Multipoint conformal integrals in D dimensions. Part I: Bipartite Mellin-Barnes representation and reconstruction

Abstract

We propose a systematic approach to calculating n-point one-loop parametric conformal integrals in D dimensions which we call the reconstruction procedure. It relies on decomposing a conformal integral over basis functions which are generated from a set of master functions by acting with the cyclic group Zn\mathbb{Z}_n. In order to identify the master functions we introduce a bipartite Mellin-Barnes representation by means of splitting a given conformal integral into two additive parts, one of which can be evaluated explicitly in terms of multivariate generalized hypergeometric series. For the box and pentagon integrals (i.e. n=4,5) we show that a computable part of the bipartite representation contains all master functions. In particular, this allows us to evaluate the parametric pentagon integral as a sum of ten basis functions generated from two master functions by the cyclic group Z5\mathbb{Z}_5. The resulting expression can be tested in two ways. First, when one of propagator powers is set to zero, the pentagon integral is reduced to the known box integral, which is also rederived through the reconstruction procedure. Second, going to the non-parametric case, we reproduce the known expression for the pentagon integral given in terms of logarithms derived earlier within the geometric approach to calculating conformal integrals. We conclude by considering the hexagon integral (n=6) for which we show that those basis functions which follow from the computable part of the bipartite representation are not enough and more basis functions are required. In the second part of our project we will describe a method of constructing a complete set of master/basis functions in the n-point case.

Full text

Multipoint conformal integrals in Ddimensions. Part I
Bipartite Mellin-Barnes representation and reconstruction
Konstantin Alkalaev and Semyon Mandrygin
I.E. Tamm Department of Theoretical Physics,
P.N. Lebedev Physical Institute, 119991 Moscow, Russia
E-mail: alkalaev@lpi.ru,semyon.mandrygin@gmail.com
Abstract: We propose a systematic approach to calculating n-point one-loop parametric
conformal integrals in Ddimensions which we call the reconstruction procedure. It relies
on decomposing a conformal integral over basis functions which are generated from a set
of master functions by acting with the cyclic group Zn. In order to identify the master
functions we introduce a bipartite Mellin-Barnes representation by means of splitting a given
conformal integral into two additive parts, one of which can be evaluated explicitly in terms
of multivariate generalized hypergeometric series.
For the box and pentagon integrals (i.e. n= 4,5) we show that a computable part of the
bipartite representation contains all master functions. In particular, this allows us to evaluate
the parametric pentagon integral as a sum of ten basis functions generated from two master
functions by the cyclic group Z5. The resulting expression can be tested in two ways. First,
when one of propagator powers is set to zero, the pentagon integral is reduced to the known
box integral, which is also rederived through the reconstruction procedure. Second, going to
the non-parametric case, we reproduce the known expression for the pentagon integral given
in terms of logarithms derived earlier within the geometric approach to calculating conformal
integrals.
We conclude by considering the hexagon integral (n= 6) for which we show that those
basis functions which follow from the computable part of the bipartite representation are
not enough and more basis functions are required. In the second part of our project we will
describe a method of constructing a complete set of master/basis functions in the n-point
case.
arXiv:2502.12127v1 [hep-th] 17 Feb 2025

Contents
1 Introduction 2
2 Multipoint parametric conformal integrals 4
2.1 Review of the Symanzik approach 4
2.2 Bipartite Mellin-Barnes representation 7
3 Constraints from the permutation invariance 12
3.1 Analytic continuation 13
3.2 Box integral 15
3.2.1 Kinematic group 16
3.2.2 Cyclic group 17
3.2.3 Consistency checks 19
3.2.4 Summary of the box integral reconstruction 21
3.3 Pentagon integral 22
3.3.1 Cyclic group 23
3.3.2 Reduction to the box integral 24
3.4 Hexagon integral 25
4 Reconstruction conjecture 27
5 Conclusion 30
A Notation and conventions 31
B Generalized hypergeometric functions 31
B.1 General definitions 32
B.2 Integral representations 34
B.3 Analytic continuation 35
C One more Mellin-Barnes representation of the conformal integral 37
D Pentagon integral 37
D.1 Basis functions 38
D.2 Non-parametric integral 40
E Kinematic group extensions 43
– 1 –

1 Introduction
Conformal integrals are an interesting specific class of integrals that arise in various areas of
quantum field theory [1]. E.g. they appear in the study of scattering multi-loop amplitudes,
where the corresponding Feynman integrals respect a dual conformal invariance [2,3] as well
as in the Fishnet CFT models [4–9] (see [10,11] for reviews). There is a number of methods
of calculating conformal integrals among which the most fruitful are the geometric approach
which treats these integrals as volumes of simplices [12–17], Yangian bootstrap [18–21], GKZ
differential equations [22–24] as well as other approaches which combine various techniques
[25–32]. Nonetheless, despite this recent remarkable progress in calculating conformal inte-
grals, explicit closed-form expressions for general conformal integrals are not yet known.
On the other hand, a more traditional way to come across conformal integrals is to use
the shadow formalism in CFT which represents correlation functions as particular integrals
possessing conformal invariance by construction [33–39]. In particular, using the shadow
formalism allows one to take advantage of various Feynman integral techniques for CFT cal-
culations.1In this paper, we instead are inspired by CFT methods for dealing with conformal
integrals. Namely, we develop the idea of asymptotic analysis of conformal integrals which
is borrowed from the conformal block decomposition of correlation functions in CFT. Recall
that the conformal blocks can be defined by doing OPEs between pairs of primary operators
inside a correlation function. This results in choosing of a particular channel, i.e. one fixes
an order and closeness of points which is known as the OPE limit. This eventually defines a
domain of convergence of the conformal block in the coordinate space. Now when the con-
formal block is represented in terms of conformal integrals, the latter are calculated in the
OPE limit to have correct asymptotic behaviour. Other asymptotics of the respective con-
formal integral describe the so-called shadow blocks [38,39]. The conformal blocks in other
coordinate domains, i.e. in other possible channels, can be obtained by means of analytic
continuation formulas.
Relying on this relation between conformal blocks and conformal integrals we can formu-
late the first point of our approach to calculating n-point conformal integrals in Ddimensions:
one fixes a convenient coordinate domain where a conformal integral can be represented in
terms of a multivariate power series and then analytically continued onto other domains.
The second point is that there is no need to evaluate conformal integral entirely, but only
partially. To this end, one decomposes a conformal integral into two additive parts which
are given by particular Mellin-Barnes integrals. This is a bipartite representation. The first
term can be evaluated explicitly in some coordinate domain, while the second one is quite
complicated and has not been computed yet. Here, we put forward our main idea that the
second term can be reconstructed from the first one by invoking permutation invariance of
the full conformal integral. Note that such a partition of conformal integral in no way is
invariant against transformations from the symmetric group Sn. Thus, the third point of our
approach is to introduce a subgroup of the symmetric group which is claimed to generate the
1Conformal integrals are also instrumental in CFT on non-trivial manifolds [40–47].
– 2 –

full conformal integral by acting on a set of master functions. We call this calculation scheme
a reconstruction.
The most intricate part of this approach is to identify both a generating group and
master functions. It turns out that a generating group can be chosen as the cyclic group
Zn⊂ Snthough this is not the only choice. A generating group is required to act on master
functions to generate a complete set of basis functions which sum up to the full conformal
integral. It follows that the generating group acts on the basis functions by reshuffling them.
To single out an independent set of master functions we claim that they are not related by
cyclic permutations.2In other words, a set of basis functions decomposes into subsets each
of which carries a representation of Zn.3
In the cases of the box and pentagon conformal integrals the corresponding master func-
tions can be naturally identified by explicitly calculating a first term of the bipartite rep-
resentation and then analytically continuing the resulting expression onto some coordinate
domain around a specified point. This procedure yields the first term as a sum of generalized
hypergeometric multivariate functions from which a number of functions can be selected as
master functions. The cyclic group generates a set of basis functions which define the second
part of the bipartite representation thereby giving the full conformal integral.
However, when considering the hexagon integral we find out that those basis functions
which come from the first part of the bipartite representations are not enough to build the
full conformal integral. Nonetheless, we put forward a conjecture how to build a complete
set of basis functions which is the subject of our forthcoming paper [48]. Essentially, there
we claim that the basis functions which we are able to identify explicitly from the bipartite
representation have a number of systematic properties that will eventually allow us to build
a desired complete set.
This paper is organized as follows. In section 2we introduce (non)-parametric n-point
conformal integrals and review the standard calculation method by Symanzik which is based
on using the Mellin-Barnes integrals. Then, in section 2.2 we propose a bipartite Mellin-
Barnes representation which splits the original conformal integral into two additive parts.
Here, we explicitly calculate the first part in some coordinate domain and discuss the second
one. In section 3we describe the action of symmetric group and consider the analytic con-
tinuation to the other coordinate domain. Section 3.2 considers the parametric box integral
which has been studied in detail in the literature and examines our reconstruction approach
using this example. Summarizing our study of the box integral, in section 3.2.4 we outline
a heuristic procedure of obtaining the full asymptotic expansions in the n-point case. Then,
in section 3.3 we apply this reconstruction machinery to the 5-point one-loop conformal in-
tegral. Here, we find the parametric pentagon integral as a sum of ten basis functions and
then perform a number of consistency checks. In section 3.4 we consider the hexagon integral
and discover that the way of choosing master functions used in the box and pentagon cases
2In this paper, a cyclic permutation will be understood only as a cycle of maximum length n, which
generates the cyclic group Zn⊂ Sn.
3In general, these are inequivalent representations of Zn.
– 3 –

is insufficient to construct a complete asymptotic expression because there should be more
master functions. Finally, in section 4, based on the considered examples, we formulate and
discuss a conjecture regarding a reconstruction of n-point one-loop conformal integral. In the
concluding section 5we summarize our results and elaborate on some perspectives for future
work.
A few appendices contain numerous technical aspects of calculations. Appendix Acollects
our notation and conventions used throughout the paper. Appendix Bdescribes various
generalized hypergeometric multivariate functions, their integral representations and analytic
continuation formulas. In appendix Cwe suggest one more Mellin-Barnes representation of
the conformal integral which can be useful in practice, e.g. for finding the conformal integral
using the computer algebra methods. Appendix Dcollects explicit expressions for the basis
functions which define the parametric pentagon conformal integral. In particular, in appendix
D.2 we examine our representation of the non-parametric (i.e. with unit propagator powers)
pentagon integral. Appendix Ediscusses an extended kinematic group which is a particular
subgroup in the symmetric group.
2 Multipoint parametric conformal integrals
The following integral over the D-dimensional Euclidean space RD
Ia
n(x) = ZRD
dDx0
πD
2
n
Y
i=1
X−ai
0i, Xij ≡Xi,j = (xi−xj)2, xi∈RD,(2.1)
where a={a1, a2, ..., an},x={x1, x2, ..., xn}, is called the n-point one-loop conformal
integral [1] provided that the propagator powers ai∈Robey the constraint
n
X
i=1
ai=D , (2.2)
which guarantees that Ia
n(x) transforms covariantly under O(D+ 1,1) conformal transforma-
tions (see e.g. [11] for details). The integrals with arbitrary aisubjected to (2.2) are called
parametric, while those with all ai= 1 in D=ndimensions are called non-parametric.
Since the paper of Symanzik [1], the conformal integrals are traditionally represented in
terms of the Mellin-Barnes integrals. In this section we first recall the Symanzik approach
and then introduce a bipartite Mellin-Barnes representation.
2.1 Review of the Symanzik approach
In order to calculate the conformal integral (2.1) one can use the Schwinger parametrization
1
Xai
0i
=1
Γ(ai)Z+∞
0
dλi
λi
λai
iexp (−λiX0i), ai>0.(2.3)
– 4 –

h2h3
h1h4

h2h3hn2hn1
h1hn
1n3
1
2
3
n
a1
a2
a3
an
1
Figure 1. The parametric conformal integral Ia
n(x) can be depicted as an n-valent vertex. The i-th
leg denotes the propagator X−ai
0iwhich is characterized by i-th position xiand propagator power ai,
the central dot denotes integration over x0.
This trick is standard and allows one to evaluate the D-dimensional integral over x0by
obtaining a set of one-dimensional integrals over λi. Then, the n-point conformal integral is
represented as
Ia
n(x) = Na
nZ+∞
0
n
Y
i=1 dλi
λi
λai
i1
|λ1,n|D
2
exp −1
|λ1,n|X
1≤i<j≤n
λiλjXij!,(2.4)
where4|λi,j|=Pj
l=iλland for brevity we defined the prefactor
Na
j=1
Γ(a1, ..., aj), j = 1, ..., n , (2.5)
for which the standard notation for the product of Γ-functions (A.3) is used.
The key observation [1] is that in (2.4) one can substitute |λ1,n| → λnas a consequence
of the conformality condition (2.2). Representing the exponential functions by means of the
Mellin-Barnes formula (B.20)
exp −λiλjXij
λn=Z+i∞
−i∞c
dsij λiλjXij
λnsij
,1≤i<j≤n−2,
exp −λkλn−1Xk,n−1
λn=Z+i∞
−i∞b
dtkλkλn−1Xk,n−1
λntk
,1≤k≤n−3,
(2.6)
one can successively calculate the integrals over λn, λn−1, ..., λ1(the integration measures are
defined in (A.4)). As a result, the conformal integral can be represented as a product
Ia
n(x) = La
n(x)Ia
n(η),(2.7)
where the leg-factor
La
n(x) = Zn−2,n−1
n−a′
n
n−1
Y
i=1 X−ai
i,n , a′
i=D
2−ai,(2.8)
4The most frequently used notation and conventions are collected in appendix A.
– 5 –

