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Homework Helper
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If anyone here has worked on the associahedron and/or amplituhedron, I would like your input. I am focusing now on the paper by Nima Arkani-Hamed et al, "Scattering Forms and the Positive Geometry of Kinematics, Color an the Worldsheet".

The key step to define the kinematic associahedron is to impose ##X_{ij} \geq 0## and also the impose the ##c_{ij} > 0 ##. They do this without giving any motivation at all. I know that it gives the correct answer in the end, but I wonder if there is any way to motivate these choices, from a physics point of view (or short of that, from a mathematical point of view). It would be nice to get some intuition about what leads to these choices, other than "because it gives the correct final answer".

Thank you!

Urs Schreiber

MathematicalPhysicist
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How do they know what is the correct answer?

Homework Helper
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How do they know what is the correct answer?
The scattering form restricted to the associahedron reproduces the correct amplitude. Or, alternatively, the volume of the dual associahedron gives the correct amplitude.

ohwilleke
Gold Member
This isn't a full fledged answer to the question but highlights some of the relevant foundation for answering it and makes a couple of relevant observations.

For reference the pre-print is at https://arxiv.org/abs/1711.09102 and the paper's authorship and abstract are as follows:

Scattering Forms and the Positive Geometry of Kinematics, Color and the Worldsheet
Nima Arkani-Hamed, Yuntao Bai, Song He, Gongwang Yan
(Submitted on 24 Nov 2017 (v1), last revised 28 Mar 2018 (this version, v2))

The search for a theory of the S-Matrix has revealed surprising geometric structures underlying amplitudes ranging from the worldsheet to the amplituhedron, but these are all geometries in auxiliary spaces as opposed to kinematic space where amplitudes live. In this paper, we propose a novel geometric understanding of amplitudes for a large class of theories. The key is to think of amplitudes as differential forms directly on kinematic space. We explore this picture for a wide range of massless theories in general spacetime dimensions. For the bi-adjoint cubic scalar, we establish a direct connection between its "scattering form" and a classic polytope--the associahedron--known to mathematicians since the 1960's. We find an associahedron living naturally in kinematic space, and the tree amplitude is simply the "canonical form" associated with this "positive geometry". Basic physical properties such as locality, unitarity and novel "soft" limits are fully determined by the geometry. Furthermore, the moduli space for the open string worldsheet has also long been recognized as an associahedron. We show that the scattering equations act as a diffeomorphism between this old "worldsheet associahedron" and the new "kinematic associahedron", providing a geometric interpretation and novel derivation of the bi-adjoint CHY formula. We also find "scattering forms" on kinematic space for Yang-Mills and the Non-linear Sigma Model, which are dual to the color-dressed amplitudes despite having no explicit color factors. This is possible due to a remarkable fact--"Color is Kinematics"--whereby kinematic wedge products in the scattering forms satisfy the same Jacobi relations as color factors. Finally, our scattering forms are well-defined on the projectivized kinematic space, a property that provides a geometric origin for color-kinematics duality.

Comments: 77 pages, 25 figures; v2, corrected discussion of worldsheet associahedron canonical form
Subjects: High Energy Physics - Theory (hep-th); Combinatorics (math.CO)
DOI: 10.1007/JHEP05(2018)096
Cite as: arXiv:1711.09102 [hep-th]
(or arXiv:1711.09102v2 [hep-th] for this version)

The pdf url is https://arxiv.org/pdf/1711.09102.pdf

We can look at the two constraints separately.

Xij ≥ 0

Xij is defined at equation 2.6 on page 8 of the body text and the assumptions about it are made are made on page 13 at equations 3.4.

It is defined to be Xi,j := si,i+1,...,j−1

So Xij is a function of sij.

sij in turn, is defined as follows on page 7 of the body text utilizing equations 2.1 and 2.2.

We begin by defining the kinematic space Kn for n massless momenta pi for i = 1, . . . , n as the space spanned by linearly independent Mandelstam variables in spacetime dimension D ≥ n−1:
sij := (pi + pj ) 2 = 2pi · pj (2.1)

For D < n−1 there are further constraints on Mandelstam variables—Gram determinant conditions—so the number of independent variables is lower. Due to the massless on-shell conditions and momentum conservation, we have n linearly independent constraints

nj=1; j≠i sij = 0 for i = 1, 2, . . . , n (2.2)

So sij (which are later described in the body text at page 13 at planar propagators are each a function of the dot product (a scalar) of massless momenta pi, which are vector quantities. And, dot products of real number space are non-negative in order to be physically meaningful.

The bold language of the definition of sij gets to the physical motivation for the assumption about Xij being non-negative.

cij> 0

cij is defined as a positive number in equation 3.5 of the body text at page 13 (so it isn't truly an assumption so much as a definition).

This is earlier discussed and motivated at body text page 5 which states:

This geometry generalizes to all n in a simple way. The full kinematic space of Mandelstam invariants is n(n − 3)/2-dimensional. A nice basis for this space is provided by all planar propagators sa,a+1,...,b−1, and there is a natural “positive region” in which all these variables are forced to be positive. There is also a natural (n−3)-dimensional subspace that is cut out by the equations −sij = cij for all non-adjacent i, j excluding the index n, where the cij > 0 are positive constants. These equalities pick out an (n − 3)-dimensional hyperplane in kinematic space whose intersection with the positive region is the associahedron polytope.

Per the link to Wikipedia in the text above:

In theoretical physics, the Mandelstam variables are numerical quantities that encode the energy, momentum, and angles of particles in a scattering process in a Lorentz-invariant fashion.

This material should allow others to more easily comment on your question and I encourage anyone who notes a mistake in my own meager contributions to point it out.

Last edited:
nrqed
Homework Helper
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This isn't a full fledged answer to the question but highlights some of the relevant foundation for answering it and makes a couple of relevant observations.

Thanks Ohwilleke for your post which nicely summarizes the key elements.

I am still trying to figure out the motivation (whether physical or mathematical) for choosing these positivity (or non-negativity) conditions on the ##s_{ij}## and ##c_{ij}##. If anyone has something to add, I would appreciate.

Have you tried contacting the authors?