aainstitutetext: LAPTh, Université Savoie Mont Blanc, CNRS, B.P. 110, F-74941
Annecy-le-Vieux, Francebbinstitutetext: Max-Planck-Institut für Physik,
Werner-Heisenberg-Institut, D-80805 München, Germany

# Pentagon Wilson loop with Lagrangian insertion at two loops in
${\mathcal{N}}=4$ super Yang-Mills theory

Dmitry Chicherin b Johannes Henn<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

We compute the two-loop result for the null pentagonal Wilson loop with a
Lagrangian insertion (normalized by the Wilson loop without insertion) in
planar, maximally supersymmetric Yang-Mills theory. This finite observable is
closely related to the Amplituhedron, and it is reminiscent of finite parts of
planar two-loop five-particle scattering amplitudes. We verify that, up to
this loop order, the leading singularities are given by the same conformally
invariant expressions that appear in all-plus pure Yang-Mills amplitudes. The
accompanying weight-four transcendental functions are expressed in terms of
the pentagon functions space known from planar two-loop five-particle
amplitudes, but interestingly only a subset of the functions appears. Being a
function of four dimensionless variables, the observable has interesting
asymptotic limits. We verify that our analytic result is consistent with soft
and collinear limits, and find an intriguingly simple pattern in the multi-
Regge limit. Thanks to the new result we can also conjecturally predict, for
general kinematics, the maximal weight piece of the planar three-loop five-
particle all-plus amplitude in pure Yang-Mills theory. Motivated by the
Amplituhedron geometry, we investigate positivity properties of the integrated
answer. Generalizing previous results at four particles, we find numerical
evidence that the two-loop five-particle result has uniform sign in a
kinematic region suggested by the loop Amplituhedron.

††preprint: LAPTH-013/22, MPP-2022-37

##  1 Introduction

The duality between scattering amplitudes and null polygonal Wilson loops in
maximally supersymmetric Yang-Mills theory (${{\mathcal{N}}=4}$ sYM)
Alday:2007hr ; Drummond:2007aua ; Brandhuber:2007yx  has led to numerous
advances in the area of scattering amplitudes. For example, in
${{\mathcal{N}}=4}$ sYM the planar loop integrand is determined from powerful
recursion relations ArkaniHamed:2010kv , or, surprisingly, via a beautiful
geometric construction Arkani-Hamed:2013jha . Scattering amplitudes with many
external particles and to high loop orders can be bootstrapped based on
educated guesses of the relevant function space, and many further insights
Dixon:2011pw ; Drummond:2014ffa ; Caron-Huot:2016owq ; Caron-Huot:2019bsq ;
Caron-Huot:2020bkp . At the same time, the novel insights into the
relationship between certain loop integrands and transcendental functions
ArkaniHamed:2010gh ; Henn:2013pwa ; Henn:2020omi , and tools for handling
these functions Goncharov:2010jf ; Duhr:2019tlz , have had a crucial impact on
computing loop integrals in QCD that are relevant for phenomenology.

However, there are two issues where further progress would be very important.
Firstly, despite the novel insights on loop integrands, only in few cases it
is actually possible to perform the integrations directly. (See
Herrmann:2019upk  for some ideas.) One important technical difficulty is the
fact that although the remainder and ratio functions Drummond:2008vq  are
infrared and ultraviolet finite, it is not known how to write them in a
manifestly finite way. A second issue is that due to dual conformal symmetry
Drummond:2007au , non-trivial amplitudes that can be studied in
${{\mathcal{N}}=4}$ sYM have at least $n\geq 6$ particles, and they depend on
$(3n-15)$ dimensionless kinematic variables. This is to be compared with
amplitudes in QCD, which depend on $(3n-11)$ kinematic variables, and where
already the cases $n=4,5$ are of enormous current interest, see for example
Czakon:2021mjy ; Caola:2021izf  and references therein. It would therefore be
desirable to have a relevant observable in ${{\mathcal{N}}=4}$ sYM that, for
the same number of particles, has similar variable dependence as QCD
amplitudes, and ideally could be computed in four dimensions, to fully exploit
the novel integrand insights.

Perhaps surprisingly, such an observable exists, and it is closely related to
the Wilson loops and scattering amplitudes usually studied. Consider an
$n$-cusp null polygonal Wilson loop with a Lagrangian insertion, normalized by
the Wilson loop without insertion Alday:2011ga ,

$\displaystyle F_{n}={\pi^{2}}\frac{\langle{W_{n}{\cal
L}(x_{0})}\rangle}{\langle{W_{n}}\rangle}\,.$ (1)

This object has a number of desirable features. All cusp divergences of the
Wilson loops cancel in the ratio, and its dual conformal symmetry is realized
without anomalies. As a result, the $n$-point observable is a function of
$(3n-11)$ dimensionless variables, just like generic QFT amplitudes.
Furthermore, one can fix a dual conformal frame where the insertion point is
sent to infinity, in which case the kinematic variables become identical to
those for $n$-particle scattering amplitudes. This means that already at four
particles, one obtains a non-trivial function of one variable, $z=t/s$. The
latter has been computed perturbatively at weak Alday:2011ga ; Alday:2012hy ;
Alday:2013ip , and at strong coupling Alday:2011ga .

The new observable is closely related to the Wilson loop / scattering
amplitude. Consider differentiating $\log\langle{W_{n}}\rangle$ in the
coupling. In this way, one obtains precisely the ratio (1) integrated over the
Lagrangian insertion point $x_{0}$. In this sense the new observable can be
thought of as an integrand of the (logarithm of the) Wilson loop / scattering
amplitude, but where all except the final $x_{0}$ integrations have been
carried out. It is the final integration over $x_{0}$ that leads to the
divergences in the Wilson loop / scattering amplitude. This fact has recently
been used to compute the four-loop non-planar cusp anomalous dimension
Henn:2019swt .

What structures does one find in the perturbative expansion of this new
quantity $F_{n}$? As we will see, thanks to its definition, it inherits many
properties of scattering amplitudes. For example, the $L$-loop contribution is
expected to be a sum over a set of leading singularities, multiplied by
transcendental functions of uniform weight $2L$. Just like for scattering
amplitudes, there are very natural questions that we can ask: What set of
leading singularities can appear? What is the space of transcendental
functions that can appear?

In a recent paper Chicherin:2022bov , the present authors studied the
structural and symmetry properties of the $n$-point observable, and gave first
answers to these questions. Building upon formulas and intuition from
scattering amplitudes, we found a simple form of the leading singularities of
$F_{n}$ in terms of a Grassmannian integral. This integral just produces one
leading singularity at four points, which can be chosen as an overall
normalization factor. The first non-trivial case is then five points, where we
find six linearly independent leading singularities. Amazingly, the leading
singularities are the same (once multiplied with an overall Parke-Taylor
factor) as those observed to appear in all-plus helicity amplitudes in QCD
Badger:2019djh ; Henn:2019mvc , and they share the same conformal symmetry
Chicherin:2022bov . This is not at all obvious, but can be shown for the
Grassmannian formula given in Chicherin:2022bov . Whether all leading
singularities are given by this formula is an open question.

In the same paper Chicherin:2022bov , the $F_{n}$ was evaluated at tree-level
and one-loop, providing evidence for the proposed leading singularity formula.
Moreover, the one-loop result gives first examples of the types of
transcendental functions one can expect. It was found that the answer can be
written in terms of finite one-loop integrals, and hence is reminiscent of
finite parts of one-loop $n$-particle scattering amplitudes. What is more, a
very curious relationship to the pure Yang-Mills all-plus amplitude was found:
$F_{n}$ at $L$ loops is equal, up to an overall normalization factors, to the
leading transcendental terms of the $(L+1)$-loop all-plus scattering
amplitudes. This was shown by explicit computation at $L=0$ and $L=1$ for any
$n$, and at $L=2$ for $n=4$.

Thanks to the connection to scattering amplitudes, one can also make progress
on the second question raised above, namely which transcendental functions can
appear. The all-loop integrand of planar ${{\mathcal{N}}=4}$ sYM
ArkaniHamed:2010kv ; Arkani-Hamed:2013jha  can be beautifully defined
geometrically in terms of the Amplituhedron Arkani-Hamed:2013jha . Very
recently, it was shown that a generalization of the Amplituhedron idea allows
one to express the new observable in terms of a sum of integrals associated to
certain negative geometries. The latter define a novel type of finite, four-
dimensional loop integrals Arkani-Hamed:2021iya . These integrals were
analyzed so far at four points, where certain infinite classes of integrals
were computed.

The Amplituhedron story also motivates to look for positivity properties of
scattering amplitudes Arkani-Hamed:2014dca ; Dixon:2016apl . Indeed, in
Arkani-Hamed:2021iya  it was found that the four-point results have definite
sign properties. It would be very interesting to study this property for
higher-point amplitudes.

Motivated by these new developments and findings, in this paper we perform a
perturbative two-loop calculation of the null pentagonal Wilson loop with
Lagrangian insertion. In this way we can test the observations of
Chicherin:2022bov  regarding leading singularities at two loops, which is
highly non-trivial. Moreover, this is the first time that this quantity is
evaluated beyond one loop beyond four points. The resulting transcendental
functions are expected to be of weight four and depend on four independent
kinematic variables, which is very interesting. For example, we test the
behaviour of the function in soft and collinear limits, and study its multi-
Regge limit. We also test positivity properties of the integrated answer.

This work is organized as follows. In Section 2, we recall the definition of
the finite observable (1), as well as perturbative results in the four-
particle case, and their key properties. In Section 3, we discuss the five-
particle observable. We present its leading singularities and recall
definitions of the pentagon functions. The latter turn out to be the
transcendental functions which describe loop corrections of the five-particle
observable. At the end of the Section we present our main result, the five-
particle observable in the two-loop approximation. In Section 4, we outline
the two-loop calculation. We construct the integrands from the five-particle
MHV amplitude integrands available in the literature, and then perform loop
integrations. We also calculate soft, collinear and multi-Regge limits of the
five-particle observable. In Section 5, we discuss positivity properties of
the five-particle perturbative corrections. In Section 6, we review the
conjectured duality between the finite observable of Section 3 and the five-
particle all-plus amplitude in pure Yang-Mills theory. Based on our results,
we predict the maximally transcendental part of the planar three-loop all-plus
amplitude in pure Yang-Mills. We mention directions for future research in
Section 7. In the Appendix, we recall definitions of the planar pentagon
functions and their basic properties.

##  2 The $n$-cusp Wilson loop with Lagrangian insertion

###  2.1 Definition and symmetry properties

Let us define the ratio of the Wilson loop correlation function with the
Lagrangian of ${\cal N}=4$ sYM and the vacuum expectation value of the Wilson
loop in the fundamental representation of the gauge group $SU(N_{c})$,

$\displaystyle\frac{1}{\pi^{2}}F_{n}(x_{1},\ldots,x_{n};x_{0})=\frac{\langle{W_{F}[x_{1},\ldots,x_{n}]\,{\cal
L}(x_{0})}\rangle}{\langle{W_{F}[x_{1},\ldots,x_{n}]}\rangle}\,,$ (2)

following the conventions of ref. Chicherin:2022bov . (For $n=4$ points, a
slightly different normalization was used in ref. Arkani-Hamed:2021iya .) The
Wilson loop contour $[x_{1},\ldots,x_{n}]$ is an $n$-cusp polygon with light-
like edges, that is

$\displaystyle(x_{i+1}-x_{i})^{2}=0,\qquad i=1,\ldots,n,$ (3)

embedded in Minkowski space, $x_{n+1}\equiv x_{1}$. In the following we work
in the planar limit $N_{c}\to\infty$. The Lagrangian in (2) is the so-called
chiral and on-shell form of the Lagrangian ${\cal L}$, which coincides with
the standard Lagrangian of ${\cal N}=4$ sYM modulo total derivatives and
equations of motion (see definition in eq. (A.13) of Eden:2011yp ). This
choice is motivated by the correlator/amplitude duality Eden:2010ce ;
Eden:2010zz ; Eden:2011yp  (see Chicherin:2022bov  for a discussion). The
ratio $F_{n}$ contains parity-odd contributions at $n>4$ because of chirality
of the Lagrangian insertion.

Being a finite quantity in $d=4$ dimensions, the ratio $F_{n}$ in eq. (2) has
a dual conformal symmetry, i.e. a conformal symmetry acting on the $x_{i}$
variables, with dual conformal weight zero at points $x_{i}$, $i\in\\{1,\ldots
n\\}$, and dual conformal weight $+4$ at the Lagrangian insertion point
$x_{0}$. The symmetry implies that $F_{n}$ can be written as a factor carrying
the dual conformal weights, multiplied by a function of $(3n-11)$ dual-
conformal cross-ratios.

The perturbative expansion of $F_{n}$ in the coupling $g^{2}=g_{\rm
YM}^{2}N_{c}/(16\pi^{2})$ is as follows

$\displaystyle F_{n}=\sum_{L\geq 0}(g^{2})^{1+L}F_{n}^{(L)}\,.$ (4)

In the following we often refer to $F^{(0)}_{n}$ as the Born-level observable,
and $F^{(L)}_{n}$ with $L>0$ as the $L$-loop observable. The expected uniform
transcendental weight property of ${\cal N}=4$ sYM theory suggests that the
terms of the perturbative expansion should be of the following form

$\displaystyle F_{n}^{(L)}=\sum_{s}R_{n,s}\,g^{(L)}_{n,s}\,.$ (5)

Here $\\{R_{n,s}\\}$ is a set of leading singularities labeled with an index
$s$, which are rational functions of $x_{1},\ldots,x_{n},x_{0}$, and the
accompanying $g^{(L)}_{n,s}$ are pure functions of transcendental weight $2L$.
In particular, at the Born-level ($L=0$) we have $g^{(0)}_{n,s}=1$, and
$F_{n}^{(0)}$ is a rational function. The leading singularities carry a dual
conformal weight $+4$ at $x_{0}$, and $g_{n,s}^{(L)}$ are functions of
$(3n-11)$ cross-ratios. Conjecturally, all planar leading singularities
$R_{n,s}$ at any $n$ are known explicitly Chicherin:2022bov .

We saw that $F_{n}$ has a dual conformal symmetry in $x$-coordinates. We can
use dual-conformal transformations, without loss of generality, to send
$x_{0}$ to space-time infinity,

$\displaystyle
f_{n}(x_{1},\ldots,x_{n}):=\lim_{x_{0}\to\infty}(x_{0}^{2})^{4}\,F_{n}(x_{1},\ldots,x_{n};x_{0})\,.$
(6)

This amounts to fixing a dual-conformal frame. This object contains the same
amount of information as $F_{n}$, and the dependence on $x_{0}$ can be easily
restored for a given $f_{n}$. We also define the leading singularities of
$F_{n}$ in this frame,

$\displaystyle
r_{n,s}(x_{1},\ldots,x_{n}):=\lim_{x_{0}\to\infty}(x_{0}^{2})^{4}R_{n,s}(x_{1},\ldots,x_{n};x_{0})\,.$
(7)

In other words, $\\{r_{n,s}\\}$ are leading singularities of $f_{n}$. The pure
functions $g^{(L)}_{n,s}$ from eq. (5) keep their form in the frame
$x_{0}\to\infty$.

