aainstitutetext: Instituto de Física Corpuscular, Universitat de València –
Consejo Superior de Investigaciones Científicas, Parc Científic, E-46980
Paterna, Valencia, Spain.bbinstitutetext: Facultad de Ciencias Físico-
Matemáticas, Universidad Autónoma de Sinaloa, Ciudad Universitaria, CP 80000
Culiacán, Mexico.ccinstitutetext: Facultad de Ciencias de la Tierra y el
Espacio, Universidad Autónoma de Sinaloa, Ciudad Universitaria, CP 80000
Culiacán, Mexico.ddinstitutetext: Dipartimento di Fisica, Università di Napoli
Federico II and INFN, Sezione di Napoli, I-80126 Napoli,
Italy.eeinstitutetext: Deutsches Elektronen-Synchrotron, DESY, Platanenallee
6, D–15738 Zeuthen, Germany.ffinstitutetext: Max-Planck-Institut für Physik,
Werner-Heisenberg-Institut, 80805 München, Germany.

# A stroll through the loop-tree duality

Jesús Aguilera-Verdugo a Félix Driencourt-Mangin b Roger J. Hernández-Pinto a
Judith Plenter c Renato Maria Prisco a,d Selomit Ramírez-Uribe a Andrés
Rentería-Olivo a Germán Rodrigo e,a German Sborlini f William J. Torres
Bobadilla c Francesco Tramontano<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

The Loop-Tree Duality (LTD) theorem is an innovative technique to deal with
multi-loop scattering amplitudes, leading to integrand-level representations
over an Euclidean space. In this article, we review the last developments
concerning this framework, focusing on the manifestly causal representation of
multi-loop Feynman integrals and scattering amplitudes, and the definition of
dual local counter-terms to cancel infrared singularities.

††preprint: IFIC/21-11, DESY 21-056, MPP-2021-65

##  1 Introduction

The most successful description of the microscopic structure of Nature is
currently given by the Standard Model (SM), which is based on quantum field
theories (QFT) with specific gauge symmetries. The strong interplay between
experiment and theory relies on confronting precise predictions with accurate
data. In the context of high-energy particle physics, the Large Hadron
Collider (LHC) and the planned future colliders Abada:2019lih ; Abada:2019zxq
; Benedikt:2018csr ; Abada:2019ono ; Blondel:2019vdq ; Bambade:2019fyw ;
Roloff:2018dqu ; CEPCStudyGroup:2018ghi  will keep on reducing the
experimental uncertainties, forcing to reach a compatible accuracy level from
the theory side. However, there are huge bottlenecks that prevent
straightforward calculations.

Due to their complexity, exact solutions of QFTs are not available for generic
scattering processes. Thus, extracting predictions from theory requires to use
approximations, whose applicability is restricted to specific situations.
Whilst lattice methods are reliable in the low-energy regime, perturbation
theory is the most suitable strategy to tackle the phenomenological
description of particle collisions at high-energies. Even if the perturbative
approach reduces the original calculation to a series expansion in the
interaction couplings, adding higher-orders is far from trivial due to the
presence of complicated phase-space and Feynman loop integrals. Moreover,
these objects feature numerical instabilities originated in threshold
configurations, as well as other singularities that cancel only after putting
all the contributions together.

In the last forty years, a huge effort has been done to develop new
methodologies for an efficient calculation of physical observables at higher-
orders. As a first step, proper regularization methods have been applied in
order to make explicit the singular structures of these objects. One of the
customary choices is Dimensional Regularization (DREG), which consists in
modifying the number of space-time dimensions to achieve integrability
tHooft:1972tcz ; Bollini:1972ui ; Cicuta:1972jf ; Ashmore:1972uj . However, in
the context of QFT, changing the number of space-time dimensions leads to
definition problems, such as the $\gamma_{5}$ Gnendiger:2017rfh ;
Bruque:2018bmy , and prevents a straightforward numerical implementation. For
this and other technical reasons, there is an ongoing effort in the high-
energy physics community to develop new strategies that locally regularise QFT
whilst keeping the standard four dimensions of the space-time Pittau:2012zd ;
Fazio:2014xea ; Mastrolia:2015maa ; Primo:2016omk ; Gnendiger:2017pys ;
TorresBobadilla:2020ekr .

Once the Feynman amplitudes and phase-space integrals are properly
regularised, we need to compute them and combine all the ingredients. On the
one hand, this requires to calculate multi-loop multi-leg Feynman integrals.
Several techniques are currently available for this purpose: Feynman
parametrisation, Mellin-Barnes transformations, IBP identities
Chetyrkin:1981qh ; Laporta:2001dd , sector decomposition Binoth:2000ps ;
Smirnov:2008py ; Carter:2010hi ; Borowka:2017idc , semi-numerical approaches
Francesco:2019yqt ; Bonciani:2019jyb ; Czakon:2008zk , among others
Heinrich:2020ybq . Very recently, algebraic geometry Larsen:2015ped ;
Bern:2017gdk ; Zeng:2017ipr ; Boehm:2017wjc ; Boehm:2018fpv ; Bendle:2019csk
was used to develop multi-loop integrand reduction algorithms Mastrolia:2011pr
; Badger:2012dp ; Zhang:2012ce ; Mastrolia:2012an ; Mastrolia:2012wf ;
Ita:2015tya ; Mastrolia:2016dhn ; Ossola:2006us , and to explore alternative
representations of Feynman integrals Baikov:1996rk ; Frellesvig:2017aai .
Also, unitarity based methods Peraro:2016wsq ; Badger:2017jhb ; Abreu:2017hqn
; Badger:2019djh  and intersection theory Mastrolia:2018uzb ;
Frellesvig:2019kgj ; Frellesvig:2019uqt ; Weinzierl:2020xyy  are being studied
to improve the reduction to master integrals and their subsequent evaluation.

On the other hand, loop contributions must be combined with real-emission
corrections and suitable counter-terms to cancel the infrared (IR) and
ultraviolet (UV) singularities, leading to a reliable prediction of the
physical cross-sections and differential distributions. There are several
strategies to achieve this cancellation, most of them based on the subtraction
methods Kunszt:1992tn ; Frixione:1995ms ; Catani:1996jh . Generally speaking,
these frameworks involve adding and subtracting counter-terms which locally
cancel the IR behaviour of the real part, but at the same time they are easy
enough to be integrated analytically and subtracted from the virtual
contribution. There are several variations of this approach: dipole
subtraction Catani:1996jh ; Catani:1996vz , $q_{T}$ subtraction/resummation
Catani:2000vq ; Catani:2013tia , antenna subtraction GehrmannDeRidder:2005cm ,
colorful subtraction Kardos:2016qol ; DelDuca:2016ily , among others. Also,
there are alternatives that rely on a different way to achieve the
cancellation of singularities, like the n-jetiness Gaunt:2015pea ;
Boughezal:2015dra  or the local analytic subtraction frameworks Magnea:2018hab
; Magnea:2019qfs .

Besides those methods, a novel strategy to tackle, simultaneously, the
efficient calculation of loop integrals and physical observables was
developed. This framework is based on the Loop-Tree Duality (LTD) theorem
Catani:2008xa ; Rodrigo:2008fp ; Bierenbaum:2010cy ; Bierenbaum:2012th , which
opens loops into trees and recasts virtual states into configurations that
resemble real-radiation processes. From the mathematical point of view, the
LTD transforms the integration domain of loop integrals into an Euclidean
space. In this way, a more intuitive understanding of the regions responsible
for the singular structure of the loop integrals emerges Buchta:2014dfa . This
knowledge can be used to explore novel techniques pointing towards more
efficient numerical implementations Buchta:2015xda ; Buchta:2015wna ,
integrand-level simplifications through asymptotic expansions Driencourt-
Mangin:2017gop ; Plenter:2019jyj ; Plenter:2020lop  and unveiling the
universal structures of scattering amplitudes at higher-orders Jurado:2017xut
; Driencourt-Mangin:2019aix ; Driencourt-Mangin:2019sfl ; Driencourt-
Mangin:2019yhu . Moreover, the LTD provides the perfect framework to handle
cross-section calculations in a fully unified way. Since the dual
representation of loop integrals is defined in Euclidean domains, the virtual
and real contributions can be directly combined at integrand-level to achieve
a fully local cancellation of IR singularities. This is the so-called _Four-
Dimensional Unsubtraction_ (FDU) approach Hernandez-Pinto:2015ysa ;
Sborlini:2016fcj ; Sborlini:2016gbr ; Sborlini:2016hat , that profits from a
local cancellation of singularities which makes it possible to bypass
additional regulators (such as DREG). Also, this formalism allow us to write
local UV counter-terms that exactly reproduce the expected results in
conventional renormalisation schemes Sborlini:2016gbr ; Sborlini:2016hat ;
Driencourt-Mangin:2019sfl ; Driencourt-Mangin:2019yhu , as well as fully local
IR dual counter-terms Prisco:2020kyb .

During the last two years, we have explored new features of the LTD approach.
In particular, we discovered that a more general analysis of the singular
structure of multi-loop multi-leg amplitudes is possible. By inquiring into
the physical and anomalous thresholds of one and two-loop amplitudes Aguilera-
Verdugo:2019kbz , we proved that it was possible to get rid of non-physical
singularities and retain only those contributions compatible with causality.
Several studies about the causal structure of scattering amplitudes are
available in the literature, using different techniques Cutkosky:1960sp ;
Tomboulis:2017rvd ; Runkel:2019yrs ; Runkel:2019zbm . In Ref. Verdugo:2020kzh
, we presented for the first time a manifestly causal integrand-level
representation inspired by the LTD theorem. This was applied to remove
unphysical threshold singularities and obtain causal integrand-level
representations of several topological families of multi-loop multi-leg
Feynman integrals Capatti:2019edf ; Capatti:2019ypt ; Verdugo:2020kzh ;
Ramirez-Uribe:2020hes ; Aguilera-Verdugo:2020kzc ; Aguilera-Verdugo:2020nrp ;
MANIFESTLYCAUSAL ; Capatti:2020ytd . Moreover, all-order causal formulae were
obtained using novel algebraic relations Bobadilla:2021rmu , leading to an
automatised framework to implement these calculations TorresBobadilla:2021dkq
. A purely geometrical interpretation Sborlini:2021owe  was also developed,
showing a complementary approach to tackle the causal structure of multi-loop
multi-leg scattering amplitudes and Feynman integrals.

The purpose of this review is to give a brief summary of the latest
developments regarding the LTD-based methods. To ease the presentation, it is
organised in two parts. In the first one, starting in Sec. 2, we settle the
notation of the LTD formalism. Special emphasis is put on the manifestly
causal representation of loop integrands in the Euclidean space, as well as
its derivation from Cauchy’s residue theorem. The basis for a rigorous
mathematical treatment of the nested residue strategy is given in Sec. 3.
Then, we discuss the advantages of the Euclidean representation of loop
integrands. In Sec. 4, we focus on the asymptotic expansion, whilst we discuss
how to obtain compact causal formulae for generic topological families of
multi-loop multi-leg Feynman integrals and scattering amplitudes in Secs. 5
and 6. In Sec. 7, a few words are given to explain novel developments
suggesting a deep connection among graph theory, algebra and the causal
structure of multi-loop scattering amplitudes.

After that, we start the second part of the review, where we center into the
computation of physical observables exploiting the LTD approach. In Sec. 8 we
discuss the basis of the FDU formalism, showing a physical examples. Then, in
Sec. 9, the construction of local counter-terms for cancelling IR
singularities is presented with more general examples, including
$\gamma^{*}\rightarrow 3\,{\rm jets}$ at NLO. Finally, we present the
conclusions and future research directions in Sec. 10.

##  2 Causality within the LTD formulation

Among the several advantages of the LTD formalism, we have recently explored
the manifestly causal representations. This constitutes a big step towards an
efficient numerical implementation of this approach, since spurious non-
physical and non-causal singularities are completely avoided. In the
following, we will introduce some useful notation and describe the basis of
the formalism. In particular, we will explain how the LTD computation through
the nested residue strategy leads to the causal representations of different
topological families to all-loop orders.

###  2.1 Dual scattering amplitudes

A generic $L$-loop scattering amplitude with $N$ external legs
$\\{p_{j}\\}_{N}$, and $n$ sets of internal lines, each set being defined by
the specific dependence on the loop momenta, is written as

$\mathcal{A}_{N}^{(L)}(1,\dots,n)=\int_{\ell_{1},\dots,\ell_{L}}\mathcal{A}_{F}^{(L)}(1,\dots,n)~{},$
(1)

with

$\mathcal{A}_{F}^{(L)}(1,\dots,n)=\mathcal{N}\big{(}\\{\ell_{s}\\}_{L},\\{p_{j}\\}_{N}\big{)}\times
G_{F}(1,\dots,n)~{}.$ (2)

This corresponds to an integral in the Minkoswki space of the $L$-loop
momenta, $\\{\ell_{s}\\}_{L}$, involving the product of Feynman propagators,
$G_{F}(q_{i})=(q_{i}^{2}-m_{i}^{2}+\imath 0)^{-1}$, and numerators,
$\mathcal{N}(\\{\ell_{s}\\}_{L},\\{p_{j}\\}_{N})$, given by the Feynman rules
of the theory considered. The $d$-dimensional integration measure in DREG
reads

$\int_{\ell_{s}}=-\imath\,\mu^{4-d}\int{\rm d}^{d}\ell_{s}/(2\pi)^{d}\,,$ (3)

and the usual Feynman propagator for a single particle is

$G_{F}(q_{i_{s}})=\frac{1}{q_{i_{s},0}^{2}-\left(q_{i_{s},0}^{(+)}\right)^{2}}\,,$
(4)

where

$q_{i_{s},0}^{(+)}=\sqrt{\mathbf{q}_{i_{s}}^{2}+m_{i_{s}}^{2}-\imath 0}~{},$
(5)

is the positive on-shell energy of the loop momentum $q_{i_{s}}$, written in
terms of its spatial components $\mathbf{q}_{i_{s}}$, its mass $m_{i_{s}}$,
and the Feynman’s infinitesimal imaginary prescription $\imath 0$. Then, we
introduce the shorthand notation

$G_{F}(1,\dots,n)=\prod_{{i_{s}}\in 1\cup\dots\cup
n}\Big{(}G_{F}\big{(}q_{i_{s}}\big{)}\Big{)}^{a_{i_{s}}}~{},$ (6)

to express the product of Feynman propagators of one set or the union of
several sets. In the former formula, $s$ represents the set of all internal
propagators with internal momenta $q_{i_{s}}=\ell_{s}+k_{i_{s}}$, that depend
on the loop momentum, $\ell_{s}$, or a specific linear combination of loop
momenta, and a combination of external momenta $k_{i_{s}}$, with $i_{s}\in s$
and $a_{i_{s}}$ arbitrary powers.

The LTD representation for scattering amplitudes is obtained by the iterative
application of the Cauchy’s residue theorem (CRT) integrating out one degree
of freedom for each loop momentum, and closing the Cauchy contour from below
the real axis, selecting the poles with negative imaginary part in the complex
plane of the loop momentum. This results in the modification of the
infinitesimal complex prescription of the Feynman propagators Catani:2008xa ;
Rodrigo:2008fp , that needs to be considered carefully in order to preserve
causality. Then, starting from Eq. (1), we set on-shell the propagators that
depend on the first loop momentum, $q_{i_{1}}$, and define

$\mathcal{A}_{D}^{(L)}(1;2,\dots,n)\equiv\sum_{i_{1}\in 1}{\rm
Res}\left(\mathcal{A}_{F}^{(L)}(1,\dots,n),\,{\rm
Im}(q_{i_{1},0}<0)\right)\,,$ (7)

where taking the residue is equivalent to integrating out the energy component
of the loop momenta. Now, we construct the nested residue iterating until the
$r$-th set as,

$\mathcal{A}_{D}^{(L)}(1,\dots,r\,;\,r+1,\dots,n)=\sum_{i_{s}\in s}{\rm
Res}\left(\mathcal{A}_{D}^{(L)}(1,\dots,r-1\,;\,r,\dots,n),\,\text{Im}(q_{i_{s},0}<0)\right)~{}.$
(8)

All sets before the semicolon contain one propagator that has been set on
shell and are linearly independent, while all the remaining propagators are
kept off-shell. Thus, this representation is equivalent to open the loop
amplitude to non-disjoint trees. Finally, the integration measure after
integrating out the energy component is modified according to

$\int_{\,\vec{\\!\ell}_{s}}\equiv-\mu^{d-4}\int\frac{{\rm
d}^{d-1}\ell_{s}}{(2\pi)^{d-1}}~{},$ (9)

transforming the $d$-dimensional Minkowski space into a $(d-1)$-dimensional
Euclidean one.

###  2.2 Multi-loop topologies through the LTD

By the properties obtained from the nested residues, Eq. (8), we are able to
construct multi-loop amplitudes from the LTD representation which are compact
and manifestly causal to all orders. The first topology under consideration is
called Maximal Loop Topology (MLT), and is characterised by $L+1$ momentum
sets where the momenta of the first $L$ sets depend on one single loop
momentum

$q_{i_{s}}=\ell_{s}+k_{i_{s}}\,,$ (10)

with $s\in\\{1,\dots,L\\}$. The momenta of the extra set, $L+1$, are given by
a linear combination of all the loop momenta, namely
$q_{i_{L+1}}=-\sum_{s=1}^{L}\ell_{s}+k_{i_{L+1}}$. Here, $k_{i_{s}}$ and
$k_{i_{L+1}}$ denote linear combinations of external momenta. The LTD
representation of the MLT topology, displayed in Fig. 1a, is presented in a
compact and symmetric form by evaluating the nested residue of Eq. (8), which
leads to

$\mathcal{A}_{\rm
MLT}^{(L)}(1,\dots,L+1)=\int_{\vec{\ell}_{1},\dots,\vec{\ell}_{L}}\sum_{i=1}^{L+1}\
\mathcal{A}_{D}^{(L)}(1,\dots,i-1,\overline{i+1},\dots,\overline{L+1}\,;\,i)~{},$
(11)

where the bars within the integrand indicate a reversal of momentum flow,
$q_{\bar{i}_{s}}=-q_{i_{s}}$, which is needed to preserve causality, and is
equivalent to select the on-shell modes with negative energy of the original
momentum flow. Each term of the sum in the integrand of Eq. (11) contains one
set $i$ with all its propagators off-shell, while the remaining $L$ sets
contain a single on-shell propagator each; a necessary condition to open
multi-loop amplitudes into disjoint trees. The dual representation of Eq. (11)
becomes singular when one or more off-shell propagators eventually become on-
shell, generating a disjoint tree dual sub-amplitude.

