Abstract
Scattering amplitudes in planar \( \mathcal{N} \) = 4 super Yang-Mills theory exhibit singularities which reflect various aspects of the cluster algebras associated to the Grassmannians Gr(4, n) and their tropical counterparts. Here we investigate the potential origins of such structures and examine the extent to which they can be recovered from the Gröbner structure of the underlying Plücker ideals, focussing on the Grassmannians corresponding to finite cluster algebras.
Starting from the Plücker ideal, we describe how the polynomial cluster variables are encoded in non-prime initial ideals associated to certain maximal cones of the positive tropical fan. Following [1] we show that extending the Plücker ideal by such variables leads to a Gröbner fan with a single maximal Gröbner cone spanned by the positive tropical rays. The associated initial ideal encodes the compatibility relations among the full set of cluster variables. Thus we find that the Gröbner structure naturally encodes both the symbol alphabet and the cluster adjacency relations exhibited by scattering amplitudes without invoking the cluster algebra at all.
As a potential application of these ideas we then examine the kinematic ideal associated to non-dual conformal massless scattering written in terms of spinor helicity variables. For five-particle scattering we find that the ideal can be identified with the Plücker ideal for Gr(3, 6) and the corresponding tropical fan contains a number of non-prime ideals which encode all additional letters of the two-loop pentagon function alphabet present in various calculations of massless five-point finite remainders.
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References
L. Bossinger et al., Families of Gröbner degenerations, Grassmannians and universal cluster algebras, SIGMA 17 (2021) 059 [arXiv:2007.14972].
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].
J. Golden, M.F. Paulos, M. Spradlin and A. Volovich, Cluster Polylogarithms for Scattering Amplitudes, J. Phys. A 47 (2014) 474005 [arXiv:1401.6446] [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. Am. Math. Soc. 15 (2001) 497 [math/0104151].
S. Fomin and A. Zelevinsky, Cluster algebras II. Finite type classification, Invent. Math. 154 (2003) 63 [math/0208229].
S. Fomin and A. Zelevinsky, Cluster algebras IV. Coefficients, Compos. Math. 143 (2007) 112 [math/0602259].
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, M. von Hippel and J. Pennington, Hexagon functions and the three-loop remainder function, JHEP 12 (2013) 049 [arXiv:1308.2276] [INSPIRE].
L.J. Dixon, J.M. Drummond, C. Duhr and J. Pennington, The four-loop remainder function and multi-Regge behavior at NNLLA in planar N = 4 super-Yang-Mills theory, JHEP 06 (2014) 116 [arXiv:1402.3300] [INSPIRE].
L.J. Dixon and M. von Hippel, Bootstrapping an NMHV amplitude through three loops, JHEP 10 (2014) 065 [arXiv:1408.1505] [INSPIRE].
L.J. Dixon, M. von Hippel and A.J. McLeod, The four-loop six-gluon NMHV ratio function, JHEP 01 (2016) 053 [arXiv:1509.08127] [INSPIRE].
S. Caron-Huot, L.J. Dixon, A. McLeod and M. von Hippel, Bootstrapping a Five-Loop Amplitude Using Steinmann Relations, Phys. Rev. Lett. 117 (2016) 241601 [arXiv:1609.00669] [INSPIRE].
S. Caron-Huot, L.J. Dixon, F. Dulat, M. von Hippel, A.J. McLeod and G. Papathanasiou, Six-Gluon amplitudes in planar \( \mathcal{N} \) = 4 super-Yang-Mills theory at six and seven loops, JHEP 08 (2019) 016 [arXiv:1903.10890] [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].
J. Drummond, J. Foster and O. Gürdoğan, Cluster Adjacency Properties of Scattering Amplitudes in N = 4 Supersymmetric Yang-Mills Theory, Phys. Rev. Lett. 120 (2018) 161601 [arXiv:1710.10953] [INSPIRE].