depends on the following combinations of squared distances,
Zij
k≡Zi,j
k=Xij
XikXj k
, i, j =k , (2.9)
which conveniently define the cross-ratios as5
(ηk)ij
ml ≡(ηk)i,j
m,l =Zij
k
Zml
k
=XijXmkXlk
XikXj kXml
.(2.10)
The splitting (2.7) is the consequence of the conformal covariance of (2.1): the leg-factor
encodes the conformal transformation properties, while the function Ia
n(η), called a bare
conformal integral, depends only on the cross-ratios (2.10). The bare conformal integral is
represented as the following n(n−3)/2-folded Mellin-Barnes integral6
Ia
n(η) = Na
nZ+i∞
−i∞Y
1≤i<j≤n−2c
dsij (ηn)i,j
n−2,n−1sij n−3
Y
k=1 b
dtk(ηn)k,n−1
n−2,n−1tk
×Γ |t1,n−3|+X
1≤i<j≤n−2
sij +a′
n!n−3
Y
l=1
Γ al+tl+
l−1
X
j=1
sjl +
n−2
X
j=l+1
slj !
×Γ αn−2,n −X
1≤i<j≤n−3
sij − |t1,n−3|!Γ αn−1,n −X
1≤i<j≤n−2
sij!.
(2.11)
Here, αi,j is defined in (A.2) and the number of cross-ratios ηequals the number of integrals
on the right-hand side of (2.11). By construction, the integrals over each variable are balanced
(see e.g. [49]), which means that the contours can be closed either to the left or to the right.
In fact, the choice of a contour is determined by a domain of convergence in which one wants
to obtain the asymptotic expansion of the conformal integral. In its turn, this determines a
specific set of cross-ratios η.
As nincreases, the evaluation of the integral (2.11) quickly becomes difficult due to the
complex pole structure of the integrand. In fact, the only well-understood example is given
by the box (n= 4) conformal integral which has been calculated by many authors in various
contexts, see e.g. [35] and references therein. Recently, however, significant progress has been
made in the computation of higher-folded Mellin-Barnes integrals involving various computer
algebra methods [50,51], in particular, the hexagon (n= 6) conformal integral was calculated
in [52,53].
We note finally that for some applications it is not necessary to know the whole integral
(2.11), but only that part of it which can be obtained by evaluating over a particular subset
of poles. E.g. this happens when calculating conformal blocks within the shadow formalism
of CFTD[38,39,44].
5The cross-ratios here are generally defined as ratios of cubic combinations of the squared distances. When
any two of upper and lower indices coincide they reduce to ratios of quadratic combinations. With a slight
abuse of terminology we will call them as quadratic and cubic cross-ratios.
6Appendix Billustrates the formalism of Mellin-Barnes integrals using examples of hypergeometric-type
functions.
– 6 –

2.2 Bipartite Mellin-Barnes representation
In what follows, we suggest a different way of representing conformal integrals (2.1) in terms
of Mellin-Barnes integrals which does not use the Symanzik conformal trick. The advantage
of our approach is that a part of the n-point conformal integral can be computed effortlessly,
while the remaining part is conjectured to be recovered by using permutation invariance of
the whole conformal integral (2.1) (see section 3).
The first step is to remove 1/|λ1,n|from the exponential in (2.4) by changing integration
variables λi→ |λ1,n|λiand obtaining
Ia
n(x)=2Na
nZ+∞
0
n
Y
i=1 dλi
λi
λai
i1
|λ1,n|D−|a1,n |exp −X
1≤i<j≤n
λiλjXij.(2.12)
Then, the contribution of |λ1,n|cancels due to the conformality constraint (2.2). Evaluating
the integral over λnyields
Ia
n(x) = 2Na
n−1Z+∞
0
n−1
Y
i=1 dλi
λi
λai
in−1
X
i=1
λiXin−an
exp −X
1≤i<j≤n−1
λiλjXij.(2.13)
Now, changing integration variables as λi→X−1
in λi(no summation over i) one finds
Ia
n(x) = 2Na
n−1
n−1
Y
i=1
X−ai
i,n Z+∞
0
n−1
Y
i=1 dλi
λi
λai
i1
|λ1,n−1|anexp −X
1≤i<j≤n−1
λiλjZij
n,(2.14)
where Zij
nare given by (2.9). Rescaling λi→λipλn−1for i= 1 , ..., n−2 and λn−1→pλn−1
one can evaluate the integral over λn−1:
Ia
n(x) = Na
n−1Γ(a′
n)
n−1
Y
i=1
X−ai
in Z+∞
0
n−2
Y
j=1 dλj
λj
λaj
j
×1 + |λ1,n−2|−an
X
1≤l<m≤n−2
λlλmZlm
n+
n−2
X
k=1
λkZk,n−1
na′
n
.
(2.15)
Introducing new variables by λi=sσifor i= 1, ..., n −3 and λn−2=s(1 − |σ|), where
|σ|≡|σ1,n−3|, the integral over sis recognized as the integral representation of the Gauss
hypergeometric function (B.18). As a result, the conformal integral is again represented as
the product (2.7) with the same leg-factor La
n(x) (2.8), but with the bare integral Ia
n(η)
written in the form
Ia
n(η) = Na
n−2Γ"a′
n−1, a′
n
D
2#ZΩ
dn−3σ
n−3
Y
i=1
σai−1
i(1 − |σ|)an−2−1
1−σ·ξa′
n2F1"a′
n−1, a′
n
D
21−ξ(σ)#,
(2.16)
– 7 –

where the integration domain is the standard orthogonal simplex Ω = {σ∈Rn−3:σj≥
0, j = 1, ..., n −3 ; |σ| ≤ 1}, and
ξ(σ) = X
1≤i<j≤n−3
σiσj(ηn)ij
n−2,n−1+ (1 − |σ|)
n−3
X
l=1
σl(ηn)l,n−2
n−2,n−1
1−σ·ξ,
σ·ξ=
n−3
X
l=1
σlξl, ξi= 1 −(ηn)i,n−1
n−2,n−1,1≤i≤n−3,
(2.17)
where (ηn)ij
n−2,n−1are the cross-ratios (2.10). Note that there are (n−3)(n−4)/2+(n−3) =
(n−3)(n−2)/2 terms in the numerator of ξ(σ), each of which contains one cross-ratio.
Adding (n−3) cross-ratios from the denominator one concludes that the bare conformal
integral (2.16) does depend on n(n−3)/2 cross-ratios of the type (2.10), as discussed below
(2.11).
Let us now express the bare integral (2.16) through the Mellin-Barnes integrals. For this
purpose, it turns out to be convenient to use the analytic continuation formula for the Gauss
hypergeometric function (B.25) that results in splitting the bare integral into two terms
Ia
n(η) = I(1),a
n(η) + I(2),a
n(η),(2.18)
where the terms on the right-hand side which we refer to as the first and second bare integrals
are given by
I(1),a
n(η) = Na
nΓ(αn−1,n , a′
n−1, a′
n)ZΩ
dn−3σ
n−3
Y
i=1
σai−1
i(1 − |σ|)an−2−1
(1 −σ·ξ)a′
n
×2F1"a′
n−1, a′
n
1−αn−1,n ξ(σ)#,
(2.19)
I(2),a
n(η) = Na
nΓ(−αn−1,n , an−1, an)ZΩ
dn−3σ
n−3
Y
i=1
σai−1
i(1 − |σ|)an−2−1
(1 −σ·ξ)a′
n
×ξ(σ)αn−1,n 2F1"an−1, an
1 + αn−1,n ξ(σ)#.
(2.20)
This is a bipartite representation of the n-point (bare) conformal integral. A few comments
are in order.
•Using a certain analytic continuation formula is a formal trick because we cannot pre-
cisely describe a domain of the functional argument (2.17). Nonetheless, this procedure
– 8 –

will eventually lead to representing the conformal integral as a multivariate powers series
in n(n−3)/2 cross-ratios which is valid for a particular arrangement of points xi.
•We note that one could directly represent the Gauss hypergeometric function in (2.16)
by means of one of its Mellin-Barnes representations (B.24). This yields a representation
of the conformal integral in terms of the Mellin-Barnes integrals given in appendix C.
However, this particular representation turns out to be less computationally efficient
than that one we derive in this section.
•Similarly, another analytic continuation of the Gauss hypergeometric function could be
used that would change the form of two terms in (2.18). Nonetheless, the convenience of
choosing this particular partitioning of the conformal integral can be seen, in particular,
in the n= 3 case, when the second bare integral vanishes since the numerator of (2.17)
has no terms, while the first bare integral leads to the star-triangle relation [1]
Ia
3(x) = S(123)
3X−a′
3
12 X−a′
2
13 X−a′
1
23 ,where S(123)
3≡Γ"a′
1, a′
2, a′
3
a1, a2, a3#,(2.21)
which is commonly represented as the following diagrammatic equality
1
2
3
a1
a2
a3
1
2
3
a0
1
a0
2
a0
3
2
=S(123)
3
1
2
3
a1
a2
a3
1
2
3
a0
1
a0
2
a0
3
2
•In what follows we restrict n≥3 so that the 1-point and 2-point conformal integrals
fall out of our consideration. These lower-point integrals diverge and require careful
regularizations which have been extensively studied, see e.g. [54–57]. Notably, the
2-point conformal integral being proportional to δ(x1−x2) is the cornerstone of the
shadow formalism of CFTD[33,58].
The term I(1),a
n.Let us now explicitly calculate the first bare integral (2.19). To integrate
over σiwe use the Mellin-Barnes representation for the Gauss hypergeometric function (B.19):
2F1"a′
n−1, a′
n
1−αn−1,n ξ(σ)#= Γ"1−αn−1,n
a′
n−1, a′
n#Z+i∞
−i∞b
dtΓ"a′
n−1+t, a′
n+t
1−αn−1,n +t#−ξ(σ)t.(2.22)
When n= 4, the numerator of (2.17) contains only one term, and thus we are left with the 1-
dimensional integral over σ1which can be transformed into the Gauss hypergeometric function
– 9 –

(B.18). At n > 4 the functional argument (2.17) is more complicated and its numerator can be
converted into the (n−3)(n−2)/2−1 Mellin-Barnes integrals by means of the Mellin-Barnes
expansion (B.21):
X
1≤i<j≤n−3
σiσj(ηn)ij
n−2,n−1+ (1 − |σ|)
n−3
X
l=1
σl(ηn)l,n−2
n−2,n−1t
=1
Γ(−t)
×Z+i∞
−i∞
n−4
Y
l=1 b
dtlZ+i∞
−i∞Y
1≤k<m≤n−3c
dskm Γ|t1,n−4|+X
1≤i<j≤n−3
sij −t
×
n−4
Y
l=1 (1 − |σ|)σl(ηn)l,n−2
n−2,n−1tlY
1≤i<j≤n−3σiσj(ηn)ij
n−2,n−1sij
×(1 − |σ|)σn−3(ηn)n−3,n−2
n−2,n−1t−|t1,n−4|−P1≤i<j≤n−3sij .
(2.23)
The integral over σiin (2.19) is recognized as the integral representation of the Lauricella
function F(n−3)
D(B.15). Thus, we obtain the first bare integral (2.19) in the following form
I(1),a
n=Na
nΓ (1 −αn−1,n , αn−1,n )Z+i∞
−i∞
dt
2πi Γ"a′
n−1+t , a′
n+t
1−αn−1,n +t#−(ηn)n−3,n−2
n−2,n−1t
×Z+i∞
−i∞
n−4
Y
l=1 b
dtl(ηn)l,n−2
n−3,n−2tlZ+i∞
−i∞Y
1≤i<j≤n−3c
dsij (ηn)ij
n−3,n−2sij 
×Γ |t1,n−4|+X
1≤i<j≤n−3
sij −t!Γ"C(1) − |B(1)|,B(1)
C(1) #F(n−3)
D"A(1),B(1)
C(1) ξ#,
(2.24)
where ξ={ξ1, ..., ξn−3}are given in (2.17), and the parameters here, including Γ(B(1))≡
Γ(B(1)
1, ..., B(1)
n−3), are expressed in terms of propagator powers aias
A(1) =a′
n+t , C(1) =|a1,n−2|+ 2t , B(1)
n−3=an−3+t− |t1,n−4| − X
1≤i<j≤n−4
sij ,
B(1)
l=al+tl+
l−1
X
j=1
sjl +
n−3
X
j=l+1
slj , l = 1, ..., n −4.
(2.25)
By construction, all the contours in (2.24) for all integrals can be closed both to the left
and right. When the contour over tis closed to the right, only the poles coming from
Γ|t1,n−4|+P1≤i<j≤n−3sij −tcontribute to the integral (see the third line in (2.24)).
Calculating the integral over tthen leads to the fact that closing contours to the right in all
the remaining integrals over tiand sij only one set of poles contributes for each integration
– 10 –

