Abstract
In this paper we study motivic amplitudes — objects which contain all of the essential mathematical content of scattering amplitudes in planar SYM theory in a completely canonical way, free from the ambiguities inherent in any attempt to choose particular functional representatives. We find that the cluster structure on the kinematic configuration space Conf n (ℙ3) underlies the structure of motivic amplitudes. Specifically, we compute explicitly the coproduct of the two-loop seven-particle MHV motivic amplitude \( \mathcal{A}_{7,2}^{\mathcal{M}} \) and find that like the previously known six-particle amplitude, it depends only on certain preferred coordinates known in the mathematics literature as cluster \( \mathcal{X} \)-coordinates on Conf n (ℙ3). We also find intriguing relations between motivic amplitudes and the geometry of generalized associahedrons, to which cluster coordinates have a natural combinatoric connection. For example, the obstruction to \( \mathcal{A}_{7,2}^{\mathcal{M}} \) being expressible in terms of classical
polylogarithms is most naturally represented by certain quadrilateral faces of the appropriate associahedron. We also find and prove the first known functional equation for the trilogarithm in which all 40 arguments are cluster \( \mathcal{X} \)-coordinates of a single algebra. In this respect it is similar to Abel’s 5-term dilogarithm identity.
Article PDF
Similar content being viewed by others
References
M.L. Mangano and S.J. Parke, Multiparton amplitudes in gauge theories, Phys. Rept. 200 (1991) 301 [hep-th/0509223] [INSPIRE].
L.J. Dixon, Calculating scattering amplitudes efficiently, hep-ph/9601359 [INSPIRE].
F. Cachazo and P. Svrček, Lectures on twistor strings and perturbative Yang-Mills theory, PoS(RTN2005)004 [hep-th/0504194] [INSPIRE].
Z. Bern, L.J. Dixon and D.A. Kosower, On-shell methods in perturbative QCD, Annals Phys. 322 (2007) 1587 [arXiv:0704.2798] [INSPIRE].
R. Roiban, M. Spradlin and A. Volovich, Scattering amplitudes in gauge theories: Progress and outlook, J.Phys. A 44 (2011) 450301.
B. Feng and M. Luo, An introduction to on-shell recursion relations, Front. Phys. ,2012,7 (5):533-575 [arXiv:1111.5759] [INSPIRE].
L. Brink, J.H. Schwarz and J. Scherk, Supersymmetric Yang-Mills theories, Nucl. Phys. B 121 (1977) 77 [INSPIRE].
F. Gliozzi, J. Scherk and D.I. Olive, Supersymmetry, supergravity theories and the dual spinor model, Nucl. Phys. B 122 (1977) 253 [INSPIRE].
A. Goncharov, Galois symmetries of fundamental groupoids and noncommutative geometry, Duke Math. J. 128 (2005) 209 [math/0208144] [INSPIRE].
O. Schlotterer and S. Stieberger, Motivic multiple Zeta values and superstring amplitudes, J. Phys. A 46 (2013) 475401 [arXiv:1205.1516] [INSPIRE].
J. Drummond and É. Ragoucy, Superstring amplitudes and the associator, JHEP 08 (2013) 135 [arXiv:1301.0794] [INSPIRE].
J. Broedel, O. Schlotterer and S. Stieberger, Polylogarithms, multiple Zeta values and superstring amplitudes, Fortsch. Phys. 61 (2013) 812 [arXiv:1304.7267] [INSPIRE].
J. Broedel, O. Schlotterer, S. Stieberger and T. Terasoma, All order α ′ -expansion of superstring trees from the Drinfeld associator, arXiv:1304.7304 [INSPIRE].
J. Drummond, J. Henn, V. Smirnov and E. Sokatchev, Magic identities for conformal four-point integrals, JHEP 01 (2007) 064 [hep-th/0607160] [INSPIRE].
