aainstitutetext: Institute of Nuclear and Particle Physics, NCSR "Demokritos",
Agia Paraskevi 15310, Greecebbinstitutetext: Department of Physics, University
of Athens,
Zographou 15784, Greececcinstitutetext: Physik-Department, Technische
Universität München,
James-Franck-Str. 1, 85748 Garching, Germany

# Planar three-loop master integrals for $2\to 2$ processes with one external
massive particle

Dhimiter D. Canko a,c and Nikolaos Syrrakos<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

We present analytic results for the two tennis-court integral families
relevant to $2\to 2$ scattering processes involving one massive external
particle and massless propagators in terms of Goncharov polylogarithms of up
to transcendental weight six. We also present analytic results for physical
kinematics for the ladder-box family and the two tennis-court families in
terms of real-valued polylogarithmic functions, making our solutions well-
suited for phenomenological applications.

###### Keywords:

Feynman integrals, QCD, N3LO Calculations

††preprint: TUM-HEP-1385/21

##  1 Introduction

Having a solid understanding of the physics of strong interactions is of vital
importance for the analysis of experimental data coming from high-energy
proton-proton colliders, such as the LHC Gehrmann:2021qex . The increasing
precision of these experiments translates to a high demand of equally precise
theoretical predictions. Such calculations are performed in the framework of
perturbative Quantum Field Theory (QFT), and in the case of strong
interactions, perturbative Quantum Chromodynamics (QCD). Obtaining more
precise predictions amounts to the computation of higher order corrections for
QCD dominated processes, which involves, among other things, the computation
of multiloop Feynman diagrams. Using the Feynman rules of QCD one can
translate these diagrams to the so-called Feynman integrals (FI), which are
then computed either numerically or analytically in the framework of
dimensional regularisation in $d=4-2\epsilon$ space-time dimensions.

It has been estimated that the scheduled High Luminosity upgrade of the LHC
will require the calculation of many scattering processes at the Next-to-Next-
to-Next-to-Leading-Order (N3LO) to reach per-cent level of precision
Amoroso:2020lgh . At that order of perturbative expansion the computation of
three-loop FI becomes a necessity, a highly non-trivial task despite the major
breakthroughs that have occurred in our understanding of their mathematical
structure and the development of many methods for their analytical and
numerical calculation Heinrich:2020ybq ; Tancredi:2021oiq . Currently, at N3LO
all FI for $2\to 2$ scattering involving massless propagators and on-shell
external particles have been computed analytically Henn:2013tua ; Henn:2020lye
, and the first phenomenological studies have appeared in the literature
Caola:2020dfu ; Caola:2021rqz ; Bargiela:2021wuy ; Caola:2021izf . These
impressive results have greatly benefited from our ability to compute FI
analytically using the method of differential equations (DE) de1 ; de2 ; de3 ;
de4 . The extension of these results to processes involving one massive
external particle, such as a vector boson decaying to 3-jets or $gg\to H+jet$
in gluon fusion, is a natural next step.

It has been twenty years since the calculation of the two-loop FI for $2\to 2$
processes involving massless propagators and one off-shell external particle
Gehrmann:2000zt ; Gehrmann:2001ck  which established the method of DE as a
powerful technique for the analytic computation of multiloop FI. These studies
developed an approach which in essence is still used today, i.e. using
Integration-By-Parts (IBP) identities Chetyrkin:1981qh ; Laporta:2000dsw  to
identify a minimal set of FI, known as master integrals (MI), deriving DE for
these integrals by differentiating with respect to kinematic invariants and,
if present, internal masses, and then using IBP identities again to recast the
derivatives of the MI in terms of MI of equal or lower number of propagators.

A few years ago, a first step was taken towards the extension of these results
at the three-loop level, with the calculation of the so-called planar ladder-
box topology involving massless propagators and one off-shell leg
DiVita:2014pza . This calculation was made possible through the adoption of
the by now established method of canonical DE Henn:2013pwa . In Henn:2013pwa
an observation was made that one can choose a good basis of MI, such that the
DE that they satisfy have only logarithmic singularities and the dimensional
regulator $\epsilon$ is fully factorised. The conjecture made in Henn:2013pwa
was that FI with constant leading singularities Arkani-Hamed:2010pyv  are good
candidates for the construction of such a basis, which is known as a pure
basis of MI.

In DiVita:2014pza  a basis of 85 MI was obtained and a canonical DE was
derived based on Magnus series expansions Argeri:2014qva . In Canko:2020gqp
we re-calculated the three-loop ladder-box topology using a variant of the
standard DE method, known as the Simplified Differential Equations (SDE)
approach Papadopoulos:2014lla . Using modern IBP tools such as FIRE6
Smirnov:2019qkx  and Kira2 Klappert:2020nbg , we obtained a set of 83 MI and
verified through the analytic results of DiVita:2014pza  that this is indeed
the minimal set of integrals needed for the computation of all Feynman
diagrams contained in the three-loop ladder-box family.

In this paper we take a step further and compute fully analytically the two
additional planar three-loop topologies, known as tennis-court topologies, in
terms of Goncharov poloylogarithms (GPLs) Goncharov:1998kja ; Duhr:2011zq ;
Duhr:2012fh ; Duhr:2014woa  of up to transcendental weight six. We also
provide expressions for all three planar integral families in all physical
regions of the phase-space, thus allowing our results to be readily available
for use in phenomenological studies. More specifically, in section 2 we
introduce the integral families treated in this paper, the SDE parametrisation
and set up basic notation, in section 3 we construct pure bases for the two
tennis-court families, derive canonical SDE for these bases and compute the
relevant boundary terms, in section 4 we discuss the analytic continuation of
our results into physical regions and present an analysis concerning the GPLs
involved in our solutions and their numerical evaluation, as well as report on
the various checks that we have performed to validate our solutions. Finally,
we give our closing remarks in section 5. In appendix A we comment on the
adjacency conditions that were recently discovered in Dixon:2020bbt ;
Chicherin:2020umh  regarding the symbol of two-loop and three-loop four-point
master integrals with one off-shell leg.

##  2 General set up

###  2.1 Integral families

We begin this section by defining the integral families that will be treated
in this paper. Even though the ladder-box topology was calculated with our
approach in Canko:2020gqp , we include it here again, since we will present
results for this family as well in physical kinematics. For the sake of
brevity we will call F1 the ladder-box topology and F2, F3 the two tennis-
court ones, as depicted in figure 1.

Figure 1: The F1 (top), F2 (bottom left) and F3 (bottom right) top-sector
diagrams. The double line represents the massive particle and all external
momenta are taken to be incoming.

The corresponding FI are defined through111Where we use the abbreviation
$q_{12}=q_{1}+q_{2}$ and $q_{123}=q_{1}+q_{2}+q_{3}$.