variable, namely, these are the poles coming from Γ(−ti) and Γ(−sij ) (cf. the integration
measure (A.4)). In this way, the whole integral (2.24) can be explicitly calculated as
I(1),a
n=Na
nΓ (1 −αn−1,n , αn−1,n )
n−3
Y
l=1
val
l
∞
X
{kl,pij }=0
n−3
Y
l=1
ukl
l
kl!Y
1≤i<j≤n−3
wpij
ij
pij!
×Γ"an−2+|k|, a′
n+|k|+|p|,A(1),B(1)
1−αn−1,n +|k|+|p|,C(1) #F(n−3)
D"A(1),B(1)
C(1) 1−v#,
(2.26)
where we used the transformation formula for the Lauricella function (B.17); the parameters
here are related to propagator powers and summation indices as follows
A(1) =a′
n−1+|k|+|p|,B(1)
l=al+kl+ 2pl,C(1) =|a1,n−2|+ 2(|k|+|p|),(2.27)
where l= 1, ..., n −3; the sums are denoted as
|p|=X
1≤i<j≤n−3
pij =
n−3
X
l=1
pl, pl=1
2l−1
X
j=1
pjl +
n−3
X
j=l+1
plj , l = 1, ..., n −3.(2.28)
The cross-ratios in (2.26) are given by
ui= (ηn−1)i,n−2
n−2,n , i = 1, ..., n −3,
wij = (ηn−1)i,j
n−2,n ,1≤i<j≤n−3,
vi= (ηn−1)i,n
n−2,n , i = 1, ..., n −3,
(2.29)
where ui, vj, and wkl are quadratic and cubic, respectively (see footnote 5). Given the mul-
tivariate power series (2.26) one can find its region of convergence, that, however, can be a
rather complicated problem. For simplicity, we suppose that such a series converges at least
when the cross-ratios (2.29) tend to specific points. In particular, since the Lauricella function
FD(B.4) is supported on the polydisk (B.5), we can claim that the multivariate power series
(2.26) converges when
ui, wkl →0 and vj→1.(2.30)
Clearly, a range of the cross-ratios is determined by a particular arrangement of points xiand
(2.30) corresponds to the coincidence limit
xn→xn−1.(2.31)
– 11 –

The term I(2),a
n.Consider the second bare integral (2.20). Repeating the same steps as in
the previous paragraph one shows that
I(2),a
n=Na
nΓ (1 + αn−1,n,−αn−1,n)Z+i∞
−i∞b
dtΓ"an−1+t, an+t
1 + αn−1,n +t, −αn−1,n −t#(ηn)n−3,n−2
n−2,n−1αn−1,n+t
×Z+i∞
−i∞
n−4
Y
l=1 b
dtl(ηn)l,n−2
n−3,n−2tlZ+i∞
−i∞Y
1≤i<j≤n−3c
dsij (ηn)i,j
n−3,n−2sij 
Γ |t1,n−4|+X
1≤i<j≤n−3
sij −t−αn−1,n!Γ"C(2) − |B(2) |,B(2)
C(2) #F(n−3)
D"A(2),B(2)
C(2) ξ#,
(2.32)
where
A(2) =an−1+t , C(2) =|a1,n−2|+ 2αn−1,n + 2t ,
B(2)
l=al+tl+
l−1
X
j=1
sjl +
n−3
X
j=l+1
slj , l = 1, ..., n −4,
B(2)
n−3=an−3+αn−1,n +t− |t1,n−4| − X
1≤i<j≤n−4
sij ;
(2.33)
the arguments of the Lauricella function F(n−3)
Dare given in (2.17).
The pole structure of the second bare integral I(2),a
n(x) is more complicated than that
of the first bare integral I(1),a
n(x) so we cannot immediately calculate the integrals in (2.32).
The reason for this difference is the use of particular analytic continuation formula for the
Gauss hypergeometric function in the integral (2.16), which makes the two terms (2.19) and
(2.20) different due to the additional factor of ξ(σ) in (2.20). In turn, when closing contours
to the right, this gives an additional set of poles over tin (2.32). Nonetheless, the problem of
computing the second bare integral I(2),a
n(x) can be addressed by invoking invariance of the
whole conformal integral under the symmetric group.
3 Constraints from the permutation invariance
The symmetric group Snnaturally acts on functions of coordinates and propagator powers
by permuting elements from x= (x1, ..., xn) and a= (a1, ..., an) as xi→xπ(i)and ai→aπ(i).
The conformal integral (2.1) is invariant under the symmetric group:
∀π∈ Sn: R(π)◦Ia
n(x) = Ia
n(x),(3.1)
where R(π) stands for the representation of π.7This invariance property is manifestly seen
on the conformal graph on fig. 1which remains invariant with respect to any permutation of
legs which are labelled by (xi, ai).
7We will use the one-line notation for elements of the symmetric group. The identical permutation will be
denoted as eand R(e) = 1.
– 12 –

On the other hand, a splitting of the conformal integral into two terms provided by the
bipartite representation is not invariant with respect to the symmetric group. In this way,
having found the first term explicitly, we expect to generate the unknown second term by
acting on the first one with particular permutations.
3.1 Analytic continuation
To simplify the study of permutation invariance, it is useful to analytically continue the known
first bare integral, which is originally defined at u,w→0 and v→1, into the other domain,
where all cross-ratios are near zero:
u,w,v→0.(3.2)
Recalling the explicit expression (2.26) we notice that this can be achieved by exploiting the
analytic continuation formulas for the Lauricella function F(n−3)
Dderived by Bezrodnykh in
[59–63] (see also appendix B). Indeed, by applying the analytic continuation formula (B.26)
one can split (2.26) into n−2 terms thereby obtaining the first bare integral in the domain
(3.2) as the following sum8
I(1),a
n(η) =
n−3
X
q=0
U(q),a
n(η),(3.3)
where the q= 0 term:
U(0),a
n=S(n−2,n−1,n)
n
n−3
Y
i=1
vai
i
∞
X
{ki,pij ,mi}=0
n−3
Y
i=1
uki
i
ki!Y
1≤i<j≤n−3
wpij
ij
pij!
n−3
Y
l=1
(vl)ml
ml!
×
n−3
Y
l=1
(al)kl+2pl+ml(a′
n−1)|k|+|p|+|m|(−)|p|
(1 −αn−1,n)|k|+|p|(1 −αn−2,n−1)|p|+|m|
,
(3.4)
the terms with q= 1, ..., n −3:
U(q),a
n=S(q,n−1,n)
nva′
n−|a1,q−1|
q
q−1
Y
l=1
val
l
∞
X
{ki,pij ,mi}=0
n−3
Y
l=1
(U(q)
l)kl
kl!Y
1≤i<j≤n−3
(W(q)
ij )pij
pij!
n−3
Y
l=1
(V(q)
l)ml
ml!
×(a′
n− |a1,q−1|)|mq|+|kq,n−3|+|pq|
(1 + a′
n− |a1,q|)|mq|+|kq+1,n−3|+|pq+1|
(−)|kq+1,n−3|+|pq+1|
×
q−1
Y
l=1
(al)kl+2pl+ml(an−2)|k|+mq
n−3
Y
l=q+1
(al)kl+2pl+ml
(1 −αn−1,n)|k|+|p|
.
(3.5)
8Here, the q-th term U(q),a
narises from the q-th term Dq(B.28).
– 13 –

Here, |p|is defined in (2.28), other notations are given in appendix A. The new cross-ratios
in (3.5) are expressed in terms of the previously introduced cross-ratios (2.29) as follows
U(q)
l=




ul, l = 1, ..., q ,
ulvq
vl
, l =q+ 1, ..., n −3,V(q)
l=










vl
vq
, l = 1, ..., q −1,
vq, l =q ,
vq
vl
, l =q+ 1, ..., n −3,
(3.6)
W(q)
ij =















wij
vq
,1≤i < j ≤q ,
wij
vj
, i = 1, ..., q , j =q+ 1, ..., n −3,
wijvq
vivj
, q + 1 ≤i<j≤n−3.
(3.7)
The overall coefficients in (3.4)-(3.5) are defined as
S(ijk)
n= Γ"|ai,j| − D
2,|aj,k| − D
2,|ak,i| − D
2
ai, aj, ak#,1≤i < j < k ≤n . (3.8)
These generalize the prefactor in the star-triangle relation (2.21). Note that the last multiplier
in the numerator contains |ak,i|with k > i, which is defined in (A.2).
It is important to stress that having the expansion (3.3) near the origin in the space of
cross-ratios, i.e. at ui, wkl , vj→0, means that other arguments must also tend to zero, i.e.
U(q)
i,W(q)
kl ,V(q)
j→0.(3.9)
Since U(q)
i,W(q)
kl ,V(q)
jare (rational) functions of ui, wkl, vj(3.6), then the condition (3.9)
determines a specific way ui, wkl, vjtend to zero.9
For future purposes, we note here that there are only two terms in the expansion (3.3)
which can be related by the action of a longest cycle Cn≡(12...n)∈ Sn. Namely,
Cn◦La
n(x) U(0),a
n(η)=La
n(x) U(1),a
n(η),(3.10)
where by Cn= R(Cn) we denote the action of Cnon functions of xand a. The proof is fairly
straightforward if we consider how Cnchanges both the leg-factor (2.8):
Cn◦La
n(x) = va′
n
1u−an−2
1
n−3
Y
j=2 w1j
vj−aj
La
n(x),(3.11)
9E.g. there are two variables such that x, y →0 and x/y →0 that means that x→0 faster than y→0.
– 14 –

and the cross-ratios (2.29):
Cn◦ui=v1
vi+1
,Cn◦un−3=v1,Cn◦wij =wi+1,j+1v1
vi+1vj+1
,
Cn◦vi=w1,i+1
vi+1
,Cn◦vn−3=u1,Cn◦wi,n−3=ui+1v1
vi+1
,
(3.12)
where i, j = 1, ..., n −4 and i < j. The relation (3.10) stems directly from the fact that
both power series U(0),a
n, U(1),a
narise from the Lauricella function F(n−3)
D(see (B.11) and
(B.28)) by means of the analytic continuation formula (B.26). Note that all other terms
(q= 0,1) in (3.3) arise from the generalized hypergeometric series G(n−3,q)(see (B.28))
which are functions of different types for different q. Hence, other possible relations among
U(q),a
ninvolving permutations different from the longest cycle cannot be obtained in such an
obvious way.
In the subsequent sections we will study the bipartite representation (2.18) supplemented
with the analytic continuation formula (3.3) and the symmetric group action on the particular
examples of the box, pentagon, and hexagon conformal integrals. This will allow us to have a
number of observations, conclusions, and exact results which will eventually be summarized
in section 4in the form of the reconstruction conjecture.
3.2 Box integral
In this case, the first bare integral I(1),a
4(u, v) is supported on some domain on the (u, v )-plane
of cross-ratios
u=X12X34
X13X24
, v =X14X23
X13X24
.(3.13)
Note that there are no cubic cross-ratios in this case. According to the general formula (2.26),
it can be explicitly calculated as the asymptotic expansion at (u, v)→(0,1):
I(1),a
4(u, v) = Na
4Γ1−α3,4, α3,4va1
×∞
X
k=0
uk
k!Γ"a1+k , a2+k , a′
3+k , a′
4+k
a1+a2+ 2k , 1−α3,4+k#2F1"a1+k , a′
3+k
a1+a2+ 2k1−v#,(3.14)
We note that an arbitrary permutation from the symmetric group S4acting on the first
bare integral I(1),a
4(3.14) produces a power series whose domain of convergence is generally
different from the original domain. For instance, a transposition (23) ∈ S4acts on the cross-
ratios as
σ23 ◦u=1
u, σ23 ◦v=v
u,(3.15)
where σij ≡R((ij)) denotes the action of (ij) on functions of xand a. Obviously, such a
permutation sends the original domain (u, v)→(0,1) to (u, v)→ ∞. In order to return back
to the original domain, the action (3.15) must be accompanied by making a suitable analytic
continuation of the resulting power series for I(1),a
4. However, analytic continuation formulas
– 15 –

