Abstract
We propose to use tensor diagrams and the Fomin-Pylyavskyy conjectures to explore the connection between symbol alphabets of n-particle amplitudes in planar \( \mathcal{N} \) = 4 Yang-Mills theory and certain polytopes associated to the Grassmannian Gr(4, n). We show how to assign a web (a planar tensor diagram) to each facet of these polytopes. Webs with no inner loops are associated to cluster variables (rational symbol letters). For webs with a single inner loop we propose and explicitly evaluate an associated web series that contains information about algebraic symbol letters. In this manner we reproduce the results of previous analyses of n ≤ 8, and find that the polytope \( {\mathcal{C}}^{\dagger}\left(4,9\right) \) encodes all rational letters, and all square roots of the algebraic letters, of known nine-particle amplitudes.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
J. Golden, A. B. Goncharov, M. Spradlin, C. Vergu and A. Volovich, Motivic amplitudes and cluster coordinates, JHEP 01 (2014) 091 [arXiv:1305.1617] [INSPIRE].
A. B. Goncharov, M. Spradlin, C. Vergu and A. Volovich, Classical polylogarithms for amplitudes and Wilson loops, Phys. Rev. Lett. 105 (2010) 151605 [arXiv:1006.5703] [INSPIRE].
S. Fomin and A. Zelevinsky, Cluster algebras I. Foundations, J. Amer. Math. Soc. 15 (2002) 497 [math/0104151].
S. Caron-Huot et al., The Steinmann cluster bootstrap for N = 4 super Yang-Mills amplitudes, PoS(CORFU2019)003 [arXiv:2005.06735] [INSPIRE].
D. Chicherin, J. M. Henn and G. Papathanasiou, Cluster algebras for Feynman integrals, Phys. Rev. Lett. 126 (2021) 091603 [arXiv:2012.12285] [INSPIRE].
S. He, Z. Li and Q. Yang, Notes on cluster algebras and some all-loop Feynman integrals, JHEP 06 (2021) 119 [arXiv:2103.02796] [INSPIRE].
N. Arkani-Hamed, T. Lam and M. Spradlin, Non-perturbative geometries for planar \( \mathcal{N} \) = 4 SYM amplitudes, JHEP 03 (2021) 065 [arXiv:1912.08222] [INSPIRE].
J. Drummond, J. Foster, O. Gürdogan and C. Kalousios, Algebraic singularities of scattering amplitudes from tropical geometry, JHEP 04 (2021) 002 [arXiv:1912.08217] [INSPIRE].
N. Henke and G. Papathanasiou, How tropical are seven- and eight-particle amplitudes?, JHEP 08 (2020) 005 [arXiv:1912.08254] [INSPIRE].
A. Herderschee, Algebraic branch points at all loop orders from positive kinematics and wall crossing, JHEP 07 (2021) 049 [arXiv:2102.03611] [INSPIRE].
D. Speyer and L. Williams, The tropical totally positive Grassmannian, J. Alg. Combin. 22 (2005) 189 [math/0312297].
N. Arkani-Hamed, S. He and T. Lam, Stringy canonical forms, JHEP 02 (2021) 069 [arXiv:1912.08707] [INSPIRE].
F. Cachazo, N. Early, A. Guevara and S. Mizera, Scattering equations: from projective spaces to tropical grassmannians, JHEP 06 (2019) 039 [arXiv:1903.08904] [INSPIRE].
F. Cachazo and J. M. Rojas, Notes on biadjoint amplitudes, Trop G(3, 7) and X(3, 7) scattering equations, JHEP 04 (2020) 176 [arXiv:1906.05979] [INSPIRE].
J. Drummond, J. Foster, O. Gürdogan and C. Kalousios, Tropical Grassmannians, cluster algebras and scattering amplitudes, JHEP 04 (2020) 146 [arXiv:1907.01053] [INSPIRE].
F. Cachazo and N. Early, Minimal kinematics: an all k and n peek into Trop+ G(k, n), SIGMA 17 (2021) 078 [arXiv:2003.07958] [INSPIRE].
S. Fomin and A. Zelevinsky, Y systems and generalized associahedra, Ann. Math. 158 (2003) 977 [hep-th/0111053] [INSPIRE].
J. Drummond, J. Foster, O. Gürdoğan and C. Kalousios, Tropical fans, scattering equations and amplitudes, JHEP 11 (2021) 071 [arXiv:2002.04624] [INSPIRE].
