aainstitutetext: Higgs Centre for Theoretical Physics, School of Physics and
Astronomy,
The University of Edinburgh, Edinburgh EH9 3FD, Scotland, UKbbinstitutetext:
Institute for Particle Physics Phenomenology, Durham University, Durham DH1
3LE, UKccinstitutetext: Institute for Theoretical Physics, ETH Zürich, 8093
Zürich, Switzerland

# Dissecting polytopes: Landau singularities and asymptotic expansions in
$2\to 2$ scattering

Einan Gardi a Franz Herzog b Stephen Jones c Yao Ma<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

Parametric representations of Feynman integrals have a key property: many,
frequently all, of the Landau singularities appear as endpoint divergences.
This leads to a geometric interpretation of the singularities as faces of
Newton polytopes, which facilitates algorithmic evaluation by sector
decomposition and asymptotic expansion by the method of regions. Here we
identify cases where some singularities appear instead as pinches in
parametric space for general kinematics, and we then extend the applicability
of sector decomposition and the method of regions algorithms to such
integrals, by dissecting the Newton polytope on the singular locus. We focus
on $2\to 2$ massless scattering, where we show that pinches in parameter space
occur starting from three loops in particular nonplanar graphs due to
cancellation between terms of opposite sign in the second Symanzik polynomial.
While the affected integrals cannot be evaluated by standard sector
decomposition, we show how they can be computed by first linearising the graph
polynomial and then splitting the integration domain at the singularity, so as
to turn it into an endpoint divergence. Furthermore, we demonstrate that
obtaining the correct asymptotic expansion of such integrals by the method of
regions requires the introduction of new regions, which can be systematically
identified as facets of the dissected polytope. In certain instances, these
hidden regions exclusively govern the leading power behaviour of the integral.
In momentum space, we find that in the on-shell expansion for wide-angle
scattering the new regions are characterised by having two or more connected
hard subgraphs, while in the Regge limit they are characterised by Glauber
modes.

††preprint: IPPP/24/47

##  1 Introduction

Evaluating Feynman integrals is an essential and central ingredient of Quantum
Field Theory in a vast range of applications. Despite impressive progress over
many years Smirnov:2004ym ; Henn:2014qga ; Weinzierl:2022eaz , it remains
notoriously difficult. While analytic approaches to Feynman integrals,
wherever available, provide highly valuable insights and eventually facilitate
fast and flexible evaluation, numerical approaches SmnTtyk09FIESTA ;
SmnSmnTtyk11FIESTA2 ; Smn14FIESTA3 ; Smn16FIESTA4 ; Smn22FIESTA5 ; KnkUeda10 ;
Carter:2010hi ; Borowka:2015mxa ; Heinrich:2023til ; pySecDec17 ; Brsk20 ;
BrskMchTld23 ; LiuMa23AMFlow ; Hdg21DiffExp ; AmdlBcnDvtRanaVcn23SeaSyde
remain important, in particular when the complexity increases, in terms of the
loop order, the number of kinematic scales, or the relevant function space.
For similar reasons, also asymptotic expansions of Feynman integrals in ratios
of external kinematic parameters become increasingly important. Specifically,
the _Method of Regions_ (MoR) BnkSmn97 ; Smirnov:2002pj ; PakSmn11 ; Jtz11 ;
JtzSmnSmn12 ; SmnvSmnSmv19 ; GrdHzgJnsMaSchlk22 ; Ma23  provides a general
strategy to perform such expansions in dimensional
regularisation111Alternatives, which avoid dimensional regularisation, also
exist, see e.g. Plenter:2020lop ., exploiting the fact that dimensional
regularisation regularises both infrared and ultraviolet divergences. This
method has been used to study a variety of kinematic limits and it provides
insight on the infrared singularity structure of integrals and amplitudes. It
also sets the basis for formulating effective field theories which capture the
relevant degrees of freedom in certain limits, such as Heavy Quark Effective
Theory (HQET) IsgWise89 ; IsgWise90 ; Nbt98book ; Mnh00book ; Grz04book  or
Soft-Collinear Effective Theory (SCET) BchBrgFrl15book ; BurFlmLk00 ;
BurPjlSwt02-1 ; BurPjlSwt02-2 ; BnkChpkDhlFdm02 ; RstStw16 .

The tremendous challenge presented by Feynman integrals is largely due to
their complex analytic structure, dictated by their singularities (see e.g.
EdenLdshfOlvPkhn02book ; Stm95book ; ClsSprStm04 ; Stmg18 ; Agarwal:2021ais ).
Given any Feynman integral, a necessary condition for the emergence of
singularities — be it infrared divergences which manifest themselves as poles
in the dimensional regulator, or branch points at particular configurations of
the external kinematic parameters such as kinematic thresholds — is given by
the Landau equations Lnd59 ; Bjorken:1959fd ; Nakanishi . In momentum
representation, the Landau equations imply that the integration contours of
some loop momenta must be _pinched_ by singularities of the integrand. The
solutions of the Landau equations then identify manifolds in the loop momentum
space where the Feynman integral possibly diverges, typically characterised by
momenta that are collinear, soft, etc. These manifolds, often referred to as
_pinch surfaces_ in literature, have been studied extensively over the past
decades ClmNtn65 ; Stm78I ; ClsStm81 ; Stm96lectures , providing the basis for
establishing factorisation properties of scattering amplitudes and cross
sections.

Landau singularities and the associated pinch surfaces also dictate the
properties of asymptotic expansions of Feynman integrals. Indeed, whenever the
resulting function is non-analytic in the limit of interest, one expects that
a straightforward expansion (a Taylor expansion of the integrand) about this
limit should not commute with the integration over the loop momenta. Instead,
one must carefully consider how the integration variables behave in the limit
considered. In the MoR, one asserts that the integration domain can be divided
into several distinct regions, such that the Feynman integral can be expressed
as a sum over integrals expanded in these regions. In each such region, there
is a distinct way in which the loop momenta behave in the limit, which in turn
dictates how the integrand should be expanded (so that the expansion converges
inside the region). Thus, each such region is governed by the vicinity of a
particular pinch surface associated with a solution of the Landau equations
GrdHzgJnsMaSchlk22 , which determines the leading behaviour of the integrand.
In practice, once the expansion is performed in each region, one simply
performs the loop momentum integration over the entire domain Smirnov:2002pj ;
Jtz11 , with the expectation that the extension beyond the domain of
convergence would cancel amongst the various regions in the sum,222This
remarkable property has been elucidated in some detail in ref. Jtz11 ,
specifying conditions under which it is guaranteed to hold. owing to the
properties of dimensional regularisation. While no general proof of the MoR is
available, this strategy has been utilised extensively, and shown to work in
every instance, provided that certain issues are handled with sufficient care
(e.g. that all integrals are regularised). A remaining challenge is the need
to identify a complete set of regions in loop momentum space at the outset,
and so far there is no algorithmic method for doing so.

In addition to the momentum representation, Feynman integrals can also be
expressed in parameter space using e.g. Feynman or Lee-Pomeransky
representation LeePmrsk13 . In these representations, the singularity
structure is encoded in the zeros of the second Symanzik polynomial
$\mathcal{F}$. One advantage of these representations is that many (sometimes
all) of the singularities manifest themselves as _endpoint divergences_. These
are fully characterised by the scalings of the parameters as they approach the
endpoint of the integration domain (typically zero or infinity). The set of
exponents defining this scaling for a particular Landau singularity are
referred to as _weight vectors_ , or _region vectors_ in the context of the
MoR. These vectors are entirely independent of the kinematic coefficients
appearing in $\mathcal{F}$, and it is therefore possible to determine them
geometrically KnkUeda10 ; MzrTln22 ; AkHmHlmMzr22 ; BrskMchTld23 ;
FvlMzrTln23prl ; FvlMzrTln23pld . To this end one first maps each monomial of
the Symanzik polynomials333Such a geometric construction can be based on the
monomials of the product of the two Symanzik polynomials, ${\cal U}\cal{F}$,
which is natural when working with Feynman parameterisation, or the monomials
of their sum, ${\cal U}+\cal{F}$, which appears in the Lee-Pomeransky
representation. See section 2 below for further details., corresponding to a
Feynman graph with $N$ edges, to an $N$-dimensional vector whose entries are
the powers of the parameters in that monomial. Regarding each of these vectors
as a point in an $N$-dimensional space, one then constructs a Newton polytope
by forming the convex hull of all these points. One then finds that the normal
vectors to the faces of the polytope are precisely the weight vectors
characterising the endpoint divergences of the integral. This geometric
construction has many applications. In particular, it facilitates the
formulation of efficient algorithms to determine Landau singularities
Brown:2009ta ; Panzer:2014caa ; FvlMzrTln23pld ; FvlMzrTln23prl ;
Dlapa:2023cvx ; Helmer:2024wax , the study of the infrared singularity
structure of Feynman integrals in dimensional regularisation AkHmHlmMzr22 ,
the numerical computation Feynman integrals using _sector decomposition_
SmnTtyk09FIESTA ; SmnSmnTtyk11FIESTA2 ; Smn14FIESTA3 ; Smn16FIESTA4 ;
Smn22FIESTA5 ; KnkUeda10 ; Carter:2010hi ; Borowka:2015mxa ; pySecDec17 ;
Heinrich:2023til , or the closely related tropical Monte Carlo integration
Brsk20 ; BrskMchTld23 , and the asymptotic expansion of Feynman integrals by
the MoR directly in parameter space, as implemented in computer codes such as
Asy2 JtzSmnSmn12  (as part of the program FIESTA SmnTtyk09FIESTA ;
SmnSmnTtyk11FIESTA2 ; Smn14FIESTA3 ; Smn16FIESTA4 ; Smn22FIESTA5 ), ASPIRE
AnthnrySkrRmn19  and pySecDec pySecDec17 ; HrchJnsSlk22 .

Our recent study GrdHzgJnsMaSchlk22  focused on the application of the
geometric MoR to the  _on-shell expansion_ of massless particle scattering in
general kinematics. Here one begins with integrals defined for off-shell
external legs and expands them about the on-shell limit. It was shown there
that region vectors correspond to pinch surfaces defined by solutions of the
Landau equations in momentum space. In particular, this paper proposed that in
the on-shell expansion, the region vectors identified using the Newton
polytope stand in one-to-one correspondence with regions in momentum space,
and that they involve just three types of modes: hard, collinear and soft. On
this basis, a purely graph-theoretical algorithm was formulated to determine
all these regions, which makes no reference to the parametric representation.
A rigorous proof for this proposition was later given in ref. Ma23 , with
extensions to other types of expansions derived (see also ref. Herzog:2023sgb
). These provide promising avenues to a more complete theoretical
understanding of the underpinnings of the MoR and infrared divergences.

To summarise, a key advantage of formulating Feynman integrals in parameter
space is the fact many of the singularities appear as endpoint divergences.
This stands in contrast to momentum space, where (infrared) singularities in
Minkowski space typically manifest themselves as pinches of the loop momenta.
As a consequence, parametric representations have a geometric interpretation
and they lend themselves naturally to algorithmic methods, providing highly-
valued tools as well as a more general theoretical understanding of asymptotic
expansion of Feynman integrals and the related singularities.

The present work is motivated by the observation that some singularities do in
fact appear as pinches in parameter space. From the Landau equations, this
occurs when both ${\cal F}$ and $\partial{\cal F}/\partial\alpha_{i}$ vanish
away from the integration boundary. While this situation has been considered
since the early days (see e.g. EdenLdshfOlvPkhn02book ), its consequences for
evaluating Feynman integrals using sector decomposition and for applying the
MoR in parameter space have not yet received much attention.444Notable
exceptions in the application of the MoR on integrals involving internal
massive propagators near threshold or in the forward limit JtzSmnSmn12 ;
AnthnrySkrRmn19  will be discussed below (see section 3.1). As we will see,
existing sector decomposition algorithms fail when a pinch singularity is
present. Similarly, algorithms based on identifying regions as facets of the
Newton polytope, fail to capture the complete set of regions when expanded
about a limit in which a pinch is present. The goal of this study is to
overcome these challenges, so as to extend the applicability of sector
decomposition and MoR algorithms to these types of integrals.

In contrast to endpoint divergences, pinch singularities appear in the Feynman
parameterisation within the interior of the domain of integration. Such
solutions of the Landau equations are sensitive to the signs of the monomials
of the Symanzik $\mathcal{F}$ polynomial, and hence cannot be identified from
the Newton polytope alone. Specifically, they arise when terms of opposite
sign in $\mathcal{F}$ and in its derivatives cancel. The simplest example is
the one-loop self-energy integral in the threshold limit (see section 3.1). In
this case,
$\mathcal{F}_{\text{bub}}=(q^{2}/4)(\alpha_{1}-\alpha_{2})^{2}+y(\alpha_{1}+\alpha_{2})^{2}$,
where $\alpha_{1,2}$ are the Feynman parameters associated with the two
propagators and $y\to 0$ at threshold. Upon expanding the integral in powers
of $y$ one encounters a pinch singularity at $\alpha_{1}=\alpha_{2}$. To
handle this, one can split the domain of integration at the singularity into
the two sectors, $\alpha_{1}>\alpha_{2}$ and $\alpha_{2}>\alpha_{1}$. One then
employs a change of integration variables, in particular in the sector
$\alpha_{1}>\alpha_{2}$ one replaces $\alpha_{1}$ by
$\tilde{\alpha}_{1}=\alpha_{1}-\alpha_{2}>0$, after which the threshold
singularity manifests itself as an endpoint singularity at
$\tilde{\alpha}_{1}\to 0$ in the new variables
$(\tilde{\alpha}_{1},\alpha_{2})$. By summing both sectors one covers the
entire integration domain of the original integral, with each sector free of
pinch singularities, rendering it amenable to the standard geometric MoR
algorithm. This example illustrates both the problem and the solution devised
by Asy2 JtzSmnSmn12  and ASPIRE AnthnrySkrRmn19 . These tools can resolve a
pinch singularity in cases where there is _a linear shift_ of the parameters
that turns the singularity into an endpoint divergence. It turns out, however,
that this is merely the simplest cancellation structure occurring, while
multi-loop integrals possess far more.

To see why the problem of pinches in parameter space might be a rather general
feature of Minkowski-space Feynman integrals, recall that according to the
graph-theoretical definition of the Symanzik ${\cal F}$ polynomial (see eq.
(2) below) all Mandelstam invariants corresponding to channels on which the
graph has cuts, enter with the same sign. However, a Feynman integral can only
be defined after momentum conservation has been imposed, i.e. all momentum-
conservation relations between the Mandelstam invariants have been employed.
This inevitably implies that certain monomials proportional to a given
Mandelstam invariant will finally appear with opposite signs.

It is easiest to demonstrate this for $2\to 2$ massless scattering (see figure
1), on which we focus in this paper. In this case, momentum conservation
implies $s+t+u=0$. For planar graphs only two of the three invariants appear
at the outset (before momentum conservation is used) so for every such
integral there is a choice of kinematic regime in which all terms in ${\cal
F}$ have the same sign – referred to here as the _same-sign regime_ – and any
other kinematic region can be reached via analytic continuation. Thus, pinch
singularities cannot occur in planar graphs for $2\to 2$ massless scattering.
However, in the nonplanar case ${\cal F}$ is bound to contain all three
invariants, $s$, $t$, and $u$, so upon using momentum conservation to
eliminate say $u$, the terms in ${\cal F}$ proportional to $s$ and to $t$
would each contain monomials with opposite signs. This exemplifies the
situation where an integral does not have a _same-sign regime_ , and it opens
up the possibility that a nontrivial cancellation in ${\cal F}$ would occur in
_general kinematics_ , leading to a pinch and potentially, an infrared
divergence of the integral to which the Newton polytope is blind.

The requirements for a pinch which would prohibit the computation of the
integral by sector decomposition, are far more strict still: not only ${\cal
F}$ but also its derivatives must vanish. Furthermore, this should occur
within the domain of integration. These requirements are not easily satisfied,
but we find that starting from three loops there are indeed massless $2\to 2$
graphs that display such a pinch (even in general kinematics), and in such
cases, application of existing sector decomposition and MoR algorithms is
doomed to fail. Although not analysed in the context of these algorithms, we
point out that graphs admitting pinches in parameter space have been
identified in the literature as interesting, and their asymptotic expansions
have been analysed by multiple authors Pkhn63I ; Pkhn63II ; IslLdsfTlr63 ;
Hld63 ; Mdst63 ; Tktpls63 ; Tktpls63nonplanar ; Hld64 ; Lam69 ; Ldsf74 ;
Cls77book ; Pkhn80book , both in wide-angle and forward kinematics.

The simplest example of a massless $2\to 2$ Feynman graph featuring a pinch in
parameter space is the 8-propagator graph $G_{\bullet\bullet}$ (see figure
2(a)), a topology sometimes referred to as the _crown_. After eliminating $u$,
the terms proportional to $s$ in the Symanzik ${\cal F}$ polynomial take the
form
$(\alpha_{a}\alpha_{b}-\alpha_{c}\alpha_{d})(\alpha_{e}\alpha_{f}-\alpha_{g}\alpha_{h})$,
and similarly for the terms proportional to $t$, albeit with a different
combination of Feynman parameters. In this case, all the
$\partial\mathcal{F}/\partial\alpha_{i}$ (and thus $\mathcal{F}$) vanish upon
satisfying three independent nonlinear relations between the $\alpha_{i}$,
which are reached within the domain of integration. This is therefore a
genuine singularity of the integral, and it occurs for any values of $s$ and
$t$. Clearly, the singularity cannot be mapped to an endpoint of the
integration domain via a linear shift. However, through a suitable rescaling
of the parameters, we are able to linearise the polynomial near the singular
locus and subsequently split the integration domain into sectors, in order to
finally map the singularity onto the endpoint of the integration domain in
each sector. In terms of the associated geometry, this operation amounts to
dissecting the Newton polytope into 24 separate polytopes, each of which is
amenable to applying the standard sector decomposition algorithm. Ultimately,
this allowed us to evaluate the integral numerically with high precision, and
check the result against the analytic computation Henn:2020lye ;
Bargiela:2021wuy .

$G$$p_{1}$$p_{3}$$p_{2}$$p_{4}$

Figure 1: General massless two-to-two scattering, $1+2\to 3+4$. The Mandelstam
invariants $s=(p_{1}+p_{2})^{2}$, $t=(p_{1}-p_{3})^{2}$, $u=(p_{1}-p_{4})^{2}$
admit $s+t+u=0$.