are known only for a limited number of generalized hypergeometric functions. Thus, to avoid
the need to use analytical continuation we first try permutations which leave the cross-ratios
invariant thereby keeping any domain of convergence intact.
3.2.1 Kinematic group
Those elements of the symmetric group S4which leave the cross-ratios invariant,
∃π∈ S4: R(π)◦u=u , R(π)◦v=v , (3.16)
form a subgroup called kinematic [64]:
Skin
4=ne, (12)(34),(13)(24),(23)(14)o≡Z2×Z2⊂ S4.(3.17)
By construction, a given element of the kinematic group generates from the first bare integral
a power series in the same domain (u, v)→(0,1). E.g. let us consider a permutation (23)(14)
and examine its action on I(1),a
4(u, v). Note that despite the cross-ratios are invariant such
an action is not trivial since the permutation also interchanges a2↔a3and a1↔a4. To this
end, one considers the invariance condition (3.1) which in this case takes the form
σ23σ14 ◦Ia
4(x) = Ia
4(x).(3.18)
Recalling the double product representation of the conformal integral (2.7) and using that
the 4-point leg-factor (2.8) transforms as
σ23σ14 ◦La
4(x) = uα3,4va1−a4La
4(x),where La
4(x) = X−a1
14 X−a′
4
23 X−α2,4
24 X−α3,4
34 ,(3.19)
one obtains that the bare conformal integrals are related as
uα3,4va1−a4σ23σ14 ◦ I(1),a
4(u, v) + σ23σ14 ◦ I(2),a
4(u, v)=I(1),a
4(u, v) + I(2),a
4(u, v).(3.20)
It can be seen that the asymptotics of the first term on the left-hand side at (u, v)→(0,1)
is different from that of I(1),a
4(3.14). It is the prefactor uα3,4va1−a4which changes the
asymptotic behaviour since by construction both I(1),a
4and σ23σ14 ◦ I(1),a
4have the same
asymptotics in the considered domain. Thus, the first term on the left-hand side must coincide
with the second term on the right-hand side which is the second bare integral I(2),a
4, i.e.
I(2),a
4(u, v) = uα3,4va1−a4σ23σ14 ◦ I(1),a
4(u, v).(3.21)
Explicitly, this gives the second bare integral as the asymptotic expansion at (u, v)→(0,1):
I(2),a
4(u, v) = Na
4Γ1−α1,2, α1,2uα3,4va1
×∞
X
k=0
uk
k!Γ"a′
1+k , a′
2+k , a3+k , a4+k
a3+a4+ 2k , 1−α1,2+k#2F1"a′
2+k , a4+k
a3+a4+ 2k1−v#,
(3.22)
– 16 –

cf. (3.14). We conclude that the second bare integral can be reconstructed from the first one
using a certain permutation. On the other hand, going back to the Mellin-Barnes represen-
tation of I(2),a
4(2.32), we note that in this case the integrals can be evaluated explicitly and
the resulting function does coincide with (3.22).
The remaining permutations from Skin
4can be considered along the same lines. One finds
out that acting with (13)(24) yields the same result, i.e. this permutation also reconstructs
the second bare integral I(2),a
4, while (12)(34) keeps each bare integral invariant, i.e.
σ12σ34 ◦La
4(x)I(m),a
4(u, v)=La
4(x)I(m),a
4(u, v), m = 1,2.(3.23)
Thus, the conformal integral can be represented as
Ia
4(x) = 1+σ23σ14 ◦La
4(x)I(1),a
4(η)
=1+σ13σ24◦La
4(x)I(1),a
4(η).
(3.24)
Here, a key concept is that the conformal integral in an appropriate coordinate domain can
be generated by acting with particular permutations on a single function which will be called
amaster function.10
It should be noted that for n > 4 the kinematic groups are all trivial, Skin
n={e}[64].
This motivates the search for other subgroups of Snwhich also reconstruct the conformal in-
tegrals from a given set of master functions. For any nthe problem looks quite complicated.
Nonetheless, going back to the n= 4 case and bearing in mind that then all the Mellin-Barnes
integrals involved into the bipartite representation can be calculated explicitly, an educated
guess is to consider the cyclic group Z4⊂ S4. In the higher-point case, this naturally general-
izes to picking Zn⊂ Snas a generating group: a subgroup of Snwhich allows reconstructing
the full conformal integral from a given set of master functions.
3.2.2 Cyclic group
The cyclic group Z4is
Z4=ne, C4,(C4)2,(C4)3o⊂ S4,(3.25)
where C4= (1234) is a longest cycle. The longest cycle permutes the two cross-ratios
C4◦u=v , C4◦v=u , (3.26)
cf. (3.15). Hence, the cycles interchange domains (u, v)→(0,1) and (u, v)→(1,0). Note,
however, that u, v = 0 is a fixed point of Z4that suggests that we first analytically continue
the first bare integral (3.14) from (u, v)→(0,1) to (u, v)→(0,0) and then consider cycles
10This observation is known in the literature on (non-)conformal integrals. E.g. a similar permutation
generated representation is valid for the non-parametric hexagon integral with three massive corners [25] or
for the non-parametric pentagon integral [15] (see also appendix D.2).
– 17 –

as generating permutations which reconstruct the full conformal integral near the origin on
the (u, v)-plane.
Thus, we can apply the general analytic continuation formula (3.3) which in this case
takes the form
I(1),a
4(u, v) = U(0),a
4(u, v)+U(1),a
4(u, v),(3.27)
where
U(0),a
4(u, v) = S(234)
4va1F4"a1, a′
3
1−α3,4,1 + α1,4u, v#,
U(1),a
4(u, v) = S(134)
4va′
4F4"a2, a′
4
1−α3,4,1−α1,4u, v#.
(3.28)
Here, F4is the fourth Appell function (B.13) which resulted from using the splitting identity
(B.14), the coefficient S(ijk)
nis defined in (3.8).
Let us now consider how Z4acts on U(0),a
4and U(1),a
4. It turns out that these two terms
are related by the cyclic permutation as follows
C4◦La
4(x) U(0),a
4(u, v)=La
4(x) U(1),a
4(u, v),(3.29)
see the general statement (3.10). Practically, this relation implies that there is only one
master function which we choose to be La
4(x) U(0),a
4(u, v). By construction, this new master
function is supported near the origin (u, v) = 0. The cyclic group Z4is of order four and
hence its elements generate from the master function three more functions which are also the
fourth Appell functions supported on the same domain. Introducing the collective notation
one can represent the resulting four functions (including the master one) as follows
φ(234)
4(x) := La
4(x) U(0),a
4(u, v) = S(234)
4V(234)
4(x)F4"a′
3, a1
1−α3,4,1−α2,3u, v#,
φ(134)
4(x) := (C4)1◦φ(234)
4(x) = S(134)
4V(134)
4(x)F4"a2, a′
4
1−α3,4,1−α1,4u, v#,
φ(124)
4(x) := (C4)2◦φ(234)
4(x) = S(124)
4V(124)
4(x)F4"a′
1, a3
1−α1,2,1−α1,4u, v#,
φ(123)
4(x) := (C4)3◦φ(234)
4(x) = S(123)
4V(123)
4(x)F4"a′
2, a4
1−α1,2,1−α2,3u, v#,
(3.30)
– 18 –

where S(ijk)
nare defined in (3.8), the leg-factors V(ijk)
4(x) are given by
V(234)
4(x) = X−a1
13 Xα14
23 Xα12
34 X−a′
3
24 , V (134)
4(x) = X−a2
24 Xα12
34 Xα23
14 X−a′
4
13 ,
V(124)
4(x) = X−a3
13 Xα23
14 Xα34
12 X−a′
1
24 , V (123)
4(x) = X−a4
24 Xα34
12 Xα14
23 X−a′
2
13 ,
(3.31)
and any two of them are related by a cyclic permutation, e.g. V(134)
4= C4◦V(234)
4(this also
holds true for S(ijk)
4, e.g. S(134)
4= C4◦S(234)
4). The functions φ(ijk)
4(x) will be called basis
functions.
One can show that in terms of the basis functions both the first and second contributions
to the conformal integral can be represented as
La
4(x)I(1),a
4(u, v) = φ(234)
4(x) + φ(134)
4(x),(3.32)
La
4(x)I(2),a
4(u, v) = φ(124)
4(x) + φ(123)
4(x).(3.33)
Here, the first line is just (3.27) in the new notation, while the second line results from
analytically continuing the second bare integral (3.22) to the domain (u, v)→0 using the
same continuation formula (B.25). In this way, one obtains yet another representation of the
full conformal integral in terms of the generalized hypergeometric series generated from the
master function by the cyclic group Z4:
Ia
4(x) = 1+ (C4)1+ (C4)2+ (C4)3◦φ(234)
4(x)
=φ(234)
4(x) + φ(134)
4(x) + φ(124)
4(x) + φ(123)
4(x).
(3.34)
This expansion in basis functions is valid within the domain of convergence of the fourth
Appell function √u+√v < 1 and reproduces the 4-point conformal integral in the form
known in the literature.11 Finally, note that the kinematic group considered in the previous
section also acts on the basis functions (3.30) so that their sum (3.34) remains invariant, see
appendix E.
3.2.3 Consistency checks
There are at least two possible non-trivial ways to test the resulting expressions. First, the
most direct way to check the n-point parametric conformal integral is to reduce a number of
points by one, i.e. n→n−1 along with aj→0 for some jfrom 1 to nand then constrain
the remaining propagator powers by the (n−1)-point conformality condition
n
X
i=1
i=j
ai=D , (3.35)
11The explicit calculation of the 4-point conformal integral has a long history which is discussed in detail in
e.g. [35].
– 19 –

cf. (2.2). The result must be given by the (n−1)-point parametric conformal integral.
Second, one can consider the non-parametric case i.e. choose all ai= 1, and then compare
the resulting expression with those known in the literature which were obtained by other
methods. These two checks are described in detail below.12
Reduction to the star-triangle. From the very definition of the 4-point integral (2.1) it
follows that if one of propagator powers equals zero, e.g. a4= 0, then the 4-point conformal
integral reduces to the 3-point conformal integral:
Ia
4(x)a4=0 =Ia1,a2,a3
3(x1, x2, x3).(3.36)
Given the 4-point conformal integral calculated in the form (3.34) one can see from (3.30)
that by setting a4= 0 all basis functions except φ(123)
4vanish due to Γ(a4→0) → ±∞ in
their denominators. The only non-vanishing contribution then leads to13
Ia
4(x)a4=0 =S(123)
4V(123)
4(x)a4=0 ,(3.37)
where we used that the Appell function F4equals 1 if one of the upper parameters is 0, see
(B.13). Substituting (3.8) and (3.31) one finds out that the left-hand side of (3.37) reproduces
the star-triangle relation (2.21) provided that a1+a2+a3=D. The remaining limiting cases
ai= 0, i= 1,2,3, can be analyzed along the same lines. The only difference is which of the
series (3.30) is non-zero in this limit. However, the numbering of basis functions is designed
in such a way that when al= 0 is imposed, a basis function φ(ijk)
4survives which does not
have lamong upper indices, i.e. i, j, k =l.
Non-parametric box. For the 4-point conformal integral, the transition from the para-
metric to non-parametric case was considered in [35,67]. Here, we reproduce a part of this
analysis for completeness. The subtlety is that when choosing ai= 1 in D= 4 (cf. (2.2))
the resulting integral diverges because the basis functions (3.30) contain divergent Γ-function
prefactors. Introducing a cutoff parameter ϵ→0 as
a1=a2=a3= 1 , a4= 1 −2ϵ⇒D= 4 −2ϵ , (3.38)
and expanding the 4-point conformal integral (3.34) around ϵ= 0 one finds
I1,1,1,1−2ϵ
4(x) = 1
X13X24
Φ(u, v)
(1 −u−v)2−4uv +O(ϵ),(3.39)
12One may also consider conformal integrals as one of points goes to infinity, e.g. xn→ ∞, that partially
breaks conformal invariance. In this limit, the 4-point conformal integral is a particular case of the triangle
integral evaluated by Boos and Davydychev [65,66]. Note that conformal integrals with partially broken
conformal invariance are instrumental within the shadow formalism for CFT on nontrivial backgrounds [44].
13Note that the product V(ijk)
4S(ijk)
4can then be thought of as a generalization of the right-hand side of the
star-triangle relation (2.21).
– 20 –

where the Bloch-Wigner function Φ(u, v) is expressed in terms of polylogarithms [67]:
Φ(u, v) = π2
3+ ln uln v+ ln 1 + u−v−λ(u, v)
2uln 1−u+v−λ(u, v)
2v
+ 2 ln 1 + u−v−λ(u, v)
2u+ 2 ln 1−u+v−λ(u, v)
2v
−2 Li2
1 + u−v−λ(u, v)
2−2 Li2
1−u+v−λ(u, v)
2,
(3.40)
where Li2is a dilogarithm. Introducing variables u=z(1 −y) and v=y(1 −z) and using
the identity Li2z+ Li2(1 −z) = π2/6−ln zln(1 −z) the Bloch-Wigner function (3.40) can
be cast into the form [35]
Φ(z, y) = ln y(1 −z)ln z
1−y+ 2 Li2(1 −z)−2 Li2y . (3.41)
Then, the non-parametric box integral (3.39) can be given in a standard form [35,67]
I1,1,1,1
4(x) = 1
X13X24
Φ(z, y)
1−z−y.(3.42)
3.2.4 Summary of the box integral reconstruction
Let us briefly formulate an emerging strategy of reconstructing a given conformal integral in
some coordinate domain. We will keep narbitrary assuming that the box integral example
can be directly generalized to the higher-point case.
•Having calculated the first bare integral in the form of multivariate hypergeometric
series (2.26), we analytically continue the resulting function from the original region to
another region around the origin in the space of cross-ratios (u,w,v= 0) by means
of the analytic continuation formula (3.3) which splits I(1),a
ninto n−2 additive terms
U(q),a
n,q= 0,1, ..., n −3.
•Among these n−2 terms we single out n−3 terms which are not related through the
action of the cyclic group (see (3.10))
Zn={e, Cn,(Cn)2, ..., (Cn)n−1}⊂Sn,(3.43)
where Cn= (123...n)∈ Snis a longest cycle.
•Combining cyclically independent functions with the leg-factor (2.8) one defines n−3
master functions
φ(q,n−1,n)
n(x) = La
n(x) U(q),a
n, q = 2,3, ..., n −3,
φ(n−2,n−1,n)
n(x) = La
n(x) U(0),a
n.(3.44)
•The idea of reconstruction: the conformal integral Ia
n(x) is supposed to be given by a
sum of functions obtained by acting with all elements of Znon the master functions.
– 21 –