The frame $x_{0}\to\infty$ has the advantage that it is easy to connect the
kinematics to familiar momentum space terminology. This is seen as follows. In
this frame, the leading singularities are Poincaré-invariant functions of $n$
points separated by light-like distances, i.e. they are invariant under space-
time shifts and Lorentz transformations. Equivalently, we can say that they
are functions of $n$ light-like momenta ($p_{i}^{2}=0$)

$\displaystyle p_{i}=x_{i}-x_{i-1}$ (8)

(with $x_{n+1}\equiv x_{1}$) which satisfy momentum conservation (see Fig. 1)

$\displaystyle\sum_{i=1}^{n}p_{i}=0\,.$ (9)

As usual, it is convenient to introduce helicity spinor parametrization of the
light-like momenta, $(p_{i})^{{\dot{\alpha}}\alpha}\equiv
p^{\mu}_{i}\sigma_{\mu}^{{\dot{\alpha}}\alpha}=\lambda_{i}^{\alpha}\tilde{\lambda}^{{\dot{\alpha}}}_{i}$.
Since we have already fixed dual conformal symmetry, this is exactly the same
kinematics as for a non-dual-conformal scattering amplitude of $n$ massless
particles. Indeed, we will see that the known results bear close resemblance
to scattering amplitudes in non-supersymmetric Yang-Mills theory. In the
following we adopt the amplitude terminology and refer to $f_{n}$ as
$n$-particle.

Figure 1: Momenta and dual-momenta assignment to the five-cusp light-like
Wilson loop contour.

In ref. Chicherin:2022bov  it was found that $F_{n}$ has a hidden conformal
symmetry (and a Yangian-like symmetry). The latter is best seen in the frame
$x_{0}\to\infty$, and when multiplying $f_{n}$ with the Parke-Taylor factors

$\displaystyle{\rm
PT}_{n}:=\frac{1}{\langle{12}\rangle\langle{23}\rangle\ldots\langle{(n-1)n}\rangle\langle{n1}\rangle}\,,$
(10)

where $\langle{ij}\rangle:=\lambda^{\alpha}_{i}\lambda_{j\alpha}$ (later on,
we will also need the conjugated spinor bracket
$[ij]:=\tilde{\lambda}_{i\,\dot{\alpha}}\tilde{\lambda}_{j}^{\dot{\alpha}}$).
It was found Chicherin:2022bov  that the leading singularities of $f_{n}$ are
conformally invariant, namely

$\displaystyle\mathbb{K}_{\alpha{\dot{\alpha}}}\left({\rm
PT}_{n}r_{n,s}\right)=0\,,$ (11)

with the conformal boost generator in momentum space of $n$-particle
scattering Witten:2003nn

$\displaystyle\mathbb{K}_{\alpha{\dot{\alpha}}}=\sum_{i=1}^{n}\frac{\partial^{2}}{\partial\lambda_{i}^{\alpha}\partial\tilde{\lambda}_{i}^{\dot{\alpha}}}\,.$
(12)

###  2.2 Review of perturbative results in the four-particle case

Let us illustrate these definitions for the known four-particle results. These
will also be useful in the following, as they serve as important consistency
checks for our novel five-particle results. Dual conformal symmetry tells us
that we can write

$\displaystyle
F_{4}=\frac{x_{13}^{2}x_{24}^{2}}{x_{01}^{2}x_{02}^{2}x_{03}^{2}x_{04}^{2}}G_{4}(g;z)\,,$
(13)

where $x_{ab}^{2}:=(x_{a}-x_{b})^{2}$ and $z$ is the unique dual-conformal
cross-ratio at $n=4$,

$\displaystyle
z=\frac{x_{13}^{2}x_{02}^{2}x_{04}^{2}}{x_{24}^{2}x_{01}^{2}x_{03}^{2}}\,.$
(14)

Equivalently, in the frame $x_{0}\to\infty$, we write the dual-conformal
cross-ratio (in momentum-space notation) as

$\displaystyle z=\frac{t}{s}\,$ (15)

in terms of the Mandelstam variables specifying the four-particle massless
kinematics,

$\displaystyle s=(p_{1}+p_{2})^{2}\;,\quad t=(p_{2}+p_{3})^{2}\,.$ (16)

There are several kinematic regions of the four-particle scattering. We will
be primarily concerned with the Euclidean region: $s<0$ and $t<0$. In this
region $z>0$.

The function $G_{4}$ has been computed to three loops in perturbation theory
Alday:2012hy ; Alday:2013ip ; Henn:2019swt . Here we reproduce the first three
terms of the expansion, writing

$\displaystyle G_{4}=\sum_{L\geq 0}(g^{2})^{1+L}G_{4}^{(L)}\,.$ (17)

The explicit polylogarithmic expressions are as follows,

$\displaystyle G^{(0)}_{4}(z)=$ $\displaystyle-1\,,$ (18) $\displaystyle
G^{(1)}_{4}(z)=$ $\displaystyle\log^{2}z+\pi^{2}\,,$ (19) $\displaystyle
G^{(2)}_{4}(z)=$
$\displaystyle-\frac{1}{2}\log^{4}z+\log^{2}z\left[\frac{2}{3}\text{Li}_{2}\left(\frac{1}{z+1}\right)+\frac{2}{3}\text{Li}_{2}\left(\frac{z}{z+1}\right)-\frac{19\pi^{2}}{9}\right]$
(20) $\displaystyle+\log
z\left[4\text{Li}_{3}\left(\frac{1}{z+1}\right)-4\text{Li}_{3}\left(\frac{z}{z+1}\right)\right]+\frac{2}{3}\left[\text{Li}_{2}\left(\frac{1}{z+1}\right)+\text{Li}_{2}\left(\frac{z}{z+1}\right)-\frac{\pi^{2}}{6}\right]^{2}$
$\displaystyle+\frac{8}{3}\pi^{2}\left[\text{Li}_{2}\left(\frac{1}{z+1}\right)+\text{Li}_{2}\left(\frac{z}{z+1}\right)-\frac{\pi^{2}}{6}\right]+8\text{Li}_{4}\left(\frac{1}{z+1}\right)+8\text{Li}_{4}\left(\frac{z}{z+1}\right)-\frac{23\pi^{4}}{18}\,.$

Each $G_{4}^{(L)}$ is a pure function of uniform transcendental degree $2L$.
The provided polylogarithmic expressions are single-valued in the Euclidean
region $z>0$.

The Amplituhedron construction provides a natural way to define the planar
integrand of $F_{4}$ directly from geometric equations Arkani-Hamed:2021iya .
In this approach, the loop corrections (18)-(20) are obtained as sums of
finite four-dimensional Feynman integrals corresponding to negative
geometries.

The Amplituhedron also suggests to look for positivity properties of
integrated functions Arkani-Hamed:2014dca ; Dixon:2016apl . The four-point
one-loop Amplituhedron corresponds to the kinematic region $z>0$, so it is
interesting to investigate the sign properties of $f_{4}^{(L)}$ in that
region. Indeed, it was found in ref. Arkani-Hamed:2021iya  that the available
four-point loop corrections are either only positive or only negative in the
Euclidean region $z>0$, and the sign alternates with the loop order,

$\displaystyle f_{4}^{(0)}(z)\bigr{|}_{z>0}<0\,,\quad
f_{4}^{(1)}(z)\bigr{|}_{z>0}>0\,,\quad
f_{4}^{(2)}(z)\bigr{|}_{z>0}<0\,,\quad\ldots$ (21)

The available perturbative data suggests that $f_{4}^{(L)}=stG_{4}^{(L)}$ is
positive within the Euclidean region at odd $L$, and it is negative at even
$L$.

Comparing to eq. (5), we see that in the four-point case, there is only one
leading singularity, namely

$\displaystyle
R_{4,0}=\frac{{x_{13}^{2}x_{24}^{2}}}{{x_{01}^{2}x_{02}^{2}x_{03}^{2}x_{04}^{2}}},$
(22)

and $g_{4,0}=G_{4}$ in that formula. It is instructive to present the leading
singularity in momentum space and to discover its hidden conformal properties.
Employing eq. (7) and passing to momentum-space notation, we have

$\displaystyle r_{4,0}=s\,t\,.$ (23)

A quick calculation shows that after normalizing by the Parke-Taylor factor
(10),

$\displaystyle{\rm
PT}_{4}\,r_{4,0}=-\frac{[12]^{2}}{\langle{34}\rangle^{2}}\,.$ (24)

In this form, the conformal invariance of eq. (11) is manifest, since the
conformal boost operator in eq. (12) annihilates any expression that depends
on $\lambda_{i}$ and $\tilde{\lambda}_{j}$ with $i\neq j$ only.

This conformal invariance is related to another surprising property of
$F_{n}$, also discovered in ref. Chicherin:2022bov , namely a duality to all-
plus amplitudes in the pure Yang-Mills theory. In particular, as discussed in
Sect. 6,

$\displaystyle A^{(1)}_{{\rm YM},4}={\rm PT}_{4}\,r_{4,0}$ (25)

is the four-particle one-loop all-plus amplitude in the pure Yang-Mills
theory.

In summary, based on the known perturbative results we can say that the
(planar) four-point observable $F_{4}$ has a number of remarkable properties,
namely:

  * •

$F^{(L)}_{4}$ has uniform transcendental weight $2L$;

  * •

positivity in the kinematic region defined by the Amplituhedron kinematics;

  * •

hidden conformal symmetry of leading singularities;

  * •

duality to four-point all-plus amplitude in pure Yang-Mills theory.

The main goal of the present paper is to investigate to which extent these
properties generalize to the five-point observable $F_{5}$. In order to do so
we compute, for the first time, the two-loop corrections to $F_{5}$.

##  3 Five-particle observable and novel two-loop result

The five-particle case is much richer than the four-particle one. Indeed,
$F_{5}$ involves four independent dual conformal cross-ratios. Consequently,
the loop corrections involve multivariable transcendental functions. We
provide evidence that they are the pentagon functions, which describe five-
particle massless scattering amplitudes in nonsupersymmetric gauge theories.

###  3.1 Five-particle kinematics

As in the previous section, we find it convenient to work in the gauge
$x_{0}\to\infty$. Let us introduce notations for the five-particle kinematics.
In order to specify the kinematic configuration, we choose five adjacent two-
particle Mandelstam variables

$\displaystyle{\bf
s}:=\\{s_{12}\,,\;s_{23}\,,\;s_{34}\,,\;s_{45}\,,\;s_{15}\\}\,,\qquad
s_{ij}:=(p_{i}+p_{j})^{2}\,,$ (26)

as well as the parity-odd Lorentz invariant

$\displaystyle\epsilon_{5}:=4\textup{i}\epsilon_{\mu\nu\rho\sigma}p_{1}^{\mu}p_{2}^{\nu}p_{3}^{\rho}p_{4}^{\sigma}\,.$
(27)

The latter is not fully independent of the Mandelstam variables (26). Indeed,
$\Delta:=(\epsilon_{5})^{2}$ is a quartic polynomial in $s_{ij}$ given by the
Gram determinant of four independent momenta,
$\Delta=\det\bigl{(}s_{ij}|^{4}_{i,j=1}\bigr{)}$. Nevertheless, the parity-odd
$\epsilon_{5}$ helps to distinguish kinematic configurations related by the
parity conjugation, $\epsilon_{5}\to-\epsilon_{5}$.

Our choice of the kinematic variables reflects the discrete symmetries of
$F_{5}$, or equivalently of $f_{5}$. Indeed, the geometry of the Wilson loop
contour in (2) is invariant under the cyclic shift $x_{i}\to x_{i+1}$ and the
reflection $x_{i}\to x_{6-i}$. In momentum space notations, the first is given
by the cyclic shift of momenta labels,

$\displaystyle\tau(p_{i}):=p_{i+1}\,,$ (28)

with $p_{6}\equiv p_{1}$, and the latter is equivalent to the reflection of
momenta labels, i.e.

$\displaystyle\rho(p_{i}):=p_{6-i}\,,\quad i=1,\ldots,5\,.$ (29)

These discrete transformations $\tau$ and $\rho$ form the dihedral group

$\displaystyle\tau^{5}=\rho^{2}=(\tau\rho)^{2}=(\rho\tau)^{2}=\mathbf{1}\,.$
(30)

Let us note that $\epsilon_{5}$ is invariant under the dihedral
transformations, $\tau(\epsilon_{5})=\rho(\epsilon_{5})=\epsilon_{5}$.

Similarly to the four-particle case, we specify the Euclidean kinematic region
by requiring that all adjacent Mandelstam invariants are negative,

$\displaystyle{\bf s}<0\;:\quad
s_{12}<0\,,\;s_{23}<0\,,\;s_{34}<0\,,\;s_{45}<0\,,\;s_{15}<0\,.$ (31)

These inequalities implicate $(\epsilon_{5})^{2}=\Delta>0$. Recalling
definition (27) of $\epsilon_{5}$, we conclude that the light-like momenta
$p_{i}^{\mu}$ are complex valued in the Euclidean region, but their scalar
products $s_{ij}$ are real-valued. The Euclidean region is invariant under the
dihedral transformations (28), (29).

###  3.2 Five-particle Amplituhedron geometry and kinematic region

We will be interested in testing positivity properties of the five-particle
loop corrections $f_{5}^{(L)}$. The question is, in what kinematic region
should one expect special sign properties of the answer? As the main
motivation is the Amplituhedron geometry, it makes sense to see what it
implies for the five-particle case.

The relevant geometry is the five-particle one-loop MHV amplituhedron, which
is carved out by the following inequalities on the four-brackets
$\langle{abcd}\rangle=\det(Z_{a},Z_{b},Z_{c},Z_{d})$ of the momentum twistors
Arkani-Hamed:2017vfh ; Herrmann:2020qlt ,

$\displaystyle\langle{ijkl}\rangle>0\,,\qquad\text{for}\quad\leq i<j<k<l\leq
5\,,$ (32) $\displaystyle\langle{ABii+1}\rangle>0\,,$ (33)
$\displaystyle\\{\langle{AB12}\rangle,\langle{AB13}\rangle,\langle{AB14}\rangle,\langle{AB15}\rangle\\}\qquad\text{has
two sign flips.}$ (34)

Here the momentum twistors $Z_{i}$, $i=1,\ldots,5$, describe the external
five-particle kinematics, and the twistor line $(AB)$ represents the loop
integration. We take $(AB)$ to be the infinity bi-twistor,
$(AB)=\epsilon^{{\dot{\alpha}}{\dot{\beta}}}$, as this corresponds to the
frame $x_{0}\to\infty$ in space-time notations. Then we have
$\langle{ABii+1}\rangle=\langle{ii+1}\rangle$ and
$\langle{ii+1jj+1}\rangle=\langle{ii+1}\rangle\langle{jj+1}\rangle
x_{ij}^{2}$. Thus, eqs. (32) and (33) imply that

$\displaystyle
s_{12}>0\,,\;s_{23}>0\,,\;s_{34}>0\,,\;s_{45}>0\,,\;s_{15}>0\,.$ (35)

This is equivalent to the Euclidean region (31) up to an (irrelevant) overall
sign flip of the Mandelstam variables.