I. Prlina, M. Spradlin and S. Stanojevic, All-loop singularities of scattering amplitudes in massless planar theories, Phys. Rev. Lett. 121 (2018) 081601 [arXiv:1805.11617] [INSPIRE].
S. Caron-Huot and K. J. Larsen, Uniqueness of two-loop master contours, JHEP 10 (2012) 026 [arXiv:1205.0801] [INSPIRE].
J. L. Bourjaily, A. J. McLeod, M. Spradlin, M. von Hippel and M. Wilhelm, Elliptic double-box integrals: massless scattering amplitudes beyond polylogarithms, Phys. Rev. Lett. 120 (2018) 121603 [arXiv:1712.02785] [INSPIRE].
W. Chang, B. Duan, C. Fraser and J.-R. Li, Quantum affine algebras and Grassmannians, Math. Zeit. 296 (2020) 1539 [arXiv:1907.13575].
S. Fomin and P. Pylyavskyy, Tensor diagrams and cluster algebras, Adv. Math. 300 (2016) 717 [arXiv:1210.1888].
S. He, Z. Li and C. Zhang, The symbol and alphabet of two-loop NMHV amplitudes from \( \overline{Q} \) equations, JHEP 03 (2021) 278 [arXiv:2009.11471] [INSPIRE].
J. Mago, A. Schreiber, M. Spradlin and A. Volovich, Symbol alphabets from plabic graphs, JHEP 10 (2020) 128 [arXiv:2007.00646] [INSPIRE].
S. He and Z. Li, A note on letters of Yangian invariants, JHEP 02 (2021) 155 [arXiv:2007.01574] [INSPIRE].
J. Mago, A. Schreiber, M. Spradlin, A. Y. Srikant and A. Volovich, Symbol alphabets from plabic graphs II: rational letters, JHEP 04 (2021) 056 [arXiv:2012.15812] [INSPIRE].
J. Mago, A. Schreiber, M. Spradlin, A. Yelleshpur Srikant and A. Volovich, Symbol alphabets from plabic graphs. Part III. N = 9, JHEP 09 (2021) 002 [arXiv:2106.01406] [INSPIRE].
A. Postnikov, Total positivity, Grassmannians, and networks, math/0609764 [INSPIRE].
N. Arkani-Hamed, J. L. Bourjaily, F. Cachazo, A. B. Goncharov, A. Postnikov and J. Trnka, Grassmannian Geometry of Scattering Amplitudes, Cambridge University Press, Cambridge U.K. (2016) [arXiv:1212.5605] [INSPIRE].
C. Fraser, T. Lam and I. Le, From dimers to webs, Trans. Amer. Math. Soc. 371 (2019) 6087 [arXiv:1705.09424].
J. D. Stasheff, Homotopy associativity of h-spaces. I, Trans. Amer. Math. Soc. 108 (1963) 275.
J. D. Stasheff, Homotopy associativity of h-spaces. II, Trans. Amer. Math. Soc. 108 (1963) 293.
N. Arkani-Hamed, Y. Bai, S. He and G. Yan, Scattering forms and the positive geometry of kinematics, color and the worldsheet, JHEP 05 (2018) 096 [arXiv:1711.09102] [INSPIRE].
S. Fomin and A. Zelevinsky, Cluster algebras IV: coefficients, Comp. Math. 143 (2007) 112 [math/0602259].
J. S. Scott, Grassmannians and cluster algebras, Proc. London Math. Soc. 92 (2006) 345.
C. Fraser, Braid group symmetries of grassmannian cluster algebras, Sel. Math. 26 (2020) 1 [arXiv:1702.00385].
G. Lusztig, Canonical bases arising from quantized enveloping algebras, J. Amer. Soc. 3 (1990) 447.
S. Cautis, J. Kamnitzer and S. Morrison, Webs and quantum skew Howe duality, Math. Ann. 360 (2014) 351 [arXiv:1210.6437].
G. Kuperberg, Spiders for rank 2 Lie algebras, Commun. Math. Phys. 180 (1996) 109 [q-alg/9712003].
D. Kim, Graphical calculus on representations of quantum Lie algebras, math/0310143.
L. Lamberti, Tensor diagrams and chebyshev polynomials, Int. Math. Res. Not. 2020 (2020) 7218 [arXiv:1609.03501].
N. Henke and G. Papathanasiou, Singularities of eight- and nine-particle amplitudes from cluster algebras and tropical geometry, JHEP 10 (2021) 007 [arXiv:2106.01392] [INSPIRE].