•The resulting expression can be checked in at least two ways:
–The n-point conformal integral reduces to the (n−1)-point one upon setting one
of propagator powers aequal to zero
Ia1,...,aj−1,aj,aj+1,...,an
n(x1, ..., xj−1, xj, xj+1, ..., xn)aj=0
=Ia1,...,aj−1,aj+1,...,an
n−1(x1, ..., xj−1, xj+1, ..., xn),
(3.45)
for any j= 1, ..., n. This reduction condition obviously follows from the very
definition (2.1).
–Going to the non-parametric case, the resulting I(1,1,...,1)
n(x) can be compared with
expressions known in the literature, if any.
In the next sections 3.3 and 3.4 we examine how this scheme works for the pentagon and
hexagon integrals and then in section 4we will be able to formulate the general reconstruction
conjecture.
3.3 Pentagon integral
For n= 5 the general formula (2.26) represents the first bare integral as the asymptotic
expansion at u1,2, w12 →0 and v1,2→1:
I(1),a
5=Na
5Γ1−α4,5, α4,5va1
1va2
2
∞
X
k1,k2,p12=0
uk1
1
k1!
uk2
2
k2!
wp12
12
p12!
×Γ"a1+k1+p12, a2+k2+p12, a3+|k|, a′
4+|k|+p12, a′
5+|k|+p12
a1+a2+a3+ 2(|k|+p12),1−α4,5+|k|+p12 #
×F1"a′
4+|k|+p12 , a1+k1+p12 , a2+k2+p12
a1+a2+a3+ 2(|k|+p12)1−v1,1−v2#,
(3.46)
where F1is the first Appell function (B.8). Among the respective cross-ratios, one is cubic
and the others are quadratic:
u1=X13X45
X14X35
, u2=X23X45
X24X35
, w12 =X12X34X45
X14X24X35
,
v1=X15X34
X14X35
, v2=X25X34
X24X35
.
(3.47)
According to the general strategy outlined in the previous section, we can try to recon-
struct the conformal integral from a set of master functions which can be found by particular
analytic continuation of the first bare integral. The analytic continuation formula (3.3)-(3.5)
splits the first bare integral (3.46) into three terms
I(1),a
5(η) = U(0),a
5(η)+U(1),a
5(η)+U(2),a
5(η),(3.48)
– 22 –

where
U(0),a
5(η) = S(345)
5va1
1va2
2
∞
X
k1,k2,p12,m1,m2=0
uk1
1
k1!
uk2
2
k2!
wp12
12
p12!
vm1
1
m1!
vm2
2
m2!
×(a1)k1+p12+m1(a2)k2+p12 +m2(a′
4)|k|+p12+|m|(−)p12
(1 −α4,5)|k|+p12 (1 −α3,4)p12+|m|
,
(3.49)
U(1),a
5(η) = S(145)
5va′
5
1
∞
X
k1,k2,p12,m1,m2=0
uk1
1
k1!
(u2v1/v2)k2
k2!
(w12/v2)p12
p12!
vm1
1
m1!
(v1/v2)m2
m2!
×(a2)k2+p12+m2(a3)|k|+m1(a′
5)|k|+p12+|m|(−)k2
(1 −α4,5)|k|+p12 (1 −α1,5)k2+|m|
,
(3.50)
U(2),a
5(η) = S(245)
5va1
1v−α15
2
∞
X
k1,k2,p12,m1,m2=0
uk1
1
k1!
uk2
2
k2!
(w12/v2)p12
p12!
(v1/v2)m1
m1!
vm2
2
m2!
×(a1)k1+p12+m1(a3)|k|+m2(−α1,5)m2+k2−m1(−)p12
(1 −α4,5)|k|+p12 (1 + α3,4)m2−m1−p12
.
(3.51)
These functions are supported near the origin of coordinates (3.2) such that the cross-ratios
satisfy the condition (3.9).
3.3.1 Cyclic group
The cyclic group is Z5=e, C5,(C5)2,(C5)3,(C5)4⊂ S5, where C5= (12345) is a longest
cycle. The general statement (3.10) about the relation of two terms in the analytic continu-
ation formula for the first bare integral now reads as
C5◦La
5(x) U(0),a
5(η)=La
5(x) U(1),a
5(η),(3.52)
where the 5-point leg-factor (2.8) is given by
La
5(x) = X−a1
15 X−a2
25 X−a′
5
34 X−α3,5
35 X−α4,5
45 .(3.53)
It means that among three functions in (3.48) there are two which are not related by cyclic
permutations, e.g. these are U(0),a
5and U(2),a
5. Then, adding the leg-factor we introduce two
master functions as
φ(345)
5(x) := La
5(x) U(0),a
5(η) = S(345)
5V(345)
5(x) P1"a1, a2, a′
4
1−α45,1−α34 ,u1, w12, u2, v1, v2#,
φ(245)
5(x) := La
5(x) U(2),a
5(η) = S(245)
5V(245)
5(x) P2"a1, a3,−α15
1−α45,1 + α34 u1,w12
v2
, u2,v1
v2
, v2#,
(3.54)
– 23 –

where we defined two generalized hypergeometric functions P1(D.1) and P2(D.2) as well as
two leg-factors
V(345)
5(x) = X−a1
14 X−a2
24 X−α45
45 X−α34
34 X−a′
4
35 ,
V(245)
5(x) = X−a1
14 X−a3
35 X−α15
24 X−α45
45 Xα34
25 ,
(3.55)
cf. (3.31).
The cyclic group Z5is of order five so that one can construct four more basis functions
from each of the master functions. In this way, we obtain ten basis functions which are listed
in appendix D. Note that our notation for the basis functions is designed in such a way
that the action of a cyclic permutation on the master function φ(ijk)
5cyclically permutes the
upper indices i, j, k. In particular, it follows that the two master functions (3.54) as well as
prefactors (3.55) are not related to each other since there are no cyclic permutations which
turn 345 into 245. Thus, the set of basis functions is split into two subsets of cyclically related
functions, i.e. into two representations of Z5.
Then, the 5-point parametric conformal integral is represented as the Z5-invariant sum
of ten basis functions:
Ia
5(x) =
4
X
j=0
(C5)j◦φ(345)
5(x) + φ(245)
5(x)
=φ(345)
5(x) + φ(145)
5(x) + φ(125)
5(x) + φ(123)
5(x) + φ(234)
5(x)
+φ(245)
5(x) + φ(135)
5(x) + φ(124)
5(x) + φ(235)
5(x) + φ(134)
5(x).
(3.56)
The expansion is defined in a domain near the origin in the space of cross-ratios. By con-
struction, both sides here are invariant under cyclic permutations.
Note that one can define a generalization of the kinematic group which we call an extended
kinematic group. Given two master functions (3.54), such a group generates the same set of
basis functions thereby providing an equivalent way of reconstructing the conformal pentagon
integral. Details are given in appendix E.
3.3.2 Reduction to the box integral
In appendix D.2 we check that going to the non-parametric case we indeed do reproduce the
formula for the non-parametric pentagon integral known in the literature [15]. Of course, we
can compare only in a given coordinate domain. Below we examine the reduction to the box
integral.
Setting one of propagator powers equal to zero, e.g. a5= 0, the pentagon integral has to
reduce to the box integral:
Ia
5(x)a5=0 =Ia1,a2,a3,a4
4(x1, x2, x3, x4).(3.57)
In fact, it is quite easy to verify this relation provided that Ia
5(x) is represented as (3.56)
At a5= 0 only those basis functions φ(ijk)
5survive which upper indices i, j, k = 5, while
– 24 –

the remaining basis functions vanish due to the presence of Γ(a5→0) → ±∞ in their
denominators, see (3.8). As a result,
Ia
5(x)a5=0 =φ(123)
5(x) + φ(234)
5(x) + φ(124)
5(x) + φ(134)
5(x)a5=0 .(3.58)
The basis function are written in terms of functions (D.1) and (D.2) which are reduced to
the fourth Appell function (B.13) when one of the upper parameters equals zero, i.e.
P1"0, A, B
C1, C2ξ#= P2"0, A, B
C1, C2ξ#=F4"A, B
C1, C2ξ3, ξ5#.(3.59)
On the other hand, the prefactors S(ijk)
5V(ijk)
5(x) are reduced as
S(123)
5V(123)
5a5=0 =S(123)
4V(123)
4, S(234)
5V(234)
5a5=0 =S(234)
4V(234)
4,
S(124)
5V(124)
5a5=0 =S(124)
4V(124)
4, S(134)
5V(134)
5a5=0 =S(134)
4V(134)
4.
(3.60)
Gathering (3.60) and (3.59) together and taking into account the symmetry properties of P1,2
(D.3)-(D.4), one can see that the right-hand side of the reduced pentagon integral (3.58) re-
produces the right-hand side of the box integral (3.34) so that the desired reduction condition
(3.57) holds true.
The other cases al= 0 for l= 1,2,3,4 can be analyzed in a similar way. The only
difference is which of basis functions φ(ijk)
5survive at al= 0, but our way of numbering
basis functions immediately indicates that these are functions with upper indices i, j, k =l.
Thus, our expression for the pentagon integral (3.56) consistently reduces to the box integral
formula (3.34).
3.4 Hexagon integral
Let us finally examine the reconstruction idea for the 6-point parametric conformal integral.
The first bare integral (2.26) is given by
I(1),a
6=Na
6Γ1−α5,6, α5,6va1
1va2
2va3
3
∞
X
{kl,pij }=0
uk1
1
k1!
uk2
2
k2!
uk3
3
k3!
wp12
12
p12!
wp13
13
p13!
wp23
23
p23!
×Γ"a4+|k|, a′
5+|k|+|p|, a′
6+|k|+|p|,B1,B2,B3
1−α5,6+|k|+|p|,C#
×F(3)
D"a′
5+|k|+|p|,B1,B2,B3
C1−v1,1−v2,1−v3#,
(3.61)
– 25 –

where |k|=k1+k2+k3,|p|=p12 +p13 +p23, and the parameters are encoded as
B1=a1+k1+p12 +p13 ,B2=a2+k2+p12 +p23 ,B3=a3+k3+p13 +p23 ,
C=a1+a2+a3+a4+ 2(|k|+|p|).
(3.62)
The nine cross-ratios in (3.61) are
u1=X14X56
X15X46
, u2=X24X56
X25X46
, u3=X34X56
X35X46
,
w12 =X12X45X56
X15X25X46
, w13 =X13X45 X56
X15X35X46
, w23 =X23X45 X56
X25X35X46
,
v1=X16X45
X15X46
, v2=X26X45
X25X46
, v3=X36X45
X35X46
.
(3.63)
In order to reconstruct the full hexagon conformal integral we represent the first bare
integral (3.61) near η=0by means of the analytic continuation formula (3.3):
I(1),a
6(η) = U(0),a
6(η)+U(1),a
6(η)+U(2),a
6(η)+U(3),a
6(η),(3.64)
where each term can be read off from (3.4)-(3.5). According to the general formula (3.10)
two of four terms here are related as
C6◦La
6(x) U(0),a
6(η)=La
6(x) U(1),a
6(η),(3.65)
where C6= (123456) ∈ S6. Again, U(0),a
6, U(2),a
6, U(3),a
6can be chosen as master functions,
while the corresponding basis functions φ(ijk)
6are generated from them by cycle permutations.
Since the respective cyclic group Z6is of order six, then the three master functions generate
a set of 18 basis functions.
At this point, we could suggest that the 6-point conformal integral is to be constructed
as the Z6-invariant sum of 18 basis functions. If so, one can consider a reduction to the
pentagon integral which in its turn has 10 terms (3.56). However, it can be shown that 18
basis functions of the hexagon integral will suffice to reproduce only 9 basis functions of the
pentagon integral. This happens if any of the hexagon propagator powers is set to zero, al= 0
for some l= 1, ..., 6.
Assuming that we are correct in our conjecture of reconstruction this means that there
should be additional master functions which are not directly seen from the analytic contin-
uation formula for the first bare integral (3.64). Indeed, one can show that available master
and basis functions turn out to possess a number of structural properties which allow one to
introduce and systematically describe a complete set of functions needed for reconstruction
[48]. This set includes both already known functions and those that are missing, as in the
hexagon case discussed above.
– 26 –