Moreover, eqs. (33) and (34) imply that the signs of
$\\{\langle{12}\rangle,\langle{13}\rangle,\langle{14}\rangle,\langle{15}\rangle\\}$
allowed by the Amplituhedron geometry are $\\{+,-,-,+\\}$, $\\{+,+,-,+\\}$ and
$\\{+,-,+,+\\}$. The configuration $\\{+,+,+,+\\}$ belongs to the Euclidean
region (31) but it is forbidden by (34). We find that $\epsilon_{5}>0$ on the
configurations $\\{+,-,-,+\\}$, $\\{+,+,-,+\\}$, $\\{+,-,+,+\\}$ and
$\epsilon_{5}<0$ on the configuration $\\{+,+,+,+\\}$. Thus, in summary, the
$\epsilon_{5}>0$ part of the five-particle Euclidean region (31) is the one-
loop MHV Amplituhedron geometry (up an irrelevant overall sign flip). This is
consistent with a related discussion in Gehrmann:2015bfy . Therefore it will
be particularly interesting to look for positivity properties of $F_{5}$ in
that region.

###  3.3 Conjecture for leading singularities

In the $n$-particle case with $n>4$, there are several leading singularities
of $F_{n}$. The leading singularities of $F_{n}$ were analyzed in
Chicherin:2022bov  in the planar limit, where it was conjectured that they are
given by certain four-point and five-point momentum twistor functions. We
denote them by $b_{ijkl}$, $b_{ijklm}$ in the frame $x_{0}\to\infty$. It turns
out that at $n=5$, one can write down ten such terms, six of which are
linearly independent. After normalizing by the Parke-Taylor prefactor ${\rm
PT}_{5}$ of eq. (10), they take the following simple form in the frame
$x_{0}\to\infty$ in spinor helicity notation,

$\displaystyle{\rm
PT}_{5}\,b_{1234}=-\frac{[23]^{2}}{\langle{45}\rangle\langle{51}\rangle\langle{14}\rangle}\;,\qquad{\rm
PT}_{5}\,b_{2345}=-\frac{[34]^{2}}{\langle{51}\rangle\langle{12}\rangle\langle{25}\rangle}\,,$
$\displaystyle{\rm
PT}_{5}\,b_{1345}=-\frac{[45]^{2}}{\langle{12}\rangle\langle{23}\rangle\langle{31}\rangle}\;,\qquad{\rm
PT}_{5}\,b_{1245}=-\frac{[51]^{2}}{\langle{23}\rangle\langle{34}\rangle\langle{42}\rangle}\,,$
$\displaystyle{\rm
PT}_{5}\,b_{1235}=-\frac{[12]^{2}}{\langle{34}\rangle\langle{45}\rangle\langle{53}\rangle}\;,\qquad{\rm
PT}_{5}\,b_{12345}=\frac{[13]^{2}}{\langle{24}\rangle\langle{45}\rangle\langle{52}\rangle}\,.$
(36)

Just as in the four-particle case, it is obvious in this form that the
conformal invariance of eq. (11) holds. Conjecturally, the six functions from
(36) are a basis for the leading singularities of the five-particle observable
$f_{5}$ to all loop orders. In the present paper, we check to two loops, by
the explicit calculation of $f_{5}^{(2)}$, that this is indeed the case.

We find it convenient to choose the following basis of six leading
singularities,

$\displaystyle r_{5,0}$ $\displaystyle=b_{1245}+b_{2345}-b_{12345}\,,$
$\displaystyle\quad r_{5,1}$ $\displaystyle=b_{2345}\,,$ $\displaystyle\quad
r_{5,2}=b_{1345}\,,$ $\displaystyle r_{5,3}$ $\displaystyle=b_{1245}\,,$
$\displaystyle\quad r_{5,4}$ $\displaystyle=b_{1235}\,,$ $\displaystyle\quad
r_{5,5}=b_{1234}\,.$ (37)

One reason for this choice is that in this way, $f_{5}=r_{5,0}$ at Born-level
is among the leading singularities. The latter expression can be also written
in a more symmetric form, avoiding spurious poles of individual $b$’s in
expression (37) for $r_{5,0}$,

$\displaystyle
r_{5,0}=-\frac{1}{2}\left(s_{12}s_{23}+s_{23}s_{34}+s_{34}s_{45}+s_{45}s_{15}+s_{15}s_{12}+\epsilon_{5}\right)\,.$
(38)

We note that it possesses the expected dihedral symmetry (recall eqs. (28),
(29)),

$\displaystyle\tau(r_{5,0})=\rho(r_{5,0})=r_{5,0}\,.$ (39)

###  3.4 Duality to all-plus pure Yang-Mills amplitudes

Similarly to the four-particle case, the five-particle $f_{5}$ is related to
the planar five-particle all-plus pure Yang-Mills scattering amplitude. This
will be discussed in more details in Sect. 6. At the lowest level in the weak-
coupling expansion, the statement is as follows. The Born-level $f_{5}$ (38)
normalized with the Parke-Taylor factor coincides with the one-loop all-plus
amplitude,

$\displaystyle A^{(1)}_{{\rm YM},5}={\rm PT}_{5}\,f_{5}^{(0)}\,.$ (40)

It is also instructive to provide expressions for the remaining five leading
singularities in terms of the Mandelstam variables. They are as follows

$\displaystyle
r_{5,1}=\frac{s_{34}}{s_{25}}\mbox{tr}_{-}(p_{2}p_{3}p_{4}p_{5})\,,\quad
r_{5,2}=\frac{s_{45}}{s_{13}}\mbox{tr}_{-}(p_{3}p_{4}p_{5}p_{1})\,,\quad
r_{5,3}=\frac{s_{15}}{s_{24}}\mbox{tr}_{-}(p_{4}p_{5}p_{1}p_{2})\,,$
$\displaystyle
r_{5,4}=\frac{s_{12}}{s_{35}}\mbox{tr}_{-}(p_{5}p_{1}p_{2}p_{3})\,,\quad
r_{5,5}=\frac{s_{23}}{s_{14}}\mbox{tr}_{-}(p_{1}p_{2}p_{3}p_{4})\,,$ (41)

where the Lorentz invariant

$\displaystyle\mbox{tr}_{-}(p_{i}p_{j}p_{k}p_{l}):=$
$\displaystyle\frac{1}{2}\mbox{tr}\left(\left(1-\gamma_{5}\right){p}_{i}{p}_{j}{p}_{k}{p}_{l}\right)$
$\displaystyle=$
$\displaystyle\langle{ij}\rangle[jk]\langle{kl}\rangle[li]\,,$ (42)

involves the parity-odd $\epsilon_{5}$. In this representation, we observe
that the five leading singularities can become singular if the scalar products
of nonadjacent momenta vanish. We also notice the five leading singularities
are related by the cyclic shifts $\tau$ and are mapped among each other by
reflection $\rho$, see eqs. (28), (29),

$\displaystyle\tau(r_{5,i})=r_{5,i+1}\,,\quad\rho(r_{5,i})=r_{5,6-i}\,,$ (43)

with $r_{5,6}\equiv r_{5,1}$.

###  3.5 Structure of loop corrections

Next, let us discuss the kinematic dependence of the pure transcendental
functions representing the loop corrections of $F_{5}$. In accordance with
(5), the $g_{5,s}$ with $s=0,1\ldots 5$ are functions of four independent dual
conformal cross-ratios, which can be chosen as

$\displaystyle{\bf
u}=\left\\{\frac{x_{25}^{2}x_{10}^{2}x_{40}^{2}}{x_{14}^{2}x_{20}^{2}x_{50}^{2}}\,,\;\frac{x_{13}^{2}x_{40}^{2}}{x_{14}^{2}x_{30}^{2}}\,,\;\frac{x_{24}^{2}x_{10}^{2}}{x_{14}^{2}x_{20}^{2}}\,,\;\frac{x_{35}^{2}x_{10}^{2}x_{40}^{2}}{x_{14}^{2}x_{30}^{2}x_{50}^{2}}\right\\}\,.$
(44)

In the dual conformal frame $x_{0}\to\infty$, the cross-ratios become ratios
of the Mandelstam variables (26),

$\displaystyle{\bf
u}=\left\\{\frac{s_{12}}{s_{15}}\,,\;\frac{s_{23}}{s_{15}}\,,\;\frac{s_{34}}{s_{15}}\,,\;\frac{s_{45}}{s_{15}}\right\\}\,.$
(45)

The explicit perturbative calculation reveals that the five-particle $f_{5}$
has the expected form of the loop corrections,

$\displaystyle f_{5}^{(L)}=$ $\displaystyle f_{5}^{(0)}\,g^{(L)}_{0}({\bf
u})+\sum_{i=1}^{5}r_{5,i}\,g^{(L)}_{i}({\bf u})\,,$ (46)

where $g_{s}^{(L)}({\bf u})$ with $s=0,1,\ldots,5$ are pure functions of
transcendental weight $2L$ of the variables (45); and $f_{5}^{(0)}=r_{5,0}$
(38). The pure functions posses the same dihedral symmetries as the
accompanying leading singularities (39) and (43). Namely, $g_{0}^{(L)}$ is
invariant under cyclic shifts and reflections

$\displaystyle\tau\left(g_{0}^{(L)}\right)=\rho\left(g_{0}^{(L)}\right)=g_{0}^{(L)}\,.$
(47)

whereas all $g_{i}^{(L)}$ are related by cyclic shifts,

$\displaystyle\tau\left(g_{i}^{(L)}\right)=g_{i+1}^{(L)}\,,\quad\rho\left(g_{i}^{(L)}\right)=g_{6-i}^{(L)}\,$
(48)

with $g_{6}^{(L)}\equiv g_{1}^{(L)}$.

Let us collect the previously known five-particle results up to one-loop
order. We have already mentioned the Born-level ($L=0$) answer $f_{5}^{(0)}$
(38), which in notations (46) is as follows,

$\displaystyle g^{(0)}_{0}=1\,,\quad{\rm and}\quad g^{(0)}_{i}=0\quad{\rm
for}\quad i=1,\ldots,5\,.$ (49)

At one loop $L=1$, we have Chicherin:2022bov ,

$\displaystyle g^{(1)}_{0}=$
$\displaystyle\log\left(\frac{s_{15}}{s_{23}}\right)\log\left(\frac{s_{12}}{s_{34}}\right)-\frac{\pi^{2}}{6}+{\rm
cyclic}\,,$ (50) $\displaystyle g^{(1)}_{1}=$ $\displaystyle-{\rm
Li}_{2}\left(1-\frac{s_{12}}{s_{34}}\right)-{\rm
Li}_{2}\left(1-\frac{s_{15}}{s_{34}}\right)-\log\left(\frac{s_{12}}{s_{34}}\right)\log\left(\frac{s_{15}}{s_{34}}\right)+\frac{\pi^{2}}{6}\,,$
(51)

and the remaining $g^{(1)}_{i}$ are cyclic shifts of $g^{(1)}_{1}$, see (48).

In this paper, we will compute $f_{5}$ to two loops $L=2$. It is in agreement
with the leading singularities and the uniform transcendentality of (46).
Before proceeding to calculations, let us discuss the relevant space of
transcendental functions that $g^{(2)}_{5,s}$ from (46) belong to.

###  3.6 Pentagon functions

We have already mentioned that the Wilson loop ratio in the frame
$x_{0}\to\infty$ is closely related to the all-plus YM scattering amplitude.
Hence, in order to describe loop corrections of the planar five-particle
$f_{5}$ we would need the same collection of transcendental functions as for
massless planar five-particle scattering in non-supersymmetric gauge theories.
They are the planar pentagon functions Gehrmann:2015bfy ; Gehrmann:2018yef .

The pentagon functions have been actively studied over the past several years.
The principal interest to them owes to two-loop calculation of QCD amplitudes.
The latter are required to provide NNLO theoretical predictions for cross-
sections measured in the ongoing collider experiments Chawdhry:2019bji . The
pentagon functions are an indispensable ingredient in these calculations.

The planar pentagon functions have been thoroughly studied in Gehrmann:2018yef
. They have been classified up to the transcendental weight four that is
required for undertaking two-loop amplitude calculations. This ref. also
provides numerical routines for their evaluations within the Euclidean region
(31) as well as in the physical scattering channels. More recently, the planar
pentagon functions have been complemented with the nonplanar ones
Chicherin:2020oor  which are required in the calculation of nonplanar
corrections of the five-particle massless scattering amplitudes. In the
present study, only planar pentagon functions will be required, and as
discussed in section 3.2, we are mostly interested in their evaluations in the
Euclidean region (31). Thus, we will rely on the planar results of ref.
Gehrmann:2018yef .

The pentagon functions are graded by their transcendental weight $w$. At
weight $w>0$, there are several linearly independent pentagon functions
$p^{(w)}_{a}$ which we label with an index $a$.111In ref. Gehrmann:2018yef ,
the pentagon functions are denoted as $f_{w,a}$ (and also $f_{w,a}^{(i)}$). We
changed notations for the pentagon functions here to avoid confusion with the
observable $f_{n}$ (6). For the sake of convenience, we also consider the
weight-zero case: $p^{(0)}=1$. Not only the pentagon functions are linearly
independent, they are also algebraically independent. Namely, any nonzero
polynomial in pentagon functions (of weight less of equal $w$) with constant
coefficients does not vanish identically.

The polylogarithmic nature of the pentagon functions is revealed upon
differentiating them in kinematic variables. Differentiation of
$p^{(w+1)}_{a}$ with $w\geq 0$ results in a linear combination (or a
polynomial) in the pentagon functions of lower weight,

$\displaystyle
dp^{(w+1)}_{a}=\sum_{k,b}A^{k}_{a,b}\,p^{(w)}_{b}d\log(W_{k})+\text{terms with
}p^{(w^{\prime})}\text{ at }w^{\prime}<w\,,$ (52)

see also (116)–(119) for more details. Here $A^{k}_{a,b}$ are rational
numbers. Summation index $b$ runs over pentagon functions of weight $w$, while
index $k$ runs over 26 letters of the planar pentagon alphabet
Gehrmann:2015bfy . We denote the alphabet letters as
$\\{W_{k}\\}_{k=1}^{20}\cup\\{W_{k}\\}_{k=26}^{31}$ following notations of
Chicherin:2017dob .222Five letters $\\{W_{k}\\}_{k=21}^{25}$ belong to the
nonplanar extension of the pentagon alphabet. We do not use them in the
present work. We recall explicit expressions for the pentagon letters as well
as more details on the pentagon functions in App. A.

Iterating eq. (52) one can obtain iterated integral representation for the
pentagon functions. It is also useful to consider the symbol projection (see
Duhr:2019wtr  for a review) of the pentagon functions. The symbol of a
pentagon functions is obtained by dropping out the transcendental constants in
the iterated integral expression. Namely, the symbol of a weight-$w$ pentagon
function is a linear combination with rational numbers of the length-$w$ words
$[W_{i_{1}},\ldots,W_{i_{w}}]$ formed from the pentagon alphabet letters. Zero
locus of the first entries are branch points of the pentagon functions. Thus,
the allowed first entries of the pentagon function symbols are five adjacent
Mandelstam variables (26). In our calculations, we work with the iterated
integrals, i.e. we keep ’beyond the symbol’ terms.