Z. Bern, M. Czakon, L.J. Dixon, D.A. Kosower and V.A. Smirnov, The four-loop planar amplitude and cusp anomalous dimension in maximally supersymmetric Yang-Mills theory, Phys. Rev. D 75 (2007) 085010 [hep-th/0610248] [INSPIRE].
L.F. Alday and J.M. Maldacena, Gluon scattering amplitudes at strong coupling, JHEP 06 (2007) 064 [arXiv:0705.0303] [INSPIRE].
J. Drummond, G. Korchemsky and E. Sokatchev, Conformal properties of four-gluon planar amplitudes and Wilson loops, Nucl. Phys. B 795 (2008) 385 [arXiv:0707.0243] [INSPIRE].
J. Drummond, J. Henn, G. Korchemsky and E. Sokatchev, On planar gluon amplitudes/Wilson loops duality, Nucl. Phys. B 795 (2008) 52 [arXiv:0709.2368] [INSPIRE].
L.F. Alday and J. Maldacena, Comments on gluon scattering amplitudes via AdS/CFT, JHEP 11 (2007) 068 [arXiv:0710.1060] [INSPIRE].
J. Drummond, J. Henn, G. Korchemsky and E. Sokatchev, Conformal Ward identities for Wilson loops and a test of the duality with gluon amplitudes, Nucl. Phys. B 826 (2010) 337 [arXiv:0712.1223] [INSPIRE].
J. Drummond, J. Henn, G. Korchemsky and E. Sokatchev, Dual superconformal symmetry of scattering amplitudes in N = 4 super-Yang-Mills theory, Nucl. Phys. B 828 (2010) 317 [arXiv:0807.1095] [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].
A. Goncharov, Geometry of configurations, polylogarithms, and motivic cohomology, Adv. Math. 114 (1995) 197.
V. Fock and A. Goncharov, Cluster ensembles, quantization and the dilogarithm, Ann. Sci. Éc. Norm. Supér. 42 (2009) 865 [math/0311245] [INSPIRE].
S. Fomin and A. Zelevinsky, Cluster algebras. I: foundations, J. Am. Math. Soc. 15 (2002) 497.
S. Fomin and A. Zelevinsky, Cluster algebras. II: finite type classification, Invent. Math. 154 (2003) 63.
N. Arkani-Hamed, F. Cachazo, C. Cheung and J. Kaplan, A duality for the S matrix, JHEP 03 (2010) 020 [arXiv:0907.5418] [INSPIRE].
N. Arkani-Hamed, F. Cachazo and C. Cheung, The grassmannian origin of dual superconformal invariance, JHEP 03 (2010) 036 [arXiv:0909.0483] [INSPIRE].
N. Arkani-Hamed, J. Bourjaily, F. Cachazo and J. Trnka, Local spacetime physics from the grassmannian, JHEP 01 (2011) 108 [arXiv:0912.3249] [INSPIRE].
N. Arkani-Hamed, J. Bourjaily, F. Cachazo and J. Trnka, Unification of residues and grassmannian dualities, JHEP 01 (2011) 049 [arXiv:0912.4912] [INSPIRE].
N. Arkani-Hamed, J.L. Bourjaily, F. Cachazo, S. Caron-Huot and J. Trnka, The all-loop integrand for scattering amplitudes in planar N = 4 SYM, JHEP 01 (2011) 041 [arXiv:1008.2958] [INSPIRE].
N. Arkani-Hamed et al., Scattering amplitudes and the positive grassmannian, arXiv:1212.5605 [INSPIRE].
S. Fomin and A. Zelevinsky, Y-systems and generalized associahedra, Ann. Math. 158 (2003) 977.
J.D. Stasheff, Homotopy associativity of H-spaces. I, Trans. Am. Math. Soc. 108 (1963) 275.
J.D. Stasheff, Homotopy associativity of H-spaces. II, Trans. Am. Math. Soc. 108 (1963) 293.