$\displaystyle G^{F1}_{a_{1}\cdots a_{15}}:=$
$\displaystyle\int\bigg{(}\prod_{l=1}^{3}e^{\gamma_{E}\epsilon}\frac{d^{d}k_{l}}{i\pi^{d/2}}\bigg{)}\frac{\left(k_{1}+q_{123}\right){}^{-2a_{11}}\left(k_{2}+q_{123}\right){}^{-2a_{12}}}{\left(k_{1}+q_{12}\right){}^{2a_{1}}\left(k_{2}+q_{12}\right){}^{2a_{2}}\left(k_{3}+q_{12}\right){}^{2a_{3}}\left(k_{1}-k_{2}\right){}^{2a_{4}}}$
$\displaystyle\times\frac{\left(k_{2}+q_{1}\right){}^{-2a_{13}}\left(k_{3}+q_{1}\right){}^{-2a_{14}}\left(k_{1}-k_{3}\right){}^{-2a_{15}}}{\left(k_{2}-k_{3}\right){}^{2a_{5}}\left(k_{3}+q_{123}\right){}^{2a_{6}}~{}k_{1}^{2a_{7}}~{}k_{2}^{2a_{8}}~{}k_{3}^{2a_{9}}\left(k_{1}+q_{1}\right){}^{2a_{10}}}$
(1)

$\displaystyle G^{F2}_{a_{1}\cdots a_{15}}:=$
$\displaystyle\int\bigg{(}\prod_{l=1}^{3}e^{\gamma_{E}\epsilon}\frac{d^{d}k_{l}}{i\pi^{d/2}}\bigg{)}\frac{\left(k_{1}+q_{123}\right){}^{-2a_{11}}~{}k_{2}^{-2a_{12}}}{\left(k_{1}+q_{12}\right){}^{2a_{1}}\left(k_{2}+q_{12}\right){}^{2a_{2}}\left(k_{2}+q_{123}\right){}^{2a_{3}}}$
$\displaystyle\times\frac{\left(k_{2}+q_{1}\right){}^{-2a_{13}}\left(k_{3}+q_{1}\right){}^{-2a_{14}}\left(k_{3}+q_{12}\right){}^{-2a_{15}}}{\left(k_{3}+q_{123}\right){}^{2a_{4}}~{}k_{3}^{2a_{5}}~{}k_{1}^{2a_{6}}\left(k_{1}+q_{1}\right){}^{2a_{7}}\left(k_{1}-k_{2}\right){}^{2a_{8}}\left(k_{1}-k_{3}\right){}^{2a_{9}}\left(k_{3}-k_{2}\right){}^{2a_{10}}}$
(2)

$\displaystyle G^{F3}_{a_{1}\cdots a_{15}}:=$
$\displaystyle\int\bigg{(}\prod_{l=1}^{3}e^{\gamma_{E}\epsilon}\frac{d^{d}k_{l}}{i\pi^{d/2}}\bigg{)}\frac{\left(k_{1}+q_{12}\right){}^{-2a_{11}}~{}k_{2}^{-2a_{12}}}{\left(k_{1}+q_{1}\right){}^{2a_{1}}\left(k_{2}+q_{1}\right){}^{2a_{2}}\left(k_{2}+q_{12}\right){}^{2a_{3}}\left(k_{3}+q_{12}\right){}^{2a_{4}}}$
$\displaystyle\times\frac{\left(k_{2}+q_{123}\right){}^{-2a_{13}}~{}k_{3}^{-2a_{14}}\left(k_{3}+q_{1}\right){}^{-2a_{15}}}{\left(k_{3}+q_{123}\right){}^{2a_{5}}\left(k_{1}+q_{123}\right){}^{2a_{6}}~{}k_{1}^{2a_{7}}\left(k_{1}-k_{2}\right){}^{2a_{8}}\left(k_{1}-k_{3}\right){}^{2a_{9}}\left(k_{3}-k_{2}\right){}^{2a_{10}}}$
(3)

with $a_{i}$ being integers and $a_{i}\leq 0$ for $i=11,\ldots,15$. The
external momenta of the considered families obey the following kinematics:
$\sum_{i=1}^{4}q_{i}=0,~{}q_{2}^{2}=m^{2},~{}q_{i}^{2}=0$ for $i=1,3,4$ and
$S_{12}=(q_{1}+q_{2})^{2},~{}S_{23}=(q_{2}+q_{3})^{2},~{}S_{13}=m^{2}-S_{12}-S_{23}$.

In this paper we employ the SDE approach Papadopoulos:2014lla  for the
analytic computation of families F2 and F3. To do so we parametrise the
external momenta by introducing a dimensionless parameter $x$ in the following
manner

$q_{1}=xp_{1},\quad q_{2}=p_{1}+p_{2}-xp_{1},\quad q_{3}=p_{3},\quad
q_{4}=p_{4}$ (4)

where the new momenta $p_{i}$ are all massless. This parametrisation produces
the following mapping for the kinematic invariants between the two momentum
configurations

$S_{12}=s_{12},\quad S_{23}=s_{23}x,\quad m^{2}=s_{12}(1-x)$ (5)

with $s_{12}=(p_{1}+p_{2})^{2},~{}s_{23}=(p_{2}+p_{3})^{2}$. The SDE approach
Papadopoulos:2014lla  is an attempt to simplify and systemize, as much as
possible, the derivation of the appropriate system of DE satisfied by the MI.
Taking $p_{i}$ to be massless, the introduction of the $x$ parameter in (4)
captures the off-shellness of the massive momentum $q_{2}$. The MI are now
dependent on $x$ through the external momenta. This allows us to derive DE by
differentiating with respect to $x$. The benefit of this approach is that,
when the poles of the DE are rational functions of $x$, the integration of the
DE naturally captures the expressibility of MI in terms of GPLs and more
importantly makes the problem independent of the number of kinematic scales
involved. Note that as $x\rightarrow 1$, the external momentum $q_{2}$ becomes
massless, which corresponds to a simpler loop integral family with one scale
less.

###  2.2 Scattering kinematics

When considering the analytic solution of multiloop FI in dimensional
regularisation, one usually solves these integrals in the Euclidean region,
where all FI are free of branch cuts, and then analytically continues the
results to the physical regions of phase-space. This is the approach that we
will follow as well.

