Causal representation and numerical evaluation of multi-loop Feynman integrals

William J. Torres Bobadilla1$\star$ 00footnotetext: Parts of the work
presented in this talk have been carried out in collaboration with Paolo
Benincasa, Germán Rodrigo and Jonathan Ronca.

1 Max-Planck-Institut für Physik, Werner-Heisenberg-Institut, 80805 München,
Germany.*<EMAIL_ADDRESS>
15th International Symposium on Radiative Corrections:
Applications of Quantum Field Theory to Phenomenology,
FSU, Tallahasse, FL, USA, 17-21 May 2021 10.21468/SciPostPhysProc.?

## Abstract

The loop-tree duality (LTD) has become a novelty alternative to bootstrap the
numerical evaluation of multi-loop scattering amplitudes. It has indeed been
found that Feynman integrands, after the application of LTD, display a
representation containing only physical information, the so-called causal
representation. In this talk, we discuss the all-loop causal representation of
multi-loop Feynman integrands, recently found in terms of features that
describe a loop topology, vertices and edges. Likewise, in order to elucidate
the numerical stability in LTD integrands, we present applications that
involve numerical evaluations of two-loop planar and non-planar triangles with
presence of several kinematic invariants.

##  1 Introduction & Motivation

In this contribution, we report on the all-loop causal representation of
multi-loop scattering amplitudes, recently proposed in Ref. [1] and automated
in the Mathematica package Lotty [2]. In this representation, that manifestly
displays the causality of scattering amplitudes [3], the structure of Feynman
integrands is cast only by physical singularities, which, within our approach,
are usually referred to as causal propagators [4, 5, 6, 7]. Alternative
approaches that aim to the generation of integrands displaying only physical
singularities have also been discussed in this conference [8, 9, 10].

Interestingly, the causal representation, obtained as by-product of the novel
formulation of the loop-tree duality (LTD) formalism [5], has allowed to have
a complete understanding on the structure of multi-loop integrands, regardless
of the loop order. In particular, since LTD relies on the direct application
of the Cauchy residue theorem on the energy component of the loop momenta [11,
12, 13, 14, 15, 16, 17, 4, 5, 18, 19, 20, 21, 22, 23, 24, 25, 8, 26, 27],
causal integrands are expressed in terms of on-shell energies, whose
functional structure turns out to be equivalent to the integrands obtained
from the Old-Fashioned-Perturbation theory (OFPT) [28, 29]. However, by
performing a term-by-term comparison between both approaches, it turns out
that the causal representation turns out to be more compact than OFPT, which
is due to redundant terms that cancel pairwise in the latter, as observed in
Ref. [8] of one-loop scalar integrands.

In view of the structure of causal integrands obtained from LTD in Ref. [6],
we notice several similarities when computing integrands whose loop topologies
have the same number of vertices but differ on the number of edges. This
observation promoted to a study in which causal integrands can be directly
generated by means of the latter features and completely overcome the
application of LTD. In order to carry out this study, we systematically
consider all causal (or threshold) singularities of a given loop topology and
organise them according to their compatibility. Likewise, as it was point in
Ref. [1], the classification of loop topologies by vertices and edges, as well
as the generation of causal integrands, corresponds to a consequence of the
well-known Steinmann relations [30, 31, 32, 33, 34, 35, 36, 37] and allowed us
to provide a conjecture for the representation of all-loop Feynman integrands.
Steinmann relations have been considered in recent works [38, 39, 40, 41, 42]
that aim at bootstrapping analytic structure of multi-loop scattering
amplitudes.

Furthermore, inspired by the recent developments of LTD to numerically
evaluate multi-loop Feynman integrals and keep understanding mathematical
properties to causal integrands, we provided the publicly Mathematica package
Lotty [2] that illustrates and automates the generation of dual and causal
integrands, allowing, in this way, to provide the scientific community with
alternative strategies to overcome the obstacles in the calculation of multi-
loop scattering amplitudes [43, 44, 45].

This contribution is organised as follows. In Sec. 2 we briefly remark the
main features of the causal representation of multi-loop scattering
amplitudes. Then, in Sec. 3, we recall numerical stability and numerical
integration of causal integrands. Then, in Sec. 4, we draw a summary of the
talk and discuss future directions.