Figure 1: Graphical representations of Maximal Loop Topology (MLT), Next-to-
Maximal Loop Topology (NMLT) and Next-to-Next-to-Maximal Loop Topology
(N2MLT). External momenta are not shown.

It is worth mentioning that the LTD representation exhibits an interesting
structure when all the contributions are added together. For example, in the
case of a MLT configuration with one propagator in each loop set and one
incoming and outgoing momentum, we obtain

${\cal
A}_{\text{MLT}}^{\left(L\right)}\left(1,\dots,L+1\right)=-\int_{\vec{\ell}_{1},\dots,\vec{\ell}_{L}}\frac{1}{x_{L+1}}\left(\frac{1}{\lambda_{1}^{-}}+\frac{1}{\lambda_{1}^{+}}\right)~{},$
(12)

with

$\displaystyle x_{L+k}=2^{L+k}\prod_{i=1}^{L+k}q_{i,0}^{(+)}~{},\quad~{}{\rm
and}~{}\quad\lambda_{1}^{\pm}=\sum_{i=1}^{L+1}q_{i,0}^{(+)}\pm k_{0,L+1}~{}.$
(13)

This expression is free of unphysical singularities, and written in terms of
the so-called _causal propagators_ , $\lambda_{1}^{\pm}$. The causal
propagators encode all the possible physical singularities that might occur.
Both $\lambda_{1}^{\pm}$ in the integrand of Eq. (12) are associated to
physical thresholds, as shown in Fig. 2. Additionally, Eq. (12) is independent
of the initial momentum configuration in the Feynman representation.

Figure 2: Representation of the causal structure of MLT topologies.

Going a step further in the topology complexity, we consider a topology
containing one extra set of momenta that depends on the sum of two loop
momenta, $q_{i_{12}}=-(\ell_{1}+\ell_{2})+k_{i_{12}}$, denoted as 12. This
configuration, as can be appreciated in Fig. 1b, is called Next-to-Maximal
Loop Topology (NMLT), characterised by $L+2$ sets of propagators, with each
set categorised by the dependence on a specific loop momentum or a linear
combination of the $L$ independent loop momenta. The LTD representation for
the NMLT is given by the compact and factorised expression

$\displaystyle\mathcal{A}_{\rm NMLT}^{(L)}(1,\dots,L+2)$ $\displaystyle=$
$\displaystyle\mathcal{A}_{\rm MLT}^{(2)}(1,2,12)~{}\otimes~{}\mathcal{A}_{\rm
MLT}^{(L-2)}(3,\dots,L+1)$ (14) $\displaystyle+$
$\displaystyle\mathcal{A}_{\rm
MLT}^{(1)}(1,2)~{}\otimes~{}\mathcal{A}^{(0)}(12)~{}\otimes~{}\mathcal{A}_{\rm
MLT}^{(L-1)}(\overline{3},\dots,\overline{L+1})~{},$

involving convolutions of MLT subtopologies. The singular structure of the
factorised subtopologies determine the causal thresholds and infrared
singularities. For example, taking $L=3$, i.e. three loops, the convolutions
within the NMLT configuration are interpreted as

$\displaystyle\mathcal{A}_{\rm MLT}^{(2)}(1,2,12)\otimes\mathcal{A}_{\rm
MLT}^{(1)}(3,4)$
$\displaystyle=\int_{\vec{\ell}_{1},\vec{\ell}_{2},\vec{\ell}_{3}}\Big{(}\mathcal{A}_{D}^{(3)}(\overline{2},\overline{12},\overline{4};1,3)+\mathcal{A}_{D}^{(3)}(1,\overline{12},\overline{4};2,3)$
$\displaystyle+\mathcal{A}_{D}^{(3)}(1,2,\overline{4};12,3)+(\overline{4}\leftrightarrow
3)\Big{)}~{},$ (15)

and

$\displaystyle\mathcal{A}_{\rm
MLT}^{(1)}(1,2)\otimes\mathcal{A}^{(0)}(12)\otimes\mathcal{A}_{\rm
MLT}^{(2)}(\overline{3},\overline{4})$
$\displaystyle=\int_{\vec{\ell}_{1},\vec{\ell}_{2},\vec{\ell}_{3}}\Big{(}\mathcal{A}_{D}^{(3)}(\overline{2},\overline{3},\overline{4};1,3)$
$\displaystyle+\mathcal{A}_{D}^{(3)}(1,\overline{3},\overline{4};2,12)\Big{)}~{},$
(16)

where the sets after the semicolon are put off shell.

Following the previous procedure, we also presented in Ref. Verdugo:2020kzh
the LTD representation for the Next-to-Next-to-Maximal Loop Topology (N2MLT),
characterised by $L+3$ sets of propagators and depicted in Fig. 1c. Again,
factorised formulae for the dual representation are obtained, namely

$\displaystyle\mathcal{A}_{\rm N^{2}MLT}^{(L)}(1,\dots,L+3,12,23)$
$\displaystyle=\mathcal{A}_{\rm
NMLT}^{(L)}(1,2,3,12,23)\otimes\mathcal{A}_{\rm MLT}^{(L-3)}(4,\dots,L+1)$
$\displaystyle+\mathcal{A}_{\rm MLT}^{(2)}(1\cup 23,2,3\cup
12)\otimes\mathcal{A}_{\rm MLT}^{(L-2)}(\overline{4},\dots,\overline{L+1})\,,$
(17)

showing a recursive construction similar to the one reported for the NMLT
configurations. Expanding the first term on the rhs of Eq. (2.2) into its
corresponding subtopologies, we have

$\displaystyle\mathcal{A}_{\rm NMLT}^{(L)}(1,2,3,12,23)$
$\displaystyle=\mathcal{A}_{\rm MLT}^{(2)}(1,2,12)\otimes\mathcal{A}_{\rm
MLT}^{(3)}(3,23)$
$\displaystyle+\int_{\vec{\ell}_{1},\vec{\ell}_{2},\vec{\ell}_{3}}\Big{(}\mathcal{A}_{D}^{(3)}(1,\overline{3},\overline{23};2,12)+\mathcal{A}_{D}^{(3)}(\overline{12},3,23;1,2)\Big{)}\,,$
(18)

where the last two terms of the rhs in the integrand remain fixed by the
condition that the sets (2,3,23) cannot generate a disjoint subtree. We can
observe that the second term in Eq. (2.2), contain a two-loop subtopology
involving five sets of momenta grouped into three sets. Therefore, the
propagators in the set 1 and 23 cannot be simultaneously off shell.

Noteworthy, there are very compact explicit formulae for the NMLT and N2MLT
configurations which make use of the causal propagators. In Sec. 5, we will
enter into more details, and provide a nice conceptual interpretation of the
manifestly causal dual representation in terms of entangled thresholds.

##  3 Mathematical properties of the nested residues

Generic scattering amplitudes are defined by integrals of rational functions.
As already explained in Sec. 2, the multi-loop LTD framework is based on the
CRT and, in this section, the formal foundations for an $L$-loop Feynman
diagram and some of their immediate consequences are explained. As it was
discussed in Ref. Aguilera-Verdugo:2020nrp , iterated residues can be easily
computed taking into account the quadratic structure of Feynman propagators.
Let us start with the space $\mathbb{C}^{\left(\mathbb{R}^{L}\right)}$ of the
functions with domain $\mathbb{R}^{L}$ and co-domain $\mathbb{C}$. If the
natural inclusion of $\mathbb{R}$ into $\mathbb{C}$ is denoted by $i$, then
the iterated residues are defined as the recursive application of the functor
$-\mathrm{Res}\circ
i:\mathbb{C}^{\left(\mathbb{R}^{n}\right)}\rightarrow\mathbb{C}^{\left(\mathbb{R}^{n-1}\right)}$,
where $\mathrm{Res}$ represents the residue of the argument for all its
negative imaginary part poles and the minus sign appears in agreement with CRT
as the integration is performed clockwise.

For the computation of multi-loop Feynman integrals and scattering amplitudes,
internal momenta are given as linear combinations of external and loop
momenta. For instance, for the scalar sunrise diagram, the associated
integrand has the form

$\mathcal{I}(p)=\frac{1}{\left(q_{1,0}^{2}-q_{1,0}^{(+)2}\right)\left(q_{2,0}^{2}-q_{2,0}^{(+)2}\right)\left((q_{1,0}+q_{2,0}+k_{3,0})^{2}-q_{3,0}^{(+)2}\right)}.$
(19)

In this case, the first iteration of the iterated residues can be developed
with respect to the variable $q_{1,0}$. This means that we extend $q_{1,0}$ to
the complex plane, and as $q_{2,0}$ is still a real variable, the poles are
located in the complex plane as shown in Fig. 3, where the dashed line
represents the poles $\pm q_{3,0}^{(+)}-q_{2,0}-k_{3,0}$. As they depend on
$q_{2,0}$, they can be located at some point along each of the dashed lines.

Figure 3: Pole structure of a rational function of two variables.

Then, after the computation of the first iterated residues, we obtain

$\begin{split}\mathrm{Res}[\mathcal{I}(p),\\{q_{2,0},\mathrm{Im}(q_{2,0})<0\\}]&=\frac{1}{2q_{1,0}^{(+)}\left(q_{2,0}^{2}-q_{2,0}^{(+)2}\right)\left(\left(q_{1,0}^{(+)}+q_{2,0}+k_{3,0}\right)^{2}-q_{3,0}^{(+)2}\right)}\\\
&+\frac{1}{2q_{3,0}^{(+)}\left(\left(q_{3,0}^{(+)}-q_{2,0}-k_{3,0}\right)^{2}-q_{1,0}^{(+)2}\right)\left(q_{2,0}^{2}-q_{2,0}^{(+)2}\right)}\,.\end{split}$
(20)

The function obtained after the first residue iteration is given as a sum of
terms. The next step corresponds to the extension of the variable $q_{2,0}$ to
the complex plane. There, the pole structure of each term of Eq. (20) is quite
similar. This can be represented pictorially as shown in the Fig. 4, where the
pole structure of each term is presented as a complex plane term. The gray
blob in both terms represents the dependence of the imaginary part of the pole
$q_{3,0}^{(+)}-q_{2,0}^{(+)}-k_{3,0}$ on the three-momenta
$\boldsymbol{q}_{2}$ and $\boldsymbol{q}_{3}$, having no definite imaginary
part sign.

Figure 4: Pole structure of the first residue of a rational function of two
variables.

These poles having no definite imaginary part sign are, in general, called
_displaced poles_ , and they happen to cancel their contributions to the
iterated residues through the relation

$\begin{split}\mathrm{Res}&\left(\mathrm{Res}\left(F(q_{i,0},q_{j,0}),\left\\{q_{i,0},q_{i,0}^{(+)}+a_{i}\right\\}\right),\left\\{q_{j,0},q_{k,0}^{(+)}-q_{i,0}^{(+)}+a_{ij}-a_{i}\right\\}\right)\\\
=-&\mathrm{Res}\left(\mathrm{Res}\left(F(q_{i,0},q_{j,0}),\left\\{q_{i,0},q_{k,0}^{(+)}-q_{j,0}+a_{ij}\right\\}\right),\left\\{q_{j,0},q_{k,0}^{(+)}-q_{i,0}^{(+)}+a_{ij}-a_{i}\right\\}\right)\end{split}$
(21)

with

$F(q_{i,0},q_{j,0})=\frac{P(q_{i,0},q_{j,0})}{\left((q_{i,0}-a_{i})^{2}-q_{i,0}^{(+)2}\right)^{\gamma_{i}}\left((q_{i,0}+q_{j,0}-a_{ij})^{2}-q_{k,0}^{(+)2}\right)^{\gamma_{k}}}\,,$
(22)

for $a_{i}$ and $a_{ij}$ linear combinations of energy components of external
momenta. Due to this cancellation, displaced poles can be ignored in the
computation, leading to the concept of _nested residues_. Furthermore, the
final result of the nested residues is independent of the order of the
iteration, although the expressions are not identical term by term. A rigorous
proof of Eq. (21) is shown in Ref. Aguilera-Verdugo:2020nrp .

As a consequence of the nested residue strategy, we can infer a general
formula for the causal structure of the MLT diagram (see Fig. 1a). This is
directly reached applying partial fractions to each term in the nested
residue, leading to

$\begin{split}G_{F}(1,\ldots,L+1)\rightarrow-\frac{1}{\prod\limits_{i=1}^{L+1}2q_{i,0}^{(+)}}\left(\frac{1}{\sum\limits_{i=1}^{L+1}q_{i,0}^{(+)}+p_{0}}+\frac{1}{\sum\limits_{i=1}^{L+1}q_{i,0}^{(+)}-p_{0}}\right).\end{split}$
(23)

Additionally, this result can be used for the insertion of an MLT diagram
within a more general topology. In this manner, it is possible to consider any
MLT insertion as a single propagator whose on-shell energy is the sum of the
on-shell energies of all the internal propagators of the MLT.

Furthermore, with the computation of the nested residues, every family of a
given topological complexity $k$, Nk-1MLT, can be re-expressed as a linear
combination of convolutions of lower topological complexity families, as it is
for the NMLT and N2MLT, both defined by their amplitudes,

$\begin{split}\mathcal{A}_{\mathrm{NMLT}}^{(L)}(1,\ldots,L+2)&\equiv\int\limits_{\ell_{1},\ldots,\ell_{L}}\mathcal{N}(\\{\ell_{i}\\}_{L},\\{p_{j}\\}_{N})\times
G_{F}(1,\ldots,L+1,12),\\\
\mathcal{A}_{\mathrm{N}^{2}\mathrm{MLT}}^{(L)}(1,\ldots,L+3)&\equiv\int\limits_{\ell_{1},\ldots,\ell_{L}}\mathcal{N}(\\{\ell_{i}\\}_{L},\\{p_{j}\\}_{N})\times
G_{F}(1,\ldots,L+1,12,23).\end{split}$ (24)

This decomposition is shown in Figs. 5 and 6 for each of these topological
families.

Figure 5: Dual decomposition of NMLT$(L)$ in terms of loop configurations with
lower topological complexity. Figure 6: Dual expansion of a N2MLT$(L)$
diagram.

It is worth mentioning that the convolution symbol does not represent a pure
factorisation, as it implies the use of the on-shell conditions to express all
the off-shell momenta. The NMLT($L$) diagram decomposes into two terms with
two MLT diagrams. One of these terms contains a MLT($L-2$) in convolution with
a MLT(2), while the other term contains a convolution of a MLT($L-2$) diagram
with all its internal lines on-shell with a MLT(1) diagram and one off-shell
momentum. In the case of the N2MLT diagram, it is also decomposed into two
terms. One of these terms is a convolution of an NMLT(3) diagram with a
MLT($L-3$) diagram, and the other term is a convolution of one fully on-shell
MLT($L-3$) diagram with a MLT(3) diagram with two external momenta inserted in
the first and third internal set. These external momenta play a fundamental
role in splitting the internal sets into two propagators each. In this manner,
the following relations are fulfilled

$\displaystyle\mathrm{NMLT}(L)$
$\displaystyle=\mathrm{MLT}(2)\otimes\mathrm{MLT}(L-2)+G_{F}(L+2)\otimes\mathrm{MLT}(1)\otimes\mathrm{MLT}(L-2)\,,$
$\displaystyle\mathrm{N}^{2}\mathrm{MLT}(L)$
$\displaystyle=\mathrm{NMLT}(3)\otimes\mathrm{MLT}(L-3)+\mathrm{MLT}(2)\otimes\mathrm{MLT}(L-3)\,,$
(25)

which justifies the factorisation relations presented in Sec. 2.

The topological complexity of the subdiagrams in the convolutions is an
additive quantity whose sum coincides with the topological complexity of the
original diagram. Recalling that the family of Feynman diagrams with
topological complexity $k$ with $L$ loops is Nk-1MLT($L$), for the NMLT(L) has
$k_{\rm NMLT}=2$ and the term in the decomposition with no fully on-shell
diagrams is given as the convolution of MLT diagrams (with $k_{\rm MLT}=1$),
thus $2k_{\rm MLT}=k_{\rm NMLT}$. For the N2MLT($L$) it is given $k_{\rm
N^{2}MLT}=3$, while the term in its decomposition with no fully on-shell
diagrams is the convolution of an NMLT($L$) and a MLT($L$) diagram, hence,
$k_{\rm NMLT}+k_{\rm MLT}=k_{\rm N^{2}MLT}$.

These results suggest a factorisation formula for each topological complexity
family. Indeed, these formulae can be reached through the nested residues and
reveal a relation between one topological family with topologies with lower
complexity. Within such an approach, every topological family can be expanded
by factorising the MLT diagrams; thus, for any topological family, its causal
structure can be expressed following Eq. (23). We profited from these
structures to derive all-order formulae for N3MLT and N4MLT families, as
explained in Sec. 6.

##  4 Asymptotic expansions within LTD

The LTD formalism is straightforwardly applicable to obtain several
simplifications of the integrand-level representations of loop amplitudes,
such as asymptotic expansions. The interest in asymptotic expansions arises
from their potential to facilitate analytic results in specific kinematic
configurations, particularly when full analytic calculations are not (yet)
possible. An asymptotic expanded result can be of great interest since it
showcases the relevant behaviour of the full amplitude in the needed kinematic
limit. There are many observables where an analytic result is not necessary
for every set of kinematics and where specific limits are the window to test
potential discrepancies between experimental measurements and SM predictions,
thus identifying new physics contributions. Furthermore, in the context of the
local cancellation of IR singularities, expanded integrands could be very
convenient to reduce computation time. While maintaining the correct analytic
structure in the divergent limit and thus allowing for the combination with
the real-emission contributions, the less complicated form of the expanded
virtual contributions is expected to evaluate faster during the point-by-point
process of numerical integration.

Since after the application of LTD an amplitude is reduced to an integral over
Euclidean three-momenta, the size of the appearing scalar products can be
unambiguously be compared to external scales. This provides a good starting
point for the development of a well-defined formalism for asymptotic
expansions of the integrand. Specific asymptotic expansions in the context of
LTD have been presented for the first time for the process
$H\rightarrow\gamma\gamma$ at one loop Driencourt-Mangin:2017gop . Recently
advances in the generalization of the formalism have been published in Refs.
Plenter:2019jyj ; Plenter:2020lop .