At first, by studying the second Symanzik polynomial Bogner:2010kv  for the
top-sector integrals222By top-sector we mean the integrals with $a_{i}=1$ for
$i=1,\ldots 10$ and $a_{i}=0$ for $i=11,\ldots 15$ in (2.1), (2.1) and (2.1).
of families F1, F2 and F3 in their Feynman parameter representation, we can
identify the Euclidean region in terms of the kinematic variables
$S_{12},~{}S_{23},~{}m^{2}$ such that

$S_{12}<0,\quad S_{23}<0,\quad m^{2}<0.$ (6)

For scattering kinematics we have three physical regions when considering
$2\to 2$ processes with one massive particle. For convenience we will denote
them as the $s,~{}t~{}\text{and}~{}u$ channels appropriately:

$\displaystyle\text{s-channel}:$ $\displaystyle~{}m^{2}>0,\quad S_{12}\geq
m^{2},\quad S_{23}\leq 0,\quad S_{13}\leq 0$ (7)
$\displaystyle\text{t-channel}:$ $\displaystyle~{}m^{2}>0,\quad S_{12}\leq
0,\quad S_{23}\geq m^{2},\quad S_{13}\leq 0$ (8)
$\displaystyle\text{u-channel}:$ $\displaystyle~{}m^{2}>0,\quad S_{12}\leq
0,\quad S_{23}\leq 0,\quad S_{13}\geq m^{2}.$ (9)

Since we will use the SDE approach for the solution of (2.1) and (2.1), we
would like to have the corresponding limits for each region of phase-space
expressed in terms of the $x,~{}s_{12},~{}s_{23}$ variables. The mapping of
(5) allows us to do so, although for reasons that will become clear at a later
stage, we define the ratio $y=\frac{s_{23}}{s_{12}}$ and use the variables
$x,~{}y,~{}s_{12}$. Our approach therefore will be to compute all MI in terms
of real-valued GPLs in the Euclidean region

$0<x<1,\quad s_{12}<0,\quad 0<y<1$ (10)

and then, using tools such as HyperIntPanzer:2014caa  and
PolyLogToolsDuhr:2019tlz , analytically continue our solutions in the physical
regions

$\displaystyle\text{s-channel}:$ $\displaystyle~{}0<x<1,\quad
s_{12}>0,\quad-1\leq y\leq 0$ (11) $\displaystyle\text{t-channel}:$
$\displaystyle~{}1<x,\quad s_{12}<0,\quad y\leq-1$ (12)
$\displaystyle\text{u-channel}:$ $\displaystyle~{}1<x,\quad s_{12}<0,\quad
y\geq 0.$ (13)

##  3 Canonical differential equations

Using IBP tools such as FIRE6 Smirnov:2019qkx  and Kira2 Klappert:2020nbg  we
identified a minimal set of 117 MI (3 at the top-sector) for family F2 and 166
MI (4 at the top sector) for family F3. Although today exist several automated
packages for the construction of a pure basis which satisfies a canonical DE
Henn:2020lye ; Meyer:2017joq ; Prausa:2017ltv ; Gituliar:2017vzm ;
Dlapa:2020cwj , finding a pure basis for a generic integral family in practise
is still a non-trivial task, assuming that such a basis even exists.

In order to construct pure bases for families F2 and F3 we used a combined
approach, attacking the problem with the following techniques. For several low
sectors, involving up to seven propagators, we used the approach based on
Magnus series expansions Argeri:2014qva , which was already successfully
applied for the construction of a pure basis for family F1 in DiVita:2014pza .
For higher sectors, involving up to nine propagators, we used the Mathematica
package DlogBasis Henn:2020lye  to identify appropriate candidates as pure
basis elements. The last but most extensively used approach is a heuristic
method based on Wasser:2018qvj , working loop-by-loop and using already known
one, two- and three-loop pure basis elements DiVita:2014pza ; Henn:2014lfa ;
Dlapa:2021qsl . For a more detailed discussion we refer the interested reader
to Canko:2021hvh .

All approaches mentioned above are valid for the determination of a candidate
pure basis. The ultimate test that such a basis has been obtained is whether
it satisfies a canonical DE. Indeed, by differentiating with respect to $x$ we
were able to obtain the following SDE in canonical form for families F2 and
F3,

$\partial_{x}\textbf{g}=\epsilon\left(\sum_{i=1}^{4}\frac{\textbf{M}_{i}}{x-l_{i}}\right)\textbf{g}$
(14)

where g is the pure basis and $\textbf{M}_{i}$ are the residue matrices
corresponding to each letter $l_{i}$. All kinematic dependence is included in
the letters $l_{i}$, leaving the matrices $\textbf{M}_{i}$ to consist solely
of rational numbers. We have found an alphabet consisting of the four
following letters

$l_{1}=0,~{}l_{2}=1,~{}l_{3}=\frac{1}{1+y},~{}l_{4}=-\frac{1}{y}.$ (15)

Notice that here we follow Papadopoulos:2015jft ; Canko:2020gqp  for the
definition of the letters of the alphabet, which is different from the
standard notation Duhr:2011zq ; Duhr:2012fh ; Duhr:2014woa . Usually the so-
called $\mathrm{d}\log$ form of a system of canonical differential equations
is given as
$\mathrm{d}\textbf{g}(\vec{x},\epsilon)=\epsilon\left(\sum_{i}\textbf{M}_{i}\mathrm{d}\log
W_{i}(\vec{x})\right)\textbf{g}(\vec{x},\epsilon)$, where the alphabet
$W_{i}(\vec{x})$ is in terms of rational or algebraic functions of the
independent variables. The standard $\mathrm{d}\log$ form is equivalent to
(14) for $W_{i}(\vec{x})=x-l_{i}$.

A few comments here are in order. Firstly, the same alphabet was found for the
F1 family in Canko:2020gqp . Secondly, we have observed that in every case
where both the standard approach, i.e. differentiating with respect to all
kinematic invariants, and the SDE approach have been applied, we have found a
reduced number of logarithmic singularities in the canonical SDE. For the
ladder topology, in DiVita:2014pza  the corresponding canonical DE was
characterised by six logarithmic singularities, i.e. six $W_{i}(\vec{x})$
functions according to the standard $\mathrm{d}\log$ notation. With the SDE
approach, we obtain a canonical DE in $x$ characterised by four
$W_{i}(\vec{x})$ functions. This reduction in the number of poles of the
corresponding canonical DE stems from the fact that we only consider the
differentiation with respect to $x$, leaving the $s_{12}$\- and $y$-dependence
of the MI to be explicitly treated in the boundary terms. Furthermore, the
fact that all three-loop planar families with one off-shell leg are
characterised by the same alphabet is also an interesting one. This was also
observed in the fully massless case, both for the planar Henn:2013tua  and the
non-planar Henn:2020lye  topologies. If this property holds true for the
three-loop non-planar families with one off-shell leg as well, then we expect
that their solution in terms of GPLs will be an achievable goal in the
foreseeable future.