We proceed by using the same technique of dissecting the polytope to perform
asymptotic expansions of integrals in this class. To verify the results and
gain another perspective, we also perform asymptotic expansions in momentum
space555To this end the work of Botts and Sterman BottsStm89  provides an
excellent starting point. and compare them to analytic results where
available. We consider two cases: wide-angle scattering and the Regge limit.
In the context of wide-angle scattering we study the on-shell expansion.
Specifically, we start with a four-point integral with massless propagators
and one off-shell external leg, with the remaining three external legs
strictly on shell, and then expand in the mass (virtuality) of the off-shell
leg. Since the expansion is performed about the on-shell limit, where a pinch
is present, the expansion is sensitive to the pinch surface. Indeed, we find
that this singularity generates a new region, which is not visible using the
original Newton polytope. We refer to this as a _hidden region_ and we show
that it appears as a facet only after dissecting the polytope as explained
above. This hidden region has a clear physical interpretation: it features two
distinct hard vertices, providing the simplest example666These contributions
are known to be suppressed in gauge theory BottsStm89SudakovEffects ;
BottsStm89 ; LbtdStm85 , but provide important examples for our study of
scalar integrals. of _Landshoff scattering_ Hld64 ; Ldsf74 ; BottsStm89 . This
stands in contrast to facet regions, which have just one hard subgraph in
wide-angle scattering GrdHzgJnsMaSchlk22 ; Ma23 . Turning to the Regge limit
of $2\to 2$ scattering, we similarly show that a hidden region emerges upon
dissecting the polytope. In this case the new region involves a loop momentum
in the _Glauber mode_. For the particular case of the 8-propagator graph
$G_{\bullet\bullet}$ in figure 2 in both the wide-angle on-shell expansion and
the Regge-limit expansion, the hidden regions alone govern the leading
asymptotic behaviour: all other regions, which are visible as facets of the
original Newton polytope, are power suppressed compared to the hidden one!

This paper is organised as follows. In section 2 we set up our notation,
briefly recalling the Feynman and Lee-Pomeransky parametric representations of
Feynman integrals, the Landau equations and the geometric approach to sector
decomposition and to the MoR. In section 3 we have two main objectives. The
first is to use the Landau equations in order to set up an algorithmic search
for graphs that are prone to having a pinch in parameter space, which we apply
to $2\to 2$ massless scattering graphs up to four loops. In the rest of the
paper we analyse the three-loop examples we identified. The second objective
is to devise a method of applying sector decomposition to integrals with a
pinch by dissecting the polytope as outlined above. We end this section with a
comparison between the numerical computation and analytic results. Next, in
section 4 we turn to study the MoR in the aforementioned class of three-loop
examples, focusing on the on-shell expansion for wide-angle scattering. We
perform a power counting analysis in both parameter space and momentum space,
demonstrating the presence of a hidden region. We also provide a numerical
analysis of the same graph near the on-shell limit. We demonstrate that this
class of integrals provides a departure from the graphical rules found in
refs. GrdHzgJnsMaSchlk22 , where only endpoint singularities were considered,
in that they involve more than one connected hard subgraph. In section 5 we
perform a similar analysis in the Regge limit, starting with the strictly on-
shell integrals. In this case we are able to verify our conclusions regarding
MoR expansion, by comparing it to the expansion of the analytic result in
powers of $-t/s$. Finally, in section 6 we summarise our conclusion and
provide an outlook.

##  2 Preliminaries

In this section, we review the concepts and definitions necessary for our
subsequent discussions of Landau singularities and the MoR. We begin in
section 2.1 by introducing the Feynman the Lee-Pomeransky representations of
Feynman integrals. In the context of the Feynman representation, we recall in
section 2.2 the Landau equations, which we later on use for identifying Landau
singularities. In section 2.3 we describe the geometric approach to sector
decomposition. Finally, using the Lee-Pomeransky representation, we briefly
recall in section 2.4 how Newton polytopes can be used to systematically
identify regions suitable for the asymptotic expansion of Feynman integrals.

###  2.1 Parametric representations

Throughout this paper, we will use $G$ to denote a Feynman graph,
$\mathcal{I}$ to denote a Feynman integral, and $N$ and $L$ to denote the
numbers of propagators and loops of $G$, respectively. We start from the
Feynman representation of $\mathcal{I}(G)\equiv\mathcal{I}(\boldsymbol{s})$,
which reads:

$\displaystyle\mathcal{I}(\boldsymbol{s})=\frac{\Gamma(\nu-LD/2)}{\prod_{e\in
G}\Gamma(\nu_{e})}\int_{0}^{\infty}\left(\prod_{e\in
G}\mathrm{d}\alpha_{e}\,\frac{\alpha_{e}^{\nu_{e}}}{\alpha_{e}}\right)\delta\left(\sum_{e\in
G}\alpha_{e}-1\right)\frac{\left[\mathcal{U}(\boldsymbol{\alpha})\right]^{\nu-(L+1)D/2}}{\left[\mathcal{F}(\boldsymbol{\alpha};\boldsymbol{s})\right]^{\nu-
LD/2}}\,,$ (1)

where $D$ is the dimension of space-time, $\nu_{e}$ is the exponent of the
denominator associated to the propagator of edge $e$, and $\nu\equiv\sum_{e\in
G}\nu_{e}$. We have used $\boldsymbol{\alpha}$ to denote the set of Feynman
parameters $\alpha_{1},\dots,\alpha_{N}$, and $\boldsymbol{s}$ to denote the
set of Mandelstam variables of $\mathcal{I}$, which consist of Lorentz
invariants formed with the external momenta $\\{p_{i}\\}$ as well as internal
squared masses $\\{m_{e}^{2}\\}$, if present.

The polynomials $\mathcal{U}(\boldsymbol{\alpha})$ and
$\mathcal{F}(\boldsymbol{\alpha};\boldsymbol{s})$, which are called the first
and second Symanzik polynomials respectively, are given by:

$\displaystyle\mathcal{U}(\boldsymbol{\alpha})=\sum_{T^{1}}\prod_{e\notin
T^{1}}\alpha_{e},\qquad\mathcal{F}(\boldsymbol{\alpha};\boldsymbol{s})=\sum_{T^{2}}(-s_{T^{2}})\prod_{e\notin
T^{2}}\alpha_{e}+\mathcal{U}(\boldsymbol{\alpha})\sum_{e}m_{e}^{2}\alpha_{e}-i\varepsilon\
,$ (2)

where $\varepsilon>0$ represents the Feynman prescription. The notations
$T^{1}$ and $T^{2}$ denote a _spanning tree_ and a _spanning 2-tree_ of the
graph $G$, respectively. The symbol $s_{T^{2}}$ is the square of the total
momentum flowing between the components of the spanning 2-tree $T^{2}$.

An alternative representation of Feynman integrals, which will be particularly
useful for applying the MoR in what follows, is the Lee-Pomeransky
representation:

$\mathcal{I}(\boldsymbol{s})=\frac{\Gamma(D/2)}{\Gamma\left((L+1)D/2-\sum_{e\in
G}\nu_{e}\right)\prod_{e\in
G}\Gamma(\nu_{e})}\cdot\int_{0}^{\infty}\left(\prod_{e\in
G}\frac{\mathrm{d}x_{e}}{x_{e}}\right)\cdot\left(\prod_{e\in
G}x_{e}^{\nu_{e}}\right)\cdot\Big{(}\mathcal{P}(\boldsymbol{x};\boldsymbol{s})\Big{)}^{-D/2}.$
(3)

Similar to above, we have used $\boldsymbol{x}$ to denote the set of Lee-
Pomeransky parameters $x_{1},\dots,x_{N}$, with $N$ the number of edges
(enumerated by $e$) of the Feynman graph $G$; $\nu_{e}$ is the exponent of the
denominator associated with the propagator $e$. The Lee-Pomeransky polynomial,
$\mathcal{P}(\boldsymbol{x};\boldsymbol{s})$, is equal to the sum over the
Symanzik polynomials:

$\displaystyle\mathcal{P}(\boldsymbol{x};\boldsymbol{s})\equiv\mathcal{U}(\boldsymbol{x})+\mathcal{F}(\boldsymbol{x};\boldsymbol{s})\,.$
(4)

Note that the Feynman and Lee-Pomeransky parameters can be related via:

$\displaystyle\alpha_{e}=\frac{x_{e}}{x_{1}+x_{2}+\dots+x_{N}}.$ (5)

Using this relation the Lee-Pomeransky representation, eq. (3), can be
translated back into the Feynman parameterisation, eq. (1) as follows. One
first inserts

$1=\int_{-\infty}^{\infty}dX\delta\left(\sum_{e\in G}x_{e}-X\right)$

into eq. (3), and then changes the variables $x_{e}=X\alpha_{e}$ according to
(5). At this point one uses the homogeneity properties of the Symanzik
polynomials in (2), to find that the $\mathcal{U}$ polynomial scales as
$X^{L}$ while the $\mathcal{F}$ polynomial scales as $X^{L+1}$, which makes it
possible to integrate over $X$ and obtain a ratio of gamma functions. The
result is exactly (1).

###  2.2 Landau equations in parameter space

In the Feynman representation, a singularity can occur when either
$\mathcal{U}$ or $\mathcal{F}$ vanish. Such singularities can occur at the
endpoint of the domain of integration, $\alpha_{i}\to 0$, forcing every
monomial to vanish independently. These are often referred to as _endpoint
singularities_ in the literature. Additionally, since the $s_{T^{2}}$ in
${\cal F}$ in (2) may have different signs, $\mathcal{F}=0$ singularities may
arise due to cancellations between different terms. Such singularities may
sometimes be avoided by analytic continuation, namely by deforming the
integration contour of the relevant $\alpha_{k}$-parameters into the complex
plane by introducing a small imaginary part $i\varepsilon_{k}$:

$\alpha_{k}\to\alpha_{k}-i\varepsilon_{k}(\boldsymbol{\alpha})\,.$ (6)

Under such a change of variables the $\mathcal{F}$ polynomial transforms as
follows

$\mathcal{F}(\boldsymbol{\alpha};\boldsymbol{s})\to\mathcal{F}(\boldsymbol{\alpha};\boldsymbol{s})-i\sum_{k}\varepsilon_{k}(\boldsymbol{\alpha})\frac{\partial\mathcal{F}(\boldsymbol{\alpha};\boldsymbol{s})}{\partial\alpha_{k}}+\mathcal{O}(\varepsilon^{2})\,.$
(7)

Thus, provided that for some $k$ (for which $\alpha_{k}\neq 0$) one has
$\frac{\partial\mathcal{F}(\boldsymbol{\alpha};\boldsymbol{s})}{\partial\alpha_{k}}\neq
0$, the contour-deformed integral avoids the singularity. This provides a
definition of the integral by analytic continuation (note that according to
the Feynman prescription in eq. (2), one should choose
$\varepsilon_{k}(\boldsymbol{\alpha})$ in eq. (6) to have the same sign as
$\frac{\partial\mathcal{F}(\boldsymbol{\alpha};\boldsymbol{s})}{\partial\alpha_{k}}$
in the neighbourhood where $\mathcal{F}$ vanishes). However, if both ${\cal
F}=0$ and $\partial\mathcal{F}/\partial\alpha_{k}=0$ for all $k$ for which
$\alpha_{k}\neq 0$, the contour becomes _pinched_. In this case the
singularity cannot be avoided, and would have an imprint on the analytic
structure of $\mathcal{I}(\boldsymbol{s})$. Singularities of this type are
referred to as _pinch singularities_.

In conclusion, on a singular surface, therefore, each $\alpha_{k}$ must either
vanish or have its contour pinched. The necessary condition for a singularity
to occur are summarised by the Landau equations, which can be expressed as
Lnd59 ; EdenLdshfOlvPkhn02book

$\displaystyle\mathcal{F}=0,\qquad\alpha_{k}\,\frac{\partial\mathcal{F}}{\partial\alpha_{k}}=0\quad\text{for
each }k\in\\{1,\dots,N\\}.$ (8)

We remark that due to the homogeneity of $\mathcal{F}$ we also have, by
Euler’s theorem,

$\sum_{k=1}^{N}\alpha_{k}\,\frac{\partial\mathcal{F}}{\partial\alpha_{k}}\,=(L+1)\mathcal{F}\,,$
(9)

where $L$ is the number of loops in $G$. Thus, the condition $\mathcal{F}=0$
is automatically fulfilled when
$\alpha_{k}\partial\mathcal{F}/\partial\alpha_{k}=0$ for all $k$.

Recently, progress has been made in solving the Landau equations
algorithmically FvlMzrTln23pld ; FvlMzrTln23prl ; Dlapa:2023cvx ;
JiangLiuXuYang24 ; Helmer:2024wax ; CrHtCrrGrx24 , which is a profoundly
difficult problem in the general setting Brown:2009ta ; Panzer:2014caa .
Solutions can be characterised by manifolds of any dimension, residing in the
large space of kinematic invariants times the $N$-dimensional _complex_
parameter space $\\{\alpha_{k}\\}$. Our interest here instead is limited to
singularities which may obstruct the computation of the integral using sector
decomposition or introduce regions in an asymptotic expansion by the MoR.
Specifically, these restrictions allow us to focus on solutions to eq. (8)
that appear within the integration domain, $\alpha_{k}\geqslant 0$.

###  2.3 Newton polytopes for sector decomposition

One of the central objects of study in this work is the Newton polytope
associated to Feynman integrals either in Feynman or Lee-Pomeransky
representation. The Newton polytope can be used to study the behaviour of
integrands in the limit that sets of parameters become small or large
simultaneously. The behaviour of the integrand near the boundaries of
parameter space can give rise to ultraviolet or infrared singularities which
appear as poles in the dimensional regulator after integration. Aside from a
potential overall ultraviolet divergence in the $\Gamma$-function prefactors,
the Newton polytope fully captures the singular structure of integrals in the
same-sign regime, where all terms in ${\cal F}$ have the same sign. Away from
the same-sign regime, or for integrals without a same-sign regime, additional
singularities can be present, for example at thresholds where intermediate
particles go on shell. Here, we briefly recap how Newton polytopes can be used
to study and blow-up, or resolve, singularities at the integration boundary
using sector decomposition.777We refer the interested reader to refs.
Bogner:2007cr ; KnkUeda10 ; Kaneko:2010kj ; Schlenk:2016cwf ; Schlenk:2016epj
; Weinzierl:2022eaz  for a more in-depth treatment. Then, in section 3, we
describe a situation in which the Newton polytope alone is insufficient to
resolve the singularities of certain integrals that do not have a same-sign
regime.

The Newton polytope corresponding to the integral
$\mathcal{I}(\boldsymbol{s})$ in Feynman parameter space is obtained as
follows. Starting from the Feynman representation given in eq. (1) the Dirac
$\delta$-distribution is first integrated out after using the Cheng-Wu theorem
to replace it by $\delta(1-\alpha_{N})$. Next, the product of the Symanzik
polynomials, $\mathcal{U}$ and $\mathcal{F}$, given in eq. (2), is constructed

$\displaystyle\mathcal{U}(\boldsymbol{\alpha})\cdot\mathcal{F}(\boldsymbol{\alpha};\boldsymbol{s})=\sum_{i=1}^{m}b_{i}(\boldsymbol{s})\,\alpha_{1}^{r_{i},1}\cdots\alpha_{N-1}^{r_{i},{N-1}}\equiv\sum_{i=1}^{m}b_{i}(\boldsymbol{s})\boldsymbol{\alpha}^{\boldsymbol{r}_{i}},$
(10)

where $\boldsymbol{r}_{i}$ are $(N-1)$ dimensional exponent vectors. The
Newton polytope, $\Delta(\mathcal{U}\cdot\mathcal{F})$, is then defined as the
convex hull of the vertices given by $\boldsymbol{r}_{i}$,

$\displaystyle\Delta(\mathcal{U}\cdot\mathcal{F})$
$\displaystyle\overset{\text{def}}{=}\mathrm{convHull}(\boldsymbol{r}_{1},\ldots,\boldsymbol{r}_{m})=\left\\{\sum_{i=1}^{m}a_{i}\boldsymbol{r}_{i}\
\Big{|}\ a_{i}\geqslant 0\land\sum_{i=1}^{m}a_{i}=1\right\\}$ (11)
$\displaystyle=\bigcap_{f\in
F}\left\\{\boldsymbol{\rho}\in\mathbb{R}^{N-1}\mid\boldsymbol{\rho}\cdot\boldsymbol{n}_{f}+a_{f}\geqslant
0\right\\},\quad a_{f}\in\mathbb{Z}\quad\forall\ f.$ (12)

In the first line, the Newton polytope is described by the vertices,
$\boldsymbol{r}_{i}$, while in the second line, the polytope is instead
defined by inward pointing vectors, $\boldsymbol{n}_{f}$, normal to the facets
of the polytope $f\in F$.

Next we describe how Newton polytope geometry is used to perform sector
decomposition of the integral. Let $S_{j}$ denote the set of facets incident
on the extremal vertex, $\boldsymbol{r}_{j}$; they define a local change of
variables,

$\displaystyle\alpha_{i}=\prod_{f\in
S_{j}}y_{f}^{\langle\boldsymbol{n}_{f},\boldsymbol{e}_{i}\rangle},$ (13)

where $\boldsymbol{e}_{i}$ are standard basis vectors in $\mathbb{R}^{N-1}$.
If more than $N-1$ facets are incident on an extremal vertex then a further
subdivision, known as triangulation, is required. This means that each
extremal vertex leads to one or more (if triangulation is required) sectors.

