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Published: 9 March 2023
Last updated: 10 March 2023

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1

Bourjaily, Jacob, and Hugh Thomas. "WHAT IS...the amplituhedron?" Notices of the American Mathematical Society 65, no. 02 (February 1, 2018): 167–69. http://dx.doi.org/10.1090/noti1630.

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2

Galashin, Pavel, and Thomas Lam. "Parity duality for the amplituhedron." Compositio Mathematica 156, no. 11 (November 2020): 2207–62. http://dx.doi.org/10.1112/s0010437x20007411.

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The (tree) amplituhedron $\mathcal {A}_{n,k,m}(Z)$ is a certain subset of the Grassmannian introduced by Arkani-Hamed and Trnka in 2013 in order to study scattering amplitudes in $N=4$ supersymmetric Yang–Mills theory. Confirming a conjecture of the first author, we show that when $m$ is even, a collection of affine permutations yields a triangulation of $\mathcal {A}_{n,k,m}(Z)$ for any $Z\in \operatorname {Gr}_{>0}(k+m,n)$ if and only if the collection of their inverses yields a triangulation of $\mathcal {A}_{n,n-m-k,m}(Z)$ for any $Z\in \operatorname {Gr}_{>0}(n-k,n)$. We prove this duality using the twist map of Marsh and Scott. We also show that this map preserves the canonical differential forms associated with the corresponding positroid cells, and hence obtain a parity duality for amplituhedron differential forms.
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3

Ferro, Livia, Tomasz Łukowski, and Matteo Parisi. "Amplituhedron meets Jeffrey–Kirwan residue." Journal of Physics A: Mathematical and Theoretical 52, no. 4 (December 28, 2018): 045201. http://dx.doi.org/10.1088/1751-8121/aaf3c3.

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4

Lam, Thomas. "Amplituhedron Cells and Stanley Symmetric Functions." Communications in Mathematical Physics 343, no. 3 (March 19, 2016): 1025–37. http://dx.doi.org/10.1007/s00220-016-2602-2.

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5

Ferro, Livia, Tomasz Łukowski, Andrea Orta, and Matteo Parisi. "Yangian symmetry for the tree amplituhedron." Journal of Physics A: Mathematical and Theoretical 50, no. 29 (June 29, 2017): 294005. http://dx.doi.org/10.1088/1751-8121/aa7594.

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6

Łukowski, Tomasz, and Robert Moerman. "Boundaries of the amplituhedron with amplituhedronBoundaries." Computer Physics Communications 259 (February 2021): 107653. http://dx.doi.org/10.1016/j.cpc.2020.107653.

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7

Ferro, L., T. Lukowski, A. Orta, and M. Parisi. "Tree-level scattering amplitudes from the amplituhedron." Journal of Physics: Conference Series 841 (May 2017): 012037. http://dx.doi.org/10.1088/1742-6596/841/1/012037.

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8

Karp, Steven N., and Lauren K. Williams. "The $m=1$ Amplituhedron and Cyclic Hyperplane Arrangements." International Mathematics Research Notices 2019, no. 5 (July 24, 2017): 1401–62. http://dx.doi.org/10.1093/imrn/rnx140.

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9

Łukowski, Tomasz. "On the boundaries of the $m=2$ amplituhedron." Annales de l’Institut Henri Poincaré D 9, no. 3 (December 22, 2022): 525–41. http://dx.doi.org/10.4171/aihpd/124.

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10

Hodges, Andrew. "Twistors and amplitudes." Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 373, no. 2047 (August 6, 2015): 20140248. http://dx.doi.org/10.1098/rsta.2014.0248.

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A brief review is given of why twistor geometry has taken a central place in the theory of scattering amplitudes for fundamental particles. The emphasis is on the twistor diagram formalism as originally proposed by Penrose, the development of which has now led to the definition by Arkani-Hamed et al. of the ‘amplituhedron’.
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11

Rao, Junjie. "All-loop Mondrian reduction of 4-particle amplituhedron at positive infinity." Nuclear Physics B 957 (August 2020): 115086. http://dx.doi.org/10.1016/j.nuclphysb.2020.115086.

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12

Ferro, Livia, and Tomasz Łukowski. "Amplituhedra, and beyond." Journal of Physics A: Mathematical and Theoretical 54, no. 3 (December 30, 2020): 033001. http://dx.doi.org/10.1088/1751-8121/abd21d.