J. Drummond, J. Foster and O. Gürdoğan, Cluster adjacency beyond MHV, JHEP 03 (2019) 086 [arXiv:1810.08149] [INSPIRE].
S. Caron-Huot, L.J. Dixon, F. Dulat, M. Von Hippel, A.J. McLeod and G. Papathanasiou, The Cosmic Galois Group and Extended Steinmann Relations for Planar \( \mathcal{N} \) = 4 SYM Amplitudes, JHEP 09 (2019) 061 [arXiv:1906.07116] [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].
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. 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].
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].
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].
Q. Yang, Schubert problems, positivity and symbol letters, JHEP 08 (2022) 168 [arXiv:2203.16112] [INSPIRE].
S. He, J. Liu, Y. Tang and Q. Yang, The symbology of Feynman integrals from twistor geometries, arXiv:2207.13482 [INSPIRE].
S. He, Z. Li and C. Zhang, A nice two-loop next-to-next-to-MHV amplitude in \( \mathcal{N} \) = 4 super-Yang-Mills, JHEP 12 (2022) 158 [arXiv:2209.10856] [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].
D. Maclagan and B. Sturmfels, Introduction to Tropical Geometry, Graduate Studies in Mathematics. Vol. 161, American Mathematical Society, Providence, U.S.A. (2015), https://doi.org/10.1090/gsm/161.
D. Speyer and B. Sturmfels, The tropical Grassmannian, Adv. Geom. 4 (2004) 389 [math/0304218].
D. Speyer and L. Williams, The tropical totally positive grassmannian, J. Algebr. Comb. 22 (2005) 189 [math/0312297].
S.B. Brodsky, C. Ceballos and J.-P. Labbé, Cluster algebras of type D4, tropical planes, and the positive tropical Grassmannian, Beitr. Algebra Geom. 58 (2017) 25 [arXiv:1511.02699].
N. Ilten, A. Nájera Chávez and H. Treffinger, Deformation Theory for Finite Cluster Complexes, arXiv:2111.02566.
T. Gehrmann, J.M. Henn and N.A. Lo Presti, Analytic form of the two-loop planar five-gluon all-plus-helicity amplitude in QCD, Phys. Rev. Lett. 116 (2016) 062001 [arXiv:1511.05409] [Erratum ibid. 116 (2016) 189903] [INSPIRE].
D. Chicherin, J. Henn and V. Mitev, Bootstrapping pentagon functions, JHEP 05 (2018) 164 [arXiv:1712.09610] [INSPIRE].
T. Gehrmann, J.M. Henn and N.A. Lo Presti, Pentagon functions for massless planar scattering amplitudes, JHEP 10 (2018) 103 [arXiv:1807.09812] [INSPIRE].
S. Abreu, L.J. Dixon, E. Herrmann, B. Page and M. Zeng, The two-loop five-point amplitude in \( \mathcal{N} \) = 4 super-Yang-Mills theory, Phys. Rev. Lett. 122 (2019) 121603 [arXiv:1812.08941] [INSPIRE].
D. Chicherin, T. Gehrmann, J.M. Henn, P. Wasser, Y. Zhang and S. Zoia, Analytic result for a two-loop five-particle amplitude, Phys. Rev. Lett. 122 (2019) 121602 [arXiv:1812.11057] [INSPIRE].
S. Badger et al., Analytic form of the full two-loop five-gluon all-plus helicity amplitude, Phys. Rev. Lett. 123 (2019) 071601 [arXiv:1905.03733] [INSPIRE].
D. Chicherin, T. Gehrmann, J.M. Henn, P. Wasser, Y. Zhang and S. Zoia, The two-loop five-particle amplitude in \( \mathcal{N} \) = 8 supergravity, JHEP 03 (2019) 115 [arXiv:1901.05932] [INSPIRE].
S. Abreu, L.J. Dixon, E. Herrmann, B. Page and M. Zeng, The two-loop five-point amplitude in \( \mathcal{N} \) = 8 supergravity, JHEP 03 (2019) 123 [arXiv:1901.08563] [INSPIRE].
