal blocks and products of OPE coefficients [19, 20].
Solving these equations in full generality, just like evaluating conformal
integrals, is a hard business and one typically restricts to kinematical
limits and uses certain assumptions. A common central assumption is that OPE
coefficients of spinning operators are reasonably well behaved in the large
spin limit, being given by analytical functions that depend logarithmically on
the spins. For instance, in [21] using this assumption together with
consistency of the conformal bootstrap equations lead to fixing the form of a
five point correlation function when neighbouring distances are taken to be
null as well as fixing the universal behaviour of these large spin OPE
coefficients.

One possible way to probe the correctness of these assumptions is by going
through an explicit computation in a specific model. One of the main goals of
this paper is to do precisely that, for the five point correlator of
$20^{\prime}$ operators in $\mathcal{N}=4$ SYM. Using the results evaluated
here for the integrals (2,3,4) we can compute this correlator in several
kinematics and extract OPE coefficients of two spinning operators for
arbitrary values of spins and polarizations.

For finite values of spin, we can compare our results with [22, 23] where some
of these structure constants were computed before. For large values of spin we
can make contact with the analytical bootstrap results of [21]. However, we
observed an important feature that first presents itself at two loops: the
large spin limit of the finite spin result is different than the expected
large spin behaviour and consequently in contradiction with the assumptions
that are usually considered in large spin analysis [24, 25, 26, 27]. Due to
our analytical control over the correlator, we were able to clean this
apparent paradox.

Another interesting type of correlation functions are the so-called simplest
correlators, introduced in [28]. They consist of operators with large values
of $R$-charge ($k_{i}\gg 1$) that factorize the correlator as the square of a
simpler object. For four point functions this object is called octagon
($\mathbb{O}$) which was bootstrapped in [28] at all orders in the t’Hooft
coupling.

The five point incarnation of such a factorized object is the decagon
($\mathbb{D}$), which was computed via integrability at two loops in the plane
kinematics [16]. By demanding that this ansatz is consistent with the
Lagragian insertion method [8, 18] at two loops we were able to lift the
decagon away from the plane into general kinematics. This is one of our main
results and it is presented in (67).

Furthermore, since the decagon is given in terms of the conformal integrals
(2,3), we can now study integrated properties of this object. More precisely,
we checked that the decagon satisfies the same properties of the octagon:
Steinmann relations and boundary data constraints such as the stampedes [29,
30], the central difference between these objects being the basis of
functions, which is far more complicated in the five point function case.

An important application of our results is the computation of these two
correlators in the limit where the five points approach the cusps of a null
polygon. We can explicitly check that the leading divergences of these
correlators are governed by two different quantities [12, 31],
$\Gamma_{\text{cusp}}$ for the $20^{\prime}$ correlator and
$\Gamma_{\text{oct}}$ for the decagon, as presented in equations (61) and (82)
respectively. Furthermore, for the $20^{\prime}$ correlator we also found a
perfect agreement with the bootstrapped results of [21], hence a match with
the null pentagon Wilson loop.

This paper is divided into two main parts. In part I we evaluate the two loop
integrals (2,3,4) in the several kinematics limits aforementioned. In part II
we study the five point function of $20^{\prime}$ operators and the decagon in
these regimes.

##  Part I Integrals

This first part is technical and details how we computed the complicated
conformal integrals (2,3,4). Some aspects of our computation can be, in
principle, applied in more general cases, being either higher loops or higher
number of points. This entire part can be skipped by readers that are not
concerned with the integrals themselves but rather with the linear combination
of integrals known as correlators.

###  2 Conformal Integrals

Five point correlators are functions that depend on five cross-ratios

$u_{1}=\frac{x_{12}^{2}x_{35}^{2}}{x_{13}^{2}x_{25}^{2}}\,,\qquad
u_{i+1}=u_{i}\big{|}_{x_{j}\rightarrow x_{j+1}}\,.$ (5)

Ideally one would like to have these integrals expressed in terms of special
functions (for example, multiple polylogarithms [32, 33]) of the cross-ratios
but in practice this is a hard task. However, there are special configurations
where the expressions of these integrals simplify. One such example is when we
set all the five points to lie on a common plane, see [16, 34]. Another
example that is of interest to this paper is the limit where some points are
approaching the light-cone of others. The goal of this section is to explain
how we evaluate these integrals in these two kinematics. We will use two
different methods to compute them: the plane configuration will be evaluated
by rewriting the integrals using the Schwinger parametrization, while the
light-cone limit will be evaluated by dividing the integrals into simpler
regions, which are further simplified by identities following from integration
by parts identities.

Before diving into the computations, we establish some notation. All the two
loop conformal integrals considered in this paper are of the form

$\displaystyle
I=\int\frac{d^{d}x_{7}d^{d}x_{8}}{x_{78}^{2}\prod_{i=1}^{5}\prod_{j=7}^{8}(x_{ij}^{2})^{a_{ij}}}$
(6)

with the constants $a_{ij}$ being integers subject to the conformal constraint
$\sum_{i}a_{ij}=d-1$. Lastly, we recall the names that these integrals usually
undergo: $\mathcal{L}$ integrals are the double-box (2), $\mathcal{S}$
integrals are the penta-box (3) and $\mathcal{D}$ integrals are the double-
penta (4).

####  2.1 Parametric integration

Schwinger parametrization is based on the following simple identity

$\displaystyle\frac{1}{A^{n}}=\frac{1}{\Gamma(n)}\int_{0}^{\infty}d\alpha\alpha^{n-1}e^{-\alpha\,A}$
(7)

and can be used to derive a representation of the integrals

$\displaystyle
I(a_{ij})=\left(\prod_{i}\int_{0}^{\infty}\alpha_{i}^{a_{i}-1}d\alpha_{i}\right)\frac{\mathcal{U}^{\omega-\frac{d}{2}}}{\mathcal{F}^{\omega}}\delta(1-\sum_{i}\alpha_{i}),\
\ \ \omega=1+\sum_{i=1}^{5}\sum_{j=7}^{8}a_{ij}-d$ (8)

where $\mathcal{U},\mathcal{F}$ are graph polynomials (see [35] for more
details) and $i$ runs over the propapagators appearing in (6) while the
$a_{i}$ are their powers.

One of the advantages of this representation is that the space time
integration of the original integral is already done. Given that the integral
(6) is conformal it is useful to send one complicated external point to
infinity resulting in a simpler parametrization. A natural choice is the point
that appears the most times in the integration or the point that appears as a
numerator. When the integral has no numerator or just one numerator like (2)
and (3) this procedure can be easily done. However, for integrals with more
than one numerator like (4) one needs to change the expression (7) to be
valid111Note that in (7) negative $n$ values correspond to poles in the gamma
function. also for integrals with numerators [35].

After sending the complicated points to infinity the the integrals ${\cal L}$
and ${\cal S}$ are ready to be evaluated, provided that we treat carefully the
effects of sending this point to infinity. We were only able to do this by
choosing a configuration where the points $x_{i}$ lie on a common plane making
the integrals ${\cal L}$ and ${\cal S}$ become linear reducible [36]. In the
end the integration over the parametric variables $\alpha_{i}$ can be
accomplished with use of a computer package HyperInt developed in [37]. The
result is expressed in terms of multiple polylogarithms multiplied by rational
functions of cross-ratios, which we attach in an auxiliary Mathematica file.

The ${\cal L}$-type integrals have a factorized cross-ratio dependence
prefactor, for example

$\displaystyle\mathcal{L}_{5,12,34}$
$\displaystyle=\frac{1}{z\bar{y}-y\bar{z}}\Bigg{(}G(0;y)G(0;\bar{z})G(1;\bar{y})G(1-y;z)+G\left(\frac{y}{1-z};\bar{y}\right)G\left(1-y,1,0;z\right)+$
$\displaystyle
G\left(\frac{y-(1-z)\bar{y}}{y},\frac{y(1-\bar{y})z}{yz+\bar{y}(1-y-z)},1-\bar{y},1;\bar{z}\right)+\dots\Bigg{)}$
(9)

where the $\dots$ stand for other multiple polylogarithms recalled in appendix
C, which contain the following list of letters

$\displaystyle\\{y,z,\bar{y},\bar{z},1-y,1-z,1-\bar{y},1-\bar{z},y-\bar{y},1-y-z,1-\bar{y}-\bar{z},z-\bar{z},$
${}^{,}y-\bar{y}(1-z)-y\bar{z},z-\bar{z}(1-y)-\bar{y}z,y\bar{y}\bar{z}-\bar{y}\bar{z}(1-z)+yz(1-\bar{y}-\bar{z}\\}$
(10)

and the cross-ratios in the plane kinematics can be related to the original
$u_{i}$ of (5) via

$\displaystyle u_{1}=z\bar{z},u_{2}$
$\displaystyle=\frac{(1-y)(1-\bar{y})(z-1)(\bar{z}-1)}{(y+z-1)(\bar{y}+\bar{z}-1)},u_{3}=\frac{y\bar{y}}{(y+z-1)(\bar{y}+\bar{z}-1)},$
$\displaystyle
u_{4}=\frac{1}{(1-y)(1-\bar{y})},u_{5}=\frac{(y+z-1)(\bar{y}+\bar{z}-1)}{(1-y)(1-\bar{y})}\,.$
(11)

These $\mathcal{L}$ integrals are by far the simplest and their overall
prefactor can be independently computed by doing an analysis of the leading
singularity of the integral, see [38]. On the other hand, the ${\cal S}$
integrals do not have a common prefactor and one cannot do such a simple
analysis for a sanity check. Due to the presence of numerators, the ${\cal D}$
integrals are more difficult to compute using this method, however all of
these integrals can be computed using the other methods explained below.

The plane kinematics is useful to obtain explicit expressions for the two loop
five point conformal integrals, but it is not enough to study some interesting
physical limits such as the OPE or sequential light-like limits. However, it
is possible to obtain corrections around plane kinematics by noticing that
both ${\cal L}$ and ${\cal S}$ integrals satisfy differential equations coming
from the four-dimensional identity

$\displaystyle\nabla_{i}\frac{1}{x_{i7}^{2}}=-4\pi^{2}\delta(x_{i7}).$ (12)

We can only apply this simple relation to integrals where the point $x_{i}$
appears once in the propagator, there are four of these equations for each
${\cal L}$ integral, two for each $S$ integral and none for the $\mathcal{D}$
integrals. As explained in [16] (see also [39]), the differential equation can
be expressed in terms of the cross-ratios (11) plus a fifth cross-ratio,
$\eta$, that controls the out of the plane kinematics, defined by

$\displaystyle
u_{2}=\frac{(1-y)(1-\bar{y})(z-1)(\bar{z}-1)}{(y+z-1)(\bar{y}+\bar{z}-1)}(1+\eta).$
(13)

It is easy to see that when $\eta=0$ one recovers the plane kinematics (11).
The differential equation, when written in these variables, has some terms
that interlink the plane and out-of-plane kinematics [16], which can be used
to generate a series expansion in $\eta$ for the integrals.

Thus, one can start with the plane expression for the integrals and, in
principle, use the differential equations to generate the full dependence in
the cross-ratios222Apart from the integrals $\mathcal{D}$, which we could not
easily compute nor lift out of the plane. In practice this method is very
inefficient and we managed to compute only the first few orders in the $\eta$
expansion. Nonetheless this gave us very non-trivial expressions for the
integrals, which were used to verify the correctness of our computations when
using the different methods described below.

####  2.2 Asymptotic expansions

Instead of sending all external points to the plane, we will compute the
integrals in a kinematic configuration that makes contact with more physical
regimes of the correlator: the single light-cone limit, where points $12$ and
$34$ become null separated of each other. We will review the method of
asymptotic expansions and explain how it can be applied efficiently to obtain
our two loop integrals in the limit $x_{12}^{2},x_{34}^{2}\rightarrow 0$, or
in term of cross-ratios when $u_{1},u_{3}\to 0$. Given that this method has
been applied extensively in integrals similar to the ones we considered here,
we will just give a lightning review and refer to following references for
more details [40, 41, 42, 43, 18].