The simplicity of the alphabet (15) in $x$ allows for a straightforward
solution of (14) in terms of GPLs. The solution can be written in the
following compact form up to weight six:

${\footnotesize\begin{split}\textbf{g}&=\epsilon^{0}\textbf{b}_{0}^{(0)}+\epsilon\left(\sum{\cal
G}_{i}\textbf{M}_{i}\textbf{b}_{0}^{(0)}+\textbf{b}_{0}^{(1)}\right)+\epsilon^{2}\left(\sum{\cal
G}_{ij}\textbf{M}_{i}\textbf{M}_{j}\textbf{b}_{0}^{(0)}+\sum{\cal
G}_{i}\textbf{M}_{i}\textbf{b}_{0}^{(1)}+\textbf{b}_{0}^{(2)}\right)+\dots\\\
&+\epsilon^{6}\left(\textbf{b}_{0}^{(6)}+\sum{\cal
G}_{ijklmn}\textbf{M}_{i}\textbf{M}_{j}\textbf{M}_{k}\textbf{M}_{l}\textbf{M}_{m}\textbf{M}_{n}\textbf{b}_{0}^{(0)}+\sum{\cal
G}_{ijklm}\textbf{M}_{i}\textbf{M}_{j}\textbf{M}_{k}\textbf{M}_{l}\textbf{M}_{m}\textbf{b}_{0}^{(1)}\right.\\\
&+\left.\sum{\cal
G}_{ijkl}\textbf{M}_{i}\textbf{M}_{j}\textbf{M}_{k}\textbf{M}_{l}\textbf{b}_{0}^{(2)}+\sum{\cal
G}_{ijk}\textbf{M}_{i}\textbf{M}_{j}\text{M}_{k}\textbf{b}_{0}^{(3)}+\sum{\cal
G}_{ij}\textbf{M}_{i}\textbf{M}_{j}\textbf{b}_{0}^{(4)}+\sum{\cal
G}_{i}\textbf{M}_{i}\textbf{b}_{0}^{(5)}\right)\end{split}}$ (16)

were $\mathcal{G}_{ab\ldots}:=\mathcal{G}(l_{a},l_{b},\ldots;x)$ represent the
GPL. The $\textbf{b}_{0}^{(i)}$ terms represent the boundary terms that need
to be determined, with $i$ indicating the corresponding weight, and consist of
Zeta functions $\zeta(i)$ and logarithms $\\{\log(-s_{12}),~{}\log(y)\\}$ of
weight $i$. Our results are presented in such a way that each coefficient of
$\epsilon^{i}$ has transcendental weight $i$. If we assign weight $-1$ to
$\epsilon$, then (16) has uniform weight zero.

###  3.1 Boundary terms

In order to determine the necessary boundary terms $\textbf{b}_{0}^{(i)}$ in
(16) we will employ the techniques developed in Canko:2020gqp ; Canko:2020ylt
, as well as showcase a new way of using them that significantly simplifies
the entire procedure.

In general we need to calculate the $x\to 0$ limit of each pure basis element.
Our first step is to exploit the canonical SDE at the limit $x\to 0$ and
define through it the resummation matrix

$\textbf{R}=\textbf{S}\mathrm{e}^{\epsilon\textbf{D}\log(x)}\textbf{S}^{-1}$
(17)

where the matrices $\textbf{S},~{}\textbf{D}$ are obtained through the Jordan
decomposition333For an earlier use of the Jordan decomposition method see
Dulat:2014mda ; Mastrolia:2017pfy . of the residue matrix for the letter
$l_{1}=0$, $\textbf{M}_{1}$,

$\textbf{M}_{1}=\textbf{S}\textbf{D}\textbf{S}^{-1}.$ (18)

On the other hand, through IBP reduction, the elements of the pure basis can
be related to a set of FI G,

$\textbf{g}=\textbf{T}\textbf{G}.$ (19)

Furthermore using the expansion by regions method Jantzen:2012mw  as
implemented in the asy code which is shipped along with FIESTA4
Smirnov:2015mct , we can obtain information for the asymptotic behaviour of
the FI in terms of which we express the pure basis of MI (19) in the limit
$x\to 0$,

${G_{i}}\mathop{=}\limits_{x\to
0}\sum\limits_{j}x^{b_{j}+a_{j}\epsilon}G^{(b_{j}+a_{j}\epsilon)}_{i}$ (20)

where $a_{j}$ and $b_{j}$ are integers and $G_{i}$ are the individual members
of the basis G of FI in (19). This analysis allows us to construct the
following relation

$\mathbf{R}\mathbf{b}=\left.\lim_{x\rightarrow
0}\mathbf{T}\mathbf{G}\right|_{\mathcal{O}\left(x^{0+a_{j}\epsilon}\right)}$
(21)

where the right-hand side implies that, apart from the terms
$x^{a_{i}\epsilon}$ coming from (20), we expand around $x=0$, keeping only
terms of order $x^{0}$. Equation (21) allows us in principle to determine all
boundary constants $\textbf{b}=\sum_{i=0}^{6}~{}\epsilon^{i}~{}b_{0}^{(i)}$.

More specifically, equation (21) produces two kinds of relations. The first
one, called pure relations in Canko:2020gqp , is in the form of linear
relations with numerical rational coefficients between boundary terms b and
the second one is in the form of linear relations that contain boundary terms
b and region-integrals $G^{(b_{j}+a_{j}\epsilon)}_{i}$ in the Feynman
parameter representation. These relations have been sufficient in past studies
Canko:2020gqp ; Canko:2020ylt ; Syrrakos:2020kba ; Syrrakos:2021nij  for the
calculation of all boundary terms. Their application for families F2 and F3
however requires the computation of several highly non-trivial region-
integrals.

To circumvent this issue, we take advantage of the fact that the $x\to 1$
limit of our solution (16) will yield the result for the fully massless
integral family Canko:2020gqp , which is already available in Henn:2013tua .
Our approach therefore is to exploit the pure relations between boundary terms
b, compute a minimal number of region-integrals and then construct the full
solution by explicitly keeping the undetermined boundary terms in the final
result. Then we extract the $x\to 1$ limit using the techniques of
Canko:2020gqp  and use the known analytic results of Henn:2013tua  to fix all
remaining unknown boundaries.