Due to the fact that scattering amplitudes are determined by their analytic
properties general considerations for integrand-level expansions should start
with the propagators as the origin of divergences. The numerator, while
playing a role in the appearance of UVdivergences, is not relevant for the
discussion of asymptotic expansions since within LTD the singular UV behaviour
is neutralised through local renormalisation before integration.

We can reparametrise the dual propagators in the following form that is more
suitable for asymptotic expansions

$\tilde{\delta}\left(q_{i}\right)\,G_{D}\left(q_{i};q_{j}\right)=\frac{\tilde{\delta}\left(q_{i}\right)}{2q_{i}\cdot
k_{ji}+\Gamma_{ij}+\Delta_{ij}-\imath 0\eta\cdot k_{ji}}~{},$ (26)

where $\Gamma_{ij}+\Delta_{ij}=k_{ji}^{2}+m_{i}^{2}-m_{j}^{2}$. If
$\Gamma_{ij}+\Delta_{ij}$ vanishes, the dual propagator is not expanded.
Otherwise the starting point for the asymptotic expansion is to demand that
the condition

$|\Delta_{ij}|\ll|2q_{i}\cdot k_{ji}+\Gamma_{ij}|$ (27)

be fulfilled for the whole range of the loop integration space except for
potentially small regions around physical divergences. Note that since dual
propagators only appear in integrands where one loop momentum has been set on
shell, the condition has to be fulfilled in the Euclidean space of the loop
three-momentum. The dual propagator can then be expanded as

$G_{D}\left(q_{i};q_{j}\right)=\sum_{n=0}^{\infty}\frac{\left(-\Delta_{ij}\right)^{n}}{\left(2q_{i}\cdot
k_{ji}+\Gamma_{ij}-i0\eta\cdot k_{ji}\right)^{n+1}}~{}.$ (28)

Further simplifications arise whenever ${\bf k}_{ji}=0$. In that case, with
the change of variables $|\vec{q_{i}}|=m_{i}/2\,(x_{i}-x_{i}^{-1})$, the
denominator of the expanded dual propagator takes the easily integrable form

$2q_{i}\cdot k_{ji}+\Gamma_{ij}-\imath 0\,\eta\cdot
k_{ji}=Q_{i}^{2}\left(x_{i}+r_{ij}\right)\left(x_{i}^{-1}+r_{ij}\right)~{}.$
(29)

The intention to rewrite the denominator in this shape determines the
parameters $\Gamma_{ij}$ and $r_{ij}$ appearing in the expansion to be
restricted by the conditions

$\displaystyle\Gamma_{ij}-\imath 0\,\eta\cdot
k_{ji}=Q_{i}^{2}\left(1+r_{ij}^{2}\right)~{},$ (30) $\displaystyle
r_{ij}=\frac{m_{i}\,k_{ji,0}}{Q_{i}^{2}}-\frac{\imath 0\,\eta\cdot
k_{ji}}{Q_{i}^{2}}~{},$ (31)

assuming $|r_{ij}|\leq 1$. In the types of limits where one hard scale $Q$ is
available, $Q_{i}^{2}=\pm Q^{2}$ can be identified. The sign is determined by
the sign of the hard scale in the expression $k_{ji}^{2}+m_{i}^{2}-m_{j}^{2}$.
This type of expansion facilitates the analytical integration based on
integrals of the form

$\int_{1}^{\infty}\frac{\mathrm{d}x_{i}}{x_{i}(x_{i}+r_{ij})(x_{i}^{-1}+r_{ij})}=\frac{\log(r_{ij})}{r_{ij}^{2}-1},\qquad~{}|r_{ij}|<1~{}.$
(32)

In addition to the relations in Eq. (30) further conditions have to be
respected by the expansion parameters to ensure that the expansion converges
both at integrand- and at integral-level. Specifically, it is fundamental that
the analytic behaviour of the dual propagator is not fundamentally changed,
i.e. that for a propagator with a singularity the expansion also has to
display that singularity, while the expansion of a non-singular propagator is
to be finite throughout the whole integration domain. The infinitesimal
imaginary prescription of $r_{ij}$ given in Eq. (30) accounts properly for the
complex prescription of the original dual propagator and therefore its causal
thresholds. A different approach has to be taken for threshold limits where a
hard scale is not identifiable. In this case the pole position of the non-
expanded propagator can be expanded to identify the correct $r_{ij}$ for the
asymptotic expansion.

The formalism of expanding the dual propagator has been developed through its
application on the locally renormalised scalar two-point function

$\mathcal{A}^{(1,R)}=\int_{\ell}\Big{[}G_{F}(q_{1};M)\,G_{F}(q_{2};m)-\left(G_{F}(\ell;\mu_{\rm
UV})\right)^{2}\Big{]}~{}.$ (33)

Applying Eq. (28) on the appearing dual propagators $G_{D}(q_{1};\ell)$ and
$G_{D}(\ell;q_{1})$ leads to a simplified expression that can be integrated
without needing to specify what type of limit is considered, giving the very
general result

$\mathcal{A}^{(1,R)}=\frac{1}{16\pi^{2}}\sum_{i,j}\Bigg{[}2+c_{0,i}\mathrm{log}\left(\frac{\mu_{\rm
UV}}{m_{i}}\right)+\sum_{n=0}^{\infty}\left(c^{(n)}_{1,i}+c^{(n)}_{2,i}\mathrm{log}\left(r_{ij}\right)\right)\Bigg{]}~{}.$
(34)

The coefficients $c$ are simple functions of the appearing scales and are
given in Ref. Plenter:2020lop  just like the parameters needed for different
limits. The expansion converges well both at integrand- and at integral-level
in the limits of one large mass, a large external momentum as well as when
approaching the physical threshold both from below and from above. Comparison
with the established method of expansion by regions Beneke:1997zp ; Pak:2010pt
; Jantzen:2011nz ; Mishima:2018olh ; Semenova:2018cwy  has shown faster
convergence as well as emphasised the advantage of decreasing degree in UV
divergence with each order in the expansion. Renormalisation within our method
thus only involves the lowest orders of the integrand-level expansion. Higher
terms are optional for increasing precision and can be added straight-
forwardly without the need to ensure further UV cancellation.

Using the same approach we have found integrand-level expansions for the
scalar three-point function

${\cal A}_{3}^{(1)}=\int_{\ell}G_{F}(q_{1},q_{2},q_{3};M)~{},$ (35)

both for the limit of a large mass

${\cal A}_{3}^{(1)}(s_{12}\ll
M^{2})=\quad-\frac{1}{16\pi^{2}}\frac{1}{2M^{2}}\left(1+\frac{r}{12}+\frac{r^{2}}{90}\right)+{\cal
O}(r^{3})~{},$ (36)

as well as for a small mass

$\displaystyle{\cal A}_{3}^{(1)}(s_{12}\gg M^{2})=$
$\displaystyle\,\int_{\ell}\frac{\tilde{\delta}\left(\ell;M\right)s_{12}}{(2\ell\cdot
p_{12})(2\ell\cdot
p_{1})}\sum_{n=0}^{\infty}\Bigg{\\{}\frac{M^{2n}}{\left(-2\ell\cdot
p_{12}+\Gamma\right)^{n+1}}+\frac{M^{2n}}{\left(2\ell\cdot
p_{12}+\Gamma\right)^{n+1}}\Bigg{\\}}~{}.$

The Euclidean structure of the dual integrand, $a(\boldsymbol{\ell})$, can be
exploited into an even more direct way by applying Taylor expansions. Here it
is important to use different assumptions on the size of the loop three-
momentum in separate parts of the integration range. For the scalar two-point
function in the large-mass limit this amounts to calculating

$\mathcal{A}=\int_{0}^{\lambda}\mathrm{d}|\boldsymbol{\ell}|~{}\mathcal{T}a(M,\infty)+\int_{\lambda}^{\infty}\mathrm{d}|\boldsymbol{\ell}|~{}\mathcal{T}a(\\{\boldsymbol{\ell},M\\},\infty)~{}.$
(38)

The expanded integrand converges well in the full range of the loop momentum
as can be seen in Fig. 7. Details on this method and on how to fix the
matching scale lambda that separates the appearing so-called dual regions can
be found in Plenter:2020lop .

Figure 7: The convergence at integrand-level of the expansion given in Eq.
(38) for the values $M=10\,m$, $p^{2}=3\,m^{2}$, and $\mu_{\rm UV}=M$. More
details about this plot can be found in the original paper Plenter:2020lop .

With the calculations above being performed in the traditional LTD
representation we will finally show an example of using the manifestly causal
representation to expand a multi-loop integral. The MLT structure offers an
ideal starting point, being void of unphysical singularities. For the large
mass limit we find

${\cal A}^{(L)}_{\rm MLT}(p^{2}\ll
m_{s}^{2})=-2\sum_{n=0}^{\infty}(p^{2})^{n}\int_{\vec{\ell}_{1},\ldots,\vec{\ell}_{L}}\frac{\left(\lambda^{0}_{L+1}\right)^{-1-2n}}{x_{L+1}}~{},$
(39)

where $\lambda^{0}_{L+1}=\sum_{s=1}^{L+1}q_{s,0}^{(+)}$ and
$x_{L+1}=\prod_{s=1}^{L+1}2q_{s,0}^{(+)}$. Notice that due to the lack of
dependence on $p_{0}$ both in $x_{L+1}$ nor in $\lambda^{0}_{L+1}$ the
asymptotic integrals on the right-hand side of Eq. (39) are a function only of
the internal masses, $m_{s}$, to all loop orders. In this way, we are
optimistic about future applications of the manifestly causal LTD
representation from Refs. Verdugo:2020kzh ; Aguilera-Verdugo:2020kzc ;
Ramirez-Uribe:2020hes ; Aguilera-Verdugo:2020nrp  to speed up the calculation
of efficient and smooth asymptotic expansions.

##  5 Manifestly causal representation and numerical efficiency

As explained in Secs. 2 and 3, the application of the nested residue strategy
leads to manifestly causal integrand-level representations of any multi-loop
multi-leg Feynman amplitude. The main advantage of such a representation is
the absence of non-physical singularities, because only terms compatible with
causality remain in the final result.

Starting from the compact LTD representations of the NMLT and N2MLT multi-loop
topologies presented in Ref. Verdugo:2020kzh  in terms of nested residues, we
reconstruct their full analytic expression in term of causal propagators only.
This operation is performed by using numerical evaluation over finite fields
vonManteuffel:2014ixa ; Peraro:2016wsq , in which we use the `C++`
implementation of the FiniteFlow Peraro:2019svx  algorithm together with its
Mathematica interface. In the following Section, we show the explicit causal
formulae for NMLT and N2MLT families for an arbitrary number of loops, and
discuss their interpretation in terms of entangled causal thresholds.
Numerical examples for 4-loop diagrams are presented, in order to provide a
comparison with available results and test the efficiency of our approach.

###  5.1 Next-to-Maximal Loop Topology (NMLT)

At three-loops, the MLT family is not enough to describe the whole set of
possible topologies. Thus we need to consider the NMLT and N2MLT topologies,
whose general dual representations were explained in Sec. 2. To simplify the
presentation, we start by considering NMLT configurations with one single
propagator in each set and no external momenta. We need to add an additional
internal line, whose momentum is given by

$\displaystyle q_{L+2}=-\ell_{1}-\ell_{2}\,.$ (40)

A pictorial representation of NMLT is provided in Fig 1b. After computing the
nested residues and adding all the contributions together, we get

$\displaystyle{\cal
A}_{\text{NMLT}}^{\left(L\right)}\left(1,2,\ldots,L+2\right)=$
$\displaystyle\int_{\vec{\ell}_{1},\ldots,\vec{\ell}_{L}}\frac{2}{x_{L+2}}\left(\frac{1}{\lambda_{1}\lambda_{2}}+\frac{1}{\lambda_{2}\lambda_{3}}+\frac{1}{\lambda_{3}\lambda_{1}}\right)\,,$
(41)

where the causal propagators are given by

$\displaystyle\lambda_{1}=\sum_{i=1}^{L+1}q_{i,0}^{\left(+\right)}\,$
$\displaystyle\lambda_{2}=q_{1,0}^{\left(+\right)}+q_{2,0}^{\left(+\right)}+q_{L+2,0}^{\left(+\right)}\,,$
$\displaystyle\lambda_{3}=\sum_{i=3}^{L+2}q_{i,0}^{\left(+\right)}\,.$ (42)

This expression was reconstructed by using numerical evaluations over finite
fields, although partial fractioning leads to the same result within a similar
computing time. In the following, we shall note that simplifications are not
straightforward when dealing with more complicated topologies.

Figure 8: Entangled causal thresholds of the NMLT topology. Products of two
causal propagators are involved.

An interesting consequence of the compact form of Eq. (41) is that it allows a
re-interpretation in in terms of entangled causal thresholds. Each
$\lambda_{i}$ is associated to a causal threshold singularity, which might
take place when the momenta flows are oriented in the same direction. Thus,
the product of causal denominators represents a configuration in which two (or
more) sets of propagators can simultaneously go on shell. The factorisation of
NMLT (and more complicated topologies) as products of MLT configurations is
the reason behind this behaviour Verdugo:2020kzh . A graphical interpretation
of the entangled causal structure of NMLT vacuum diagrams is provided in Fig.
8, with the dashed lines indicating the propagators that can be set
simultaneously on shell.

###  5.2 Next-to-Next-to-Maximal Topology (N2MLT)

As mentioned before, reaching a full description of NkMLT with $k\leq 2$ is
enough to obtain the causal representation of up to three-loop scattering
amplitudes. The so-called N2MLT can be built recursively from the NMLT by
adding an additional line with momentum

$\displaystyle q_{L+3}=-\ell_{2}-\ell_{3}\,.$ (43)

The minimal example of such topology is the Mercedes-Benz diagram ($L=3$)
shown in Fig. 1c. For the sake of simplicity, we restrict here to the case
without external momenta. Using the LTD representation in Ref. Verdugo:2020kzh
, we can add together all the contributions and obtain

$\displaystyle{\cal
A}_{\text{N}^{2}\text{MLT}}^{\left(L\right)}\left(1,2,\ldots,L+3\right)$
$\displaystyle=\int_{\vec{\ell}_{1},\ldots,\vec{\ell}_{L}}\frac{1}{x_{L+3}}\frac{{\cal
N}(\\{q_{i,0}^{(+)}\\})}{\prod_{i=1}^{7}\lambda_{i}}\,,$ (44)

with $\lambda_{1}$ through $\lambda_{3}$ defined in Eq. (42),

$\displaystyle\lambda_{4}=q_{2,0}^{\left(+\right)}+q_{3,0}^{\left(+\right)}+q_{L+3,0}^{\left(+\right)}\,,$
$\displaystyle\lambda_{6}=q_{1,0}^{\left(+\right)}+q_{3,0}^{\left(+\right)}+q_{L+2,0}^{\left(+\right)}+q_{L+3,0}^{\left(+\right)}\,,$
$\displaystyle\lambda_{5}=q_{1,0}^{\left(+\right)}+q_{L+3,0}^{\left(+\right)}+\sum_{i=4}^{L+1}q_{i,0}^{\left(+\right)}\,,$
$\displaystyle\lambda_{7}=q_{2,0}^{\left(+\right)}+\sum_{i=4}^{L+3}q_{i,0}^{\left(+\right)}\,,$
(45)

and ${\cal N}(\\{q_{i,0}^{(+)}\\})$ a degree-four polynomial in
$q_{i,0}^{(+)}$. Compared to the NMLT case, the complexity of the polynomial
in the denominator makes it highly non-trivial to unveil a formula similar to
Eq. (41). Thus, we rely on the analytic reconstruction over finite fields to
obtain

$\displaystyle{\cal
A}_{\text{N}^{2}\text{MLT}}^{\left(L\right)}\left(1,2,\ldots,L+3\right)=$
$\displaystyle-\int_{\vec{\ell}_{1},\cdots,\vec{\ell}_{L}}\frac{2}{x_{L+3}}\Bigg{[}\frac{1}{\lambda_{1}}\left(\frac{1}{\lambda_{2}}+\frac{1}{\lambda_{3}}\right)\left(\frac{1}{\lambda_{4}}+\frac{1}{\lambda_{5}}\right)$
$\displaystyle+\frac{1}{\lambda_{6}}\left(\frac{1}{\lambda_{2}}+\frac{1}{\lambda_{4}}\right)\left(\frac{1}{\lambda_{3}}+\frac{1}{\lambda_{5}}\right)+\frac{1}{\lambda_{7}}\left(\frac{1}{\lambda_{2}}+\frac{1}{\lambda_{5}}\right)\left(\frac{1}{\lambda_{3}}+\frac{1}{\lambda_{4}}\right)\Bigg{]}\,.$
(46)

It is worth appreciating that the package Lotty was used to efficiently reach
these results TorresBobadilla:2021dkq . As in the case of NMLT topologies, it
is possible to interpret this result by using entangled thresholds. This time,
there are products of three causal propagators. Notice that not all the
combinations of causal propagators are allowed. This is because causal
propagators exhibit some compatibility issues, which can be explained by
digging into graph theory. More details about this issue and the connection
with Cutkosky’s rules can be found in Ref. Sborlini:2021owe .

Figure 9: Entangled causal thresholds of the N2MLT topology. Products of three
causal propagators are involved.