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13

Karp, Steven, Lauren Williams, and Yan Zhang. "Decompositions of amplituhedra." Annales de l’Institut Henri Poincaré D 7, no. 3 (August 22, 2020): 303–63. http://dx.doi.org/10.4171/aihpd/87.

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14

He, Song, Chia-Kai Kuo, and Yao-Qi Zhang. "The momentum amplituhedron of SYM and ABJM from twistor-string maps." Journal of High Energy Physics 2022, no. 2 (February 2022). http://dx.doi.org/10.1007/jhep02(2022)148.

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Abstract We study remarkable connections between twistor-string formulas for tree amplitudes in $$ \mathcal{N} $$ N = 4 SYM and $$ \mathcal{N} $$ N = 6 ABJM, and the corresponding momentum amplituhedron in the kinematic space of D = 4 and D = 3, respectively. Based on the Veronese map to positive Grassmannians, we define a twistor-string map from G+(2, n) to a (2n−4)-dimensional subspace of the 4d kinematic space where the momentum amplituhedron of SYM lives. We provide strong evidence that the twistor-string map is a diffeomorphism from G+(2, n) to the interior of momentum amplituhedron; the canonical form of the latter, which is known to give tree amplitudes of SYM, can be obtained as pushforward of that of former. We then move to three dimensions: based on Veronese map to orthogonal positive Grassmannian, we propose a similar twistor-string map from the moduli space $$ {\mathrm{\mathcal{M}}}_{0,n}^{+} $$ ℳ 0 , n + to a (n−3)-dimensional subspace of 3d kinematic space. The image gives a new positive geometry which conjecturally serves as the momentum amplituhedron for ABJM; its canonical form gives the tree amplitude with reduced supersymmetries in the theory. We also show how boundaries of compactified $$ {\mathrm{\mathcal{M}}}_{0,n}^{+} $$ ℳ 0 , n + map to boundaries of momentum amplituhedra for SYM and ABJM corresponding to factorization channels of amplitudes, and in particular for ABJM case the map beautifully excludes all unwanted channels.
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15

Arkani-Hamed, Nima, and Jaroslav Trnka. "The Amplituhedron." Journal of High Energy Physics 2014, no. 10 (October 2014). http://dx.doi.org/10.1007/jhep10(2014)030.

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16

Dian, Gabriele, and Paul Heslop. "Amplituhedron-like geometries." Journal of High Energy Physics 2021, no. 11 (November 2021). http://dx.doi.org/10.1007/jhep11(2021)074.

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Abstract We consider amplituhedron-like geometries which are defined in a similar way to the intrinsic definition of the amplituhedron but with non-maximal winding number. We propose that for the cases with minimal number of points the canonical form of these geometries corresponds to the product of parity conjugate amplitudes at tree as well as loop level. The product of amplitudes in superspace lifts to a star product in bosonised superspace which we give a precise definition of. We give an alternative definition of amplituhedron-like geometries, analogous to the original amplituhedron definition, and also a characterisation as a sum over pairs of on-shell diagrams that we use to prove the conjecture at tree level. The union of all amplituhedron-like geometries has a very simple definition given by only physical inequalities. Although such a union does not give a positive geometry, a natural extension of the standard definition of canonical form, the globally oriented canonical form, acts on this union and gives the square of the amplitude.
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17

Herrmann, Enrico, Cameron Langer, Jaroslav Trnka, and Minshan Zheng. "Positive geometry, local triangulations, and the dual of the Amplituhedron." Journal of High Energy Physics 2021, no. 1 (January 2021). http://dx.doi.org/10.1007/jhep01(2021)035.