Whenever we have a parameter, like these null distances, which is sent to zero
we can apply the asymptotic expansion method. The main idea behind the method
is to divide each integration variable into two regions: one where the
integration variable is small enough to be comparable to the small parameter
and other where it is large enough such that the parameter can be ignored. We
will take the limits sequentially, i.e. first we take $x_{12}^{2}\rightarrow
0$ and only then we take $x_{34}^{2}\rightarrow 0$. In general, for an $l$
loop integral there are $2^{l}$ regions when we take a distance to be light-
like, in particular, for the two loop case we have the following four regions

$1.\,\,x_{7}\ll 1\,,x_{8}\ll 1\,;\quad 2.\,\,x_{7}\ll 1\,,x_{8}\gg 1\,;\quad
3.\,\,x_{7}\gg 1\,,x_{8}\ll 1\,;\quad 4.\,\,x_{7}\gg 1\,,x_{8}\gg 1\,.$ (14)

The advantage of doing this separation is that we are allowed to simplify the
integrand accordingly in each region by using the identities

$\displaystyle\frac{1}{(x_{ij}^{2})^{b}}=\frac{1}{(x_{1j}^{2})^{b}}\sum_{n_{b}=0}^{\infty}{{-b}\choose{n_{b}}}\frac{(x_{1i}^{2}-2x_{1i}\cdot
x_{1j})^{n_{b}}}{(x_{1j}^{2})^{n_{b}}}$ (15)

where we assumed $x_{j}$ to be large and $x_{i}$ to be small. Consider for
example, the integral $\mathcal{L}_{5,13,24}$ and let us apply the
prescription above

$\displaystyle\mathcal{L}_{5,13,24}=\int\frac{d^{4}x_{7}d^{4}x_{8}}{x_{17}^{2}x_{37}^{2}x_{28}^{2}x_{48}^{2}x_{78}^{2}}=I_{1}+I_{2}+I_{3}+I_{4},$
(16) $\displaystyle I_{1}=\sum_{n,m=0}^{\infty}$
$\displaystyle\int\frac{d^{d}x_{7}d^{d}x_{8}\,(2x_{13}\cdot
x_{37}-x_{17}^{2})^{n}(2x_{14}\cdot
x_{48}-x_{18}^{2})^{m}}{(x_{13}^{2})^{1+n}(x_{14}^{2})^{1+m}x_{17}^{2}x_{18}^{2}x_{28}^{2}x_{78}^{2}},$
$\displaystyle I_{3}=\sum_{n,m=0}^{\infty}$
$\displaystyle\int\frac{d^{d}x_{7}d^{d}x_{8}\,(2x_{14}\cdot
x_{48}-x_{18}^{2})^{m}\,(2x_{17}\cdot
x_{18}-x_{18}^{2})^{n}}{(x_{14}^{2})^{1+m}x_{17}^{2}x_{18}^{2}x_{28}^{2}x_{37}^{2}(x_{17}^{2})^{1+n}},\,$
(17) $\displaystyle
I_{4}=\sum_{n=0}^{\infty}\int\frac{d^{d}x_{7}d^{d}x_{8}(2x_{12}\cdot
x_{18}-x_{12}^{2})^{n}}{x_{17}^{2}(x_{18}^{2})^{2+n}x_{37}^{2}x_{48}^{2}x_{78}^{2}}.$

where we assumed $x_{12}^{2}\rightarrow 0$ and have taken $x_{5}$ to infinity.
Let us note we only have written explicitly $I_{3}$ and not $I_{2}$ because
the latter does not contribute - it evaluates to zero, since it is
scaleless[44]. One important point is that this method allows to
divide/regulate the dependence on the null distance $x_{12}^{2}$. For example,
region 4 is effectively the original integral evaluated exactly at
$x_{12}^{2}=0$, while the remaining regions are the ones that can contain a
non-trivial dependence in $x_{12}^{2}$ and in particular can generate
divergent terms such as $\ln x_{12}^{2}$.

It should also be evident from the integrals above that each $I_{i}$ is
simpler than the original integral. The idea is that now we can sytematically
identify the dependence on the vanishing distance and drop it. Moreover, one
can apply the asymptotic expansion method again to each individual region, but
this time considering that the distance $x_{34}^{2}$ is becoming null. This
will give rise to several simpler regions that we were able to evaluate. In
some cases it was not even necessary to consider the $x_{34}^{2}\to 0$ regime
since the integrals became simple enough to be computed, as is the case of
$I_{1}$ above.

As mentioned before, the ${\cal D}$ integrals cannot be easily computed using
the method of parametric integration. For this reason, we will devote the
remainder of this section to this particular type of integral. This should
also serve as a paradigmatic example that features all the details of our
computation and that can be applied to all other simpler integrals. Our
specific example will be

${\cal
D}_{24,135}=\int\frac{d^{d}x_{7}d^{d}x_{8}x_{27}^{2}x_{48}^{2}}{x_{17}^{2}x_{18}^{2}x_{28}^{2}x_{37}^{2}x_{38}^{2}x_{47}^{2}x_{57}^{2}x_{58}^{2}x_{78}^{2}}\,,$
(18)

when considering the null limits $x_{12}^{2},x_{34}^{2}\to 0$, only three non-
trivial regions contribute at leading order in the null distances, they are

$\displaystyle\qquad\quad\quad{\cal D}_{24,135}=I_{3}+I_{4,2}+I_{4,4},$ (19)
$\displaystyle I_{3}=\sum_{n,m=0}^{\infty}$
$\displaystyle\int\frac{d^{d}x_{7}d^{d}x_{8}\,(x_{17}^{2}-2x_{12}\cdot
x_{17})(x_{14}^{2}-2x_{14}\cdot x_{18})(2x_{13}\cdot x_{18})^{n}(2x_{17}\cdot
x_{18})^{m}}{(x_{13}^{2})^{1+n}x_{18}^{2}x_{28}^{2}(x_{17}^{2})^{2+m}x_{37}^{2}x_{47}^{2}}$
$\displaystyle I_{4,2}=\sum_{n,m=0}^{\infty}\int$
$\displaystyle\left.\frac{d^{d}x_{7}d^{d}x_{8}\,(x_{23}^{2}-2x_{32}\cdot
x_{37})\,(x_{38}^{2}-2x_{38}\cdot x_{34})(2x_{13}\cdot
x_{37})^{n}(2x_{38}\cdot
x_{37})^{m}}{(x_{13}^{2})^{1+n}x_{18}^{2}\,x_{37}^{2}\,x_{28}^{2}\,x_{47}^{2}\,(x_{38}^{2})^{2+m}}\right|_{x_{12}^{2}=0}$
$\displaystyle\qquad\quad
I_{4,4}=\int\left.\frac{d^{d}x_{7}d^{d}x_{8}x_{27}^{2}x_{48}^{2}}{x_{17}^{2}x_{18}^{2}x_{28}^{2}x_{37}^{2}x_{38}^{2}x_{47}^{2}x_{78}^{2}}\right|_{x_{12}^{2}=x_{34}^{2}=0}.$

where the index indicates from which region they originated (14). For example,
$I_{4,2}$ comes from the region $4$ of the asymptotic expansion of
$x_{12}^{2}\to 0$ and region $2$ of the asymptotic expansion of $x_{34}^{2}\to
0$ while $I_{3}$ comes from the region $3$ of the $x_{12}^{2}\to 0$ expansion,
and can already be evaluated without the need to take any further limit in
$x_{34}^{2}$.

The first two regions have a similar structure and can be simplified using the
same strategy. Let’s focus on the first one, where the integration over the
point $x_{8}$ can be done in $d=4-\epsilon$ regularization333The conformal
integrals do not depend on the regulator $\epsilon$ and checking that this
factor cancels after summing all the regions is a non trivial sanity check one
always performs.

$\displaystyle\int\frac{d^{d}x_{8}(x_{13}\cdot x_{18})^{n}\,(x_{17}\cdot
x_{78})^{m}}{x_{18}^{2}x_{28}^{2}}=\frac{(x_{12}\cdot x_{13})^{n}(x_{17}\cdot
x_{12})^{m}}{\epsilon(x_{12}^{2})^{\epsilon}}\frac{(1-\epsilon)_{m+n}}{(2-2\epsilon)_{m+n}}+\dots$
(20)

where the $\dots$ represent subleading terms in $x_{12}^{2}$. The integration
over $x_{7}$ can also be simplified by the following identity

$\displaystyle\int\frac{d^{d}x_{7}\,(2x_{17}\cdot
x_{12})^{n}}{(x_{17}^{2})^{1+n}x_{37}^{2}x_{47}^{2}}=\frac{1}{n!}\left(z\cdot\frac{d}{dx_{1}}\right)^{n}\int\frac{d^{d}x_{7}}{x_{17}^{2}x_{37}^{2}x_{47}^{2}}\bigg{|}_{z=x_{12}}.$
(21)

Given that (21) is a convergent integral one needs to series expand (20) up to
subleading order in $\epsilon$

$\displaystyle\frac{(1-\epsilon)_{n}}{(2-2\epsilon)_{n}}=\frac{1}{1+n}+\frac{\epsilon}{(1+n)^{2}}\left((1+n)S_{1+n}-(2n+1)\right).$
(22)

By combining (20), (21) and (22) in the region integral $I_{3}$, one can
integrate each term of the sum in $n$ and $m$ and write this integral as some
power law expansion in the cross-ratios. However there is a way to rewrite
this integral using the identity

$\displaystyle\int_{0}^{1}\,dt\,t^{n+a_{1}\epsilon}(1-t)^{\epsilon}=\frac{1+n-\epsilon(1+a_{1}+(1+n)S_{n})}{(1+n)^{2}}+O(\epsilon^{2}).$
(23)

The advantage of introducing this integral is that now the sum over $n_{38}$
and $n_{78}$ can be performed effortlessly through the integral representation
form of the one loop three point integral

$\displaystyle\int\frac{d^{d}x_{7}}{x_{17}^{2}x_{37}^{2}x_{47}^{2}}=\frac{1}{(x_{13}^{2})^{3-\frac{d}{2}}}\bigg{[}f_{0}\left(\frac{x_{34}^{2}}{x_{13}^{2}},\frac{x_{14}^{2}}{x_{13}^{2}}\right)+\epsilon\,f_{1}\left(\frac{x_{34}^{2}}{x_{13}^{2}},\frac{x_{14}^{2}}{x_{13}^{2}}\right)\bigg{]}\,,$
(24)

where $f_{0}$ and $f_{1}$ are simple combination of polylogarithms recalled in
equation (84) of appendix A. Then each term in the series expansion in
$\epsilon$ that appears in the summand can be recast in an integral
representation. For example, the most complicated one, which involves the
harmonic number $S_{n}$, can be written as

$\displaystyle I_{3}$
$\displaystyle=x_{14}^{2}\sum_{n,m}\frac{S_{1+n+m}(2x_{12}\cdot
x_{13})^{n}}{(1+n+m)m!}\left(-z\cdot\frac{d}{dx_{1}}\right)^{m}\int\frac{d^{d}x_{7}}{x_{17}^{2}x_{37}^{2}x_{47}^{2}}\bigg{|}_{z=x_{12}}+\dots$
(25) $\displaystyle=$ $\displaystyle-\int
dt\frac{\ln(1-t)f_{0}\left(\frac{u_{3}u_{5}}{u_{4}(1-t(1-u_{2}u_{5}))},\frac{t(1-u_{5})}{u_{4}(1-t(1-u_{2}u_{5}))}\right)}{x_{13}^{2}(1-t(1-u_{2}u_{5}))^{2}}+\dots$
(26)

where the $\dots$ represent the other terms in the $\epsilon$ expansion that
can be handled similarly, in the end we obtain a simple integral
representation for the region $I_{3}$, which we refrain from writing here due
to its size.

The region $I_{4,2}$ can be done in a similar manner, minding some small
differences. Since the one loop three point integral over $x_{8}$ is being
evaluated exactly on the light-cone $x_{12}^{2}=0$, there is an extra
divergence in $\epsilon$ which, in turn, requires us to expand (22) to the
next higher order in $\epsilon$. This by no means spoils our strategy since we
can still represent the extra terms of the $\epsilon$ expansion as an integral
representation of the type (23).

The last region, $I_{4,4}$, computes the original integral with the
constraints $x_{12}^{2}=x_{34}^{2}=0$. This effectively makes it simpler to
deal with than the original integral, but this is still by far the hardest
region to compute. There is simple no way to compute it using the same
strategy we described above. Therefore, to proceed we used integration-by-
parts identities, also known as IBP reduction [45, 46]. These identities allow
us to rewrite any Feynman integral as a linear combination of a finite set of
integrals, which are usually called master integrals. The number of master
integrals depends crucially on the number of loops, the number of external
points, and the kinematics. While the original integral ${\cal D}_{24,135}$
can be expressed in terms of $74$ master integrals, the one from region
$I_{4,4}$ is given by a linear combination of only $28$ master integrals

$\displaystyle I_{4,4}=\sum_{i=1}^{28}\alpha_{i}g_{i}$ (27)

where $\alpha_{i}$ are known rational functions of the dimension $d$ and the
three cross ratios $u_{2},u_{4},u_{5}$ and $g_{i}$ are the master integrals,
written explicitly in appendix A. The problem, then, is to evaluate these $28$
functions.

The key point is that this system is closed: all the derivatives with respect
to the remaining three cross ratios of each one of these $28$ master integrals
can also be expressed as a linear combination (with rational coefficients) of
the same $28$ integrals, forming then a system of first order differential
equations that we need to solve, which we also attach in the auxiliary
Mathematica file.

We were able to compute $16$ master integrals as explicit combination of
multiple polylogarithms of the cross-ratios $u_{2},u_{4},u_{5}$. However, for
the remaining $12$ we could not solve the differential equations while keeping
arbitrary values of the cross-ratios. Instead, we considered $u_{4}$ and
$u_{5}$ generic, while $u_{2}$ is expanded around one. This is not a random
point to expand around, it is related to the OPE limit of correlators, which
we treat carefully in the next section.

The most non-trivial part of solving this system of equations is to find the
solution for general $u_{4}$ and $u_{5}$ with $u_{2}=1$. With the dependence
on two cross-ratios worked out, it is trivial to expand the differential
equations as powers in the remaining cross-ratios $u_{2}$, and solve for each
one of the master integrals $g_{i}$ systematically. For example, we did this
up to order $\mathcal{O}((1-u_{2})^{10})$ which will allow us to access plenty
of perturbative data in the correlator analysis.

Finally, we can sum each one of the regions we computed and reconstruct the
wanted integral ${\cal D}_{24,135}$. In order to evaluate all the integrals
that appear in the correlator, one follows the same recipe: take the integral
and break it into regions. Some regions are simple and can be computed like
$I_{3}$, while others are hard are expressed in terms of master integrals like
$I_{4,4}$. Plugging in the computed $g_{i}$ computes the hard part and summing
this result with the simple part yields the expression for the original
integral. These results are in the attached Mathematica file.

