Scattering amplitudes and simple canonical forms for simple polytopes

Salvatori, Giulio; Stanojevic, Stefan

doi:10.1007/jhep03(2021)067

Title: Scattering amplitudes and simple canonical forms for simple polytopes

Abstract

We provide an efficient recursive formula to compute the canonical forms of arbitrary d-dimensional simple polytopes, which are convex polytopes such that every vertex lies precisely on d facets. For illustration purposes, we explicitly derive recursive formulae for the canonical forms of Stokes polytopes, which play a similar role for a theory with quartic interaction as the Associahedron does in planar bi-adjoint φ³ theory. As a by-product, our formula also suggests a new way to obtain the full planar amplitude in φ⁴ theory by taking suitable limits of the canonical forms of constituent Stokes polytopes.

Authors:

Salvatori, Giulio ^[1]; Stanojevic, Stefan ^[1]

Brown Univ., Providence, RI (United States). Dept. of Physics

Publication Date:: Fri Mar 05 00:00:00 EST 2021

Research Org.:: Brown Univ., Providence, RI (United States)

Sponsoring Org.:: USDOE Office of Science (SC); Simons Foundation

OSTI Identifier:: 1851281

Grant/Contract Number:: SC0010010; 376208

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: 2021; Journal Issue: 3; Journal ID: ISSN 1029-8479

Publisher:: Springer Nature

Country of Publication:: United States

Language:: English

Subject:: 71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; Physics; Scattering Amplitudes; Differential and Algebraic Geometry

Citation Formats


                    Salvatori, Giulio, and Stanojevic, Stefan. Scattering amplitudes and simple canonical forms for simple polytopes.  United States: N. p., 2021. 
Web.  doi:10.1007/jhep03(2021)067.

Copy to clipboard


                    Salvatori, Giulio, & Stanojevic, Stefan. Scattering amplitudes and simple canonical forms for simple polytopes.  United States.  https://doi.org/10.1007/jhep03(2021)067

Copy to clipboard


                    Salvatori, Giulio, and Stanojevic, Stefan. Fri .  
"Scattering amplitudes and simple canonical forms for simple polytopes".  United States.  https://doi.org/10.1007/jhep03(2021)067.  https://www.osti.gov/servlets/purl/1851281.

Copy to clipboard


                    
@article{osti_1851281,

  title        = {Scattering amplitudes and simple canonical forms for simple polytopes},

  author       = {Salvatori, Giulio and Stanojevic, Stefan},

  abstractNote = {We provide an efficient recursive formula to compute the canonical forms of arbitrary d-dimensional simple polytopes, which are convex polytopes such that every vertex lies precisely on d facets. For illustration purposes, we explicitly derive recursive formulae for the canonical forms of Stokes polytopes, which play a similar role for a theory with quartic interaction as the Associahedron does in planar bi-adjoint φ3 theory. As a by-product, our formula also suggests a new way to obtain the full planar amplitude in φ4 theory by taking suitable limits of the canonical forms of constituent Stokes polytopes.},

  doi          = {10.1007/jhep03(2021)067},

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

  number       = 3,

  volume       = 2021,

  place        = {United States},

  year         = {Fri Mar 05 00:00:00 EST 2021},

  month        = {Fri Mar 05 00:00:00 EST 2021}

}

Copy to clipboard

Journal Article:

Free Publicly Available Full Text

Accepted Manuscript (DOE)

Publisher's Version of Record

https://doi.org/10.1007/jhep03(2021)067

Other availability

Search WorldCat to find libraries that may hold this journal

Save / Share:

Export Metadata

Save to My Library

Works referenced in this record:

1-loop amplitudes from the Halohedron
journal, December 2019

Salvatori, Giulio
Journal of High Energy Physics, Vol. 2019, Issue 12
DOI: 10.1007/JHEP12(2019)074

An etude on recursion relations and triangulations
journal, May 2019

He, Song; Yang, Qinglin
Journal of High Energy Physics, Vol. 2019, Issue 5
DOI: 10.1007/JHEP05(2019)040

Scattering forms and the positive geometry of kinematics, color and the worldsheet
journal, May 2018

Arkani-Hamed, Nima; Bai, Yuntao; He, Song
Journal of High Energy Physics, Vol. 2018, Issue 5
DOI: 10.1007/JHEP05(2018)096

The positive geometry for 𝜙p interactions
journal, October 2019

Raman, Prashanth
Journal of High Energy Physics, Vol. 2019, Issue 10
DOI: 10.1007/JHEP10(2019)271

The Amplituhedron
journal, October 2014

Arkani-Hamed, Nima; Trnka, Jaroslav
Journal of High Energy Physics, Vol. 2014, Issue 10
DOI: 10.1007/JHEP10(2014)030

Geometric Realizations of the Accordion Complex of a Dissection
journal, May 2018

Manneville, Thibault; Pilaud, Vincent
Discrete & Computational Geometry, Vol. 61, Issue 3
DOI: 10.1007/s00454-018-0004-2

Positive geometries and canonical forms
journal, November 2017

Arkani-Hamed, Nima; Bai, Yuntao; Lam, Thomas
Journal of High Energy Physics, Vol. 2017, Issue 11
DOI: 10.1007/JHEP11(2017)039

On positive geometries of quartic interactions: Stokes polytopes, lower forms on associahedra and world-sheet forms
journal, April 2020

Aneesh, P. B.; Banerjee, Pinaki; Jagadale, Mrunmay
Journal of High Energy Physics, Vol. 2020, Issue 4
DOI: 10.1007/JHEP04(2020)149