After applying the change of variables for each sector, the integral can now
be written as the sum,

$\displaystyle\mathcal{I}(\boldsymbol{s})=\sum_{\sigma}|\sigma|\int_{0}^{1}\left(\prod_{f\in\sigma}\frac{\mathrm{d}y_{f}}{y_{f}}y_{f}^{\langle\boldsymbol{n}_{f},\boldsymbol{\nu}\rangle-(t_{\mathcal{U}}+t_{\mathcal{F}})a_{f}}\right)\,\left(\tilde{\mathcal{U}}_{\sigma}(\boldsymbol{y}_{f})\right)^{t_{\mathcal{U}}}\left(\tilde{\mathcal{F}}_{\sigma}(\boldsymbol{y}_{f};\boldsymbol{s})\right)^{t_{\mathcal{F}}}.$
(14)

Here $\sigma$ indexes the simplicial cones associated with the extremal
vertices, $|\sigma|$ is the Jacobian determinant associated to the change of
variables in each sector, $\boldsymbol{\nu}$ is a vector of propagator
exponents built from $\nu_{e}$, and $t_{\mathcal{U}}=\nu-(L+1)D/2$ with
$t_{\mathcal{F}}=-\nu+LD/2$. The polynomials $\tilde{\mathcal{U}}_{\sigma}$
and $\tilde{\mathcal{F}}_{\sigma}$ can be obtained by substituting the change
of variables of eq. (13) and factoring out an overall monomial, giving

$\displaystyle\mathcal{U}(\boldsymbol{y}_{f})=\left(\prod_{f\in\sigma}y_{f}^{-a_{f}}\right)\tilde{\mathcal{U}}_{\sigma}(\boldsymbol{y}_{f}),\qquad\qquad\mathcal{F}(\boldsymbol{y}_{f};\boldsymbol{s})=\left(\prod_{f\in\sigma}y_{f}^{-a_{f}}\right)\tilde{\mathcal{F}}_{\sigma}(\boldsymbol{y}_{f};\boldsymbol{s})\,.$
(15)

The polynomials $\tilde{\mathcal{U}}_{\sigma}$ and
$\tilde{\mathcal{F}}_{\sigma}$ are guaranteed to each contain a constant term.
The singularities appearing at the boundaries of integration now appear as an
overall factor in eq. (14) and the remaining polynomials,
$\tilde{\mathcal{U}}_{\sigma}$ and $\tilde{\mathcal{F}}_{\sigma}$, are
guaranteed to be finite for any subset of $y_{f}\rightarrow 0$.

Recently, there has been a resurgence of interest in using geometric concepts
to understand the singularity structure of Feynman integrals, including both
dimensionally regulated and kinematic singularities. For example, a similar
geometrical construction has been used to obtain a general formula for
predicting the leading singularities in the $\epsilon$ expansion of generic
Feynman integrals AkHmHlmMzr22 . In refs. FvlMzrTln23prl ; FvlMzrTln23pld , a
geometrical construction was used to find solutions of the Landau equations
depending on the kinematic invariants. In another development, concepts from
tropical geometry, including _generalised permutahedra_ , were used to devise
a procedure for efficiently integrating finite Feynman integrals Brsk20 ;
BrskMchTld23 . Also in the context of tropical geometry, very recent work has
focused on studying subtraction formulae using the facets of the Newton
polytopes Svtr24 . Symmetries of Feynman integrals have also been investigated
using the symmetries of Newton polytopes dlCrz24 .

###  2.4 Newton polytopes for the MoR

In parameter space, the scaleful regions contributing to the MoR expansion can
be found algorithmically using Newton polytopes. The approach is believed to
capture all relevant regions, provided the terms of the $\mathcal{F}$ or,
equivalently, $\mathcal{P}$ polynomials do not cancel each other in the
positive hyperoctant JtzSmnSmn12 ; Smirnov:2002pj ; Smirnov:2004ym  leading to
a pinch in the limit about which we expand. Here we briefly review the Newton
polytope approach to the MoR in our notation, assuming such cancellations do
not occur. Then, in the remainder of this paper, we examine the situation
where such cancellations do occur and are relevant for both the evaluation and
expansion of Feynman integrals.

The MoR begins by assuming that a subset of invariants
$\boldsymbol{t}\subset\boldsymbol{s}$ are small compared to the rest, and that
we are interested in the asymptotic expansion of eq. (3) around the kinematic
limit in which they vanish. To this end, it is convenient to introduce an
$|\boldsymbol{s}|$-dimensional scaling vector
$\boldsymbol{w}=(w_{1},w_{2},w_{3},\dots)$, with $w_{i}=1$ if
$s_{i}\in\boldsymbol{t}$ and $w_{i}=0$ otherwise, and introduce a parameter
$\lambda\ll 1$ such that $s_{i}\sim\lambda^{w_{i}}Q^{2}$, with $Q^{2}$ some
fixed nonzero scale. A specific class of expansions we would be interested in,
following GrdHzgJnsMaSchlk22 , is the on-shell expansion in massless
scattering, where the set $\boldsymbol{t}$ consists exclusively of the
virtualities $p_{i}^{2}$ of some external legs. A more detailed definition is
given in eq. (56) below. We will also discuss the Regge limit in on-shell
$2\to 2$ massless scattering where the set $\boldsymbol{t}$ consists
exclusively of the Mandelstam invariant $t$, while $s$ remains large.

The Newton polytope of interest for applying the MoR can be obtained using the
Lee-Pomeransky polynomial888Alternatively, we can obtain the region vectors
using the _product_
$\mathcal{U}(\boldsymbol{\alpha})\cdot\mathcal{F}(\boldsymbol{\alpha};\boldsymbol{s})$
rather than the _sum_ appearing in
$\mathcal{P}(\boldsymbol{x};\boldsymbol{s})$. As shown in ref. SmnvSmnSmv19 ,
the resulting region vectors are in one-to-one correspondence and differ only
by a constant shift, see also refs. HrchJnsSlk22 ; GrdHzgJnsMaSchlk22 .,
defined in eq. (4), with the Mandelstam invariants rescaled as
$\boldsymbol{s}\to\lambda^{\boldsymbol{w}}\boldsymbol{s}$,

$\displaystyle\mathcal{P}(\boldsymbol{x};\lambda^{\boldsymbol{w}}\boldsymbol{s})=\sum_{i=1}^{m}c_{i}(\boldsymbol{s})\,x_{1}^{r_{i,1}}\dots
x_{N}^{r_{i,N}}\,\lambda^{r_{i,N+1}},$ (16)

where, for Feynman integrals, the powers $r_{i,j}\in\\{0,1,2\\}$. For each
$i$, we define the $(N+1)$-dimensional exponent vector
$\boldsymbol{r}_{i}\equiv(r_{i,1},\ldots,r_{i,N};r_{i,N+1})\equiv(\hat{\boldsymbol{r}}_{i};r_{i,N+1})$.
The Newton polytope $\Delta(\mathcal{P})$ is then the _convex hull_ of the
exponent vectors,

$\displaystyle\Delta(\mathcal{P})$
$\displaystyle\overset{\text{def}}{=}\mathrm{convHull}(\boldsymbol{r}_{1},\ldots,\boldsymbol{r}_{m})=\left\\{\sum_{i}^{m}a_{i}\boldsymbol{r}_{i}\
\Big{|}\ a_{i}\geqslant 0\land\sum_{i}^{m}a_{i}=1\right\\}$ (17)
$\displaystyle=\bigcap_{f\in
F}\left\\{\boldsymbol{\rho}\in\mathbb{R}^{N+1}\mid\boldsymbol{\rho}\cdot\boldsymbol{v}_{f}+a_{f}\geqslant
0\right\\},\quad a_{f}\in\mathbb{Z}\quad\forall\ f,$ (18)

where $\boldsymbol{v}_{f}$ are inward pointing vectors normal to the facets of
the polytope $f\in F$. The Newton polytope $\Delta(\mathcal{P})$ is an
$(N+1)$-dimensional polytope, the first $N$ dimensions correspond to the Lee-
Pomeransky parameters $x_{1},\ldots,x_{N}$, which are integrated over, while
the $(N+1)$-th dimension corresponds to the expansion parameter $\lambda$.

The _lower facets_ of $\Delta(\mathcal{P})$ are defined to be those facets
which have an inward pointing normal vector $\boldsymbol{v}_{f}$ with a
positive $(N+1)$-th component, $v_{N+1}>0$. Let us call the set of lower
facets $F_{+}$, with $F_{+}\subset F$. The lower facets capture the behaviour
of $\mathcal{P}$ where $\lambda\rightarrow 0$ and describe the regions
relevant for the MoR. The integrand of each region can by obtained by
rescaling the integrand eq. (3) as
$\boldsymbol{s}\to\lambda^{\boldsymbol{w}}\boldsymbol{s}$ and
$\boldsymbol{x}\to\lambda^{\boldsymbol{u}_{R}}\boldsymbol{x}$,
$\lambda\to\lambda^{v_{R,N+1}}$, where
$\boldsymbol{v}_{R}=(\boldsymbol{u}_{R},v_{R,N+1})$ and $R\in F_{+}$.

Given a scaling vector $\boldsymbol{u}_{R}$ for some region $R$, the scaling
rule of each Lee-Pomeransky parameter $x_{e}\in\boldsymbol{x}$ can be directly
translated into the scaling of rule of the corresponding edge momentum $l_{e}$
as follows GrdHzgJnsMaSchlk22 ; Engel:2022kde :

$x_{e}\sim\frac{1}{l_{e}^{2}}\sim\lambda^{u_{R,e}}.$ (19)

This relates regions in parameter space to regions in momentum space.
Specifically, in the on-shell expansion scaling vectors $\boldsymbol{u}_{R}$
involve just three types of entries: $u_{R,e}\in\\{0,-1,-2\\}$, which
correspond respectively to the following three momentum modes: _hard_ ,
_collinear_ and _soft_ GrdHzgJnsMaSchlk22 ; Ma23 . Note that in momentum space
the virtuality does not always uniquely fix the scaling of individual
components of the loop momentum (for example a jet loop momentum can be
collinear to different external vectors). In the on-shell expansion, upon
imposing momentum conservation this mapping is uniquely determined.

As described in ref. GrdHzgJnsMaSchlk22 , the lower facets of the Newton
polytope capture the solutions of the Landau equations that are relevant for
infrared regions, as well as the hard region. However, the Newton polytope is
completely insensitive to the coefficients $c_{i}(\boldsymbol{s})$ appearing
in eq. (16). If we allow the coefficients to differ in sign, the terms of
$\mathcal{P}$ may cancel each other, potentially leading to further _pinch
solutions_ of the Landau equations not captured by the facet analysis
presented above. These solutions can give rise to additional scaleful regions
that must be included in the MoR analysis. We refer to the regions overlooked
by the Newton polytope procedure as _hidden regions_.

##  3 Computing integrals with pinched contours in parametric representation

As already mentioned, our prime motivation for this work stems from the MoR,
and specifically from the geometric approach to the MoR. In contrast to
momentum space, the geometric approach offers an algorithmic procedure of
identifying regions as (lower) facets of the Newton polytope defined by the
Symanzik graph polynomials PakSmn11 ; HrchJnsSlk22 . In the Lee-Pomeransky
representation this polytope is defined as the convex hull of all monomials in
$\mathcal{P}(\boldsymbol{x};\boldsymbol{s})$ of eq. (4) in the space defined
by the _powers_ of the parameters $x_{i}$. Importantly, the coefficients of
these monomials are not encoded in the polytope. For integrals defined in
Euclidean space, all monomials in $\mathcal{P}(\boldsymbol{x};\boldsymbol{s})$
have the same sign, and then the information contained in the polytope is
sufficient to identify all regions. In Minkowski space, the situation is
different in that the signs of distinct monomials appearing in the ${\cal P}$
polynomial may play an important role in dictating the singularity structure
of the integral: additional singularities may emerge due to particular
cancellations between monomials of opposite sign. In certain cases, these
singularities may be essential for identifying MoR regions. Indeed, it is well
known that some regions, which are necessary for obtaining the correct
asymptotic expansion of certain Feynman integrals JtzSmnSmn12 ;
AnthnrySkrRmn19 , do not correspond to facets of the polytope associated with
$\mathcal{P}(\boldsymbol{x};\boldsymbol{s})$. In this work we refer to such
regions as _hidden regions_.

A key step in developing the geometric approach to the MoR was taken in ref.
JtzSmnSmn12 , which was the first to propose a method to systematically
identify such hidden regions. Ref. JtzSmnSmn12  presented a Mathematica code,
asy2.m, which is designed to identify hidden regions associated with a
_linear_ cancellation between monomials. In such cases it separates the
integration domain at the singularity locus, and maps each sector back onto
the standard integration domain by a suitable shift of the integration
parameters. The newly defined polynomial in each such sector, which is free of
the aforementioned cancellation, gives rise to a new Newton polytope whose
facets correspond to regions. This technique has been demonstrated JtzSmnSmn12
; SmnvSmnSmv19 ; AnthnrySkrRmn19  to identify all regions in certain classes
of integrals. We proceed by briefly recalling the method and then reviewing
known examples where it applies.

###  3.1 Motivating examples

The method to resolve hidden regions, known as PreResolve in asy2, works by
selecting a variable appearing in a positive monomial, $\alpha_{i}$, and a
variable appearing in a negative monomial, $\alpha_{j}$, then making a linear
change of variables,

$\displaystyle\alpha_{i}$ $\displaystyle\rightarrow
m_{ii}\,\alpha_{i}+m_{ij}\,\alpha_{j},$ (20a) $\displaystyle\alpha_{j}$
$\displaystyle\rightarrow m_{ji}\,\alpha_{i}+m_{jj}\,\alpha_{j}$ (20b)

with the parameters $m$ chosen (if possible) to eliminate a negative monomial.
The above replacements are tried on all pairs of variables appearing in
opposite sign monomials and iterated provided the number of negative monomials
is not increased by the substitution. The resolution procedure is considered
successful if, after a finite number of iterations, the modified $\mathcal{F}$
polynomial has only positive monomials. A similar procedure was introduced in
the paper of Ananthanarayan et al. AnthnrySkrRmn19  which works by applying
the above procedure to the Gröbner basis of the $\mathcal{F}$ polynomial and
its first order derivatives, rather than directly on $\mathcal{F}$ itself. In
the remainder of this section we will consider some examples where the above
procedures work and discuss the interpretation of the output. In sections 4
and 5, we will discuss in detail situations in which the above procedures
fail.

The first example for which PreResolve works is given by a one-loop massive
bubble close to threshold,

$\displaystyle{\cal I}_{\text{bub}}(q^{2},y)$
$\displaystyle=\int\frac{d^{D}k}{i\pi^{D/2}}\frac{1}{(k^{2}-m^{2})\big{(}(k-q)^{2}-m^{2}\big{)}}$
$\displaystyle=\Gamma(\epsilon)\int
d\alpha_{1}d\alpha_{2}\frac{\delta(1-\alpha_{1}-\alpha_{2})(\alpha_{1}+\alpha_{2})^{-2+2\epsilon}}{\Big{(}\mathcal{F}_{\text{bub}}(\alpha_{1},\alpha_{2};q^{2},y)\Big{)}^{\epsilon}}$
(21)

with $D=4-2\epsilon$ and

$\mathcal{F}_{\text{bub}}(\alpha_{1},\alpha_{2};q^{2},y)=(q^{2}/4)(\alpha_{1}-\alpha_{2})^{2}+y(\alpha_{1}+\alpha_{2})^{2}\,\qquad\quad\text{and}\quad\quad
y=m^{2}-q^{2}/4.$

In the threshold limit $y\to 0$, the integrand develops a singularity due to
the vanishing of the term $(q^{2}/4)(\alpha_{1}-\alpha_{2})^{2}$ along the
line $\alpha_{1}=\alpha_{2}$. To understand the nature of this singularity it
is instructive to consider the Landau equations:

$\alpha_{1}=0\quad\text{or}\quad\frac{d\mathcal{F}_{\text{bub}}}{d\alpha_{1}}=0=(q^{2}/2)(\alpha_{1}-\alpha_{2})+2y(\alpha_{1}+\alpha_{2})\,,$
(22)
$\alpha_{2}=0\quad\text{or}\quad\frac{d\mathcal{F}_{\text{bub}}}{d\alpha_{2}}=0=(q^{2}/2)(\alpha_{2}-\alpha_{1})+2y(\alpha_{1}+\alpha_{2})\,.$
(23)

At $y=0$ the equations

$\frac{d\mathcal{F}_{\text{bub}}}{d\alpha_{1}}=\frac{d\mathcal{F}_{\text{bub}}}{d\alpha_{2}}=0$
(24)

are satisfied on the line $\alpha_{1}=\alpha_{2}$, identifying this as a pinch
singularity. The $\delta$ constraint fixes its location at
$\alpha_{1}=\frac{1}{2}=\alpha_{2}$. Note that for nonzero $y$ the only
solution is $\alpha_{1}=0=\alpha_{2}$ which is an endpoint singularity rather
than a pinch, which is, however, not supported by the $\delta$-constraint, and
is therefore discarded.

The pinch can also be described from the perspective of an expansion in $y$ as
follows. When $y$ approaches zero, the solution of the condition
$\mathcal{F}_{\text{bub}}(\alpha_{1},\alpha_{2};q^{2},y)=0$ is:

$\alpha_{2}=1-\alpha_{1},\qquad\alpha_{1}\rightarrow\frac{1}{2}\pm\frac{i\sqrt{y}}{\sqrt{q^{2}}}.$
(25)

Thus, for nonzero $y$ there are two complex roots in $\alpha_{1}$, at which
$\mathcal{F}_{\text{bub}}$ vanishes. In the limit $y\to 0$ these roots merge
at $\alpha_{1}=1/2=\alpha_{2}$, pinching the contour on the real axis from
above and below. This is the very point where the derivative conditions (24)
are satisfied.

The program asy2 has a feature to resolve such singularities, and to find the
underlying regions within the MoR approach by sector decomposing over the two
sectors:

  1. 1)

$\alpha_{1}<\alpha_{2}$ :
$(\alpha_{1},\alpha_{2})\to(\alpha_{1}/2,\alpha_{1}/2+\alpha_{2})$

  2. 2)

$\alpha_{1}>\alpha_{2}$ :
$(\alpha_{1},\alpha_{2})\to(\alpha_{1}+\alpha_{2}/2,\alpha_{2}/2)$

In each of these sectors the boundaries are mapped back onto the Feynman
simplex ($1=\alpha_{1}+\alpha_{2}$) via the indicated mappings, and as a
result the line singularity is mapped onto the boundary. For example, in
sector 1 the resulting integral is

$\displaystyle{\cal I}_{\text{bub}}^{(1)}(q^{2},y)$
$\displaystyle=\frac{\Gamma(\epsilon)}{2}\int
d\alpha_{1}d\alpha_{2}\frac{\delta(1-\alpha_{1}-\alpha_{2})(\alpha_{1}+\alpha_{2})^{-2+2\epsilon}}{\Big{(}(q^{2}/4)(\alpha_{2})^{2}+y(\alpha_{1}+\alpha_{2})^{2}\Big{)}^{\epsilon}}\,,$
(26)

where no further singularities occur in the domain of integration. Thus, in
the new formulation all regions can be identified using the Newton polytope
approach.