After using such complex methods to compute these integrals it is good to
perform some checks. Each one of the regions as well as the master integrals
have $\epsilon$ divergences coming from dimension regularization, but once we
combine then in a conformal integral all the $\epsilon$ divergences must
cancel. This was essential for us to debug several steps of our computations.
In the end, when possible, we also do a sanity check by comparing the results
with the plane expressions from parametric integrations.

####  2.3 Logarithm terms

While the two loop integrals we are considering may be complicated, the terms
that diverge with the distances that are taken to null are easier to compute.
This divergent terms come only from the simple part of the integrals, which we
can explicitly integrate (after some tricks) while power law corrections come
from the hard part which are given by a combination of master integrals. If we
are interested only in the leading behaviour of the correlator, which is often
the case, we can neglect this hard part and focus only in the more treatable
simple part of the integrals.

The integrands of $\mathcal{L}$, $\mathcal{S}$ and ${\cal D}$ are rational
functions of $x_{ij}^{2}$. Since there are two integrations over internal
points and we take the limit $x_{12}^{2},x_{34}^{2}\to 0$, the resulting
integral can have at most a quadratic divergence in logarithms. Therefore, in
this regime a generic two loop conformal integral $I$ can be expressed as

$I=c_{20}\ln^{2}(u_{1})+c_{11}\ln(u_{1})\ln(u_{3})+c_{02}\ln^{2}(u_{3})+c_{10}\ln(u_{1})+c_{01}\ln(u_{3})+c_{00}$
(28)

where the $c_{ij}$ are rational functions of $u_{2},u_{4},u_{5}$. In general,
the expression for the integral has uniform transcendentality, which at this
loop order is four. Hence, the functions multiplying logs will have lower
transcendentality (for example, $c_{20}$ has transcendentality two) and thus
are easier to evaluate.

In order to compute these coefficients, we turn to the method of asymptotic
expansions detailed earlier. As an example, let’s evaluate the coefficient
$c_{20}$ of the following integral

$\displaystyle\mathcal{L}_{1;23;45}=$
$\displaystyle\int\frac{d^{4}x_{7}d^{4}x_{8}}{x_{17}^{2}x_{18}^{2}x_{27}^{2}x_{37}^{2}x_{48}^{2}x_{78}^{2}x_{58}^{2}},,$
(29)

in the limit $x_{12}^{2}\to 0$, this integral can be expressed as the sum of
the following regions

$\displaystyle I_{1}=$
$\displaystyle\sum_{m_{i}=0}^{\infty}\frac{2^{m_{1}+m_{2}}}{\left(x_{14}^{2}\right)^{m_{2}+1}\left(x_{13}^{2}\right)^{m_{1}+1}}\int\frac{d^{d}x_{6}d^{d}x_{7}\left(x_{13}\cdot
x_{16}\right)^{m_{1}}\left(x_{14}\cdot
x_{17}\right)^{m_{2}}}{x_{16}^{2}x_{17}^{2}x_{26}^{2}x_{67}^{2}}$ (30)
$\displaystyle I_{2}=$
$\displaystyle\sum_{m_{i}=0}^{\infty}\frac{2^{m_{1}+m_{2}}}{\left(x_{13}^{2}\right)^{m_{1}+1}}\int\frac{d^{d}x_{6}d^{d}x_{7}\left(x_{13}\cdot
x_{16}\right)^{m_{1}}\left(x_{16}\cdot
x_{17}\right)^{m_{2}}}{x_{16}^{2}\left(x_{17}^{2}\right)^{m_{2}+2}x_{26}^{2}x_{47}^{2}},$
(31)

where region 3 integrates to zero and region 4 can be neglected since it does
not generate $\ln{x^{2}_{12}}$ terms. If we wanted to compute the coefficient
of other logarithm such as $c_{02}$, we could not immediately neglect these
other regions. One must again perform the asymptotic expansion method of
taking $x^{2}_{34}\to 0$ and consider the subregions of those integrals that
could generate logarithm divergences. After performing this analysis on our
example, it turns out that this particular integral has no $\ln(x_{34}^{2})$
dependence and therefore it has coefficients $c_{02}=c_{11}=c_{01}=0$. Thus
all log dependence is encoded in the regions $I_{1}$ and $I_{2}$ written
above, which can be evaluated using equation (30) of [41], yielding

$\displaystyle I_{1}=$
$\displaystyle\sum_{m_{i}=0}^{\infty}\frac{\left(2x_{13}\cdot
x_{12}\right)^{m_{1}}\left(2x_{14}\cdot
x_{12}\right)^{m_{2}}H(1,1,m_{2},0)H(3-d/2,1,m_{1}+m_{2},0)}{\left(x_{14}^{2}\right)^{m_{2}+1}\left(x_{13}^{2}\right)^{m_{1}+1}(x_{12}^{2})^{4-d}}$
(32) $\displaystyle I_{2}=$
$\displaystyle\sum_{m_{i}=0}^{\infty}\frac{\left(2x_{13}\cdot
x_{12}\right)^{m_{1}}\left(2x_{12}\cdot
x_{14}\right)^{m_{2}}H(1,1,m_{1}+m_{2},0)H(m_{2}+2,1,m_{2},0)}{(x_{12}^{2})^{2-d/2}\left(x_{13}^{2}\right)^{m_{1}+1}\left(x_{14}^{2}\right)^{m_{2}+3-d/2}}$
(33)

where the functions $H$ are given by a ratio of gamma functions

$H(\alpha,\beta,s,i)=\frac{\Gamma(d-2)\Gamma\left(d/2+i-\beta\right)\Gamma\left(\alpha+\beta+i-d/2\right)\Gamma\left(d/2-i+s-\alpha\right)}{(2-d/2)\Gamma(\alpha)\Gamma(\beta)\Gamma\left(2-d/2\right)\Gamma\left(d/2-1\right)^{2}\Gamma(d+s-\alpha-\beta)}.$
(34)

Expanding around $\epsilon=0$, it is easy to isolate the logarithm
contributions, for instance the $\ln^{2}(x_{12}^{2})$ dependence of each
region is given by

$\displaystyle I_{1}\sim$
$\displaystyle\sum_{m_{i}=0}^{\infty}\frac{\left(2x_{13}\cdot
x_{12}\right)^{m_{1}}\left(2x_{14}\cdot
x_{12}\right)^{m_{2}}}{(1+m_{2})(1+m_{1}+m_{2})\left(x_{14}^{2}\right)^{m_{2}+1}\left(x_{13}^{2}\right)^{m_{1}+1}}\ln^{2}(x_{12}^{2}),$
(35) $\displaystyle I_{2}\sim$
$\displaystyle\sum_{m_{i}=0}^{\infty}\frac{-\left(2x_{13}\cdot
x_{12}\right)^{m_{1}}\left(2x_{12}\cdot
x_{14}\right)^{m_{2}}}{2(1+m_{2})(1+m_{1}+m_{2})\left(x_{13}^{2}\right)^{m_{1}+1}\left(x_{14}^{2}\right)^{m_{2}+1}}\ln^{2}(x_{12}^{2}),$
(36)

combining the contribution of these two integrals allows us to write the
coefficient $c_{20}$ as the following sum

$c_{20}=\sum_{m_{i}=0}^{\infty}\frac{\left(2x_{13}\cdot
x_{12}\right)^{m_{1}}\left(2x_{14}\cdot
x_{12}\right)^{m_{2}}}{2(1+m_{2})(1+m_{1}+m_{2})\left(x_{14}^{2}\right)^{m_{2}+1}\left(x_{13}^{2}\right)^{m_{1}+1}}\,.$
(37)

By making use of the identity $(1+a)^{-1}=\int_{0}^{1}t^{a}dt$ in the
expression of $c_{20}$, one can recast this coefficient as a simple integral
representation

$c_{20}=\int_{0}^{1}\int_{0}^{1}\frac{dt_{1}dt_{2}}{2(x_{13}^{2}-2t_{2}x_{13}\cdot
x_{12})(x_{14}^{2}-2t_{1}t_{2}x_{14}\cdot x_{12})}$ (38)

which can be explicitly integrated

$\displaystyle c_{20}=$
$\displaystyle\frac{u_{4}}{4(1-u_{5})x_{13}^{2}}(2\,\text{Li}_{2}\left(u_{2}\right)-2\,\text{Li}_{2}\left(u_{2}u_{5}\right)+\ln\left(u_{5}\right){}^{2}+2\ln\left(u_{2}\right)\ln\left(u_{5}\right)+$
(39) $\displaystyle-2$
$\displaystyle\ln\left(1-u_{2}u_{5}\right)\ln\left(u_{5}\right)+2\ln\left(1-u_{2}\right)\ln\left(u_{2}\right)-2\ln\left(u_{2}\right)\ln\left(1-u_{2}u_{5}\right))$
(40)

where we can see the appearance of the expected transcendentality two
functions. It is then easy to take further limits of this expressions, such as
more null limits or OPE limits to study the correlator, as we discuss below.

The other coefficients, such as $c_{10}$ can be calculated in a similar
fashion, with the difference being that the summand will also contain harmonic
numbers coming from the expansion of the gamma functions. We can, however, use
(23) to write the sums as integral representations and perform the integrals
explicitly obtaining similar expressions as (40). Using this method, we
managed to extract the logarithm coefficients of all two loop integrals, which
are also in the attached Mathematica file.

##  Part II Correlators

After evaluating the relevant conformal integrals we can focus to the study of
the five point correlation functions. All the cases we will study involve
correlators of identical operators, which can be written as

$G_{5}(u_{1},\dots,u_{5})=(x_{12}^{2}x_{34}^{2})^{\Delta}\left(\frac{x_{15}^{2}x_{35}^{2}}{x_{13}^{2}}\right)^{\frac{\Delta}{2}}\langle\mathcal{O}_{\Delta}(x_{1})\dots\mathcal{O}_{\Delta}(x_{5})\rangle\,.$
(41)

where $\Delta$ is their scale dimension and $u_{i}$ are cross-ratios defined
in (5). We will focus on two particular choices of charge for the external
operators in $\mathcal{N}=4$ SYM: first when the external operators are the
lightest ($\Delta=2$) and second when the external operators are heavy
($\Delta\to\infty$), as depicted in figure 1. Physically, these two
correlators are very different. The later forms a thick perimeter of
propagators that effectively decouples space into two regions: inside and
outside [28]. Thus, the correlator is given by the square of a more
fundamental object, the decagon, see [16, 15]. Meanwhile the light correlator
sits in the opposite spectrum of correlators and in perturbation theory
maximally couples these inside and outside regions.

Figure 1: On the left the correlation function of five weight two operators
($20^{\prime}$). On the right the correlation function of large weight
operators, the decagon.

The next two sections deal with these very different objects. In section 3 we
study the lightest correlator under several kinematical limits that are
governed by small and large spin dynamics. In both cases, we compare our
results with the literature, the small spins being compared with the
perturbative computations [22] and the large spin being matched with the
constraints coming from analytic conformal bootstrap [21]. In section 4 we
analyse the decagon, by checking its physical properties such as Steinmann
relations, null limit constrains like stampedes [30] and the plane kinematics
results [16].

###  3 Lightest correlator

Any five point correlation function in a conformal field theory can be
decomposed into a product of kinematical and dynamical terms

$G_{5}(u_{1},\dots,u_{5})=\sum_{J_{1},J_{2},\ell}P_{J_{1}J_{2}\ell}\,\mathcal{F}(u_{1},\dots,u_{5})\,,$
(42)

where $\mathcal{F}(u_{i})$ is the kinematical function known as the conformal
block which is fully fixed by conformal symmetry, and $P_{J_{1}J_{2}\ell}$ is
the OPE data that encodes all the theory dependent information of the theory.

Figure 2: Two kinematical limits of the correlator. Black lines represent
general kinematics while solid coloured lines represent Lorenzian OPE (points
become null separated) and dashed lines represent Euclidean OPE (points
approach each other). The dotted lines on top represent the OPE channels we
chose to decompose the correlator while the colors indicate the physical
implication of the limits for the operators being exchanged. We arrive in both
limits by first considering the single light-cone OPE displayed on the top.