S. Herrmann, A. Jensen, M. Joswig and B. Sturmfels, How to draw tropical planes, arXiv:0808.2383.
F. Cachazo, A. Guevara, B. Umbert and Y. Zhang, Planar Matrices and Arrays of Feynman Diagrams, arXiv:1912.09422 [INSPIRE].
S. He, L. Ren and Y. Zhang, Notes on polytopes, amplitudes and boundary configurations for Grassmannian string integrals, JHEP 04 (2020) 140 [arXiv:2001.09603] [INSPIRE].
A. Felikson, M. Shapiro and P. Tumarkin, Skew-symmetric cluster algebras of finite mutation type, J. Eur. Math. Soc. 14 (2012) 1135 [arXiv:0811.1703].
M. Gekhtman, M. Shapiro and A. Vainshtein, Cluster algebras and Poisson geometry, American Mathematical Society, U.S.A. (2010) [math/0208033].
B. Keller, Cluster algebras, quiver representations and triangulated categories, arXiv:0807.1960.
Z. Bern, L. J. Dixon, D. C. Dunbar and D. A. Kosower, One loop n point gauge theory amplitudes, unitarity and collinear limits, Nucl. Phys. B 425 (1994) 217 [hep-ph/9403226] [INSPIRE].
Z. Bern, L. J. Dixon, D. C. Dunbar and D. A. Kosower, Fusing gauge theory tree amplitudes into loop amplitudes, Nucl. Phys. B 435 (1995) 59 [hep-ph/9409265] [INSPIRE].
L. J. Dixon, J. M. Drummond and J. M. Henn, Bootstrapping the three-loop hexagon, JHEP 11 (2011) 023 [arXiv:1108.4461] [INSPIRE].
L. J. Dixon, J. M. Drummond and J. M. Henn, Analytic result for the two-loop six-point NMHV amplitude in N = 4 super Yang-Mills theory, JHEP 01 (2012) 024 [arXiv:1111.1704] [INSPIRE].
S. Caron-Huot and S. He, Jumpstarting the all-loop S-matrix of planar N = 4 super Yang-Mills, JHEP 07 (2012) 174 [arXiv:1112.1060] [INSPIRE].
Z. Bern, V. Del Duca, L. J. Dixon and D. A. Kosower, All non-maximally-helicity-violating one-loop seven-gluon amplitudes in N = 4 super-Yang-Mills theory, Phys. Rev. D 71 (2005) 045006 [hep-th/0410224] [INSPIRE].
S. Caron-Huot, Superconformal symmetry and two-loop amplitudes in planar N = 4 super Yang-Mills, JHEP 12 (2011) 066 [arXiv:1105.5606] [INSPIRE].
J. M. Drummond, G. Papathanasiou and M. Spradlin, A symbol of uniqueness: the cluster bootstrap for the 3-loop MHV heptagon, JHEP 03 (2015) 072 [arXiv:1412.3763] [INSPIRE].
L. J. Dixon, J. Drummond, T. Harrington, A. J. McLeod, G. Papathanasiou and M. Spradlin, Heptagons from the Steinmann cluster bootstrap, JHEP 02 (2017) 137 [arXiv:1612.08976] [INSPIRE].
J. Drummond, J. Foster, O. Gürdoğan and G. Papathanasiou, Cluster adjacency and the four-loop NMHV heptagon, JHEP 03 (2019) 087 [arXiv:1812.04640] [INSPIRE].
R. Britto, F. Cachazo and B. Feng, Generalized unitarity and one-loop amplitudes in N = 4 super-Yang-Mills, Nucl. Phys. B 725 (2005) 275 [hep-th/0412103] [INSPIRE].
Z. Bern, L. J. Dixon and D. A. Kosower, All next-to-maximally-helicity-violating one-loop gluon amplitudes in N = 4 super-Yang-Mills theory, Phys. Rev. D 72 (2005) 045014 [hep-th/0412210] [INSPIRE].
S. He, Z. Li and C. Zhang, Two-loop octagons, algebraic letters and \( \overline{Q} \) equations, Phys. Rev. D 101 (2020) 061701 [arXiv:1911.01290] [INSPIRE].
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
ArXiv ePrint: 2106.01405
Rights and permissions
Open Access . This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.
About this article
Cite this article
Ren, L., Spradlin, M. & Volovich, A. Symbol alphabets from tensor diagrams. J. High Energ. Phys. 2021, 79 (2021). https://doi.org/10.1007/JHEP12(2021)079
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP12(2021)079