To exemplify this approach we will present here the computation of all
boundary terms for the F3 family, assuming we have already solved family F2.
Starting with the construction of equation (21), we determine the so-called
pure relations. These relations usually determine the boundary terms for the
top-sector basis elements, which are the most non-trivial to compute since
they involve the most complicated region-integrals. For the case of family F3
these pure relations for the four top-sector basis elements are

$\displaystyle b_{163}=$
$\displaystyle-\frac{371b_{1}}{132}-\frac{8b_{2}}{11}+\frac{12b_{4}}{11}-\frac{637b_{5}}{33}+32b_{10}+\frac{64b_{11}}{3}+\frac{37b_{12}}{6}-15b_{13}+\frac{3b_{15}}{22}-15b_{18}$
$\displaystyle+12b_{19}-\frac{9b_{22}}{4}-\frac{9b_{23}}{2}+3b_{24}-\frac{15b_{25}}{22}-\frac{b_{26}}{22}+12b_{29}+8b_{30}-6b_{31}+\frac{75b_{35}}{11}+6b_{36}$
(22)
$\displaystyle-\frac{84b_{38}}{11}+9b_{43}+36b_{52}-18b_{53}-21b_{54}+6b_{57}-\frac{6b_{77}}{11}-b_{81}-12b_{87}+3b_{88}-2b_{123}$
$\displaystyle-6b_{130}+2b_{137}+6b_{144}-2b_{152}-2b_{159},$ $\displaystyle
b_{164}=$
$\displaystyle-\frac{745b_{1}}{264}-\frac{13b_{2}}{66}+\frac{6b_{4}}{11}-\frac{359b_{5}}{22}+\frac{92b_{10}}{3}+\frac{52b_{11}}{3}+\frac{31b_{12}}{6}-\frac{89b_{13}}{6}+\frac{51b_{15}}{22}$
$\displaystyle-\frac{97b_{18}}{6}+12b_{19}-\frac{9b_{22}}{8}-\frac{9b_{23}}{2}+3b_{24}-\frac{15b_{25}}{44}+\frac{5b_{26}}{22}+10b_{29}+\frac{20b_{30}}{3}-6b_{31}$
(23)
$\displaystyle+\frac{75b_{35}}{22}+5b_{36}-\frac{42b_{38}}{11}+9b_{43}+36b_{52}-18b_{53}-\frac{33b_{54}}{2}+6b_{57}-\frac{3b_{77}}{11}-b_{81}$
$\displaystyle-12b_{87}+3b_{88}-2b_{123}-6b_{130}+2b_{137}+4b_{144}-3b_{152}-3b_{159},$
$\displaystyle b_{165}=$ $\displaystyle 0,$ (24) $\displaystyle b_{166}=$
$\displaystyle-\frac{1531b_{1}}{4752}-\frac{128b_{2}}{297}+\frac{47b_{4}}{33}-\frac{1891b_{5}}{396}+\frac{74b_{10}}{9}+\frac{20b_{11}}{3}+\frac{7b_{12}}{3}-\frac{127b_{13}}{36}-\frac{415b_{15}}{264}$
$\displaystyle+\frac{13b_{16}}{8}+\frac{10b_{17}}{3}-\frac{47b_{18}}{36}-2b_{19}+\frac{5b_{20}}{6}-\frac{21b_{22}}{16}-\frac{11b_{23}}{6}+\frac{5b_{24}}{12}-\frac{35b_{25}}{132}-\frac{6b_{26}}{11}$
$\displaystyle+\frac{16b_{29}}{3}+\frac{32b_{30}}{9}-\frac{10b_{31}}{3}+\frac{581b_{35}}{132}+\frac{29b_{36}}{18}-\frac{197b_{38}}{33}+\frac{3b_{43}}{2}-\frac{14b_{49}}{3}+7b_{52}$
(25)
$\displaystyle-5b_{53}-\frac{89b_{54}}{12}+\frac{13b_{57}}{3}-\frac{8b_{60}}{3}-\frac{b_{61}}{6}+2b_{62}-\frac{7b_{77}}{33}-\frac{b_{81}}{6}+3b_{83}-\frac{b_{84}}{2}-\frac{13b_{87}}{6}$
$\displaystyle+\frac{7b_{88}}{12}-\frac{2b_{89}}{3}+\frac{5b_{97}}{6}-\frac{b_{108}}{3}-\frac{2b_{123}}{3}-b_{130}+\frac{2b_{137}}{3}+2b_{144}-\frac{4b_{152}}{3}-\frac{2b_{159}}{3}.$

Similar relations are obtained for 109 basis elements in total, leaving the
following basis elements undetermined,

$\displaystyle\big{\\{}$ $\displaystyle
b_{1},b_{2},b_{4},b_{5},b_{7},b_{10},b_{11},b_{12},b_{13},b_{15},b_{16},b_{17},b_{18},b_{19},b_{20},b_{22},b_{23},b_{24},b_{25},b_{26},b_{29},b_{30},b_{31},b_{35},$
$\displaystyle
b_{36},b_{38},b_{43},b_{49},b_{50},b_{52},b_{53},b_{54},b_{57},b_{59},b_{60},b_{61},b_{62},b_{77},b_{81},b_{83},b_{84},b_{85},b_{87},b_{88},b_{89},b_{97},b_{108},$
$\displaystyle
b_{116},b_{123},b_{130},b_{135},b_{137},b_{144},b_{151},b_{152},b_{157},b_{159}\big{\\}}.$
(26)

From the above list of boundary terms, most of them are known either from
family F1 or from family F2. The only genuinely unknown boundary terms
therefore are

$\big{\\{}b_{108},b_{123},b_{135},b_{144},b_{157},b_{159}\big{\\}}.$ (27)

Furthermore, $b_{108}$ can be easily calculated through a direct integration
of the region-integrals that appear in its result coming from (21),

$\displaystyle b_{108}=$
$\displaystyle-2b_{19}+\frac{3b_{21}}{4}+s_{12}^{2}\epsilon^{5}G_{111101012000000}^{(-2\epsilon)}+4s_{12}\epsilon^{4}G_{1022010110-10000}^{(-\epsilon)}$
$\displaystyle-3s_{12}^{2}\epsilon^{4}G_{112201001000000}^{(-\epsilon)}+6s_{12}\epsilon^{5}G_{011101012000000}^{(0)}.$
(28)

As you can see, the boundary term $b_{108}$ involves four region-integrals
with the most non-trivial having seven propagators, i.e requiring the
integration of seven Feynman parameters - in general a non-trivial, but in
this case achievable task. This leaves us with the following five undetermined
boundary terms $\big{\\{}b_{123},b_{135},b_{144},b_{157},b_{159}\big{\\}}$. If
we were to continue with this approach for the rest of the undetermined
boundary terms, we would have to compute several highly non-trivial region-
integrals, involving the direct integration of up to nine Feynman parameters.
Such calculations can be extremely difficult, even if the result of the
integration is rather simple.

To obtain the remaining boundary terms, we will exploit a feature of the SDE
approach which is that by taking the $x\to 1$ limit of our results, we recover
the solution for the massless momentum configuration, as can be seen from (4)
and (5). Our approach involves the following steps:

  1. 1.

Construct solution (16) using a simple ansatz for the undetermined boundary
terms, i.e. $b_{i}=\sum_{k=0}^{6}a(i,k)\epsilon^{k}$.

  2. 2.