Even though it is completely fixed, the five point conformal block for general
kinematics is a complicated object not known in closed form, [14, 19, 20]. In
order to study these higher point functions one usually turns to kinematical
limits of the blocks where several simplifications occur. Here will be no
different. We will start by considering the light-cone OPE between points 12
and points 34, see top of figure 2 (in terms of cross-ratios $u_{1},u_{3}\to
0$). As explained in [21], this limit projects into leading twist operators
being exchanged in the OPE channels and allows to write the conformal block as
an integral representation

$\displaystyle\mathcal{F}(u_{i})$
$\displaystyle=\frac{\Gamma\left(2J_{1}+\tau_{1}\right)\Gamma\left(2J_{2}+\tau_{2}\right)}{\Gamma\left(J_{1}+\frac{\tau_{1}}{2}\right)^{2}\Gamma\left(J_{2}+\frac{\tau_{2}}{2}\right)^{2}}u_{1}^{\frac{\tau_{1}}{2}}u_{3}^{\frac{\tau_{2}}{2}}u_{4}^{\frac{\tau_{1}-\Delta}{2}+\ell}u_{5}^{\frac{\tau_{2}}{2}+\ell}(1-u_{2})^{\ell}\times$
(43)
$\displaystyle\times\int_{0}^{1}\int_{0}^{1}dt_{1}dt_{2}(t_{1}(1-t_{1}))^{J_{1}-1+\frac{\tau_{1}}{2}}(t_{2}(1-t_{2}))^{J_{2}-1+\frac{\tau_{2}}{2}}\times$
$\displaystyle\times\frac{(1-u_{4}-t_{1}(1-u_{4}-u_{5}+u_{2}u_{4}u_{5}))^{J_{2}-\ell}(1-u_{5}-t_{2}(1-u_{4}-u_{5}+u_{2}u_{4}u_{5}))^{J_{1}-\ell}}{(1-t_{1}(1-u_{5})-t_{2}(1-u_{4})+t_{1}t_{2}(1-u_{4}-u_{5}+u_{2}u_{4}u_{5}))^{J_{1}+J_{2}-\frac{\Delta-\tau_{1}-\tau_{2}}{2}}}$

where $\\{J_{1},\tau_{1}\\}$ and $\\{J_{2},\tau_{2}\\}$ parametrize the spin
and the twist of the operators being exchanged in the channels 12 and 34,
respectively.

With the conformal block defined we can turn to the dynamical object appearing
in the OPE decomposition, $P_{J_{1}J_{2}\ell}$. As can be seen in figure 2,
this quantity is given by a combination of structure constants of one and two
spinning operators

$P_{J_{1}J_{2}\ell}=C^{\bullet\circ\circ}_{J_{1}}C^{\bullet\circ\circ}_{J_{2}}C^{\bullet\bullet\circ}_{J_{1},J_{2},\ell}$
(44)

where the polarization $\ell$ accounts for the several tensor structures [47]
that appear in the three point function of two spinning operators recalled in
appendix B.

From now on we will consider the case where the external operators are the
lightest operators ($\Delta=2$), the ubiquitous $20^{\prime}$ operators of
$\mathcal{N}=4$ SYM. The five point function integrand of such operators at
two loops was already worked out in [18] and now we can study its integrated
properties. The plan is very simple, compare the integrated correlator with
the block expansion in several kinematic limits and extract physical
information about the structure constants of two spinning operators.

#### Finite spin

After taking the single light-cone limit we consider the Euclidean OPE between
the remaining points of the five point correlator, as depicted in the right of
figure 2. This projects into small spin and small polarizations being
exchanged in the operator product expansion. For the cross-ratios, this limit
amounts to start with $u_{1},u_{3}\to 0$ and then take the remaining $u_{i}\to
1$. In the conformal blocks side, one can easily expand the integrand of (43)
and trivially perform the integration in the auxiliary variables $t_{i}$. On
the other hand, with the conformal integrals now evaluated we can also compute
the integrated correlator at two loops. By matching these two expressions we
can just read off the structure constants (44).

Although perfect valid, the method described above is inefficient to extract
huge amounts of data. What in turn is used is the Casimir extraction method,
described in Appendix E of [25]. In a nutshell, this method consist of
iteratively apply the quadratic Casimir on the correlator and use the fact
that the conformal blocks are eigenfunctions of this differential operator to
systematically read off the structure constants of the correlator.

We were able to extract more than a thousand new structure constants between
two spinning operators. For polarizations $\ell=0,\dots,6$ we extracted
data444For $\ell=0$ the integrals are simpler and we can extract a lot of
data, we went up to $J=50$ but it is easier to generate much more. To
illustrate this, in the file we also attach data for $J_{1}=2$ and
$J_{2}=0,\dots,200$. for two spinning operators up to spin $J=36$. At this
point the two loop data is give by the rational numbers built from integers
with almost 100 digits. This data is attached in the supplemental Mathematica
file, and examples are

$\displaystyle\hat{C}^{\bullet\bullet\circ}_{8,12,3}$
$\displaystyle=1-\lambda\left(\frac{48563133305437}{2650343295600}\right)+\lambda^{2}\left(\frac{359608758568933328334171885613}{1376766638568246319954560000}+\frac{1526}{165}\zeta_{3}\right)$
$\displaystyle\hat{C}^{\bullet\bullet\circ}_{10,10,4}$
$\displaystyle=1-\lambda\left(\frac{793855867937}{49997493000}\right)+\lambda^{2}\left(\frac{286981761743641772075609}{1333199630018692800000}\right)\,,$

where $\hat{C}$ is the tree-level normalized structure constant recalled in
appendix B.

Despite the abundant data, we were not able to guess a closed expression for
the structure constant of two spinning operators at two loops. However, we
observed that the coefficient of $\zeta_{3}$ in the data is independent of the
polarization and follows a simple pattern

$\hat{C}^{\bullet\bullet\circ}_{J_{1},J_{2},\ell}=\dots+24\zeta_{3}\lambda^{2}|S_{1}(J_{1})-S_{1}(J_{2})|+\dots$
(45)

which was previously observed in an independent computation by Bianchi in [22]
and suggests the following naive large spin limit

$\hat{C}^{\bullet\bullet\circ}_{J_{1},J_{2},\ell}\xrightarrow{\text{naive}}\dots+24\zeta_{3}\lambda^{2}|\ln(J_{1})-\ln(J_{2})|+\dots$
(46)

which is drastically different than the large spin and polarization expression
bootstrapped in [21]

$\hat{C}^{\bullet\bullet\circ}_{J_{1},J_{2},\ell}\xrightarrow{\text{correct}}\mathcal{N}e^{-\frac{f(\lambda)}{4}(\ln^{2}\ell+\ln
4\ln(J_{1},J_{2}))-\frac{g(\lambda)}{2}\ln\ell}\,.$ (47)

The correct expression is analytic in spins and polarizations, while the naive
expression depends non-analytically as the modulus of the spins. Below we
clean this apparent paradox by tracking how the structure constants go from
small to large values of spin and understanding how such non-analytic
contributions get washed way in the large spin limit.

#### Large spin

Here we consider the limit depicted on the left of figure 2, which we will
denote by open box limit. After taking the sigle light-cone limit we then send
the distances between the points 15 and 45 to be null, thus projecting into
large spin operators being exchanged. Lastly, we make points 23 approach each
other projecting further into small polarizations, see [21]. In terms of
cross-ratios the open box limits consists in taking $u_{1},u_{3}\to 0$, then
$u_{4},u_{5}\to 0$ and finally $u_{2}\to 1$.

More precisely, in this limit the spins are going to infinity and the cross-
ratios are vanishing, but the quantities $x=2J_{2}\sqrt{u_{4}}$ and
$y=2J_{1}\sqrt{u_{5}}$ are held fixed. The block in this limit was computed in
[21], and it reads:

$\displaystyle\mathcal{F}(u_{1},\dots,u_{5})$
$\displaystyle=2^{2(2+J_{1}+J_{2})+\gamma_{1}+\gamma+2}u_{1}^{\tau_{1}/2}u_{3}^{\tau_{2}/2}(1-u_{2})^{\ell}u_{4}^{\frac{2\ell+\gamma_{1}}{4}}u_{5}^{\frac{4+2\ell+\gamma_{2}}{4}}\times$
$\displaystyle\times
K_{\ell+\frac{\gamma_{1}}{2}}\left(2J_{2}\sqrt{u_{4}}\right)K_{\ell+\frac{\gamma_{2}}{2}}\left(2J_{1}\sqrt{u_{5}}\right)\,.$
(48)

It is then trivial to write the five point correlation function (42) as

$\displaystyle G_{5}(u_{1},\dots,u_{5})$
$\displaystyle=\left(\frac{u_{1}u_{3}}{u_{4}}\right)\sum_{\ell=0}^{\infty}\int_{0}^{\infty}dxdy\left(\frac{u_{1}^{\frac{\gamma_{1}}{2}}u_{3}^{\frac{\gamma_{2}}{2}}u_{4}^{\frac{\gamma_{1}}{4}}u_{5}^{\frac{\gamma_{2}}{4}}}{2^{2\ell-\gamma_{1}-\gamma_{2}}}\frac{(1-u_{2})^{\ell}}{\Gamma(1+\ell)^{2}}\right)\times$
$\displaystyle\times\hat{P}\left(\frac{x}{2\sqrt{u_{4}}},\frac{y}{2\sqrt{u_{5}}},\ell\right)x^{1+\ell}K_{\ell+\frac{\gamma_{1}}{2}}(x)y^{1+\ell}K_{\ell+\frac{\gamma_{2}}{2}}(y)$
(49)

where we traded sum over spins by integrals and factored out the tree-level
value of the large spin structure constants

$P_{J_{1}J_{2}\ell}\to\frac{2^{-2J_{1}-2J_{2}}\pi
J_{1}^{\ell+\frac{1}{2}}J_{2}^{\ell+\frac{1}{2}}}{\Gamma(\ell+1)^{2}}\hat{P}(J_{1},J_{2},\ell)$
(50)

where the quantity $\hat{P}$ is the tree-level normalized combination of
structure constants that encodes all the perturbative corrections to the OPE
data, and $\gamma_{i}$ is the anomalous dimension of the twist two operators
at large spin [48],

$\gamma(J)=f(\lambda)\ln{J}+g(\lambda)$ (51)

where $f(\lambda)$ and $g(\lambda)$ are the cusp and collinear anomalous
dimensions that govern the large spin dynamics.

Following the same ideas of [24, 21] we assume that in the large spin limit
the structure constants have the form

$\hat{P}(J_{1},J_{2},\ell)=\sum_{a,b,c}p_{\ell}(a,b,c)\lambda^{a}\ln^{b}{J_{1}}\ln^{c}{J_{2}}$
(52)

where $p_{\ell}(a,b,c)$ are unknown coefficients. Using the conformal
integrals computed here, we can evaluate the correlator at two loops in the
open box limit and compare with the block expansion. Doing that fixes these
coefficients and yields the following expression for the large spin, small
polarization structure constants of two spinning operators

$C^{\bullet\bullet\circ}_{J_{1},J_{2},\ell}=\mathcal{N}_{\ell}e^{-\frac{\ln{2}}{2}f(\lambda)\ln{J_{1}J_{2}}}$
(53)

of course the expression above can only be trusted at two loops, but it is
tempting to conjecture the all loop expression above since our expressions
have the same spin dependence as the large spin and large polarizations
results derived in [21] and presented in (47). The only difference is that the
large polarization information gets washed away into the normalization
$\mathcal{N}_{\ell}$. The bootstrap arguments of [21] could not fix the spin-
independent normalization. However, our direct comparison allows us to
explicitly compute it to be

$\mathcal{N}_{\ell}=\left(2^{-g(\lambda)}e^{-2\gamma_{E}g(\lambda)+\gamma_{E}^{2}f(\lambda)-8\zeta_{2}\lambda+96\zeta_{4}\lambda^{2}}\right)\times\left(e^{-2(S_{1}(\ell)^{2}-S_{2}(\ell))\lambda+\lambda^{2}n_{\ell}}\right)$
(54)

The normalization is given by the product of two terms. The first is
polarization independent while the second depends on the polarization $\ell$.
At one loop we were able to express this dependence as some simple harmonic
numbers, but at two loops we could not find an analogous expression555Where
$\gamma_{E}$ is the Euler’s constant and the $n_{\ell}$ are a rational
functions that we can explicit compute but could not find a pattern for:
$n_{0}=0$, $n_{1}=22$, $n_{2}=303/8$, $n_{3}=32639/648$ $\dots$.. Opposed to
the structure constants, we could also not write the normalization factor in a
way that hints a lift to an all loop expression. However, at one loop where we
do have an analytic expression in terms of polarizations we can do a nice
sanity check: by further taking the limit $\ell\to\infty$ in (53) we can
recover the bootstraped result of [21] for large spins and polarizations.

At this point we turn to the apparent contradiction between the correct (53)
and the naive (45) expressions for the large spin finite polarization
structure constants

$\displaystyle\hat{C}^{\bullet\bullet\circ}_{J_{1},J_{2},\ell}$
$\displaystyle\xrightarrow{\text{naive}}\dots+24\zeta_{3}\lambda^{2}|\ln(J_{1})-\ln(J_{2})|+\dots$
(55) $\displaystyle\hat{C}^{\bullet\bullet\circ}_{J_{1},J_{2},\ell}$
$\displaystyle\xrightarrow{\text{correct}}\dots+24\zeta_{3}\lambda^{2}(2\gamma_{E}-\ln{2})+\dots$
(56)

Since this absolute value feature is shared among all finite polarizations,
let’s focus in the simplest case of $\ell=0$, where it is sufficient to
consider the correlator at $u_{2}=1$. In this regime, the two loop correlator
can be written as the sum of two terms

$G_{5}=1+\lambda(\dots)+\lambda^{2}\left(\zeta_{3}\frac{u_{5}(1+u_{5})(1-u_{4})^{3}\ln{u_{4}}-u_{4}(1+u_{4})(1-u_{5})^{3}\ln{u_{5}}}{(u_{4}-u_{5})(1-u_{4}u_{5})}+R\right)$
(57)

where the first term has an explicit $\zeta_{3}$ dependence and the second
term is a complicated combination of multiple polylogarithms, which in
principle have no $\zeta_{3}$ dependence

$R=-3(9+u_{4})(1+u_{5})\text{Li}_{1}(1-u_{4})\text{Li}_{3}(1-u_{4})+12(1-u_{4})(1+u_{5})\text{Li}_{1}(1-u_{5})\text{Li}_{3}(1-u_{4})+\dots$
(58)

and we ignore the one loop part for our analysis since it has no $\zeta_{3}$
and it has no problems whatsoever when analytic continuing to large spins.

