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JHEP08(2019)042

Published for SISSA by Springer

Received: June 13, 2019

Accepted: July 28, 2019

Published: August 8, 2019

The momentum amplituhedron

David Damgaard,

Livia Ferro,

a,b

Tomasz Lukowski

and Matteo Parisi

Arnold-Sommerfeld-Center for Theoretical Physics,

Ludwig-Maximilians-Universit

at,

Theresienstraße 37, 80333 M

unchen, Germany

School of Physics, Astronomy and Mathematics,

University of Hertfordshire,

Hatﬁeld, Hertfordshire, AL10 9AB, U.K.

Mathematical Institute, University of Oxford,

Andrew Wiles Building, Radcliﬀe Observatory Quarter,

Woodstock Road, Oxford, OX2 6GG, U.K.

E-mail: d.damgaard@lmu.de, livia.ferro@lmu.de, t.lukowski@herts.ac.uk,

parisi@maths.ox.ac.uk

Abstract: In this paper we deﬁne a new object, the momentum amplituhedron, which

is the long sought-after positive geometry for tree-level scattering amplitudes in N =

4 super Yang-Mills theory in spinor helicity space. Inspired by the construction of the

ordinary amplituhedron, we introduce bosonized spinor helicity variables to represent our

external kinematical data, and restrict them to a particular positive region. The momentum

amplituhedron M

n,k

is then the image of the positive Grassmannian via a map determined

by such kinematics. The scattering amplitudes are extracted from the canonical form with

logarithmic singularities on the boundaries of this geometry.

Keywords: Scattering Amplitudes, Supersymmetric Gauge Theory

ArXiv ePrint: 1905.04216

Open Access,

 The Authors.

Article funded by SCOAP

https://doi.org/10.1007/JHEP08(2019)042

JHEP08(2019)042

Contents

1 Introduction 1

2 The deﬁnition 2

2.1 The ordinary amplituhedron 2

2.2 The momentum amplituhedron 4

2.3 Momentum amplituhedron volume form 8

3 Examples 10

3.1 MHV/MHV amplitudes 10

3.2 NMHV

amplitude 12

4 Conclusions and outlook 16

A Orthogonal complements 17

B Proof of the relation (Xpq) = hY pqi 18

C Momentum conservation 18

D Positive Mandelstam variables for k = 2 19

1 Introduction

In the past decade a new, geometric picture has emerged for scattering amplitudes in pla-

nar N = 4 super Yang-Mills (SYM) theory. It originated from the observation that the

tree-level amplitudes and loop-level integrands of n-point amplitudes for all helicity sectors

can be computed using integrals over the Grassmannian space [1, 2]. In such formula-

tion, amplitudes can be extracted from a Grassmannian integral over a suitable contour

which selects a particular sum of residues. Building upon this idea, novel studies revealed

the interrelation between the rich combinatorial structure of positive Grassmannians and

the physical properties of amplitudes [3]. From this point of view, the aforementioned

residues are associated with positroid cells, which are particular subvarieties inside the

positive Grassmannian. The proper combination of cells is selected by using the Britto-

Cachazo-Feng-Witten (BCFW) recursion relations [4, 5]. However, the cells contributing

to a particular amplitude are seemingly not related to each other inside the positive Grass-

mannian. Nevertheless, via a map deﬁned by a positive matrix of bosonized momentum

twistor variables, they assemble in a convex-like object. The image of the positive Grass-

mannian through such map is a geometric space, the amplituhedron [6], and the union of the

cell images provides a particular triangulation. The amplituhedron became eventually the

ﬁrst example of a vast family of the so-called positive geometries [7], which nowadays pro-

vide a geometric description for various quantities in theoretical physics: see, for instance,

– 1 –

JHEP08(2019)042

the kinematic associahedron [8], the cosmological polytope [9], and positive geometries in

CFT [10, 11].

Nevertheless, despite the name, the amplituhedron is more naturally suited to describe

the dual Wilson loop rather than the amplitude itself, being deﬁned in the momentum

twistor space. In particular, the employment of these variables restricts the possible gener-

alization of this geometry to scattering amplitudes in other models, since it is based on the

Amplitude/Wilson loop duality which is present only in planar N = 4 SYM. Therefore it

limits the possibility of ﬁnding positive geometries for scattering amplitudes in less super-

symmetric models and beyond the planar sector. It is then desirable to ﬁnd a geometric

description directly in the ordinary twistor space or, even better, in the spinor helicity space

(λ

). The ﬁrst attempt in this direction was made in [12], where it was suggested that

the amplituhedron in momentum space should be the image of the twistor-string world-

sheet [13] through the Roiban-Spradlin-Volovich (RSV) equations [14]. In particular, it

was conjectured that the space should have proper sign ﬂips for both λ

and

, as well

as the additional assumption that the planar Mandelstam variables should be positive. In

this paper we will show that a suitable positive geometry with such characteristics exists

and provides the proper expressions for the amplitude when written in the non-chiral su-

perspace (λ

, η

, ˜η

). In order to achieve our goal, we will ﬁrst introduce its bosonized

version: (Λ

). By assuming that the external kinematic data Λ and

Λ satisfy particular

positivity conditions, we will reproduce the sign ﬂips postulated in [12]. Additionally, fur-

ther constraints entangling Λ and

Λ will enforce positivity of Mandelstam variables. Then,

we will deﬁne the momentum amplituhedron as the image of the positive Grassmannian

through a map determined by this positive external data. We will show that such space is

indeed a positive geometry, whose canonical logarithmic diﬀerential form encodes scattering

amplitudes in spinor helicity variables.