4 Reconstruction conjecture
Summing up our discussion in the previous sections we can now describe the reconstruction
procedure in more detail.
•Bipartite representation. By formal identical transformations the conformal integral
Ia
n(x) is split into two integral parts that defines the bipartite Mellin-Barnes represen-
tation:
Ia
n(x) = La
n(x)I(1),a
n(η) + La
n(x)I(2),a
n(η).(4.1)
The first term here is found in the form of multivariate generalized hypergeometric
series (2.26), while the second term is represented through the (n−3)(n−2)/2–folded
Mellin-Barnes integral (2.32) with a complex structure of poles, which currently is not
known in analytic form. It is the reconstruction procedure that suggests avoiding any
further integration because the second bare integral can be recovered from the first one
by invoking the permutation invariance of the full conformal integral.
•Coordinate domains. The function Ia
n(x) is defined on a domain Dn⊂(RD×)n, which
can be found by first calculating the conformal integral explicitly in a particular coor-
dinate domain and then using analytic continuation formulas.14
The two terms in (4.1) are supported on their own domains which are generally wider
than Dn. The domain of the first term D(1)
nis such that Dn⊂D(1)
n, it is also hard to
identify explicitly. We denote the respective domain in the space of cross-ratios Hnas
b
D(1)
n⊂ Hn, i.e. b
D(1)
n=η(D(1)
n).
–There is a smaller domain b
Anwhich is the convergence region of I(1),a
ncalculated
as a multivariate power series near a particular point:
b
D(1)
n⊃b
An∋(u= 0,w= 0,v= 1) .(4.2)
–Using the known analytic continuation formulas one can define I(1),a
non a different
domain b
Bncontaining the origin of coordinates in Hn:
b
D(1)
n⊃b
Bn∋(u= 0,w= 0,v= 0) .(4.3)
–In fact, an exact shape of either domain remains unclear. Instead, we operate with
small enough neighbourhoods of concrete points in Hnlike 0,1,∞.
•Symmetric group. By construction, the conformal integral is invariant under action of
the symmetric group
∀π∈ Sn: R(π)◦Ia
n(x) = Ia
n(x).(4.4)
14This can be contrasted with other possible calculation schemes such as the geometric approach where the
conformal integral is given by volume of a simplex in a space of constant curvature which sign depends on a
particular kinematic regime, i.e. on a particular coordinate domain, see e.g. [16].
– 27 –

There are maps of the (co)domains induced by elements of the symmetric group Sn
acting on x= (x1, ..., xn). In the space of cross-ratios Hnthe symmetric group acts by
transformations
∀π∈ Sn:η→η′= R(π)◦η=Hπ(η),(4.5)
where Hπare homogeneous rational functions. One can single out various subgroups of
Snwhich act in Hnin some special way.
–E.g., one can consider a stabilizer (kinematic) group,
Skin
n:= {π∈ Sn: R(π)◦η=η,∀η∈ Hn}⊂Sn.(4.6)
It is clear that the stabilizer leaves any domain in Hninvariant. However, one can
show that the stabilizers are all trivial for n > 4 [64].
–Other subgroups change chosen domains. Given a domain b
Xn⊂ Hnone can
consider a subgroup
Gn⊂ Sn:b
X′
n= R(Gn)◦b
Xnsuch that b
Xn∩b
X′
n=∅.(4.7)
E.g., there is a cyclic subgroup generated by Cn= (123...n)∈ Sn, i.e.
Zn={e, Cn,(Cn)2, ..., (Cn)n−1}⊂Sn.(4.8)
–One may consider extended kinematic groups b
Skin
nwhich are nontrivial for n > 4
and coincide with Skin
4when n= 4 (see appendix E). However, at our current level
of understanding, b
Skin
nare derived objects, i.e. these subgroups can be defined only
in terms of the cyclic group and its action on basis functions that is not useful in
practice.
•Basis functions. One can choose a set of basis functions φ(ijk)
nsupported on a common
domain b
Xn⊂b
Dn. The considered cases n= 4,5,6 allow us to emphasize that the basis
functions share a number of common features:
–Γ-function prefactor S(ijk)
n;
–Leg-factor V(ijk)
n(x) ;
–Hypergeometric-type power series in n(n−3)/2 cross-ratios.
Moreover, each of these elements and, therefore, a given basis function is determined
by choosing 3 pairs (xi, ai) out of nsuch pairs.
•Master functions. The conformal integral Ia
n(x) must be invariant under the symmetric
group Sn. However, the basis functions φ(ijk)
n(x) are supported on a particular coordi-
nate domain that implies that the full symmetric group is broken down to a particular
generating subgroup of the type (4.7) which acts on the basis functions in a specific
– 28 –

way: among all basis functions one can single out a number of master functions which
generate all other basis functions by acting with elements of the generating group. By
construction, the generating group reshuffles a set of basis functions. It is important to
stress that a set of basis functions depends on the choice of a coordinate domain that
in its turn defines a suitable generating group.
•Consistency checks. In order to check the resulting expressions for the conformal inte-
grals obtained by the reconstruction method one can compare them with those known
in the literature. There are two possible checks: (1) reducing a number of points,
n→n−1; (2) going to the non-parametric case, ai→1.
If one chooses the domain as b
Xn=b
Bnand the generating group as Zn, then the conformal
integral can explicitly be evaluated by means of the following
Reconstruction conjecture. The n-point conformal integral (2.1)is the Zn-invariant sum
of master functions
Ia
n(x)nr
=
n−1
X
m=0
(Cn)m◦X
(ijk)∈Tn
φ(ijk)
n(x),(4.9)
where Tnis a set of index triples (ijk)which are not related to each other by cyclic per-
mutations, 1≤i < j < k ≤n. A cardinal number of Tnequals the number of master
functions,
|Tn|=






(n−2)(n−1)
3! ,if n
3/∈N,
(n−2)(n−1) + 4
3! ,if n
3∈N.
(4.10)
A total number of basis functions is equal to the binomial coefficient
n
3=n(n−1)(n−2)
3! ,(4.11)
which determines a number of all possible index triples. The symbol nr
=in (4.9)means that
among all basis functions produced by acting with cycles on master functions one keeps only
non-repeating ones which number equals (4.11).
As drafted, this conjecture shifts the focus to finding master functions as well as describing
their properties. A few comments are in order.
•Some of master functions are contained directly in the calculated part La
n(x)I(1),a
n.
Since among n−2 candidate master functions delivered by the analytic continuation
formula (3.3) the first two functions are cyclically dependent (see (3.10)), there remain
n−3 master functions which we explicitly know from evaluating the first bare integral.
However, a number of master functions is conjectured to grow quadratically (4.10) and
n= 6 is the first time when the n−3 functions turn out to be insufficient for the full
reconstruction.
– 29 –

•The number of master and basis functions (4.10) and (4.11) in the considered cases
n= 4,5,6 equals 1, 2, 4 and 4, 10, 20, respectively. In the hexagon case we see that in
addition to 18 basis functions coming from the first bare integral there are two more basis
functions produced from the additional master function15 and now their total number
is sufficient to reproduce the pentagon integral when checking n→n−1 reduction.
•We claim that unknown part of master functions can be derived from already known
basis functions. The point is that the proposed parameterization by index triples allows
one to reveal a number of remarkable properties of master/basis functions which can be
effectively described by means of some diagrammatics. This will enable us eventually
to build a complete set of functions explicitly, thereby reconstructing the full conformal
integral in a given coordinate domain [48].
5 Conclusion
In this paper we have proposed a new method of calculating n-point parametric one-loop
conformal integrals and elaborated two examples of the box and pentagon (non)-parametric
integrals. Essentially, there are three main points: (1) a bipartite Mellin-Barnes represen-
tation which allows one to evaluate an additive part of the conformal integral explicitly;
(2) using asymptotic expansions near particular points and analytic continuation on other
domains; (3) permutation invariance can be used to recover the remaining unknown part.
This reconstruction procedure operates with a set of master functions which generate a
wider set of basis functions by means of permutations from the cyclic group. The resulting
expression for the conformal integral supported on a particular coordinate domain is given
simply by summing all basis functions. For the box and pentagon integrals we have shown
that the basis functions are directly defined from the explicitly found part of the bipartite rep-
resentation. For the hexagon and higher-point integrals a complete set of basis functions will
be described in our forthcoming paper [48] in which will finalize the reconstruction method.
It would be natural to find an appropriate modification of the reconstruction method for
at least two types of Feynman integrals possessing conformal symmetry. These are multi-loop
higher-point conformal integrals which are currently being considered by other methods, see
e.g. [28,68–71] as well as massive conformal integrals studied in [16,72,73]. The latter have
recently been shown to play an essential role when calculating contact Witten diagrams [74].
Besides possible generalizations, the already obtained results can be immediately applied
in CFTDby means of the shadow formalism (see our discussion in the Introduction). Espe-
cially, the one-loop conformal integrals are useful in CFTDon non-trivial backgrounds, where
one can show that the problem of calculating n-point thermal conformal blocks boils down to
knowing (2n+ 2)-point one-loop conformal integrals explicitly (see [44] for 1-point thermal
blocks).
15This means that the respective representation of the cyclic group Z6on basis functions is two-dimensional,
and not six-dimensional as in the case of other hexagon master functions. Such a shortening happens every
time when n/3∈N.
– 30 –

Acknowledgements. We would like to thank Leonid Bork, Sergey Derkachev, Alexey Isaev
for discussions about related topics and Marcus Spradlin for correspondence. S.M. is also
very grateful to Ekaterina Semenova for her support at all stages of this project as well as for
her help in drawing diagrams.
Our work was supported by the Foundation for the Advancement of Theoretical Physics
and Mathematics “BASIS”.
A Notation and conventions
Let g={g1, ..., gN}denote a set of elements such as parameters of functions, powers, inte-
gration variables, etc. Their various sums will be denoted as
|gi,j|=
j
X
l=i
gl,|g|=
N
X
l=1
gl,|gq|=|gq,N |−|g1,q−1|.(A.1)
In the case of propagator powers a= (a1, ..., an) we also use
|ai,j|=












j
X
l=i
al, i < j ,
n
X
l=i
al+
j
X
l=1
al, i > j ,
a′
i=D
2−ai, αi,j =ai+aj−D
2≡ai−a′
j,(A.2)
where Dis the dimension of Euclidean space RD.
The products of Γ-functions are denoted as
Γ"a1, ..., aM
b1, ..., bK#=Γ(a1, ..., aM)
Γ(b1, ..., bK),Γ(a1, ..., aM) =
M
Y
i=1
Γ(ai).(A.3)
In the Mellin-Barnes integrals it is convenient to use the following modified measure:
b
dt=dt
2πi Γ(−t).(A.4)
B Generalized hypergeometric functions
In this appendix we discuss various generalized hypergeometric functions focusing on their
integral representations, identities as well as analytic continuation formulas. We closely follow
the review [63]. See also [49,62,75,76].
– 31 –

B.1 General definitions
Following the Horn’s classification scheme, a N-point multivariate power series
+∞
X
m1,...,mN=−∞
A(m1, ..., mN)ξm1
1... ξmN
N≡X
m∈ZN
A(m)ξm(B.1)
is a generalized hypergeometric function if
A(m+ej)
A(m)=Pj(m)
Qj(m),∀j= 1, ... , n , (B.2)
where Pj(m) and Qj(m) are some polynomials in m={m1, ..., mN}and ej={0, ..., 1, ..., 0}
is a vector with all components equal to zero except j-th one. In other words, a given
generalized hypergeometric function (B.1) have expansion coefficients such that their ratios
are rational functions of parameters.
The relation (B.2) treated as an equation for A(m1, ..., mN) admits many solutions. In
particular, when
Pj(m)=(a+|m|)(bj+mj), Qj(m)=(c+|m|)(1 + mj),(B.3)
where m,b, a, c are constant complex-valued parameters (see (A.1) for notation), the solution
to (B.2) is the N-point Lauricella function D:
F(N)
D"a , b
cξ#=∞
X
|m|=0
(a)|m|(b)m
(c)|m|
ξm
m!,(B.4)
which is convergent in the unit polydisk
UN=ξ∈CN:|ξj|<1, j = 1, ..., N .(B.5)
The (rising) factorials in (B.4) are defined as
(a)m=Γ(a+m)
Γ(a),(b)m= (b1)m1... (bN)mN,m! = m1!... mN!.(B.6)
At N= 1 and N= 2 the Lauricella function D(B.4) becomes the Gauss hypergeometric
function 2F1and the first Appell function F1, respectively:
2F1"a , b
cξ#=∞
X
m=0
(a)m(b)m
(c)m
ξm
m!,(B.7)
F1"a, b1, b2
cξ1, ξ2#=∞
X
m1,m2=0
(a)m1+m2(b1)m1(b2)m2
(c)m1+m2
ξm1
1
m1!
ξm2
2
m2!.(B.8)
– 32 –