###  5.3 Adding external momenta and higher-powers

For realistic multi-loop scattering amplitudes, we need to take into account
external legs. The analytic reconstruction algorithm used to obtain compact
formulae for vacuum diagrams can be also applied to topologies with external
momenta. We want to highlight that the insertion of external momenta does not
affect the causal physical behaviour of these integrals: the difference w.r.t.
the vacuum case is that the entangled configurations are duplicated, according
to the direction of the energy flow for external particles. For instance, a
generic N2MLT with external legs inserted in the vertices is given by

$\displaystyle{\cal A}_{\text{N}^{2}\text{MLT}}^{\left(L\right)}\big{(}1,2$
$\displaystyle,\ldots,\left(L+1\right)_{-p_{13}},\left(L+2\right)_{p_{2}},\left(L+3\right)_{-p_{3}}\big{)}=-\int_{\vec{\ell}_{1},\cdots,\vec{\ell}_{L}}\frac{1}{x_{L+3}}$
$\displaystyle\times$
$\displaystyle\Bigg{[}\frac{1}{\lambda_{1}^{+}}\left(\frac{1}{\lambda_{2}^{-}}+\frac{1}{\lambda_{3}^{-}}\right)\left(\frac{1}{\lambda_{4}^{+}}+\frac{1}{\lambda_{5}^{+}}\right)+\frac{1}{\lambda_{6}^{+}}\left(\frac{1}{\lambda_{3}^{-}}+\frac{1}{\lambda_{5}^{-}}\right)\left(\frac{1}{\lambda_{2}^{+}}+\frac{1}{\lambda_{4}^{+}}\right)$
$\displaystyle+\frac{1}{\lambda_{7}^{+}}\left(\frac{1}{\lambda_{3}^{-}}+\frac{1}{\lambda_{4}^{-}}\right)\left(\frac{1}{\lambda_{2}^{+}}+\frac{1}{\lambda_{5}^{+}}\right)+\left(\lambda_{i}^{+}\leftrightarrow\lambda_{i}^{-}\right)\Bigg{]}\,.$
(47)

More details about the algorithms used to perform this computations can be
found in Refs. Aguilera-Verdugo:2020kzc ; TorresBobadilla:2021dkq .

Besides that, it was shown in Refs. Hernandez-Pinto:2015ysa ; Sborlini:2015uia
; Sborlini:2016gbr ; Sborlini:2016hat  that the presence of self-energy
insertions or generic scattering amplitudes, as well as some local UV counter-
terms, might require to consider propagators with higher-powers. As explained
in Refs. Aguilera-Verdugo:2020kzc ; Aguilera-Verdugo:2020nrp , the causal
structure of these amplitudes can be obtained by applying a differential
operator. Explicitly, we can raise the power of the propagators by taking
derivatives w.r.t. $q_{i,0}^{\left(+\right)}$. For instance,

$\displaystyle\left(G_{F}\left(q_{i}\right)\right)^{\alpha_{i}}=\frac{1}{\left(\alpha_{i}-1\right)!}\frac{\partial^{\alpha_{i}-1}}{\partial\left((q_{i,0}^{\left(+\right)})^{2}\right)^{\alpha_{i}-1}}\,G_{F}(q_{i})\,,$
(48)

which suggests the definition of the operator

$\displaystyle\frac{\partial}{\partial(q_{i,0}^{\left(+\right)})^{2}}\,\bullet$
$\displaystyle=\frac{1}{2q_{i,0}^{\left(+\right)}}\frac{\partial}{\partial(q_{i,0}^{\left(+\right)})}\,\bullet\,.$
(49)

The iterated application of this operator to the causal representation of
scattering amplitudes with single powers of the denominators leads to causal
representation of the corresponding multi-power amplitude, as carefully
explained in Ref. Aguilera-Verdugo:2020nrp .

Figure 10: Four-dimensional MLT, NMLT and N2MLT at four loops, as a function
of the internal masses $m_{4}^{2}$ and $m_{5}^{2}$. Whilst the solid lines
corresponds to the results obtained within the LTD formalism, the dots are the
numerical results obtained with Fiesta 4.2.

###  5.4 Numerical implementations

Finally, we would like to highlight that the causal representation of multi-
loop multi-leg Feynman integrals and scattering amplitudes leads to a very
smooth numerical integration. In Ref. Aguilera-Verdugo:2020kzc , we studied
several examples and compared the performance of the LTD-inspired approach
with other standard frameworks (such as Fiesta 4.2 and SecDec 3.0). We
considered the generic scalar amplitude

$\displaystyle{\cal
A}_{\text{N}^{k-1}\text{MLT}}^{\left(L\right)}\big{(}1^{2},2^{2},\ldots,L^{2}$
$\displaystyle,L+1,\dots,L+k\big{)}$
$\displaystyle\,=\prod_{i=1}^{L}\frac{\partial}{\partial(q_{i,0}^{\left(+\right)})^{2}}{\cal
A}_{\text{N}^{k-1}\text{MLT}}^{\left(L\right)}\left(1,2,\ldots,L+1,\ldots,L+k\right)\,,$
(50)

with $k=\\{0,1,2\\}$, at three and four loops, and changing the dimensionality
of the space-time from $d=2$ to $d=4$. Here, we applied the differential
operator detailed in Sec. 5.3 in order to raise the powers of the denominators
and achieve integrability in the UV region. In Fig. 10, we show a few examples
of Eq. (50) for $d=4$, doing a scan in $m_{4}^{2}$, by fixing $m_{5}^{2}$.

The inclusion of arbitrary masses does not introduce any additional
complication within the LTD-based framework. Also, only $(d-1)$ integrations
for each loop are needed, whilst any methodology based on the traditional
Feynman parametrisation approaches scales with the number of propagators.
Moreover, we explicitly checked that the absence of non-causal singularities
leads to a very efficient integration. The technology to perform these
calculations was included in the Wolfram Mathematica package Lotty, recently
published by one of the authors TorresBobadilla:2021dkq .

##  6 Universal opening at four-loops

The multi-loop topologies that appear for the first time at four loops are
characterised by multi-loop diagrams with $L+4$ and $L+5$ sets of propagators
which correspond to the next-to-next-to-next-to maximal loop topology (N3MLT)
and next-to-next-to-next-to-next-to maximal loop topology (N4MLT). In fact,
N4MLT naturally includes all Nk-1MLT configurations, with $k\leq 4$.

The N4MLT family is represented with three main topologies, which were checked
with `QGRAF` Nogueira:1991ex  and are shown in Fig. 11. Based in the
similarity of these topologies with the insertion of a four-point subamplitude
with trivalent vertices in a larger topology, a unified description is
proposed. The three N4MLT topologies are interpreted as the $t$-, $s$\- and
$u$-kinematic channels, respectively, of a universal topology.

Figure 11: Diagrams of the $\rm N^{4}MLT$ family. The diagram on the l.h.s.
corresponds to the $t$ channel, the diagram on the center is the $s$ channel
and the diagram on the r.h.s. corresponds to the $u$ channel. External
particles are not shown.

The three topologies contain $L+4$ common sets of propagators, and one extra
set which is different for each of them. Each of the first $L$ sets depends on
one characteristic loop momentum $\ell_{s}$, the remaining four common sets
are established as linear combinations of the loop momenta. The extra sets are
the distinctive key to each of the channels in the universal topology where
the momenta of their propagators is identified as different linear
combinations of $\ell_{2}$, $\ell_{3}$ and $\ell_{4}$.

To assemble the three N4MLT channels into a single topology, a current $J$ is
defined as the union of three sets that characterise each channel,

$J\equiv 23\cup 34\cup 24~{}.$ (51)

Due to momentum conservation, the three subsets do not contribute to the same
individual Feynman diagram but they all contribute at amplitude level. Based
on the described framework, the Feynman representation of the N4MLT universal
topology is expressed as

$\displaystyle\mathcal{A}_{\rm
N^{4}MLT}^{(L)}(1,\ldots,L+1,12,123,234,J)=\int_{\ell_{1},\dots,\ell_{L}}\mathcal{A}_{F}^{(L)}(1,\dots,L+1,12,123,234,J)~{}.$
(52)

Figure 12: Diagrammatic representation for the factorised opening of the
multiloop N4MLT universal topology. Only the on-shell cut of the last MLT-like
subtopology with reversed momentum flow is shown.

The dual opening of this topology fulfills a factorisation identity similar to
those presented in Ref. Verdugo:2020kzh  for NMLT and N2MLT

$\displaystyle{\cal A}^{(L)}_{\rm N^{4}MLT}$
$\displaystyle(1,\ldots,L+1,12,123,234,J)$ $\displaystyle={\cal A}^{(4)}_{\rm
N^{4}MLT}(1,2,3,4,12,123,234,J)\otimes{\cal A}_{\rm MLT}^{(L-4)}(5,\dots,L+1)$
$\displaystyle+{\cal A}^{(3)}_{\rm N^{2}MLT}(1\cup 234,2,3,4\cup
123,12,J)\otimes{\cal A}_{\rm
MLT}^{(L-3)}(\overline{5},\dots,\overline{L+1})~{},$ (53)

also valid regardless of the internal configuration. This factorisation
identity is depicted in Fig. 12 and is called the universal identity in view
of the fact that it is the only master expression required to open to
nondisjoint trees any scattering amplitude of up to four loops. It also
enables to infer the causal structure of the complete topology by exploring
the causal behaviour of its subtopologies.

The term $\mathcal{A}^{(4)}_{\rm N^{4}MLT}$ on the r.h.s. of Eq. (6) considers
all possible configurations with four on-shell propagators in the sets
$\\{1,2,3,4,12,123,234,J\\}$, while ${\cal A}^{(3)}_{\rm N^{2}MLT}$ in the
second term assumes three on-shell conditions. The terms ${\cal A}_{\rm
MLT}^{(L-4)}(5,\dots,L+1)$ and ${\cal A}_{\rm
MLT}^{(L-3)}(\overline{5},\dots,\overline{L+1})$ are computed according to
Ref. Verdugo:2020kzh ; Aguilera-Verdugo:2020nrp .

The four-loop subtopology in Eq. (6) is opened through a factorisation
identity which is written in terms of known subtopologies,

$\displaystyle\mathcal{A}_{\rm N^{4}MLT}^{(4)}(1,2,3,4,12,123,234,J)$
$\displaystyle=\mathcal{A}_{\rm{N^{2}MLT}}^{(4)}(1,2,3,4,12,123,234)\otimes\mathcal{A}^{(0)}(J)$
$\displaystyle+\sum_{{\bf s}\in
J}\mathcal{A}_{D}^{(4)}(1,2,3,4,12,123,234,{\bf s})\,.$ (54)

Exhibiting a similar structure to Eq. (6), the three-loop subtopology in Eq.
(6) is given by

$\displaystyle{\cal A}^{(3)}_{\rm N^{2}MLT}(1\cup 234,2,3,4\cup 123,12,J)$
$\displaystyle=\mathcal{A}_{\rm{NMLT}}^{(3)}(1\cup 234,2,3,4\cup
123,12)\otimes\mathcal{A}^{(0)}(J)$ $\displaystyle+\sum_{{\bf s}\in
J}\mathcal{A}_{D}^{(3)}(1,2,3,4,12,123,234,{\bf s})\,,$ (55)

where the bold symbol ${\bf s}$ is used to indicate that these contributions
are those containing on-shell propagators in the $J$-sets.

Figure 13: Causal configurations of the ${\rm N^{3}MLT}$ topology. The set
number $5$ accounts for all the propagators in the sets $5$ to $L+1$.

The first term on the r.h.s. of Eq. (6) and Eq. (6) consists of a four-loop
N2MLT and three-loop NMLT subtopology respectively. These terms are describing
dual trees where all the propagators with momenta in $J$ remain off shell. The
second term on the r.h.s. of Eq. (6) and Eq. (6) collects contributions that
characterise the $s$, $t$ or $u$ channel shown in Fig. 11. To obtain these
contributions a propagator in either set $23$, $34$ or $24$ are set on shell.

The explicit expressions for the $s$, $t$ and $u$ channel arising from Eq. (6)
and Eq. (6) are presented in Ref. Ramirez-Uribe:2020hes  where all the results
are consistent with the absence of disjoint trees. Repeated propagators from
self-energy insertions were treated as single propagators raised to specific
powers and are not considered to generate disjoint trees when the repeated
propagator is set on shell.

Notice that the number of trees in the LTD forest can also be computed through
the combinatorial exercise of selecting, from the full list of sets, all
possible subsets of $L$ elements that cannot generate disjoint trees.
Nevertheless, the momentum flows of the on-shell propagators can only be
determined through the nested residues. Also, it is important to mention that
the number of terms for a given Nk-1MLT topology in Eq. (6) scales with the
number of loops and linearly with the number of propagators per loop set, but
the sum over residues, equivalently over internal propagators, is implicitly
accounted in this expression.

###  6.1 Causal representations

To confirm the causal conjecture for the N4MLT family, the strategy proposed
in Ref. Aguilera-Verdugo:2020kzc  is applied to the multi-loop N3MLT, $t$, $s$
and $u$ channels. Each topology considers a configuration with one internal
propagator in each loop set, four external momenta for N3MLT and six external
particles for $t$, $s$ and $u$ channels.

As a first step, the LTD representation is obtained for each of the selected
topologies through the universal N4MLT expression in Eq. (6). After computing
the nested residues and adding them all together a causal expression is found.
The integrand of the causal dual representation reads in terms of on-shell
energies, the energy components of the linear combination and causal
denominators. Let us remember and emphasise that these causal denominators are
constructed from sums of on-shell energies exclusively, and they represent
potential singular configuration.

Figure 14: Extra causal configurations of the $t$-channel of the ${\rm
N^{4}MLT}$ topology.

The results given by the straightforward application of LTD leads directly to
a manifestly causal expression, however, the resulting numerator is a lengthy
polynomial in the on-shell and external energies Ramirez-Uribe:2020hes .
Therefore, a way to get a more suitable causal expression is to reinterpret it
in terms of a number of entangled thresholds equal to the difference between
the number of propagators and the number of loops by, e.g., analytical
reconstruction from numerical evaluation over finite fields
vonManteuffel:2014ixa ; Peraro:2016wsq ; Peraro:2019svx  as defined in Ref.
Aguilera-Verdugo:2020kzc .

The discussion was stated only for scalar integrals given that they fully
encode all the compatible causal combinations. In general, tensor reduction
commutes with LTD and can be used to deal with tensor integrals. Another
feature exploited is that external momenta attached to interaction vertices
that connect different loop sets do not alter the number of internal
propagators and therefore the complexity of the causal representation.
Therefore, the full causal expressions with external momenta can be deduced
from the causal representation of the vacuum configuration by matching the
momentum flows of the entangled thresholds.

Starting with the multi-loop N3MLT, there are thirteen causal denominators
which are depicted in Fig. 13. The analytically reconstruction was done by
matching all combinations of four thresholds that are causally compatible to
each other.

Going forward to the causal representation of the N4MLT family, we have to
consider all the entangled configurations with the presence of five causal
thresholds. The $t$ channel depends on the causal denominators already defined
for the N3MLT configuration and additional nine extra causal denominators that
depend on $q_{23,0}^{(+)}$ where the corresponding configurations are shown in
Fig. 14.

The $s$-channel can be obtained through the structure of the $t$-channel. The
causal denominators are obtained just by a clockwise rotation of the
$t$-channel and therefore by a permutation of the arguments of the causal
denominators that are channel specific. Similar to the $s$-channel, the
$u$-channel is obtained from the $t$-channel through the substitution
$23\rightarrow 24$ and by the exchange $3\leftrightarrow 4$ or
$2\leftrightarrow 234$ ($123$ remains invariant). There are, however, three
new configurations that arise because the $u$-channel is nonplanar. These new
configurations are shown in Fig. 15.

Figure 15: Extra causal configurations of the $u$-channel of the ${\rm
N^{4}MLT}$ topology due to nonplanarity.

One advantage of the causal representation is that the number of terms for a
given Nk-1MLT topology is independent of the number of loops in the causal
representation but requires to specify additional causal thresholds and
additional causal entanglements when more internal propagators are considered.
Furthermore, the most significant advantage of the causal representation with
respect to the LTD representation stems from the core difference between them,
the presence or absence of noncausal singularities. The straightforward
application of the nested residue generates multiple threshold singularities,
nevertheless, with a clever analytical rearrangement, the absence of noncausal
singularities is achieved and leads to a causal representation which is more
efficient and stable numerically in all the integration domain Ramirez-
Uribe:2020hes .

##  7 Novel developments on causality

The applications of the LTD-inspired methods have been spreading very fast in
the last years. In particular, it turned out to be an excellent tool to unveil
the structure of causal singularities of multi-loop multi-leg Feynman
integrals and scattering amplitudes. There have been very recent findings
regarding general all-loop formulae to describe any NkMLT amplitude, by using
clever algebraic relations among them Bobadilla:2021rmu . This computational
technology has been implemented in the package Lotty TorresBobadilla:2021dkq ,
which allows to automatically obtain the causal representation of multi-loop
multi-loop Feynman integrals and scattering amplitudes.

Figure 16: Example of two incompatible causal propagators with non-disjoint
sets of cusps (left) and incompatible momenta orientation (right).

On the other hand, investigations inquiring on the geometrical aspects of
multi-loop diagrams were performed. In Ref. Sborlini:2021owe , we used
concepts from graph theory to describe all the possible causal propagators and
the allowed entangled thresholds associated to a given diagram. Describing
diagrams in terms of cusps (i.e. interaction vertices) and edges (i.e. sets of
propagators connecting two fixed vertices) was also proposed in Ref.
Bobadilla:2021rmu . This description leads to the _cusp matrix_ , a
geometrical object that allows one to completely characterise the causal
structure of a given diagram. Explicitly, we realised that the causal
propagators are associated to all the possible connected binary partitions of
cusps. This has a very nice connection to a generalised geometrical
interpretation of the Cutkosky’s rules Cutkosky:1960sp . Also, we implemented
a computational algorithm that benefits from graph theory tools to efficiently
obtain all the causal propagators. We tested it against the direct nested
residue calculation and subsequent identification of $\lambda$’s: the
geometric algorithm was, at least, a factor ten faster.

The information contained in the cusp matrix allow us to exactly reconstruct
all the possible entangled thresholds by imposing geometrical selection
criteria. These can be summarised in the following list:

  1. 1.

_All the lines are cut_ : Each possible entangled combination of causal
denominators must involve the on-shell energies of all the propagators.

  2. 2.

_Absence of crossings_ : Causal denominators are associated to connected
partitions of vertices. Only those partitions involving disjoint sets of
vertices, or those that are strictly included, can be successfully entangled.
Alternatively, if we associate a line connecting the cut edges to each
$\lambda$, then two $\lambda$’s can be entangled if their lines do no cross.

  3. 3.

_Consistent flux orientation_ : A set of causal propagators is compatible (or
can be entangled) if the associated binary partitions can consistently be
oriented, i.e. if all the internal lines contribute to the entangled cuts with
the same orientation.

  4. 4.

_Causal propagator orientation_ : When external particles are present, the
relative signs among aligned on-shell energies (i.e. $\sum q_{i,0}^{(+)}$) and
the energy component of the external momenta is determined by the _orientation
matrix_ , which can be built from the cusp matrix.