The paper is structured as follows. We start in section 2 by reviewing the formulation

of the original amplituhedron in the bosonized momentum twistor space. We proceed by

deﬁning the momentum amplituhedron, i.e. the positive geometry in the bozonized spinor

helicity variables. Afterwards, we show how to ﬁnd the logarithmic diﬀerential form on the

momentum amplituhedron and how to extract the scattering amplitudes from it. Section 3

consists of examples which show in detail how to use the construction from section 2.

We end the paper with Conclusions and Outlook, and a few appendices containing more

technical details of our construction.

2 The deﬁnition

2.1 The ordinary amplituhedron

We start by recalling the construction of the amplituhedron in momentum twistor space.

In the past few years there has been a lot of progress on diﬀerent descriptions of the am-

plituhedron [7, 15]. We will focus here on two of them, which will be relevant for our

construction of the momentum amplituhedron: the original deﬁnition introduced in [6] and

the description based on the sign ﬂips presented in [15]. The ﬁrst states that the amplituhe-

dron A

(m=4)

n,k

can be described on the space of bosonized supertwistors Z

, A = 1, . . . , 4+k

– 2 –

JHEP08(2019)042

which specify the kinematic data for the n-particle N

MHV amplitude. The components of

bosonized supertwistors include the bosonic part of the momentum supertwistors (λ

, ˜µ

˙a

a, ˙a = 1, 2, and the bosonized version of the fermionic components ξ

= φ

, α = 1, . . . k

where φ

are auxiliary Grassmann-odd parameters and R = 1, . . . , 4 is the R-symmetry

index. As already explored in the literature, there exists a straightforward generalization

of bosonized variables beyond the case relevant for physics, m = 4, and in the following we

will allow any values for the label m. We start by demanding that the matrix of bosonized

variables Z = (Z

) ∈ M

(m + k

, n) is positive, i.e. all its ordered maximal minors are

positive. Then the amplituhedron A

(m)

n,k

is deﬁned as the image of the map

: G

, n) → G(k

, k

+ m) , (2.1)

given by

= c

αi

∈ G(k

, k

+ m) , C = (c

αi

) ∈ G

, n) . (2.2)

Here, G

, n) is the positive Grassmannian, i.e. the space of all positive matrices modulo

GL(k

) transformations. For each amplituhedron A

(m)

n,k

, one can deﬁne a (k

·m)-dimensional

diﬀerential form Ω

(m)

n,k

, called the volume form, which has logarithmic singularities on all

boundaries (of all dimensions) of the amplituhedron space A

(m)

n,k

. In particular, the vol-

ume form Ω

(m=4)

n,k

encodes the N

MHV tree-level amplitude in N = 4 SYM. The geo-

metric space A

(m)

n,k

together with the form Ω

(m)

n,k

describe a positive geometry, as deﬁned

in [7]. Throughout the years, various methods to ﬁnd the volume form Ω

(m)

n,k

have been

proposed [7, 16–18]. One can, for example, triangulate the amplituhedron A

(m)

n,k

by e.g.

ﬁnding a collection of positroid cells T = {∆

} of dimension k

· m in G

, n) such that

the images of these cells through the function Φ

do not overlap and cover the ampli-

tuhedron. To each positroid cell one can associate a canonical form ω

with logarithmic

singularities on all its boundaries, see [3]. The volume form Ω

(m)

n,k

is found by evaluating

the push-forward of the canonical forms ω

via the function Φ

and then summing over

all positroid cells in the triangulation

Ω

(m)

n,k

∆

∈T

(Φ

)

∗

. (2.3)

The result of the push-forward is a logarithmic diﬀerential form on G(k

, k

+ m) which can

be written as

Ω

(m)

n,k

∆

∈T

log α

(Y, Z) ∧ d

log α

(Y, Z) ∧ . . . ∧ d

log α

(Y, Z) , (2.4)

where α

(Y, Z) are the canonical positive coordinates parametrizing the cell ∆

. The

explicit expressions for various values of parameters (n, m, k

) can be found e.g. in [7].

An alternative way to ﬁnd the volume form is to introduce a volume function Ω

(m)

n,k

deﬁned by

Ω

(m)

n,k

α=1

. . . Y

i Ω

(m)

n,k

, (2.5)

– 3 –

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