In the five-particle setting the parity conjugation is nontrivial. The
pentagon functions are defined to have definite parity, i.e. they are either
even or odd upon $\epsilon_{5}\to-\epsilon_{5}$.

###  3.7 Novel five-particle two-loop result

In this Section, we present the five-particle $f_{5}$ in the two-loop
approximation. It is the result of an explicit two-loop calculation provided
in the next Section. We find that it agrees with the expected form (46) of the
leading singularities,

$\displaystyle f_{5}^{(2)}=$ $\displaystyle f_{5}^{(0)}\,g^{(2)}_{0}({\bf
u})+\sum_{i=1}^{5}r_{5,i}\,g^{(2)}_{i}({\bf u})\,.$ (53)

The loop corrections $g^{(2)}_{0}$ and $g^{(2)}_{i}$ with $i=1,\ldots,5$ are
pure functions of the transcendental weight four, which are polynomials in the
planar pentagon functions of Gehrmann:2015bfy . Namely, $g^{(2)}_{0}$ and
$g^{(2)}_{i}$ are linear combinations with rational coefficients of the
monomials

$\displaystyle
p^{(4)}_{a},\;p^{(3)}_{a}p^{(1)}_{b},\;p^{(2)}_{a}p^{(2)}_{b},\;p^{(2)}_{a}p^{(1)}_{b}p^{(1)}_{c},\;p^{(1)}_{a}p^{(1)}_{b}p^{(1)}_{c}p^{(1)}_{d},\;\zeta_{3}p^{(1)}_{a},\;\pi^{2}p^{(2)}_{a},\;\pi^{2}p^{(1)}_{a}p^{(1)}_{b}$
(54)

and two weight-4 transcendental constants $\phi^{(4)}_{0}$ and
$\phi^{(4)}_{1}$. Note that despite the fact that the planar pentagon
functions Gehrmann:2018yef  are not dimensionless, they are organized in
dimensionless expressions in $g^{(2)}_{0}$ and $g^{(2)}_{i}$.

The one-loop $g^{(1)}_{0},\,g^{(1)}_{i}$ are parity even, while the two-loop
functions involve parity-odd pentagon functions. Namely, $g^{(2)}_{0}$ is
parity-even and $g^{(2)}_{i}$ contains both parity-odd and parity-even terms.

We collected the pentagon function expressions for the two-loop (as well as
one-loop) $f_{5}$ in computer-readable ancillary files. This representation
can be easily evaluated numerically using the public code provided with
Gehrmann:2018yef . We also provide iterated integral expressions (and
consequently the symbol expressions) for one-loop and two-loop $f_{5}$ in the
ancillary files.

We provide numerical values of $f_{5}$,

$\displaystyle f_{5}^{(0)}=-7.81495\,,\quad f_{5}^{(1)}=96.3895\,,\quad
f_{5}^{(2)}=-1398.9113\,,$ (55)

at the reference point (with $\epsilon_{5}>0$)

$\displaystyle
s_{12}=-1\,,\;s_{23}=-\frac{31}{11}\,,\;s_{34}=-\frac{59}{43}\,,\;s_{45}=-\frac{23}{13}\,,\;s_{15}=-\frac{47}{61}\,.$
(56)

Let us now discuss some features of the result. The iterated integral
expression of $g_{0}^{(2)}$ involves 25 letters of the planar pentagon
alphabet, namely the letter $W_{31}$ is absent. Similarly, planar pentagon
letters $W_{6},W_{8},W_{10},W_{31}$ are absent in the iterated integral
expression of $g_{1}^{(2)}$, so it involves 22 letters. The remaining
functions $g^{(2)}_{i}$ ($i=2,\ldots,5$) are cyclic permutations of
$g_{1}^{(2)}$, so they also involve 22 letters and the missing letters are the
cyclic permutations of $W_{6},W_{8},W_{10},W_{31}$. Previously, it was noticed
that the letter $W_{31}$ is absent from a properly defined finite hard part of
the planar (as well as nonplanar) five-particle two-loop amplitudes in
maximally supersymmetric Yang-Mills theory Abreu:2018aqd ; Chicherin:2018yne
and supergravity Chicherin:2019xeg ; Abreu:2019rpt . The same is also true for
the hard part of QCD amplitudes with gluons and quarks in the external states
Abreu:2019odu ; Abreu:2020cwb . This was investigated in the context of
cluster algebras in Chicherin:2020umh .

We have already noticed that the first entries involve only five letters since
they describe the branching points of the functions. The last entries of the
symbol are revealed upon differentiation, see (52). We observe that the last
entries of $g^{(2)}_{0}$ symbol involve 15 linearly independent combinations
of 20 letters, and the last entries of $g^{(2)}_{1}$ symbol involve 14
linearly independent combinations of 19 letters. In the context of remainder
and ratio functions of planar amplitudes in ${\cal N}=4$ sYM Drummond:2008vq ,
powerful constraints on the last entries of their symbol expressions follow
from the $\bar{Q}$-equation Caron-Huot:2011dec , which describes the broken
super-conformal symmetry of this finite observable. This differential equation
provided crucial input for the hexagon bootstrap calculations of multi-loop
scattering amplitudes in ${\cal N}=4$ sYM. The reduced number of letters in
the last entries we observe here suggests to us that a $\bar{Q}$-equation for
a supersymmetrised $f_{5}$ could be useful in multiloop calculations. This is
currently under investigation CHT .

##  4 Details of the two-loop calculation and singular limits

###  4.1 From amplitude integrands to integrands of $F_{n}$

The Wilson loop correlators coming in the definition of $F_{n}$ (2) are not
well-defined in $d=4$ dimensions because of the cusp divergences
Drummond:2007au . However, the ratio $F_{n}$ is a finite observable in $d=4$
dimensions. Thus, we would like to bypass regularization issues as much as
possible staying in four dimensions. More precisely, the aim is to find four-
dimensional integrands ${\cal I}^{(L)}$ which represent the loop-corrections
$F^{(L)}_{n}$ in (4),

$\displaystyle F_{n}^{(L)}=\frac{1}{(\textup{i}\pi^{2})^{L}L!}\int
d^{4}y_{1}\ldots d^{4}y_{L}\,{\cal
I}^{(L)}_{n}(x_{1},\ldots,x_{n};x_{0}|y_{1},\ldots,y_{L})\,.$ (57)

We assume that the integrands are symmetrized over all $L!$ permutations of
the integration points $y_{1},\ldots,y_{L}$. Let us stress that since
$F_{n}^{(L)}$ is finite, the loop integrations in (57) are well-defined in
$d=4$.

We take a shortcut to concise expressions for the planar integrand ${\cal
I}^{(L)}$ employing results about integrands of the scattering amplitude in
${\cal N}=4$ sYM available in the literature. Indeed, the planar MHV
scattering amplitude is equivalent (up to regularization scheme) to the vacuum
expectation value of the null polygonal Wilson loop at weak as well as at
strong coupling (see Alday:2008yw  for a review),

$\displaystyle\langle{W_{n}}\rangle\sim M_{n}\equiv\frac{A^{\rm
MHV}_{n}}{A^{\rm MHV}_{\rm tree}}\,,$ (58)

provided that coordinates (i.e. dual momenta) of the Wilson loop cusps and
momenta of scattered particles are related as in (8). One of the possible ways
to make the duality in (58) between two divergent at weak coupling objects
exact is to consider their integrands. Namely, properly defined four-
dimensional $L$-loop integrand of the MHV amplitude and of the Wilson loop are
identical Alday:2010zy ; Eden:2010zz ; Adamo:2011dq .

Following this path, we consider the perturbative expansion of the planar
$n$-particle MHV amplitude

$\displaystyle M_{n}=1+\sum_{L\geq 1}(g^{2})^{L}M_{n}^{(L)}$ (59)

where $M_{n}^{(L)}$ is formally represented by the planar $L$-loop integrand
$I^{(L)}_{n}$ written in the dual momenta (8),

$\displaystyle M_{n}^{(L)}=\frac{1}{(\textup{i}\pi^{\frac{d}{2}})^{L}L!}\int
d^{d}y_{1}\ldots d^{d}y_{L}\,I^{(L)}_{n}(x_{1},\ldots,x_{n}|y_{1}\ldots
y_{L})\,.$ (60)

The four-dimensional amplitude integrands are very well studied Arkani-
Hamed:2012zlh . They can be constructed recursively via the loop BCFW Arkani-
Hamed:2010zjl , and via the twistor space Feynman rules Mason:2010yk . They
allow for geometric description as dlog forms on the Amplituhedron Arkani-
Hamed:2013jha . They have also been constructed in manifestly local form
Arkani-Hamed:2010pyv  to high loop orders via from soft and collinear
consistency relations Bourjaily:2011hi , via a connection to correlation
functions Eden:2011we ; Eden:2012tu , and via generalized unitarity.

A systematic way to define the Wilson loop integrand—and via the duality
$M_{n}\sim\langle{W_{n}}\rangle$ (58) the integrand of the planar MHV
amplitude—avoiding ill-defined four-dimensional loop integrations (60) relies
on the Lagrangian insertion formula. According to it, differentiation of a
correlation function in the coupling results in insertion of the Lagrangian,

$\displaystyle g^{2}\partial_{g^{2}}\langle{W_{n}}\rangle=-\textup{i}\int
d^{d}x_{0}\,\langle{W_{n}{\cal L}(x_{0})}\rangle\,.$ (61)

Expanding both sides of the previous relation in the coupling, we find that
the correlator $\langle{W_{n}{\cal L}}\rangle$ evaluated at the lowest
perturbative level equals to the one-loop integrand of
$\langle{W_{n}}\rangle$. Iterating this procedure, one finds that the $L$-loop
integrand $I_{n}^{(L)}$ of the Wilson loop equals to the correlation function
of the Wilson loop $W_{n}$ with $L$ Lagrangians insertions evaluated at the
lowest perturbative order. The latter is a finite rational function in four
dimensions.

Similarly, the insertion formula (61) implies that the finite ratio $F_{n}$
(2) can be considered as the integrand of the logarithm of the Wilson loop,

$\displaystyle
g^{2}\partial_{g^{2}}\log\langle{W_{n}}\rangle=\int\frac{d^{d}x_{0}}{\textup{i}\pi^{\frac{d}{2}}}\,F_{n}(x_{1},\ldots,x_{n};x_{0})\,.$
(62)

Namely, $F_{n}$ is a finite function, and only integration over $x_{0}$ in
(62) results in cusp divergences of $\log\langle{W_{n}}\rangle$. In the
dimensional regularization $d=4-2\epsilon$, they manifest themselves as poles
$1/\epsilon^{2}$. On the contrary, each of the dimensionally regularized loop
integration in $M_{n}^{(L)}$ (60) brings in $1/\epsilon^{2}$ pole.

Eq. (62) and the duality $M_{n}\sim\langle{W_{n}}\rangle$ enables us to relate
the planar integrands ${\cal I}^{(L)}_{n}$ of $F_{n}^{(L)}$ from (57) with
planar integrands $I^{(L)}_{n}$ of the MHV amplitude $M_{n}$ from (60). We
expand $\log M_{n}$ in the coupling, see (59),

$\displaystyle g^{2}\partial_{g^{2}}\log\langle{W_{n}}\rangle=$ $\displaystyle
g^{2}M_{n}^{(1)}+g^{4}\left[2M^{(2)}_{n}-\left(M^{(1)}_{n}\right)^{2}\right]$
$\displaystyle+g^{6}\left[3M^{(3)}_{n}-3M^{(1)}_{n}M^{(2)}_{n}+\left(M^{(1)}_{n}\right)^{3}\right]+{\cal
O}\left(g^{8}\right),$ (63)

and we match the integrands (57) and (60) on both sides of the previous
equation,

$\displaystyle{\cal I}^{(0)}(x_{0})=I^{(1)}(x_{0})\,,$ (64)
$\displaystyle{\cal
I}^{(1)}(x_{0}|y_{1})=I^{(2)}(x_{0},y_{1})-I^{(1)}(x_{0})I^{(1)}(y_{1})\,,$
(65) $\displaystyle{\cal
I}^{(2)}(x_{0}|y_{1},y_{2})=I^{(3)}(x_{0},y_{1},y_{2})-I^{(1)}(x_{0})I^{(2)}(y_{1},y_{2})-I^{(1)}(y_{1})I^{(2)}(x_{0},y_{2})$
$\displaystyle\hskip
85.35826pt-I^{(1)}(y_{2})I^{(2)}(x_{0},y_{1})+2I^{(1)}(x_{0})I^{(1)}(y_{1})I^{(2)}(y_{2})\,,$
(66)

where we omit dependence of the integrands on the external kinematics
$x_{1},\ldots,x_{n}$ for the sake of brevity. Let us note that the integrands
${\cal I}^{(L)}$ of $F_{n}^{(L)}$ are symmetric in permutations of
$x_{0},y_{1},\ldots,y_{L}$, namely $x_{0}$ appears in the integrand on the
same footing with the integration points $y_{1},\ldots,y_{L}$. This is not
surprising since ${\cal I}^{(L)}$ is the $(L+1)$-loop integrand of
$\log\langle{W_{n}}\rangle$ with integration over dual momenta
$x_{0},y_{1},\ldots,y_{L}$.

We have already noticed that loop integrations in $F_{n}$ (57) are well
defined in four dimensions in contrast with ill-defined loop integrations in
$M_{n}$ (60). This property can be observed at the level of the integrand as
well. Namely, the integrands ${\cal I}^{(L)}$ are less singular than the
amplitude integrands $I^{(L)}$. The integrand (57) has the following form with
local poles

$\displaystyle{\cal
I}^{(L)}_{n}(x_{0}|y_{1},\ldots,y_{L}):=\frac{N^{(L)}_{n}(x_{1},\ldots,x_{n};x_{0}|y_{1},\ldots,y_{L})}{\prod\limits_{\begin{subarray}{c}1\leq
i<j\leq n\\\
1<|i-j|<n-1\end{subarray}}x_{ij}^{2}\cdot\prod\limits_{i=1}^{n}x_{i0}^{2}\cdot\prod\limits_{i=1}^{n}\prod\limits_{j=1}^{L}(x_{i}-y_{j})^{2}}$
(67)

where the numerator $N_{n}^{(L)}$ is a polynomial in all dual momenta. The
amplitude integrand $I^{(L)}$ has an analogous local form, but its numerator
is different. The numerator $N^{(L)}$ of ${\cal I}^{(L)}$ in (67) suppresses
potential divergences of the loop integrations. The cusp divergences of the
Wilson loop can be attributed to the regions of loop integrations where one
(or several) of the loop variables $y_{j}$ approaches any light-like edge
$[x_{i};x_{i+1}]$. The numerator of the integrand ${\cal I}^{(L)}$ vanishes in
this regime,

$\displaystyle N^{(L)}\bigl{(}x_{0}|y_{1},\ldots,y_{j}=\alpha
x_{i}+(1-\alpha)x_{i+1},\ldots,y_{L}\bigr{)}=0\,,\quad 0\leq\alpha\leq 1\,.$
(68)

Because of the permutation symmetry of $x_{0},y_{1},\ldots,y_{L}$, the
numerator $N^{(L)}$ also vanishes if $x_{0}$ belongs to any light-like edge,
$x_{0}\in[x_{i};x_{i+1}]$. This property of the numerator serves as a helpful
cross-check of the integrand.