A. Hodges, Eliminating spurious poles from gauge-theoretic amplitudes, JHEP 05 (2013) 135 [arXiv:0905.1473] [INSPIRE].
R. Penrose, Twistor algebra, J. Math. Phys. 8 (1967) 345 [INSPIRE].
R. Penrose and M.A. MacCallum, Twistor theory: an approach to the quantization of fields and space-time, Phys. Rept. 6 (1972) 241 [INSPIRE].
E. Witten, Perturbative gauge theory as a string theory in twistor space, Commun. Math. Phys. 252 (2004) 189 [hep-th/0312171] [INSPIRE].
L. Mason and D. Skinner, Dual superconformal invariance, momentum twistors and grassmannians, JHEP 11 (2009) 045 [arXiv:0909.0250] [INSPIRE].
E. Witten, An interpretation of classical Yang-Mills theory, Phys. Lett. B 77 (1978) 394 [INSPIRE].
N. Beisert and C. Vergu, On the geometry of null polygons in full N = 4 superspace, Phys. Rev. D 86 (2012) 026006 [arXiv:1203.0525] [INSPIRE].
N. Beisert, S. He, B.U. Schwab and C. Vergu, Null polygonal Wilson loops in full N = 4 superspace, J. Phys. A 45 (2012) 265402 [arXiv:1203.1443] [INSPIRE].
Z. Bern, L. Dixon, D. Kosower, R. Roiban, M. Spradlin et al., The two-loop six-gluon MHV amplitude in maximally supersymmetric Yang-Mills theory, Phys. Rev. D 78 (2008) 045007 [arXiv:0803.1465] [INSPIRE].
J. Drummond, J. Henn, G. Korchemsky and E. Sokatchev, Hexagon Wilson loop = six-gluon MHV amplitude, Nucl. Phys. B 815 (2009) 142 [arXiv:0803.1466] [INSPIRE].
V. Del Duca, C. Duhr and V.A. Smirnov, An analytic result for the two-loop hexagon Wilson loop in N = 4 SYM, JHEP 03 (2010) 099 [arXiv:0911.5332] [INSPIRE].
V. Del Duca, C. Duhr and V.A. Smirnov, The two-loop hexagon Wilson loop in N = 4 SYM, JHEP 05 (2010) 084 [arXiv:1003.1702] [INSPIRE].
A. Beilinson, Height pairing between algebraic cycles, in K-theory, arithmetic and geometry, Y.I. Manin ed., Springer, Berlin Germany (1987).
P. Deligne and A.B. Goncharov, Groupes fondamentaux motiviques de Tate mixte, Ann. Sci. École Norm. Sup. 38 (2005) 1.
S.J. Bloch, Higher regulators, algebraic K-theory, and zeta functions of elliptic curves, American Mathematical Society, Providence U.S.A. (2000).
A. Suslin, K 3 of a field and the Bloch group, Proc. Steklov Inst. Math. 183 (1990) 217.
D. Zagier, Polylogarithms, Dedekind zeta functions, and the algebraic K-theory of fields, in Arithmetic algebraic geometry, J.L. Colliot-Thelene et al., Boston U.S.A. (1991).
A.B. Goncharov, Polylogarithms and motivic Galois groups, in Motives (Seattle, WA, 1991), American Mathematical Sociesty, Providence U.S.A. (1994).
A. Goncharov, Deninger’s conjecture on L-functions of elliptic curves at s = 3, J. Math. Sci., New York 81 (1996) 2631 [alg-geom/9512016].
S. Caron-Huot, Superconformal symmetry and two-loop amplitudes in planar N = 4 super Yang-Mills, JHEP 12 (2011) 066 [arXiv:1105.5606] [INSPIRE].
S. Fomin and A. Zelevinsky, The Laurent phenomenon, Adv. Appl. Math. 28 (2002) 119.
J.S. Scott, Grassmannians and cluster algebras, Proc. Lond. Math. Soc. III Ser. 92 (2006) 345.
M. Gekhtman, M. Shapiro, and A. Vainshtein, Cluster algebras and Poisson geometry, Mosc. Math. J. 3 (2003) 899.
B. Keller, Cluster algebras, quiver representations and triangulated categories, in Triangulated categories, Cambridge University Press, Cambridge U.K. (2010).