In Fig. 16, we show examples of causal propagators which can not be entangled
for the Mercedes-Benz diagram (i.e. a particular case of N2MLT or a four-cusp
topology). On the left, the criteria 2 is not fulfilled, since the associated
partitions of cusps are not disjoint. On the right, we consider three causal
propagators which can not be consistently oriented (criteria 3): the edge
connecting the cusps 1 and 3 does not allow a compatible orientation.

Thus, given a diagram, determining its cusp matrix and imposing the selection
criteria 1-4, we can obtain a formula describing its causal structure. In
accordance with the results presented in Ref. Bobadilla:2021rmu , we found
that

$\displaystyle{\cal
A}_{N}^{\left(L\right)}(1,\ldots,L+k)=\sum_{\sigma\in\Sigma}\,\int_{\vec{\ell}_{1},\cdots,\vec{\ell}_{L}}\frac{{\cal
N}_{\sigma}(\\{q_{r,0}^{(+)}\\},\\{p_{j,0}\\})}{x_{L+k}}\,\times\prod_{i=1}^{k}\frac{1}{-\lambda_{\sigma(i)}}\,+\,(\sigma\leftrightarrow\bar{\sigma})\,,$
(56)

describes the causal structure of any multi-loop multi-leg Feynman diagram or
collection of Feynman diagrams with topological complexity $k-1$. The set
$\Sigma$ contains all the subsets of products of $k$ causal propagators
$\lambda_{i}^{\pm}$ fulfilling the selection criteria 1-4, and ${\cal
N}_{\sigma}$ is given by the application of an operator depending on the
subset of $\sigma$. Thus, this result confirms that multi-loop multi-leg
amplitudes can be purely described in terms of cusps and edges, regardless the
number of loops or external lines.

##  8 Four-Dimensional Unsubtration (FDU)

The standard computation of accurate predictions at Next-to-Leading Order
(NLO) in QFT requires to deal with non trivial integrands. These integrands
carry multiple singularities due to the low (IR) and high (UV) energy regimes.
The cancellation of those divergent quantities, typically occurs at integral-
level. By considering the Kinoshita-Lee-Nauenberg (KLN) theorem
Kinoshita:1962ur ; Lee:1964is , soft and collinear divergences (IR
singularities) are removed, and by adding suitable ultraviolet counter-terms,
UV singularities are renormalised, rendering the cross section finite. This
algorithm for cancelling singularities can be extended to Next-to-Next-to-
Leading order (NNLO) and beyond; however, since the complexity of the
integrands is higher at higher orders, this procedure is reaching a bottleneck
because cancellation of singularities must be achieved at integral level,
i.e., the $\epsilon\rightarrow 0$ limit is taken once the cross-section is
known in DREG. Since LTD transform loop integrals in phase-space integrals,
among other properties discussed in this document, the cancellation of
singularities could occurs at integrand level. Hence, based on the LTD
theorem, the Four Dimensional Unsubtraction Hernandez-Pinto:2015ysa ;
Sborlini:2015uia ; Sborlini:2016fcj ; Sborlini:2016gbr ; Sborlini:2016hat
(FDU) method presents a new paradigm to compute observables in four
dimensions, since the definition of cross-section is free of IR and UV
singularities. The first application of the LTD at cross-section level was
done in a toy model based on the simplest $\phi^{3}$ theory. In Ref.
Hernandez-Pinto:2015ysa  it was shown for the first time that LTD is powerful
to build physical observables in four dimensions by a proper mapping between
real and virtual kinematical variables.

###  8.1 Local cancellation of infrared singularities within FDU

Let us start the discussion of the FDU approach by analysing the cancellation
of soft and collinear divergences. The simplest scenario is the decay process
$1\rightarrow n$ where the Born level cross-section is given by,

$\displaystyle\sigma^{(0)}=\int{\rm dPS}^{1\rightarrow
n}\,|\mathcal{M}_{n}^{(0)}|^{2}\,\mathcal{S}_{0}(\\{p_{i}\\})\,,$ (57)

with $|\mathcal{M}_{n}^{(0)}|^{2}$ the LO contribution and $\mathcal{S}_{0}$
is the IR-safe measure function. The virtual correction to the LO cross-
section is computed as

$\displaystyle\sigma^{(1)}_{V}=\int{\rm dPS}^{1\rightarrow n}\int_{\ell}2{\rm
Re}\,\langle\mathcal{M}_{n}^{(0)}|\mathcal{M}_{n}^{(1)}\rangle\,\mathcal{S}_{0}(\\{p_{i}\\})\,,$
(58)

where $\langle\mathcal{M}_{n}^{(0)}|\mathcal{M}_{n}^{(1)}\rangle$ represents
the interference between the Born level and one-loop amplitudes. It is
important to emphasise at this point that FDU requires also to maintain self-
energy contributions because they contain both IR and UV divergences that will
contribute to the full cancellation of singularities. The IR singularities
from the virtual component are cancelled against real radiation contributions.

For the sake of simplicity, let us consider that the process under study only
has final-state radiation singularities. Hence, the real contribution given by

$\displaystyle\sigma^{(1)}_{R}=\int{\rm dPS}^{1\rightarrow
n+1}\,|\mathcal{M}_{n+1}^{(0)}|^{2}\,\mathcal{S}_{1}(\\{p^{\prime}_{i}\\})\,,$
(59)

contains all Feynman diagrams with one extra particle in the final state in
$\mathcal{M}_{n+1}^{(0)}$ and $\mathcal{S}_{1}$ is the measure function for
$n+1$ particles. It is important to recall at this moment that primed
variables are used to describe the momenta of real matrix elements while
unprimed variables are labelling Born and loop level momenta.

The FDU starts with the analysis of the virtual cross-section which can be
decomposed as a sum of dual contributions as,

$\displaystyle\sigma^{(1)}_{V}$ $\displaystyle=\int{\rm dPS}^{1\rightarrow
n}\,\sum_{i=1}^{N}\int_{\vec{\ell}}I_{i}(q_{i})\,\mathcal{S}_{0}(\\{p_{i}\\})$
$\displaystyle\equiv\int{\rm dPS}^{1\rightarrow
n}\,\sum_{i=1}^{N}\sigma_{D,i}^{(1)}\,,$ (60)

with $I_{i}(q_{i})$ the $N$ dual integrals that arise from the direct
application of the LTD theorem. We notice that each $I_{i}(q_{i})$ has set one
internal particle on shell and it is characterised by $q_{i}$. Then, the loop
integral has been transformed to phase-space integrals, therefore each dual
integral behaves as the contribution of one real particle emitted. In order to
achieve the complete cancellation of IR singularities, each
$\sigma_{D,i}^{(1)}$ must be paired, at integrand level with a corresponding
real radiation twin component. The real radiation is sliced as,

$\displaystyle\sigma^{(1)}_{R,i}=\int{\rm dPS}^{1\rightarrow
n+1}\,d\sigma_{R}^{(1)}\,\mathcal{R}_{i}\,,$ (61)

where $\mathcal{R}_{i}$ represents a partition of the full integral, such that
$\sum_{i}\sigma_{R,i}^{(1)}=\sigma_{R}^{(1)}$ and considering that only one IR
divergent configuration is allowed in each $\mathcal{R}_{i}$. Finally, in
order to merge both integrals, a mapping between the kinematical variables is
implemented. On the one hand, each partition in $\sigma_{D,i}^{(1)}$ contains
$n$ external momenta plus one extra integration variable corresponding to
$q_{i}$, therefore, each $\sigma_{D,i}^{(1)}$ is mapped into its real sliced
contribution as,

$\displaystyle\mathcal{T}_{i}(\\{p_{1},\dots,p_{n},q_{i}\\})\rightarrow\\{p^{\prime}_{1},\dots,p^{\prime}_{n+1}\\}\,,$
(62)

where $\mathcal{T}_{i}$ is a bijective transformation among real and virtual
variables. The momentum mapping between variables is analogous to the one used
by the dipole method Catani:1996jh ; Catani:1996vz , since the singularities
are associated to soft emissions or the double collinear limit Kosower:1999rx
; Sborlini:2013jba . These singular IR configurations are associated to
specific contributions in the virtual part, by selecting two massless partons
per partition. On one hand the spectator and on the other hand the emitter.
Hence, the four momenta of the emitter and spectator in addition with the loop
three-momentum are used to reconstruct the kinematical phase-space of the real
emission cross-section where there is a similar configuration, i.e., an
emitter decaying into two partons in a soft or collinear regime. Explicitly,
if $p_{i}$ is the momentum of the final-state emitter, $q_{i}$ the internal
on-shell momentum prior the emitter and $p_{j}$ is the momentum of the final-
state spectator, we apply the momentum mapping

$\displaystyle p_{r}^{\prime\mu}$ $\displaystyle=q_{i}^{\mu}\,,$
$\displaystyle p_{i}^{\prime\mu}$
$\displaystyle=p_{i}^{\mu}-q_{i}^{\mu}+\alpha_{i}\,p_{j}^{\mu}\,,$
$\displaystyle p_{j}^{\prime\mu}$
$\displaystyle=(1-\alpha_{i})\,p_{j}^{\mu}\,,$ $\displaystyle
p_{k}^{\prime\mu}$ $\displaystyle=\,p_{k}^{\mu}\,,\qquad\qquad\qquad k\neq
i,j\,,$ (63)

with $p_{r}^{\prime}$ the momentum of the extra radiation of the process and,
in the massless scenario,
$\alpha_{i}=(q_{i}-p_{i})^{2}/(2p_{j}\cdot(q_{i}-p_{i}))$. Furthermore, this
mapping preserves momentum conservation since $p_{i}+p_{j}+\sum_{k\neq
i,j}p_{k}=p_{i}^{\prime}+p_{j}^{\prime}+p_{r}^{\prime}+\sum_{k\neq
i,j}p^{\prime}_{k}$ is fulfilled. A similar mapping is used when considering
massive particles and it was shown in Ref Sborlini:2016hat . The extension of
this formalism to higher-orders (i.e. NNLO and beyond) will require additional
kinematical transformations which take care of the singular behaviour of
scattering amplitudes in the multiple-collinear limit Catani:1999ss ;
Catani:2003vu ; Sborlini:2014eib ; Sborlini:2014mpa ; Sborlini:2014kla ;
DelDuca:2019ggv ; DelDuca:2020vst .

###  8.2 Self-energy insertions and renormalisation

Before moving forward with IR cancellation, let us review the local
renormalisation of UV singularities. The standard cancellation of UV
singularities requires the renormalisation of field wave-functions and
couplings. In the FDU formalism, this feature is obtained through the
construction of local UV counter-terms. At one-loop, the scenario where
massless and massive particles are propagating in the loop has been studied
Sborlini:2016gbr ; Sborlini:2016hat . Remarkably, it was shown that a smooth
transition between massless and massive renormalisation constants takes place
and UV singularities are well understood. Let us highlight a crucial
difference between the standard renormalisation constant in DREG and LTD.
Wave-function renormalisation constants are obtained from self-energy
diagrams. In particular, massless bubble diagrams in DREG are neglected in the
renormalisation procedure since IR and UV divergences are considered as equal
and they cancel out. However, the same divergent poles in the FDU formalism
contribute separately. Specifically, it means that IR singularities of loop
diagrams vanish with only the IR poles of the real radiation and there are no
mixed cancellation between UV and IR poles. Therefore, the remaining UV
divergences must be removed when renormalisation is implemented in the FDU
scheme. Hence, we stress that integrands in the FDU are separated into the IR
and UV domains, and this identification is crucial to render cross-sections
free of singularities in the FDU formalism.

We analyze the construction of the renormalisation constants in the FDU
framework. By using standard Feynman rules, the unintegrated massive wave-
function renormalisation constant, in the Feynman gauge, at one-loop is given
by,

$\displaystyle\Delta Z_{2}(p_{1};M)=-g_{\mathrm{S}}^{2}\,C_{\rm
F}\,\int_{\ell}G_{F}(q_{1})\,G_{F}(q_{3})\,\left[(d-2)\frac{q_{1}\cdot
p_{2}}{p_{1}\cdot p_{2}}\right.+4\,M^{2}\left.\left(1-\frac{q_{1}\cdot
p_{2}}{p_{1}\cdot p_{2}}\right)G_{F}(q_{3})\right]\,.$

Eq. (8.2) is the most general wave-function renormalisation constant since it
includes the massless and massive case and, as it was previously mentioned,
the transition to the massless case, $\Delta Z_{2}(p,0)$, is straightforward.
The function $\Delta Z_{2}(p_{1};M)$ contains singularities associated to the
UV domain, therefore we must find the UV component of Eq. (8.2), $\Delta
Z_{2}^{\rm UV}$, and subtract it in order to find a UV-free wave-function
renormalisation constant, $\Delta Z_{2}^{\rm IR}$. The UV part is extracted by
performing an expansion of the integrand around the UV propagator
$G_{F}(q_{\rm UV})=(q^{2}_{\rm UV}-\mu_{\rm UV}^{2}+\imath 0)^{-1}$. In
particular, for Eq. (8.2), it is found

$\displaystyle\Delta Z_{2}^{\rm UV}(p_{1})$
$\displaystyle=(2-d)\,g_{\mathrm{S}}^{2}\,C_{\rm
F}\,\int_{\ell}\big{[}G_{F}(q_{\text{UV}})\big{]}^{2}\,\left(1+\frac{q_{\text{UV}}\cdot
p_{2}}{p_{1}\cdot
p_{2}}\right)\Big{[}1\\!-\\!G_{F}(q_{\text{UV}})(2\,q_{\text{UV}}\cdot
p_{1}+\mu^{2}_{\text{UV}})\Big{]}\,.$

Hence, we define $\Delta Z_{2}^{\rm IR}$ as

$\displaystyle\Delta Z_{2}^{\text{IR}}=\Delta Z_{2}-\Delta
Z_{2}^{\text{UV}}\,,$ (66)

which is free of UV singularities and the IR singularities poles are those
needed to remove the remaining singularities from the virtual and real mapping
of the cross-section.

To remove UV singularities it is important to build proper UV counter-terms
that can be extracted by the direct study of the UV properties of physical
amplitudes. After the combination of real and virtual matrix elements and
renormalisation, the remaining amplitude has only UV singularities, in the
following, we consider generic amplitudes with only UV singularities. At two
loops, a generic two loop amplitude can be written as,

$\displaystyle\mathcal{A}^{(2)}=\int_{\ell_{1}}\int_{\ell_{2}}\mathcal{I}(\ell_{1},\ell_{2})\,,$
(67)

where the integrand is a function of the loop variables $\ell_{1}$ and
$\ell_{2}$. UV divergences shall appear when the three-momenta
$|\vec{\ell}_{1}|$ and $|\vec{\ell}_{2}|$ tend to infinity. In the two-loop
scenario, there are three possible UV limits: i)
$|\vec{\ell}_{1}|\rightarrow\infty$ and $|\vec{\ell}_{2}|$ remains fixed, ii)
$|\vec{\ell}_{2}|\rightarrow\infty$ and $|\vec{\ell}_{1}|$ is fixed and, iii)
$|\vec{\ell}_{1,2}|\rightarrow\infty$. In order to extract the UV behaviour of
the integrand, we impose the replacement,

$\displaystyle\mathcal{S}_{j,\rm UV}:\\{\ell_{j}^{2}|\ell_{j}\cdot
k_{i}\\}\rightarrow\\{\lambda^{2}q_{j,\rm UV}^{2}+(1-\lambda^{2})\mu_{\rm
UV}^{2}|\lambda\,q_{j,\rm UV}\cdot k_{i}\\}\,,$ (68)

for a given loop momentum $\ell_{j}$ and then, we expand the expression up to
logarithmic order around the UV propagator. This action is represented by the
$L_{\lambda}$ operator. Notice, however, that the result shall generate a
finite part after integration that has to be fixed to find the right value of
the integral. Therefore, the first counter-terms is computed by

$\displaystyle\mathcal{A}_{j,\rm
UV}^{(2)}=L_{\lambda}\left(\mathcal{A}^{(2)}\Big{|}_{\mathcal{S}_{j,\rm
UV}}\right)-d_{j,\rm UV}\,\mu_{\rm UV}^{2}\int_{\ell_{j}}(G_{F}(q_{j,\rm
UV}))^{3}\,,$ (69)

with $d_{j,\rm UV}$ the fixing parameter which makes the finite part of
integral to be zero in the $\overline{\rm MS}$ scheme.

The remaining divergences shall occur when both $|\vec{\ell}_{1}|$ and
$|\vec{\ell}_{2}|$ approach to infinity simultaneously. Then, to build a
counter-term that mimics this behaviour, the following replacement is
implemented,

$\displaystyle\mathcal{S}_{\rm UV^{2}}:$
$\displaystyle\\{\ell_{j}^{2}|\ell_{j}\cdot\ell_{k}|\ell_{j}\cdot
k_{i}\\}\rightarrow$ $\displaystyle\\{\lambda^{2}q_{j,\rm
UV}^{2}+(1-\lambda^{2})\mu_{\rm UV}^{2}|\lambda^{2}q_{j,\rm UV}\cdot q_{k,\rm
UV}+(1-\lambda^{2})\,\mu_{\rm UV}^{2}/2|\lambda\,q_{j,\rm UV}\cdot k_{i}\\}\,$
(70)

on the subtracted integrand. Similarly to the previous UV counter-term, the
application of the $L_{\lambda}$ operation shall produce a finite piece that
has to be fixed to build properly the counter-term, $\mathcal{A}^{(2)}_{\rm
UV^{2}}$. Explicitly,

$\displaystyle\mathcal{A}_{\rm
UV^{2}}^{(2)}=L_{\lambda}\left(\left(\mathcal{A}^{(2)}-\sum_{j=1,2}\mathcal{A}^{(2)}_{j,\rm
UV}\right)\Bigg{|}_{\mathcal{S_{\rm UV^{2}}}}\right)-d_{\rm UV^{2}}\,\mu_{\rm
UV}^{4}\int_{\ell_{1}}\int_{\ell_{2}}(G_{F}(q_{1,\rm UV}))^{3}(G_{F}(q_{12,\rm
UV}))^{3}\,,$ (71)

with $d_{\rm UV^{2}}$ the fixing parameter of the double UV limit. Having all
UV divergences under control, the renormalised amplitude at two-loops,
$\mathcal{A}_{\rm R}^{(2)}$, can be constructed by the subtraction of all UV
counter-terms, such that

$\displaystyle\mathcal{A}_{\rm R}^{(2)}=\mathcal{A}^{(2)}-\mathcal{A}_{1,\rm
UV}^{(2)}-\mathcal{A}_{2,\rm UV}^{(2)}-\mathcal{A}_{\rm UV^{2}}^{(2)}\,,$ (72)

is free of IR and UV singularities.