Another example of this kind, also discussed in ref. JtzSmnSmn12 , is the
following pentagon integral for a forward scattering kinematic configuration

$\displaystyle\begin{split}{\cal
I}_{\text{penta}}(Q^{2},m^{2})&=\int\frac{d^{D}k}{i\pi^{D/2}}\frac{1}{(k^{2}-m^{2})(k^{2}-2p\cdot
k)(k^{2}+2p\cdot k)(k^{2}-2q\cdot k)(k^{2}+2k\cdot q)}\\\
&=-\Gamma(3+\epsilon)\int d\alpha_{1}\dots
d\alpha_{5}\,\delta(1-\alpha_{1}\dots-\alpha_{5})\,\times\\\ &\hskip
100.0pt\Big{(}\mathcal{U}_{\text{penta}}(\\{\boldsymbol{\alpha}\\})\Big{)}^{1+2\epsilon}\,\Big{(}\mathcal{F}_{\text{penta}}(\\{\boldsymbol{\alpha}\\};Q^{2},m^{2})\Big{)}^{-3-\epsilon}\end{split}$
(27)

where

$\displaystyle\mathcal{U}_{\text{penta}}(\\{\boldsymbol{\alpha}\\})$
$\displaystyle=$
$\displaystyle\alpha_{1}+\alpha_{2}+\alpha_{3}+\alpha_{4}+\alpha_{5}\,,$ (28a)
$\displaystyle\mathcal{F}_{\text{penta}}(\\{\boldsymbol{\alpha}\\};Q^{2},m^{2})$
$\displaystyle=$
$\displaystyle\alpha_{1}\mathcal{U}_{\text{penta}}(\\{\boldsymbol{\alpha}\\})m^{2}+(\alpha_{2}-\alpha_{3})(\alpha_{4}-\alpha_{5})Q^{2}\,.$
(28b)

The Landau conditions are

$\displaystyle\alpha_{1}=0\quad\text{or}\quad\frac{d\mathcal{F}_{\text{penta}}}{d\alpha_{1}}=0=(\alpha_{1}+\mathcal{U}_{\text{penta}})m^{2}\,,$
(29a)
$\displaystyle\alpha_{2}=0\quad\text{or}\quad\frac{d\mathcal{F}_{\text{penta}}}{d\alpha_{2}}=0=(\alpha_{1})m^{2}+(\alpha_{4}-\alpha_{5})Q^{2}\,,$
(29b)
$\displaystyle\alpha_{3}=0\quad\text{or}\quad\frac{d\mathcal{F}_{\text{penta}}}{d\alpha_{3}}=0=(\alpha_{1})m^{2}-(\alpha_{4}-\alpha_{5})Q^{2}\,,$
(29c)
$\displaystyle\alpha_{4}=0\quad\text{or}\quad\frac{d\mathcal{F}_{\text{penta}}}{d\alpha_{4}}=0=(\alpha_{1})m^{2}+(\alpha_{2}-\alpha_{3})Q^{2}\,,$
(29d)
$\displaystyle\alpha_{5}=0\quad\text{or}\quad\frac{d\mathcal{F}_{\text{penta}}}{d\alpha_{5}}=0=(\alpha_{1})m^{2}-(\alpha_{2}-\alpha_{3})Q^{2}\,.$
(29e)

To satisfy (29a) we must require $\alpha_{1}=0$ since there exists no solution
to $\frac{d\mathcal{F}_{\text{penta}}}{d\alpha_{1}}=0$. Alternatively, we can
consider the limit $m^{2}/Q^{2}\to 0$ in which
$\frac{d\mathcal{F}_{\text{penta}}}{d\alpha_{1}}=0$ is automatically
satisfied. In either of these cases eqs. (29b)-(29e) allow for a pinch
singularity located at

$\alpha_{4}-\alpha_{5}=0=\alpha_{2}-\alpha_{3}\,.$ (30)

Note that other singularities are also possible, e.g.
$\alpha_{2}=0=\alpha_{3}$ solves the Landau equations for any value of
$\alpha_{4}$ and $\alpha_{5}$, since the corresponding derivatives of
$\mathcal{F}_{\text{penta}}$ vanish. This singularity should, however, be
interpreted as an endpoint singularity in the sense that each monomial
vanishes independently and no cancellation takes place. Note also that if we
had only $\alpha_{2}=\alpha_{3}$ then $\mathcal{F}_{\text{penta}}$ would still
vanish, but $\alpha_{4},\alpha_{5}$ could be used to avoid the singularity by
deforming the contour into the complex plain. Thus, the only pinch singularity
appears subject to the two conditions, $\alpha_{2}=\alpha_{3}$ and
$\alpha_{4}=\alpha_{5}$, assuming either $\alpha_{1}\to 0$ or $m^{2}\to 0$.
Given that the singularity is linear in the difference of two parameters, a
similar resolution strategy as described above can map these singularities to
the boundary, allowing the Newton polytope approach to be used sector by
sector to reveal the regions.

In the next sections we will see that more general situations of cancellation
structures may occur, involving higher degree polynomials, e.g.
$\alpha_{i}\alpha_{j}=\alpha_{k}\alpha_{l}$ for some $i,j,k,l$, similarly
leading to pinch singularities, which obstruct both the evaluation of the
integral and its expansion with available strategies.

###  3.2 Searching for pinch singularities in parameter space

Given a Feynman integral, the faces of the corresponding Newton polytope
suffice to identify all the Landau singularities as long as there exists an
analytic continuation to a same-sign regime where $\mathcal{F}$ is positive
definite. When there is no same-sign regime, Landau singularities may arise
from the cancellation of $\mathcal{F}$ terms with opposite signs. To
understand such cancellation structures, let us define the polynomials
$\mathcal{F}_{+}$ and $\mathcal{F}_{-}$ at a given kinematic point as the
positive and negative monomials of $\mathcal{F}$, respectively, with
$\mathcal{F}=\mathcal{F}_{+}+\mathcal{F}_{-}$. The second Landau equation then
becomes

$\displaystyle\alpha_{e}\frac{\partial\mathcal{F}_{+}}{\partial\alpha_{e}}+\alpha_{e}\frac{\partial\mathcal{F}_{-}}{\partial\alpha_{e}}=0,\quad\text{for
}e\in\\{1,\dots,N\\}.$ (31)

Since $\alpha_{e}$ are all non-negative, the sign of a given monomial in
$\mathcal{F}$ is directly linked to that of the Mandelstam invariant it
contains. Importantly, before examining $\mathcal{F}$ using the Landau
equations, one must impose all constraints relating kinematic invariants based
on momentum conservation. In particular, the separation of the monomials in
${\cal F}$ into $\mathcal{F}_{+}$ and $\mathcal{F}_{-}$ must be done after all
such relations are taken into account, so ${\cal F}$ is written in terms of a
minimal set of _independent_ kinematic variables. Having done that, one may
distinguish between two types of cancellation scenarios which may arise. One
scenario arises when terms with the same invariant cancel between them,
independently of the kinematics. The second scenario is when the cancellation
involves terms associated with more than one invariant, leading to a potential
Landau singularity at a particular kinematic configuration. In this paper, we
focus exclusively on the first type of cancellation, which occur for general
kinematics (under certain restrictions, e.g. on-shell conditions) potentially
giving rise to infrared divergences. With this in mind, we first treat the
coefficient of each independent invariant
$s_{ij}\in\boldsymbol{s}\setminus\boldsymbol{t}$ as a separate ${\cal F}$
polynomial, which we denote by ${\cal F}^{(s_{ij})}$ and search for
cancellations among its terms, such that (31) admits a nontrivial solution. If
such a solution exists for all independent invariants in ${\cal
F}(\boldsymbol{\alpha},\boldsymbol{s})$, we investigate if a compatible
solution exists. If it does, we have identified a pinch singularity.

Our objective is to identify a necessary condition for the existence of pinch
singularities in parameter space within the domain of integration. This
situation is characterised by a solution of the Landau equations (8) for
positive $\alpha_{i}$, leading to the following cancellation requirement:
_there exists at least one Feynman parameter $\alpha_{i}\neq 0$ with
$\frac{\partial\mathcal{F}}{\partial\alpha_{i}}$ containing non-vanishing
terms of both signs that cancel on the solution._ This requirement is
satisfied in the motivating examples of eqs. (3.1) and (27) above.

This requirement allows for the possibility that a set of parameters
$\\{\alpha_{j}\\}$ strictly vanish. The pinch then corresponds to a solution
of the Landau equations which is also present in a sub-topology of the Feynman
graph under consideration. When some first derivatives
$\frac{\partial\mathcal{F}}{\partial\alpha_{j}}$ are either positive definite
or negative definite, the Landau equations imply that each monomial in
$\alpha_{j}\frac{\partial\mathcal{F}}{\partial\alpha_{j}}$ must strictly
vanish. The cancellation requirement is then that after all such parameters
$\\{\alpha_{j}\\}$ are set to zero, there is still some $\alpha_{i}\neq 0$
such that $\frac{\partial\mathcal{F}}{\partial\alpha_{i}}$, which vanishes on
the solution, receives contributions from both $\mathcal{F}_{+}$ and
$\mathcal{F}_{-}$.

Note that this implies that the form of an
$\mathcal{F}(\boldsymbol{\alpha};\boldsymbol{s})$ polynomial that is prone to
having pinch singularities is rather constrained. For example, the following
expressions

$\displaystyle\mathcal{F}(\boldsymbol{\alpha};\boldsymbol{s})=(\alpha_{a}-\alpha_{b})f(\boldsymbol{\alpha};\boldsymbol{s}),\quad\text{
or
}\quad\mathcal{F}(\boldsymbol{\alpha};\boldsymbol{s})=(\alpha_{a}\alpha_{b}-\alpha_{c}\alpha_{d})f(\boldsymbol{\alpha};\boldsymbol{s}),$
(32)

with $f(\boldsymbol{\alpha};\boldsymbol{s})$ positive or negative definite, do
not satisfy the cancellation requirement. To see this, let us consider the
first equation of (32) as an example. If the cancellation requirement is to be
fulfilled, there must exist some dependence on $\alpha_{e}$ in
$f(\boldsymbol{\alpha};\boldsymbol{s})$ with $e\neq a,b$, such that
$\alpha_{e}\frac{\partial\mathcal{F}_{+}}{\partial\alpha_{e}}=\alpha_{a}\alpha_{e}\frac{\partial
f}{\partial\alpha_{e}}$ and
$\alpha_{e}\frac{\partial\mathcal{F}_{-}}{\partial\alpha_{e}}=-\alpha_{b}\alpha_{e}\frac{\partial
f}{\partial\alpha_{e}}$ are non-vanishing and cancel on the solution. Now we
show that in fact these terms vanish independently. To this end, consider the
Landau equations (8) with respect to $\alpha_{a}$ and $\alpha_{b}$, which
imply that each monomial in
$\alpha_{a}\frac{\partial\mathcal{F}}{\partial\alpha_{a}}$ and
$\alpha_{b}\frac{\partial\mathcal{F}}{\partial\alpha_{b}}$ strictly vanishes
on the solution. We note, however, that all the monomials of
$\alpha_{e}\frac{\partial\mathcal{F}}{\partial\alpha_{e}}$ are included in
$\mathcal{F}=\alpha_{a}\frac{\partial\mathcal{F}}{\partial\alpha_{a}}+\alpha_{b}\frac{\partial\mathcal{F}}{\partial\alpha_{b}}$,
all of which must vanish on the solution. Hence there is no way that the
cancellation requirement could be satisfied. We conclude that having
$\mathcal{F}$ terms of distinct signs is not enough.

As pointed out in the past by several authors Pkhn63II ; Tktpls63nonplanar ;
Hld64 ; Lam69 ; Ldsf74 , a typical expression of
$\mathcal{F}(\boldsymbol{\alpha};\boldsymbol{s})$ for $2\to 2$ scattering that
features a pinch singularity includes terms proportional to

$(\alpha_{a}\alpha_{b}-\alpha_{c}\alpha_{d})(\alpha_{e}\alpha_{f}-\alpha_{g}\alpha_{h}).$
(33)

We now present an algorithm which searches for integrals potentially
satisfying the cancellation requirement above. We will see that diagrams with
cancellation patterns such as (33) are identified by the algorithm.

Algorithm to identify potential pinch singularities in ${\cal F}^{(s_{ij})}$

Based on the cancellation requirement stated above, we now construct an
algorithm which identifies all the graphs that potentially contain a pinch in
parameter space, under the assumption that
$\mathcal{F}=\mathcal{F}^{(s_{ij})}(\\{\alpha_{k}\\})$, i.e. $\mathcal{F}$ is
proportional to a single Mandelstam invariant $s_{ij}$, and hence a function
of the Feynman parameters alone. We then follow three simple steps:

  * Step 1. Compute $\mathcal{F}_{+}$ and $\mathcal{F}_{-}$. If either of them vanishes, exit the algorithm outputting that there are no pinch Landau singularities. Otherwise go to Step 2.

  * Step 2. Compute $\partial\mathcal{F}_{+}/\partial\alpha_{i}$ and $\partial\mathcal{F}_{-}/\partial\alpha_{i}$ for all the $\alpha_{i}$ that $\mathcal{F}$ depends on. If none of these derivatives vanish, exit the algorithm outputting that there are pinch singularities (which may or may not be within the integration domain). Otherwise go to Step 3.

  * Step 3. Identify the $i$ for which $\partial\mathcal{F}_{+}/\partial\alpha_{i}=0$ or $\partial\mathcal{F}_{-}/\partial\alpha_{i}=0$, replace $\mathcal{F}$ by $\mathcal{F}|_{\alpha_{i}=0}$, and return to Step 1.

Application to massless on-shell $2\to 2$ wide-angle scattering

Here we focus on singularities in $2\to 2$ massless wide-angle scattering. In
order to apply the above algorithm to this case we must consider the interplay
of the three channels $s$, $t$ and $u$ represented in the ${\cal F}$
polynomial, which satisfy the momentum conservation and on-shell conditions,
i.e. $s+t+u=0$. Given that we have, for example, eliminated $u=-s-t$ from
${\cal F}$, then the complete ${\cal F}$ polynomial takes the form999Here and
below we use the subscript $u$ to indicate that we eliminated it from ${\cal
F}$, setting $u=-s-t$.:

${\cal F}_{u}(\boldsymbol{\alpha};s,t)\equiv{\cal
F}^{(s)}(\boldsymbol{\alpha};s)+{\cal F}^{(t)}(\boldsymbol{\alpha};t)\,.$ (34)

We now apply the above algorithm separately on ${\cal
F}^{(s)}(\boldsymbol{\alpha};s)$ and on ${\cal
F}^{(t)}(\boldsymbol{\alpha};t)$. If we cannot exclude a pinch in either of
the two, we go ahead and look for a solution of the _combined set_ of Landau
equations:

$\frac{\partial{\cal
F}^{(s)}(\boldsymbol{\alpha};s)}{\partial\alpha_{i}}=0,\qquad\frac{\partial{\cal
F}^{(t)}(\boldsymbol{\alpha};t)}{\partial\alpha_{i}}=0,\qquad\forall i$ (35)

If such a solution exists, we have a true candidate for a pinch singularity.

A rather obvious potential caveat could be that while the polynomial ${\cal
F}_{u}$, obtained by eliminating $u$ in favour of $-s-t$ seems to satisfy the
conditions for a pinch, upon considering the same graph using an alternative
set of independent invariants, say by $t=-s-u$, the resulting polynomial

${\cal F}_{t}(\boldsymbol{\alpha};s,u)\equiv{\cal
F}^{(s)}(\boldsymbol{\alpha};s)+{\cal F}^{(u)}(\boldsymbol{\alpha};u)$ (36)

has no pinch solution. Such a scenario is indeed realised for planar graphs.
Of course the conclusion is that this graph has no pinch (in fact, in the case
of planar graphs ${\cal F}$ has a same-sign regime). Such false candidate
graphs for a pinch can be easily eliminated upon running the algorithm on all
sets of independent invariants, i.e. on ${\cal
F}_{t}(\boldsymbol{\alpha};s,u)$, on ${\cal F}_{u}(\boldsymbol{\alpha};s,t)$
and on ${\cal F}_{s}(\boldsymbol{\alpha};t,u)$. If a pinch singularity is
excluded in any of them, then there is no such solution.

By running this algorithm through all massless $2\to 2$ graphs with three- and
four-point vertices and up to four loops, we find that for _both_ wide-angle
scattering kinematics and nearly-forward scattering kinematics, there are no
one-loop or two-loop graphs exhibiting pinch Landau singularities. At the
three-loop order, there are 10 graphs, shown in figure 2, which potentially
contain pinch Landau singularities. We have used the symbols $\bullet$, $s$,
$t$, and $u$ to denote different internal scattering topologies in these
graphs: $\bullet$ represents scattering at a four-point vertex, while $s$,
$t$, and $u$ represent scattering in the $s$, $t$, and $u$ channels,
respectively. In section 3.3 we show how knowledge of the pinch singularity
facilitates the numerical evaluation of the integral, by enabling us to
identify a polytope dissection that maps the singularity to the boundary of
the domain of integration. In sections 4 and 5 we discuss how the pinch
singularity leads to new regions, referred to as _hidden regions_ , that were
not visible as a facet of the Newton polytope of the integral in wide-angle
scattering and forward scattering kinematics, respectively.

At four-loop order, the algorithm identifies 1097 graphs potentially
containing pinch Landau singularities. A key observation is that 1081 of these
graphs can be obtained by adding one extra propagator to the graphs of figure
2. Indeed, we explicitly checked, using Maple’s graph isomorphism command
Maple , whether for each of these 1097 graphs we could identify a subgraph
isomorphic to one of the graphs in figure 2, after deleting any one edge. This
check failed only for 16 graphs. Therefore, the three-loop graphs potentially
containing pinch Landau singularities appear as subgraphs of 1081 four-loop
graphs. Three examples from the set of 1081 four-loop integrals are shown in
figure 3, they contain $G_{\bullet\bullet}$, $G_{\bullet s}$, and $G_{ss}$ as
subgraphs, respectively. Of the remaining 16 four-loop integrals, 4 integrals
contain a four-point vertex, have twelve propagators, and are related to each
other by crossing; an example is $\mathfrak{G}^{\prime}$ shown in figure 4.
The final 12 four-loop integrals each have thirteen propagators and contain
one of these 4 four-loop integrals as a subgraph.