##  2 All-loop causal representation of Feynman integrands

In order to explicitly illustrate the causal representation of multi-loop
Feynman integrands, let us consider, for the sake of simplicity, the most
general loop topology built from five vertices and all possible connections
among edges (or internal lines). A pictorial representation of this six-loop
topology is depicted in Fig. 1, in which grey lines do not cross among
themselves. Hence, taking into account this topology, one can consider a five-
point six-loop Feynman integral with ten internal lines,

$\displaystyle\mathcal{A}_{5}^{\left(6\right)}\left(1,2,\ldots,10\right)$
$\displaystyle=\int_{\ell_{1},\ldots,\ell_{6}}\mathcal{N}\left(\left\\{\ell_{i}\right\\}_{6},\left\\{p_{j}\right\\}_{5}\right)\times
G_{F}\left(1,\ldots,10\right)\,,$ (1)

where $\int_{\ell_{s}}\equiv-\imath\mu^{4-d}\int
d^{d}\ell_{s}/\left(2\pi\right)^{d}$ and Feynman propagators $G_{F}$ are
compactly represented as follows,

$\displaystyle G_{F}\left(1,\ldots,r\right)$
$\displaystyle=\prod_{i=1}^{r}\left(G_{F}\left(q_{i}\right)\right)^{\alpha_{i}}\,,$
(2)

where $\alpha_{i}$ are powers of the propagators and, motivated by LTD, one
explicitly pulls out the dependence on the energy components of the loop
momenta in each internal line,

$\displaystyle G_{F}\left(q_{i}\right)$
$\displaystyle=\frac{1}{q_{i}^{2}-m_{i}^{2}+\imath
0}=\frac{1}{\left(q_{i,0}+q_{i,0}^{\left(+\right)}\right)\left(q_{i,0}-q_{i,0}^{\left(+\right)}\right)}\,.$
(3)

The latter corresponds to the usual Feynman propagator of a one single
particle, with $m_{i}$ its mass, $+\imath 0$ the infinitesimal imaginary
Feynman prescription and,

$\displaystyle q_{i,0}^{\left(+\right)}$
$\displaystyle=\sqrt{\boldsymbol{q}_{i}^{2}+m_{i}^{2}-\imath 0}\,,$ (4)

the on-shell energy of the loop momentum $q_{i}$ written in terms of the
spatial components $\boldsymbol{q}_{i}$.

Figure 1: Multi-loop topology with five vertices and ten internal lines.

Then, to describe the loop topology of Fig. 1, we work with the propagators,

$\displaystyle q_{i}=$ $\displaystyle\begin{cases}\ell_{i},\,&i=1,\ldots,6\\\
\ell_{1}-\ell_{4}-\ell_{5}-p_{1},&i=7\\\
\ell_{2}-\ell_{1}-\ell_{6}-p_{2},&i=8\\\
\ell_{3}-\ell_{2}+\ell_{5}-p_{3},&i=9\\\
\ell_{4}-\ell_{3}+\ell_{6}-p_{4}&i=10\end{cases}\quad\,,$ (5)