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Abstract We initiate the systematic study of local positive spaces which arise in the context of the Amplituhedron construction for scattering amplitudes in planar maximally supersymmetric Yang-Mills theory. We show that all local positive spaces relevant for one-loop MHV amplitudes are characterized by certain sign-flip conditions and are associated with surprisingly simple logarithmic forms. In the maximal sign-flip case they are finite one-loop octagons. Particular combinations of sign-flip spaces can be glued into new local positive geometries. These correspond to local pentagon integrands that appear in the local expansion of the MHV one-loop amplitude. We show that, geometrically, these pentagons do not triangulate the original Amplituhedron space but rather its twin “Amplituhedron-Prime”. This new geometry has the same boundary structure as the Amplituhedron (and therefore the same logarithmic form) but differs in the bulk as a geometric space. On certain two-dimensional boundaries, where the Amplituhedron geometry reduces to a polygon, we check that both spaces map to the same dual polygon. Interestingly, we find that the pentagons internally triangulate that dual space. This gives a direct evidence that the chiral pentagons are natural building blocks for a yet-to-be discovered dual Amplituhedron.
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18

Herrmann, Enrico, Cameron Langer, Jaroslav Trnka, and Minshan Zheng. "Positive geometry, local triangulations, and the dual of the Amplituhedron." Journal of High Energy Physics 2021, no. 1 (January 2021). http://dx.doi.org/10.1007/jhep01(2021)035.

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Abstract We initiate the systematic study of local positive spaces which arise in the context of the Amplituhedron construction for scattering amplitudes in planar maximally supersymmetric Yang-Mills theory. We show that all local positive spaces relevant for one-loop MHV amplitudes are characterized by certain sign-flip conditions and are associated with surprisingly simple logarithmic forms. In the maximal sign-flip case they are finite one-loop octagons. Particular combinations of sign-flip spaces can be glued into new local positive geometries. These correspond to local pentagon integrands that appear in the local expansion of the MHV one-loop amplitude. We show that, geometrically, these pentagons do not triangulate the original Amplituhedron space but rather its twin “Amplituhedron-Prime”. This new geometry has the same boundary structure as the Amplituhedron (and therefore the same logarithmic form) but differs in the bulk as a geometric space. On certain two-dimensional boundaries, where the Amplituhedron geometry reduces to a polygon, we check that both spaces map to the same dual polygon. Interestingly, we find that the pentagons internally triangulate that dual space. This gives a direct evidence that the chiral pentagons are natural building blocks for a yet-to-be discovered dual Amplituhedron.
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19

Damgaard, David, Livia Ferro, Tomasz Lukowski, and Matteo Parisi. "The momentum amplituhedron." Journal of High Energy Physics 2019, no. 8 (August 2019). http://dx.doi.org/10.1007/jhep08(2019)042.

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20

Arkani-Hamed, Nima, and Jaroslav Trnka. "Into the amplituhedron." Journal of High Energy Physics 2014, no. 12 (December 2014). http://dx.doi.org/10.1007/jhep12(2014)182.

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21

Franco, Sebastián, Daniele Galloni, Alberto Mariotti, and Jaroslav Trnka. "Anatomy of the amplituhedron." Journal of High Energy Physics 2015, no. 3 (March 2015). http://dx.doi.org/10.1007/jhep03(2015)128.

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22

Ferro, Livia, Tomasz Lukowski, Andrea Orta, and Matteo Parisi. "Towards the amplituhedron volume." Journal of High Energy Physics 2016, no. 3 (March 2016). http://dx.doi.org/10.1007/jhep03(2016)014.

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23

Kojima, Ryota, and Junjie Rao. "Triangulation-free trivialization of 2-loop MHV amplituhedron." Journal of High Energy Physics 2020, no. 10 (October 2020). http://dx.doi.org/10.1007/jhep10(2020)140.

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Abstract This article introduces a new approach to implement positivity for the 2-loop n-particle MHV amplituhedron, circumventing the conventional triangulation with respect to positive variables of each cell carved out by the sign flips. This approach is universal for all linear positive conditions and hence free of case-by-case triangulation, as an application of the trick of positive infinity first introduced in [6] for the multi-loop 4-particle amplituhedron. Moreover, the proof of 2-loop n-particle MHV amplituhedron in [4] is revised, and we explain the nontriviality and difficulty of using conventional triangulation while the results have a simple universal pattern. A further example is presented to tentatively explore its generalization towards handling multiple positive conditions at 3-loop and higher.
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24

Łukowski, Tomasz, Robert Moerman, and Jonah Stalknecht. "On the geometry of the orthogonal momentum amplituhedron." Journal of High Energy Physics 2022, no. 12 (December 1, 2022). http://dx.doi.org/10.1007/jhep12(2022)006.