Construct a resummation matrix from the $\textbf{M}_{2}$ residue matrix and
using the shuffle properties of GPLs extract the part of the solution (16)
which is regular at $x=1$. These two ingredients allow us to obtain the $x\to
1$ limit of (16) Canko:2020gqp .

  3. 3.

Map the $x\to 1$ limit of (2.1), i.e. the massless tennis-court, to the same
family defined in Henn:2013tua .

  4. 4.

Map the pure basis for the massless tennis-court of Henn:2013tua  to the pure
basis that can be obtained from the $x\to 1$ limit of the pure basis of F3
using the techniques of Canko:2020gqp .

The mapping between our results and that of Henn:2013tua  yields that the
variable $y=\frac{s_{23}}{s_{12}}$ that we introduced earlier is equal to the
dimensionless parameter $x$ used therein444Not to be confused with the SDE
parameter $x$ that we use throughout this paper.. In the case of the F2 family
the respective relation is $y=\frac{1}{x}$. This exemplifies the reason we
introduced $y$ in the first place.

The above steps allow us to fix all remaining boundary terms in a purely
analytical way. We find that a general ansatz for all boundary terms can be
constructed,

$\displaystyle b_{i}$
$\displaystyle=c(i,0)+\epsilon\log(y)c(i,1)+\epsilon^{2}\left(\frac{1}{2}\log^{2}(y)c(i,2,2)+\frac{1}{6}\pi^{2}c(i,2,1)\right)$
$\displaystyle+\epsilon^{3}\left(\frac{1}{6}\log^{3}(y)c(i,3,2)+\frac{1}{6}\pi^{2}\log(y)c(i,3,1)+\zeta(3)c(i,3,3)\right)$
$\displaystyle+\epsilon^{4}\left(\zeta(3)\log(y)c(i,4,4)+\frac{1}{24}\log^{4}(y)c(i,4,3)+\frac{1}{12}\pi^{2}\log^{2}(y)c(i,4,2)+\frac{1}{90}\pi^{4}c(i,4,1)\right)$
$\displaystyle+\epsilon^{5}\bigg{(}\frac{1}{2}\zeta(3)\log^{2}(y)c(i,5,5)+\frac{1}{120}\log^{5}(y)c(i,5,3)+\frac{1}{36}\pi^{2}\log^{3}(y)c(i,5,2)$
(29)
$\displaystyle+\frac{1}{90}\pi^{4}\log(y)c(i,5,1)+\zeta(5)c(i,5,6)+\frac{1}{6}\pi^{2}\zeta(3)c(i,5,4)\bigg{)}$
$\displaystyle+\epsilon^{6}\bigg{(}\frac{1}{6}\zeta(3)\log^{3}(y)c(i,6,6)+\log(y)\left(\frac{1}{6}\pi^{2}\zeta(3)c(i,6,5)+\zeta(5)c(i,6,8)\right)+\frac{1}{720}\log^{6}(y)c(i,6,4)$
$\displaystyle+\frac{1}{144}\pi^{2}\log^{4}(y)c(i,6,3)+\frac{1}{180}\pi^{4}\log^{2}(y)c(i,6,2)+\zeta(3)^{2}c(i,6,7)+\frac{1}{945}\pi^{6}c(i,6,1)\bigg{)}$

where we have multiplied by $(-s_{12})^{(3\epsilon)}$ and expanded up to
$\epsilon^{6}$ to make the ansatz more compact. This is of course the most
general form that is compatible with the scaling and universal
transcendentality of the boundaries. If the same alphabet holds for some or
all the non-planar families one could envision obtaining the boundary terms by
fitting numerically this ansatz for the remaining families of integrals using
the PSLQ algorithm pslq .

##  4 Results

###  4.1 Analytic continuation

As already mentioned, our results are expressed in terms of GPLs up to weight
six and thus can be numerically computed at high precision using automated
tools, like GinaC Vollinga:2004sn . The weight W=$1\ldots 6$ is identified as
the number of letters $l_{i}$ in $\mathcal{G}(l_{i},\ldots;x)$. For the
evaluations to be fast and efficient the GPLs should not contain letters
$l_{i}$ along the integration path connecting the origin and the argument $x$,
i.e. $l_{i}\notin[0,x]$. Although this holds true in the Euclidean region,
this is not the case in the physical regions anymore. One can get over this
problem by using fibration-basis techniques Duhr:2019tlz ; Panzer:2014caa  for
changing appropriately the arguments of GPLs at each physical region and
making them real-valued. We followed this procedure using HyperInt
Panzer:2014caa , aiming to obtain results in terms of real-valued GPLs which
can be efficiently evaluated numerically, thus making them well-suited for
phenomenological applications.

Regions | Letters | Argument | Letters | Argument
---|---|---|---|---
Euclidean | $\\{0,1,-1/y,1/(1+y)\\}$ | $x$ | $-$ | $-$
s-channel | $\\{0,1,-1/y,1/(1+y)\\}$ | $x$ | $-$ | $-$
t-channel | $\\{0,1,1+y,-y\\}$ | $1/x$ | $\\{0,1\\}$ | $-1/y$
u-channel | $\\{0,1,1+y,-y\\}$ | $1/x$ | $\\{0,-1\\}$ | $y$
Table 1: Structure of GPLs appearing in each of the 4 kinematic regions.

In order to make more clear the form of our solutions, we present in table 1
the arguments and the letters of the GPLs in the Euclidean and each of the
physical regions after using HyperInt. Note that prior to the use of
fibration-basis techniques, no GPLs with arguments depending on $y$ are
present. However, casting each GPL in a fibration basis which contains real-
valued GPLs and explicit imaginary terms, produces GPLs with $y$-dependent
argument as well.

In table 2 we provide an analysis regarding the number of GPLs that appear in
each transcendental weight, for each kinematic region. In the same table we
quote also the total number of GPLs and the timing for their numerical
evaluation using GinaC. All the families have the same GPLs at every region
except for the F1 family, which in the Euclidean region (and in the s-channel)
contains 42 less GPLs, all of them of weight 6. The number of GPLs is the same
for the Euclidean and the s-channel due to the fact that the GPLs of the
Euclidean region are already real valued in the s-channel and thus their is no
need of using fibration-basis techniques for them.

Regions | $W=1$ | $W=2$ | $W=3$ | $W=4$ | $W=5$ | $W=6$ | Total | Timings (sec)
---|---|---|---|---|---|---|---|---
Euclidean | 4 | 14 | 50 | 124 | 367 | 734 | 1293 | 39.0225769
s-channel | 4 | 14 | 50 | 124 | 367 | 734 | 1293 | 39.2172529
t-channel | 6 | 18 | 58 | 155 | 419 | 603 | 1259 | 62.0567800
u-channel | 5 | 16 | 54 | 147 | 403 | 572 | 1197 | 55.1049640
Table 2: Number of GPLs per transcendental weight and per kinematic region,
and timings for the numerical evaluation of the total GPLs. The quoted timings
are obtained using the Ginsh command of PolyLogTools, running 1-core in a
personal laptop (i7 processor, 8-core, 16GB RAM). The phase-space points that
were used are (30) for the Euclidean region and (31), (32), (33) for the
correspoing physical regions.

