# Scattering equations: real solutions and particles on a line

@article{Cachazo2016ScatteringER, title={Scattering equations: real solutions and particles on a line}, author={Freddy Cachazo and Sebastian Mizera and Guojun Zhang}, journal={Journal of High Energy Physics}, year={2016}, volume={2017}, pages={1-22} }

A bstractWe find n(n − 3)/2-dimensional regions of the space of kinematic invariants, where all the solutions to the scattering equations (the core of the CHY formulation of amplitudes) for n massless particles are real. On these regions, the scattering equations are equivalent to the problem of finding stationary points of n − 3 mutually repelling particles on a finite real interval with appropriate boundary conditions. This identification directly implies that for each of the (n − 3…

## Figures from this paper

## 34 Citations

Bootstrapping solutions of scattering equations

- MathematicsJournal of High Energy Physics
- 2019

A bstractThe scattering equations are a set of algebraic equations connecting the kinematic space of massless particles and the moduli space of Riemann spheres with marked points. We present an…

Properties of scattering forms and their relation to associahedra

- Mathematics
- 2017

A bstractWe show that the half-integrands in the CHY representation of tree amplitudes give rise to the definition of differential forms — the scattering forms — on the moduli space of a Riemann…

Scattering and Strebel graphs

- Mathematics
- 2021

We consider a special scattering experiment with n particles in Rn−3,1. The scattering equations in this set-up become the saddle-point equations of a Penner-like matrix model, where in the large n…

Scattering forms and the positive geometry of kinematics, color and the worldsheet

- Mathematics
- 2017

A bstractThe search for a theory of the S-Matrix over the past five decades has revealed surprising geometric structures underlying scattering amplitudes ranging from the string worldsheet to the…

Hyperbolic geometry and amplituhedra in 1+2 dimensions

- MathematicsJournal of High Energy Physics
- 2018

A bstractRecently, the existence of an Amplituhedron for tree level amplitudes in the bi-adjoint scalar field theory has been proved by Arkani-Hamed et al. We argue that hyperbolic geometry…

Aspects of Scattering Amplitudes and Moduli Space Localization

- Mathematics
- 2020

We propose that intersection numbers of certain cohomology classes on the moduli space of genus-zero Riemann surfaces with $n$ punctures, $\mathcal{M}_{0,n}$, compute tree-level scattering amplitudes…

Notes on worldsheet-like variables for cluster configuration spaces

- Mathematics
- 2021

We continue the exploration of various appearances of cluster algebras in scattering amplitudes and related topics in physics. The cluster configuration spaces generalize the familiar moduli…

1-loop amplitudes from the Halohedron

- PhysicsJournal of High Energy Physics
- 2019

Abstract
We recently proposed the Halohedron to be the 1-loop Amplituhedron for planar 𝜙3 theory. Here we prove this claim by showing how it is possible to extract the integrand for the partial…

Scattering equations: from projective spaces to tropical grassmannians

- MathematicsJournal of High Energy Physics
- 2019

A bstractWe introduce a natural generalization of the scattering equations, which connect the space of Mandelstam invariants to that of points on ℂℙ1, to higher-dimensional projective spaces ℂℙk − 1.…

Compatible cycles and CHY integrals

- MathematicsJournal of High Energy Physics
- 2019

Abstract
The CHY construction naturally associates a vector in ℝ(n−3)! to every 2- regular graph with n vertices. Partial amplitudes in the biadjoint scalar theory are given by the inner product…

## References

SHOWING 1-10 OF 49 REFERENCES

General solution of the scattering equations

- Mathematics
- 2015

A bstractThe scattering equations, originally introduced by Fairlie and Roberts in 1972 and more recently shown by Cachazo, He and Yuan to provide a kinematic basis for describing tree amplitudes for…

The polynomial form of the scattering equations

- Mathematics
- 2014

A bstractThe scattering equations, recently proposed by Cachazo, He and Yuan as providing a kinematic basis for describing tree amplitudes for massless particles in arbitrary space-time dimension…

Scattering Amplitudes and the Positive Grassmannian

- Mathematics
- 2012

We establish a direct connection between scattering amplitudes in planar four-dimensional theories and a remarkable mathematical structure known as the positive Grassmannian. The central physical…

Ambitwistor strings and the scattering equations

- Mathematics
- 2014

A bstractWe show that string theories admit chiral infinite tension analogues in which only the massless parts of the spectrum survive. Geometrically they describe holomorphic maps to spaces of…

Scattering of massless particles in arbitrary dimensions.

- MathematicsPhysical review letters
- 2014

A compact formula for the complete tree-level S-matrix of pure Yang-Mills and gravity theories in arbitrary spacetime dimensions is presented and Gauge invariance is completely manifest as it follows from a simple property of the Pfaffian.

Scattering in three dimensions from rational maps

- Mathematics
- 2013

A bstractThe complete tree-level S-matrix of four dimensional $ \mathcal{N}=4 $ super Yang-Mills and $ \mathcal{N}=8 $ supergravity has compact forms as integrals over the moduli space of certain…

Characterizing the solutions to scattering equations that support tree-level NkMHV gauge/gravity amplitudes

- Mathematics
- 2016

A bstractIn this paper we define, independent of theories, two discriminant matrices involving a solution to the scattering equations in four dimensions, the ranks of which are used to divide the…

Scattering of massless particles: scalars, gluons and gravitons

- Mathematics
- 2014

A bstractIn a recent note we presented a compact formula for the complete tree-level S-matrix of pure Yang-Mills and gravity theories in arbitrary spacetime dimension. In this paper we show that a…

The Grassmannian origin of dual superconformal invariance

- Mathematics, Physics
- 2010

A dual formulation of the S Matrix for $$ \mathcal {N} $$ = 4 SYM has recently been presented, where all leading singularities of n-particle Nk−2MHV amplitudes are given as an integral over the…

Scattering equations and matrices: from Einstein to Yang-Mills, DBI and NLSM

- Mathematics
- 2014

A bstractThe tree-level S-matrix of Einstein’s theory is known to have a representation as an integral over the moduli space of punctured spheres localized to the solutions of the scattering…