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

## Abstract

The 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 amplituhedron, but these are all geometries in auxiliary spaces as opposed to the kinematical space where amplitudes actually live. Motivated by recent advances providing a reformulation of the amplituhedron and planar N = 4 SYM amplitudes directly in kinematic space, we propose a novel geometric understanding of amplitudes in more general theories. The key idea is to think of amplitudes not as functions, but rather as differential forms on kinematic space. We explore the resulting picture for a wide range of massless theories in general spacetime dimensions. For the bi-adjoint Φ3 scalar theory, 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 level amplitude is simply the “canonical form” associated with this “positive geometry”. Fundamental physical properties such as locality and unitarity, as well as novel “soft” limits, are fully determined by the combinatorial geometry of this polytope. Furthermore, the moduli space for the open stringmore »

- Authors:

- Inst. for Advanced Study, Princeton, NJ (United States). School of Natural Sciences
- Princeton Univ., NJ (United States). Dept. of Physics
- Chinese Academy of Sciences (CAS), Beijing (China). Inst. of Theoretical Physics, Key Lab. of Theoretical Physics; Univ. of Chinese Academy of Sciences, Beijing (China)
- Tsinghua Univ., Beijing (China). Inst. for Advanced Study; Chinese Academy of Sciences (CAS), Beijing (China). Inst. of Theoretical Physics, Key Lab. of Theoretical Physics

- Publication Date:

- Research Org.:
- Inst. for Advanced Study, Princeton, NJ (United States)

- Sponsoring Org.:
- USDOE Office of Science (SC)

- OSTI Identifier:
- 1502485

- Grant/Contract Number:
- [SC0009988]

- Resource Type:
- Accepted Manuscript

- Journal Name:
- Journal of High Energy Physics (Online)

- Additional Journal Information:
- [Journal Name: Journal of High Energy Physics (Online); Journal Volume: 2018; Journal Issue: 5]; Journal ID: ISSN 1029-8479

- Publisher:
- Springer Berlin

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS; Scattering Amplitudes Differential and Algebraic Geometry

### Citation Formats

```
Arkani-Hamed, Nima, Bai, Yuntao, He, Song, and Yan, Gongwang. Scattering forms and the positive geometry of kinematics, color and the worldsheet. United States: N. p., 2018.
Web. doi:10.1007/jhep05(2018)096.
```

```
Arkani-Hamed, Nima, Bai, Yuntao, He, Song, & Yan, Gongwang. Scattering forms and the positive geometry of kinematics, color and the worldsheet. United States. doi:10.1007/jhep05(2018)096.
```

```
Arkani-Hamed, Nima, Bai, Yuntao, He, Song, and Yan, Gongwang. Tue .
"Scattering forms and the positive geometry of kinematics, color and the worldsheet". United States. doi:10.1007/jhep05(2018)096. https://www.osti.gov/servlets/purl/1502485.
```

```
@article{osti_1502485,
```

title = {Scattering forms and the positive geometry of kinematics, color and the worldsheet},

author = {Arkani-Hamed, Nima and Bai, Yuntao and He, Song and Yan, Gongwang},

abstractNote = {The 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 amplituhedron, but these are all geometries in auxiliary spaces as opposed to the kinematical space where amplitudes actually live. Motivated by recent advances providing a reformulation of the amplituhedron and planar N = 4 SYM amplitudes directly in kinematic space, we propose a novel geometric understanding of amplitudes in more general theories. The key idea is to think of amplitudes not as functions, but rather as differential forms on kinematic space. We explore the resulting picture for a wide range of massless theories in general spacetime dimensions. For the bi-adjoint Φ3 scalar theory, 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 level amplitude is simply the “canonical form” associated with this “positive geometry”. Fundamental physical properties such as locality and unitarity, as well as novel “soft” limits, are fully determined by the combinatorial geometry of this polytope. 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 the interior of this old “worldsheet associahedron” and the new “kinematic associahedron”, providing a geometric interpretation and simple conceptual derivation of the bi-adjoint CHY formula. We also find “scattering forms” on kinematic space for Yang-Mills theory and the Non-linear Sigma Model, which are dual to the fully 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, all our scattering forms are well-defined on the projectivized kinematic space, a property which can be seen to provide a geometric origin for color-kinematics duality.},

doi = {10.1007/jhep05(2018)096},

journal = {Journal of High Energy Physics (Online)},

number = [5],

volume = [2018],

place = {United States},

year = {2018},

month = {5}

}

*Citation information provided by*

Web of Science

Web of Science

#### Figures / Tables:

Works referenced in this record:

##
Infinite tension limit of the pure spinor superstring

journal, March 2014

- Berkovits, Nathan
- Journal of High Energy Physics, Vol. 2014, Issue 3

##
Combinatorics and topology of Kawai-Lewellen-Tye relations

journal, August 2017

- Mizera, Sebastian
- Journal of High Energy Physics, Vol. 2017, Issue 8

##
Scattering in three dimensions from rational maps

journal, October 2013

- Cachazo, Freddy; He, Song; Yuan, Ellis Ye
- Journal of High Energy Physics, Vol. 2013, Issue 10

##
A relation between tree amplitudes of closed and open strings

journal, May 1986

- Kawai, H.; Lewellen, D. C.; Tye, S. -H. H.
- Nuclear Physics B, Vol. 269, Issue 1

##
New BCJ representations for one-loop amplitudes in gauge theories and gravity

journal, May 2018

- He, Song; Schlotterer, Oliver; Zhang, Yong
- Nuclear Physics B, Vol. 930

##
Perturbative Gauge Theory as a String Theory in Twistor Space

journal, October 2004

- Witten, Edward
- Communications in Mathematical Physics, Vol. 252, Issue 1-3

##
Complete N-point superstring disk amplitude II. Amplitude and hypergeometric function structure

journal, August 2013

- Mafra, Carlos R.; Schlotterer, Oliver; Stieberger, Stephan
- Nuclear Physics B, Vol. 873, Issue 3

##
Multigluon cross sections and 5-jet production at hadron colliders

journal, January 1989

- Kleiss, Ronald; Kuijf, Hans
- Nuclear Physics B, Vol. 312, Issue 3

##
Infrared behaviour of the one-loop scattering equations and supergravity integrands

journal, April 2015

- Casali, Eduardo; Tourkine, Piotr
- Journal of High Energy Physics, Vol. 2015, Issue 4

##
Loop Integrands for Scattering Amplitudes from the Riemann Sphere

journal, September 2015

- Geyer, Yvonne; Mason, Lionel; Monteiro, Ricardo
- Physical Review Letters, Vol. 115, Issue 12

##
New color decompositions for gauge amplitudes at tree and loop level

journal, April 2000

- Del Duca, Vittorio; Dixon, Lance; Maltoni, Fabio
- Nuclear Physics B, Vol. 571, Issue 1-2

##
Ambitwistor strings and the scattering equations at one loop

journal, April 2014

- Adamo, Tim; Casali, Eduardo; Skinner, David
- Journal of High Energy Physics, Vol. 2014, Issue 4

##
Manifestly crossing-invariant parametrization of n-meson amplitude

journal, September 1969

- Koba, Ziro; Nielsen, Holger Bech
- Nuclear Physics B, Vol. 12, Issue 3

##
Factorization of chiral string amplitudes

journal, September 2016

- Huang, Yu-tin; Siegel, Warren; Yuan, Ellis Ye
- Journal of High Energy Physics, Vol. 2016, Issue 9

##
Scattering of Massless Particles in Arbitrary Dimensions

journal, October 2014

- Cachazo, Freddy; He, Song; Yuan, Ellis Ye
- Physical Review Letters, Vol. 113, Issue 17

##
One-loop corrections from higher dimensional tree amplitudes

journal, August 2016

- Cachazo, Freddy; He, Song; Yuan, Ellis Ye
- Journal of High Energy Physics, Vol. 2016, Issue 8

##
Scattering equations: real solutions and particles on a line

journal, March 2017

- Cachazo, Freddy; Mizera, Sebastian; Zhang, Guojun
- Journal of High Energy Physics, Vol. 2017, Issue 3

##
Explicit BCJ numerators from pure spinors

journal, July 2011

- Mafra, Carlos R.; Schlotterer, Oliver; Stieberger, Stephan
- Journal of High Energy Physics, Vol. 2011, Issue 7

##
The Amplituhedron

journal, October 2014

- Arkani-Hamed, Nima; Trnka, Jaroslav
- Journal of High Energy Physics, Vol. 2014, Issue 10

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

journal, July 2015

- Cachazo, Freddy; He, Song; Yuan, Ellis Ye
- Journal of High Energy Physics, Vol. 2015, Issue 7

##
CHY loop integrands from holomorphic forms

journal, March 2017

- Gomez, Humberto; Mizera, Sebastian; Zhang, Guojun
- Journal of High Energy Physics, Vol. 2017, Issue 3