In the following, we consider the five-particle case $n=5$. Thus, in order to
construct integrands ${\cal I}_{5}^{(L)}$ according to (64)–(66), the five-
particle planar MHV amplitude integrands are required. The latter have been
calculated in Ambrosio:2013pba  up to the six-loop order with the help of the
amplitude/correlator duality Eden:2010ce ; Eden:2010zz ; Eden:2011yp . They
are explicitly dual conformal functions with local poles. Since we aim at the
two-loop calculation of $F_{5}$, we use the MHV amplitude integrands of
Ambrosio:2013pba  up to the three-loop order, see (66). For example, the one-
loop integrand of the five-particle MHV amplitude, see (64),

$\displaystyle{\cal I}^{(0)}_{5}(x_{0})=$
$\displaystyle-\frac{1}{2}\biggl{[}\frac{x_{24}^{2}x_{35}^{2}}{x_{20}^{2}x_{30}^{2}x_{40}^{2}x_{50}^{2}}+\frac{x_{14}^{2}x_{35}^{2}}{x_{10}^{2}x_{30}^{2}x_{40}^{2}x_{50}^{2}}+\frac{x_{14}^{2}x_{25}^{2}}{x_{10}^{2}x_{20}^{2}x_{40}^{2}x_{50}^{2}}+\frac{x_{13}^{2}x_{25}^{2}}{x_{10}^{2}x_{20}^{2}x_{30}^{2}x_{50}^{2}}$
$\displaystyle+\frac{x_{13}^{2}x_{24}^{2}}{x_{10}^{2}x_{20}^{2}x_{30}^{2}x_{40}^{2}}+\frac{\epsilon(x_{1},x_{2},x_{3},x_{4},x_{5},x_{0})}{x_{10}^{2}x_{20}^{2}x_{30}^{2}x_{40}^{2}x_{50}^{2}}\biggr{]}$
(69)

contains only local poles, and its dual conformal invariance (with weight $+4$
at point $x_{0}$) is explicit. The integrand contains the parity-odd term
which is proportional to the totally antisymmetric tensor

$\displaystyle\epsilon(x_{1},x_{2},x_{3},x_{4},x_{5},x_{0}):=-\frac{i}{6}\sum_{\sigma\in{\cal
S}_{6}}(-1)^{\sigma}x_{\sigma_{1}}^{2}\,\epsilon(x_{\sigma_{2}},x_{\sigma_{3}},x_{\sigma_{4}},x_{\sigma_{5}})\,,$
(70)

where
$\epsilon(a,b,c,d)=\epsilon_{\mu\nu\rho\sigma}a^{\mu}b^{\nu}c^{\rho}d^{\sigma}$,
$(-1)^{\sigma}$ stands for the signature of $\sigma$, and where the summation
runs over all permutations of $\\{1,2,3,4,5,0\\}$. It is dual-conformally
covariant with weight $-1$ with respect to each of the points Ambrosio:2013pba
.

The higher-loop MHV amplitude integrands $I^{(L)}_{5}$ of Ambrosio:2013pba ,
and consequently ${\cal I}^{(L)}_{5}$, have a form similar to (69). They also
involve the parity-odd $\epsilon(x_{1},x_{2},x_{3},x_{4},x_{5},y_{k})$ with
$k=1,\ldots,L$. All totally antisymmetric parity-odd tensors can be reduced to
a single one and distances squared $x_{ij}^{2}$ due to the following identity,

$\displaystyle\epsilon(x_{1},x_{2},x_{3},x_{4},x_{5},x_{0})\,\epsilon(x_{1},x_{2},x_{3},x_{4},x_{5},x_{0^{\prime}})=-\det\left(x_{ij}^{2}\bigr{|}^{i=0,1,\ldots,5}_{j=0^{\prime},1,\ldots,5}\right).$
(71)

Summarizing, we have at our disposal four-dimensional $(6+L)$-point integrands
${\cal I}^{(L)}_{5}(x_{0})$ which are explicitly dual-conformal and contain
only local poles.

###  4.2 Integrands in the conformal frame

Figure 2: The two-loop pentabox topology (l.h.s.) and the one-loop pentagon
topology (r.h.s.) in momentum space (drawn by blue dotted lines) and their
dual momentum representation (solid black lines). Integration vertices of the
dual graph are denoted by black blobs. Dashed black lines denote numerators.
Labels $\sigma_{1},\sigma_{2},\ldots,\sigma_{5}$ of the external momenta and
the dual momenta are cyclically ordered, such that
$x_{\sigma_{i}}-x_{\sigma_{i-1}}=p_{\sigma_{i}}$.

The integrands ${\cal I}^{(L)}_{5}(x_{0})$ are functions of $6+L$ points. In
the frame $x_{0}\to\infty$, the dual-conformal integrand simplifies to a
Poincaré-invariant function of $5+L$ points,

$\displaystyle\iota_{5}^{(L)}(x_{1},\ldots,x_{5}|y_{1},\ldots,y_{L}):=\lim_{x_{0}\to\infty}(x_{0}^{2})^{4}\,{\cal
I}^{(L)}_{5}(x_{1},\ldots,x_{5};x_{0}|y_{1},\ldots,y_{L})\,.$ (72)

For example, the integrand (69) with $L=0$ is the Born-level $f_{5}$, see
(38),

$\displaystyle\iota_{5}^{(0)}(x_{1},\ldots,x_{5})=f_{5}^{(0)}=r_{5,0}\,.$ (73)

The parity-odd totally antisymmetric tensor from (70) simplifies to
$\epsilon_{5}$ (27) in the frame $x_{0}\to\infty$,

$\displaystyle\lim_{x_{0}\to\infty}\frac{1}{x_{0}^{2}}\epsilon(x_{1},x_{2},x_{3},x_{4},x_{5},x_{0})=\epsilon_{5}\,.$
(74)

In view of the previous relation and (71), we find that all parity-odd terms
of the integrand are proportional to $\epsilon_{5}$ in the frame
$x_{0}\to\infty$. Thus, we decompose the integrand into parity-even and
parity-odd terms as follows,

$\displaystyle\iota_{5}^{(L)}=\iota_{5}^{(L)+}+\epsilon_{5}\,\iota_{5}^{(L)-}$
(75)

where $\iota_{5}^{(L)\pm}$ are rational functions of $x_{ij}^{2}$,
$y_{kl}^{2}$, $(x_{i}-y_{k})^{2}$ with $k,l=1,\ldots,L$ and $i,j=1,\ldots,5$.
They contain only simple local poles in loop variables, namely $y_{kl}^{2}$
and $(x_{i}-y_{k})^{2}$ in the denominators. Along with simple local poles
$x_{ij}^{2}$ in the external momenta they also contain spurious nonlocal poles
$\Delta$ and $\Delta^{2}$. The latter cancel out among each other in the final
result.

###  4.3 From Feynman loop integrals to pentagon functions

Having the five-particle integrands at our disposal, we proceed to loop
integrations. Both parity components of the integrand, $\iota_{5}^{(L)\pm}$,
can be integrated in four dimensions. However, individual terms of the
integrands would result into divergent loop integrations. In order to
implement loop integrations of $\iota_{5}^{(L)\pm}$ term by term, we introduce
dimensional regularization $d=4-2\epsilon$. We are mostly concerned with the
two-loop calculation, i.e. $L=2$,

$\displaystyle\frac{1}{2\left(\textup{i}\pi^{\frac{d}{2}}\right)^{2}}\int
d^{d}y_{1}d^{d}y_{2}\,\iota_{5}^{(2)\pm}(x_{1},\ldots,x_{5}|y_{1},y_{2})\,.$
(76)

Each term of the integrand $\iota_{5}^{(2)\pm}$ results into a loop integral
which is either a genuine two-loop integral of the pentabox topology or a
product of one-loop pentagon topologies, see Fig. 2. We employ the amplitude
terminology naming the topologies which refers to the momentum representation.
Each of the loop integrals is a cyclic permutation of planar topologies, i.e.
labels $\sigma_{1},\ldots,\sigma_{5}$ of the external momenta (and dual
momenta) in Fig. 2 are a cyclic permutation of $1,2,\ldots,5$.

The Feynman integrals of the pentabox topology are as follows (see l.h.s. of
Fig. 2)

$\displaystyle\frac{1}{\left(\textup{i}\pi^{\frac{d}{2}}\right)^{2}}\int\frac{d^{d}y_{1}d^{d}y_{2}}{\prod\limits_{i=1}^{5}(y_{1}-x_{\sigma_{i}})^{2\alpha_{i}}\;\prod\limits_{i=1}^{5}(y_{2}-x_{\sigma_{i}})^{2\beta_{i}}\;(y_{1}-y_{2})^{2}}$
(77)

where integer propagator powers $\alpha_{i},\beta_{i}$ satisfy
$\alpha_{i},\beta_{i}\leq 1$ and $\alpha_{4},\beta_{1},\beta_{2}\leq 0$;
namely, they contain up to eight simple propagators and three irreducible
scalar products forming the numerator. The product of two pentagon topologies
also contain only simple propagators, i.e. $\alpha_{i},\beta_{i}\leq 1$,

$\displaystyle\int\frac{d^{d}y_{1}}{\textup{i}\pi^{\frac{d}{2}}}\frac{1}{\prod\limits_{i=1}^{5}(y_{1}-x_{\sigma_{i}})^{2\alpha_{i}}}\int\frac{d^{d}y_{2}}{\textup{i}\pi^{\frac{d}{2}}}\frac{1}{\prod\limits_{i=1}^{5}(y_{2}-x_{\sigma_{i}})^{2\beta_{i}}}\,.$
(78)

The two-loop pentabox integrals and the one-loop pentagon integrals are
expressible in the basis of 61 and 11 master integrals, respectively, due to
the IBP relations Chetyrkin:1981qh . The basis of pure master integrals is a
distinguished one Henn:2013pwa , since the pure master integrals evaluate to a
particularly compact expressions. For the planar pentabox topology, it was
constructed in Gehrmann:2015bfy ; Gehrmann:2018yef . We implement IBP
reductions of the scalar Feynman integrals (77) and (78) to the pure basis by
means of FIRE6 Smirnov:2019qkx .

The pure master integrals are Laurent series in the dimensional regularization
parameter $\epsilon$. The coefficients of the series are polynomials (over
rational numbers) in the pentagon functions (see Sect. 3.6) and several
transcendental constants Gehrmann:2018yef . Upon assigning transcendental
weight $-1$ to $\epsilon$, the Laurent series expansion of the pure master
integrals is of homogeneous transcendental weight.

The pure master integrals are soft/collinear divergent that is manifested as
poles in the dimensional regularization parameter $\epsilon$. Individual terms
of (76) contain poles up to order $1/\epsilon^{4}$. The poles cancel out in
the sum of all terms in agreement with finiteness of the quantity
$f_{5}^{(2)}$ in four dimensions, and we can put $\epsilon=0$.

Finally, we find $f_{5}^{(2)}$ has six leading singularities in agreement with
(53). The accompanying them six pure functions $g^{(2)}_{0}$ and $g^{(2)}_{i}$
are polynomials in the planar pentagon functions and transcendental constants
$\pi,\zeta_{3}$ (as well as $\phi^{(4)}_{0}$ and $\phi^{(4)}_{1}$) of
homogeneous transcendental weight four (see Sect. 3.7 and ancillary files for
the explicit expressions).

###  4.4 Validity checks and singular limits

$\displaystyle\begin{array}[]{c}\includegraphics[height=62.59596pt]{4WLcontourColl.eps}\end{array}\;\xleftarrow{p_{4}||p_{5}}\;\begin{array}[]{c}\includegraphics[height=76.82234pt]{5WLcontour.eps}\end{array}\;\xrightarrow{p_{5}\to
0}\;\begin{array}[]{c}\includegraphics[height=62.59596pt]{4WLcontour.eps}\end{array}$
Figure 3: Soft $p_{5}\to 0$ and collinear $p_{4}||p_{5}$ limit of the five-
cusp Wilson loop contour.

Scattering amplitudes and Wilson loops are known to simplify in a number of
singular regimes. The soft, collinear, and (multi)-Regge limits are of special
interest for amplitudes, since they have physical relevance. In the soft and
collinear limits a five-particle amplitude reduces to a four-particle
amplitude. In these limits, the five-cusp Wilson loop contour turns into a
four-cusp contour, see Fig. 3. We would like to study these limits of the
five-particle perturbative data $f_{5}^{(L)}$. Previously studied four-
particle $f_{4}$ does not allow for undertaking these limits. We expect that
the finite observable $f_{5}$ smoothly reduces in the soft and collinear
limits to the four-particle $f_{4}$ in analogy with the finite remainder of
the MHV amplitude in ${\cal N}=4$ sYM.

Indeed, in the soft limit with one of the particle momenta vanishing, $f_{5}$
reduces to its four-particle counterpart $f_{4}$,

$\displaystyle f_{5}^{(L)}(p_{1},\ldots,p_{5})\to
f_{4}^{(L)}(p_{1},\ldots,p_{4})\qquad\text{at}\quad p_{5}\to 0\,.$ (79)

We verify (79) for the available perturbative data $L=0,1,2$. We introduce a
parametrization of the kinematics in order to approach smoothly the soft limit
preserving the momentum conservation. The following leading singularities take
a very simple form in the soft limit $p_{5}\to 0$,

$\displaystyle r_{5,2},\,r_{5,3}\to 0\,,\quad r_{5,0},\,r_{5,5}\to-st$ (80)

where $s,\,t$ (16) are Mandelstam invariants of the four-particle kinematics,
while $r_{5,1}$ and $r_{5,4}$ are two linearly independent functions which
depend on the direction in which the soft limit $p_{5}\to 0$ is approached.
The accompanying pure functions vanish, and

$\displaystyle g^{(L)}_{1},\,g^{(L)}_{4}\to 0$ (81)

so the soft limit takes the following form,

$\displaystyle f^{(L)}_{5}\to-st\lim_{p_{5}\to
0}\left(g^{(L)}_{0}+g^{(L)}_{5}\right)=st\,G_{4}^{(L)}(z).$ (82)

We checked that it matches the four-particle perturbative data (18)–(20). In
the $L=2$ case we checked the statement at the symbol level.