In the multi-loop scenario, multiple ultraviolet poles will appear, since all
loops could tend to infinity at different speed.

###  8.3 $A^{*}\rightarrow q\bar{q}$ at NLO in QCD

The FDU method has been studied in different processes, for the sake of
simplicity, we recall the application of the algorithm in $A^{*}\rightarrow
q\bar{q}$ a NLO in QCD, with $A=H,\gamma,Z$.

At NLO, virtual correction $A^{*}(p)\rightarrow q(p_{1})+\bar{q}(p_{2})$ must
be mapped into the real radiation process $A^{*}(p)\rightarrow
q(p^{\prime}_{1})+\bar{q}(p^{\prime}_{2})+g(p^{\prime}_{r})$. As it was
explained in previous subsections, we must also keep all virtual contributions
since self-energy contributions are crucial to cancel out all IR
singularities. In Fig. 17, we present the complete set of Feynman diagrams of
the real and virtual processes.

Figure 17: Momentum configuration of NLO QCD corrections to the process
$A^{*}\rightarrow q\bar{q}(q)$, assuming that the decaying particle does not
couple to gluons.

It is important to recall that in this example we consider massive quarks,
i.e., $p_{1}^{2}=p_{2}^{2}=M^{2}$ and $p_{1}^{\prime 2}=p_{2}^{\prime
2}=M^{2}$ and $p_{r}^{\prime 2}=0$. It is convenient to write the momenta
$p_{1}$ and $p_{2}$ in terms of massless four vectors, $\hat{p}_{1}$ and
$\hat{p}_{2}$, such as

$\displaystyle
p_{1}^{\mu}=\beta_{+}\hat{p}_{1}^{\mu}+\beta_{-}\hat{p}_{2}^{\mu}\,,\qquad
p_{2}^{\mu}=\beta_{-}\hat{p}_{1}^{\mu}+\beta_{+}\hat{p}_{2}^{\mu}\,,$ (73)

with $\beta_{\pm}=(1\pm\beta)/2$, $\beta=\sqrt{1-m^{2}}$ and
$m=2M/\sqrt{s_{12}}$. Therefore, since the momentum of the gluon is labelled
in the virtual contribution as $q_{1}$ and LTD shall set it on-shell,
$q_{1}^{2}=0$, the momentum mapping proposed for the merging of virtual and
real cross-sections is

$\displaystyle p_{r}^{\prime\mu}$ $\displaystyle=q_{1}^{\mu}\,,$
$\displaystyle p_{1}^{\prime\mu}$
$\displaystyle=(1-\alpha_{1})\,\hat{p}_{1}^{\mu}+(1-\gamma_{1})\,\hat{p}_{2}^{\mu}-q_{1}^{\mu}\,,$
$\displaystyle p_{2}^{\prime\mu}$
$\displaystyle=\alpha_{1}\,\hat{p}_{1}^{\mu}+\gamma_{1}\,\hat{p}_{2}^{\mu}\,,$
(74)

where $\alpha_{1}$ and $\gamma_{1}$ are parameters determined by imposing on-
shellness. Notice that Eq. (8.3) fulfills momentum conservation. This mapping
will be used to remove some IR singularities of the integral related to
$\tilde{\delta}(q_{1})$, the remaining IR singularities shall cancel through
renormalisation. Now, since there are other singular IR behaviours in the real
that can be recognised in the virtual, when $q_{2}^{2}=M^{2}$, the second
momentum mapping is proposed to be,

$\displaystyle p_{r}^{\prime\mu}$
$\displaystyle=(1-\gamma_{2})\,\hat{p}_{1}^{\mu}+(1-\alpha_{2})\,\hat{p}_{2}^{\mu}-q_{2}^{\mu}\,,$
$\displaystyle p_{1}^{\prime\mu}$
$\displaystyle=\gamma_{2}\,\hat{p}_{1}^{\mu}+\alpha_{2}\,\hat{p}_{2}^{\mu}\,,$
$\displaystyle p_{2}^{\prime\mu}$ $\displaystyle=q_{2}^{\mu}\,,$ (75)

with $\alpha_{2}$ and $\gamma_{2}$ parameters which are also determined by the
on-shell conditions. In virtue of soft and collinear divergences in the real
cross-section being only in this two LTD cuts, we proceed to slice the full
real element with the general function,

$\displaystyle\mathcal{R}_{i}=\\{y^{\prime}_{ir}<\min(y^{\prime}_{jk})\\}\,,\qquad\sum\mathcal{R}_{i}=1\,,$
(76)

which in this particular case are explicitly,

$\displaystyle\theta(y^{\prime}_{2r}<y^{\prime}_{1r})+\theta(y^{\prime}_{1r}<y^{\prime}_{2r})=1\,,$
(77)

with $y^{\prime}_{ir}=2p_{i}^{\prime}\cdot p_{r}^{\prime}/s_{12}$. The
integration domain is well determined and it is shown in Fig. 18.

Figure 18: The dual integration regions in the loop three-momentum space, with
$\xi_{\perp}=\sqrt{\xi^{2}_{x}+\xi^{2}_{y}}$.

We remark that this arrangement of the real element phase-space is general and
it can be applied to any $1\rightarrow 3$ real cross-section that is mapped
into the $1\rightarrow 2$ virtual cross-section.

Let us now compute the virtual decay rate of $A^{*}\rightarrow q\bar{q}$. The
LTD representation is given by,

$\displaystyle\Gamma_{{\rm V},A}^{(1,{\rm
R})}=\frac{1}{2\sqrt{s_{12}}}\sum_{i=1}^{3}\int{\rm dPS}^{1\rightarrow
2}\,2{\rm Re}\,\langle\mathcal{M}_{A}^{(0)}|\mathcal{M}_{A}^{(1,{\rm
R})}(\widetilde{\delta}(q_{i}))\rangle$ (78)

with the renormalised one loop amplitude computed as,

$\displaystyle|\mathcal{M}_{A}^{(1,{\rm
R})}\rangle=|\mathcal{M}^{(1)}_{A}\rangle-|\mathcal{M}_{A}^{(1,{\rm
UV})}\rangle+\frac{1}{2}(\Delta Z_{2}^{{\rm IR}}(p_{1})+\Delta Z_{2}^{{\rm
IR}}(p_{2}))|\mathcal{M}_{A}^{(0)}\rangle\,,$ (79)

and $|\mathcal{M}_{A}^{(1,{\rm UV})}\rangle$ is the unintegrated UV counter-
term of the one-loop vertex correction. The real element, after the splitting
of the phase-space in two domains, is written as,

$\displaystyle\widetilde{\Gamma}_{{\rm
R},A,i}^{(1)}=\frac{1}{2\sqrt{s_{12}}}\int{\rm dPS}^{1\rightarrow
3}\,|\mathcal{M}_{A^{*}\rightarrow
q\bar{q}g}^{(0)}|^{2}\,\mathcal{R}_{i}(y^{\prime}_{ir}<y^{\prime}_{jr})\,,\qquad
i,j=\\{1,2\\}\,.$ (80)

The sum of the virtual and real contributions in Eq. (78) and Eq. (80) is a
single integral in the loop-three momentum. It is UV and IR finite locally and
can be calculated numerically with $\epsilon=0$. The results, normalised to
the LO decay rate $\Gamma_{A}^{(0)}$, are presented in Fig. 19. The agreement
with the analytical predictions is excellent in all cases. Furthermore, we
notice that the massless quark scenario is recovered smoothly in the FDU
however, in DREG this is not the case since individual contributions are not
smoothly defined in that limit.

Figure 19: NLO total decay rate for $A^{*}\rightarrow q\bar{q}$ normalised to
its corresponding LO decay rate. The solid lines correspond to DREG analytic
result while dots were computed numerically in the FDU formalism for the
standard Higgs boson (left panel) and off-shell vector bosons (right panel).

##  9 Local dual counter-terms for cross-sections

In the following, we will focus on the NLO QCD corrections to cross-sections
involving massless colored particles and any number of other colourless SM
particles. The extension of the following ideas to the massive case has been
worked out following the same reasoning, which is presented in Ref.
Prisco:2020kyb . The reduction of any one-loop amplitude to a combination of
scalar integrals is an easy task compared to the two-loop case. Of course,
such a reduction does not alter the structure of the divergences of the
amplitude but to properly reproduce all the IR singularities, in addition to
the wave function renormalisation, one has to keep also the massless bubble
integrals that are formally discarded usually (as already explained in Sec.
8). Once all the box functions are expressed in terms of their corresponding
$6-$dimensional version, it turns out that the IR divergences of a
renormalised one-loop amplitude interfered with the leading order are given by
the triangle scalar functions with two external partons connected by a
massless propagator times the corresponding color connected tree level
amplitude multiplied $s_{ij}=2\,p_{i}\cdot p_{j}$, the massless bubbles times
(1/2) the Born for every external (gluon) quark external particle multiplied
times the corresponding Casimir, and the external wave function
renormalisation constant time the Born. The above set of IR singularities is
universal and, in fact, it reproduces the known formula for the poles of any
one-loop amplitude in QCD. Inspired by the LTD theorem, we considered the cuts
of these loop integrals that capture their infrared and collinear divergences.
As a result of this procedure one obtains a novel subtraction scheme. For the
sake of simplicity we will restrict the discussion to the case of a colourless
initial state and so we will consider colored radiation exchanged only among
the final state partons. Our general formula for the dual counter-term in the
case of a quark emitting a gluon is the following:

$\sigma_{qg,b}^{DS}\equiv 8\pi\alpha_{S}\mathcal{N}_{in}\int
d\Phi_{m}\,{}_{m}\langle
1,\dots,m|\mathbf{T}_{ac}\mathbf{T}_{b}|1,\dots,m\rangle_{m}\left[V_{q_{a}g_{c},b}(p_{i},p_{j})+G_{q_{a}g_{c},b}(p_{i},p_{j})\right]\,,$
(81)

where $\mathcal{N}_{in}$ represents all the non-QCD factors, including the
symmetry factor for identical partons in the final state,
$|1,\dots,m\rangle_{m}$ is the $m$-parton Born amplitude and
$\mathbf{T}_{ac}\mathbf{T}_{b}$ is a colour-charge operator acting on the
colour indices of the partons $ac$ and $b$, respectively. The functions
$V_{qg,b}$ and $G_{qg,b}$ in Eq.(81) are defined by

$\displaystyle V_{qg,b}$
$\displaystyle\equiv\int_{q_{1}}\tilde{\delta}(q_{1})\,\mathcal{R}_{1}\left[-\frac{2s_{ij}}{(-2q_{1}\cdot
p_{i})(2q_{1}\cdot p_{j})}+\frac{2}{(-2q_{1}\cdot p_{i})}\right]\,,$
$\displaystyle G_{qg,b}$
$\displaystyle\equiv-\frac{(1-\varepsilon)}{s_{ij}}\int_{q_{1}}\tilde{\delta}(q_{1})\,\mathcal{R}_{1}\frac{2q_{1}\cdot
p_{j}}{(-2q_{1}\cdot p_{i})}\,,$ (82)

where $V_{qg,b}$ groups together the dual contributions from the reduction of
the virtual amplitude and $G_{qg,b}$ is the contribution of the quark wave
function renormalisation. In the above formulas $p_{i}$ and $p_{j}$ represent
the momentum of the emitter (with color index $ac$) and spectator (color index
$b$) in the Born kinematic respectively, while $q_{1}$ is the loop momentum
that is associated to the gluon in the real radiation kinematic. In the case a
gluon is radiated collinear to another gluon the corresponding dual counter-
term $\sigma^{DS}_{gg,b}$ is given by

$\sigma^{DS}_{gg,b}\equiv
8\pi\alpha_{S}\mathcal{N}_{in}\int\\!d\Phi_{m}\,{}_{m}\langle
1,\dots,m|\mathbf{T}_{ac}\mathbf{T}_{b}\left[V^{\mu\nu}_{gg,b}(p_{i},p_{j})+G^{\mu\nu}_{gg,b}(p_{i},p_{j})\right]|1,\dots,m\rangle_{m}\,,$
(83)

where the symmetrisation $p^{\prime}_{a}\leftrightarrow p^{\prime}_{c}$ has to
be performed on the right-hand side. Once again, in Eq. (83) we have extracted
only the dual contributions with $q_{1}$ on-shell, obtaining

$\displaystyle V^{\mu\nu}_{gg,b}\\!\\!$
$\displaystyle\equiv\\!-\\!\\!\int_{q_{1}}\tilde{\delta}(q_{1})\mathcal{R}_{1}\left[-\frac{2s_{ij}}{(-2q_{1}\cdot
p_{i})(2q_{1}\cdot p_{j})}+\frac{1}{(-2q_{1}\cdot p_{i})}\right]g^{\mu\nu}\,,$
(84a) $\displaystyle G^{\mu\nu}_{gg,b}\\!\\!$
$\displaystyle\equiv\\!-\\!\\!\int_{q_{1}}\\!\\!\frac{\tilde{\delta}(q_{1})\mathcal{R}_{1}}{(-2q_{1}\cdot
p_{i})}\\!\left[g^{\mu\nu}\\!+\frac{d-2}{(-2q_{1}\cdot
p_{i})}\\!\\!\left(\frac{q_{1}\cdot p_{j}}{p_{i}\cdot
p_{j}}-1\right)\\!\\!\left(q_{1}^{\mu}-\frac{q_{1}\cdot p_{i}}{p_{i}\cdot
p_{j}}p_{j}^{\mu}\right)\\!\\!\left(q_{1}^{\nu}-\frac{q_{1}\cdot
p_{i}}{p_{i}\cdot p_{j}}p_{j}^{\nu}\right)\\!\right]\,\,.$ (84b)

Finally, the case of a gluon splitting into a collinear quark-antiquark pair
receive contributions only from the wave function contribution so that the
corresponding dual counter-term $\sigma^{DS}_{q\bar{q},b}$ is given by

$\sigma^{DS}_{q\bar{q},b}\equiv
8\pi\alpha_{S}\mathcal{N}_{in}\int\\!d\Phi_{m}\,{}_{m}\langle
1,\dots,m|\mathbf{T}_{ac}\mathbf{T}_{b}\,G^{\mu\nu}_{q\bar{q},b}(p_{i},p_{j})|1,\dots,m\rangle_{m}\,,$
(85)

with

$G^{\mu\nu}_{q\bar{q},b}\\!\\!\equiv\\!\\!\\!\int_{q_{1}}\\!\\!\frac{\tilde{\delta}(q_{1})\mathcal{R}_{1}T_{R}N_{f}}{C_{A}(-2q_{1}\cdot
p_{i})}\\!\left[g^{\mu\nu}\\!\\!+\\!\frac{4}{(-2q_{1}\cdot
p_{i})}\\!\\!\left(\frac{q_{1}\cdot p_{j}}{p_{i}\cdot
p_{j}}-1\right)\\!\\!\left(q_{1}^{\mu}-\frac{q_{1}\cdot p_{i}}{p_{i}\cdot
p_{j}}p_{j}^{\mu}\right)\\!\\!\left(q_{1}^{\nu}-\frac{q_{1}\cdot
p_{i}}{p_{i}\cdot p_{j}}p_{j}^{\nu}\right)\\!\right].$ (86)

Once a proper momentum mapping is considered, these counter-terms can be
analytically integrated over the region in which $p_{a}\cdot p_{c}<p_{b}\cdot
p_{c}$ in the real phase space and added back to the virtual contribution.
This last operation corresponds to extract all the infrared and collinear
divergences out of the virtual matrix element. We have used the mapping
adopted in the seminal work by Catani and Seymour Catani:1996vz .