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Abstract In this paper we focus on the orthogonal momentum amplituhedron $$ \mathcal{O} $$ O k, a recently introduced positive geometry that encodes the tree-level scattering amplitudes in ABJM theory. We generate the full boundary stratification of $$ \mathcal{O} $$ O k for various k and conjecture that its boundaries can be labelled by so-called orthogonal Grassmannian forests (OG forests). We determine the generating function for enumerating these forests according to their dimension and show that the Euler characteristic of the poset of these forests equals one. This provides a strong indication that the orthogonal momentum amplituhedron is homeomorphic to a ball. This paper is supplemented with the Mathematica package orthitroids which contains useful functions for exploring the structure of the positive orthogonal Grassmannian and the orthogonal momentum amplituhedron.
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25

Damgaard, David, Livia Ferro, Tomasz Łukowski, and Robert Moerman. "Momentum amplituhedron meets kinematic associahedron." Journal of High Energy Physics 2021, no. 2 (February 2021). http://dx.doi.org/10.1007/jhep02(2021)041.

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Abstract In this paper we study a relation between two positive geometries: the momen- tum amplituhedron, relevant for tree-level scattering amplitudes in $$ \mathcal{N} $$ N = 4 super Yang-Mills theory, and the kinematic associahedron, encoding tree-level amplitudes in bi-adjoint scalar φ3 theory. We study the implications of restricting the latter to four spacetime dimensions and give a direct link between its canonical form and the canonical form for the momentum amplituhedron. After removing the little group scaling dependence of the gauge theory, we find that we can compare the resulting reduced forms with the pull-back of the associahedron form. In particular, the associahedron form is the sum over all helicity sectors of the reduced momentum amplituhedron forms. This relation highlights the common sin- gularity structure of the respective amplitudes; in particular, the factorization channels, corresponding to vanishing planar Mandelstam variables, are the same. Additionally, we also find a relation between these canonical forms directly on the kinematic space of the scalar theory when reduced to four spacetime dimensions by Gram determinant constraints. As a by-product of our work we provide a detailed analysis of the kinematic spaces relevant for the four-dimensional gauge and scalar theories, and provide direct links between them.
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26

Arkani-Hamed, Nima, Andrew Hodges, and Jaroslav Trnka. "Positive amplitudes in the amplituhedron." Journal of High Energy Physics 2015, no. 8 (August 2015). http://dx.doi.org/10.1007/jhep08(2015)030.

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27

Arkani-Hamed, Nima, Hugh Thomas, and Jaroslav Trnka. "Unwinding the amplituhedron in binary." Journal of High Energy Physics 2018, no. 1 (January 2018). http://dx.doi.org/10.1007/jhep01(2018)016.

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28

Srikant, Akshay Yelleshpur. "Emergent unitarity from the amplituhedron." Journal of High Energy Physics 2020, no. 1 (January 2020). http://dx.doi.org/10.1007/jhep01(2020)069.

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29

Bern, Zvi, Enrico Herrmann, Sean Litsey, James Stankowicz, and Jaroslav Trnka. "Evidence for a nonplanar amplituhedron." Journal of High Energy Physics 2016, no. 6 (June 2016). http://dx.doi.org/10.1007/jhep06(2016)098.

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30

Dennen, T., I. Prlina, M. Spradlin, S. Stanojevic, and A. Volovich. "Landau singularities from the amplituhedron." Journal of High Energy Physics 2017, no. 6 (June 2017). http://dx.doi.org/10.1007/jhep06(2017)152.

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31

Lukowski, Tomasz, and Jonah Stalknecht. "The hypersimplex canonical forms and the momentum amplituhedron-like logarithmic forms." Journal of Physics A: Mathematical and Theoretical, March 30, 2022. http://dx.doi.org/10.1088/1751-8121/ac62ba.

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Abstract In this paper we provide a formula for the canonical differential form of the hypersimplex Δ_{k,n} for all n and k. We also study the generalization of the momentum amplituhedron M_{n,k} to m=2, which has been conjectured to share many properties with the hypersimplex, and we provide counterexamples for these conjectures. Nevertheless, we find interesting momentum amplituhedron-like logarithmic differential forms in the m=2 version of the spinor helicity space, that have the same singularity structure as the hypersimplex canonical forms.
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32

Damgaard, David, Livia Ferro, Tomasz Łukowski, and Robert Moerman. "Kleiss-Kuijf relations from momentum amplituhedron geometry." Journal of High Energy Physics 2021, no. 7 (July 2021). http://dx.doi.org/10.1007/jhep07(2021)111.