Similarly, in the limit of collinear $p_{4}$ and $p_{5}$, the five-particle
$f_{5}$ smoothly reduces to $f_{4}$,

$\displaystyle f_{5}^{(L)}(p_{1},p_{2},p_{3},p_{4},p_{5})\to
f_{4}^{(L)}(p_{1},p_{2},p_{3},P)\qquad\text{at}\quad p_{4}\to
xP,\;p_{5}\to(1-x)P$ (83)

where $x$ is the splitting parameter and $p_{4}+p_{5}=P$. The leading
singularities are as follows in the collinear limit

$\displaystyle r_{5,1},\,r_{5,2},\,r_{5,3}\to 0\,,\quad
r_{5,0},\,r_{5,4},\,r_{5,5}\to-st\,,$ (84)

and the relation between the five-particle and four-particle (18)–(20)
perturbative data is as follows

$\displaystyle f_{5}^{(L)}\to-
st\lim_{p_{4}||p_{5}}\left(g^{(L)}_{0}+g^{(L)}_{4}+g^{(L)}_{5}\right)=stG_{4}^{(L)}(z)\,.$
(85)

In the $L=2$ case we checked the statement at the symbol level.

The leading singularities are divergent if a Mandelstam variable $s_{ij}$ with
nonadjacent $i$ and $j$ vanishes, namely $r_{i,5}$ is singular at
$s_{i-1\,i+1}=0$ with $i=1,\ldots,5$. This singularity is suppressed by the
pure function $g^{(L)}_{i}$, namely

$\displaystyle g^{(L)}_{i}({\bf u})\Bigr{|}_{s_{i-1\,i+1}=0}=0\,,$ (86)

so their product $r_{5,i}\,g^{(L)}_{i}$ is finite at this kinematic point. We
explicitly checked this statement for the perturbative data $L=1,2$ (at the
symbol level for $L=2$). Let us note that a particular case of the spurious
singularity $s_{i-1\,i+1}=0$ is a kinematical configuration with collinear
nonadjacent momenta $p_{i-1}||p_{i+1}$.

Finally, let us recall that the perturbative calculation of $f_{5}^{(L)}$ with
$L=1,2$ involves nontrivial cancellations of soft/collinear poles among the
contributing five-particle Feynman integrals, see (76). This is a self-
consistency check of our calculation. Moreover, the result turned out to be
consistent with the maximal transcendentality principle Kotikov:2002ab  that
is common for observables in ${\cal N}=4$ sYM.

###  4.5 Multi-Regge limit

Figure 4: Five-particle multi-Regge kinematics.

The observable $f_{5}$ depends on the same set of variables as five-particle
scattering amplitudes, and is closely related to scattering amplitudes.
Scattering amplitudes are known to simplify and exhibit a lot of structure in
a number of asymptotic regimes, such as soft, collinear and high-energy
(Regge) limits. Therefore it is interesting to study $f_{5}$ in such limits.
We have already considered the soft and collinear limits of $f_{5}$. Let us
investigate high-energy scattering next. Five-particle amplitudes are known to
simplify in the multi-Regge kinematics Kuraev:1976ge ; DelDuca:1995hf . In
this regime, the final-state particles are strongly ordered in rapidity and
have comparable transverse momenta.

We consider the scattering channel $12\to 345$. The multi-Regge limit is
reached via the following parametrization of the kinematics (26), together
with the limit $\delta\to 0$ (see Fig. 4),

$\displaystyle s_{12}=\frac{s_{1}s_{2}}{\kappa\delta^{2}}\,,\quad
s_{23}=-z_{1}z_{2}\kappa\,,\quad s_{34}=\frac{s_{1}}{\delta}\,,$
$\displaystyle s_{45}=\frac{s_{2}}{\delta}\,,\quad
s_{15}=-(1-z_{1})(1-z_{2})\kappa\,.$ (87)

The parity-odd tensor (27) is as follows

$\displaystyle\epsilon_{5}=\frac{s_{1}s_{2}}{\delta^{2}}(z_{1}-z_{2})+{\cal
O}(1/\delta)\,.$ (88)

In the physical scattering regime $12\to 345$, we have $(z_{1})^{*}=z_{2}$ and
$s_{1},s_{2},\kappa>0$. Whereas in the Euclidean region (31), $z_{1},z_{2}$
are real and independent (either $z_{1}>1,\,z_{2}<0$ or $z_{1}<0,\,z_{2}>1$)
and $s_{1},s_{2},\kappa<0$. In the following we consider the Euclidean region
of the multi-Regge kinematics.

Since the loop correction of $f_{5}$ are described by the planar pentagon
functions, we apply the multi-Regge limit to the $26$-letter alphabet of
planar pentagon function (see Sect. 3.6). We find that it factorizes into four
independent alphabets Chicherin:2018yne ; Chicherin:2019xeg ; Caron-
Huot:2020vlo ,

$\displaystyle\\{\delta\\}\,,\\{\kappa\\}\,,\\{s_{1}\,,s_{2}\,,s_{1}-s_{2}\,,s_{1}+s_{2}\\}\,,\\{z_{1}\,,z_{2}\,,1-z_{1}\,,1-z_{2}\,,z_{1}-z_{2}\,,1-z_{1}-z_{2}\\}\,.$
(89)

It tells us that any pentagon function, in the multi-Regge limit (87), turns
into (a sum of) functions that take the following form,

$\displaystyle(\log\delta)^{a}\times(\log\kappa)^{b}\times
H_{s}(s_{1},s_{2})\times H_{z}(z_{1},z_{2})\,,$ (90)

for some non-negative integers $a,b$, and where $H_{s}$ are Harmonic
Polylogarithms (HPL) Remiddi:1999ew  described by the $s_{1},s_{2}$-dependent
alphabet letters given in Eq. (89) only, and similarly $H_{z}$ are 2dHPL
functions Gehrmann:2000zt  with the $z_{1},z_{2}$-dependent alphabet letters.
In fact, we find that the leading term of the multi-Regge asymptotics takes a
much simpler form than (90) at $L=1,2$. It is given by five logarithmic
functions, see (94).

The leading singularities $r_{5,0}$ and $r_{5,4}$ are dominant in the multi-
Regge asymptotics (87). Their leading terms coincide,

$\displaystyle
r_{5,0},\,r_{5,4}=-z_{1}(1-z_{2})\frac{s_{1}s_{2}}{\delta^{2}}+{\cal
O}\left(\delta^{-1}\right),$ (91)

while the remaining four leading singularities are subleading in the multi-
Regge limit,

$\displaystyle r_{5,1},\,r_{5,2},\,r_{5,3},\,r_{5,5}={\cal
O}\left(\delta^{-1}\right).$ (92)

Thus, it reasonable to normalize the loop corrections of $f_{5}$ (46) by the
Born-level $f_{5}^{(0)}$ (38). The ratio contains only logarithmic divergences
at $\delta\to 0$,

$\displaystyle\frac{f_{5}^{(L)}}{f_{5}^{(0)}}$
$\displaystyle=g^{(L)}_{0}+g^{(L)}_{4}+{\cal
O}\left(\delta\log^{2L}(\delta)\right)$
$\displaystyle=\sum_{k=0}^{2L}\log^{k}(\delta)\,h_{k}^{(L)}(z_{1},z_{2},s_{1},s_{2},\kappa)+{\cal
O}\left(\delta\log^{2L}(\delta)\right),$ (93)

where $h_{k}^{(L)}$ is of transcendental weight $2L-k$. We calculated that the
leading term of the multi-Regge asymptotics $h_{k}^{(L)}$ at the one-loop and
two-loop order (i.e. $L=1,2$) relying on the approach of Caron-Huot:2020vlo .
We find that in the Euclidean region they are given by polynomials in
logarithms

$\displaystyle\log\left(\frac{s_{1}}{\kappa}\right)\,,\;\log\left(\frac{s_{2}}{\kappa}\right)\,,\;\log(-z_{1}z_{2})\,,\;\log\left(-(1-z_{1})(1-z_{2})\right)$
(94)

and the transcendental constants $\pi^{2}$, $\zeta_{3}$. The explicit
expressions are provided in the ancillary files.

Finally, based on the available perturbative data, we notice that both the
logarithm of the four-particle and five-particle observables have a very
simple and similar expressions in the Regge and multi-Regge limits,
respectively. The leading logarithmic (LL) and next-to-leading logarithmic
(NLL) terms of the four-particle observable (13) in the high-energy limit
$z\to 0$ are as follows,

$\displaystyle\log\left(\frac{f_{4}}{g^{2}f_{4}^{(0)}}\right)=$
$\displaystyle\left(-g^{2}+\pi^{2}g^{4}+{\cal O}\left(g^{6}\right)\right){\bf
L}^{2}+\left(-4\zeta_{3}g^{4}+{\cal O}\left(g^{6}\right)\right){\bf L}+{\cal
O}\left({\bf L}^{0}\right)\,$ (95)

where ${\bf L}=\log(z)$, whereas the LL and NLL terms of the five-particle
observable in the multi-Regge limit $\delta\to 0$,

$\displaystyle\log\left(\frac{f_{5}}{g^{2}f_{5}^{(0)}}\right)=$
$\displaystyle\left(-g^{2}+\pi^{2}g^{4}+{\cal O}\left(g^{6}\right)\right){\bf
L}^{2}+\Bigl{(}\left(g^{2}-\pi^{2}g^{4}\right)\log\left(\frac{s_{1}^{2}s_{2}^{2}}{\kappa^{4}z_{1}z_{2}(1-z_{1})(1-z_{2})}\right)$
$\displaystyle-4\zeta_{3}g^{4}+{\cal O}\left(g^{6}\right)\Bigr{)}{\bf L}+{\cal
O}\left({\bf L}^{0}\right)$ (96)

with ${\bf L}=\log(\delta^{2})$. In particular, we observe that the leading
logarithmic terms in the four-particle (95) and five-particle (96) cases are
identical. It would be very interesting to understand the patterns observed in
eqs. (95) and (96) better.

##  5 Positivity properties of the five-particle observable

###  5.1 Positivity hypothesis

The Amplituhedron provides a geometric construction of the finite loop
integrands of $F_{n}$ associating them to negative geometries that was
demonstrated in the four-particle case Arkani-Hamed:2021iya . A related
interesting question is about positivity properties of the integrated
quantities. In section 2.2, we reviewed the known positivity properties found
in the four-particle case. Here we wish to perform a similar analysis at five
points. As discussed in section 3.2, the Amplituhedron geometry suggests to us
to focus on the Euclidean region, with $\epsilon_{5}>0$. Therefore, the
hypothesis that we wish to test, using the available perturbative data, is

$\displaystyle f_{5}^{(L)}\bigr{|}_{\rm Eucl^{+}}<0\;\text{ at even
}L\,,\quad{\rm and}\quad f_{5}^{(L)}\bigr{|}_{\rm Eucl^{+}}>0\;\text{ at odd
}L\,,$ (97)

where the subscript stands for the Euclidean region, eq. (31), with
$\epsilon_{5}>0$.

Note that compared to the four-particle case, the five-particle $f_{5}^{(L)}$
depends on four cross-ratios ${\bf u}$ (45). Indeed, the five-particle loop
corrections $L>0$ involve pentagon functions which are multivariable functions
of the polylogarithmic type. This makes the five-particle positivity
hypothesis (97) even more nontrivial than the four-particle positivity (21).

###  5.2 Consistency of positivity hypothesis with singular limits

Let us begin by verifying that the conditions (97) are satisfied in singular
limits. In the soft and collinear limits $f_{5}^{(L)}$ reduces to
$f_{4}^{(L)}$, see (79) and (83), so the five-particle positivity (103)
reduces to the four-particle positivity (21).

Another simplified setting for testing the positivity properties of $f_{5}$ is
provided by the multi-Regge asymptotics, which we have calculated in Sect. 4.5
at $L\leq 2$. The sign of the ratio (93) is dominated by the leading
logarithmic term

$\displaystyle\frac{f_{5}^{(L)}}{f_{5}^{(0)}}=h^{(L)}_{2L}\left(\log\delta\right)^{2L}+{\cal
O}\left(\left(\log\delta\right)^{2L-1}\right)$ (98)

at sufficiently small $\delta$. The one-loop and two-loop data,
$h^{(1)}_{2}=-4$ and $h^{(2)}_{4}=8$, agree with the positivity conjecture
(97).

Next, we elaborate on testing the conjecture (97) in the full kinematic
setting. We find it convenient to study separately the sign properties of the
leading singularities on the one hand, and of the loop corrections (pentagon
functions) on the other hand.

###  5.3 Amplituhedron subregions

subregion | $s_{25}$ | $s_{13}$ | $s_{24}$ | $s_{35}$ | $s_{14}$
---|---|---|---|---|---
$\mathrm{(A)}$ | $+$ | $+$ | $+$ | $+$ | $+$
$\mathrm{(B)}$ | $+$ | $+$ | $-$ | $+$ | $+$
$\mathrm{(C)}$ | $-$ | $+$ | $+$ | $+$ | $-$
Table 1: Subregions of the five-particle Euclidean region (31) which are
specified by the signs of the nonadjacent bi-particle Mandelstam variables
$s_{i\,i+2}$. For subregions $\mathrm{(B)}$ and $\mathrm{(C)}$ there exist
four similar regions respectively that are obtained by cyclic permutations.

Prior to presenting the positivity properties of the five-particle leading
singularities we need to have a better understanding of the Mandelstam
invariants signs in the Euclidean region (31). The two-particle Mandelstam
invariants with adjacent indices are negative, $s_{i\,i+1}<0$, but $s_{ij}$
with nonadjacent $i$ and $j$, namely $s_{25},s_{13},s_{24},s_{35},s_{14}$, do
not have a definite sign in the Euclidean region (31). Specifying signs of all
$s_{ij}$ we fix a particular subregion of the Euclidean region (31). The signs
of $s_{ij}$ cannot be chosen arbitrarily. They are correlated in a certain
way. We observe that Mandelstam invariants $s_{i\,i+2}$ and $s_{i+2\,i+4}$
cannot be negative simultaneously. Indeed, the five-particle momentum
conservation implies

$\displaystyle s_{i\,i+2}+s_{i+1\,i+2}+s_{i+2\,i+3}+s_{i+2\,i+4}=0\,.$ (99)

Since $s_{i+1\,i+2}<0$ and $s_{i+2\,i+3}<0$ in the Euclidean region (31) then
at least one of $s_{i\,i+2}$ and $s_{i+2\,i+4}$ is positive. Summarizing, we
find 11 subregions of the Euclidean region:

  * •

One subregion with all five $s_{25},s_{13},s_{24},s_{35},s_{14}$ being
positive;

  * •

Five subregions with one of $s_{ii+2}$ being negative and others being
positive;

  * •

Five subregions with $s_{i\,i+2}$ and $s_{i+1\,i+3}$ being negative and others
being positive.

Since $f_{5}$ is invariant under dihedral symmetry, it is sufficient to
consider the three cases given in Tab. 1.

subregion | $r_{5,0}$ | $r_{5,1}$ | $r_{5,2}$ | $r_{5,3}$ | $r_{5,4}$ | $r_{5,5}$
---|---|---|---|---|---|---
$\mathrm{(A)}$ | $-$ | $+$ | $+$ | $+$ | $+$ | $+$
$\mathrm{(B)}$ | $-$ | $+$ | $+$ | $-$ | $+$ | $+$
$\mathrm{(C)}$ | $-$ | $-$ | $+$ | $+$ | $+$ | $-$
Table 2: Signs of the leading singularities (37) in different kinematic
regions.