###  9.1 $\gamma^{*}\rightarrow 3$ jets at NLO

As an example of application, we consider the NLO correction to the three-jet
production in $e^{+}e^{-}$ annihilation $\gamma^{*}\\!\rightarrow
q(p_{1})\,\bar{q}(p_{2})\,g(p_{3})$. For simplicity we will restrict to the
case of gluon radiation, namely $\gamma^{*}\rightarrow
q(p^{\prime}_{1})\,\bar{q}(p^{\prime}_{2})\,g(p^{\prime}_{3})\,g(p^{\prime}_{4})$.
The case of four quark production can be treated along the same lines. By
inspecting the virtual sector, we find six emitter-spectator pairs for the
process $\gamma^{*}\rightarrow q(p_{1})\,\bar{q}(p_{2})\,g(p_{3})$, that are
$\\{q\bar{q},qg,\bar{q}g\\}$ and the three pairs where the emitter and the
spectator switch. Once a real kinematic configuration
$(p^{\prime}_{1},p^{\prime}_{2},p^{\prime}_{3},p^{\prime}_{4})$ is generated,
the six kinematic invariants $s_{ij}$ are analyzed and as a consequence six
dual counter-terms are activated. To each emitter-spectator pair, we associate
the following dual counter-terms:

$\displaystyle q\bar{q}\quad$
$\displaystyle\longrightarrow\quad\left\\{\quad\begin{aligned}
&&g(p^{\prime}_{3})\,\,\text{as
radiation:}&&\quad\sigma^{DS}_{13,2}\quad\text{if}\,\,s_{13}<s_{23},&&\quad\sigma^{DS}_{23,1}\quad\text{if}\,\,s_{13}>s_{23}\\\
&&g(p^{\prime}_{4})\,\,\text{as
radiation:}&&\quad\sigma^{DS}_{14,2}\quad\text{if}\,\,s_{14}<s_{24},&&\quad\sigma^{DS}_{24,1}\quad\text{if}\,\,s_{14}>s_{24}\end{aligned}\right.$
$\displaystyle qg\quad$
$\displaystyle\longrightarrow\quad\left\\{\quad\begin{aligned}
&&g(p^{\prime}_{3})\,\,\text{as
radiation:}&&\quad\sigma^{DS}_{13,4}\quad\text{if}\,\,s_{13}<s_{34},&&\quad\sigma^{DS}_{43,1}\quad\text{if}\,\,s_{13}>s_{34}\\\
&&g(p^{\prime}_{4})\,\,\text{as
radiation:}&&\quad\sigma^{DS}_{14,3}\quad\text{if}\,\,s_{14}<s_{34},&&\quad\sigma^{DS}_{34,1}\quad\text{if}\,\,s_{14}>s_{34}\end{aligned}\right.$
$\displaystyle\bar{q}g\quad$
$\displaystyle\longrightarrow\quad\left\\{\quad\begin{aligned}
&&g(p^{\prime}_{3})\,\,\text{as
radiation:}&&\quad\sigma^{DS}_{23,4}\quad\text{if}\,\,s_{23}<s_{34},&&\quad\sigma^{DS}_{43,2}\quad\text{if}\,\,s_{24}>s_{34}\\\
&&g(p^{\prime}_{4})\,\,\text{as
radiation:}&&\quad\sigma^{DS}_{24,3}\quad\text{if}\,\,s_{24}<s_{34},&&\quad\sigma^{DS}_{34,2}\quad\text{if}\,\,s_{24}>s_{34}\end{aligned}\right.$
(87)

Since the process involves only three colored partons, the colour algebra
factorises and one has $\mathbf{T}_{1}\mathbf{T}_{2}=C_{A}/2-C_{F}$ and
$\mathbf{T}_{1}\mathbf{T}_{3}=\mathbf{T}_{2}\mathbf{T}_{3}=-C_{A}/2$. In the
virtual sector, we use the integrated version of the dual counter-terms that
are given by

$\displaystyle\sigma^{DS}_{q,\bar{q}}$
$\displaystyle\sim\frac{\alpha_{S}}{2\pi}\frac{(4\pi)^{\varepsilon}}{\Gamma(1-\varepsilon)}\left(\frac{C_{A}}{2}-C_{F}\right)\int
d\Phi_{2}\left(\frac{\mu^{2}}{s_{12}}\right)^{\varepsilon}|A^{(0)}_{q\bar{q}g}|^{2}\left[\frac{1}{\varepsilon^{2}}+\frac{3}{2\varepsilon}+3+4\log(2)-\frac{\pi^{2}}{2}\right]$
$\displaystyle\sigma^{DS}_{\bar{q},q}$
$\displaystyle\sim\frac{\alpha_{S}}{2\pi}\frac{(4\pi)^{\varepsilon}}{\Gamma(1-\varepsilon)}\left(\frac{C_{A}}{2}-C_{F}\right)\int
d\Phi_{2}\left(\frac{\mu^{2}}{s_{12}}\right)^{\varepsilon}|A^{(0)}_{q\bar{q}g}|^{2}\left[\frac{1}{\varepsilon^{2}}+\frac{3}{2\varepsilon}+3+4\log(2)-\frac{\pi^{2}}{2}\right]$
$\displaystyle\sigma^{DS}_{q,g}$
$\displaystyle\sim-\frac{\alpha_{S}}{2\pi}\frac{(4\pi)^{\varepsilon}}{\Gamma(1-\varepsilon)}\frac{C_{A}}{2}\int
d\Phi_{2}\left(\frac{\mu^{2}}{s_{13}}\right)^{\varepsilon}|A^{(0)}_{q\bar{q}g}|^{2}\left[\frac{1}{\varepsilon^{2}}+\frac{3}{2\varepsilon}+3+4\log(2)-\frac{\pi^{2}}{2}\right]$
$\displaystyle\sigma^{DS}_{g,q}$
$\displaystyle\sim-\frac{\alpha_{S}}{2\pi}\frac{(4\pi)^{\varepsilon}}{\Gamma(1-\varepsilon)}\frac{C_{A}}{2}\int
d\Phi_{2}\left(\frac{\mu^{2}}{s_{13}}\right)^{\varepsilon}|A^{(0)}_{q\bar{q}g}|^{2}\left[\frac{1}{\varepsilon^{2}}+\frac{11}{6\varepsilon}+\frac{55}{18}+\frac{14}{3}\log(2)-\frac{\pi^{2}}{2}\right]$
$\displaystyle\sigma^{DS}_{\bar{q},g}$
$\displaystyle\sim-\frac{\alpha_{S}}{2\pi}\frac{(4\pi)^{\varepsilon}}{\Gamma(1-\varepsilon)}\frac{C_{A}}{2}\int
d\Phi_{2}\left(\frac{\mu^{2}}{s_{23}}\right)^{\varepsilon}|A^{(0)}_{q\bar{q}g}|^{2}\left[\frac{1}{\varepsilon^{2}}+\frac{3}{2\varepsilon}+3+4\log(2)-\frac{\pi^{2}}{2}\right]$
$\displaystyle\sigma^{DS}_{g,\bar{q}}$
$\displaystyle\sim-\frac{\alpha_{S}}{2\pi}\frac{(4\pi)^{\varepsilon}}{\Gamma(1-\varepsilon)}\frac{C_{A}}{2}\int
d\Phi_{2}\left(\frac{\mu^{2}}{s_{23}}\right)^{\varepsilon}|A^{(0)}_{q\bar{q}g}|^{2}\left[\frac{1}{\varepsilon^{2}}+\frac{11}{6\varepsilon}+\frac{55}{18}+\frac{14}{3}\log(2)-\frac{\pi^{2}}{2}\right]$
(88)

where $|A^{(0)}_{q\bar{q}g}|^{2}$ is the Born amplitude for the process
$\gamma^{*}\rightarrow q\bar{q}g$ and we have neglected terms of order
$\mathcal{O}(\varepsilon)$.

Figure 20: Rapidity distribution of the most energetic jet in
$e^{+}e^{-}\rightarrow\gamma^{*}\rightarrow 3$ jets at
$\sqrt{s}=125\,\text{GeV}$. Jets are clustered using the Durham algorithm with
$y_{\text{cut}}=0.05$ Only the NLO corrections are plotted. Results obtained
using Dual Subtraction (DS) results are shown in red, while the ones obtained
using Catani-Seymour (CS) dipoles are shown in blue. The error bars correspond
to the statistical error given by the Monte Carlo integration.

In Fig. 20 we show the differential rapidity distribution of the most
energetic jet in $e+e-$ annihilation at $\sqrt{s}=125\,\text{GeV}$. Jets are
clustered using the Durham jet algorithm with resolution parameter set at
$y_{\text{cut}}=0.05$. A perfect agreement with the same computation performed
using Catani-Seymour dipoles is observed. The same level of agreement is
observed for any other differential distribution.

In this way, we have shown that a fully local implementation of an LTD-
inspired method for computing cross-sections is reliable. These studies
complement the ones presented in Sec. 8, indicating that the FDU framework
constitutes a powerful strategy for reaching higher-order corrections with
fully numerical methods. An extension of this technology to calculate N2LO
corrections for IR-safe observables is being actively studied.

##  10 Outlook and further developments

In this manuscript, we reported on the recent developments in the numerical
evaluation of multi-loop scattering amplitudes through the application of the
Loop-Tree Duality (LTD) formalism. Based on the original approach, at one and
two loops Catani:2008xa ; Bierenbaum:2010cy , we presented a novel formulation
of LTD Verdugo:2020kzh , which allowed to bootstrap the treatment of Feynman
integrals, regardless of the loop order, and shed light to a complete
automation. In effect, the decomposition of two- and three-loop scattering
amplitudes in terms of dual integrands was obtained by introducing the
concepts of maximal (MLT), next-to-maximal (NMLT) and next-to-next-to-maximal
(N2MLT) loop topologies. These families of diagrams share general properties
that allow allows for the generalization of the dual representation at any
loop order.

Since LTD heavily relies on the application of the Cauchy residue theorem, we
studied in details the treatment of multivariate rational functions in Ref.
Aguilera-Verdugo:2020nrp , giving, in this way, a mathematical support to the
novel formulation of Ref. Verdugo:2020kzh . Likewise, we proved all
conjectures that were provided in the latter, thus, rendering in a well
organised formalism ready to focus on physical applications. Hence, in view of
the LTD decompositions of up-to three-loop scattering amplitudes, the natural
extension, along the lines of Ref. Verdugo:2020kzh , was considering the four-
loop case Ramirez-Uribe:2020hes . In effect, the complexity in the treatment
of four-loop topologies increased because, differently from the two- and
three-loop cases, this loop order was not described by only one loop topology,
as in the lower cases. Hence, we introduced new configurations that appears at
four loops: the N3MLT and N4MLT families. In this way, our formalism allowed a
complete understanding and decomposition of the dual representation of any up-
to four-loop scattering amplitude.

In the spirit of profiting from the dual representation of integrands through
LTD, we observed that the complete sum of the latter, for a given Feynman
integral, led to a representation of an integrand that only manifests physical
information. In other words, unphysical singularities or _pseudo-thresholds_
cancel out when all dual integrands are summed up. This is a remarkable
feature of LTD that was originally observed in Refs. Driencourt-Mangin:2019aix
; Aguilera-Verdugo:2019kbz ; Verdugo:2020kzh  and extensively studied in Ref.
Aguilera-Verdugo:2020kzc  for MLT, NMLT and N2MLT configurations. In fact, the
work performed in Ref. Aguilera-Verdugo:2020kzc  made a comparison between the
numerical stability of dual and causal integrands. While in the former
numerical instabilities, originated from pseudo-thresholds, were present, the
latter displayed a smooth behaviour: the numerical evaluation was much more
stable and this allowed a straightforward computation of the Feynman
integrals. In particular, we considered the numerical integration of
ultraviolet finite integrals at three and four loops with presence of several
kinematic variables.

The causal representation of scattering amplitudes has recently been a very
active topic due to the simple structure of these integrands in terms of
causal (physical) thresholds. In fact, an understanding of the structure of
the latter started to appear in the literature by means of geometric
properties Sborlini:2021owe  and is yet under consideration. On top of the
former approach, it was recently conjectured a breathtaking formulation of the
all-loop causal representation of multi-loop scalar integrands, obtained from
the features that characterised a topology, cusps and edges Bobadilla:2021rmu
. This formulation provides, regardless of the loop order, the most symmetric
causal representation of Feynman integrands, in a complementary and
independent way to the nested residue calculation used by the LTD formalism.
Thus, to give a support to these conjectures, we provide the Mathematica
package Lotty to automate both dual and causal representation of scattering
amplitudes TorresBobadilla:2021dkq .

From the above-mentioned discussion, we claim that a treatment of Feynman
integrands at an arbitrary number of loops is under control. Moreover, this is
not the end of the story when comparing theoretical predictions with data
derived by collider experiments. In effect, one yet has to deal with
ultraviolet (UV) and infrared (IR) singularities. While the framework at next-
to-leading order (NLO) is completely understood by means of the Four-
Dimensional Unsubtraction scheme Hernandez-Pinto:2015ysa ; Sborlini:2016gbr ;
Sborlini:2016hat  and the dual subtractions Prisco:2020kyb , a clear path
towards calculations at higher orders (NNLO and beyond) needs to be devised.
Preliminary studies of IR-safe scattering amplitudes has been recently
considered in Refs. Driencourt-Mangin:2019aix ; Driencourt-Mangin:2019yhu .
Besides, the extension to the calculation of cross sections at NNLO is under
study, in particular, focusing on a careful _local_ treatment of IR
singularities.

We expect that the ideas presented in this manuscript will allow to elucidate
in detail the state-of-the-art of the LTD formalism and the causal
representation of scattering amplitudes by means of the latter. Likewise, the
ideas that we have developed will certainly supply the scientific community
with a powerful strategy to tackle the calculation of multi-loop Feynman
integrals, in which phenomenological and formal applications can efficiently
be carried out.

## Acknowledgements

This research was supported in part by COST Action CA16201 (PARTICLEFACE).
This work is supported by the Spanish Government (Agencia Estatal de
Investigación) and ERDF funds from European Commission (Grants No.
FPA2017-84445-P), Generalitat Valenciana (Grant No. PROMETEO/2017/053). JP
acknowledges support from "la Caixa" Foundation (No. 100010434,
LCF/BQ/IN17/11620037), and the European Union’s H2020-MSCA Grant Agreement No.
713673; SRU from CONACyT and Universidad Autónoma de Sinaloa; JJAV from
Generalitat Valenciana (GRISOLIAP/2018/101); AERO from the Spanish Government
(PRE2018-085925); and RJHP from the project A1-S-33202 (Ciencia Básica),
Ciencia de Frontera 2021-2042 and Sistema Nacional de Investigadores.