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Abstract In recent years, it has been understood that color-ordered scattering amplitudes can be encoded as logarithmic differential forms on positive geometries. In particular, amplitudes in maximally supersymmetric Yang-Mills theory in spinor helicity space are governed by the momentum amplituhedron. Due to the group-theoretic structure underlying color decompositions, color-ordered amplitudes enjoy various identities which relate different orderings. In this paper, we show how the Kleiss-Kuijf relations arise from the geometry of the momentum amplituhedron. We also show how similar relations can be realised for the kinematic associahedron, which is the positive geometry of bi-adjoint scalar cubic theory.
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33

Bai, Yuntao, and Song He. "The amplituhedron from momentum twistor diagrams." Journal of High Energy Physics 2015, no. 2 (February 2015). http://dx.doi.org/10.1007/jhep02(2015)065.

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34

Langer, Cameron, and Akshay Yelleshpur Srikant. "All-loop cuts from the Amplituhedron." Journal of High Energy Physics 2019, no. 4 (April 2019). http://dx.doi.org/10.1007/jhep04(2019)105.

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35

Kojima, Ryota, and Cameron Langer. "Sign flip triangulations of the amplituhedron." Journal of High Energy Physics 2020, no. 5 (May 2020). http://dx.doi.org/10.1007/jhep05(2020)121.

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36

Chicherin, Dmitry, and Johannes Henn. "Pentagon Wilson loop with Lagrangian insertion at two loops in $$ \mathcal{N} $$ = 4 super Yang-Mills theory." Journal of High Energy Physics 2022, no. 7 (July 2022). http://dx.doi.org/10.1007/jhep07(2022)038.

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Abstract We compute the two-loop result for the null pentagonal Wilson loop with a Lagrangian insertion (normalized by the Wilson loop without insertion) in planar, maximally supersymmetric Yang-Mills theory. This finite observable is closely related to the Amplituhedron, and it is reminiscent of finite parts of planar two-loop five-particle scattering amplitudes. We verify that, up to this loop order, the leading singularities are given by the same conformally invariant expressions that appear in all-plus pure Yang-Mills amplitudes. The accompanying weight-four transcendental functions are expressed in terms of the pentagon functions space known from planar two-loop five-particle amplitudes, but interestingly only a subset of the functions appears. Being a function of four dimensionless variables, the observable has interesting asymptotic limits. We verify that our analytic result is consistent with soft and collinear limits, and find an intriguingly simple pattern in the multi-Regge limit. Thanks to the new result we can also conjecturally predict, for general kinematics, the maximal weight piece of the planar three-loop five-particle all-plus amplitude in pure Yang-Mills theory. Motivated by the Amplituhedron geometry, we investigate positivity properties of the integrated answer. Generalizing previous results at four particles, we find numerical evidence that the two-loop five-particle result has uniform sign in a kinematic region suggested by the loop Amplituhedron.
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37

Huang, Yu-tin, Ryota Kojima, Congkao Wen, and Shun-Qing Zhang. "The orthogonal momentum amplituhedron and ABJM amplitudes." Journal of High Energy Physics 2022, no. 1 (January 2022). http://dx.doi.org/10.1007/jhep01(2022)141.

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Abstract In this paper, we introduce the momentum space amplituhedron for tree-level scattering amplitudes of ABJM theory. We demonstrate that the scattering amplitude can be identified as the canonical form on the space given by the product of positive orthogonal Grassmannian and the moment curve. The co-dimension one boundaries of this space are simply the odd-particle planar Mandelstam variables, while the even-particle counterparts are “hidden” as higher co-dimension boundaries. Remarkably, this space can be equally defined through a series of “sign flip” requirements of the projected external data, identical to “half” of four-dimensional $$ \mathcal{N} $$ N = 4 super Yang-Mills theory (sYM). Thus in a precise sense the geometry for ABJM lives on the boundary of $$ \mathcal{N} $$ N = 4 sYM. We verify this relation through eight-points by showing that the BCFW triangulation of the amplitude tiles the amplituhedron. The canonical form is naturally derived using the Grassmannian formula for the amplitude in the $$ \mathcal{N} $$ N = 4 formalism for ABJM theory.
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38

Heslop, Paul, and Alastair Stewart. "The twistor Wilson loop and the amplituhedron." Journal of High Energy Physics 2018, no. 10 (October 2018). http://dx.doi.org/10.1007/jhep10(2018)142.