When we OPE expand the expression above, namely when we expand $u_{4}$ and
$u_{5}$ around one, the polylogarithms in the $R$-term indeed never generate a
factor with $\zeta_{3}$. All the $\zeta_{3}$ dependence of the structure
constant comes from the first term of (57). Moreover, by comparing with the
block expansion one can explicitly check that the OPE expansion of this term
indeed generates the observed $\zeta_{3}|S_{1}(J_{1})-S_{1}(J_{2})|$ behaviour
of the data.

Now let’s consider what happens at large spin, or in terms of cross-ratios
when we take $u_{4}\to 0$ and $u_{5}\to 0$. First, due to its denominator, the
first term in (57) has an order of limit issue, which can be understood as the
large spin incarnation of the non-analytical absolute value. However, in the
limit of the cross-ratios going to zero the multiple polylogarithms in the $R$
part do give a $\zeta_{3}$ dependence, for example the two terms written in
(58) give

$R=27\zeta_{3}\ln{u_{4}}-12\zeta_{3}\ln{u_{5}}+\dots$ (59)

combining all the new $\zeta_{3}$ dependence that arises in the null limit of
the $R$ term with the starting $\zeta_{3}$ dependence of the two loop
correlator precisely cancels the order of limits issue and yields a simple
open-box limit of the correlator

$G_{5}=-12\zeta_{3}(\ln{u_{1}}+\ln{u_{3}}+\ln{u_{4}}+\ln{u_{5}})$ (60)

which in turn gets translated to the simple large spin behaviour of the
structure constants observed in (53).

This simple two loop analysis of spinning OPE coefficients proved to be
crucial to clarify a potential problem in the assumptions that are usually
done in analytical conformal bootstrap studies [24, 21, 25]. Obviously, it
would be even better to put these assumptions on a firmer footing by deriving
a generalization of the, by now well established, Lorentzian inversion formula
[49]. On the other hand it also gives motivation to delve into the extremely
non-trivial case of two loop six point functions which we leave to the future.

Another interesting kinematical limit of the correlator is the null polygon
limit, where we take all the cross-ratios to zero. Following the procedure of
2.3 we were able to evaluate the log divergences of the conformal integrals,
and consequently compute the leading behaviour of the correlator, it reads

$\ln{G_{5}}=\mathcal{C}_{0}+\sum_{i=1}^{5}(-2\lambda+4\lambda^{2}\zeta_{2})\ln{u_{i}}\ln{u_{i+1}}+4\zeta_{2}\lambda^{2}(\ln{u_{i}}\ln{u_{i+2}}+\ln^{2}{u_{i}})-12\zeta_{3}\lambda^{2}\ln{u_{i}}$
(61)

being the constant
$\mathcal{C}_{0}=-8\zeta_{2}\lambda+126\zeta_{4}\lambda^{2}$. This expression
for the null correlator is the same as the two loop expansion of the
bootstrapped result presented in (17) of [21], up to the constant
$\mathcal{C}_{0}$, which could not be fixed by bootstrap arguments.

In the null polygon limit all the cross-ratios are going to zero, while both
spins and polarizations are going to infinity. Under these circumstances the
conformal block exponentiates [21] and we can write the five point correlator
in a very simple form

$\displaystyle G_{5}(u_{i})$
$\displaystyle=\frac{u_{1}u_{3}}{u_{2}u_{4}u_{5}}\int
dJ_{1}dJ_{2}d\ell\,\frac{u_{1}^{\frac{\gamma_{1}}{2}}u_{3}^{\frac{\gamma_{2}}{2}}}{2^{-2-\gamma_{1}-\gamma_{2}}}\frac{J_{1}^{1+\frac{\gamma_{2}}{2}}J_{2}^{1+\frac{\gamma_{1}}{2}}}{\ell^{2-\frac{\gamma_{1}}{2}-\frac{\gamma_{2}}{2}}}\hat{P}(J_{1},J_{2},\ell)\,e^{-\ell
u_{2}-\frac{J_{2}^{2}}{\ell}u_{4}-\frac{J_{1}^{2}}{\ell}u_{5}}\,,$ (62)

in order to study the structure constants for large spin and large
polarizations we again start by making an ansatz for the expression of these
OPE coefficients

$\hat{P}(J_{1},J_{2},\ell)=\sum_{a,b,c}p(a,b,c,d)\lambda^{a}\ln^{b}{J_{1}}\ln^{c}{J_{2}}\ln^{d}{\ell}\,,$
(63)

and by comparing the block expansion (62) with the null correlator (61) we can
fix all these $p$-coefficients. In the end we recover the bootstrapped result
of [21] for the large spin and polarization structure constants presented in
(53) being the only difference that now we have access to the overall
normalization of the structure constants

$\mathcal{N}=2^{-g(\lambda)}e^{-8\zeta_{2}\lambda+101\zeta_{4}\lambda^{2}+\frac{5\gamma_{E}}{4}(\gamma_{E}f(\lambda)-2g(\lambda))}$
(64)

which is correct up to two loops, but we also were not able to write in a way
that hints a lift to an all loop expression.

###  4 Large weight correlator

Now we shift gears and consider correlators built by large charge operators.
We start by picking specific R-charge polarizations for the operators in the
five point correlator and define the decagon as the large charge limit
($k\to\infty$) of the following correlation function

$\displaystyle\langle\textrm{tr}(X^{2k})(x_{1})\textrm{tr}(\bar{X}^{k}\bar{Y}^{k})(x_{2})\textrm{tr}(\bar{Z}^{k}Y^{k})(x_{3})\textrm{tr}(Z^{2k})(x_{4})\textrm{tr}(\bar{Z}^{k}\bar{X}^{k})(x_{5})\rangle=\frac{\mathbb{D}^{2}(u_{i})}{(x_{12}^{2}x_{23}^{2}x_{34}^{2}x_{45}^{2}x_{15}^{2})^{k}}$
(65)

where $X,Y,Z$ and their bar counterparts are the six scalar operators of
$\mathcal{N}=4$ SYM, see figure 3. This is the five point generalization of
the famous simplest four point correlation function

$\displaystyle\langle\textrm{tr}(\bar{Z}^{k}\bar{X}^{k})(x_{1})\textrm{tr}(X^{2k})(x_{2})\textrm{tr}(\bar{Z}^{k}\bar{X}^{k})(x_{3})\textrm{tr}(Z^{2k})(x_{4})\rangle=\frac{\mathbb{O}^{2}(z,\bar{z})}{(x_{12}^{2}x_{23}^{2}x_{34}^{2}x_{14}^{2})}\,,$
(66)

introduced in [28] and bootstraped by demanding the following three axioms

  * •

The basis of functions are built from four point ladder integrals.

  * •

There is log truncation in the double trace OPE channel (Steinmann relations).

  * •

Constraints from the null square limit.

The decagon, on the other hand, cannot be bootstrapped at arbitrary loop
orders since the very first step used in the octagon fails: at high loop
orders the basis of functions that govern the decagon is unknown, making it
impossible to systematically study the constrains coming from OPE and null
limits. However, at two loops we have all the ingredients to study the
decagon. A simple ansatz inspired by the ideas of [18] together with result of
the plane [16] is enough to obtain the integrand of the decagon666By cyclic
permutations we mean taking $u_{i}\to u_{i+1}$ and shift the indices of the
integrals by one unity in a cyclic way, as presented in (71) and (72). for
general kinematics

$\displaystyle\mathbb{D}=1-\lambda\left(1+u_{1}u_{4}-u_{5}\right)\Phi^{(1)}\left(u_{1}u_{4},u_{5}\right)+2\lambda^{2}\left(2\left(1+u_{1}u_{4}-u_{5}\right)\Phi^{(2)}\left(u_{1}u_{4},u_{5}\right)+\right.$
$\displaystyle\left.+\left(u_{1}u_{4}+u_{2}u_{5}-u_{1}-u_{5}\right)\mathbb{L}_{1,23,45}+\left(1-u_{3}+u_{2}u_{4}\right)\mathbb{S}_{1,25,34}\right)+\textrm{cyclic}+O(\lambda^{6})$
(67)

being and $\Phi^{(L)}$ the well-known ladder integrals [50] reminded in
appendix C and the other conformal integrals are recalled below

$\displaystyle\mathbb{L}_{{\color[rgb]{0.92578125,0,0.04296875}1};{\color[rgb]{0,0.640625,0.95703125}2}{\color[rgb]{0.66796875,0.015625,0.83203125}3};{\color[rgb]{1,0.6015625,0}4}{\color[rgb]{0.17578125,0.87890625,0}5}}$
$\displaystyle=x_{{\color[rgb]{0.92578125,0,0.04296875}1}{\color[rgb]{0.66796875,0.015625,0.83203125}3}}^{2}x_{{\color[rgb]{0.92578125,0,0.04296875}1}{\color[rgb]{1,0.6015625,0}4}}^{2}x_{{\color[rgb]{0,0.640625,0.95703125}2}{\color[rgb]{0.17578125,0.87890625,0}5}}^{2}\int\frac{d^{4}x_{7}d^{4}x_{8}}{x_{{\color[rgb]{0.92578125,0,0.04296875}1}7}^{2}x_{{\color[rgb]{0.92578125,0,0.04296875}1}8}^{2}x_{{\color[rgb]{0,0.640625,0.95703125}2}7}^{2}x_{{\color[rgb]{0.66796875,0.015625,0.83203125}3}7}^{2}x_{{\color[rgb]{1,0.6015625,0}4}8}^{2}x_{{\color[rgb]{0.17578125,0.87890625,0}5}8}^{2}x_{78}^{2}}$
(68)
$\displaystyle\mathbb{S}_{{\color[rgb]{0.92578125,0,0.04296875}1},{\color[rgb]{0,0.640625,0.95703125}2}{\color[rgb]{0.17578125,0.87890625,0}5},{\color[rgb]{0.66796875,0.015625,0.83203125}3}{\color[rgb]{1,0.6015625,0}4}}$
$\displaystyle=x_{{\color[rgb]{0,0.640625,0.95703125}2}{\color[rgb]{1,0.6015625,0}4}}^{2}x_{{\color[rgb]{0,0.640625,0.95703125}2}{\color[rgb]{0.17578125,0.87890625,0}5}}^{2}x_{{\color[rgb]{0.66796875,0.015625,0.83203125}3}{\color[rgb]{0.17578125,0.87890625,0}5}}^{2}\int\frac{d^{4}x_{7}\,d^{4}x_{8}x_{{\color[rgb]{0.92578125,0,0.04296875}1}7}^{2}}{x_{{\color[rgb]{0.92578125,0,0.04296875}1}8}^{2}x_{{\color[rgb]{0,0.640625,0.95703125}2}7}^{2}x_{{\color[rgb]{0.17578125,0.87890625,0}5}7}^{2}x_{{\color[rgb]{0.17578125,0.87890625,0}5}8}^{2}x_{{\color[rgb]{0.66796875,0.015625,0.83203125}3}7}^{2}x_{{\color[rgb]{1,0.6015625,0}4}8}^{2}x_{{\color[rgb]{1,0.6015625,0}4}7}^{2}x_{78}^{2}}\,.$
(69)

We will take advantage of the integrals that were computed in the first part
of the paper to verify that the decagon (67) satisfies the last two bootstrap
axioms listed above. After confirming that the axioms hold, we can also invert
the logic and ask: how much of the decagon one can fix by demanding the
bootstrap constrains? This is what we turn to next.

Figure 3:  On the left the Decagon with canonical ordering and on the right
the Decagon with non-canonical ordering of the operators. The red lines
represent $X$ propagators, the blue lines $Y$ propagators and the black lines
$Z$ propagators.

#### Steinmann

The second axiom states that the octagon, when expanded in the double trace
OPE, is linear in logs of the cross ratios that are approaching zero. This
property can be equivalently recast in terms of Steinmann relations that
appear frequently in scattering amplitudes. This property is easier to express
in terms of dual momenta variables $p_{i}\equiv x_{i}-x_{i+1}$, meanwhile the
usual Mandelstam invariants $s_{ij}=(p_{i}+p_{j})^{2}$ can be written in terms
of $x_{i}$ as

$\displaystyle s_{12}=x_{13}^{2},\ \ \ s_{23}=x_{24}^{2},\ \ \
s_{34}=x_{35}^{2},\ \ \ s_{45}=x_{14}^{2},\ \ \ s_{15}=x_{25}^{2}$ (70)

Steinmann relation is the statement that the double discontinuity in
overlapping channels is zero. For example, the action of
$\textrm{disc}_{s_{12}}\textrm{disc}_{s_{23}}$ on the decagon should vanish.
Equivalently, it states that there should be no $\ln x_{13}^{2}\ln x_{24}^{2}$
in the decagon777Note that the limit $x_{13}^{2},x_{24}^{2}\rightarrow 0$ can
be mapped, under permutation, to the limit $x_{12}^{2},x_{34}^{2}\rightarrow
0$ discussed in the previous sections or equivalently we could have used non-
conventional labelling of the poitns depicted on the right of figure 3 where
Steinmann would translate into the absence of $\ln u_{1}\ln u_{3}$ in the
decagon in the limit $u_{1},u_{3}\rightarrow 0$. in the limit
$x_{13}^{2},x_{24}^{2}\rightarrow 0$.