The Lauricella function F(N)
Dbelongs to the family of generalized hypergeometric func-
tions G(N,j)labelled by j= 1, ..., N + 1 (see (A.1) for notation):
G(N,j)"a , b
cξ#=∞
X
|m|=0
(a)|mj|(b)m
(c)|mj|
ξm
m!,(B.9)
which converge in the unit polydisk (B.5). The subscripts of the Pochhammer symbols in
(B.9) can be negative, in which case the Pochhammer symbol reads as
(a)−m=(−1)m
(1 −a)m
, m ∈Z+.(B.10)
The Lauricella function F(N)
Dfollows from (B.9) at j= 1 and j=N+ 1:
F(N)
D"a , b
cξ#=G(N,1) "a , b
cξ#, F (N)
D"1−c , b
1−aξ#=G(N,N +1)"a , b
cξ#.(B.11)
The Horn function G2follows from (B.9) at j= 2 and N= 2:
G(2,2)"a, b1, b2
cξ1, ξ2#=G2b1, b2, a, 1−c−ξ1,−ξ2
=∞
X
m1,m2=0
(b1)m1(b2)m2(a)m2−m1(1 −c)m1−m2
(−ξ1)m1
m1!
(−ξ2)m2
m2!.
(B.12)
Finally, note that given a N-point generalized hypergeometric function one can construct
a (N+m)-point hypergeometric function through the so-called splitting identities. E.g., the
fourth Appell function F4defined by the double power series
F4"a1, a2
c1, c2ξ1, ξ2#=∞
X
m1,m2=0
(a1)m1+m2(a2)m2+m1
(c1)m1(c2)m2
ξm1
1
m1!
ξm2
2
m2!,(B.13)
and converging when √ξ1+√ξ2<1 can be represented in terms of the Gauss hypergeometric
function (B.7) by means of the following splitting identity
F4"a1, a2
c1, c2ξ1, ξ2#=∞
X
m1=0
ξm1
1
m1!
(a)m1(a2)m1
(c)m1
2F1"a1+m1, a2+m2
c2ξ2#.(B.14)
– 33 –

B.2 Integral representations
Generalized hypergeometric functions can also be defined through the Euler-type integrals.
E.g., for Re bi>0,Re(c− |b|)>0 the Lauricella function F(N)
D(B.4) can be represented as
F(N)
D"a , b
cξ#= Γ"c
c− |b|,b#ZΩ
dNσσb−1(1 − |σ|)c−|b|−1
(1 −σ·ξ)a,(B.15)
where σ·ξ=σ1ξ1+...+σNξN, and the integration domain is the standard orthogonal simplex
Ω = {σ∈RN:σj≥0, j = 1, ..., N ;|σ| ≤ 1}(see (A.3) for notation).
Euler-type integrals are useful in finding transformation formulas for generalized hyperge-
ometric functions. This can be illustrated by another integral representation of the Lauricella
function F(N)
D:
F(N)
D"a , b
cξ#= Γ"c
c−a, a #Z∞
0
dt ta−1(1 −t)c−a−1
N
Y
j=1
(1 −t ξj)−bj,(B.16)
valid when Re a > 0,Re(c−a)>0 and |arg(1 −ξj)|<Π, ∀j= 1, ..., N . Changing the
integration variable as t→1−tone can establish the following transformation
F(N)
D"a , b
cξ#= (1−ξ)−bF(N)
D"c−a , b
cP(ξ)#,P(ξ) = ξ1
ξ1−1, ..., ξN
ξN−1.(B.17)
At N= 1 it is the Pfaff transformation of the Gauss hypergeometric function.
Moreover, the integral representations can be useful in finding analytic continuation for-
mulas. E.g. the Gauss hypergeometric function 2F1has the following integral representation
2F1"a, b
1 + a+b−c1−ξ#= Γ"1 + a+b−c
a, b #Z∞
0
dssb−1(1 + s)c−b−1
(1 + s ξ)a,(B.18)
which extends the original power series definition to a domain around 1.
Mellin-Barnes integrals. In order to deal with the problem of analytic continuation of
generalized hypergeometric functions, it is useful to use the Mellin-Barnes integral technique.
E.g. the Mellin-Barnes representation for the Gauss hypergeometric function is given by
2F1"a, b
cξ#= Γ"c
a, b #Z+i∞
−i∞
ds
2πi Γ"a+s, b +s, −s
c+s#(−ξ)s,(B.19)
where the integration contour is chosen such that the poles of the Γ-functions s(1) =−a−k
and s(2) =−b−klie to the left of the contour and the poles s(3) =klie to the right (k∈Z+).
– 34 –

The Mellin-Barnes integrals are also useful in representing elementary functions, e.g. the
exponential function
exp (−X) = 1
2πi Z+i∞
−i∞
dsΓ(−s)Xs,(B.20)
and the power function
1
(λ1+... +λN)a=1
Γ(a)
1
(2πi)N−1Z+i∞
−i∞
ds1···Z+i∞
−i∞
dsN−1Γ(−s1)···Γ(−sN−1)
×Γ(s1+... +sN−1+a)λs1
1···λsN−1
N−1λ−s1−...−sN−1−a
n.
(B.21)
In the last formula the integration contours for siare chosen such that poles s(1)
i=−Si−ki
are to the left, where Si=s1+... +sN+a−siand poles s(2)
i=−kiare to the right,
ki∈Z+,∀i= 1, ..., N −1.
B.3 Analytic continuation
When calculating Mellin-Barnes integrals, e.g. (B.19), there is an ambiguity in closing the
contour to the left or right. However, it can be fixed by specifying domains of argument
ξand parameters a, b, c. More precisely, when closing a contour to the right, the integral
(B.19) can be calculated as a sum of residues at the poles s(3) =k∈Z+, yielding the Gauss
hypergeometric series (B.7) which converges for |ξ|<1. On the other hand, when closing
a contour to the left, the same integral is a sum of residues at the poles s(1) =−k−aand
s(2) =−k−bwith k∈Z+. In this way, one obtains the analytic continuation formula for the
Gauss hypergeometric series (B.7) in the domain |ξ|>1:
2F1"a , b
cξ#= Γ"c, b −a
b, c −a#(−ξ)−a2F1"a, 1 + a−c
1 + a−b
1
ξ#
+ Γ"c, a −b
a, c −b#(−ξ)−b2F1"b, 1 + b−c
1 + b−a
1
ξ#.
(B.22)
Another analytic continuation formula can be derived by means of the first Barnes lemma:
Z+i∞
−i∞
dt
2πi Γ(a+t, b +t, s −t, c −a−b−t)=Γ"a+s, b +s, c −b, c −a
c+s#.(B.23)
Substituting this relation into (B.19) one obtains the following integral representation
2F1"a, b
cξ#= Γ"c
a, b, c −a, c −b#Z+i∞
−i∞
dt
2πi Γ(a+t, b +t, c −a−b−t, −t)(1 −ξ)t,(B.24)
– 35 –

where we also used (B.21) to evaluate the integral over s. Computing the integral as a sum
over residues at the poles t(1) =k∈Z+and t(2) =c−a−b+k, we find the following analytic
continuation formula for the Gauss hypergeometric function
2F1"a, b
cξ#= Γ"c, c −a−b
c−a, c −b#2F1"a, b
1 + a+b−c1−ξ#
+ Γ"c, a +b−c
a, b #(1 −ξ)c−a−b2F1"c−a, c −b
1 + c−a−b1−ξ#.
(B.25)
By extending this procedure to generalized hypergeometric functions we also find the analytic
continuation formula for the Lauricella function F(N)
D(B.4):16
F(N)
D"a , b
cξ#=
N
X
q=0
AqDq"a , b
cξ#,(B.26)
with a new domain of convergence
KN=ξ∈CN: 0 <|1−ξ1|< ... < |1−ξN|<1 ; |arg(1 −ξj)|< π , j = 1, ..., N,(B.27)
cf. (B.5). The functions Dqare given by
D0"a , b
cξ#=F(N)
D"a , b
1 + a+|b| − c1−ξ#,
Dq"a , b
cξ#= (1 −ξq)c−a−|b1,q |N
Y
l=q+1
(1 −ξl)−blG(N,q)"c−a− |b1,q−1|,e
bq
1 + c−a− |b1,q|Ξ(q)#,
(B.28)
where q= 1, ..., N and G(N ,q)is defined in (B.9). The parameters and arguments in (B.28)
are packed into the following combinations:
e
bq={b1, ..., bq−1, c − |b|, bq+1 , ..., bN},
Ξ(q)=1−ξ1
1−ξq
, ..., 1−ξq−1
1−ξq
,1−ξq,1−ξq
1−ξq+1
, ..., 1−ξq
1−ξN.(B.29)
The coefficients in (B.26) are given by
A0= Γ"c, c −a− |b|
c− |b|, c −a#, Aq= Γ"c, c − |b1,q−1| − a, a +|b1,q | − c
a, bq, c −a#.(B.30)
16For N= 2, the formula (B.26) which was first derived in [76], provides an analytic continuation of the
first Appell function F1.
– 36 –

It is worth noting that the right-hand side of (B.26) contains two Lauricella functions. The
first one is given by D0, while the term D1is proportional to the Lauricella function by virtue
of (B.11).
C One more Mellin-Barnes representation of the conformal integral
In this section, we describe an alternative way of representing the conformal integral in terms
of Mellin-Barnes integrals. To this end, one steps back to (2.16) and represents the Gauss
hypergeometric function 2F1therein by means of the analytic continuation formula (B.24):
2F1"a′
n−1, a′
n
D
21−ξ(σ)#=Z+i∞
−i∞b
dtΓ"D
2, a′
n−1+t, a′
n+t, αn−1,n −t
an−1, an, a′
n−1, a′
n#ξ(σ)t,(C.1)
where ξ(σ) is given in (2.17). Applying to the numerator of ξ(σ) the same procedure as in
(2.23), one can obtain the following representation of the n-point conformal integral:
Ia
n=Na
nZ+i∞
−i∞
dt
2πi Γa′
n−1+t , a′
n+t , αn−1,n −t(ηn)n−3,n−2
n−2,n−1t
×Z+i∞
−i∞
n−4
Y
i=1 b
dti(ηn)i,n−2
n−3,n−2tiZ+i∞
−i∞Y
1≤i<j≤n−3c
dsij (ηn)ij
n−3,n−2sij 
×Γ |t1,n−4|+X
1≤i<j≤n−3
sij −t!Γ"C(1) − |B(1)|,B(1)
C(1) #F(n−3)
D"A(1),B(1)
C(1) ξ#,
(C.2)
where A(1),B(1) and C(1) are defined in (2.25). Although the parameters here are the same
as in the first bare integral (2.24), the pole structure is more similar to that of the second
bare integral (2.32). In particular, when closing a contour over tto the right, there are two
sets of poles coming from Γ(αn−1,n −t) and Γ(|t1,n−4|+P1≤i<j≤n−3sij −t). Therefore, in
the main text we focus on the bipartite representation of the conformal integral which allows
one to evaluate the first bare integral explicitly.
D Pentagon integral
Here, we collect the corresponding master and basis functions and then verify that the ob-
tained expression for the parametric pentagon integral correctly reproduces the known ex-
pression in the non-parametric case.
– 37 –

D.1 Basis functions
A basis function for the pentagon integral can be expressed in terms of two generalized
hypergeometric functions which are defined as17
P1"A1, A2, B
C1, C2ξ#=X
li=0
(−)l2(A1)l1+l2+l4(A2)l2+l3+l5(B)l1+l2+l3+l4+l5
(C1)l1+l2+l3(C2)l2+l4+l5
5
Y
i=1
ξli
i
li!,(D.1)
P2"A1, A2, B
C, E ξ#=X
li=0
(−)l2(A1)l1+l2+l4(A2)l1+l3+l5(B)l5+l3−l4
(C)l1+l2+l3(E)l5−l4−l2
5
Y
i=1
ξli
i
li!.(D.2)
The functions obey obvious symmetry relations
P1"A1, A2, B
C1, C2ξ1, ξ2, ξ3, ξ4, ξ5#= P1"A2, A1, B
C1, C2ξ1, ξ4, ξ5, ξ2, ξ3#
= P1"A1, A2, B
C2, C1ξ4, ξ2, ξ5, ξ1, ξ3#,
(D.3)
P2"A1, A2, B
C, E ξ1, ξ2, ξ3, ξ4, ξ5#= P2"A2, A1,1−E
C, 1−Bξ1, ξ3, ξ2, ξ5, ξ4#.(D.4)
To prove (D.4) one uses (B.10).
The action of C5∈Z5on the cross-ratios (3.47) is summarized in the table
u1u2w12 v1v2
C5v1
v2v1u2v1
v2
w12
v2u1
(C5)2w12
u1v2
w12
v2
w12v1
u1v2
u2v1
u1v2
v1
v2
(C5)3u2
u1
u2v1
u1v2
w12u2
u1v2
w12
u1
w12
u1v2
(C5)4v2w12
u1
u2v1
u1u2u2
u1
The pentagon integral Ia
5(x) is the sum of 10 terms (3.56) which are divided into two groups
depending on two master function φ(345)
5(x) and φ(245)
5(x) (3.54) used to generate them
through the cyclic permutations.
17Note that P1is the Srivastava-Daoust hypergeometric function [77,78], which also arises in the analysis
of the hexagon integral [18]. The pentagon example shows that there is also another type of function, i.e. P2,
which to the best of our knowledge has not been considered in the literature.
– 38 –