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39

Herrmann, Enrico, and Jaroslav Trnka. "The SAGEX Review on Scattering Amplitudes, Chapter 7: Positive Geometry of Scattering Amplitudes." Journal of Physics A: Mathematical and Theoretical, August 4, 2022. http://dx.doi.org/10.1088/1751-8121/ac8709.

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Abstract Scattering amplitudes are both a wonderful playground to discover novel ideas in Quantum Field Theory and simultaneously of immense phenomenological importance to make precision predictions for e.g.~particle collider observables and more recently also for gravitational wave signals. In this review chapter, we give an overview of some of the exciting recent progress on reformulating QFT in terms of mathematical, geometric quantities, such as polytopes, associahedra, Grassmanians, and the amplituhedron. In this novel approach, standard notions of locality and unitarity are derived concepts rather than fundamental ingredients in the construction which might give us a handle on a number of open questions in QFT that have evaded an answer for decades. We first give a basic summary of positive geometry, before discussing the associahedron---one of the simplest physically relevant geometric examples---and its relation to tree-level scattering amplitudes in bi-adjoint $\phi^3$ theory. Our second example is the amplituhedron construction for scattering amplitudes in planar maximally supersymmetric Yang-Mills theory.
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40

Arkani-Hamed, N., S. He, G. Salvatori, and H. Thomas. "Causal diamonds, cluster polytopes and scattering amplitudes." Journal of High Energy Physics 2022, no. 11 (November 10, 2022). http://dx.doi.org/10.1007/jhep11(2022)049.

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Abstract 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, remarkably yielding a finite cluster polytope for the case of Dynkin quivers. The $$ \mathcal{A} $$ A n−3,$$ \mathcal{B} $$ B n−1/$$ \mathcal{C} $$ C n−1 and $$ \mathcal{D} $$ D n polytopes are the amplituhedra for n-point tree amplitudes, one-loop tadpole diagrams, and full integrand of one-loop amplitudes. We also introduce a polytope $$ \overline{\mathcal{D}} $$ D ¯ n, which chops the $$ \mathcal{D} $$ D n polytope in half along a symmetry plane, capturing one-loop amplitudes in a more efficient way.
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41

Bai, Yuntao, Song He, and Thomas Lam. "The amplituhedron and the one-loop Grassmannian measure." Journal of High Energy Physics 2016, no. 1 (January 2016). http://dx.doi.org/10.1007/jhep01(2016)112.

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42

An, Yang, Yi Li, Zhinan Li, and Junjie Rao. "All-loop Mondrian diagrammatics and 4-particle amplituhedron." Journal of High Energy Physics 2018, no. 6 (June 2018). http://dx.doi.org/10.1007/jhep06(2018)023.

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43

Ferro, Livia, Tomasz Łukowski, and Robert Moerman. "From momentum amplituhedron boundaries to amplitude singularities and back." Journal of High Energy Physics 2020, no. 7 (July 2020). http://dx.doi.org/10.1007/jhep07(2020)201.

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44

Kojima, Ryota. "Triangulation of 2-loop MHV amplituhedron from sign flips." Journal of High Energy Physics 2019, no. 4 (April 2019). http://dx.doi.org/10.1007/jhep04(2019)085.

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45

Karp, Steven N. "Defining amplituhedra and Grassmann polytopes." Discrete Mathematics & Theoretical Computer Science DMTCS Proceedings, 28th... (April 22, 2020). http://dx.doi.org/10.46298/dmtcs.6356.