Besides the four point ladder integrals which are already known to satisfy
Steinmann relations, the only five point integrals that satisfy the Steinmann
relation independently are

$\displaystyle\mathbb{L}_{{\color[rgb]{0.92578125,0,0.04296875}1},{\color[rgb]{0,0.640625,0.95703125}2}{\color[rgb]{0.66796875,0.015625,0.83203125}3},{\color[rgb]{1,0.6015625,0}4}{\color[rgb]{0.17578125,0.87890625,0}5}},\
\ \
\mathbb{L}_{{\color[rgb]{0,0.640625,0.95703125}2},{\color[rgb]{0.92578125,0,0.04296875}1}{\color[rgb]{0.17578125,0.87890625,0}5},{\color[rgb]{0.66796875,0.015625,0.83203125}3}{\color[rgb]{1,0.6015625,0}4}},\
\ \
\mathbb{L}_{{\color[rgb]{0.66796875,0.015625,0.83203125}3},{\color[rgb]{0.92578125,0,0.04296875}1}{\color[rgb]{0,0.640625,0.95703125}2},{\color[rgb]{1,0.6015625,0}4}{\color[rgb]{0.17578125,0.87890625,0}5}},\
\ \
\mathbb{L}_{{\color[rgb]{1,0.6015625,0}4},{\color[rgb]{0.92578125,0,0.04296875}1}{\color[rgb]{0.17578125,0.87890625,0}5},{\color[rgb]{0,0.640625,0.95703125}2}{\color[rgb]{0.66796875,0.015625,0.83203125}3}},\
\ \
\mathbb{L}_{{\color[rgb]{0.17578125,0.87890625,0}5},{\color[rgb]{0.92578125,0,0.04296875}1}{\color[rgb]{0,0.640625,0.95703125}2},{\color[rgb]{0.66796875,0.015625,0.83203125}3}{\color[rgb]{1,0.6015625,0}4}}$
(71)
$\displaystyle\mathbb{S}_{{\color[rgb]{0.92578125,0,0.04296875}1},{\color[rgb]{0,0.640625,0.95703125}2}{\color[rgb]{0.17578125,0.87890625,0}5},{\color[rgb]{0.66796875,0.015625,0.83203125}3}{\color[rgb]{1,0.6015625,0}4}},\
\ \
\mathbb{S}_{{\color[rgb]{0,0.640625,0.95703125}2},{\color[rgb]{0.92578125,0,0.04296875}1}{\color[rgb]{0.66796875,0.015625,0.83203125}3},{\color[rgb]{1,0.6015625,0}4}{\color[rgb]{0.17578125,0.87890625,0}5}},\
\ \ \
\mathbb{S}_{{\color[rgb]{0.66796875,0.015625,0.83203125}3},{\color[rgb]{0,0.640625,0.95703125}2}{\color[rgb]{1,0.6015625,0}4},{\color[rgb]{0.92578125,0,0.04296875}1}{\color[rgb]{0.17578125,0.87890625,0}5}},\
\ \
\mathbb{S}_{{\color[rgb]{1,0.6015625,0}4},{\color[rgb]{0.66796875,0.015625,0.83203125}3}{\color[rgb]{0.17578125,0.87890625,0}5},{\color[rgb]{0.92578125,0,0.04296875}1}{\color[rgb]{0,0.640625,0.95703125}2}},\
\ \ \
\mathbb{S}_{{\color[rgb]{0.17578125,0.87890625,0}5},{\color[rgb]{0.92578125,0,0.04296875}1}{\color[rgb]{1,0.6015625,0}4},{\color[rgb]{0,0.640625,0.95703125}2}{\color[rgb]{0.66796875,0.015625,0.83203125}3}}\,,$
(72)

which are precisely the only integrals that appear in the plane computation of
[16]. One could imagine that some strange combination of integrals with
rational prefactors could, in principle, satisfy Steinmann relations. At
leading order in the OPE expansion this is indeed the case, but by requiring
absence of double discontinuities at every order in the OPE expansion makes it
so that no such combination exists888We can explicitly check this in the case
of the two loop five point integrals here considered. Therefore, only the
self-satisfying Steinmann integrals of (71) and (72) can contribute.

This two loop fulfillment of Steinmann relations also suggests that they might
be satisfied at all loops for the decagon, just as is the case for the
octagon. It would be interesting to use this property to severely reduce the
possible integrals that can contribute to the decagon at higher loops.
Although difficult, this task could be treatable, since these log terms of the
integrals are precisely the easiest ones to evaluate using the asymptotic
expansion method of section 2.3.

#### Stampedes

The third axiom for the octagon can be recast in the so-called stampedes
constraints [29], which were later extended for other polygons such as the
decagon [30]. The stampedes limit is the combination of the null pentagon
limit $u_{i}\rightarrow 0$ with perturbation theory, $\lambda\to 0$, where the
cusp times are held fixed

$t_{k}^{2}=-\lambda\log{u_{k-1}}\log{u_{k}}\,.$ (73)

When these special polygons (octagon, decagon, etc..) do not have internal
diagonals, the stampedes limit predicts an exponentiation. For instance, the
decagon is conjectured to be

$\displaystyle\mathbb{D}=e^{-(t_{1}^{2}+t_{2}^{2}+t_{3}^{2}+t_{4}^{2}+t_{5}^{2})}\,.$
(74)

The leading log behavior can be easily computed via the asymptotic expansions
methods of section 2.3, and gives the following expression for the stampedes
limits of the integrals

$\displaystyle\lambda\Phi^{(1)}(u_{1}u_{4},u_{5})$
$\displaystyle=-(t_{1}^{2}+t_{5}^{2})$ (75)
$\displaystyle\lambda^{2}\Phi^{(2)}(u_{1}u_{4},u_{5})$
$\displaystyle=\frac{(t_{1}^{2}+t_{5}^{2})^{2}}{4}$ (76)
$\displaystyle\lambda^{2}\mathbb{L}_{1,23,45}$
$\displaystyle=\frac{t_{1}^{4}}{4}+\frac{t_{1}^{2}t_{2}^{2}}{2}+\frac{t_{1}^{2}t_{5}^{2}}{2}+t_{2}^{2}t_{5}^{2}$
(77) $\displaystyle\lambda^{2}\mathbb{S}_{1,25,34}$
$\displaystyle=\frac{t_{2}^{2}t_{3}^{2}}{2}+\frac{t_{4}^{2}t_{5}^{2}}{2}+t_{1}^{2}t_{4}^{2}+t_{1}^{2}t_{3}^{2}$
(78)

It is then trivial to plug these results back into the expression for the
decagon (67) and check that they give the same as the two loop expansion of
the stampedes result (74).

Similar to Steinmann relations, the stampedes also depend only on easier to
compute log terms. Both constraints are expected to hold at all loops, which
motivates the bootstrap analysis of how constraining are these conditions.

#### Bootstrap

Any five point correlator (1) can be written as a combination of the two loop
four and five point integrals: $\Phi^{(L)}$, $\mathcal{L}_{i,jk,lm}$,
$\mathcal{S}_{i,jk,lm}$ and $\mathcal{D}_{ij,klm}$, see [18, 51]. By demanding
Steinmann relations we severely reduce the basis of integrals to be only four
point integrals, (71) and (72). At this point one can write that the most
general two loop decagon function is given by

$\mathbb{D}_{\texttt{2-loops}}=\lambda^{2}\left(a_{1}(u_{i})\Phi^{(2)}\left(u_{1}u_{4},u_{5}\right)+b_{1}(u_{i})\mathbb{L}_{1,23,45}+c_{1}(u_{i})\mathbb{S}_{1,25,34}\right)+\textrm{cyclic}$
(79)

where $a_{i},b_{i},c_{i}$ are unknown rational functions of the cross-ratios
related to each other by cyclic symmetry

$a_{i}|_{u_{i}\to u_{i+1}}=a_{i+1}\,,\quad b_{i}|_{u_{i}\to
u_{i+1}}=b_{i+1}\,,\quad c_{i}|_{u_{i}\to u_{i+1}}=c_{i+1}\,.$ (80)

Demanding that the putative decagon (79) matches the stampedes prediction (74)
completely fixes the null limit of these polynomials

$a_{i}(u_{k}\to 0)=4,\quad b_{i}(u_{k}\to 0)=0,\quad c_{i}(u_{k}\to 0)=2\,.$
(81)

where the vanishing of the $b_{i}$ coefficients was already anticipated in
[31].

The combination of Steinmann and stampedes relations is sufficient to fix the
decagon in the limit where all the neighbouring distances are becoming null.
The decagon in this regime reads

$\ln\mathbb{D}=\mathcal{C}_{0}+\sum_{i=1}^{5}\left((-2\lambda+8\zeta_{2}\lambda^{2})\ln{u_{i}}\ln{u_{i+1}}+4\zeta_{2}\lambda^{2}(\ln{u_{i}}\ln{u_{i+2}}+\ln^{2}{u_{i}})\right)$
(82)

where999This constant is the hardest to get, since it depends on the
complicated part of the asymptotic regions of the integrals, the one that is
given in terms of master integrals and do not generate logs.
$\mathcal{C}_{0}=-10\zeta_{2}\lambda+135\zeta_{4}\lambda^{2}$. Unfortunately,
using only these two constraints is not sufficient to fix the decagon for
general kinematics. At this stage, the full constraint of the decagon can only
be achieved by restricting our expression to the plane kinematics and
comparing with the expressions of [16]. This fixes all the polynomials
$a_{i},b_{i},c_{i}$ as rational functions of cross-ratios and yields the
result (67), but it is not a promising strategy for fixing higher loops or
higher polygons, since it requires performing complicated integrability
computations.

###  5 Discussion

It has been known for more than a decade that correlation functions of the
lightest scalars in planar gauge theories are dual, when the points approach
the cusps of a null polygon, to a Wilson loop evaluated in the null polygon
path [2]. In $\mathcal{N}=4$ SYM, this relation is enhanced to an even more
interesting triality since null polygonal Wilson loops are also related to
gluon scattering amplitudes [4].

More recently, this triality was promoted, using conformal bootstrap
methods[24, 21, 25], to a quadrality with the inclusion of correlation
functions of large spinning operators in the relations. A crucial point in the
bootstrap derivation relies on the assumption that the correlators of spinning
operators have a simple behaviour in the large spin limit. The Lorentzian
inversion formula[49] puts this assumption on firm ground for the simplest
case when only one spinning operator is involved. Unfortunately, for two or
more spinning operator a generalization of the Lorentzian inversion formula is
not available yet and one has to resort to specific theories to get support
for these assumptions. This is precisely what we have accomplished in this
paper by looking into $\mathcal{N}=4$ SYM at two loops for the next to
simplest correlator, i.e. three point functions of two spinning operators and
one scalar (which should be dual to a null polygon Wilson loop with five edges
[21]). This two loop exercise is particularly important, given the existence
of a partial result in the literature [22] that appears to be in contradiction
with the usual assumptions of the analytic conformal bootstrap methods.

We have computed this two loop three point function, resolved the apparent
contradiction and showed that the assumptions of the conformal bootstrap
methods hold (at least for this model and at two loop order). These OPE
coefficients involving two spinning operators were extracted by doing
conformal block decomposition of a five point correlation function. Besides
the large spin structure, we also obtained more than a thousand three point
functions for for several finite spins. More importantly, our method can be
used to obtain even more if they are needed.

The main difficulty in carrying out the computation was that the five point
correlator was expressed in terms of some conformal integrals that were
previously unknown. So a by-product of our endeavor was the evaluation of two
loop five point conformal integrals in particular limits, obtained using two
different methods. The computation of these integrals is important per se as
they appear in other physical quantities. Moreover, we have observed
interesting properties of these two loop conformal integrals. For example, all
two loop five point integrals can be expressed in terms of multiple
polylogarithms (in the plane kinematics) or that in the open-box limit the
result can be expressed in terms of harmonic polylogarithms.

A natural and interesting next step would be to check if these properties
continue to hold for higher point functions or higher loops. In fact, studying
these type of conformal integrals at six points would be interesting from the
WL/correlation function function duality since WL is not only given by the
BDS-ansatz and starts to depend on finite cross-ratios [54]. Testing these
relations beyond one loop would be extremely non-trivial check of the duality
and could be useful to establish integrability/bootstrap understanding of this
web of relations.

Still at five points we considered null polygonal configurations of a
different type of correlator, the decagon. A remarkable difference between the
decagon and the light correlator in the null pentagon limit can be seen when
comparing equations (61) and (82). While the divergent part of the light
correlator is governed by the usual
$\Gamma_{\text{cusp}}=8\lambda-16\zeta_{2}\lambda^{2}+\dots$ the divergent
part of the decagon is governed by
$\Gamma_{\text{oct}}=8\lambda-32\zeta_{2}\lambda^{2}+\dots$, as anticipated by
[31]. These distinct UV divergences of the correlator get translated into
distinct IR divergences of the amplitudes. As explained in [12] this
difference can be understood in terms of Coulomb branch regularizations of the
amplitudes [55], depending on the order that one sends the external and
internal masses of the Coulomb branch amplitudes to zero.

It would be fascinating to extend this analysis of the decagon to higher loop
orders, or perhaps to fully bootstrap this object as the octagon. The central
problem with this approach is the lack of a simple computable basis. Using the
results of our integrals we reconstructed the decagon in several kinematics,
but in none of them we were able to come up with a simple basis of integrals
or functions (akin to the ladders of the octagon) that we could kick start the
bootstrap games. In fact, this is not surprising given that even the octagon
at two loops would also not give a strong hint of what the all loop basis
could be. Higher loops results played a much more decisive role in hinting
what the all loop basis turned out to be.