in which, after applying LTD, one recognises the set of causal propagators,

$\displaystyle\lambda_{1}^{\pm}=q_{\left(1,4,5,7\right),0}^{\left(+\right)}\pm
p_{1,0}\,,$
$\displaystyle\lambda_{12}^{\pm}=q_{\left(2,4,5,6,7,8\right),0}^{\left(+\right)}\pm
p_{12,0}\,,$
$\displaystyle\lambda_{24}^{\pm}=q_{\left(1,2,3,4,8,10\right),0}^{\left(+\right)}\pm
p_{24,0}\,,$
$\displaystyle\lambda_{2}^{\pm}=q_{\left(1,2,6,8\right),0}^{\left(+\right)}\pm
p_{2,0}\,,$
$\displaystyle\lambda_{13}^{\pm}=q_{\left(1,2,3,4,7,9\right),0}^{\left(+\right)}\pm
p_{13,0}\,,$
$\displaystyle\lambda_{35}^{\pm}=q_{\left(2,3,5,7,8,10\right),0}^{\left(+\right)}\pm
p_{35,0}\,,$
$\displaystyle\lambda_{3}^{\pm}=q_{\left(2,3,5,9\right),0}^{\left(+\right)}\pm
p_{3,0}\,,$
$\displaystyle\lambda_{23}^{\pm}=q_{\left(1,3,5,6,8,9\right),0}^{\left(+\right)}\pm
p_{23,0}\,,$
$\displaystyle\lambda_{34}^{\pm}=q_{\left(2,4,5,6,9,10\right),0}^{\left(+\right)}\pm
p_{34,0}\,,$
$\displaystyle\lambda_{4}^{\pm}=q_{\left(3,4,6,10\right),0}^{\left(+\right)}\pm
p_{4,0}\,,$
$\displaystyle\lambda_{45}^{\pm}=q_{\left(3,4,6,7,8,9\right),0}^{\left(+\right)}\pm
p_{45,0}\,,$
$\displaystyle\lambda_{25}^{\pm}=q_{\left(1,2,6,7,9,10\right),0}^{\left(+\right)}\pm
p_{25,0}\,,$
$\displaystyle\lambda_{5}^{\pm}=q_{\left(7,8,9,10\right),0}^{\left(+\right)}\pm
p_{5,0}\,,$
$\displaystyle\lambda_{14}^{\pm}=q_{\left(1,3,5,6,7,10\right),0}^{\left(+\right)}\pm
p_{14,0}\,,$
$\displaystyle\lambda_{15}^{\pm}=q_{\left(1,4,5,8,9,10\right),0}^{\left(+\right)}\pm
p_{15,0}\,.$ (6)

The causal representation of a topology built from five vertices is expected
to admit the decomposition in terms of the following summands,

$\displaystyle\frac{1}{\lambda_{i}^{h_{i}}\lambda_{j}^{h_{j}}\lambda_{k}^{h_{k}}\lambda_{l}^{h_{l}}}\,,$
(7)

where $i,j,k,l\in\\{1,2,3,4,5,12,13,14,15,23,24,25,34,35\\}$ and
$h_{\\{i,j,k,l\\}}\in\\{+,-\\}$. Nevertheless, not all products of
$\lambda_{i}$’s in (6) will appear in the representation, since there will be
products of two $\lambda_{i}^{\pm}$’s, that cannot appear because of the
incompatible alignment of internal and external momenta. To illustrate this
subtlety, let us consider a possible entanglement between
$\lambda_{12}^{\pm}$, $\lambda_{14}^{\pm}$ and $\lambda_{12}^{\pm}$,
$\lambda_{34}^{\pm}$. In Fig. 2 (left), we note that the causal thresholds
$\lambda_{12}^{\pm}$, $\lambda_{14}^{\pm}$ cannot be considered together when
providing a candidate for the causal representation, whereas,
$\lambda_{12}^{\pm}$, $\lambda_{34}^{\pm}$, in Fig. 2 (right), appear with
combinations of other $\lambda_{i}^{\pm}$’s. The very same analysis can be
applied to tuples of length three and four.

Figure 2: Loop topologies with (left:) two overlapped and (right:) two
entangled thresholds.

Interestingly, the above-mentioned features of overlapped and entangled causal
thresholds may be related to Steinmann relations, since they state that double
discontinuities in overlapping cuts vanish,

$\displaystyle\text{Disc}_{s_{I}}\left(\text{Disc}_{s_{J}}\mathcal{A}_{N}^{\left(L\right)}\right)=0\,,\quad\text{where
}\begin{cases}I\not\subset J\\\ J\not\subset I\end{cases}\,,$ (8)

where $\text{Disc}_{s_{I}}\mathcal{A}_{N}^{\left(L\right)}$ represents the
discontinuity of the scattering amplitude (or Feynman integral) of Eq. (1) at
the kinematic invariant $s_{I}=\left(\sum_{i\in I}p_{i}\right)^{2}$ generated
by the external momenta that belong to the set $I$. A naive interpretation of
the Steinmann relations within our framework may be given by promoting the
kinematic invariants $s_{I}$ to the sum of energies of external momenta
present in the causal propagators.