In the following, we ignore lower dimensional subregions with one or several
$s_{ij}$ vanishing. In the full-dimensional subregions $(\mathrm{A})$,
$(\mathrm{B})$, $(\mathrm{C})$ of the Euclidean region either $\epsilon_{5}>0$
or $\epsilon_{5}<0$. We will restrict our considerations to the half of the
Euclidean region with $\epsilon_{5}>0$, as motivated in section 3.2.

###  5.4 Sign properties of the leading singularities and one-loop functions

Let us now study sign properties of the leading singularities in the regions
introduced in the previous subsection. We can immediately see that $r_{5,0}$
(38) is negative in the Euclidean region (31) with $\epsilon_{5}>0$. In order
to infer the sign of the remaining leading singularities (41), we note that

$\displaystyle\mbox{tr}_{-}(p_{i}p_{i+1}p_{i+2}p_{i+3})<0\,,\qquad\text{at}\quad\epsilon_{5}>0\quad\text{and}\quad{\bf
s}<0\,.$ (100)

Then the sign of $r_{5,i}$ is correlated with the sign of $s_{i+1\,i+4}$,

$\displaystyle
r_{5,i}\,s_{i+1\,i+4}>0\,,\qquad\text{at}\quad\epsilon_{5}>0\quad\text{and}\quad{\bf
s}<0\,.$ (101)

Thus, the leading singularities have a definite sign inside each of the
subregions $(\mathrm{A})$, $(\mathrm{B})$, and $(\mathrm{C})$, as summarized
in Tab. 2.

subregion | $g^{(1)}_{0}$ | $g^{(1)}_{1}$ | $g^{(1)}_{2}$ | $g^{(1)}_{3}$ | $g^{(1)}_{4}$ | $g^{(1)}_{5}$
---|---|---|---|---|---|---
$\mathrm{(A)}$ | $-$ | $+$ | $+$ | $+$ | $+$ | $+$
$\mathrm{(B)}$ | $*$ | $+$ | $+$ | $-$ | $+$ | $+$
$\mathrm{(C)}$ | $*$ | $-$ | $+$ | $+$ | $+$ | $-$
Table 3: Signs of the pure one-loop functions (51) in different kinematic
regions. The symbol $*$ means that our numerical analysis found both positive
and negative signs within the region.

We study numerically positivity properties of the one-loop $L=1$ pure
functions (50) and (51) in the Euclidean region (31). Numerical evaluations of
the one-loop functions, which are given by logarithms and dilogarithms, does
not pose any problem. They can be rapidly evaluated with high numerical
precision. We probed ${\cal O}(10^{9})$ random points in the Euclidean region.
Our findings are as follows:

  * •

Tab. 3 shows the sign patterns of the different one-loop functions that are
suggested by the numerical analysis.

  * •

Comparing to Tab. 2, one notices that the signs of $r_{5,i}$ and $g^{(1)}_{i}$
with $i=1,\ldots,5$ match. The signs of each of the six terms in eq. (46) are
collected in Tab. 4.

  * •

Tab. 4 shows that one the six terms can take negative values in regions
$(\mathrm{B})$ and $(\mathrm{C})$.333Interestingly, we notice that
$g_{0}^{(1)}>0$ occurs if the kinematic point is close to the boundaries of
the Euclidean region, and $g_{0}^{(1)}<0$ inside the bulk away from the
boundaries. However, our numerical evaluations suggest that their sum is
always positive, so that $f_{5}^{(1)}\bigr{|}_{\rm Eucl^{+}}>0$. This is in
agreement with analysis in ref. Gehrmann:2015bfy .

Let us provide more information on how the sign of the loop functions and the
leading singularities could conspire. The leading singularity $r_{5,i}$
changes sign crossing the boundary $s_{i-1\,i+1}=0$ of the Euclidean
subregions, and it is singular on the boundary. However, this singularity is
spurious for the observable $f_{5}$. The loop function $g_{i}^{(L)}$ vanishes
on the the boundary $s_{i-1\,i+1}=0$ (recall (86)) and it changes sign
crossing the boundary. Then the product $r_{5,i}\,g_{i}^{(L)}$ is smooth and
has the same sign on both sides of the boundary $s_{i-1\,i+1}=0$ (in its
vicinity).

It is worth noting that the constraints $\epsilon_{5}>0$ is necessary for
positivity of $f_{5}^{(1)}$. In the subregions $(\mathrm{A})$, $(\mathrm{B})$
and $(\mathrm{C})$ of the Euclidean region with $\epsilon_{5}<0$, all six
leading singularities are negative-valued

$\displaystyle r_{5,0}<0\,,\qquad r_{5,i}<0\,,\qquad
i=1,2,\ldots,5\,,\qquad\text{at}\quad\epsilon_{5}<0\quad\text{and}\quad{\bf
s}<0\,.$ (102)

At the same time, the one-loop functions (50)–(51) do not depend on the sign
of $\epsilon_{5}$. We observe that $f_{5}^{(1)}$ does not have a definite sign
at $\epsilon_{5}<0$.

subregion | $r_{5,0}\,g^{(1)}_{0}$ | $r_{5,1}\,g^{(1)}_{1}$ | $r_{5,2}\,g^{(1)}_{2}$ | $r_{5,3}\,g^{(1)}_{3}$ | $r_{5,4}\,g^{(1)}_{4}$ | $r_{5,5}\,g^{(1)}_{5}$
---|---|---|---|---|---|---
$\mathrm{(A)}$ | $+$ | $+$ | $+$ | $+$ | $+$ | $+$
$\mathrm{(B)},\,\mathrm{(C)}$ | $*$ | $+$ | $+$ | $+$ | $+$ | $+$
Table 4: Signs of the six terms in the one-loop ($L=1$) expression (46) of
$f_{5}^{(1)}$ in different kinematic regions.

###  5.5 Sign properties of the two-loop functions

Exploring sign properties of the two-loop functions is more intricate, as we
need to evaluate weight three and four pentagon functions. In order to do so,
we rely on the computer code Gehrmann:2018yef , which is designed to evaluate
pentagon functions (of transcendental weight up to four) by doing one-
dimensional numerical integrations.444 We also constructed one-fold integral
representations directly for the two-loop functions $g_{0}^{(2)}$ and
$g^{(2)}_{i}$ bypassing the pentagon function basis. We implemented these one-
fold integrations in a Mathematica code in order to cross-check evaluations of
$g_{0}^{(2)}$ and $g^{(2)}_{i}$ based on pentagon function evaluations with
the code of Gehrmann:2018yef .

We evaluated the pentagon functions at ${\cal O}(10^{4})$ phase-space points
from each of the three subregions $(\mathrm{A})$, $(\mathrm{B})$,
$(\mathrm{C})$. We generated random phase-space points with uniform
distribution of the five ${\bf s}$ (26) in the range $(-5;0)$. Moreover, we
separately generated another ${\cal O}(10^{4})$ random phase-space points with
one or several $s_{ij}$ close to zero in each of the subregions
$(\mathrm{A})$, $(\mathrm{B})$, $(\mathrm{C})$. These phase-space points are
close to the boundaries of Euclidean region and to the spurious surfaces in
the Euclidean region.

As compared to the one-loop situation, we find that the two-loop pure
functions provide more sign patterns, see Tab. 5.

Combining this with the signs of the leading singularities, the key features
we find, based on the numerical samplings above, are:

  * •

subregion $(\mathrm{A})$: the signs of two-loop pure functions are opposite to
the signs of the accompanying leading singularities;

  * •

subregion $(\mathrm{B})$: if $r_{5,i}<0$ then $g^{(2)}_{i}>0$, whereas the
remaining two-loop pure functions can take positive or negative values. We
managed to identify 20 different sign patterns in this case (see the ancillary
files);

  * •

subregion $(\mathrm{C})$: all two-loop pure functions can take positive or
negative values. We identified 23 sign patterns in this case (see the
ancillary files).

Despite of the fact that six terms of $f^{(2)}_{5}$ (46) are not necessarily
all negative in subregions $(\mathrm{B})$ and $(\mathrm{C})$555Similarly to
the one-loop observation, see footnote 3, our numerical experiments suggest
that for the phase space points which are not close to the boundaries
$s_{i\,i+1}=0$ of the Euclidean region, the sings of the two-loop functions
are opposite to those of the leading singularities. Namely the sign patterns
$(+|--+--)$ and $(+|+---+)$ in the second line and the third lines of Tab. 5,
respectively, are dominant in the bulk of the Euclidean region. The remaining
sign patterns of the two-loop functions are observed in vicinity of the
boundaries. (second and third line of Tab. 5), we observe that their sum is
always negative in the Euclidean region with $\epsilon_{5}>0$, i.e.
$f^{(2)}_{5}\bigr{|}_{\rm Eucl^{+}}<0$. The above analysis underlines how non-
trivial this property is.

In summary, we have provided numerical evidence that the positivity conjecture
(97) holds up to $L=2$,

$\displaystyle f_{5}^{(0)}\bigr{|}_{\rm Eucl^{+}}<0\,,\quad
f_{5}^{(1)}\bigr{|}_{\rm Eucl^{+}}>0\,,\quad f_{5}^{(2)}\bigr{|}_{\rm
Eucl^{+}}<0\,.$ (103)

subregion | $g^{(2)}_{0}$ | $g^{(2)}_{1}$ | $g^{(2)}_{2}$ | $g^{(2)}_{3}$ | $g^{(2)}_{4}$ | $g^{(2)}_{5}$
---|---|---|---|---|---|---
$\mathrm{(A)}$ | $+$ | $-$ | $-$ | $-$ | $-$ | $-$
$\mathrm{(B)}$ | $*$ | $*$ | $*$ | $+$ | $*$ | $*$
$\mathrm{(C)}$ | $*$ | $*$ | $*$ | $*$ | $*$ | $*$
Table 5: Observed signs of two-loop pure functions in kinematic regions.
Correlations among the signs $*$ are shown in more detail in the ancillary
files.

##  6 Prediction for the three-loop all-plus pure Yang-Mills amplitude

The scattering amplitude of $n$ gluons in the all-plus helicity configuration
in the pure Yang-Mills theory on the one hand, and the Lagrangian insertion in
the $n$-cusp Wilson loop in ${\cal N}=4$ sYM on the other hand, are closely
related. In reference Chicherin:2022bov  we conjectured that the maximal
transcendentality parts of the two quantities agree in the planar limit. Let
us briefly review this duality in the general, $n$-point case, and then
specialize to the five-point amplitude.

The all-plus amplitude vanishes at the tree level. The one-loop $n$-particle
color-ordered all-plus partial amplitude $A^{(1)}_{{\rm YM},n}$ is a rational
function Bern:1993qk . It equals to the Born-level $n$-particle observable
$f_{n}$ (6) normalised with the Parke-Taylor factor (10),

$\displaystyle A^{(1)}_{{\rm YM},n}={\rm PT}_{n}\,f^{(0)}_{n}\,.$ (104)

We have already mentioned the four-particle (24) and five-particle (40)
instances of this duality relation.

Higher orders of the all-plus amplitude $A_{n}^{\rm YM}$ perturbative
expansion contain divergences. We consider them in the dimensional
regularization with $d=4-2\epsilon$. The leading color two-loop all-plus
amplitude is calculated in Dunbar:2016cxp ; Dunbar:2017nfy . The duality is
formulated in terms of the finite remainders ${\cal H}_{n}^{\rm MHV}$ and
${\cal H}_{n}^{\rm YM}$ of the four-particle planar leading-color amplitudes.
They refer to the MHV amplitude of ${\cal N}=4$ sYM and the all-plus YM
amplitude, respectively,

$\displaystyle{\cal H}_{n}^{\rm MHV}=\frac{A_{n}^{\rm MHV}}{A_{n,{\rm
tree}}^{\rm MHV}{\cal Z}_{\rm IR}^{\rm MHV}}\;,\qquad{\cal H}_{n}^{\rm
YM}=\frac{A_{n}^{\rm YM}}{g^{2}A^{(1)}_{{\rm YM},n}{\cal Z}_{\rm IR}^{\rm
YM}}\,.$ (105)

The infrared renormalization constants ${\cal Z}_{\rm IR}$ minimally subtract
divergences, namely $\epsilon$-expansion of $\log{\cal Z}_{\rm IR}$ contains
$\epsilon$-pole terms but it does not contain finite terms,

$\displaystyle{\cal Z}_{\rm
IR}=1-g^{2}\left(\frac{n}{\epsilon^{2}}+\frac{1}{\epsilon}\sum_{i=1}^{n}\log\left(\frac{\mu^{2}}{-s_{i\,i+1}}\right)\right)+{\cal
O}(g^{4})\,.$ (106)

The maximally transcendental parts of the renormalization constants coincide
in both theories Kotikov:2002ab , ${\cal Z}_{\rm IR}^{\rm MHV}\sim{\cal
Z}_{\rm IR}^{\rm YM}$, that we denote with $\sim$. Let us note that $\log{\cal
H}_{n}^{\rm MHV}+\log{\cal Z}^{\rm MHV}_{\rm IR}$ is known at any loop order
at $n\leq 5$ due to the ABDK/BDS ansatz Anastasiou:2003kj ; Bern:2005iz
666Usually the ABDK/BDS ansatz is presented in a form with non-minimal
infrared subtraction.. Then, the conjecture expresses the maximally
transcendental part of the finite all-plus amplitude remainder ${\cal
H}_{n}^{\rm YM}$ in the planar leading-color approximation in terms of the
finite $n$-particle observables in ${\cal N}=4$ sYM theory,

$\displaystyle\log{\cal H}_{n}^{\rm
YM}\sim\log\left(\frac{f_{n}}{g^{2}f_{n}^{(0)}}\right)+\log{\cal H}_{n}^{\rm
MHV}+{\cal O}(\epsilon)\,.$ (107)

The appearance of the MHV amplitude in the duality relation (107) is explained
by the well-known duality $A_{n}^{\rm MHV}\sim\langle{W_{n}}\rangle$ between
the planar MHV amplitudes and the polygonal Wilson loops (see Alday:2008yw ;
Henn:2020omi  for reviews).

Roughly speaking, the conjecture (107) relates $(L+1)$-loop all-plus amplitude
and $L$-loop $f_{n}$. In the four-particle case $n=4$, the duality takes the
form

$\displaystyle\log{\cal H}_{4}^{\rm YM}\sim\log\left(1-\sum_{L\geq
1}g^{2L}G_{4}^{(L)}\right)+\log{\cal H}_{4}^{\rm MHV}+{\cal O}(\epsilon)\,,$
(108)

with $G_{4}$ defined in eq. (17) that is validated in ref. Chicherin:2022bov
for the three-loop all-plus amplitude calculated in Jin:2019nya ;
Caola:2021izf .