## References

  * (1) FCC collaboration, A. Abada et al., _FCC Physics Opportunities: Future Circular Collider Conceptual Design Report Volume 1_ , _Eur. Phys. J. C_ 79 (2019) 474.
  * (2) FCC collaboration, A. Abada et al., _FCC-ee: The Lepton Collider: Future Circular Collider Conceptual Design Report Volume 2_ , _Eur. Phys. J. ST_ 228 (2019) 261–623.
  * (3) FCC collaboration, A. Abada et al., _FCC-hh: The Hadron Collider: Future Circular Collider Conceptual Design Report Volume 3_ , _Eur. Phys. J. ST_ 228 (2019) 755–1107.
  * (4) FCC collaboration, A. Abada et al., _HE-LHC: The High-Energy Large Hadron Collider: Future Circular Collider Conceptual Design Report Volume 4_ , _Eur. Phys. J. ST_ 228 (2019) 1109–1382.
  * (5) A. Blondel, J. Gluza, S. Jadach, P. Janot and T. Riemann, eds., _Theory for the FCC-ee: Report on the 11th FCC-ee Workshop Theory and Experiments_ , vol. 3/2020 of _CERN Yellow Reports: Monographs_ , (Geneva), CERN, 5, 2019.  10.23731/CYRM-2020-003.
  * (6) P. Bambade et al., _The International Linear Collider: A Global Project_ , 1903.01629.
  * (7) CLIC, CLICdp collaboration, P. Roloff, R. Franceschini, U. Schnoor, A. Wulzer et al., _The Compact Linear e +e- Collider (CLIC): Physics Potential_, 1812.07986.
  * (8) CEPC Study Group collaboration, M. Dong et al., _CEPC Conceptual Design Report: Volume 2 - Physics & Detector_, 1811.10545.
  * (9) G. ’t Hooft and M. J. G. Veltman, _Regularization and Renormalization of Gauge Fields_ , _Nucl. Phys._ B44 (1972) 189–213.
  * (10) C. G. Bollini and J. J. Giambiagi, _Dimensional Renormalization: The Number of Dimensions as a Regularizing Parameter_ , _Nuovo Cim._ B12 (1972) 20–26.
  * (11) G. M. Cicuta and E. Montaldi, _Analytic renormalization via continuous space dimension_ , _Lett. Nuovo Cim._ 4 (1972) 329–332.
  * (12) J. F. Ashmore, _A Method of Gauge Invariant Regularization_ , _Lett. Nuovo Cim._ 4 (1972) 289–290.
  * (13) C. Gnendiger and A. Signer, _$\gamma_{5}$ in the four-dimensional helicity scheme_, _Phys. Rev. D_ 97 (2018) 096006, [1710.09231].
  * (14) A. M. Bruque, A. L. Cherchiglia and M. Pérez-Victoria, _Dimensional regularization vs methods in fixed dimension with and without $\gamma_{5}$_, _JHEP_ 08 (2018) 109, [1803.09764].
  * (15) R. Pittau, _A four-dimensional approach to quantum field theories_ , _JHEP_ 1211 (2012) 151, [1208.5457].
  * (16) R. A. Fazio, P. Mastrolia, E. Mirabella and W. J. Torres Bobadilla, _On the Four-Dimensional Formulation of Dimensionally Regulated Amplitudes_ , _Eur. Phys. J._ C74 (2014) 3197, [1404.4783].
  * (17) P. Mastrolia, A. Primo, U. Schubert and W. J. Torres Bobadilla, _Off-shell currents and color–kinematics duality_ , _Phys. Lett._ B753 (2016) 242–262, [1507.07532].
  * (18) A. Primo and W. J. Torres Bobadilla, _BCJ Identities and $d$-Dimensional Generalized Unitarity_, _JHEP_ 04 (2016) 125, [1602.03161].
  * (19) C. Gnendiger et al., _To ${d}$, or not to ${d}$: recent developments and comparisons of regularization schemes_, _Eur. Phys. J._ C77 (2017) 471, [1705.01827].
  * (20) W. Torres Bobadilla et al., _May the four be with you: Novel IR-subtraction methods to tackle NNLO calculations_ , 2012.02567.
  * (21) K. G. Chetyrkin and F. V. Tkachov, _Integration by Parts: The Algorithm to Calculate beta Functions in 4 Loops_ , _Nucl. Phys._ B192 (1981) 159–204.
  * (22) S. Laporta, _High precision calculation of multiloop Feynman integrals by difference equations_ , _Int. J. Mod. Phys._ A15 (2000) 5087–5159, [hep-ph/0102033].
  * (23) T. Binoth and G. Heinrich, _An automatized algorithm to compute infrared divergent multiloop integrals_ , _Nucl. Phys. B_ 585 (2000) 741–759, [hep-ph/0004013].
  * (24) A. Smirnov and M. Tentyukov, _Feynman Integral Evaluation by a Sector decomposiTion Approach (FIESTA)_ , _Comput. Phys. Commun._ 180 (2009) 735–746, [0807.4129].
  * (25) J. Carter and G. Heinrich, _SecDec: A general program for sector decomposition_ , _Comput. Phys. Commun._ 182 (2011) 1566–1581, [1011.5493].
  * (26) S. Borowka, G. Heinrich, S. Jahn, S. Jones, M. Kerner, J. Schlenk et al., _pySecDec: a toolbox for the numerical evaluation of multi-scale integrals_ , _Comput. Phys. Commun._ 222 (2018) 313–326, [1703.09692].
  * (27) F. Moriello, _Generalised power series expansions for the elliptic planar families of Higgs + jet production at two loops_ , _JHEP_ 01 (2020) 150, [1907.13234].
  * (28) R. Bonciani, V. Del Duca, H. Frellesvig, J. Henn, M. Hidding, L. Maestri et al., _Evaluating a family of two-loop non-planar master integrals for Higgs + jet production with full heavy-quark mass dependence_ , _JHEP_ 01 (2020) 132, [1907.13156].
  * (29) M. Czakon, _Tops from Light Quarks: Full Mass Dependence at Two-Loops in QCD_ , _Phys. Lett. B_ 664 (2008) 307–314, [0803.1400].
  * (30) G. Heinrich, _Collider Physics at the Precision Frontier_ , 2009.00516.
  * (31) K. J. Larsen and Y. Zhang, _Integration-by-parts reductions from unitarity cuts and algebraic geometry_ , _Phys. Rev. D_ 93 (2016) 041701, [1511.01071].
  * (32) Z. Bern, M. Enciso, H. Ita and M. Zeng, _Dual Conformal Symmetry, Integration-by-Parts Reduction, Differential Equations and the Nonplanar Sector_ , _Phys. Rev. D_ 96 (2017) 096017, [1709.06055].
  * (33) M. Zeng, _Differential equations on unitarity cut surfaces_ , _JHEP_ 06 (2017) 121, [1702.02355].
  * (34) J. Böhm, A. Georgoudis, K. J. Larsen, M. Schulze and Y. Zhang, _Complete sets of logarithmic vector fields for integration-by-parts identities of Feynman integrals_ , _Phys. Rev. D_ 98 (2018) 025023, [1712.09737].
  * (35) J. Böhm, A. Georgoudis, K. J. Larsen, H. Schönemann and Y. Zhang, _Complete integration-by-parts reductions of the non-planar hexagon-box via module intersections_ , _JHEP_ 09 (2018) 024, [1805.01873].
  * (36) D. Bendle, J. Böhm, W. Decker, A. Georgoudis, F.-J. Pfreundt, M. Rahn et al., _Integration-by-parts reductions of Feynman integrals using Singular and GPI-Space_ , _JHEP_ 02 (2020) 079, [1908.04301].
  * (37) P. Mastrolia and G. Ossola, _On the Integrand-Reduction Method for Two-Loop Scattering Amplitudes_ , _JHEP_ 11 (2011) 014, [1107.6041].
  * (38) S. Badger, H. Frellesvig and Y. Zhang, _Hepta-Cuts of Two-Loop Scattering Amplitudes_ , _JHEP_ 1204 (2012) 055, [1202.2019].
  * (39) Y. Zhang, _Integrand-Level Reduction of Loop Amplitudes by Computational Algebraic Geometry Methods_ , _JHEP_ 09 (2012) 042, [1205.5707].
  * (40) P. Mastrolia, E. Mirabella, G. Ossola and T. Peraro, _Scattering Amplitudes from Multivariate Polynomial Division_ , _Phys. Lett._ B718 (2012) 173–177, [1205.7087].
  * (41) P. Mastrolia, E. Mirabella, G. Ossola and T. Peraro, _Integrand-Reduction for Two-Loop Scattering Amplitudes through Multivariate Polynomial Division_ , _Phys. Rev._ D87 (2013) 085026, [1209.4319].
  * (42) H. Ita, _Two-loop Integrand Decomposition into Master Integrals and Surface Terms_ , _Phys. Rev. D_ 94 (2016) 116015, [1510.05626].
  * (43) P. Mastrolia, T. Peraro and A. Primo, _Adaptive Integrand Decomposition in parallel and orthogonal space_ , _JHEP_ 08 (2016) 164, [1605.03157].
  * (44) G. Ossola, C. G. Papadopoulos and R. Pittau, _Reducing full one-loop amplitudes to scalar integrals at the integrand level_ , _Nucl. Phys._ B763 (2007) 147–169, [hep-ph/0609007].
  * (45) P. Baikov, _Explicit solutions of the three loop vacuum integral recurrence relations_ , _Phys. Lett. B_ 385 (1996) 404–410, [hep-ph/9603267].
  * (46) H. Frellesvig and C. G. Papadopoulos, _Cuts of Feynman Integrals in Baikov representation_ , _JHEP_ 04 (2017) 083, [1701.07356].
  * (47) T. Peraro, _Scattering amplitudes over finite fields and multivariate functional reconstruction_ , _JHEP_ 12 (2016) 030, [1608.01902].
  * (48) S. Badger, C. Brønnum-Hansen, H. B. Hartanto and T. Peraro, _First look at two-loop five-gluon scattering in QCD_ , _Phys. Rev. Lett._ 120 (2018) 092001, [1712.02229].
  * (49) S. Abreu, F. Febres Cordero, H. Ita, B. Page and M. Zeng, _Planar Two-Loop Five-Gluon Amplitudes from Numerical Unitarity_ , _Phys. Rev. D_ 97 (2018) 116014, [1712.03946].
  * (50) S. Badger, D. Chicherin, T. Gehrmann, G. Heinrich, J. Henn, T. Peraro et al., _Analytic form of the full two-loop five-gluon all-plus helicity amplitude_ , _Phys. Rev. Lett._ 123 (2019) 071601, [1905.03733].
  * (51) P. Mastrolia and S. Mizera, _Feynman Integrals and Intersection Theory_ , _JHEP_ 02 (2019) 139, [1810.03818].
  * (52) H. Frellesvig, F. Gasparotto, S. Laporta, M. K. Mandal, P. Mastrolia, L. Mattiazzi et al., _Decomposition of Feynman Integrals on the Maximal Cut by Intersection Numbers_ , _JHEP_ 05 (2019) 153, [1901.11510].
  * (53) H. Frellesvig, F. Gasparotto, M. K. Mandal, P. Mastrolia, L. Mattiazzi and S. Mizera, _Vector Space of Feynman Integrals and Multivariate Intersection Numbers_ , _Phys. Rev. Lett._ 123 (2019) 201602, [1907.02000].
  * (54) S. Weinzierl, _On the computation of intersection numbers for twisted cocycles_ , 2002.01930.
  * (55) Z. Kunszt and D. E. Soper, _Calculation of jet cross-sections in hadron collisions at order alpha-s**3_ , _Phys. Rev._ D46 (1992) 192–221.
  * (56) S. Frixione, Z. Kunszt and A. Signer, _Three jet cross-sections to next-to-leading order_ , _Nucl. Phys._ B467 (1996) 399–442, [hep-ph/9512328].
  * (57) S. Catani and M. Seymour, _The Dipole formalism for the calculation of QCD jet cross-sections at next-to-leading order_ , _Phys.Lett._ B378 (1996) 287–301, [hep-ph/9602277].
  * (58) S. Catani and M. Seymour, _A General algorithm for calculating jet cross-sections in NLO QCD_ , _Nucl.Phys._ B485 (1997) 291–419, [hep-ph/9605323].
  * (59) S. Catani, D. de Florian and M. Grazzini, _Universality of nonleading logarithmic contributions in transverse momentum distributions_ , _Nucl. Phys. B_ 596 (2001) 299–312, [hep-ph/0008184].
  * (60) S. Catani, L. Cieri, D. de Florian, G. Ferrera and M. Grazzini, _Universality of transverse-momentum resummation and hard factors at the NNLO_ , _Nucl. Phys. B_ 881 (2014) 414–443, [1311.1654].
  * (61) A. Gehrmann-De Ridder, T. Gehrmann and E. W. N. Glover, _Antenna subtraction at NNLO_ , _JHEP_ 09 (2005) 056, [hep-ph/0505111].
  * (62) A. Kardos, _Jet production in the $\small{CoLoRFul}$NNLO framework_, _PoS_ RADCOR2015 (2016) 047.
  * (63) V. Del Duca, C. Duhr, A. Kardos, G. Somogyi, Z. Szőr, Z. Trócsányi et al., _Jet production in the CoLoRFulNNLO method: event shapes in electron-positron collisions_ , _Phys. Rev. D_ 94 (2016) 074019, [1606.03453].
  * (64) J. Gaunt, M. Stahlhofen, F. J. Tackmann and J. R. Walsh, _N-jettiness Subtractions for NNLO QCD Calculations_ , _JHEP_ 09 (2015) 058, [1505.04794].
  * (65) R. Boughezal, F. Caola, K. Melnikov, F. Petriello and M. Schulze, _Higgs boson production in association with a jet at next-to-next-to-leading order_ , _Phys. Rev. Lett._ 115 (2015) 082003, [1504.07922].
  * (66) L. Magnea, E. Maina, G. Pelliccioli, C. Signorile-Signorile, P. Torrielli and S. Uccirati, _Local analytic sector subtraction at NNLO_ , _JHEP_ 12 (2018) 107, [1806.09570].
  * (67) L. Magnea, E. Maina, G. Pelliccioli, C. Signorile-Signorile, P. Torrielli and S. Uccirati, _Local analytic sector subtraction for final state radiation at NNLO_ , _PoS_ RADCOR2019 (2019) 057, [1912.09368].
  * (68) S. Catani, T. Gleisberg, F. Krauss, G. Rodrigo and J.-C. Winter, _From loops to trees by-passing Feynman’s theorem_ , _JHEP_ 09 (2008) 065, [0804.3170].
  * (69) G. Rodrigo, S. Catani, T. Gleisberg, F. Krauss and J.-C. Winter, _From multileg loops to trees (by-passing Feynman’s Tree Theorem)_ , _Nucl. Phys. B Proc. Suppl._ 183 (2008) 262–267, [0807.0531].
  * (70) I. Bierenbaum, S. Catani, P. Draggiotis and G. Rodrigo, _A Tree-Loop Duality Relation at Two Loops and Beyond_ , _JHEP_ 10 (2010) 073, [1007.0194].
  * (71) I. Bierenbaum, S. Buchta, P. Draggiotis, I. Malamos and G. Rodrigo, _Tree-Loop Duality Relation beyond simple poles_ , _JHEP_ 03 (2013) 025, [1211.5048].
  * (72) S. Buchta, G. Chachamis, P. Draggiotis, I. Malamos and G. Rodrigo, _On the singular behaviour of scattering amplitudes in quantum field theory_ , _JHEP_ 11 (2014) 014, [1405.7850].
  * (73) S. Buchta, _Theoretical foundations and applications of the Loop-Tree Duality in Quantum Field Theories_.  PhD thesis, Valencia U., 2015.  1509.07167.
  * (74) S. Buchta, G. Chachamis, P. Draggiotis and G. Rodrigo, _Numerical implementation of the loop–tree duality method_ , _Eur. Phys. J._ C77 (2017) 274, [1510.00187].
  * (75) F. Driencourt-Mangin, G. Rodrigo and G. F. Sborlini, _Universal dual amplitudes and asymptotic expansions for $gg\rightarrow H$ and $H\rightarrow\gamma\gamma$ in four dimensions_, _Eur. Phys. J. C_ 78 (2018) 231, [1702.07581].
  * (76) J. Plenter, _Asymptotic Expansions Through the Loop-Tree Duality_ , _Acta Phys. Polon. B_ 50 (2019) 1983–1992.
  * (77) J. Plenter and G. Rodrigo, _Asymptotic expansions through the loop-tree duality_ , 2005.02119.
  * (78) J. L. Jurado, G. Rodrigo and W. J. Torres Bobadilla, _From Jacobi off-shell currents to integral relations_ , _JHEP_ 12 (2017) 122, [1710.11010].
  * (79) F. Driencourt-Mangin, G. Rodrigo, G. F. R. Sborlini and W. J. Torres Bobadilla, _Universal four-dimensional representation of $H\rightarrow\gamma\gamma$ at two loops through the Loop-Tree Duality_, _JHEP_ 02 (2019) 143, [1901.09853].
  * (80) F. Driencourt-Mangin, _Four-dimensional representation of scattering amplitudes and physical observables through the application of the Loop-Tree Duality theorem_.  PhD thesis, U. Valencia (main), 2019.  1907.12450.
  * (81) F. Driencourt-Mangin, G. Rodrigo, G. F. Sborlini and W. J. Torres Bobadilla, _On the interplay between the loop-tree duality and helicity amplitudes_ , 1911.11125.
  * (82) R. J. Hernandez-Pinto, G. F. R. Sborlini and G. Rodrigo, _Towards gauge theories in four dimensions_ , _JHEP_ 02 (2016) 044, [1506.04617].
  * (83) G. F. Sborlini, _Loop-tree duality and quantum field theory in four dimensions_ , _PoS_ RADCOR2015 (2016) 082, [1601.04634].
  * (84) G. F. R. Sborlini, F. Driencourt-Mangin, R. Hernandez-Pinto and G. Rodrigo, _Four-dimensional unsubtraction from the loop-tree duality_ , _JHEP_ 08 (2016) 160, [1604.06699].
  * (85) G. F. R. Sborlini, F. Driencourt-Mangin and G. Rodrigo, _Four-dimensional unsubtraction with massive particles_ , _JHEP_ 10 (2016) 162, [1608.01584].
  * (86) R. M. Prisco and F. Tramontano, _Dual Subtractions_ , 2012.05012.
  * (87) J. J. Aguilera-Verdugo, F. Driencourt-Mangin, J. Plenter, S. Ramírez-Uribe, G. Rodrigo, G. F. Sborlini et al., _Causality, unitarity thresholds, anomalous thresholds and infrared singularities from the loop-tree duality at higher orders_ , _JHEP_ 12 (2019) 163, [1904.08389].
  * (88) R. E. Cutkosky, _Singularities and discontinuities of Feynman amplitudes_ , _J. Math. Phys._ 1 (1960) 429–433.
  * (89) E. Tomboulis, _Causality and Unitarity via the Tree-Loop Duality Relation_ , _JHEP_ 05 (2017) 148, [1701.07052].
  * (90) R. Runkel, Z. Szőr, J. P. Vesga and S. Weinzierl, _Causality and loop-tree duality at higher loops_ , _Phys. Rev. Lett._ 122 (2019) 111603, [1902.02135].
  * (91) R. Runkel, Z. Szőr, J. P. Vesga and S. Weinzierl, _Integrands of loop amplitudes within loop-tree duality_ , _Phys. Rev. D_ 101 (2020) 116014, [1906.02218].
  * (92) J. J. Aguilera-Verdugo, F. Driencourt-Mangin, R. J. Hernandez Pinto, J. Plenter, S. Ramirez-Uribe, A. E. Renteria Olivo et al., _Open loop amplitudes and causality to all orders and powers from the loop-tree duality_ , _Phys. Rev. Lett._ 124 (2020) 211602, [2001.03564].
  * (93) Z. Capatti, V. Hirschi, D. Kermanschah, A. Pelloni and B. Ruijl, _Numerical Loop-Tree Duality: contour deformation and subtraction_ , _JHEP_ 04 (2020) 096, [1912.09291].
  * (94) Z. Capatti, V. Hirschi, D. Kermanschah and B. Ruijl, _Loop-Tree Duality for Multiloop Numerical Integration_ , _Phys. Rev. Lett._ 123 (2019) 151602, [1906.06138].
  * (95) S. Ramírez-Uribe, R. J. Hernández-Pinto, G. Rodrigo, G. F. R. Sborlini and W. J. Torres Bobadilla, _Universal opening of four-loop scattering amplitudes to trees_ , _JHEP_ 04 (2021) 129, [2006.13818].
  * (96) J. J. Aguilera-Verdugo, R. J. Hernandez-Pinto, G. Rodrigo, G. F. R. Sborlini and W. J. Torres Bobadilla, _Causal representation of multi-loop Feynman integrands within the loop-tree duality_ , _JHEP_ 01 (2021) 069, [2006.11217].
  * (97) J. Jesús Aguilera-Verdugo, R. J. Hernández-Pinto, G. Rodrigo, G. F. R. Sborlini and W. J. Torres Bobadilla, _Mathematical properties of nested residues and their application to multi-loop scattering amplitudes_ , _JHEP_ 02 (2021) 112, [2010.12971].
  * (98) J. Aguilera-Verdugo, R. Hernandez-Pinto, S. Ramirez-Uribe, G. Rodrigo, G. Sborlini and W. Torres Bobadilla, _Manifestly Causal Scattering Amplitudes_ , in _SnowMass 2021_ , August, 2020.
  * (99) Z. Capatti, V. Hirschi, D. Kermanschah, A. Pelloni and B. Ruijl, _Manifestly Causal Loop-Tree Duality_ , 2009.05509.
  * (100) W. J. Torres Bobadilla, _Loop-tree duality from vertices and edges_ , _JHEP_ 04 (2021) 183, [2102.05048].
  * (101) W. J. Torres Bobadilla, _Lotty – The loop-tree duality automation_ , 2103.09237.
  * (102) G. F. R. Sborlini, _A geometrical approach to causality in multi-loop amplitudes_ , 2102.05062.
  * (103) M. Beneke and V. A. Smirnov, _Asymptotic expansion of Feynman integrals near threshold_ , _Nucl. Phys._ B522 (1998) 321–344, [hep-ph/9711391].
  * (104) A. Pak and A. Smirnov, _Geometric approach to asymptotic expansion of Feynman integrals_ , _Eur. Phys. J. C_ 71 (2011) 1626, [1011.4863].
  * (105) B. Jantzen, _Foundation and generalization of the expansion by regions_ , _JHEP_ 12 (2011) 076, [1111.2589].
  * (106) G. Mishima, _High-Energy Expansion of Two-Loop Massive Four-Point Diagrams_ , _JHEP_ 02 (2019) 080, [1812.04373].
  * (107) T. Y. Semenova, A. V. Smirnov and V. A. Smirnov, _On the status of expansion by regions_ , _Eur. Phys. J. C_ 79 (2019) 136, [1809.04325].
  * (108) A. von Manteuffel and R. M. Schabinger, _A novel approach to integration by parts reduction_ , _Phys. Lett. B_ 744 (2015) 101–104, [1406.4513].
  * (109) T. Peraro, _FiniteFlow: multivariate functional reconstruction using finite fields and dataflow graphs_ , _JHEP_ 07 (2019) 031, [1905.08019].
  * (110) G. F. R. Sborlini, R. Hernández-Pinto and G. Rodrigo, _From dimensional regularization to NLO computations in four dimensions_ , _PoS_ EPS-HEP2015 (2015) 479, [1510.01079].
  * (111) P. Nogueira, _Automatic Feynman graph generation_ , _J.Comput.Phys._ 105 (1993) 279–289.
  * (112) T. Kinoshita, _Mass singularities of Feynman amplitudes_ , _J. Math. Phys._ 3 (1962) 650–677.
  * (113) T. D. Lee and M. Nauenberg, _Degenerate Systems and Mass Singularities_ , _Phys. Rev._ 133 (1964) B1549–B1562.
  * (114) D. A. Kosower and P. Uwer, _One loop splitting amplitudes in gauge theory_ , _Nucl. Phys. B_ 563 (1999) 477–505, [hep-ph/9903515].
  * (115) G. F. R. Sborlini, D. de Florian and G. Rodrigo, _Double collinear splitting amplitudes at next-to-leading order_ , _JHEP_ 01 (2014) 018, [1310.6841].