##
Color Dual Form for Gauge-Theory Amplitudes

journal, August 2011

- Bern, Z.; Dennen, T.
- Physical Review Letters, Vol. 107, Issue 8

##
New Relations for Gauge-Theory and Gravity Amplitudes at Loop Level

journal, April 2017

- He, Song; Schlotterer, Oliver
- Physical Review Letters, Vol. 118, Issue 16

##
Multiple zeta values and periods of moduli spaces $\overline{\mathfrak{M}}_{0,n}$

journal, January 2009

- Brown, Francis C. S.
- Annales scientifiques de l'École normale supérieure, Vol. 42, Issue 3

##
Labelled tree graphs, Feynman diagrams and disk integrals

journal, November 2017

- Gao, Xiangrui; He, Song; Zhang, Yong
- Journal of High Energy Physics, Vol. 2017, Issue 11

##
One-loop amplitudes on the Riemann sphere

journal, March 2016

- Geyer, Yvonne; Mason, Lionel; Monteiro, Ricardo
- Journal of High Energy Physics, Vol. 2016, Issue 3

##
A duality for the S matrix

journal, March 2010

- Arkani-Hamed, N.; Cachazo, F.; Cheung, C.
- Journal of High Energy Physics, Vol. 2010, Issue 3

##
Scattering of massless particles: scalars, gluons and gravitons

journal, July 2014

- Cachazo, Freddy; He, Song; Yuan, Ellis Ye
- Journal of High Energy Physics, Vol. 2014, Issue 7

##
Properties of scattering forms and their relation to associahedra

journal, March 2018

- de la Cruz, Leonardo; Kniss, Alexander; Weinzierl, Stefan
- Journal of High Energy Physics, Vol. 2018, Issue 3

##
New ambitwistor string theories

journal, November 2015

- Casali, Eduardo; Geyer, Yvonne; Mason, Lionel
- Journal of High Energy Physics, Vol. 2015, Issue 11

##
A current algebra for some gauge theory amplitudes

journal, November 1988

- Nair, V. P.
- Physics Letters B, Vol. 214, Issue 2

##
Unwinding the amplituhedron in binary

journal, January 2018

- Arkani-Hamed, Nima; Thomas, Hugh; Trnka, Jaroslav
- Journal of High Energy Physics, Vol. 2018, Issue 1

##
Worldsheet geometries of ambitwistor string

journal, June 2015

- Ohmori, Kantaro
- Journal of High Energy Physics, Vol. 2015, Issue 6

##
On the null origin of the ambitwistor string

journal, November 2016

- Casali, Eduardo; Tourkine, Piotr
- Journal of High Energy Physics, Vol. 2016, Issue 11

##
Perturbative Quantum Gravity as a Double Copy of Gauge Theory

journal, August 2010

- Bern, Zvi; Carrasco, John Joseph M.; Johansson, Henrik
- Physical Review Letters, Vol. 105, Issue 6

##
Scattering Amplitudes from Intersection Theory

journal, April 2018

- Mizera, Sebastian
- Physical Review Letters, Vol. 120, Issue 14

##
Positive geometries and canonical forms

journal, November 2017

- Arkani-Hamed, Nima; Bai, Yuntao; Lam, Thomas
- Journal of High Energy Physics, Vol. 2017, Issue 11

##
The amplituhedron from momentum twistor diagrams

journal, February 2015

- Bai, Yuntao; He, Song
- Journal of High Energy Physics, Vol. 2015, Issue 2

##
The irreducibility of the space of curves of given genus

journal, January 1969

- Deligne, P.; Mumford, D.
- Publications mathématiques de l'IHÉS, Vol. 36, Issue 1

##
Ambitwistor strings and the scattering equations

journal, July 2014

- Mason, Lionel; Skinner, David
- Journal of High Energy Physics, Vol. 2014, Issue 7

##
Eliminating spurious poles from gauge-theoretic amplitudes

journal, May 2013

- Hodges, Andrew
- Journal of High Energy Physics, Vol. 2013, Issue 5

##
Monoïdes préordonnés et chaînes de Malcev

journal, January 1951

- Tamari, Dov
- Bulletin de la Société mathématique de France, Vol. 79

Works referencing / citing this record:

##
Bootstrapping solutions of scattering equations

journal, February 2019

- Liu, Zhengwen; Zhao, Xiaoran
- Journal of High Energy Physics, Vol. 2019, Issue 2

Figures / Tables found in this record:

*Figures/Tables have been extracted from DOE-funded journal article accepted manuscripts.*