Direct Proof of the Tree-Level Scattering Amplitude Recursion Relation in Yang-Mills Theory
journal, May 2005

Britto, Ruth; Cachazo, Freddy; Feng, Bo
Physical Review Letters, Vol. 94, Issue 18
DOI: 10.1103/PhysRevLett.94.181602

Grassmannian Geometry of Scattering Amplitudes
book, May 2016

Arkani-Hamed, Nima; Bourjaily, Jacob; Cachazo, Freddy
DOI: 10.1017/CBO9781316091548

Similar Records in DOE PAGES and OSTI.GOV collections:

Causal diamonds, cluster polytopes and scattering amplitudes

Journal Article Arkani-Hamed, N. ; He, S. ; Salvatori, G. ; ... - Journal of High Energy Physics (Online)

The “amplituhedron” for tree-level scattering amplitudes in the bi-adjoint φ³ theory is given by the ABHY associahedron in kinematic space, which has been generalized to give a realization for all finite-type cluster algebra polytopes, labelled by Dynkin diagrams. In this letter we identify a simple physical origin for these polytopes, associated with an interesting (1 + 1)-dimensional causal structure in kinematic space, along with solutions to the wave equation in this kinematic “spacetime” with a natural positivity property. The notion of time evolution in this kinematic spacetime can be abstracted away to a certain “walk”, associated with any acyclic quiver,more »« less
https://doi.org/10.1007/jhep11(2022)049

Full Text Available
Scattering forms and the positive geometry of kinematics, color and the worldsheet

Journal Article Arkani-Hamed, Nima ; Bai, Yuntao ; He, Song ; ... - Journal of High Energy Physics (Online)

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. Wemore »« less
Cited by 133
https://doi.org/10.1007/jhep05(2018)096

Full Text Available
Stringy canonical forms

Journal Article Arkani-Hamed, Nima ; He, Song ; Lam, Thomas - Journal of High Energy Physics (Online)

Canonical forms of positive geometries play an important role in revealing hidden structures of scattering amplitudes, from amplituhedra to associahedra. In this paper, we introduce “stringy canonical forms”, which provide a natural definition and extension of canonical forms for general polytopes, deformed by a parameter α'. They are defined by real or complex integrals regulated with polynomials with exponents, and are meromorphic functions of the exponents, sharing various properties of string amplitudes. As α'→ 0, they reduce to the usual canonical form of a polytope given by the Minkowski sum of the Newton polytopes of the regulating polynomials, or equivalentlymore »« less
https://doi.org/10.1007/jhep02(2021)069

Full Text Available
On positive geometry and scattering forms for matter particles

Journal Article Herderschee, Aidan ; He, Song ; Teng, Fei ; ... - Journal of High Energy Physics (Online)

We initiate the study of positive geometry and scattering forms for tree-level amplitudes with matter particles in the (anti-)fundamental representation of the color/flavor group. As a toy example, we study the bi-color scalar theory, which supplements the bi-adjoint theory with scalars in the (anti-)fundamental representations of both groups. Using a recursive construction we obtain a class of unbounded polytopes called open associahedra (or associahedra with certain facets at infinity) whose canonical form computes amplitudes in bi-color theory, for arbitrary number of legs and flavor assignments. In addition, we discuss the duality between color factors and wedge products, or "color ismore »« less
https://doi.org/10.1007/jhep06(2020)030

Full Text Available
Symbol alphabets from tensor diagrams

Journal Article Ren, Lecheng ; Spradlin, Marcus ; Volovich, Anastasia - Journal of High Energy Physics (Online)

We propose to use tensor diagrams and the Fomin-Pylyavskyy conjectures to explore the connection between symbol alphabets of n-particle amplitudes in planarmore »« less
https://doi.org/10.1007/jhep12(2021)079

Full Text Available

Similar Records

Title: Scattering amplitudes and simple canonical forms for simple polytopes

Abstract

Citation Formats

1-loop amplitudes from the Halohedron journal, December 2019

An etude on recursion relations and triangulations journal, May 2019

Scattering forms and the positive geometry of kinematics, color and the worldsheet journal, May 2018

The positive geometry for 𝜙p interactions journal, October 2019

The Amplituhedron journal, October 2014

Geometric Realizations of the Accordion Complex of a Dissection journal, May 2018

Positive geometries and canonical forms journal, November 2017

On positive geometries of quartic interactions: Stokes polytopes, lower forms on associahedra and world-sheet forms journal, April 2020

Direct Proof of the Tree-Level Scattering Amplitude Recursion Relation in Yang-Mills Theory journal, May 2005

Grassmannian Geometry of Scattering Amplitudes book, May 2016

1-loop amplitudes from the Halohedron
journal, December 2019

An etude on recursion relations and triangulations
journal, May 2019

Scattering forms and the positive geometry of kinematics, color and the worldsheet
journal, May 2018

The positive geometry for 𝜙p interactions
journal, October 2019

The Amplituhedron
journal, October 2014

Geometric Realizations of the Accordion Complex of a Dissection
journal, May 2018

Positive geometries and canonical forms
journal, November 2017

On positive geometries of quartic interactions: Stokes polytopes, lower forms on associahedra and world-sheet forms
journal, April 2020

Direct Proof of the Tree-Level Scattering Amplitude Recursion Relation in Yang-Mills Theory
journal, May 2005

Grassmannian Geometry of Scattering Amplitudes
book, May 2016