Hence, by constructing an Ansatz that accounts for the Steinmann relations and
reconstructing the analytic expressions for the causal integrand by means of
finite fields arithmetic [46, 47, 48, 49], we find for the six-loop five-point
scalar integral,

$\displaystyle\mathcal{A}_{5}^{\left(6\right)}=\int_{\ell_{1},\ldots,\ell_{6}}\frac{1}{x_{10}}\sum_{\begin{subarray}{c}i=1\\\
j=i+1\end{subarray}}^{5}\ \sum_{\begin{subarray}{c}k=1\\\ l=k+1\\\ k,l\neq
i,j\end{subarray}}^{5}L_{ij}^{+}\,L_{kl}^{-}\,,$ $\displaystyle
L_{ij}^{\pm}=\frac{1}{\lambda_{ij}^{\pm}}\left(\frac{1}{\lambda_{i}^{\pm}}+\frac{1}{\lambda_{j}^{\pm}}\right)\,,$
(9)

with $x_{n}=\prod_{i=1}^{n}2\,q_{i,0}^{\left(+\right)}$.

Notice that, in view of this very compact structure of the causal
representation of a six-loop integrand with five vertices, we proposed in Ref.
[1] an algorithm to generate causal representation of topologies constructed
from $k+2$ cusps. In fact, due to symmetry in the structure of topologies when
all possible connections, edges, between vertices are considered, we
conjectured a close formula when all possible edges are taken into account,

$\displaystyle\mathcal{F}_{L+k}=$
$\displaystyle\sum_{\begin{subarray}{c}i_{1}\ll i_{N_{i}}\\\ j_{1}\ll
j_{N_{j}}\end{subarray}}^{k+2}\Omega_{\vec{i}}^{\vec{j}}\ L_{i_{1}i_{2}\ldots
i_{N_{i}}}^{+}L_{j_{1}j_{2}\ldots j_{N_{j}}}^{-}\,,$ (10)

with,

$\displaystyle\Omega_{\vec{i}}^{\vec{j}}$
$\displaystyle=\begin{cases}1&\text{If }\vec{i}\cap\vec{j}=\emptyset\\\
0&\text{otherwise}\end{cases}\,,$ (11)

where $\vec{i}=\\{i_{1},i_{2},\ldots,i_{N_{i}}\\}$,
$\vec{j}=\\{j_{1},j_{2},\ldots,j_{N_{j}}\\}$, $i_{1}\ll i_{N_{i}}$ is the
lexicographic ordering $i_{1}<i_{2}<\cdots<i_{N_{i}}$,
$N_{i}=\left[k/2\right]+1$ and $N_{j}=k-\left[k/2\right]$. The functions
$L_{ijk\ldots}^{\pm}$ contain the causal information of the integrand,

$\displaystyle L_{i_{1}i_{2}\ldots
i_{N}}^{\pm}=\frac{1}{\lambda_{i_{1}i_{2}\ldots
i_{N}}^{\pm}}\sum_{\begin{subarray}{c}j_{1}\ll j_{N-1}\\\
\vec{j}\subset\vec{i}\end{subarray}}L_{j_{1}j_{2}\ldots j_{N-1}}^{\pm}\,,$
$\displaystyle L_{i_{1}}^{\pm}=\frac{1}{\lambda_{i_{1}}^{\pm}}\,.$ (12)

The close formula (10), together with the definitions of $L^{\pm}$, have been
implemented in the built-in routines of the Mathematica package Lotty (see
Sec. 3).

On top of decomposition (10), we would like to remark that an extensive study
of various representations of scattering amplitudes, including the ones
presented in this talk (LTD and causal), are currently under consideration
[50], where we elucidate a clear pattern to go from one representation to
another along the lines of Ref. [51].

##  3 Numerical evaluation of Feynman integrals

Figure 3: Numerical stability of causal integrands of two-loop planar (left)
and non-planar (right) scalar triangles as a function of $x_{1}$ and $x_{2}$
with fixed values for the angles $\theta_{ij}$.

In Refs. [2], we provided the Mathematica package Lotty that automates dual
and causal representation of LTD along the lines of Refs. [5] and [1],
respectively. This package is publicly available and can be downloaded from
the Bitbucket repository:

$\textrm{\href https://bitbucket.org/wjtorresb/lotty/src/master/}\,.$

The main motivation to provide this automation was to give the scientific
community with an illustrative and pedestrian overview of LTD by means of a
Mathematica package. In effect, to recreate the advantage of employing the
causal representation of multi-loop scattering amplitudes, we provide the
evaluation of planar ($I_{\text{p}}^{(2)}$) and non-planar
($I_{\text{np}}^{(2)}$) three-point integrals at two loops.