S. Fomin and N. Reading, Root systems and generalized associahedra, in Geometric combinatorics, American Mathematical Society, Providence U.S.A. (2007).
C. Anastasiou et al., Two-loop polygon Wilson loops in N = 4 SYM, JHEP 05 (2009) 115 [arXiv:0902.2245] [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].
L.J. Dixon, J.M. Drummond and J.M. Henn, Bootstrapping the three-loop hexagon, JHEP 11 (2011) 023 [arXiv:1108.4461] [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].
V. Del Duca, C. Duhr and V.A. Smirnov, A two-loop octagon Wilson loop in N = 4 SYM, JHEP 09 (2010) 015 [arXiv:1006.4127] [INSPIRE].
P. Heslop and V.V. Khoze, Analytic results for MHV Wilson loops, JHEP 11 (2010) 035 [arXiv:1007.1805] [INSPIRE].
P. Heslop and V.V. Khoze, Wilson loops 3-loops in special kinematics, JHEP 11 (2011) 152 [arXiv:1109.0058] [INSPIRE].
T. Goddard, P. Heslop and V.V. Khoze, Uplifting amplitudes in special kinematics, JHEP 10 (2012) 041 [arXiv:1205.3448] [INSPIRE].
L. Ferro, Differential equations for multi-loop integrals and two-dimensional kinematics, JHEP 04 (2013) 160 [arXiv:1204.1031] [INSPIRE].
J. Bartels, L. Lipatov and A. Prygarin, Collinear and Regge behavior of 2 → 4 MHV amplitude in N = 4 super Yang-Mills theory, arXiv:1104.4709 [INSPIRE].
A. Prygarin, M. Spradlin, C. Vergu and A. Volovich, All two-loop MHV amplitudes in multi-Regge kinematics from applied symbology, Phys. Rev. D 85 (2012) 085019 [arXiv:1112.6365] [INSPIRE].
J. Bartels, A. Kormilitzin, L. Lipatov and A. Prygarin, BFKL approach and 2 → 5 maximally helicity violating amplitude in \( \mathcal{N} \) = 4 super-Yang-Mills theory, Phys. Rev. D 86 (2012) 065026 [arXiv:1112.6366] [INSPIRE].
L. Lipatov, A. Prygarin and H.J. Schnitzer, The multi-Regge limit of NMHV amplitudes in N =4 SYM theory, JHEP 01 (2013) 068 [arXiv:1205.0186] [INSPIRE].
L.J. Dixon, C. Duhr and J. Pennington, Single-valued harmonic polylogarithms and the multi-Regge limit, JHEP 10 (2012) 074 [arXiv:1207.0186] [INSPIRE].
J. Pennington, The six-point remainder function to all loop orders in the multi-Regge limit, JHEP 01 (2013) 059 [arXiv:1209.5357] [INSPIRE].
L.F. Alday, D. Gaiotto, J. Maldacena, A. Sever and P. Vieira, An operator product expansion for polygonal null Wilson loops, JHEP 04 (2011) 088 [arXiv:1006.2788] [INSPIRE].
D. Gaiotto, J. Maldacena, A. Sever and P. Vieira, Bootstrapping null polygon Wilson loops, JHEP 03 (2011) 092 [arXiv:1010.5009] [INSPIRE].
D. Gaiotto, J. Maldacena, A. Sever and P. Vieira, Pulling the straps of polygons, JHEP 12 (2011) 011 [arXiv:1102.0062] [INSPIRE].
B. Basso, A. Sever and P. Vieira, Space-time S-matrix and flux-tube S-matrix at finite coupling, Phys. Rev. Lett. 111 (2013) 091602 [arXiv:1303.1396] [INSPIRE].
N. Beisert et al., Review of AdS/CFT integrability: an overview, Lett. Math. Phys. 99 (2012) 3 [arXiv:1012.3982] [INSPIRE].