$p_{1}$$p_{3}$$p_{2}$$p_{4}$

(a) $G_{\bullet\bullet}$

$p_{1}$$p_{3}$$p_{2}$$p_{4}$

(b) $G_{\bullet s}$

$p_{1}$$p_{3}$$p_{2}$$p_{4}$

(c) $G_{\bullet t}$

$p_{1}$$p_{3}$$p_{2}$$p_{4}$

(d) $G_{\bullet u}$

$p_{1}$$p_{3}$$p_{2}$$p_{4}$

(e) $G_{ss}$

$p_{1}$$p_{3}$$p_{2}$$p_{4}$

(f) $G_{tt}$

$p_{1}$$p_{3}$$p_{2}$$p_{4}$

(g) $G_{uu}$

$p_{1}$$p_{3}$$p_{2}$$p_{4}$

(h) $G_{st}$

$p_{1}$$p_{3}$$p_{2}$$p_{4}$

(i) $G_{su}$

$p_{1}$$p_{3}$$p_{2}$$p_{4}$

(j) $G_{tu}$

Figure 2: All the massless four-point three-loop graphs with a possible pinch
Landau singularity in parameter space.

Upon analysing the integral $\mathfrak{G}^{\prime}$, shown in figure 4, we
find that when considering the coefficients of the Mandelstam invariants
individually there is a pinch solution of the Landau equations. This implies
that a hidden region may exist in the Regge limit described in section 5,
where a single Mandelstam invariant becomes small relative to the others.
However, no simultaneous solution of the Landau equations is possible when two
nonzero Mandelstam invariants are present in the $\mathcal{F}$ polynomial.
This implies that a pinch solution does not exist for this on-shell integral
in general kinematics, and that no hidden region is present in the on-shell
expansion in wide-angle scattering, described in section 4.

$p_{1}$$p_{3}$$p_{2}$$p_{4}$

(a) $\mathfrak{G}_{\bullet\bullet}$

$p_{1}$$p_{3}$$p_{2}$$p_{4}$

(b) $\mathfrak{G}_{\bullet s}$

$p_{1}$$p_{3}$$p_{2}$$p_{4}$

(c) $\mathfrak{G}_{ss}$

Figure 3: Example four-loop graphs with possible pinch Landau singularities in
parameter space, due to one appearing in a three-loop subgraph. They are
labelled $\mathfrak{G}_{\bullet\bullet}$, $\mathfrak{G}_{\bullet s}$, and
$\mathfrak{G}_{ss}$, because they contain $G_{\bullet\bullet}$, $G_{\bullet
s}$, and $G_{ss}$ of figure 2 respectively as subgraphs.

$p_{1}$$p_{3}$$p_{2}$$p_{4}$

Figure 4: A four-loop graph example $\mathfrak{G}^{\prime}$ that was
identified by the algorithm as possibly having a pinch Landau singularity in
parameter space, but which does not contain any of the three-loop structures
in figure 2 as a subgraph.

###  3.3 Identifying the pinch and dissecting the polytope

Before further discussing the interpretation of the pinch Landau singularities
identified above, let us first consider how they impact the evaluation of
integrals in parameter space. Outside the same-sign regime (where all ${\cal
F}$ terms have the same sign) Feynman integrals can develop simple poles on
the real axis of the parameters which must be avoided using a contour
compatible with the causal $i\varepsilon$ Feynman prescription. Specifically,
the imaginary part of the $\mathcal{F}$ polynomial must be _negative_ in the
vicinity of the pole. As described in section 2.2, such an integration contour
can be realised by giving a small imaginary part to the integration variables.
A commonly used prescription for the deformation is given by,

$\displaystyle\alpha_{k}\rightarrow\widetilde{\alpha}_{k}(\boldsymbol{\alpha)}=\alpha_{k}-i\tau_{k}(\boldsymbol{\alpha}),$
$\displaystyle\tau_{k}(\boldsymbol{\alpha})=\lambda_{k}\alpha_{k}(1-\alpha_{k})\frac{\partial\mathcal{F}(\boldsymbol{\alpha};\boldsymbol{s})}{\partial\alpha_{k}}.$
(37)

The factors $\alpha_{k}(1-\alpha_{k})$ in $\tau_{k}$ ensure that the contour
deformation vanishes at the endpoints of integration ($\alpha_{k}=0$ and
$\alpha_{k}=1$). Upon inserting the new variables into $\mathcal{F}$ and
expanding in a power series in $\tau$ we obtain,

$\displaystyle\begin{split}\mathcal{F}(\widetilde{\boldsymbol{\alpha}};\boldsymbol{s})&=\mathcal{F}(\boldsymbol{\alpha})-i\sum_{j}\tau_{j}(\boldsymbol{\alpha})\frac{\partial\mathcal{F}(\boldsymbol{\alpha};\boldsymbol{s})}{\partial\alpha_{j}}+\mathcal{O}(\tau^{2})\\\
&=\mathcal{F}(\boldsymbol{\alpha})-i\sum_{j}\lambda_{j}\alpha_{j}(1-\alpha_{j})\left(\frac{\partial\mathcal{F}(\boldsymbol{\alpha};\boldsymbol{s})}{\partial\alpha_{j}}\right)^{2}+\mathcal{O}(\tau^{2}).\end{split}$
(38)

The reason for choosing the contour deformation of $\alpha_{k}$ to be
proportional to the derivative of the ${\cal F}$ polynomial with respect to
$\alpha_{k}$ in eq. (37) now becomes clear: this prescription ensures that the
first correction term in the expansion of
$\mathcal{F}(\widetilde{\boldsymbol{\alpha}};\boldsymbol{s})$ does not depend
on the sign of $\partial\mathcal{F}/\partial\alpha_{j}$. For sufficiently
small $\lambda_{k}\geq 0$, such that terms of order $\tau^{3}$ can be
neglected, this deformation therefore ensures that
$\mathcal{F}(\widetilde{\boldsymbol{\alpha}};\boldsymbol{s})$ has a negative
imaginary part, as required by the Feynman prescription. This contour
deformation prescription101010Or an equivalent prescription for
$\mathcal{V}(\boldsymbol{\alpha};\boldsymbol{s})=\mathcal{F}(\boldsymbol{\alpha};\boldsymbol{s})/\mathcal{U}(\boldsymbol{\alpha})$.
Soper:1999xk ; Binoth:2005ff ; Nagy:2006xy ; Anastasiou:2007qb ;
Lazopoulos:2007ix ; Beerli:2008zz ; Borowka:2012yc ; Borowka:2014aaa  is used
in programs which numerically evaluate Feynman integrals, including FIESTA
SmnTtyk09FIESTA ; SmnSmnTtyk11FIESTA2 ; Smn14FIESTA3 ; Smn16FIESTA4 ;
Smn22FIESTA5 , pySecDec pySecDec17  and feyntrop BrskMchTld23 . Examining the
deformation, one may expect that the prescription would fail if there exists a
point inside the integration domain for which $\mathcal{F}=0$ and
$\alpha_{j}(1-\alpha_{j})\partial\mathcal{F}/\partial\alpha_{j}=0$ for all $j$
simultaneously, which is closely related to the Landau condition in eq. (8).
The conclusion is that contour deformation is likely to be ineffective if a
pinch solution to the Landau equations appears within the domain of
integration.

Starting from three-loop order, for $2\rightarrow 2$ massless scattering,
examples do exist which satisfy the criteria $\mathcal{F}=0$ and
$\alpha_{j}\partial\mathcal{F}/\partial\alpha_{j}=0$ for all $j$, for which
the above prescription indeed fails. Note that this obstacle is a fundamental
property of the integral and not an artifact of our specific choice for the
contour deformation prescription.

One example is the Feynman integral corresponding to the crown diagram
$G_{\bullet\bullet}$, shown in figure 2(a). We may express the integral as,

$\displaystyle\mathcal{I}_{G_{\bullet\bullet}}(\boldsymbol{s})=$
$\displaystyle\int\frac{d^{D}k_{1}}{i\pi^{D/2}}\frac{d^{D}k_{2}}{i\pi^{D/2}}\frac{d^{D}k_{3}}{i\pi^{D/2}}\frac{1}{D_{0}^{n_{0}}\ldots
D_{7}^{n_{7}}},$ $\displaystyle=$ $\displaystyle\Gamma(2+3\epsilon)\int
d\alpha_{0}\dots
d\alpha_{7}\,\,\delta(1-\alpha_{0}\dots-\alpha_{7})\,\Big{(}\mathcal{U}(\boldsymbol{\alpha})\Big{)}^{4\epsilon}\,\Big{(}\mathcal{F}(\boldsymbol{\alpha};\boldsymbol{s})\Big{)}^{-2-3\epsilon}.$
(39)

The inverse propagators are $D_{j}=q_{j}^{2}+i\varepsilon$ ($j=0,\dots,7$),
with $q_{j}$ the $j$-th element of the set

$\displaystyle\Big{\\{}k_{1},\ p_{1}-k_{1},\ k_{2},\ p_{2}-k_{2},\ k_{3},\
p_{3}-k_{3},\ k_{1}+k_{2}-k_{3},\ p_{4}-k_{1}-k_{2}+k_{3}\Big{\\}}.$ (40)

The Symanzik polynomials $\mathcal{U}$ and $\mathcal{F}$ read (under the
parameterisation in figure 6(a)):

$\displaystyle\begin{split}&\mathcal{U}(\boldsymbol{\alpha})=(\alpha_{0}+\alpha_{1})(\alpha_{2}+\alpha_{3})(\alpha_{4}+\alpha_{5})+(\alpha_{0}+\alpha_{1})(\alpha_{2}+\alpha_{3})(\alpha_{6}+\alpha_{7})\\\
&\qquad\qquad+(\alpha_{0}+\alpha_{1})(\alpha_{4}+\alpha_{5})(\alpha_{6}+\alpha_{7})+(\alpha_{2}+\alpha_{3})(\alpha_{4}+\alpha_{5})(\alpha_{6}+\alpha_{7}),\\\
&\mathcal{F}(\boldsymbol{\alpha};\boldsymbol{s})=(-s_{12})(\alpha_{1}\alpha_{4}-\alpha_{0}\alpha_{5})(\alpha_{3}\alpha_{6}-\alpha_{2}\alpha_{7})+(-s_{13})(\alpha_{1}\alpha_{2}-\alpha_{0}\alpha_{3})(\alpha_{5}\alpha_{6}-\alpha_{4}\alpha_{7})\,,\end{split}$
(41)

where $s_{12}\equiv(p_{1}+p_{2})^{2}$, $s_{13}\equiv(p_{1}-p_{3})^{2}$, and
$s_{14}\equiv(p_{1}-p_{4})^{2}$, and we have used the relation
$s_{12}+s_{13}+s_{23}=0$ to eliminate $s_{23}$, which leads to monomials of
different sign multiplying the remaining invariants $s_{12}$ and $s_{13}$.
Note that, as a consequence of this, there is no analytic continuation
prescription of the form $s_{ij}\rightarrow s_{ij}\pm i\varepsilon$ which
would reproduce the overall $i\varepsilon$ in eq. (2) for all values of the
Feynman parameters $\alpha_{i}\geqslant 0$.

Possible pinch singularities in $\mathcal{F}$ of eq. (41), could be associated
to the vanishing of subsets of the following polynomials:

$\displaystyle\begin{split}v_{1}=\alpha_{1}\alpha_{4}-\alpha_{0}\alpha_{5}\,,\qquad
v_{2}=\alpha_{3}\alpha_{6}-\alpha_{2}\alpha_{7}\,,\\\
v_{3}=\alpha_{1}\alpha_{2}-\alpha_{0}\alpha_{3}\,,\qquad
v_{4}=\alpha_{5}\alpha_{6}-\alpha_{4}\alpha_{7}\,.\end{split}$ (42)

Computing the derivatives of the
$\mathcal{F}(\boldsymbol{\alpha};\boldsymbol{s})$ polynomial with respect to
each Feynman parameter we obtain,

$\displaystyle\frac{\partial\mathcal{F}(\boldsymbol{\alpha};\boldsymbol{s})}{\partial\alpha_{0}}$
$\displaystyle=s_{12}x_{5}(\alpha_{3}\alpha_{6}-\alpha_{2}x_{7})+s_{13}\alpha_{3}(\alpha_{5}\alpha_{6}-\alpha_{4}\alpha_{7}),$
(43a) $\displaystyle\hskip 85.35826pt\vdots$
$\displaystyle\frac{\partial\mathcal{F}(\boldsymbol{\alpha};\boldsymbol{s})}{\partial\alpha_{7}}$
$\displaystyle=s_{12}\alpha_{2}(\alpha_{1}\alpha_{4}-\alpha_{0}\alpha_{5})+s_{13}\alpha_{4}(\alpha_{1}\alpha_{2}-\alpha_{0}\alpha_{3}).$
(43h)

We see that the Landau equations can be satisfied with all $\alpha_{i}>0$ for
generic $s_{12},s_{13}$ if and only if $v_{1}=v_{2}=v_{3}=v_{4}=0$,
corresponding to,

$\displaystyle\alpha_{2}=\frac{\alpha_{0}\alpha_{3}}{\alpha_{1}},$
$\displaystyle\alpha_{4}=\frac{\alpha_{0}\alpha_{5}}{\alpha_{1}},$
$\displaystyle\alpha_{6}=\frac{\alpha_{0}\alpha_{7}}{\alpha_{1}}.$ (44)

This relations define the _pinch surface_. Note that the vanishing of any
subset of three $v_{i}$ in eq. (42) implies the vanishing of the fourth.

Attempting to evaluate $\mathcal{I}_{G_{\bullet\bullet}}$ by numerically
integrating over the parameters and using the contour deformation given in eq.
(38) to avoid singularities within the integration domain will fail. This is
because all
$\partial{\mathcal{F}(\boldsymbol{\alpha};\boldsymbol{s})}/\partial{\alpha_{i}}$
can vanish precisely on the hypersurface $\mathcal{F}=0$, where deformation is
needed. We conclude that there exists no contour deformation that can render
eq. (3.3) free of singularities in the domain of integration.

One way of avoiding this issue is to split, or _dissect_ , the integral such
that the Landau singularity present inside the integration domain – on the
pinch surface given by eq. (44) – is mapped to a boundary in parameter space.
In the new variables, the singularity will then appear as an endpoint
singularity rather than a pinch singularity. To obtain a suitable
decomposition of the integral, we can first try to apply the PreResolve
procedure of asy2, described in section 3.1. However, we observe that a linear
change of variables (even if iterated) will not result in a polynomial with
monomials of opposite sign eliminated. The singularities present go beyond the
pairwise line singularities discussed in ref. JtzSmnSmn12  and are therefore
not currently treatable by asy2. Indeed, running asy2 on this example, it
correctly identifies that iterated linear changes of variables are
insufficient to resolve the singularity and reports that preresolution has
failed.

Instead, we first find it useful to linearise the cancellation, i.e. we
perform a blowup that decreases the degree of the polynomial defining the
variety of $\mathcal{F}$. Under the following change of variables,

$\displaystyle\alpha_{0}=y_{0}\cdot\alpha_{1},\qquad\alpha_{2}=y_{2}\cdot\alpha_{3},\qquad\alpha_{4}=y_{4}\cdot\alpha_{5},\qquad\alpha_{6}=y_{6}\cdot\alpha_{7},$
(45)

the Symanzik polynomials become,

$\displaystyle\begin{split}&\mathcal{U}(\boldsymbol{\alpha})=(y_{0}+1)(y_{2}+1)(y_{4}+1)\alpha_{1}\alpha_{3}\alpha_{5}+(y_{0}+1)(y_{2}+1)(y_{6}+1)\alpha_{1}\alpha_{3}\alpha_{7}\\\
&\qquad\qquad+(y_{0}+1)(y_{4}+1)(y_{6}+1)\alpha_{1}\alpha_{5}\alpha_{7}+(y_{2}+1)(y_{4}+1)(y_{6}+1)\alpha_{3}\alpha_{5}\alpha_{7},\end{split}$
(46a)
$\displaystyle\begin{split}\mathcal{F}(\boldsymbol{x};\boldsymbol{s})=(-s_{12})(y_{4}-y_{0})(y_{6}-y_{2})\alpha_{1}\alpha_{3}\alpha_{5}\alpha_{7}+(-s_{13})(y_{2}-y_{0})(y_{6}-y_{4})\alpha_{1}\alpha_{3}\alpha_{5}\alpha_{7}.\end{split}$
(46b)

We see that the polynomials defining the zero of $\mathcal{F}$ occurring due
to cancellation become simpler, for example, up to overall factors
$(\alpha_{1}\alpha_{4}-\alpha_{0}\alpha_{5})\rightarrow(y_{4}-y_{0})$. Next,
we dissect the integral by imposing a strict hierarchy between the even-
numbered parameters. For example, assuming $y_{0}\geqslant y_{2}\geqslant
y_{4}\geqslant y_{6}$ changes the boundaries of integration from $[0,\infty]$
in each variable to new boundaries depending on the variables. We can then
change variables according to,

$\displaystyle y_{0}$ $\displaystyle=z_{0}+z_{2}+z_{4}+z_{6},$ (47a)
$\displaystyle y_{2}$ $\displaystyle=z_{2}+z_{4}+z_{6},$ (47b) $\displaystyle
y_{4}$ $\displaystyle=z_{4}+z_{6},$ (47c) $\displaystyle y_{6}$
$\displaystyle=z_{6},$ (47d)

so the boundaries of integration for the new variables are again then mapped
to $[0,\infty]$. Considering all possible hierarchies will split the integral
into $4!=24$ new integrals, each of which will have polynomials of definite
sign multiplying the invariants $s_{12}$ and $s_{13}$. Note also that the
change of variables does not introduce any sign into the $\mathcal{U}$
polynomial. The 24 new integrals have the following $\mathcal{F}$ polynomials,

$\displaystyle\mathcal{F}_{1}(\boldsymbol{\alpha};\boldsymbol{s})$
$\displaystyle=\alpha_{1}\alpha_{3}\alpha_{5}\alpha_{7}\left[-s_{12}(\alpha_{0}+\alpha_{2})(\alpha_{2}+\alpha_{4})-s_{13}(\alpha_{0}\alpha_{4})\right],$
(48a) $\displaystyle\mathcal{F}_{2}(\boldsymbol{\alpha};\boldsymbol{s})$
$\displaystyle=\alpha_{1}\alpha_{3}\alpha_{5}\alpha_{7}\left[-s_{12}(\alpha_{2})(\alpha_{0}+\alpha_{2}+\alpha_{6})+s_{13}(\alpha_{0}\alpha_{6})\right],$
(48b) $\displaystyle\mathcal{F}_{3}(\boldsymbol{\alpha};\boldsymbol{s})$
$\displaystyle=\alpha_{1}\alpha_{3}\alpha_{5}\alpha_{7}\left[-s_{12}(\alpha_{0}\alpha_{2})-s_{13}(\alpha_{0}+\alpha_{4})(\alpha_{2}+\alpha_{4})\right],$
(48c) $\displaystyle\mathcal{F}_{4}(\boldsymbol{\alpha};\boldsymbol{s})$
$\displaystyle=\alpha_{1}\alpha_{3}\alpha_{5}\alpha_{7}\left[s_{12}(\alpha_{0}\alpha_{6})-s_{13}(\alpha_{4})(\alpha_{0}+\alpha_{4}+\alpha_{6})\right],$
(48d) $\displaystyle\mathcal{F}_{5}(\boldsymbol{\alpha};\boldsymbol{s})$
$\displaystyle=\alpha_{1}\alpha_{3}\alpha_{5}\alpha_{7}\left[s_{12}(\alpha_{6})(\alpha_{0}+\alpha_{2}+\alpha_{6})+s_{13}(\alpha_{0}+\alpha_{6})(\alpha_{2}+\alpha_{6})\right],$
(48e) $\displaystyle\mathcal{F}_{6}(\boldsymbol{\alpha};\boldsymbol{s})$
$\displaystyle=\alpha_{1}\alpha_{3}\alpha_{5}\alpha_{7}\left[s_{12}(\alpha_{0}+\alpha_{6})(\alpha_{4}+\alpha_{6})+s_{13}(\alpha_{6})(\alpha_{0}+\alpha_{4}+\alpha_{6})\right],$
(48f)

with the remaining 18 integrals equal to the above 6 up to a relabelling of
the Feynman parameters. Each of the new integrals is free of pinch
singularities within the integration domain.