Furthermore, because of the causal representation of multi-loop scalar
integrands, conjectured in Ref. [1] and briefly summarised in Sec. 2, one can
simply make use of the built-in Lotty routine `AllCausal` and get the most
compact representation of the latter in terms of physical singularities,

  * In[1]:=

AllCausal[5]

  * Out[1]=

(L[-][{3,4}] + L[-][{3,5}] + L[-][{4,5}]) L[+][{1,2}]\+ (L[-][{2,4}] +
L[-][{2,5}] + L[-][{4,5}]) L[+][{1,3}]\+ (L[-][{2,3}] + L[-][{2,5}] +
L[-][{3,5}]) L[+][{1,4}]\+ (L[-][{2,3}] + L[-][{2,4}] + L[-][{3,4}])
L[+][{1,5}]\+ (L[-][{1,4}] + L[-][{1,5}] + L[-][{4,5}]) L[+][{2,3}]\+
(L[-][{1,3}] + L[-][{1,5}] + L[-][{3,5}]) L[+][{2,4}]\+ (L[-][{1,3}] +
L[-][{1,4}] + L[-][{3,4}]) L[+][{2,5}]\+ (L[-][{1,2}] + L[-][{1,5}] +
L[-][{2,5}]) L[+][{3,4}]\+ (L[-][{1,2}] + L[-][{1,4}] + L[-][{2,4}])
L[+][{3,5}]\+ (L[-][{1,2}] + L[-][{1,3}] + L[-][{2,3}]) L[+][{4,5}]

where, as mentioned in Sec. 2, each $L[\pm][\ldots]$ is expressed in terms of
physical singularities only. Besides, to explicitly represent causal
representations for planar and non-planar triangles in terms of on-shell
energies, one can yet make use of the additional Lotty routines:
`RefineCausal` and `Lamb2qij`.

Let us remark that within our automation in Mathematica we are not intending
to provide a full numerical integration of multi-loop Feynman integrals.
Moreover, we can profit from the latter and naively integrate our causal
integrands through the built-in function of Lotty for the change of variables,
e.g. a two-loop integrand in $d=4$ dimensions:

  * In[2]:=

dim = 4; (* set the dimension *)LoopMom = {l1, l2};(* list of loop momenta *)

  * In[3]:=

LoopToSC[LoopMom, dim]meassure[LoopMom, dim]intvars[LoopMom, dim,
"MaxValueR"->max]

  * Out[3]=

{l1 -> {r1 Cos[$\displaystyle\theta$11], r1 Cos[$\displaystyle\theta$12]
Sin[$\displaystyle\theta$11], r1 Sin[$\displaystyle\theta$11]
Sin[$\displaystyle\theta$12] }, l2 -> {r2 Cos[$\displaystyle\theta$21], r2
Cos[$\displaystyle\theta$22] Sin[$\displaystyle\theta$21], r2
Sin[$\displaystyle\theta$21] Sin[$\displaystyle\theta$22] } }

  * Out[4]=

$\displaystyle\text{r1}^{\text{2}}$ $\displaystyle\text{r2}^{\text{2}}$
Sin[$\displaystyle\theta$11] Sin[$\displaystyle\theta$21]

  * Out[5]=

{{r1,0,max}, {r2,0,max}, {$\displaystyle\theta$11,0,$\displaystyle\pi$},
{$\displaystyle\theta$12,0, 2$\displaystyle\pi$},
{$\displaystyle\theta$21,0,$\displaystyle\pi$},
{$\displaystyle\theta$22,0,2$\displaystyle\pi$}}

where, by default in the integration limits of each radius,
"MaxValueR"->$10^{5}$. However, to avoid considering the integration domain
$[0,\infty)$, one can map $[0,\infty)$ to the segment $(0,1]$, with the change
of variables, $r_{i}=(1-x_{i})/x_{i}$ . Additionally, expressions for on-shell
energies ($q_{i,0}^{(+)}$) can be also generated by Lotty, taking into account
the momentum components of the external kinematics. For instance, in the
three-point loop topology, one may choose, without loss of generality, the
center-of-mass frame (COMF),