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International audience The totally nonnegative Grassmannian Gr≥0 k,n is the set of k-dimensional subspaces V of Rn whose nonzero Plucker coordinates all have the same sign. In their study of scattering amplitudes in N = 4 supersym- metric Yang-Mills theory, Arkani-Hamed and Trnka (2013) considered the image (called an amplituhedron) of Gr≥0 k,n under a linear map Z : Rn → Rr, where k ≤ r and the r × r minors of Z are all positive. One reason they required this positivity condition is to ensure that the map Gr≥0 k,n → Grk,r induced by Z is well defined, i.e. it takes everynelement of Gr≥0 k,n to a k-dimensional subspace of Rr. Lam (2015) gave a sufficient condition for the induced map Gr≥0 k,n → Grk,r to be well defined, in which case he called the image a Grassmann polytope. (In the case k = 1, Grassmann polytopes are just polytopes, and amplituhedra are cyclic polytopes.) We give a necessary and sufficient condition for the induced map Gr≥0 k,n → Grk,r to be well defined, in terms of sign variation. Using previous work we presented at FPSAC 2015, we obtain an equivalent condition in terms of the r × r minors of Z (assuming Z has rank r).
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46

Arkani-Hamed, Nima, Johannes Henn, and Jaroslav Trnka. "Nonperturbative negative geometries: amplitudes at strong coupling and the amplituhedron." Journal of High Energy Physics 2022, no. 3 (March 2022). http://dx.doi.org/10.1007/jhep03(2022)108.

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Abstract The amplituhedron determines scattering amplitudes in planar $$ \mathcal{N} $$ N = 4 super Yang-Mills by a single “positive geometry” in the space of kinematic and loop variables. We study a closely related definition of the amplituhedron for the simplest case of four-particle scattering, given as a sum over complementary “negative geometries”, which provides a natural geometric understanding of the exponentiation of infrared (IR) divergences, as well as a new geometric definition of an IR finite observable $$ \mathcal{F} $$ F (g, z) — dually interpreted as the expectation value of the null polygonal Wilson loop with a single Lagrangian insertion — which is directly determined by these negative geometries. This provides a long-sought direct link between canonical forms for positive (negative) geometries, and a completely IR finite post-loop-integration observable depending on a single kinematical variable z, from which the cusp anomalous dimension Γcusp(g) can also be straightforwardly obtained. We study an especially simple class of negative geometries at all loop orders, associated with a “tree” structure in the negativity conditions, for which the contributions to $$ \mathcal{F} $$ F (g, z) and Γcusp can easily be determined by an interesting non-linear differential equation immediately following from the combinatorics of negative geometries. This lets us compute these “tree” contributions to $$ \mathcal{F} $$ F (g, z) and Γcusp for all values of the ‘t Hooft coupling. The result for Γcusp remarkably shares all main qualitative characteristics of the known exact results obtained using integrability.
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47

Arkani-Hamed, Nima, Cameron Langer, Akshay Yelleshpur Srikant, and Jaroslav Trnka. "Deep Into the Amplituhedron: Amplitude Singularities at All Loops and Legs." Physical Review Letters 122, no. 5 (February 7, 2019). http://dx.doi.org/10.1103/physrevlett.122.051601.

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48

Rao, Junjie. "4-particle amplituhedron at 3-loop and its Mondrian diagrammatic implication." Journal of High Energy Physics 2018, no. 6 (June 2018). http://dx.doi.org/10.1007/jhep06(2018)038.

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49

Vaid, Deepak, and Devadharsini Suresh. "Coherent states and particle scattering in loop quantum gravity." European Physical Journal C 82, no. 8 (August 19, 2022). http://dx.doi.org/10.1140/epjc/s10052-022-10701-6.

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AbstractQuantum field theory provides us with the means to calculate scattering amplitudes. In recent years a dramatic new development has lead to great simplification of such calculations. This is based on the discovery of the “amplituhedron” in the context of scattering of massless gauge bosons in Yang–Mills theory. One of the main challenges facing Loop Quantum Gravity is the lack of a clear description of particle scattering processes and a connection to flat space QFT. Here we show a correspondence between the space of kinematic data of the scattering N massless particles and U(N) coherent states in LQG. This correspondence allows us to provide the outlines of a theory of quantum gravity based upon the dynamics of excitations living on the the positive Grassmannian.
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50

He, Song, Chia-Kai Kuo, Zhenjie Li, and Yao-Qi Zhang. "All-Loop Four-Point Aharony-Bergman-Jafferis-Maldacena Amplitudes from Dimensional Reduction of the Amplituhedron." Physical Review Letters 129, no. 22 (November 23, 2022). http://dx.doi.org/10.1103/physrevlett.129.221604.

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