###  3.4 Numerically evaluating the dissected integral

Before dissection, the direct application of the sector decomposition
algorithm, as described in section 2.3, fails to resolve the singularity
associated to the pinch singularity and yields integrands unsuitable for
numerical evaluation. Turning now to the dissected integrals, for given values
of $s_{12}$ and $s_{13}$ some of the integrals will have a same-sign regime
and can be evaluated without contour deformation. However, for any possible
choice of values of the Mandelstam invariants, not all integrals will be
simultaneously in a same-sign regime. Nevertheless, these integrals can now be
evaluated with the contour deformation given in eq. (38). Furthermore, it is
also possible to find a prescription of the form $s_{ij}\rightarrow s_{ij}\pm
i\varepsilon$ for each integral which allows them to be analytically
continued.

Although the above procedure avoids the pinch singularity, the direct
numerical evaluation of this three-loop 4-point integral is still challenging.
Each of the sectors has a $1/\epsilon^{5}$ pole that cancels in the sum of
sectors. Choosing the phase-space point $s_{12}=1,s_{13}=-1/5$ we obtain,

$\displaystyle\mathcal{I}_{1}/C_{\epsilon}=$
$\displaystyle\,\epsilon^{-5}\left[0.55555538\mathbf{27}\right]$
$\displaystyle+$
$\displaystyle\,\epsilon^{-4}\left[-3.884290\mathbf{14}+5.235983\mathbf{13}\,i\right]$
$\displaystyle+$ $\displaystyle\,\mathcal{O}(\epsilon^{-3}),$ (49a)
$\displaystyle\mathcal{I}_{2}/C_{\epsilon}=$
$\displaystyle\,\epsilon^{-5}\left[2.222232\mathbf{11}\right]$
$\displaystyle+$
$\displaystyle\,\epsilon^{-4}\left[-7.92923\mathbf{11}+20.94388\mathbf{18}\,i\right]$
$\displaystyle+$ $\displaystyle\,\mathcal{O}(\epsilon^{-3}),$ (49b)
$\displaystyle\mathcal{I}_{3}/C_{\epsilon}=$
$\displaystyle\,\epsilon^{-5}\left[-2.7777888\mathbf{83}\right]$
$\displaystyle+$
$\displaystyle\,\epsilon^{-4}\left[18.519682\mathbf{69}-15.708041\mathbf{67}\,i\right]$
$\displaystyle+$ $\displaystyle\,\mathcal{O}(\epsilon^{-3}),$ (49c)
$\displaystyle\mathcal{I}_{4}/C_{\epsilon}=$
$\displaystyle\,\epsilon^{-5}\left[2.2222211\mathbf{19}\right]$
$\displaystyle+$
$\displaystyle\,\epsilon^{-4}\left[-13.294002\mathbf{23}\right]$
$\displaystyle+$ $\displaystyle\,\mathcal{O}(\epsilon^{-3}),$ (49d)
$\displaystyle\mathcal{I}_{5}/C_{\epsilon}=$
$\displaystyle\,\epsilon^{-5}\left[-2.7777713\mathbf{46}\right]$
$\displaystyle+$
$\displaystyle\,\epsilon^{-4}\left[12.74345\mathbf{17}-23.56186\mathbf{15}\,i\right]$
$\displaystyle+$ $\displaystyle\,\mathcal{O}(\epsilon^{-3}),$ (49e)
$\displaystyle\mathcal{I}_{6}/C_{\epsilon}=$
$\displaystyle\,\epsilon^{-5}\left[0.55555546\mathbf{19}\right]$
$\displaystyle+$
$\displaystyle\,\epsilon^{-4}\left[-4.0702347\mathbf{61}\right]$
$\displaystyle+$ $\displaystyle\,\mathcal{O}(\epsilon^{-3}),$ (49f)

where $C_{\epsilon}=\Gamma(2+3\epsilon)$ is the integral prefactor given in
eq. (3.3), and the bold digits are used to report the error on two stated
digits preceding them. The precision stated above is obtained using contour
deformation as implemented in pySecDec to evaluate the dissected integrals.
Alternatively, it is possible to dramatically improve the numerical precision
using the techniques described in ref. TomLoopsLegsProc . Summing over the
sectors using the higher-precision evaluation111111Directly summing over the
results in eq. (49) we would obtain the value
$\mathcal{I}_{G_{\bullet\bullet}}^{\mathrm{numeric}}=\epsilon^{-4}\left[8.34150\mathbf{87}-52.3601\mathbf{10}\,i\right]+\mathcal{O}(\epsilon^{-3})$.,
the full numeric result is given by,

$\displaystyle\mathcal{I}_{G_{\bullet\bullet}}^{\mathrm{numeric}}$
$\displaystyle=4\
(\mathcal{I}_{1}+\mathcal{I}_{2}+\mathcal{I}_{3}+\mathcal{I}_{4}+\mathcal{I}_{5}+\mathcal{I}_{6})$
$\displaystyle=\epsilon^{-4}\left[8.3400403920\mathbf{28}-52.35987755983\mathbf{47}I\right]+\mathcal{O}(\epsilon^{-3}),$
(50)

this agrees, within the numerical integration error, with the analytic result
obtained in ref. Bargiela:2021wuy , with $x=-s_{13}/s_{12}$,

$\displaystyle\mathcal{I}_{G_{\bullet\bullet}}^{\mathrm{analytic}}(x)$
$\displaystyle=-\frac{8}{3}\left[\frac{i\pi}{x(1-x)}+\frac{\log
x}{1-x}+\frac{\log(1-x)}{x}\right]\frac{1}{\epsilon^{4}}+\mathcal{O}(\epsilon^{-3})$
$\displaystyle=\epsilon^{-4}\left[8.3400403922-52.3598775598I\right]+\mathcal{O}(\epsilon^{-3}).$
(51)

Dissecting the original Newton polytope of $G_{\bullet\bullet}$, as described
in section 3.3, has mapped the pinch singularity, which was originally within
the domain of integration, to the boundary of integration. With the
singularity now at the boundary, we have established that it is possible to
resolve it using the method of sector decomposition, enabling the numerical
evaluation of $G_{\bullet\bullet}$. In the following sections, we will
investigate how the presence of the pinch singularity for the on-shell
integral impacts the regions relevant for the MoR when the integral is
expanded about a limit where the pinch is present.

##  4 On-shell expansion for wide-angle scattering

In this section we study the implication of pinch Landau singularities in
parameters space from the perspective of the MoR, focusing on the on-shell
expansion of wide-angle scattering, the expansion discussed in refs.
GrdHzgJnsMaSchlk22 ; Ma23 . Generally, one starts with a graph with massless
propagators contributing to an off-shell Green’s function with $M$ external
legs, out of which $K_{0}$ are strictly on-shell, while $K_{\lambda}$ are
expanded about the on-shell limit,

$\displaystyle\left\\{\begin{array}[]{ll}p_{i}^{2}=0&\qquad i=1,\dots,K_{0}\\\
p_{i}^{2}\sim\lambda Q^{2}&\qquad i=K_{0}+1,\dots,K,\\\ q_{i}^{2}\sim
Q^{2}&\qquad i=K+1,\dots,M\\\ p_{i_{1}}\cdot p_{i_{2}}\sim
Q^{2}&\qquad\forall\,\,i_{1}\neq i_{2}\,,\end{array}\right.$ (56)

where the total number of massless jets, once the limit $\lambda\to 0$ is
taken, is given by $K\equiv K_{0}+K_{\lambda}$. Note that the wide-angle
condition is incorporated in $p_{i_{1}}\cdot p_{i_{2}}\sim Q^{2}$, implying
that while the jets $p_{i}$ are nearly lightlike, the angle between their
three-momenta $\boldsymbol{p}_{i_{1}}$ and $\boldsymbol{p}_{i_{2}}$ is
$\mathcal{O}(1)$ for any $i_{1}$ and $i_{2}$.

Given any graph with the kinematics of (56), the complete list of _facet_
regions can be described by the hard-collinear-soft picture in figure 5, as
proposed in ref. GrdHzgJnsMaSchlk22  and proved in ref. Ma23 . According to
this picture, any region in the on-shell expansion of wide-angle scattering
graphs consists of three types of modes:

$\displaystyle\begin{cases}\text{hard:}&k_{H}^{\mu}\sim Q(1,1,1);\\\
\text{collinear to }{p_{i}}:&k_{C_{i}}^{\mu}\sim
Q(1,\lambda,\lambda^{1/2});\\\ \text{soft:}&k_{S}^{\mu}\sim
Q(\lambda,\lambda,\lambda).\end{cases}$ (57)

In the scaling above, we have used the $p_{i}$ lightcone coordinate, for
$i=1,\dots,K$. Namely, $k^{\mu}=(k\cdot\overline{\beta}_{i},\
k\cdot\beta_{i},\ k\cdot\beta_{i\perp})$, where $\beta_{i}^{\mu}$ is a null
vector in the direction of $p_{i}^{\mu}$, defined as
$\beta_{i}^{\mu}=\frac{1}{\sqrt{2}}(1,\boldsymbol{p}_{i}/|\boldsymbol{p}_{i}|)$.
For each $\beta_{i}^{\mu}$, we also define
$\overline{\beta}_{i}^{\mu}\equiv\frac{1}{\sqrt{2}}(1,-\boldsymbol{p}_{i}/|\boldsymbol{p}_{i}|)$,
so that $\beta_{i}\cdot\overline{\beta}_{i}=1$. The hard ($H$), the $i$-th jet
($J_{i}$), and the soft ($S$) subgraphs in figure 5 consist respectively of
propagators carrying momenta $k_{H}^{\mu}$, $k_{C_{i}}^{\mu}$, and
$k_{S}^{\mu}$, which scale according to (57). All the off-shell external
momenta $q_{i}$ attach to $H$, and each on-shell external momentum $p_{i}$
($i\in\\{1,\dots,K\\}$) attaches to the corresponding $J_{i}$. Note that $H$,
$J_{1}$, …, $J_{K}$ are all connected, while $S$ can consist of multiple
components, each of which is connected to two or more jets.121212Further
restrictions on the soft subgraph arise in cases where some external momenta
are strictly on shell. See ref. Ma23  for more detail. Let us emphasise that
the connectedness of the hard subgraph $H$ is a key property of facet regions.
As we will see below, this property no longer holds for the hidden regions,
i.e. regions that are associated with a pinch singularity in the limit
$\lambda\to 0$, which therefore do not correspond to facets of the Newton
polytope.

$H$$S$$J_{1}$$J_{2}$$J_{K}$$q_{K+1}$$q_{M}$$p_{1}$$p_{2}$$p_{K}$

Figure 5: The general configuration of facet regions in the on-shell expansion
of wide-angle scattering, where the graph $G$ is the union of the hard
subgraph $H$, the jet subgraphs $J_{1},\dots,J_{K}$, and the soft subgraph
$S$. All the off-shell external momenta $q_{K+1},\dots,q_{M}$ attach to $H$,
and each on-shell external momentum $p_{i}$ attaches to a corresponding
$J_{i}$. Note that for facet regions, each of the subgraphs
$H,J_{1},\dots,J_{K}$ must be connected, while each connected component of $S$
must be connected to two or more jets.

To study this expansion we focus in this section exclusively on the simplest
case of relevance, namely $2\to 2$ massless scattering. In sections 4.1 to 4.3
we consider the general on-shell expansion, where $p_{i}^{2}\sim\lambda Q^{2}$
for all $i=1\ldots 4$, with $p_{i_{1}}\cdot p_{i_{2}}\sim Q^{2}$. In sections
4.4 and 4.5, we further specialise to the case where only starts off shell,
$p_{1}^{2}\sim\lambda Q^{2}$, with the remaining three jets $i=2,3,4$ are
strictly on shell ($p_{2}^{2}=p_{3}^{2}=p_{4}^{2}=0$) at the outset. In either
of these scenarios the limit about which we are expanding, where all external
legs are on shell, exposes the pinch singularity we studied in section 3.3. As
we shall see, this is sufficient to introduce a hidden region into these
asymptotic expansions.

###  4.1 Facet regions and hidden regions in $2\to 2$ scattering

As can be checked by the algorithm proposed in section 3.2, there are no pinch
Landau singularities in parameter space in on-shell $2\to 2$ massless
scattering for any one- or two-loop graphs. Based on the conjectured
correspondence between Landau singularities and regions GrdHzgJnsMaSchlk22  we
are therefore led to exclude the possibility of hidden regions in such graphs,
and trust the asymptotic expansion obtained based on facet regions alone.

The situation is rather different at three loops, where, based on the Landau
equations we concluded in the previous section that certain graphs (figure 2)
may feature pinch singularities. In section 3.3 we explicitly identified this
pinch singularity in the simplest graph of this type, the three-loop
8-propagator crown graph $G_{\bullet\bullet}$. Then, by dissecting the
polytope at the singularity we were able to evaluate the on-shell integral
using sector decomposition.

$p_{1}$$p_{3}$$p_{2}$$p_{4}$$1$$0$$4$$5$$7$$6$$2$$3$

(a) A choice of labelling the 8 propagators of the graph $G_{\bullet\bullet}$.

$p_{1}$$p_{3}$$p_{2}$$p_{4}$$H_{1}$$H_{2}$$k_{1}$$k_{2}$$k_{3}$$k_{4}$

(b) The Landshoff scattering with hard vertices $H_{1}$ and $H_{2}$.

Figure 6: The graph $G_{\bullet\bullet}$ with the propagators labelled
$e_{0},e_{1},\dots,e_{7}$, and its associated (Landshoff scattering) loop-
momentum configuration where the hard subgraph has two distinct components,
$H_{1}$ and $H_{2}$. Note that the collinear line momenta attached to $H_{1}$,
which are labelled as $k_{1},\dots,k_{4}$, depend on each other via momentum
conservation, because this graph has three loops in total.

Considering now the on-shell _expansion_ of $G_{\bullet\bullet}$ where one or
more of the external lines is initially off shell, $p_{i}^{2}={\cal O}(\lambda
Q^{2})$, and expanding in powers of $\lambda$, we expect a hidden region to
emerge, which is induced by the aforementioned pinch singularity. Let us then
consider this putative hidden region in some detail.

By labelling the propagators of $G_{\bullet\bullet}$ as in figure 6(a), the
corresponding Symanzik polynomials are:

$\displaystyle\begin{split}&\mathcal{U}(\boldsymbol{x})=(x_{0}+x_{1})(x_{2}+x_{3})(x_{4}+x_{5})+(x_{0}+x_{1})(x_{2}+x_{3})(x_{6}+x_{7})\\\
&\qquad\qquad+(x_{0}+x_{1})(x_{4}+x_{5})(x_{6}+x_{7})+(x_{2}+x_{3})(x_{4}+x_{5})(x_{6}+x_{7}),\end{split}$
(58a)
$\displaystyle\begin{split}&\mathcal{F}(\boldsymbol{x};\boldsymbol{s})=(-p_{1}^{2})\left\\{x_{0}x_{1}\big{[}(x_{2}+x_{3})(x_{4}+x_{5})+(x_{2}+x_{3})(x_{6}+x_{7})+(x_{4}+x_{5})(x_{6}+x_{7})\big{]}\right.\\\
&\phantom{\mathcal{F}(\boldsymbol{x};\boldsymbol{s})=(-p_{1}^{2})\\{}\left.+(x_{1}x_{2}x_{4}+x_{0}x_{3}x_{5})(x_{6}+x_{7})\right\\}\\\
&\qquad\qquad+(-p_{2}^{2})\left\\{x_{2}x_{3}\big{[}(x_{0}+x_{1})(x_{4}+x_{5})+(x_{4}+x_{5})(x_{6}+x_{7})+(x_{0}+x_{1})(x_{6}+x_{7})\big{]}\right.\\\
&\phantom{\mathcal{F}(\boldsymbol{x};\boldsymbol{s})=(-p_{1}^{2})\\{}\left.+(x_{1}x_{4}x_{7}+x_{0}x_{5}x_{6})(x_{2}+x_{3})\right\\}\\\
&\qquad\qquad+(-p_{3}^{2})\left\\{x_{4}x_{5}\big{[}(x_{0}+x_{1})(x_{2}+x_{3})+(x_{2}+x_{3})(x_{6}+x_{7})+(x_{0}+x_{1})(x_{6}+x_{7})\big{]}\right.\\\
&\phantom{\mathcal{F}(\boldsymbol{x};\boldsymbol{s})=(-p_{1}^{2})\\{}\left.+(x_{0}x_{3}x_{6}+x_{1}x_{2}x_{7})(x_{4}+x_{5})\right\\}\\\
&\qquad\qquad+(-p_{4}^{2})\left\\{x_{6}x_{7}\big{[}(x_{0}+x_{1})(x_{4}+x_{5})+(x_{4}+x_{5})(x_{2}+x_{3})+(x_{0}+x_{1})(x_{2}+x_{3})\big{]}\right.\\\
&\phantom{\mathcal{F}(\boldsymbol{x};\boldsymbol{s})=(-p_{1}^{2})\\{}\left.+(x_{2}x_{4}x_{7}+x_{3}x_{5}x_{6})(x_{0}+x_{1})\right\\}\\\
&\qquad\qquad+(-s_{12})(x_{0}x_{5}-x_{1}x_{4})(x_{2}x_{7}-x_{3}x_{6})+(-s_{13})(x_{0}x_{3}-x_{1}x_{2})(x_{4}x_{7}-x_{5}x_{6}),\end{split}$
(58b)

where we have used the relation between the Mandelstam variables to eliminate
$s_{14}$ from $\mathcal{F}(\boldsymbol{x};\boldsymbol{s})$ in favour of
$-s_{12}-s_{13}+\sum_{i=1}^{4}p_{i}^{2}$. This readily introduces explicit
minus signs into the last line of eq.
(LABEL:eq:8_propagator_wideangle_Symanzik_polynomialsF), leading to potential
cancellation as discussed in section 3.2. The facet regions of
$G_{\bullet\bullet}$, as one can check, are all compatible with the
configuration of hard, jet, and soft modes shown in figure 5. More information
on these regions will be provided in section 4.4 below (see table 1 there).