The only alternative at the moment is to continue to work with the basis of
all possible five point conformal integrals. If one can compute the log
divergence of these integrals then, in principle, one could play the same
bootstrap games as we did. First use Steinmann relations to reduce the number
of integrals and then stampedes to fix the coefficients of these integrals,
ending up with an expression for the null limit of the decagon. At two loops
this is enough to fix all the coefficients, but there is no guarantee that
this is sufficient to constrain the coefficients at higher loops.

Both Steinmann and stampedes can be easily extended to include more points,
therefore this bootstrap games could be also extended to higher point
functions. For instance, for six point functions one could start to probe even
more complicated objects such as the dodecagon. Where many intriguing aspects
such as connections to amplitudes and origins limits could be studied [56]. We
are starting to explore the complicated world of higher point functions at
higher orders in perturbation theory, and we are intrigued to see if our one
loop intuition remains correct.

### Acknowledgements

We would like to thank Till Bargheer, Frank Coronado, Albert Bekov and Antonio
Antunes for illuminating discussions. Centro de Física do Porto is partially
funded by Fundação para a Ciência e a Tecnologia (FCT) under the grant
UID04650-FCUP. The work of CB was funded by the Deutsche
Forschungsgemeinschaft (DFG, German Research Foundation) – 460391856. V.G. is
supported by Simons Foundation grants #488637 (Simons collaboration on the
non-perturbative bootstrap) and Fundacao para a Ciencia e Tecnologia (FCT)
under the grant CEECIND/03356/2022. B.F. is supported by Simons Foundation
grant #488637 (Simons collaboration on the non-perturbative bootstrap) and by
Fundação para a Ciência e Tecnologia, under the IDPASC doctoral program, under
the grand PRT/BD/154692/2022.

###  Appendix A Master integral and form of the differential equation

Here we list the 28 master integrals $g_{i}$ appearing in equation (27). The
differential equation that they satisfy has this form

$\displaystyle\frac{d}{du_{j}}g_{i}=\sum_{m}A^{j}_{im}g_{m}$ (83)

where the matrices $A^{j}_{im}$ are rational functions of the cross ratios and
$\epsilon$ and are given in an auxiliary file. The master integrals $g_{i}$
are defined by

$\displaystyle
g_{1}=\int\frac{d^{d}x_{6}d^{d}x_{7}}{x_{17}^{2}x_{18}^{2}x_{37}^{2}x_{38}^{2}},$
$\displaystyle g_{2}=$
$\displaystyle\int\frac{d^{d}x_{6}d^{d}x_{7}}{x_{17}^{2}x_{28}^{2}x_{37}^{2}x_{38}^{2}},$
$\displaystyle g_{3}=$
$\displaystyle\int\frac{d^{d}x_{6}d^{d}x_{7}}{x_{17}^{2}x_{18}^{2}x_{38}^{2}x_{47}^{2}},$
$\displaystyle
g_{4}=\int\frac{d^{d}x_{6}d^{d}x_{7}}{x_{28}^{2}x_{37}^{2}x_{78}^{2}},$
$\displaystyle g_{5}=$
$\displaystyle\int\frac{d^{d}x_{6}d^{d}x_{7}}{x_{18}^{2}x_{47}^{2}x_{78}^{2}},$
$\displaystyle g_{6}=$
$\displaystyle\int\frac{d^{d}x_{6}d^{d}x_{7}}{x_{18}^{2}x_{38}^{2}x_{47}^{2}x_{78}^{2}},$
$\displaystyle
g_{7}=\int\frac{d^{d}x_{6}d^{d}x_{7}}{x_{17}^{2}x_{28}^{2}x_{37}^{2}x_{78}^{2}},$
$\displaystyle g_{8}=$
$\displaystyle\int\frac{d^{d}x_{6}d^{d}x_{7}}{x_{17}^{2}x_{28}^{2}x_{37}^{2}x_{38}^{2}x_{78}^{2}},$
$\displaystyle g_{9}=$
$\displaystyle\int\frac{d^{d}x_{6}d^{d}x_{7}}{x_{17}^{2}x_{18}^{2}x_{38}^{2}x_{47}^{2}x_{78}^{2}},$
$\displaystyle
g_{10}=\int\frac{d^{d}x_{6}d^{d}x_{7}}{x_{28}^{2}x_{38}^{2}x_{47}^{2}x_{78}^{2}},$
$\displaystyle g_{11}=$
$\displaystyle\int\frac{d^{d}x_{6}d^{d}x_{7}}{x_{18}^{2}x_{37}^{2}x_{78}^{2}},$
$\displaystyle g_{12}=$
$\displaystyle\int\frac{d^{d}x_{6}d^{d}x_{7}}{x_{28}^{2}x_{47}^{2}x_{78}^{2}},$
$\displaystyle
g_{13}=\int\frac{d^{d}x_{6}d^{d}x_{7}}{x_{17}^{2}x_{28}^{2}x_{38}^{2}x_{78}^{2}},$
$\displaystyle g_{14}=$
$\displaystyle\int\frac{d^{d}x_{6}d^{d}x_{7}}{x_{17}^{2}x_{28}^{2}x_{38}^{2}x_{47}^{2}},$
$\displaystyle g_{15}=$
$\displaystyle\int\frac{d^{d}x_{6}d^{d}x_{7}}{x_{1,7}^{2}x_{3,8}^{2}x_{4,7}^{2}x_{7,8}^{2}},$
$\displaystyle
g_{16}=\int\frac{d^{d}x_{6}d^{d}x_{7}}{x_{17}^{2}x_{28}^{2}x_{47}^{2}x_{78}^{2}},$
$\displaystyle g_{17}=$
$\displaystyle\int\frac{d^{d}x_{6}d^{d}x_{7}}{x_{17}^{2}x_{28}^{2}x_{38}^{2}x_{47}^{2}(x_{78}^{2})^{2}},$
$\displaystyle g_{18}=$
$\displaystyle\int\frac{d^{d}x_{6}d^{d}x_{7}}{x_{17}^{2}x_{28}^{2}x_{38}^{2}(x_{47}^{2})^{2}x_{78}^{2}},$
$\displaystyle
g_{19}=\int\frac{d^{d}x_{6}d^{d}x_{7}}{x_{17}^{2}(x_{28}^{2})^{2}x_{38}^{2}x_{47}^{2}x_{78}^{2}},$
$\displaystyle g_{20}=$
$\displaystyle\int\frac{d^{d}x_{6}d^{d}x_{7}}{(x_{17}^{2})^{2}x_{28}^{2}x_{38}^{2}x_{47}^{2}x_{78}^{2}},$
$\displaystyle g_{21}=$
$\displaystyle\int\frac{d^{d}x_{6}d^{d}x_{7}}{x_{17}^{2}x_{28}^{2}x_{38}^{2}x_{47}^{2}x_{78}^{2}},$
$\displaystyle
g_{22}=\int\frac{d^{d}x_{6}d^{d}x_{7}}{x_{17}^{2}x_{18}^{2}x_{28}^{2}x_{38}^{2}x_{47}^{2}x_{78}^{2}},$
$\displaystyle g_{23}=$
$\displaystyle\int\frac{d^{d}x_{6}d^{d}x_{7}}{x_{18}^{2}x_{28}^{2}x_{37}^{2}x_{47}^{2}x_{78}^{2}},$
$\displaystyle g_{24}=$
$\displaystyle\int\frac{d^{d}x_{6}d^{d}x_{7}}{x_{17}^{2}x_{28}^{2}x_{37}^{2}x_{38}^{2}x_{47}^{2}x_{78}^{2}},$
$\displaystyle
g_{25}=\int\frac{d^{d}x_{6}d^{d}x_{7}}{x_{17}^{2}x_{28}^{2}x_{37}^{2}x_{47}^{2}x_{78}^{2}},$
$\displaystyle g_{26}=$
$\displaystyle\int\frac{d^{d}x_{6}d^{d}x_{7}}{x_{18}^{2}x_{28}^{2}x_{38}^{2}x_{47}^{2}x_{78}^{2}},$
$\displaystyle g_{27}=$
$\displaystyle\int\frac{d^{d}x_{6}d^{d}x_{7}}{x_{17}^{2}x_{18}^{2}x_{28}^{2}x_{37}^{2}x_{38}^{2}x_{47}^{2}x_{78}^{2}},$
$\displaystyle
g_{28}=\int\frac{d^{d}x_{6}d^{d}x_{7}}{x_{17}^{2}x_{18}^{2}x_{28}^{2}(x_{37}^{2})^{2}x_{38}^{2}x_{47}^{2}x_{78}^{2}}.$

One important one loop master integral is given by

$\displaystyle\int\frac{d^{d}x_{8}}{x_{18}^{2}x_{38}^{2}x_{48}^{2}}=\frac{12\text{Li}_{2}(1-v)+6\ln
u\ln
v}{6(1-v)}+\epsilon\bigg{[}\frac{2\text{Li}_{3}\left(\frac{v-1}{v}\right)-2\text{Li}_{3}(1-v)}{(v-1)}$
(84) $\displaystyle+\frac{12\text{Li}_{2}(1-v)(\ln u+2)+3\ln
u\ln^{2}v+3\ln^{2}u\ln v+12\ln u\ln v-2\ln^{3}v}{6(v-1)}\bigg{]}+O(u)$

where we have taken $x_{13}^{2}=1$ and used $u=x_{34}^{2},v=x_{14}^{2}$. Note
that we have only written the leading order in $u\rightarrow 0$ limit since we
only needed this order.

###  Appendix B Perturbative data

Here we review what is known about the perturbative expressions of structure
constants of one and two spin operators up to two loops, as well as establish
conventions used in throughout the paper.

The structure constants for one spinning operator have been studied
extensively in the literature. The most efficient way of computing these three
point functions are via the conformal block decomposition of a four point
function. The first non trivial computation of these three point functions in
$\mathcal{N}=4$ was done at one loop level in [57] and later extended for up
to three loops and any spin in [41] and for spin two operators is known up to
five loops [43]. Here we recall their expressions up to two loops

$\displaystyle C^{\bullet\circ\circ}_{J}$
$\displaystyle=C^{\bullet\circ\circ}_{\texttt{tree}}(1+\lambda\hat{C}^{\bullet\circ\circ}_{\texttt{1-loop}}+\lambda^{2}\hat{C}^{\bullet\circ\circ}_{\texttt{2-loop}}+\dots)$
(85) $\displaystyle C^{\bullet\circ\circ}_{\texttt{tree}}$
$\displaystyle=\frac{J!}{\sqrt{(2J)!}}$ (86) $\displaystyle
C^{\bullet\circ\circ}_{\texttt{1-loop}}$
$\displaystyle=4S_{1}(J)^{2}-4S_{1}(J)S_{1}(2J)-2S_{2}(J)$ (87) $\displaystyle
C^{\bullet\circ\circ}_{\texttt{2-loop}}$
$\displaystyle=20S_{-4}(J)+8S_{-2}(J)^{2}+8S_{-3}(J)S_{1}(J)-16S_{-2}(J)S_{1}(J)^{2}+8S_{1}(J)^{4}+$
$\displaystyle+8S_{-3}(J)S_{1}(2J)+16S_{-2}(J)S_{1}(J)S_{1}(2J)-16S_{1}(J)^{3}S_{1}(2J)+8S_{1}(J)^{2}S_{1}(2J)^{2}+$
$\displaystyle+8S_{-2}(J)S_{2}(J)-32S_{1}(J)^{2}S_{2}(J)+24S_{1}(J)S_{1}(2J)S_{2}(J)+16S_{1}(J)^{2}S_{2}(2J)+$
$\displaystyle+6S_{2}(J)^{2}+8S_{1}(J)S_{3}(J)+8S_{1}(2J)S_{3}(J)+20S_{4}(J)+16S_{1}(J)S_{-2,1}(J)+$
$\displaystyle-16S_{-3,1}(J)+8S_{-2,2}(J)-16S_{1,3}(J)+24\zeta_{3}S_{1}(J)$
(88)

where the $S_{i_{1},\dots,i_{n}}(J)$ are the harmonic sums.