J.M. Drummond, J.M. Henn and J. Plefka, Yangian symmetry of scattering amplitudes in N =4 super Yang-Mills theory, JHEP 05 (2009) 046 [arXiv:0902.2987] [INSPIRE].
L.F. Alday, D. Gaiotto and J. Maldacena, Thermodynamic bubble ansatz, JHEP 09 (2011) 032 [arXiv:0911.4708] [INSPIRE].
L.F. Alday, J. Maldacena, A. Sever and P. Vieira, Y-system for scattering amplitudes, J. Phys. A 43 (2010) 485401 [arXiv:1002.2459] [INSPIRE].
L. Ferro, T. Lukowski, C. Meneghelli, J. Plefka and M. Staudacher, Harmonic R-matrices for scattering amplitudes and spectral regularization, Phys. Rev. Lett. 110 (2013) 121602 [arXiv:1212.0850] [INSPIRE].
V. Del Duca et al., The one-loop six-dimensional hexagon integral with three massive corners, Phys. Rev. D 84 (2011) 045017 [arXiv:1105.2011] [INSPIRE].
C. Duhr, H. Gangl and J.R. Rhodes, From polygons and symbols to polylogarithmic functions, JHEP 10 (2012) 075 [arXiv:1110.0458] [INSPIRE].
M. Bullimore and D. Skinner, Descent equations for superamplitudes, arXiv:1112.1056 [INSPIRE].
A. Brandhuber, G. Travaglini and G. Yang, Analytic two-loop form factors in N = 4 SYM, JHEP 05 (2012) 082 [arXiv:1201.4170] [INSPIRE].
C. Bogner and F. Brown, Symbolic integration and multiple polylogarithms, PoS(LL2012) 053 [arXiv:1209.6524] [INSPIRE].
A.E. Lipstein and L. Mason, From the holomorphic Wilson loop to ‘d log’ loop-integrands for super-Yang-Mills amplitudes, JHEP 05 (2013) 106 [arXiv:1212.6228] [INSPIRE].
S.G. Naculich, H. Nastase and H.J. Schnitzer, All-loop infrared-divergent behavior of most-subleading-color gauge-theory amplitudes, JHEP 04 (2013) 114 [arXiv:1301.2234] [INSPIRE].
J. Drummond et al., Leading singularities and off-shell conformal integrals, JHEP 08 (2013) 133 [arXiv:1303.6909] [INSPIRE].
C. Duhr, Hopf algebras, coproducts and symbols: an application to Higgs boson amplitudes, JHEP 08 (2012) 043 [arXiv:1203.0454] [INSPIRE].
A. von Manteuffel and C. Studerus, Top quark pairs at two loops and Reduze 2, PoS(LL2012) 059 [arXiv:1210.1436] [INSPIRE].
T. Gehrmann, L. Tancredi and E. Weihs, Two-loop QCD helicity amplitudes for g g → Z g and g g → Zγ, JHEP 04 (2013) 101 [arXiv:1302.2630] [INSPIRE].
C. Anastasiou, C. Duhr, F. Dulat and B. Mistlberger, Soft triple-real radiation for Higgs production at N3LO, JHEP 07 (2013) 003 [arXiv:1302.4379] [INSPIRE].
J.M. Henn, Multiloop integrals in dimensional regularization made simple, Phys. Rev. Lett. 110 (2013) 251601 [arXiv:1304.1806] [INSPIRE].
C.F. Gauss, Pentagramma mirificum, in Werke, Band III, Göttingen Germany (1863).
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.
Author information
Authors and Affiliations
Corresponding author
Additional information
ArXiv ePrint: 1305.1617
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License (https://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
About this article
Cite this article
Golden, J.K., Goncharov, A.B., Spradlin, M. et al. Motivic amplitudes and cluster coordinates. J. High Energ. Phys. 2014, 91 (2014). https://doi.org/10.1007/JHEP01(2014)091
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP01(2014)091