As compared to the four-particle all-plus amplitude, the perturbative data for
the five-particle all-plus amplitude is available only up to the two-loop
order Gehrmann:2015bfy . Thus, we can use the duality relation (107) to
provide predictions about the three-loop five-particle all-plus amplitude.
Indeed, the finite remainder of the five-particle all-plus amplitude defined
in (105) has the following perturbative expansion,

$\displaystyle{\cal H}_{5}^{\rm YM}:=1+g^{2}{\cal H}_{{\rm
YM},5}^{(1)}+g^{4}{\cal H}_{{\rm YM},5}^{(2)}+{\cal O}(g^{6})$ (109)

where ${\cal H}_{{\rm YM},5}^{(L)}$ describe the $(L+1)$-loop corrections of
the all-plus amplitude. Expanding the five-particle duality relation (107)
perturbatively in the coupling, we find,

$\displaystyle{\cal H}_{{\rm YM},5}^{(1)}=$
$\displaystyle\sum_{i=1}^{5}\frac{r_{5,i}}{f_{5}^{(0)}}g_{i}^{(1)}-\frac{1}{2}\mathbf{L}_{2}+\frac{5\pi^{2}}{12}\,,$
(110) $\displaystyle{\cal H}_{{\rm YM},5}^{(2)}=$
$\displaystyle\sum_{i=1}^{5}\frac{r_{5,i}}{f_{5}^{(0)}}\left[g_{i}^{(2)}+g_{i}^{(1)}\left(-g_{0}^{(1)}+\frac{5\pi^{2}}{12}-\frac{1}{2}\mathbf{L}_{2}\right)\right]$
$\displaystyle+g^{(2)}_{0}-\frac{1}{2}\left(g_{0}^{(1)}\right)^{2}+\frac{\pi^{2}}{3}g_{0}^{(1)}+\zeta_{3}\mathbf{L}_{1}+\left(\mathbf{L}_{2}-\frac{\pi^{2}}{6}\right)^{2}-\frac{\pi^{4}}{12}\,,$
(111)

where the one-loop pure functions $g^{(1)}_{0}$ and $g^{(1)}_{i}$ are given in
(50) and (51), respectively. Expressions for the two-loop pure functions
$g^{(2)}_{0}$ and $g^{(2)}_{i}$ in the pentagon function basis are provided in
the ancillary files (see discussion around eq. (54)). We also use shorthand
notations

$\displaystyle\mathbf{L}_{1}:=\sum_{i=1}^{5}\log\left(\frac{\mu^{2}}{s_{i\,i+1}}\right)\,,\qquad\mathbf{L}_{2}:=\sum_{i=1}^{5}\log^{2}\left(\frac{\mu^{2}}{s_{i\,i+1}}\right)\,.$
(112)

The expression (110) is a special case of the $n$-particle two-loop duality
relation that we verified in Chicherin:2022bov . However, eq. (111) is new,
and it predicts the maximally transcendental part of the five-particle three-
loop all-plus planar amplitude.

It would be extremely interesting to test our conjecture about the maximally
transcendental part of the planar five-particle all-plus amplitude by an
explicit three-loop Feynman graph calculation. Furthermore, a three-loop
calculation of $f_{5}$ will provide a conjecture for the maximally
transcendental part of the planar four-loop five-particle all-plus amplitude.

##  7 Discussion and outlook

In this paper, we studied a null pentagonal Wilson loop with a Lagrangian
insertion, $f_{5}$, and computed for the first time its two-loop corrections.
The two-loop result is written as a sum of six conformal leading singularities
(as conjectured in Chicherin:2022bov ), each multiplied by a transcendental
function of weight four. The functions needed are pentagon functions that are
familiar from two-loop planar five-particle scattering amplitudes in QCD.

We noticed that similarly to appropriately-defined finite parts of scattering
amplitudes Chicherin:2020umh , the alphabet needed to express the answer does
not require the full planar pentagon alphabet. In particular, the letter
$W_{31}$ is not needed. Moreover, we observe that the last entries of the
weight-four functions we compute depend on a subset of alphabet letters only.
We interpret this as a hint that an analogue of the $\bar{Q}$-equation Caron-
Huot:2011dec , which was instrumental in the bootstrap of scattering
amplitudes in ${\mathcal{N}}=4$ sYM, could play a role here as well, for a
supersymmetrised version of $f_{5}$. This exciting question is currently under
investigation CHT .

We studied the analytic result in several singular limits, finding consistency
with previous four-particle results in the soft and collinear limits, and
providing new data in the multi-Regge limit. These limits played and important
role in bootstrap approaches to scattering amplitudes in ${\mathcal{N}}=4$
sYM. We hope that the data we provide can be valuable for similar analyses for
the Wilson loops with Lagrangian insertions. Note that the expressions in the
limit that we spell out in the paper contain all logarithmically enhanced and
finite terms. If required, power-suppressed terms can readily be obtained from
the full analytic result that we provided in terms of pentagon functions,
along the lines of Caron-Huot:2020vlo .

In reference Arkani-Hamed:2021iya , it was found that the four-cusp Wilson
loop with Lagrangian insertion has definite sign at each loop order $L$ (the
overall sign alternates with $L$) in the Euclidean region. We analyzed $f_{5}$
numerically in the Euclidean region. We focused on the subset with
$\epsilon_{5}>0$, which is natural from the point of view of the one-loop
Amplituhedron. At one loop, special sign patterns of the integrated answer had
already been observed in that region Gehrmann:2015bfy . In the present paper,
we provided numerical evidence that the two-loop correction has a definite
sign in that region, despite that fact that individual terms can have
alternating signs throughout the region. We take this as strong evidence for a
positivity conjecture of the observable $f_{5}$. It would be very interesting
to understand how the positivity property follows from properties of the loop
integrands.

Our computation in the present paper used state-of-the-art scattering
amplitude and Feynman integral methods for the loop integrations. However, it
was shown in Arkani-Hamed:2021iya  that the observable can be written in terms
of manifestly finite, four-dimensional integrals. The new representations are
motivated by the Amplituhedron geometry, using a novel decomposition of the
observable in terms of certain ‘negative’ geometries. This offers a new
perspective and approach to studying the observables. It would be very
exciting to harness the power of the four-dimensional approach and apply it to
the $n$-particle observable ACHT2022progress .

## Acknowledgements

It is a pleasure to thank Nima Arkani-Hamed, Jaroslav Trnka, and Simone Zoia
for stimulating discussions and helpful comments on the draft. This project
received funding from the European Union’s Horizon 2020 research and
innovation programme under ERC grant Novel structures in scattering amplitudes
(grant agreement No 725110). D.C. is supported by the French National Research
Agency in the framework of the Investissements d’avenir program
(ANR-15-IDEX-02).

##  Appendix A Pentagon functions

In this Appendix, we provide more details on the pentagon functions, the
underlying pentagon alphabet, and the iterated integral representation for the
pentagon functions.

The letters of the planar pentagon alphabet
$\\{W_{k}\\}_{k=1}^{20}\cup\\{W_{k}\\}_{k=26}^{31}$ are naturally organized in
several groups by the cyclic symmetry. The first 20 letters are parity-even,
and they are given by linear combinations of the Mandelstam variables,

$\displaystyle W_{1}=s_{12}\;,$ $\displaystyle\;W_{1+i}=\tau^{i}(W_{1})\,,$
$\displaystyle W_{6}=s_{34}+s_{45}\;,$
$\displaystyle\;W_{6+i}=\tau^{i}(W_{6})\,,$ $\displaystyle
W_{11}=s_{12}-s_{45}\;,$ $\displaystyle\;W_{11+i}=\tau^{i}(W_{11})\,,$
$\displaystyle W_{16}=s_{13}\;,$ $\displaystyle\;W_{16+i}=\tau^{i}(W_{16})\,,$
(113)

with $i=1,\ldots,4$. The letter $W_{31}=\epsilon_{5}$ is parity even and
invariant under cyclic shifts $\tau$. The remaining five letters are parity-
odd, namely $d\log(W_{26+i})\to-d\log(W_{26+i})$ under parity conjugation.
They have more complicated expressions,

$\displaystyle
W_{26}=\frac{\mbox{tr}_{-}(p_{4}p_{5}p_{1}p_{2})}{\mbox{tr}_{+}(p_{4}p_{5}p_{1}p_{2})}\;,\qquad
W_{26+i}=\tau^{i}(W_{26})\;,\qquad i=1,\ldots,4\,.$ (114)

Let us note that five of the leading singularities of $f_{5}$ allow for
factorization in the alphabet letters, e.g.

$\displaystyle(r_{5,1})^{2}=W_{2}(W_{3})^{3}W_{4}\frac{W_{29}}{W_{20}}\,,$
(115)

and similarly for its cyclic shifts (43).

The pentagon functions $p_{a}^{(w)}$ are defined as a linear combination of
multi-fold iterated integrals over the pentagon alphabet.777In ref.
Gehrmann:2018yef , the planar pentagon functions of the given weight $w$ are
grouped into cyclic orbits. Each orbit contains either a single function,
which is invariant under cyclic shifts $\tau$ of the particle labels $p_{i}\to
p_{i+1}$, or five pentagon functions which are related among each other by the
cyclic shift. In the latter case, an index $i=1,\ldots,5$ distinguishes
functions from the same cyclic orbit. So, the pentagon functions of
Gehrmann:2018yef  are labeled as follows. Weight-one: $f^{(i)}_{1,1}$; weight-
two: $f^{(i)}_{2,1}$; weight-three: $f^{(i)}_{3,\alpha}$ with $\alpha=1,2,3$
and $f_{3,4}$; weight-four: $f^{(i)}_{4,\alpha}$ with $\alpha=1,\ldots,11$ and
$f_{4,12}$. The present notations match with those of Gehrmann:2018yef  as
follows: $f_{w,a}=p^{(w)}_{a}$ and $f_{w,a}^{(i)}=p^{(w)}_{a,i}$. We also
imply that $p^{(w)}_{a}$ refers to both $p^{(w)}_{a}$ and $p^{(w)}_{a,i}$.
Despite the fact that the loop corrections in the five-particle kinematics
nontrivially depend only on four dimensionless variables, e.g. the ratios
(45), the pentagon functions $p^{(w)}_{a}=p^{(w)}_{a}({\bf s})$ are defined as
functions of the five Mandelstam variables (26). In this way the cyclic
symmetry is not broken.

The collection of pentagon functions, supplemented with transcendental
constants $\pi^{2}$ and $\zeta_{3}$, is closed upon differentiation (52).
Taking into account the transcendental weight of the constants, the
differentiation decrements the weight by one unit,

$\displaystyle dp^{(1)}_{a}=$ $\displaystyle\sum_{k}A^{k}_{a}\,d\log W_{k}\,,$
(116) $\displaystyle dp^{(2)}_{a}=$
$\displaystyle\sum_{k}A^{k}_{a,b}\,p^{(1)}_{b}d\log W_{k}\,,$ (117)
$\displaystyle dp^{(3)}_{a}=$
$\displaystyle\sum_{k}\left[A^{k}_{a,b}\,p^{(2)}_{b}+A^{k}_{a,b,c}\,p^{(1)}_{b}p^{(1)}_{c}+A^{k}_{a}\,\pi^{2}\right]d\log
W_{k}\,,$ (118) $\displaystyle dp^{(4)}_{a}=$
$\displaystyle\sum_{k}\Bigl{[}A^{k}_{a,b}\,p^{(3)}_{b}+A^{k}_{a,b,c}\,p^{(2)}_{b}p^{(1)}_{c}+A^{k}_{a,b,c,d}\,p^{(1)}_{b}p^{(1)}_{c}p^{(1)}_{d}$
$\displaystyle+A^{\prime
k}_{a,b}\,\pi^{2}p^{(1)}_{b}+A^{k}_{a}\,\zeta_{3}\Bigr{]}d\log W_{k}\,,$ (119)

where $A^{k}_{a,b},A^{k}_{a,b,c},\ldots$ are rational numbers and we omit
summation over repeated labels of the pentagon functions.

Differential equations (116)–(119) can be straightforwardly integrated. In
order to do so, one chooses a base point ${\bf s}_{0}$ in the Euclidean region
(31), e.g.

$\displaystyle{\bf s}_{0}\;:\quad s_{12}=s_{23}=s_{34}=s_{45}=s_{15}=-1\,,$
(120)

and connects it with any other point ${\bf s}$ of the Euclidean region by a
path $\gamma({\bf s}_{0},{\bf s})$. Provided pentagon function values at ${\bf
s}_{0}$ are given,888All planar pentagon functions (up to weight four) of
Gehrmann:2015bfy  are defined to vanish at the Euclidean base point ${\bf
s}_{0}$ except for $p^{(3)}_{4}$. The latter is given by a weight-3
transcendental constant $p^{(3)}_{4}({\bf s}_{0})=\frac{2}{3}d_{37,3}$ defined
in Gehrmann:2015bfy . we evaluate them at ${\bf s}$, e.g.

$\displaystyle p^{(2)}_{a}({\bf
s})=\sum_{k}A_{a,b}^{k}\,[p^{(1)}_{b},W_{k}]({\bf s})$ (121)

where we use shorthand notations

$\displaystyle[p_{a}^{(w)},W_{k}]({\bf s}):=\int\limits_{\gamma({\bf
s}_{0},{\bf s})}d\log(W_{k})\,p^{(w)}_{a}\,.$ (122)

Iterating this procedure, we express $p^{(w)}_{a}$ as a linear combination of
$w^{\prime}$-fold iterated integrals with $w^{\prime}\leq w$. For example, the
one-loop functions $g^{(1)}_{0}$ and $g^{(1)}_{1}$ (51) are the following two-
fold iterated integrals,

$\displaystyle g^{(1)}_{0}=$
$\displaystyle\left[\frac{W_{1}}{W_{3}},\frac{W_{5}}{W_{2}}\right]+\left[\frac{W_{5}}{W_{2}},\frac{W_{1}}{W_{3}}\right]-\frac{\pi^{2}}{6}+{\rm
cyclic}\,,$ (123) $\displaystyle g^{(1)}_{1}=$
$\displaystyle\left[\frac{W_{1}}{W_{3}},\frac{W_{13}}{W_{5}}\right]+\left[\frac{W_{5}}{W_{3}},\frac{W_{15}}{W_{1}}\right]+\frac{\pi^{2}}{6}\,.$
(124)

If we omit transcendental constants in the iterated integral expressions, we
obtain their symbol counterparts. So, the symbol of a weight-$w$ pentagon
function is a linear combination (with rational coefficients) of length-$w$
words $[W_{i_{1}},W_{i_{2}},\ldots,W_{i_{w}}]$ formed from the letters of the
pentagon alphabet.

The letters of the pentagon alphabet have definite parity, i.e.
$\\{W_{i}\\}_{i=26}^{30}$ (114) are parity-odd and $\\{W_{i}\\}_{i=1}^{20}$
(113) and $W_{31}$ are parity even. The parity grading is inherited by the
symbols as well, namely a parity-odd word
$[W_{i_{1}},W_{i_{2}},\ldots,W_{i_{w}}]$ involves odd number of parity-odd
letters. The planar pentagon functions and the corresponding iterated
integrals have definite parity as well.999Among the planar pentagon functions
of Gehrmann:2015bfy  the parity-odd are $p^{(3)}_{4},p^{(4)}_{12}$ and
$p^{(4)}_{10,i}$ with $i=1,\ldots,5$. We would also need to assign odd parity
to the transcendental constant $d_{37,3}$ in order the iterated integrals have
definite parity.

## References

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