J. Herzog and T. Hibi, Monomial Ideals, Graduate Texts in Mathematics. Vol. 260, Springer-Verlag London, London, U.K. (2011), https://doi.org/10.1007/978-0-85729-106-6.
T. Mora and L. Robbiano, The Gröbner fan of an ideal, J. Symb. Comput. 6 (1988) 183.
M. Einsiedler and S. Tuncel, When does a polynomial ideal contain a positive polynomial?, J. Pure Appl. Algebra 164 (2001) 149.
J. Drummond, J. Foster, O. Gürdoğan and C. Kalousios, Tropical fans, scattering equations and amplitudes, JHEP 11 (2021) 071 [arXiv:2002.04624] [INSPIRE].
A.N. Jensen, Gfan, a software system for Gröbner fans and tropical varieties, available at http://home.imf.au.dk/jensen/software/gfan/gfan.html.
S. Herrmann, A. Jensen, M. Joswig and B. Sturmfels, How to Draw Tropical Planes, Electron. J. Combin. 16 (2009) R6 [arXiv:0808.2383].
A.N. Jensen, Planes in five dimensional tropical projective space, https://users-math.au.dk/jensen/Research/G3_7/grassmann3_6.html.
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].
A.N. Jensen, Planes in six dimensional tropical projective space, https://users-math.au.dk/jensen/Research/G3_7/grassmann3_7.html.
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].
K. Kaveh and C. Manon, Khovanskii bases, higher rank valuations, and tropical geometry, SIAM J. Appl. Algebra Geom. 3 (2019) 292.
L. Bossinger, Full-rank valuations and toric initial ideals, Int. Math. Res. Not. 2021 (2020) 7715.
L. Bossinger, Tropical totally positive cluster varieties, arXiv:2208.01723.
S. Fomin, L. Williams and A. Zelevinsky, Introduction to Cluster Algebras. Chapter 6, arXiv:2008.09189.
S. Abreu, B. Page, E. Pascual and V. Sotnikov, Leading-Color Two-Loop QCD Corrections for Three-Photon Production at Hadron Colliders, JHEP 01 (2021) 078 [arXiv:2010.15834] [INSPIRE].
S. Badger, C. Brønnum-Hansen, H.B. Hartanto and T. Peraro, Analytic helicity amplitudes for two-loop five-gluon scattering: the single-minus case, JHEP 01 (2019) 186 [arXiv:1811.11699] [INSPIRE].
S. Abreu, J. Dormans, F. Febres Cordero, H. Ita, B. Page and V. Sotnikov, Analytic Form of the Planar Two-Loop Five-Parton Scattering Amplitudes in QCD, JHEP 05 (2019) 084 [arXiv:1904.00945] [INSPIRE].
Z. Bern, L.J. Dixon and D.A. Kosower, One loop corrections to five gluon amplitudes, Phys. Rev. Lett. 70 (1993) 2677 [hep-ph/9302280] [INSPIRE].
J.M. Henn, A. Matijašić and J. Miczajka, One-loop hexagon integral to higher orders in the dimensional regulator, JHEP 01 (2023) 096 [arXiv:2210.13505] [INSPIRE].
Acknowledgments
We are grateful to the Isaac Newton Institute and the organizers of the hybrid workshop “Interdisciplinary applications of cluster algebras” held in 2021 where the discussions that lead to this article started. Furthermore L.B. acknowledges support of the PAPIIT project IA100122, Dirección General de Asuntos del Personal Académico, Universidad Nacional Autónoma de México 2022.
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Bossinger, L., Drummond, J.M. & Glew, R. Adjacency for scattering amplitudes from the Gröbner fan. J. High Energ. Phys. 2023, 2 (2023). https://doi.org/10.1007/JHEP11(2023)002
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DOI: https://doi.org/10.1007/JHEP11(2023)002