$\displaystyle p_{1}^{\mu}=E_{\text{cm}}/2\\{1,1,0,0\\}$ $\displaystyle
p_{2}^{\mu}=E_{\text{cm}}/2\\{1,-1,0,0\\}\,,$ (13)

where $E_{\text{cm}}$ is the energy of COMF. In the two-loop planar triangle,
this amounts to,

  * Out[6]=

{$\displaystyle\text{q[1]}_{\text{0}}^{\text{(+)}}$ ->
$\displaystyle\sqrt{\text{$\displaystyle\text{r1}^{\text{2}}$\ +\
$\displaystyle\text{m[1]}^{\text{2}}$}}$,
$\displaystyle\text{q[2]}_{\text{0}}^{\text{(+)}}$ ->
$\displaystyle\frac{\text{1}}{\text{2}}$$\displaystyle\sqrt{\text{$\displaystyle\text{Ecm}^{\text{2}}$\
+\ 4\ Ecm\ r1\ Cos[$\displaystyle\theta$11]\ +\ 4\
($\displaystyle\text{r1}^{\text{2}}$\ +\
$\displaystyle\text{m[2]}^{\text{2}}$)}}$,
$\displaystyle\text{q[3]}_{\text{0}}^{\text{(+)}}$ ->
$\displaystyle\sqrt{\text{$\displaystyle\text{r1}^{\text{2}}$\ +\
$\displaystyle\text{m[3]}^{\text{2}}$}}$,
$\displaystyle\text{q[4]}_{\text{0}}^{\text{(+)}}$ ->
$\displaystyle\sqrt{\text{$\displaystyle\text{r2}^{\text{2}}$\ +\
$\displaystyle\text{m[4]}^{\text{2}}$}}$,
$\displaystyle\text{q[5]}_{\text{0}}^{\text{(+)}}$ ->
$\displaystyle\sqrt{\text{$\displaystyle\text{r2}^{\text{2}}$\ +\
$\displaystyle\text{m[5]}^{\text{2}}$}}$,
$\displaystyle\text{q[6]}_{\text{0}}^{\text{(+)}}$ ->
$\displaystyle\sqrt{\text{$\displaystyle\text{r1}^{\text{2}}$+$\displaystyle\text{r2}^{\text{2}}$+$\displaystyle\text{m[6]}^{\text{2}}$+2r1r2\
(Cos[$\displaystyle\theta$11]Cos[$\displaystyle\theta$21]\ +\
Cos[$\displaystyle\theta$12-$\displaystyle\theta$22]Sin[$\displaystyle\theta$11]Sin[$\displaystyle\theta$21])}}$
}

The numerical stability of causal integrands for planar and non-planar two-
loop triangles in terms of the compactified variable $x_{i}$ and angles
$\theta_{ij}$ is displayed in Fig. 3.

| Planar triangle | Non-planar triangle
---|---|---
$\frac{s}{m^{2}}$ | LTD ($10^{-6}$) |  SecDec 3.0 ($10^{-6}$) | LTD ($10^{-6}$) |  SecDec 3.0 ($10^{-6}$)
$-\frac{1}{4}$ | 9.48(5) | 9.4647(9) | 4.461(3) | 4.4606(4)
$-1$ | 8.10(5) | 8.0885(8) | 4.101(3) | 4.1012(4)
$-\frac{9}{4}$ | 6.49(3) | 6.4760(6) | 3.627(5) | 3.6276(3)
$-4$ | 5.02(2) | 5.0188(5) | 3.15(5) | 3.1334(3)
$+\frac{1}{4}$ | 10.68(6) | 10.651(1) | 4.743(3) | 4.7436(4)
$1$ | 13.11(8) | 13.070(1) | 5.259(3) | 5.2590(5)
$+\frac{9}{4}$ | 20.81(1) | 20.748(2) | 6.533(3) | 6.5331(6)
$+\frac{25}{16}$ | 15.74(9) | 15.700(1) | 5.748(3) | 5.7474(6)
Table 1: Numerical integration with NIntegrate of planar and non-planar two-
loop triangles in the causal representation. The values of integrations are
compared with SecDec 3.0.