###  4.2 Numerical checks

For the validation of our results we have performed various numerical checks
in the Euclidean and physical regions. More specifically, for the Euclidean
region we compared numerically every MI of each family with pySecDec
Borowka:2017idc  and Fiesta Smirnov:2015mct  for the point

$s_{12}\rightarrow-7,\,y\rightarrow 3/7,\,x\rightarrow 1/4$ (30)

finding perfect agreement within the numerical accuracy provided by these
programs.

Regarding our results for physical regions, obtaining numerical results for
all MI using pySecDec or Fiesta can be rather challenging, and in some cases
even impossible. To that end, in order to check that the analytic continuation
was correct, we determined a specific set of low-sector MI (with up to 7
propagators), which contained the total set of GPLs at each region and we
numerically compared them with pySecDec and Fiesta for the points

$\displaystyle\text{s-channel}:~{}s_{12}\rightarrow
2,\,y\rightarrow-1/2,\,x\rightarrow 1/4$ (31)
$\displaystyle\text{t-channel}:~{}s_{12}\rightarrow-2,\,y\rightarrow-3/2,\,x\rightarrow
5/3$ (32) $\displaystyle\text{u-channel}:~{}s_{12}\rightarrow-2,\,y\rightarrow
3/2,\,x\rightarrow 5/3.$ (33)

After properly choosing the branch of the analytic continuation, i.e. fixing
the values of the auxiliary functions
$\\{\delta(1/x),\delta(-1/y),\delta(y)\\}$ coming from HyperInt, perfect
agreement was found for every MI that we checked.

###  4.3 Analytic checks

For all MI of families F2 and F3 we performed analytic checks at the limit
$x\to 1$ against the results of Henn:2013tua  in the Euclidean region. Even
though this limit was used to obtain some of the necessary boundary terms,
ensuring that all MI of families F2 and F3 correctly reproduce the results of
Henn:2013tua  at the limit $x\to 1$ constitutes a non-trivial check.

###  4.4 Ancillary files

Several of our results are available in gitfiles  in Mathematica readable
format. More specifically, we include the following files:

  * •

The file input.m contains the propagators from (2.1), (2.1), (2.1) as well as
their kinematics, the mapping (5) and the alphabet (15) which is the same for
all three integral families.

  * •

The directories Fi, with $i=1,2,3$, which contain:

    * –

Fibasis.m: the pure basis.

    * –

FiDE.m: the canonical DE.

    * –

FiBounds.m: boundary terms for solving the canonical DE in the Euclidean
region.

    * –

FiE.m: the solution in the Euclidean region.

    * –

Fis.m: the solution in the s-channel.

    * –

Fit.m: the solution in the t-channel.

    * –

Fiu.m: the solution in the u-channel.

We also provide the files Fibasis.m, FiDE.m and FiBounds.m with the arXiv
submission of our paper.

##  5 Conclusion

In this paper we presented the analytic computation of the remaining planar
three-loop tennis-court families for $2\to 2$ processes involving massless
propagators and one off-shell leg, in terms of GPLs of up to transcendental
weight six. We also presented results for all three-loop planar families, the
so-called ladder-box topology with one off-shell leg as well as the two
tennis-courts, for scattering kinematics in terms of real-valued
polylogarithmic functions, making our results well-suited for phenomenological
studies of processes such as vector boson decaying to 3-jets or $gg\to H+jet$
in gluon fusion, at N3LO precision.

We constructed pure bases for the two tennis-court families and derived
canonical DE in the SDE approach, obtaining a simple four-letter alphabet
(15), which was identical to the one found for the ladder-box topology in
Canko:2020gqp . It would be interesting to apply the standard DE approach to
the pure bases of the two tennis-court families, i.e. differentiate with
respect to all kinematic invariants, and see if their alphabets match the one
of the ladder-box topology obtained in DiVita:2014pza .

A natural next step is the consideration of the non-planar three-loop families
with one massive leg. For the massless case Henn:2020lye , all integral
families satisfied canonical DE with the same alphabet as in the planar
families Henn:2013tua . It will be interesting to see if the non-planar
families with one massive leg exhibit the same behaviour as well. Constructing
pure bases for all of them is a challenging, yet we believe not insurmountable
task.

The mathematical simplicity of the canonical DE (14) for all planar families
was instrumental in obtaining results in terms of real-valued GPLs for all
physical regions, using existing tools Panzer:2014caa ; Duhr:2019tlz  for
their analytic continuation. Should such simplicity be found for the non-
planar families, we are confident that their solution in terms of real-valued
GPLs for all physical regions of phase-space would be an achievable goal.

###### Acknowledgements.

All figures have been drawn using JaxoDraw Binosi:2008ig . We would like to
thank Federico Gasparotto and Luca Mattiazzi for their help with the Magnus
series expansions method at the first stages of this project and for many
fruitful discussions. The research work of DC was supported by the Hellenic
Foundation for Research and Innovation (HFRI) under the HFRI Ph.D. Fellowship
grant (Fellowship Number: 554). NS was supported by the Excellence Cluster
ORIGINS funded by the Deutsche Forschungsgemeinschaft (DFG, German Research
Foundation) under Germany’s Excellence Strategy - EXC-2094 - 390783311.

##  Appendix A Adjacency conditions in the SDE approach

In this section we comment on the interpretation of the adjacency conditions
discovered in Dixon:2020bbt ; Chicherin:2020umh , within the SDE approach. In
order to do this, firstly, we make a connection between the standard DE method
and the SDE approach.

Figure 2: The ladderbox (F1) family using the notation followed in
Chicherin:2020umh . The double line represents the massive particle and all
external momenta are taken to be incoming.

