Abstract
We comment on the status of “Steinmann-like” constraints, i.e. all-loop constraints on consecutive entries of the symbol of scattering amplitudes and Feynman integrals in planar \( \mathcal{N} \) = 4 super-Yang-Mills, which have been crucial for the recent progress of the bootstrap program. Based on physical discontinuities and Steinmann relations, we first summarize all possible double discontinuities (or first-two-entries) for (the symbol of) amplitudes and integrals in terms of dilogarithms, generalizing well-known results for n = 6, 7 to all multiplicities. As our main result, we find that extended-Steinmann relations hold for all finite integrals that we have checked, including various ladder integrals, generic double-pentagon integrals, as well as finite components of two-loop NMHV amplitudes for any n; with suitable normalization such as minimal subtraction, they hold for n = 8 MHV amplitudes at three loops. We find interesting cancellation between contributions from rational and algebraic letters, and for the former we have also tested cluster-adjacency conditions using the so-called Sklyanin brackets. Finally, we propose a list of possible last-two-entries for MHV amplitudes up to 9 points derived from \( \overline{Q} \) equations, which can be used to reduce the space of functions for higher-point MHV amplitudes.
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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].
N. Arkani-Hamed and J. Trnka, The Amplituhedron, JHEP 10 (2014) 030 [arXiv:1312.2007] [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, C. Duhr, M. von Hippel and J. Pennington, Bootstrapping six-gluon scattering in planar N = 4 super-Yang-Mills theory, PoS LL2014 (2014) 077 [arXiv:1407.4724] [INSPIRE].
L. J. Dixon and M. von Hippel, Bootstrapping an NMHV amplitude through three loops, JHEP 10 (2014) 065 [arXiv:1408.1505] [INSPIRE].
J. M. Drummond, G. Papathanasiou and M. Spradlin, A Symbolof Uniqueness: The Cluster Bootstrap for the 3-Loop MHV Heptagon, JHEP 03 (2015) 072 [arXiv:1412.3763] [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].
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].
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].
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].
L. J. Dixon and Y.-T. Liu, Lifting Heptagon Symbols to Functions, JHEP 10 (2020) 031 [arXiv:2007.12966] [INSPIRE].
J. Golden and A. J. McLeod, The two-loop remainder function for eight and nine particles, JHEP 06 (2021) 142 [arXiv:2104.14194] [INSPIRE].
S. Caron-Huot, L. J. Dixon, J. M. Drummond, F. Dulat, J. Foster, O. Gürdoğan et al., The Steinmann Cluster Bootstrap for N = 4 Super Yang-Mills Amplitudes, PoS CORFU2019 (2020) 003 [arXiv:2005.06735] [INSPIRE].
Z. Bern, L. J. Dixon and V. A. Smirnov, Iteration of planar amplitudes in maximally supersymmetric Yang-Mills theory at three loops and beyond, Phys. Rev. D 72 (2005) 085001 [hep-th/0505205] [INSPIRE].
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. Hodges, Eliminating spurious poles from gauge-theoretic amplitudes, JHEP 05 (2013) 135 [arXiv:0905.1473] [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].
C. Duhr, H. Gangl and J. R. Rhodes, From polygons and symbols to polylogarithmic functions, JHEP 10 (2012) 075 [arXiv:1110.0458] [INSPIRE].
D. Gaiotto, J. Maldacena, A. Sever and P. Vieira, Pulling the straps of polygons, JHEP 12 (2011) 011 [arXiv:1102.0062] [INSPIRE].
O. Steinmann, Über den zusammenhang zwischen den wightmanfunktionen und den retardierten kommutatoren, Helv. Phys. Acta 33 (1960) 257.
O. Steinmann, Wightman-funktionen und retardierte kommutatoren. II, Helv. Phys. Acta 33 (1960) 347.
L. F. Alday, D. Gaiotto and J. Maldacena, Thermodynamic Bubble Ansatz, JHEP 09 (2011) 032 [arXiv:0911.4708] [INSPIRE].
J. Golden and A. J. Mcleod, Cluster Algebras and the Subalgebra Constructibility of the Seven-Particle Remainder Function, JHEP 01 (2019) 017 [arXiv:1810.12181] [INSPIRE].
J. L. Bourjaily, H. Hannesdottir, A. J. McLeod, M. D. Schwartz and C. Vergu, Sequential Discontinuities of Feynman Integrals and the Monodromy Group, JHEP 01 (2021) 205 [arXiv:2007.13747] [INSPIRE].
L. J. Dixon, A. J. McLeod and M. Wilhelm, A Three-Point Form Factor Through Five Loops, JHEP 04 (2021) 147 [arXiv:2012.12286] [INSPIRE].
P. Benincasa, A. J. McLeod and C. Vergu, Steinmann Relations and the Wavefunction of the Universe, Phys. Rev. D 102 (2020) 125004 [arXiv:2009.03047] [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 and S. He, Jumpstarting the All-Loop S-matrix of Planar N = 4 Super Yang-Mills, JHEP 07 (2012) 174 [arXiv:1112.1060] [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].
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].
D. Speyer and L. Williams, The tropical totally positive grassmannian, J. Algebr. Comb. 22 (2005) 189.
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. Henke and G. Papathanasiou, How tropical are seven- and eight-particle amplitudes?, JHEP 08 (2020) 005 [arXiv:1912.08254] [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. Arkani-Hamed, T. Lam and M. Spradlin, Positive configuration space, Commun. Math. Phys. 384 (2021) 909 [arXiv:2003.03904] [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].