Hence, one can naively make use of the built-in Mathematica routine
`NIntegrate` to integrate our expressions. A few representative results are
collected in Table 1, where we emphasise that no optimisation or educated use
of the latter routine is carried out. Likewise, to compare our numerical
integrations, we make use of publicly automated codes that evaluates the
latter by means of the sector decomposition algorithm [52, 53, 54, 55], SecDec
3.0 [56] and Fiesta 4.2 [57].

In view of the interesting compact structure obtained for multi-loop causal
integrands, it is worth dedicating further studies to numerical integrations.
The natural extension is having a stable integrator that systematically allows
to improve precision in evaluations. To this end, one can rely on publicly
available implementations for multi-dimensional numerical integrations, e.g.
Cuba library [58, 59], and interface it with Lotty to, thus, keep control of
the numerical evaluation from the symbolic side [60].

##  4 Discussion

The loop-tree duality (LTD) formulation has become a novel alternative
approach to numerically evaluate multi-loop scattering amplitudes, which in
the current era is one of the main obstacles to go further in theoretical
predictions. However, dual integrands, or expressions obtained from the direct
application of LTD on multi-loop Feynman integrands, are not the ultimate
representation, since it yet contains so-called spurious contributions but
they can yet be employed to understand LTD decompositions in scattering
amplitudes, regardless of the loop order [5].

In this talk, we introduced the concept of causal representation of multi-loop
scattering amplitudes, which in a nutshell corresponds to having an integrand
displaying only physical singularities. Remarkably, the Old-Fashioned-
Perturbation theory (OFPT) aims at displaying a very same pattern for Feynman
integrands. However, these OFPT integrands turn out to be explicitly
classified according to number of vertices and not accounting for number of
edges (or internal lines). In the causal representation, generated from LTD,
the situation is slightly different, the structure of causal integrands
accounts for both vertices and edges. In effect, as elucidated in Ref. [1], it
is possible to generate a representation for all-loop scalar integrands, by
accounting for the features of the loop topology, given by vertices and edges.

Furthermore, we remarked that the all-loop causal representation for multi-
loop Feynman integrands has been conjectured by keeping into account the
Steinmann relations. Moreover, to give support to this conjecture, we provided
a Mathematica implementation of the latter by means of the automated package
Lotty [2], in which a direct comparison between LTD and causal integrands can
straightforwardly be performed.

Different representations of scattering amplitudes are currently appearing,
the main question to address is: what is then the best representation?
Perhaps, there is no a unique answer and, on the contrary, one could think of
asking further and being more specific. For instance, what would we like to
extract from a given representation? In this case, one could put on the table
all available representations and extract the most convenient one for the
desirable study [50].

In the spirit of profiting from the compact structure of the causal
representation of multi-loop Feynman integrands, we observed a smooth
behaviour in their numerical evaluation. Therefore, to integrate the latter,
this is a very important insight since one can only focus on the treatment of
physical singularities. We illustrated that by simply using the built-in
Mathematica routines with no optimisation a preliminary numerical integration
can yet be carried out. However, it is worth investigating further the
interplay between analytic expressions provided by Lotty and the various
implementations that aim at having full control on multi-dimensional numerical
integration [60].

## Acknowledgements

The author wishes to thank Nima Arkani-Hamed, Andrea Banfi, Martin Beneke,
Johannes Henn, Jonas Lindert, Sebastian Mizera, and Tiziano Peraro for
enlightening discussions.

## References

  * [1] W. J. Torres Bobadilla,  _Loop-tree duality from vertices and edges_ ,  JHEP 04, 183 (2021),  10.1007/JHEP04(2021)183,  2102.05048.
  * [2] W. J. Torres Bobadilla,  _Lotty – The loop-tree duality automation_ ,  Eur. Phys. J. C 81(6), 514 (2021),  10.1140/epjc/s10052-021-09235-0,  2103.09237.
  * [3] J. Aguilera-Verdugo, R. Hernández-Pinto, S. Ramírez-Uribe, G. Rodrigo, G. Sborlini and W. J. Torres Bobadilla,  _Manifestly causal scattering amplitudes_ ,  Letter of Intention submitted to SnowMass (2021).
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