Our hypothesis is that in addition to the facet regions, $G_{\bullet\bullet}$
features a hidden region whose properties we now explore. Based on our
analysis in Feynman parameter space, the pinch occurs at (44), _within_ the
domain of integration, rather than on its boundary. Specifically, using the
$\delta(1-\sum_{k=0}^{7}\alpha_{k})$ constraint and the symmetry of the graph,
it is easy to see that the solution is at $\alpha_{k}=\frac{1}{8}$ for all
$k$. Converting this to Lee-Pomeransky parameters according to (5) we can
deduce that the only consistent scaling vectors are ones where all parameters
$x_{k}$ (for $k=0$ to $7$) scale in the same way. In momentum space this
corresponds (via eq. (19)) to all propagators having the same virtuality
scaling GrdHzgJnsMaSchlk22 . Given the nearly-lightlike external kinematics,
it is natural to expect that all propagators would be jet like, that is, every
line momentum is collinear to one external momentum, with virtuality
$\mathcal{O}(\lambda Q^{2})$, such that in figure 6 propagators 0 and 1 are
both collinear to $p_{1}$, propagators 2 and 3 are collinear to $p_{2}$, 4 and
5 are collinear to $p_{3}$, and 6 and 7 are collinear to $p_{4}$. Given the
collinear scaling law of all propagators, along with the usual correspondence
(19), we deduce that each Lee-Pomeransky parameter $x_{k}$ must be
$\mathcal{O}(\lambda^{-1})$, yielding the corresponding region vector:

$\displaystyle\boldsymbol{v}_{\text{Landshoff}}=(-1,-1,-1,-1,-1,-1,-1,-1;1)\,,$
(59)

where we named the hidden region vector $\boldsymbol{v}_{\text{Landshoff}}$ in
order to capture its physical interpretation as one corresponding to Landshoff
scattering, where there are multiple – in this case two – separate hard
connected components $H_{1}$ and $H_{2}$, as shown in figure 6(b).

We stress that the momentum-space interpretation of the hidden region as
corresponding to a scattering process with two _disjoint_ hard subgraphs
clarifies why it cannot be associated with a facet of the Newton polytope: all
facet region in the on-shell expansion in wide-angle scattering involve a
single connected hard subgraph GrdHzgJnsMaSchlk22 ; Ma23 , consistent with
figure 5. Further analysis of the momentum configuration in the Landshoff
scattering region will be presented in section 4.3.

Given the scaling rule (59) of the parameters, each individual term in the
last line of eq. (LABEL:eq:8_propagator_wideangle_Symanzik_polynomialsF),
namely

$\displaystyle(-s_{12})(x_{0}x_{5}-x_{1}x_{4})(x_{2}x_{7}-x_{3}x_{6})+(-s_{13})(x_{0}x_{3}-x_{1}x_{2})(x_{4}x_{7}-x_{5}x_{6}),$
(60)

is $\mathcal{O}(\lambda^{-4})$, while all the other terms in $\mathcal{P}$
(including all of the terms in ${\cal U}$) are $\mathcal{O}(\lambda^{-3})$,
and hence can naïvely be discarded. Below we shall call the individual terms
in eq. (60) _superleading terms_. If we simply use them to substitute the Lee-
Pomeransky polynomial $\mathcal{P}(\boldsymbol{x};\boldsymbol{s})$, the
integral would be scaleless. As a result, a naïve conclusion would be that the
vector $\boldsymbol{v}_{\text{Landshoff}}$ does not correspond to any region.
This conclusion is however false, as it ignores the possibility of mutual
cancellation between these terms, which would imply that their individual
$\mathcal{O}(\lambda^{-4})$ scaling does not represent the actual scaling of
the combinations in eq. (60). Indeed, if these terms conspire to cancel, i.e.
they admit131313Note that the fourth condition follows from the first three.

$\frac{x_{0}x_{5}}{x_{1}x_{4}}-1\ \sim\ \frac{x_{2}x_{7}}{x_{3}x_{6}}-1\ \sim\
\frac{x_{0}x_{3}}{x_{1}x_{2}}-1\ \sim\ \frac{x_{4}x_{7}}{x_{5}x_{6}}-1\ \sim\
{\cal O}(\sqrt{\lambda})$ (61)

then the two combinations in eq. (60) behave as

$\displaystyle(x_{0}x_{5}-x_{1}x_{4})(x_{2}x_{7}-x_{3}x_{6})\sim(x_{0}x_{3}-x_{1}x_{2})(x_{4}x_{7}-x_{5}x_{6})\sim\lambda^{-3},$
(62)

in which case the last line of eq.
(LABEL:eq:8_propagator_wideangle_Symanzik_polynomialsF) would be of the _same_
order of magnitude as the remaining terms in $\mathcal{P}$. This would lead to
a scaleful integral, and we hence discovered a _hidden region_ with the
scaling vector of (59) in which all the terms in the ${\cal P}$ polynomial
survive!

To verify this and exclude the existence of additional hidden regions, which
might not correspond to the scaling of eq. (59), in section 4.4 we will
dissect the Newton polytope according to section 3.3 and examine the lower
facets of the new polytope in each sector. Before doing so, in sections 4.2
and 4.3 we shall perform two independent power-counting analyses in parameter
space and in momentum space respectively, demonstrating that the leading
asymptotic behaviour of the graph $G_{\bullet\bullet}$ is exclusively from
this hidden region, namely,

$\mathcal{I}(s,t,\lambda
p_{i}^{2})\sim\lambda^{\mu[G_{\bullet\bullet},\text{Landshoff}]},\qquad\text{with}\qquad\mu[G_{\bullet\bullet},\text{Landshoff}]=-\frac{1}{2}-3\epsilon,$
(63)

and that this hidden region therefore dominates the asymptotic expansion of
this integral.

###  4.2 Power counting in parameter space

Let us first carry out the power counting analysis in Lee-Pomeransky
representation. To begin with, according to eq. (3), the integral takes the
form:

$\mathcal{I}_{G_{\bullet\bullet}}(s,t,\lambda
p_{i}^{2})=\frac{\Gamma(D/2)}{\Gamma\left(2D-8\right)}\int_{0}^{\infty}\,\prod_{e=0}^{7}\mathrm{d}x_{e}\,\cdot\Big{(}\mathcal{P}(\boldsymbol{x};s,t,\lambda
p_{i}^{2})\Big{)}^{-D/2}.$ (64)

where, in the Landshoff region, according to eqs. (60) and (62), the integrand
scales as

$\displaystyle\left(\mathcal{P}^{(\text{Landshoff})}(\boldsymbol{x};s,t,\lambda
p_{i}^{2})\right)^{D/2}\sim\lambda^{(-3)\cdot(-D/2)}=\lambda^{6-3\epsilon}.$
(65)

To obtain the contribution to the degree of divergence from the integration
measure, we shall change the variables $\\{x_{0}$,$x_{2}$,$x_{4}$,$x_{6}\\}$
into $\\{y_{0}$,$y_{2}$,$y_{4}$,$y_{6}\\}$ respectively, such that

$\displaystyle x_{0}=y_{0}\cdot x_{1},\qquad x_{2}=y_{2}\cdot x_{3},\qquad
x_{4}=y_{4}\cdot x_{5},\qquad x_{6}=y_{6}\cdot x_{7}.$ (66)

It then follows that $y_{i}$ is $\mathcal{O}(1)$ for every $i=0,2,4,6$, and
from eq. (62), they further satisfy

$\displaystyle(y_{4}-y_{0})(y_{6}-y_{2})x_{1}x_{3}x_{5}x_{7}\sim\lambda^{-3},$
(67a)
$\displaystyle(y_{2}-y_{0})(y_{6}-y_{4})x_{1}x_{3}x_{5}x_{7}\sim\lambda^{-3}.$
(67b)

As $x_{1}x_{3}x_{5}x_{7}\sim\lambda^{-4}$, it is clear that
$(y_{2}-y_{0})(y_{6}-y_{4})\sim\lambda$ and
$(y_{4}-y_{0})(y_{6}-y_{2})\sim\lambda$. Furthermore, it can be deduced that
for any $i,j\in\\{0,2,4,6\\}$, the dominant contribution arises from
$y_{i}-y_{j}\sim\lambda^{1/2}$. To see this, let us set

$y_{2}-y_{0}\sim\lambda^{1/2+a}\qquad\text{and}\qquad
y_{6}-y_{4}\sim\lambda^{1/2-a}$

for some $a\in\mathbb{R}$, consistently with eq. (67b). If $a>0$, then
$\lambda^{1/2-a}\gg\lambda^{1/2+a}$, and either $y_{4}$ or $y_{6}$ can be
varied freely in a range of $\mathcal{O}(\lambda^{1/2-a})$. So at a fixed
$y_{0}$, either of the following two scenarios holds:

$\displaystyle y_{4}-y_{0}\sim\lambda^{1/2-a}\qquad\qquad\text{or}$
$\displaystyle y_{6}-y_{2}=\underbrace{(y_{2}-y_{0})}_{{\cal
O}({\lambda^{1/2+a}})}-\underbrace{(y_{6}-y_{0})}_{{\cal
O}({\lambda^{1/2-a}})}\sim\lambda^{1/2-a}\,.$

However, in each of these cases one can use eq. (67a) to deduce the scaling of
the remaining differences, namely

$\displaystyle\text{if}\quad y_{4}-y_{0}\sim\lambda^{1/2-a}$ then
$\displaystyle y_{6}-y_{2}\sim\lambda^{1/2+a}$ and then $\displaystyle
y_{6}-y_{0}=(y_{6}-y_{2})+(y_{2}-y_{0})\sim\lambda^{1/2+a}\,,$
$\displaystyle\text{if}\quad y_{6}-y_{2}\sim\lambda^{1/2-a}$ then
$\displaystyle y_{4}-y_{0}\sim\lambda^{1/2+a}\,.$

In either case it follows that the integration measure for all the $y$
variables can then be rewritten as:

$\displaystyle\begin{split}dy_{0}dy_{2}dy_{4}dy_{6}&=dy_{0}\,d(y_{2}-y_{0})\,d(y_{4}-y_{0})\,d(y_{6}-y_{0})\\\
&\sim\lambda^{1/2+a}\cdot\lambda^{1/2-a}\cdot\lambda^{1/2+a}\\\
&=\lambda^{3/2+a}\qquad\qquad\text{for}\,\,a>0.\end{split}$

Using the same reasoning, one obtains
$dy_{0}dy_{2}dy_{4}dy_{6}\sim\lambda^{3/2-a}$ for $a<0$. Therefore, the
measure $dy_{0}dy_{2}dy_{4}dy_{6}$ attains its maximum $\lambda^{3/2}$ at
$a=0$, namely, $y_{i}-y_{j}\sim\lambda^{1/2}$ for any $i,j\in\\{0,2,4,6\\}$.
Therefore, the leading behaviour of the integral subject to the constraints
(67a) and (67b) is governed by $a=0$. This implies that there is a hidden
region with the scaling vector (59) in the original variables, with $a=0$,
i.e. with all $y_{i}-y_{j}\sim\lambda^{1/2}$. Note, however, that this
analysis does not exclude the existence of other, power-suppressed, scaleful
hidden regions, which share the same scaling in the original variables, but
with $a\neq 0$. In fact, in section 3.4 we shall explicitly encounter such
regions. For the $a=0$ case, which dominates the asymptotic expansion, the
whole integration measure is then:

$\displaystyle\prod_{i=0}^{7}dx_{i}=\left(\prod_{i=1,3,5,7}x_{i}dx_{i}\right)\cdot
dy_{0}dy_{2}dy_{4}dy_{6}\sim\lambda^{-8}\cdot\lambda^{3/2}=\lambda^{-13/2}.$
(68)

The degree of divergence for this hidden region, according to eqs. (64), (65)
and (68), is then

$\displaystyle\mu[G_{\bullet\bullet},\text{Landshoff}]=-\frac{13}{2}+(-3)\cdot\left(-\frac{D}{2}\right)=-\frac{1}{2}-3\epsilon\,,$
(69)

as anticipated in eq. (63) above.

###  4.3 Power counting in momentum space

To check the result in eq. (69) and gain further physical insight into its
origin, we shall analyse the asymptotic behaviour of this region directly in
momentum space, following the work of Botts and Sterman BottsStm89 . We
consider the center-of-mass frame for the $1+2\to 3+4$ process, with

$p_{1}+p_{2}=p_{3}+p_{4}\,.$ (70)

We denote the scattering angle as $\theta$, and define $v_{i}^{\mu}$ to be the
unit lightlike vector representing the direction of the $p_{i}^{\mu}$ jet.
Namely,

$\displaystyle p_{i}^{2}=\lambda Q^{2},\qquad p_{i}\cdot v_{i}\sim\lambda
Q,\qquad p_{i}\cdot\overline{v}_{i}\sim Q,\qquad p_{i}\cdot
v_{i\perp}\sim\sqrt{\lambda}Q\,;$ (71a) $\displaystyle v_{i}^{2}=0,\qquad
v_{1}\cdot v_{2}=v_{3}\cdot v_{4}=1,\,\qquad v_{1}\cdot v_{3}=v_{2}\cdot
v_{4}=\frac{1}{2}(1-\textup{cos}\ \theta)\,,$ (71b)

where for later convenience, for each $i\in\\{1,\dots,4\\}$, we have
introduced $\overline{v}_{i}^{\mu}$ as the unit lightlike vector that is back-
to-back with $v_{i}^{\mu}$. Furthermore, we define $u_{i}^{\mu}$ to be the
unit vector in the scattering plane which is normal to both $v_{i}^{\mu}$ and
$\overline{v}_{i}^{\mu}$, and $n_{i}^{\mu}$ to be unit vectors in the
remaining $D-3$ dimensional space normal to the scattering plane. It then
follows that

$\displaystyle\overline{v}_{1}^{\mu}=v_{2}^{\mu},\qquad\overline{v}_{2}^{\mu}=v_{1}^{\mu},\qquad\overline{v}_{3}^{\mu}=v_{4}^{\mu},\qquad\overline{v}_{4}^{\mu}=v_{3}^{\mu}.$
(72a) $\displaystyle u_{1}^{\mu}=u_{2}^{\mu},\qquad
u_{3}^{\mu}=u_{4}^{\mu},\qquad u_{i}^{2}=n_{i}^{2}=1,$ (72b) $\displaystyle
v_{i}\cdot u_{i}=\overline{v}_{i}\cdot u_{i}=0,\quad v_{i}\cdot
n_{j}=\overline{v}_{i}\cdot n_{j}=u_{i}\cdot n_{j}=0.$ (72c)

Based on these notations, we define the three independent loop momenta
$k_{1}$, $k_{2}$, and $k_{3}$ to be the line momenta between the external
$p_{i}$ vertices for $i=1,2,3$, respectively, and the four-point vertex
$H_{1}$ shown in figure 6(b). It is convenient to define in a similar way a
fourth loop momentum (for the line connecting $p_{4}$ and $H_{1}$), such that
momentum conservation at $H_{1}$ implies

$k_{1}+k_{2}=k_{3}+k_{4}\,,$ (73)

where eqs. (70) and (73) readily imply momentum conservation at the vertex
$H_{2}$. We parameterise the loop momenta as follows:

$\displaystyle
k_{i}^{\mu}=Q\left(\xi_{i}v_{i}^{\mu}+\lambda\kappa_{i}\overline{v}_{i}^{\mu}+\sqrt{\lambda}\tau_{i}u_{i}^{\mu}+\sqrt{\lambda}\nu_{i}n^{\mu}_{i}\right),\qquad
i=1,2,3,4.$ (74)

As $k_{i}^{\mu}$ is collinear to $p_{i}^{\mu}$, we have the scaling parameter
$\lambda\to 0$ while $\xi_{i},\kappa_{i},\tau_{i},\nu_{i}\sim 1$. In
particular, the large component varies over the whole range $0<\xi_{i}<1$.

The integral corresponding to figure 6 takes the form

$\displaystyle\displaystyle{\mathcal{I}_{G_{\bullet\bullet}}(s,t,\lambda
p_{i}^{2})=\int\frac{d^{D}k_{1}}{i\pi^{D/2}}\frac{d^{D}k_{2}}{i\pi^{D/2}}\frac{d^{D}k_{3}}{i\pi^{D/2}}\underbrace{\prod_{i=1}^{4}\frac{1}{(k_{i}^{2}+i\varepsilon)((p_{i}-k_{i})^{2}+i\varepsilon)}}_{\sim\lambda^{-8}},}$
(75)

where used the collinear scaling of the propagators in eq. (74) to determine
the scaling of integrand.

The integration measure over the three independent loop momenta can then be
written using the parameterisation of (74) as:

$\displaystyle\int
d^{D}k_{1}d^{D}k_{2}d^{D}k_{3}=Q^{3D}\int\prod_{i=1}^{3}d\xi_{i}d\kappa_{i}d\tau_{i}d\nu_{i}\,\nu_{i}^{D-4}d\Omega_{i}^{(D-3)}\,,$
(76)

where we introduced polar coordinates for the $D-3$ dimensional normal space
parameterised by the projections $\nu_{i}$ parallel to $n_{i}$ and the
differential solid angle $d\Omega_{i}^{(D-3)}$. In (76) we have not included
the $\lambda$ scaling. Doing so requires some care, as we now explain.