L. Ren, M. Spradlin and A. Volovich, Symbol alphabets from tensor diagrams, JHEP 12 (2021) 079 [arXiv:2106.01405] [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 III: N = 9, JHEP 09 (2021) 002 [arXiv:2106.01406] [INSPIRE].
N. Arkani-Hamed, J. L. Bourjaily, F. Cachazo and J. Trnka, Local Integrals for Planar Scattering Amplitudes, JHEP 06 (2012) 125 [arXiv:1012.6032] [INSPIRE].
J. M. Drummond, J. M. Henn and J. Trnka, New differential equations for on-shell loop integrals, JHEP 04 (2011) 083 [arXiv:1010.3679] [INSPIRE].
S. Caron-Huot, L. J. Dixon, M. von Hippel, A. J. McLeod and G. Papathanasiou, The Double Pentaladder Integral to All Orders, JHEP 07 (2018) 170 [arXiv:1806.01361] [INSPIRE].
J. Henn, E. Herrmann and J. Parra-Martinez, Bootstrapping two-loop Feynman integrals for planar \( \mathcal{N} \) = 4 SYM, JHEP 10 (2018) 059 [arXiv:1806.06072] [INSPIRE].
J. L. Bourjaily, A. J. McLeod, M. von Hippel and M. Wilhelm, Rationalizing Loop Integration, JHEP 08 (2018) 184 [arXiv:1805.10281] [INSPIRE].
E. Herrmann and J. Parra-Martinez, Logarithmic forms and differential equations for Feynman integrals, JHEP 02 (2020) 099 [arXiv:1909.04777] [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. 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, All Master Integrals for Three-Jet Production at Next-to-Next-to-Leading Order, Phys. Rev. Lett. 123 (2019) 041603 [arXiv:1812.11160] [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].
D. Chicherin, J. Henn and V. Mitev, Bootstrapping pentagon functions, JHEP 05 (2018) 164 [arXiv:1712.09610] [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].
S. He, Z. Li and Q. Yang, Truncated cluster algebras and Feynman integrals with algebraic letters, JHEP 12 (2021) 110 [arXiv:2106.09314] [INSPIRE].
J. M. Drummond, J. Henn, V. A. Smirnov and E. Sokatchev, Magic identities for conformal four-point integrals, JHEP 01 (2007) 064 [hep-th/0607160] [INSPIRE].
J. M. Drummond, G. P. Korchemsky and E. Sokatchev, Conformal properties of four-gluon planar amplitudes and Wilson loops, Nucl. Phys. B 795 (2008) 385 [arXiv:0707.0243] [INSPIRE].
S. He, Z. Li, Y. Tang and Q. Yang, The Wilson-loop d log representation for Feynman integrals, JHEP 05 (2021) 052 [arXiv:2012.13094] [INSPIRE].
S. He, Z. Li, Q. Yang and C. Zhang, Feynman Integrals and Scattering Amplitudes from Wilson Loops, Phys. Rev. Lett. 126 (2021) 231601 [arXiv:2012.15042] [INSPIRE].
J. L. Bourjaily, A. J. McLeod, C. Vergu, M. Volk, M. Von Hippel and M. Wilhelm, Rooting Out Letters: Octagonal Symbol Alphabets and Algebraic Number Theory, JHEP 02 (2020) 025 [arXiv:1910.14224] [INSPIRE].
Z. Li and C. Zhang, The three-loop MHV octagon from \( \overline{Q} \) equations, JHEP 12 (2021) 113 [arXiv:2110.00350] [INSPIRE].
J. Golden, A. J. McLeod, M. Spradlin and A. Volovich, The Sklyanin Bracket and Cluster Adjacency at All Multiplicity, JHEP 03 (2019) 195 [arXiv:1902.11286] [INSPIRE].
E. K. Sklyanin, Some algebraic structures connected with the Yang-Baxter equation. Representations of quantum algebras, Funct. Anal. Appl. 17 (1983) 273 [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].
J. L. Bourjaily, S. Caron-Huot and J. Trnka, Dual-Conformal Regularization of Infrared Loop Divergences and the Chiral Box Expansion, JHEP 01 (2015) 001 [arXiv:1303.4734] [INSPIRE].
N. I. Usyukina and A. I. Davydychev, Exact results for three and four point ladder diagrams with an arbitrary number of rungs, Phys. Lett. B 305 (1993) 136 [INSPIRE].
D. J. Broadhurst and A. I. Davydychev, Exponential suppression with four legs and an infinity of loops, Nucl. Phys. B Proc. Suppl. 205-206 (2010) 326 [arXiv:1007.0237] [INSPIRE].
Z. Li and C. Zhang, The three-loop MHV octagon from \( \overline{Q} \) equations, JHEP 12 (2021) 113 [arXiv:2110.00350] [INSPIRE].
S. Oh, A. Postnikov, and D. Speyer, Weak separation and plabic graphs, Proc. Lond. Math. Soc. 110 (2015) 721.
S. Caron-Huot, Superconformal symmetry and two-loop amplitudes in planar N = 4 super Yang-Mills, JHEP 12 (2011) 066 [arXiv:1105.5606] [INSPIRE].
A. Kristensson, M. Wilhelm and C. Zhang, Elliptic Double Box and Symbology Beyond Polylogarithms, Phys. Rev. Lett. 127 (2021) 251603 [arXiv:2106.14902] [INSPIRE].
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He, S., Li, Z. & Yang, Q. Comments on all-loop constraints for scattering amplitudes and Feynman integrals. J. High Energ. Phys. 2022, 73 (2022). https://doi.org/10.1007/JHEP01(2022)073
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DOI: https://doi.org/10.1007/JHEP01(2022)073