Basis functions generated from φ(345)
5(x):
φ(345)
5(x) = S(345)
5V(345)
5(x) P1"a1, a2, a′
4
1−α45,1−α34 ,u1, w12, u2, v1, v2#,(D.5)
φ(145)
5(x) = S(145)
5V(145)
5(x) P1"a2, a3, a′
5
1−α15,1−α45 ,
v1
v2
,u2v1
v2
, v1,w12
v2
, u1#,(D.6)
φ(125)
5(x) = S(125)
5V(125)
5(x) P1"a3, a4, a′
1
1−α12,1−α15 ,
w12
u1v2
,w12v1
u1v2
,w12
v2
,u2v1
u1v2
,v1
v2#,(D.7)
φ(123)
5(x) = S(123)
5V(123)
5(x) P1"a4, a5, a′
2
1−α23,1−α12 ,
u2
u1
,w12u2
u1v2
,u2v1
u1v2
,w12
u1
,w12
u1v2#,(D.8)
φ(234)
5(x) = S(234)
5V(234)
5(x) P1"a5, a1, a′
3
1−α34,1−α23 ,v2,u2v1
u1
,w12
u1
, u2,u2
u1#.(D.9)
Basis functions generated from φ(245)
5(x):
φ(245)
5(x) = S(245)
5V(245)
5(x) P2"a1, a3,−α15
1−α45,1 + α34 u1,w12
v2
, u2,v1
v2
, v2#,(D.10)
φ(135)
5(x) = S(135)
5V(135)
5(x) P2"a2, a4,−α12
1−α15,1 + α45 
v1
v2
,u2v1
u1v2
, v1,w12
u1v2
, u1#,(D.11)
φ(124)
5(x) = S(124)
5V(124)
5(x) P2"a3, a5,−α23
1−α12,1 + α15 
w12
u1v2
,w12
u1
,w12
v2
,u2
u1
,v1
v2#,(D.12)
φ(235)
5(x) = S(235)
5V(235)
5(x) P2"a4, a1,−α34
1−α23,1 + α12 
u2
u1
, u2,u2v1
u1v2
, v2,w12
u1v2#,(D.13)
φ(134)
5(x) = S(134)
5V(134)
5(x) P2"a5, a2,−α45
1−α34,1 + α23 v2, v1,w12
u1
, u1,u2
u1#.(D.14)
– 39 –

D.2 Non-parametric integral
In what follows we compare our asymptotic expansion of the non-parametric pentagon integral
with the exact formula proposed in [15].
Geometric approach. Within the ambient space approach the integral is defined as
I(5) =Zd5x0
iπ5/2
5
Y
i=1
1
(xi−x0)2=Zd5Q
iπd/2
5
Y
i=1
1
(−2Pi·Q),(D.15)
where PA
i=PA(xi), QA=QA(x0), A= 1, ..., d + 2, are n+ 1 null vectors in R1,d+1 such that
local coordinates xµ∈Rdare introduced as XA(x) = (1, x2, xµ). The volume of a particular
4-simplex in a constant curvature 4-dimensional space calculates the pentagon integral (D.15)
[12–14,16]:
I(5) =25
2Γ5
2
p|det Pij|V(4) ,(D.16)
where Pij ≡ −2Pi·Pj= (xi−xj)2and the volume can be calculated by means of the Schl¨afli
formula given by
V(4) =π
6X
1≤i<j≤5
log
Wi·Wj−q(Wi·Wj)2−W2
iW2
j
Wi·Wj+q(Wi·Wj)2−W2
iW2
j
.(D.17)
In this formula nvectors Wiare uniquely defined through Wi·Pj=δij (for review see e.g.
[79]). Introducing the rescaled integral
I(5) = (P13P14P24P25P35 )−1/2˜
I(5) ,(D.18)
after identical transformations one finds that [15]
˜
I(5) =π3
2
2p−∆(5) (1 + g+g2+g3+g4)(log r−p−∆(5)
r+p−∆(5) ! s−p−∆(5)
s+p−∆(5) !),(D.19)
where
r=(1 −t2)(1 −t5)−t1(2 −t3−t4)−t3t5−t2t4+t1t3t4
2√t1
,
s=(1 −t5)(1 −t2t5)−t1(1 + t5−2t3t5+t4+t2t4t5−t1t4)
2√t1t5
,
(D.20)
∆(5) = 1 −t1(1 −t3(1 + t4) + t2t2
4) + cyclic+t1t2t3t4t5,(D.21)
the cross-ratios are chosen as
t1=P14 P23
P13 P24
, t2=P25 P34
P24 P35
, t3=P13 P45
P14 P35
, t4=P15 P24
P14 P25
, t5=P12 P35
P13 P25
,(D.22)
and g:ti→ti+1 is a cyclic permutation which says that the pentagon formula (D.19) has
a manifest cyclic permutation symmetry. The function ∆(5) ≷0 depends on the choice a
specific kinematical regime, i.e. on values of variables Pi. The resulting function (D.19) has
10 terms which number matches 10 terms in the asymptotic expansion (3.56).
– 40 –

Redefinitions. Introducing simplified notation, from (2.7) for the non-parametric pentagon
integral one has
I5=LI5,(D.23)
where the leg-factor is
L= (X15X25X34X35X45 )−1(D.24)
and I5is expressed in terms of the cross-ratios (3.47):
u1=X13X45
X14X35
, u2=X23X45
X24X35
, w12 =X12X34X45
X14X24X35
,
v1=X15X34
X14X35
, v2=X25X34
X24X35
.
(D.25)
Since Xij ≡Pij, cf. (2.1), one finds that (D.22) and (D.25) are related as
t1=u2
u1
, t2=v2, t3=u1, t4=v1
v2
, t5=w12
u1v2
;
u1=t3, u2=t1t3, w12 =t2t3t5, v1=t2t4, v2=t2.
(D.26)
On the other hand, the two definitions of the conformal integral (D.15) and (D.23) are related
as I5=iI(5). Thus, one has the relation
I5=iL−1˜
L˜
I(5) =i t3/2
2t−1/2
3t4˜
I(5) ,(D.27)
where ˜
I(5) is given by (D.19) and the leg-factor ˜
L= (X13X14X24X25X35 )−1/2in (D.18) is
related to (D.24) as ˜
L=t3/2
2t−1/2
3t4L.
Unit propagator powers. Contrary to the case of unit parameters in the box integral
considered in section 3.2.3, the non-parametric pentagon integral has no divergent prefactors
so that no regularization is required. Choosing ai= 1 we introduce the simplified notation
for two functions (D.1)-(D.2):
P1z1, z2, z3, z4, z5≡P1"1,1,3
2
3
2,3
2z1, z2, z3, z4, z5#=∞
X
{ni}=0
zn1
1
n1!
zn2
2
n2!
zn3
3
n3!
zn4
4
n4!
zn5
5
n5!
×(−)n2(n1+n2+n4)!(n2+n3+n5)! 3
2n1+n2+n3+n4+n5
3
2n1+n2+n33
2n2+n4+n5
;
(D.28)
P2z1, z2, z3, z4, z5≡P2"1,1,1
2
3
2,1
2z1, z2, z3, z4, z5#=∞
X
{ni}=0
zn1
1
n1!
zn2
2
n2!
zn3
3
n3!
zn4
4
n4!
zn5
5
n5!
×(−)n2(n1+n2+n4)!(n1+n3+n5)! 1
2n3−n4+n5
3
2n1+n2+n31
2n5−n2−n4
.
(D.29)
– 41 –

Then, the symmetry properties (D.3)-(D.4) take the form
P1z1, z2, z3, z4, z5=P1z1, z4, z5, z2, z3=P1z4, z2, z5, z1, z3,
P2z1, z2, z3, z4, z5=P2z1, z3, z2, z5, z4.
(D.30)
The asymptotic expansion of the pentagon integral I5from (D.23) reads as
[2π3
2]−1I5=v1v2P1u1, w12, u2, v1, v2−v1v
1
2
2P2hu1,w12
v2
, u2,v1
v2
, v2i
+v
3
2
1P1hu1,u2v1
v2
,w12
v2
, v1,v1
v2i−v
3
2
1u−1
2
1P2hv1
v2
,v1u2
u1v2
, v1,w12
u1v2
, u1i
+v
3
2
1v−1
2
2u−1
1w
1
2
12 P1hw12
u1v2
,w12v1
u1v2
,w12
v2
,v1u2
u1v2
,v1
v2i−v1w
1
2
12u−1
1P2hw12
u1v2
,w12
u1
,w12
v2
,u2
u1
,v1
v2i
+v1u−3
2
1u
1
2
2w
1
2
12 P1hu2
u1
,w12u2
u1v2
,v1u2
u1v2
,w12
u1
,w12
u1v2i−v1v
1
2
2u−1
1u
1
2
2P2hu2
u1
, u2,v1u2
u1v2
, v2,w12
u1v2i
+v1v2u−1
1u
1
2
2P1hv2,u2v1
u1
,w12
u1
, u2,u2
u1i−v1v2u−1
2
1P2hv2, v1,w12
u1
, u1,u2
u1i.
(D.31)
Comparing two functions by Wolfram Mathematica. The power series representation
(D.31) is suitable for checking the asymptotic expansion of the analytic formula (D.19) at
η→0 (3.2) and (3.9). The issue is that a function of many variables can tend to its value
in parametric way. This means that one can choose a particular path in the five-dimensional
space of cross-ratios H5and then compare expansions of (D.31) and (D.19).
•Consider how the multivariate power series (D.31) tends to zero. To this end, one
collectively denotes the cross-ratios (D.25) as η={u1, u2, w12, v1, v2}and scales them
near η= 0 as ηi→ηiλαi, where {αi>0, i = 1, ..., 5} ≡ α, the proper time parameter
λ∈[0,1]. This means that one approaches η= 0 along a particular way defined by
fixing powers α.
•Choose a set αwith not too large αi(in this case the Mathematica computation takes
less time) which is consistent with sending η→0 (3.2) and (3.9).
•Single out the leading order in λ, i.e. I5=a0+a1λγ+O(λγ+...) with some power
γ=γ(αi), where expansion coefficients a0,1=a0,1(η) are some rational functions of
cross-ratios.
•The resulting coefficients a0,1are to be compared with those ones arising when expand-
ing the exact formula for I(5) using the same scaling pattern, i.e. I(5) =i(b0+b1λγ
+O(λγ+...)). To this end, one expresses the cross-ratios ti(D.22) via ηby means of
relations (D.26) and then sends η→0 along the same path in H5, i.e. using the same
powers α.
– 42 –

234 134
124123
(12)(34)
e
(13)(24)
(14)(23)
Figure 2. Action of the kinematic group Skin
4(3.17) on the basis functions (3.30). Each node denotes a
basis function φ(ijk)
4(x) which indices are shown inside a node. The color lines represent permutations
listed on the right (the identity permutation acts on each node trivially). The diagram also reflects the
group multiplication law, i.e. each triangle formed by color lines represents a multiplication gi◦gj=gk,
where gi,j,k ∈Skin
4.
•The exact formula and the asymptotic expansion of the pentagon integral may coincide
in a given coordinate domain iff a0,1=b0,1. One can further expand in λand compare
higher-order expansion coefficients.
Following this procedure one fixes the powers e.g. as α={3,1,4,1,2}. Obviously, there
are infinitely many choices of αi, but here we are choosing those ones with not too high
integer values that simplifies computations. Near η= 0 the function ∆(5) >0 and the
exact formula (D.19) has under the logarithm a unimodular complex-valued function, i.e. the
logarithm is pure imaginary. However, the final expression for the conformal integral is real
due to the prefactor p−∆(5) which is also pure imaginary in this case. Using simple Wolfram
Mathematica functions we explicitly expand both functions up to O(λ11/2) and find that the
two resulting series have the same form.
E Kinematic group extensions
The kinematic group Skin
4=Z2×Z2also acts on the expansion in basis functions (3.34)
obtained by means of the cyclic group Z4. One can show that Z2×Z2acts on the basis
functions by rearranging them according to the diagram rule shown on fig. 2. Any two nodes
are related by a group element from Z2×Z2that means that there is a single master function.
This observation allows one to give a complementary definition of the kinematic group Skin
4
which replaces the original definition as a stabilizer group of cross-ratios for n > 4. Thus,
it is possible to overcome the difficulty that Skin
n={e}for n > 4 since this new extended
kinematic group b
Skin
nis non-trivial for n > 4.
One can introduce an extended kinematic group by following a number of steps: (0) one
analytically continues the first bare integral to a domain around the origin of coordinates
in the space of cross-ratios; (1) one acts with the cyclic group Znon the master functions
supported on the same domain; (2) one represents the conformal integral as a sum of basis
functions; (3) one identifies all permutations /∈Znwhich rearrange basis functions while
keeping their domains around the origin.
– 43 –

(13)(45)
(14)(23)
(25)(34)
(15)(24)
(12)(35)
345
145
125123
234
245
135
124235
134
Figure 3. In the 5-point case, the ten basis functions (D.5)-(D.14) can be equivalently obtained by
acting with b
Skin
5. These can be grouped into two pentagon diagrams. The identical permutation leaves
each node invariant.
As can be seen, such a working definition is largely based on the cyclic group, and
currently there are no guidelines that could help define an extended kinematic group inde-
pendently. In the 4-point case we have Skin
4=b
Skin
4. In the 5-point case, one can show that
an extended kinematic group b
Skin
5is generated by the following elements
e, (14)(23),(15)(24),(25)(34),(12)(35),(13)(45)⊂ S5.(E.1)
b
Skin
5acts on the ten basis functions as shown on fig. 3.
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