In traditional power counting for a given pinch surface, a key step is to
identify the _normal coordinates_ and the _internal coordinates_ LbyStm78 ;
Stm78I ; Stm95book . The former are by definition the ones that tend to zero
with the scaling parameter $\lambda$. As can be seen in the parameterisation
of the jet loop momenta (74), among all components, the only internal
coordinate is the one proportional to $v_{i}^{\mu}$, i.e. the large momentum
component governed by $\xi_{i}$; all the others are suppressed in the limit
$\lambda\to 0$, and are therefore normal coordinates. However, in the
Landshoff scattering picture we study here, as soon as the jet-scaling
condition is imposed, momentum conservation (73) implies that $\xi_{1}$,
$\xi_{2}$, and $\xi_{3}$ cannot anymore all be regarded as internal
coordinates simultaneously, since their magnitudes are constrained. Indeed,
inserting the parameterisation (74) into (73), it is direct to check that
$\xi_{i}$ for $i\neq 1$ can be expressed as

$\displaystyle\xi_{2}=\xi_{1}-\frac{1}{2}\sqrt{\lambda}\cos^{2}(\theta)\left(\tan\left(\frac{\theta}{2}\right)\Delta\tau-\cot\left(\frac{\theta}{2}\right)\Sigma\tau\right)+\lambda(\kappa_{2}-\kappa_{1}),$
(77a)
$\displaystyle\xi_{3}=\xi_{1}+\frac{1}{2}\sqrt{\lambda}\tan\left(\frac{\theta}{2}\right)\Delta\tau+\lambda(\kappa_{2}-\kappa_{4}),$
(77b)
$\displaystyle\xi_{4}=\xi_{1}-\frac{1}{2}\sqrt{\lambda}\cot\left(\frac{\theta}{2}\right)\Sigma\tau+\lambda(\kappa_{2}-\kappa_{3}),$
(77c)

where we defined

$\Delta\tau\equiv\tau_{1}+\tau_{2}-\tau_{3}-\tau_{4}\,,\qquad\quad\Sigma\tau=\tau_{1}+\tau_{2}+\tau_{3}+\tau_{4}.$

A first observation from eq. (77) is that all the $\xi_{i}$ are equal up to
$\mathcal{O}(\sqrt{\lambda})$ terms. Note that this statement is valid only
for generic $\theta$; for forward or backward scattering ($\theta\to 0$ or
$\pi$), it is invalidated. We will revisit this aspect in section 5.1.1, while
here we proceed assuming that $\theta$ is indeed generic.

The relations (77) also allow us to perform the change of integration
variables $\\{\xi_{2},\xi_{3}\\}\to\\{\kappa_{4},\tau_{4}\\}$, with the
Jacobian,

$\mathrm{det}\left(\frac{\partial(\xi_{2},\xi_{3})}{\partial(\kappa_{4},\tau_{4})}\right)=\lambda^{3/2}\cos(\theta)\cot(\theta)\,.$
(78)

Once this is done the only internal coordinate is $\xi_{1}$, while all the
other integration variables are normal coordinates. The scaling of the
integration measure in eq. (76) can now be obtained:

$\displaystyle\int\prod_{i=1}^{3}d\xi_{i}\
d\kappa_{i}d\tau_{i}d\nu_{i}\,\nu_{i}^{D-4}=\int_{0}^{1}d\xi_{1}\underbrace{\left(\int\prod_{i=1}^{3}\
(\lambda
d\kappa_{i})(\lambda^{\frac{1}{2}}d\tau_{i})(\lambda^{\frac{1}{2}-\epsilon}d\nu_{i}\,\nu_{i}^{-2\epsilon})\right)}_{\sim\lambda^{6-3\epsilon}}$
$\displaystyle\hskip 199.16928pt\cdot\int
d\kappa_{4}d\tau_{4}\,\,\underbrace{\mathrm{det}\left(\frac{\partial(\xi_{2},\xi_{3})}{\partial(\kappa_{4},\tau_{4})}\right)}_{\sim\lambda^{3/2}}\,.$
(79)

Eq. (79) fully captures the constraints due to momentum conservation, subject
to the assumption that $k_{i}$ are jet-like momenta admitting the scaling rule
of eq. (74).

Therefore, the corresponding degree of divergence follows from eqs. (75) and
(79):

$\displaystyle\mu[G_{\bullet\bullet},\text{Landshoff}]=6-3\epsilon+\frac{3}{2}-8=-\frac{1}{2}-3\epsilon,$
(80)

which coincides with the result in parameter space, eq. (69). Note that this
result regarding the asymptotic behaviour of figure 6(a) is also consistent
with the literature, e.g. Hld64 ; BottsStm89 .

###  4.4 Obtaining the complete set of regions via polytope dissection

In sections 4.1-4.3, we argued and provided evidence, in both parameter space
and momentum space, that $G_{\bullet\bullet}$ contains a Landshoff region,
which does not appear as a facet of the Newton polytope, i.e. it is a hidden
region. With the polytope dissection discussed in section 3.3 in hand, it is
now possible to systematically identify that such a region does exist and to
check for other potentially hidden regions using the new polytopes.

$\boldsymbol{v}_{\mathrm{R}}$ ($x_{0},x_{1},\ldots,x_{7};\lambda$) | degree of divergence
---|---
$(-2,-1,-2,-1,-2,-1,-2,-1;1)$ | $-6\epsilon$
$(-1,-2,-1,-2,-1,-2,-1,-2;1)$ | $-6\epsilon$
$(-1,-1,-1,0,-1,0,-1,0;1)$ | $1-3\epsilon$
$(-1,-1,0,-1,0,-1,0,-1;1)$ | $1-3\epsilon$
$(-1,-1,0,0,0,0,0,0;1)$ | $-\epsilon$
$(0,0,0,0,0,0,0,0;1)$ | $0$
Table 1: Regions obtained by directly applying the MoR to the on-shell
expansion of graph $G_{\bullet\bullet}$ with $p_{1}^{2}\sim\lambda Q^{2}$ and
$p_{i}^{2}=0\ (i=2,3,4)$.

Firstly, let us directly apply the MoR to $G_{\bullet\bullet}$. We consider
the case with a single leg, $p_{1}$, slightly off-shell, $p_{1}^{2}\sim\lambda
Q^{2}$, and all other legs strictly on-shell $p_{i}^{2}=0$ for $i=2,3,4$. The
MoR identifies the six regions shown in table 1. We observe that the Landshoff
region, having a uniform scaling vector (59), is not a facet of the original
integrand and is therefore absent.141414As explained in section 4.1, the
Landshoff region _cannot_ be characterised by a facet of the original
polytope, as it must admit the relations (61), in addition to the scaling of
$x_{i}$. According to the table, the leading power behaviour of the facet
regions of the integral in the small $\lambda$ limit is
$\mathcal{O}(\lambda^{0})$.

$\boldsymbol{v}_{\mathrm{R}}$ ($y_{0},x_{1},y_{2},x_{3},y_{4},x_{5},y_{6},x_{7};\lambda$) | $\boldsymbol{v}_{\mathrm{R}}$ ($x_{0},x_{1},\ldots,x_{7};\lambda$) | degree of
---|---|---
divergence
$(1/2,-1,1/2,-1,1/2,-1,0,-1;1)$ | $(-2,-2,-2,-2,-2,-2,-2,-2;2)$ | $-1/2-3\epsilon$
$(0,-1,1,-1,1,-1,0,-1;1)$ | $(-1,-1,-1,-1,-1,-1,-1,-1;1)$ | $-3\epsilon$
$(1,-1,1,-1,0,-1,0,-1;1)$ | $(-1,-1,-1,-1,-1,-1,-1,-1;1)$ | $-3\epsilon$
$(-1,-1,-1,-1,-1,-1,-1,-1;1)$ | $(-2,-1,-2,-1,-2,-1,-2,-1;1)$ | $-6\epsilon$
$(1,-2,1,-2,1,-2,1,-2;1)$ | $(-1,-2,-1,-2,-1,-2,-1,-2;1)$ | $-6\epsilon$
$(0,-1,0,0,0,0,0,0;1)$ | $(-1,-1,0,0,0,0,0,0;1)$ | $-\epsilon$
$(0,0,0,0,0,0,0,0;1)$ | $(0,0,0,0,0,0,0,0;1)$ | $0$
Table 2: On-shell expansion of $G_{\bullet\bullet}$, with
$p_{1}^{2}\sim\lambda Q^{2}$ and $p_{i}^{2}=0\ (i=2,3,4)$, first integral
dissection $\mathcal{I}_{1}$.

Next, we will dissect the polytope using the procedure described in section
3.3 and compare the regions we obtain from the new set of integrals. Examining
eq. (LABEL:eq:8_propagator_wideangle_Symanzik_polynomialsF), we note that
after applying the change of variables given in eq. (45) and dissecting the
integral according to eq. (47), we obtain a set of 24 new integrals,
$\mathcal{I}_{1},\ldots,\mathcal{I}_{24}$, one for each of the orderings of
$\\{y_{0},y_{2},y_{4},y_{6}\\}$. These integrals have $\mathcal{F}$
polynomials similar to eq. (48), but with additional terms proportional to
$(-p_{1}^{2})$ multiplied by non-negative polynomials of the parameters. Each
of the new integrals has a same-sign regime for some choice of the value of
$p_{1}^{2},s$ and $t$ (although not the same choice for all integrals). We
therefore expect that each integral separately can be analytically continued
from a same-sign regime to the region of interest, and the application of the
MoR should now truly identify all scaleful regions: there are no hidden
regions in the new integrals. In conclusion, we expect that all the regions of
the original integral, including hidden regions, would be obtained as facet
regions of the new polytopes.

Applying the MoR to each of the 24 new integrals, we obtain 24 sets of
regions. In table 2 we display the regions for the first integral dissection,
$\mathcal{I}_{1}$, in terms of both the new parameters,
$y_{0},x_{1},y_{2},x_{3},y_{4},x_{5},y_{6},x_{7}$, in which it has been
computed, and after translating back to the original parameters,
$x_{0},\ldots,x_{7}$. We find that the Landshoff region, which did not appear
as a facet of the Newton polytope of the original integrand, now appears as a
facet of the dissected integrand polytopes, and is detected geometrically
using the straightforward application of the MoR algorithm as described in
section 2.4.

We note in passing that there are some other interesting changes in the set of
region vectors upon replacing the original integral by the 24 new ones. In
particular, the region $(-1,-1,-1,0,-1,0,-1,0;1)$ in the original polytope
(see table 1), is absent in $\mathcal{I}_{1}$ but is present in other integral
dissections, e.g. $\mathcal{I}_{10}$. The region $(-1,-1,0,-1,0,-1,0,-1;1)$ in
the original polytope, is entirely absent in the integral dissections. Its
absence should not be a problem for recovering the correct asymptotic
expansion: importantly, non-analytic behaviour of the form
$\lambda^{-3\epsilon}$ is of course still present.

The Landshoff region is present in all 24 integral dissections, ${\cal
I}_{1},\ldots,{\cal I}_{24}$ and has a degree of divergence of
$\mu=-1/2-3\epsilon$, making it the leading region in the small $\lambda$
limit for $G_{\bullet\bullet}$. We observe that there is a many-to-one map
when going from the new variables to the old variables and, therefore, there
can be several regions in the new variables that have the same scaling in the
old variables. Comparing to the analysis presented in section 4.2, we note
that indeed the first region vector reported in table 2 is the Landshoff
region, compatible with the one predicted in eq. (59) (the factor of $2$ in
their normalisation is immaterial), and this region drives the behaviour of
the integral at small $\lambda$. Examining the scaling of the new parameters
for this region vector, they are each precisely as predicted in section 4.2,
and the degree of divergence corresponds to the case $a=0$ in the analysis
presented there. The second and third regions reported in the table also
correspond to region vectors with uniform scaling of the propagators in the
original variables. We find that they correspond to the case
$a=\pm\frac{1}{2}$ in section 4.2.

In summary, we have shown that the Landshoff region is the unique hidden
region in the on-shell expansion of $G_{\bullet\bullet}$ when a single leg is
taken slightly off-shell. Repeating the above analysis for the case where all
legs are taken slightly off-shell, $p_{i}^{2}\sim\lambda Q^{2}$ for
$i=1,\ldots,4$, we obtain a total of 73 facet regions by applying the MoR to
the original integral, and as expected, we observe that the Landshoff region
is missed. After dissecting the polytope and obtaining regions for each of the
24 new integrals, we find that only 71 regions are present. The Landshoff
region is present in each of the 24 new integral dissections. It is the
leading-power region in the small-$\lambda$ limit and has a degree of
divergence $\mu=-1/2-3\epsilon$, as expected.

###  4.5 Numerical evidence and discussion

The power-counting technique, which we have applied to the Landshoff
scattering region, can also be implemented to other facet regions of
$G_{\bullet\bullet}$. One can check that the degree of divergence
$\mu\geqslant 0$ for these regions, which implies that the infrared
singularities are at most logarithmic. In other words, the Landshoff region
encodes the entire power divergence of $G_{\bullet\bullet}$. Therefore, we
have a perfect example where the hidden region not only exists, but also plays
the most important role in this asymptotic expansion; without it even the
leading asymptotic behaviour would not be captured.

$G$$p_{1}$$p_{3}$$p_{2}$$p_{4}$

Figure 7: The generic picture for expanding one external momentum ($p_{1}$) in
the on-shell expansion, all others ($p_{2}$, $p_{3}$, and $p_{4}$) exactly on-
shell.

Below we show some numerical evidence for the statements above by considering
the kinematics that the external momenta $p_{2}$, $p_{3}$, and $p_{4}$ are put
exactly on shell, while $p_{1}^{2}\neq 0$ (see figure 7). By running pySecDec,
one can evaluate both the entire original integral
$\mathcal{I}(G_{\bullet\bullet})$, and the leading term in the power expansion
of the Landshoff scattering region, for a given $p_{1}^{2}$. By approaching
the limit $p_{1}^{2}\to 0$, their asymptotic behaviours can be compared.

We take $G=G_{\bullet\bullet}$ in figure 7, with results shown in figure 8.
The integral $\mathcal{I}(G_{\bullet\bullet})$, as a function of $p_{1}^{2}$,
is depicted by the blue (real part) and red (imaginary part) curves,
respectively. In contrast, the contribution from the Landshoff scattering
region is depicted by the black (real part) and yellow (imaginary part)
curves, respectively. The following observations are straightforward from the
plots. First, the contribution from the Landshoff scattering region exhibits a
power-like behaviour, which is a good approximation to the integral
$\mathcal{I}(G_{\bullet\bullet})$ at small $p_{1}^{2}$. Second, the imaginary
part of the Landshoff scattering region is negligible compared to the real
part, which implies that the leading contribution to the asymptotic expansion
is purely real when normalised as in eq. (3.3). Third, the imaginary part of
$\mathcal{I}(G_{\bullet\bullet})$, as well as the difference between the real
parts of $\mathcal{I}(G_{\bullet\bullet})$ and the Landshoff scattering
region, originate from the contributions of other regions, which are
subleading in the asymptotic expansion; they become important only as
$p_{1}^{2}$ increases.

Figure 8: Numerical evaluation of the ${\cal O}(\epsilon^{-3})$ contribution
to ${\cal I}(G_{\bullet\bullet})$ with one off-shell leg (leg 1), plotted as a
function of $p_{1}^{2}$, which is gradually taken small, down to
$p_{1}^{2}/s_{12}=10^{-5}$. The value of the integral ${\cal
I}(G_{\bullet\bullet})$ at ${\cal O}(\epsilon^{-3})$ is shown on a linear
vertical axis on the left panel and on a logarithmic vertical axis on the
right panel. The blue and red points correspond to the full integral, while
the black and yellow ones to the leading term in the asymptotic expansion of
the integral according to the MoR, i.e. the one scaling as $\lambda^{-1/2}$.

$p_{1}$$p_{3}$$p_{2}$$p_{4}$

(a) $G_{\bullet\bullet}$

$p_{1}$$p_{3}$$p_{2}$$p_{4}$

(b) $G_{\bullet s}$

$p_{1}$$p_{3}$$p_{2}$$p_{4}$

(c) $G_{\bullet t}$

$p_{1}$$p_{3}$$p_{2}$$p_{4}$

(d) $G_{\bullet u}$

$p_{1}$$p_{3}$$p_{2}$$p_{4}$

(e) $G_{ss}$

$p_{1}$$p_{3}$$p_{2}$$p_{4}$

(f) $G_{tt}$

$p_{1}$$p_{3}$$p_{2}$$p_{4}$

(g) $G_{uu}$

$p_{1}$$p_{3}$$p_{2}$$p_{4}$

(h) $G_{st}$

$p_{1}$$p_{3}$$p_{2}$$p_{4}$

(i) $G_{su}$

$p_{1}$$p_{3}$$p_{2}$$p_{4}$

(j) $G_{tu}$

Figure 9: The Landshoff scattering pictures corresponding to the graphs in
figure 2.

Having understood ${\cal I}(G_{\bullet\bullet})$ in momentum space as
Landshoff scattering, namely collinear split of each of the four jet momenta
subject to momentum conservation at the two hard vertices, it is readily clear
that the same (Landshoff) momentum configuration defines a region in each of
the remaining graphs. We describe these regions in figure 9 by colouring the
edges of the graphs, such that the four jets are coloured in green, lime,
olive, and teal respectively, while the hard subgraph is coloured in blue. It
is evident that each of these coloured graphs feature two disconnected hard
subgraphs (clearly, upon contraction of these hard subgraphs one recovers
$G_{\bullet\bullet}$).

We now argue that each Landshoff scattering region in figure 9 is the unique
hidden region of the corresponding graph of figure 2. Thus, figure 9 exhausts
in fact the entire set of hidden regions for massless $2\to 2$ scattering at
the three-loop level. To see this, we first observe that for any such region,
the superleading terms, each of which is $\mathcal{O}(\lambda^{-4})$, are
always identical to those of $G_{\bullet\bullet}$, namely, eq. (60). It then
allows us to employ the same change of variables shown in eq. (45) and divide
the integration range into the same sectors as eq. (47). The new 24 integrals
correspond to $\mathcal{F}$ polynomials that are identical to those in eq.
(48) up to $\mathcal{O}(\lambda^{-3})$ terms (and terms that are further
suppressed) involving the Feynman parameters of the propagators which do not
feature in $G_{\bullet\bullet}$. While these additional terms in $\mathcal{F}$
modify the integral, they do not affect the location of the pinch, and hence
the dissection of the polytope. Finally, it is straightforward to check that
all the corresponding new integrals are free from pinch singularities within
the integration domain.

As a result of dissecting the polytope, it is a natural possibility that
additional facet regions can appear that correspond neither to a hidden region
associated with cancellation between monomials in ${\cal F}$, nor to facets of
the original polytope (that is, after mapping the vectors obtained from facets
of the new polytope back into the original parameters). For example, the
regions of $G_{\bullet s}$, in the original parameters, are listed in table 3,