To set the coupling convention we also recall the anomalous dimension of these
operators

$\displaystyle\gamma(J)$
$\displaystyle=\lambda\gamma_{\texttt{1-loop}}+\lambda^{2}\gamma_{\texttt{2-loop}}+\dots$
(89) $\displaystyle\gamma_{\texttt{1-loop}}$ $\displaystyle=8S_{1}(J)$ (90)
$\displaystyle\gamma_{\texttt{2-loop}}$
$\displaystyle=-16S_{-3}(J)-32S_{-2}(J)S_{1}(J)-32S_{1}(J)S_{2}(J)-16S_{3}(J)+32S_{-2,1}(J)$
(91)

so at large spin we have the following cusp and collinear anomalous dimension

$\displaystyle f(\lambda)$
$\displaystyle=8\lambda-16\zeta_{2}\lambda^{2}+\dots$ (92) $\displaystyle
g(\lambda)$
$\displaystyle=8\gamma_{E}\lambda-(24\zeta_{3}-16\zeta_{2}\gamma_{E})\lambda^{2}+\dots$
(93)

For spinning operators we will use the usual index free notation, which
consist in contracting the spinning operator with an auxiliary null
polarization vector $\epsilon$

$\mathcal{O}_{J}(x,\epsilon)=\epsilon_{\mu_{1}}\dots\epsilon_{\mu_{J}}\mathcal{O}^{\mu_{1}\dots\mu_{J}}(x)$
(94)

The three point function of two such operators is parametrized as

$\langle\mathcal{O}_{J_{1}}(x_{1},\epsilon_{1}),\mathcal{O}_{J_{2}}(x_{2},\epsilon_{2}),\mathcal{O}(x_{3})\rangle=\frac{V_{1,23}^{J_{1}}V_{2,31}^{J_{2}}}{x_{12}^{\kappa_{1}+\kappa_{2}-\kappa_{3}}x_{23}^{\kappa_{2}+\kappa_{3}-\kappa_{1}}x_{31}^{\kappa_{3}+\kappa_{1}-\kappa_{2}}}\sum_{\ell=0}^{\ell_{\text{max}}}C^{\bullet\bullet\circ}_{J_{1},J_{2},\ell}T^{\ell}$
(95)

where $\kappa_{i}=\Delta_{i}+J_{i}$ is the conformal spin and

$V_{i,jk}=\frac{\epsilon_{i}\cdot x_{ik}x_{ij}^{2}-\epsilon_{i}\cdot
x_{ij}x_{ik}^{2}}{x^{2}_{jk}},\quad H_{ij}=(\epsilon_{i}\cdot
x_{ij})(\epsilon_{j}\cdot
x_{ij})-\tfrac{1}{2}x^{2}_{ij}(\epsilon_{i}\cdot\epsilon_{j})\,,\quad
T=\frac{H_{12}}{V_{1,23}V_{2,31}}$ (96)

are a basis of conformal covariant tensors defined in [47]. The three point
function is a polynomial in this tensor structures with the degree fixed by
conformal symmetry to be $\ell_{\text{max}}=\text{min}(J_{1},J_{2})$.

The structure constants of two spinning operators appearing in this three
point function were also worked out up to one loop [22, 25].

$\displaystyle C^{\bullet\bullet\circ}_{J_{1},J_{2},\ell}$
$\displaystyle=C^{\bullet\bullet\circ}_{\texttt{tree}}(1+\lambda\hat{C}^{\bullet\bullet\circ}_{\texttt{1-loop}}+\lambda^{2}\hat{C}^{\bullet\bullet\circ}_{\texttt{2-loop}}+\dots)$
$\displaystyle C^{\bullet\bullet\circ}_{\texttt{tree}}$
$\displaystyle=\frac{J_{1}!}{\sqrt{(2J_{1})!}}\frac{J_{2}!}{\sqrt{(2J_{2})!}}\binom{J_{1}}{\ell}\binom{J_{2}}{\ell}$
$\displaystyle C^{\bullet\bullet\circ}_{\texttt{1-loop}}$
$\displaystyle=4\left(S_{1}(J_{1})+S_{1}(J_{2})\right)\sum_{i=1}^{\ell}\frac{\binom{J_{1}+J_{2}+1}{i}\binom{\ell}{i}}{i\binom{J_{1}}{i}\binom{J_{2}}{i}}-4\frac{(J_{1}+1)(J_{2}+1)}{J_{1}+J_{2}+2}\sum_{i=1}^{\ell}\frac{\binom{J_{1}+J_{2}+2}{i}a_{i}^{\ell}}{i\binom{J_{1}}{i}\binom{J_{2}}{i}}+$
$\displaystyle+4S_{1}(J_{1})^{2}+-4S_{1}(J_{1})S_{1}(2J_{1})-2S_{2}(J_{1})+4S_{1}(J_{2})^{2}-4S_{1}(J_{2})S_{1}(2J_{2})-2S_{2}(J_{2})$
$\displaystyle C^{\bullet\bullet\circ}_{\texttt{2-loop}}$
$\displaystyle=\dots+24\zeta_{3}|S_{1}(J_{1})-S_{1}(J_{2})|$

where the constants $a_{i}^{\ell}$ are given by

$a_{i}^{\ell}=\frac{(-1)^{i}}{2}\left(-S_{1}(\ell)^{2}-S_{2}(\ell)+2\sum_{k=2}^{i}\frac{(-1)^{k}\binom{l}{k-1}}{k-1}\left(S_{1}(k-2)+S_{1}(\ell)\right)\right)\,.$
(97)

and the two loop structure constants are unknown up to the $\zeta_{3}$
coefficient.

In [58] a new basis for the three point functions was introduced, it reads

$\langle\mathcal{O}_{J_{1}}(x_{1},\epsilon_{1}),\mathcal{O}_{J_{2}}(x_{2},\epsilon_{2}),\mathcal{O}(x_{3})\rangle=\frac{V_{1,23}^{J_{1}}V_{2,31}^{J_{2}}}{x_{12}^{\kappa_{1}+\kappa_{2}-\kappa_{3}}x_{23}^{\kappa_{2}+\kappa_{3}-\kappa_{1}}x_{31}^{\kappa_{3}+\kappa_{1}-\kappa_{2}}}\sum_{\ell=0}^{\ell_{\text{max}}}A^{\bullet\bullet\circ}_{J_{1},J_{2},\ell}T^{\ell}(1-T)^{\ell_{\text{max}}-\ell}$
(98)

we denominate it as the integrability basis, since it makes trivial the
relation of the integrable hexagons [59] with the spinning structure
constants, see [58]. The map between the usual basis and integrability basis
is also vey simple to work out.

$A(J_{1},J_{2},\ell)=\sum_{k=0}^{\ell}\frac{\binom{\ell_{\text{max}}}{\ell}\binom{\ell}{k}}{\binom{\ell_{\text{max}}}{k}}C(J_{1},J_{2},k)$
(99)
$C(J_{1},J_{2},\ell)=\sum_{k=0}^{\ell}\frac{\binom{\ell_{\text{max}}}{\ell}\binom{\ell}{k}}{\binom{\ell_{\text{max}}}{k}}(-1)^{\ell-k}A(J_{1},J_{2},k)$
(100)

One can also work out the two spin structure constants in teh integrability
basis

$\displaystyle A^{\bullet\bullet\circ}_{J_{1},J_{2},\ell}$
$\displaystyle=A^{\bullet\bullet\circ}_{\texttt{tree}}(1+\lambda\hat{A}^{\bullet\bullet\circ}_{\texttt{1-loop}}+\lambda^{2}\hat{A}^{\bullet\bullet\circ}_{\texttt{2-loop}}+\dots)$
$\displaystyle A^{\bullet\bullet\circ}_{\texttt{tree}}$
$\displaystyle=\frac{J_{1}!}{\sqrt{(2J_{1})!}}\frac{J_{2}!}{\sqrt{(2J_{2})!}}\binom{J_{\text{min}}}{\ell}\binom{J_{\text{max}}+\ell}{\ell}$
$\displaystyle A^{\bullet\bullet\circ}_{\texttt{1-loop}}$
$\displaystyle=6S_{1}({J_{\text{min}}})^{2}-4S_{1}({J_{\text{min}}})S_{1}(2{J_{\text{min}}})-4S_{1}({J_{\text{max}}})S_{1}(2{J_{\text{max}}})-4S_{1}({J_{\text{max}}})S_{1}({J_{\text{min}}}-{\ell})+$
$\displaystyle-2S_{1}({J_{\text{min}}}-{\ell})^{2}-4S_{1}({J_{\text{min}}})S_{1}({\ell})+4S_{1}({J_{\text{min}}}-{\ell})S_{1}({\ell})-2S_{1}({\ell})^{2}-2S_{2}({\ell})+$
$\displaystyle+4S_{1}({J_{\text{min}}})S_{1}({J_{\text{max}}}+{\ell})+4S_{1}({J_{\text{max}}})S_{1}({J_{\text{max}}}+{\ell})-2S_{2}({J_{\text{max}}})-2S_{2}({J_{\text{min}}}-{\ell})$
$\displaystyle A^{\bullet\bullet\circ}_{\texttt{2-loop}}$
$\displaystyle=\dots+24\zeta_{3}\binom{J_{\text{min}}+1}{\ell}(S_{1}(J_{\text{max}})-S_{1}(J_{\text{min}}))$

It is amazing to see that this simple combination of binomials can transform
the ugly sums of the usual basis into an expression that has only nice
harmonic numbers of the new basis. It also has similar features as [60], since
polarizations and spins are in the same footing one can expect analyticity of
the structure constants in these quantities in large spin and polarizations
limits like the Regge limit studied in [60].

###  Appendix C Examples of conformal integrals

We start by recalling the definition of the multiple polylogarithms [32] via
the recursive iterated integral

$G(a_{1},\dots,a_{n};z)=\int_{0}^{z}\frac{dt}{t-a_{1}}G(a_{2},\dots,a_{n};t)$
(101)

with $G(z)=1$ and $a_{i},z$ are complex variables.

We also consider four point integrals at one and two loops, they are defined
as

$\displaystyle\Phi^{(1)}(u_{{\color[rgb]{0.92578125,0,0.04296875}1}}u_{{\color[rgb]{1,0.6015625,0}4}},u_{{\color[rgb]{0.17578125,0.87890625,0}5}})$
$\displaystyle=x_{{\color[rgb]{0.92578125,0,0.04296875}1}{\color[rgb]{1,0.6015625,0}4}}^{2}x_{{\color[rgb]{0,0.640625,0.95703125}2}{\color[rgb]{0.17578125,0.87890625,0}5}}^{2}\int\frac{d^{4}x_{6}}{x_{{\color[rgb]{0.92578125,0,0.04296875}1}6}^{2}x_{{\color[rgb]{0,0.640625,0.95703125}2}6}^{2}x_{{\color[rgb]{1,0.6015625,0}4}6}^{2}x_{{\color[rgb]{0.17578125,0.87890625,0}5}6}^{2}}$
(102)
$\displaystyle\Phi^{(2)}(u_{{\color[rgb]{0.92578125,0,0.04296875}1}}u_{{\color[rgb]{1,0.6015625,0}4}},u_{{\color[rgb]{0.17578125,0.87890625,0}5}})$
$\displaystyle=x_{{\color[rgb]{0.92578125,0,0.04296875}1}{\color[rgb]{1,0.6015625,0}4}}^{4}x_{{\color[rgb]{0,0.640625,0.95703125}2}{\color[rgb]{0.17578125,0.87890625,0}5}}^{2}\int\frac{d^{4}x_{6}d^{4}x_{7}}{x_{{\color[rgb]{0.92578125,0,0.04296875}1}6}^{2}x_{{\color[rgb]{0,0.640625,0.95703125}2}6}^{2}x_{{\color[rgb]{0.17578125,0.87890625,0}5}6}^{2}x_{{\color[rgb]{0,0.640625,0.95703125}2}7}^{2}x_{{\color[rgb]{1,0.6015625,0}4}7}^{2}x_{{\color[rgb]{0.17578125,0.87890625,0}5}7}^{2}x_{67}^{2}}$
(103)

and can be easily integrated

$\displaystyle\Phi^{(L)}(u,v)$
$\displaystyle=\frac{1}{\bar{z}-z}F^{(L)}\left(\frac{z}{1-z},\frac{\bar{z}}{1-\bar{z}}\right)\,,$
(104) $\displaystyle F^{(L)}(z,\bar{z})$
$\displaystyle=\sum_{k=0}^{L}\frac{(-1)^{k}(2L-k)!}{k!(L-k)!L!}\ln^{k}(z\bar{z})\left(\text{Li}_{2L-k}(z)-\text{Li}_{2L-r}(\bar{z})\right)$
(105)

being $z,\bar{z}$ are the usual four point cross-ratios $z\bar{z}=u$ and
$(1-z)(1-\bar{z})=v$.

In general we cannot write in such small expressions the five point two loop
integrals. However in the kinematical regimes described in the figure 2 we
can. For example, in the open-box limit depicted on the left of figure 2 one
of the integral reads

$\mathbb{L}_{2,13,45}=\frac{7\pi^{4}}{180}+\frac{\pi^{2}}{2}\ln^{2}{u_{1}}+\frac{\pi^{2}}{6}\ln{u_{1}}\ln{u_{4}}-3\zeta_{3}\ln{u_{1}}-3\zeta_{3}\ln{u_{4}}+\dots$
(106)

where the dots represent subleading contributions. In the OPE limit depicted
on the right of figure 2 another example of integral is given by

$\displaystyle\mathbb{D}_{13,245}$
$\displaystyle=4+U_{4}-U_{5}+\frac{11}{6}U_{4}U_{5}-\zeta_{3}U_{4}U_{5}+\ln{u_{1}}\left(-2+\frac{3}{2}U_{4}-\frac{U_{5}}{2}+\frac{U_{4}U_{5}}{2}\right)+$
$\displaystyle+\ln{u_{3}}\left(-2+\frac{U_{4}}{2}-\frac{U_{5}}{2}+\frac{U_{4}U_{5}}{2}\right)+\ln{u_{1}}\ln{u_{3}}\left(1+\frac{U_{4}}{2}+\frac{U_{5}}{2}-\frac{U_{4}U_{5}}{4}\right)+\dots$
(107)

where $\mathbb{D}_{13,245}=x^{2}_{24}x^{2}_{25}x^{2}_{45}\mathcal{D}_{13,245}$
is defined using (4) to be a function only of cross-ratios and
$U_{i}=(1-u_{i})$ are the cross-ratios that are expanded around zero in the
OPE expansion.

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