For the standard DE method we are going to use the notation used in
Chicherin:2020umh  (see figure 2). In this article, the Lorentz invariants are
defined as follows

$z_{1}=\frac{2p^{\prime}_{1}\cdot
p^{\prime}_{2}}{P^{2}},\,\,\,\,z_{2}=\frac{2p^{\prime}_{2}\cdot
p^{\prime}_{3}}{P^{2}}\,\,\,\,\text{and}\,\,\,\,z_{3}=1-z_{1}-z_{2}.$ (34)

The alphabet of this family (using the standard notation Duhr:2011zq ;
Duhr:2012fh ; Duhr:2014woa ), as it is derived by applying the standard DE
method, is

$\\{z_{1},z_{2},z_{3},1-z_{1},1-z_{2},1-z_{3}\\}\to\\{z_{1},z_{2},1-z_{1}-z_{2},1-z_{1},1-z_{2},z_{1}+z_{2}\\}$
(35)

and the pure basis of master integrals G satisfies the differential equation

$\begin{split}\mathrm{d}\textbf{G}&=\epsilon\bigg{(}\textbf{A}_{z_{1}}\mathrm{d}\log(z_{1})+\textbf{A}_{z_{2}}\mathrm{d}\log(z_{2})+\textbf{A}_{1-z_{1}-z_{2}}\mathrm{d}\log(1-z_{1}-z_{2})\\\
&+\textbf{A}_{1-z_{1}}\mathrm{d}\log(1-z_{1})+\textbf{A}_{1-z_{2}}\mathrm{d}\log(1-z_{2})+\textbf{A}_{z_{1}+z_{2}}\mathrm{d}\log(z_{1}+z_{2})\bigg{)}\textbf{G}\end{split}$
(36)

The $C_{2}$ adjacency conditions Chicherin:2020umh  imply that the letters
$\\{1-z_{i},1-z_{j}\\}$ for $i,j=1,2,3$ and $i\neq j$ never appear next to
each other in a symbol, or equivalently (in terms of the matrices of the DE),
that the matrices of the DE satisfy the following conditions

$\textbf{A}_{1-z_{1}}\textbf{A}_{1-z_{2}}=0,\,\,\,\,\textbf{A}_{1-z_{1}}\textbf{A}_{z_{1}+z_{2}}=0\,\,\,\,\text{and}\,\,\,\,\textbf{A}_{1-z_{2}}\textbf{A}_{z_{1}+z_{2}}=0.$
(37)

As we have already discussed, in our treatment of the three-loop planar
families with one off-shell leg, we use the notation as is depicted in figure
1, while the kinematic invariants are defined as in (5) with the extra
introduction of the ratio $y=s_{23}/s_{12}$ for convenience. The pure basis of
master integrals satisfies the differential equation (14), which in more
detail can be written as

$\frac{\mathrm{d}}{\mathrm{d}x}\textbf{G}=\epsilon\left(\frac{\textbf{M}_{0}}{x}+\frac{\textbf{M}_{1}}{x-1}+\frac{\textbf{M}_{-1/y}}{x+1/y}+\frac{\textbf{M}_{1/(1+y)}}{x-1/(1+y)}\right)\textbf{G}.$
(38)

From figures 1 and 2 (using also (4)) we observe that the momenta in the two
different notations are related via the transformation

$P\to p_{1}+p_{2}-xp_{1},\,\,\,\,p^{\prime}_{1}\to
xp_{1},\,\,\,\,p^{\prime}_{2}\to
p_{4},\,\,\,\,\text{and}\,\,\,\,p^{\prime}_{3}\to p_{3}.$ (39)

Thus using (39) in (34) we find that the transformation that takes us from the
$\\{z_{1},z_{2}\\}$ invariants to the $\\{x,y\\}$ ones, is

$z_{1}=\frac{xy}{1-x}\,\,\,\,\text{and}\,\,\,\,z_{2}=\frac{1}{1-x}$ (40)

while the reverse transformation reads

$x=\frac{z_{2}-1}{z_{2}}\,\,\,\,\text{and}\,\,\,\,y=\frac{z_{1}}{z_{2}-1}.$
(41)

In addittion, using (40) the alphabet in (35) can be written in terms of
$\\{x,y\\}$ as (keeping the same ordering for the letters)

$\left\\{\frac{xy}{1-x},\frac{1}{1-x},\frac{-x(1+y)}{1-x},-\left(\frac{1+y}{1-x}\right)\left(x-\frac{1}{1+y}\right),\frac{-x}{1-x},\left(\frac{y}{1-x}\right)\left(x+\frac{1}{y}\right)\right\\}.$
(42)

Having established the connection between the standard DE method and the SDE
approach, via the information stated above, we can see now how the adjacency
conditions are interpreted within the latter. More specifically, expressing
the letters in (36) via the relation (42) and differentiating with respect to
$x$ we obtain

$\begin{split}\frac{\mathrm{d}}{\mathrm{d}x}\textbf{G}&=\epsilon\left(\frac{\textbf{A}_{1-z_{2}}+\textbf{A}_{z_{1}}+\textbf{A}_{1-z_{1}-z_{2}}}{x}+\frac{\textbf{A}_{z_{1}+z_{2}}}{x+1/y}+\frac{\textbf{A}_{1-z_{1}}}{x-1/(1+y)}\right.\\\
&\left.-\frac{\textbf{A}_{z_{1}}+\textbf{A}_{z_{2}}+\textbf{A}_{1-z_{1}-z_{2}}+\textbf{A}_{1-z_{1}}+\textbf{A}_{1-z_{2}}+\textbf{A}_{z_{1}+z_{2}}}{x-1}\right)\textbf{G}\end{split}$
(43)

Comparing equations (38) and (43), we find that following relations arise
among the residue matrices of the standard DE method and the SDE approach

$\begin{split}&\textbf{A}_{1-z_{1}}=\textbf{M}_{1/(1+y)}\\\
&\textbf{A}_{z_{1}+z_{2}}=\textbf{M}_{-1/y}\\\
&\textbf{A}_{z_{2}}=-\textbf{M}_{0}-\textbf{M}_{1}-\textbf{M}_{1/(1+y)}-\textbf{M}_{1/(1+y)}\\\
&\textbf{A}_{1-z_{2}}=\textbf{M}_{0}-\textbf{A}_{z_{1}}-\textbf{A}_{1-z_{1}-z_{2}}\end{split}$
(44)

From (44) we see that the matrices
$\\{\textbf{A}_{1-z_{1}},\textbf{A}_{z_{1}+z_{2}\\}}$ are completely
determined by the residue matrices of SDE, while $\textbf{A}_{1-z_{2}}$
remains unknown. Thus the only adjacency relation that can be passed from the
DE method to the SDE residue matrices is the

$\textbf{A}_{1-z_{1}}\textbf{A}_{z_{1}+z_{2}}=0\Rightarrow\textbf{M}_{1/(1+y)}\textbf{M}_{-1/y}=0$
(45)

Having the residue matrices $\\{\textbf{M}_{1/(1+y)},\textbf{M}_{-1/y}\\}$ for
the families F1, F2 and F3 we checked and we found out that indeed the above
relation holds true for all of them555For the F1 family this was obvious, as
the adjacency conditions have been already studied for this family.. Moreover,
assuming that (as in the SDE approach) the alphabet of the two tennis-court
families is the same with the one of the ladder-box family, this result can be
viewed as a first indication of the validity of the adjacency conditions for
the tennis-court families.

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