Abstract
We translate between different formulations of Yangian invariants relevant for the computation of tree-level scattering amplitudes in \( \mathcal{N} \) = 4 super-Yang-Mills theory. While the R-operator formulation allows to relate scattering amplitudes to structures well known from integrability, it can equally well be connected to the permutations encoded by on-shell graphs.
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References
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].
P.S. Howe, K.S. Stelle and P.K. Townsend, Miraculous Ultraviolet Cancellations in Supersymmetry Made Manifest, Nucl. Phys. B 236 (1984) 125 [INSPIRE].
L. Brink, O. Lindgren and B.E.W. Nilsson, N = 4 Yang-Mills Theory on the Light Cone, Nucl. Phys. B 212 (1983) 401 [INSPIRE].
L. Brink, O. Lindgren and B.E.W. Nilsson, The Ultraviolet Finiteness of the N = 4 Yang-Mills Theory, Phys. Lett. B 123 (1983) 323 [INSPIRE].
S. Mandelstam, Light Cone Superspace and the Ultraviolet Finiteness of the N = 4 Model, Nucl. Phys. B 213 (1983) 149 [INSPIRE].
J.M. Drummond, J. Henn, G.P. 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].
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].
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 J. Kaplan, What is the Simplest Quantum Field Theory?, JHEP 09 (2010) 016 [arXiv:0808.1446] [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, 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, Unification of Residues and Grassmannian Dualities, JHEP 01 (2011) 049 [arXiv:0912.4912] [INSPIRE].
N. Arkani-Hamed et al., Scattering Amplitudes and the Positive Grassmannian, arXiv:1212.5605 [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].
L. Ferro, T. Lukowski, C. Meneghelli, J. Plefka and M. Staudacher, Spectral Parameters for Scattering Amplitudes in N = 4 Super Yang-Mills Theory, JHEP 01 (2014) 094 [arXiv:1308.3494] [INSPIRE].
N. Beisert, J. Broedel and M. Rosso, On Yangian-invariant regularisation of deformed on-shell diagrams in N = 4 super-Yang-Mills theory, arXiv:1401.7274 [INSPIRE].
D. Chicherin, S. Derkachov and R. Kirschner, Yang-Baxter operators and scattering amplitudes in N = 4 super-Yang-Mills theory, Nucl. Phys. B 881 (2014) 467 [arXiv:1309.5748] [INSPIRE].
D. Chicherin and R. Kirschner, Yangian symmetric correlators, Nucl. Phys. B 877 (2013) 484 [arXiv:1306.0711] [INSPIRE].
D. Nandan and C. Wen, Generating All Tree Amplitudes in N = 4 SYM by Inverse Soft Limit, JHEP 08 (2012) 040 [arXiv:1204.4841] [INSPIRE].
R. Frassek, N. Kanning, Y. Ko and M. Staudacher, Bethe Ansatz for Yangian Invariants: Towards Super Yang-Mills Scattering Amplitudes, Nucl. Phys. B 883 (2014) 373 [arXiv:1312.1693] [INSPIRE].
N. Kanning, T. Lukowski and M. Staudacher, A Shortcut to General Tree-level Scattering Amplitudes in N = 4 SYM via Integrability, arXiv:1403.3382 [INSPIRE].
V.P. Nair, A Current Algebra for Some Gauge Theory Amplitudes, Phys. Lett. B 214 (1988) 215 [INSPIRE].
E. Witten, Perturbative gauge theory as a string theory in twistor space, Commun. Math. Phys. 252 (2004) 189 [hep-th/0312171] [INSPIRE].
R. Britto, F. Cachazo and B. Feng, New recursion relations for tree amplitudes of gluons, Nucl. Phys. B 715 (2005) 499 [hep-th/0412308] [INSPIRE].
R. Britto, F. Cachazo, B. Feng and E. Witten, Direct proof of tree-level recursion relation in Yang-Mills theory, Phys. Rev. Lett. 94 (2005) 181602 [hep-th/0501052] [INSPIRE].
A. Brandhuber, P. Heslop and G. Travaglini, A Note on dual superconformal symmetry of the N = 4 super Yang-Mills S-matrix, Phys. Rev. D 78 (2008) 125005 [arXiv:0807.4097] [INSPIRE].
A. Postnikov, Total positivity, Grassmannians and networks, math/0609764 [INSPIRE].
N.J. MacKay, Introduction to Yangian symmetry in integrable field theory, Int. J. Mod. Phys. A 20 (2005) 7189 [hep-th/0409183] [INSPIRE].
N. Beisert and B.U.W. Schwab, Bonus Yangian Symmetry for the Planar S-matrix of N = 4 Super Yang-Mills, Phys. Rev. Lett. 106 (2011) 231602 [arXiv:1103.0646] [INSPIRE].
N. Beisert and M. de Leeuw, The RTT-Realization for the Deformed gl(2—2) Yangian, arXiv:1401.7691 [INSPIRE].
V.G. Drinfeld, Hopf algebras and the quantum Yang-Baxter equation, Sov. Math. Dokl. 32 (1985) 254 [INSPIRE].
V.G. Drinfeld, Quantum groups, J. Math. Sci. 41 (1988) 898.
S.M. Khoroshkin and V.N. Tolstoi, Yangian double and rational R matrix, hep-th/9406194 [INSPIRE].
L.A. Takhtajan and L.D. Faddeev, The quantum method of the inverse problem and the Heisenberg XYZ model, Russ. Math. Surveys 34 (1979) 11.
P.P. Kulish and E.K. Sklyanin, On the solution of the Yang-Baxter equation, J. Sov. Math. 19 (1982) 1596.
L.D. Faddeev, N. Yu. Reshetikhin and L. Takhtajan, Quantization of Lie groups and Lie algebras, Leningrad Math. J. 1 (1990) 193.
S. Khoroshkin, D. Lebedev and S. Pakuliak, Intertwining operators for the central extension of the Yangian double, q-alg/9602030 [INSPIRE].
S.M. Khoroshkin, Central extension of the Yangian double, q-alg/9602031 [INSPIRE].
L.J. Dixon, J.M. Henn, J. Plefka and T. Schuster, All tree-level amplitudes in massless QCD, JHEP 01 (2011) 035 [arXiv:1010.3991] [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].
L.J. Mason and D. Skinner, Dual Superconformal Invariance, Momentum Twistors and Grassmannians, JHEP 11 (2009) 045 [arXiv:0909.0250] [INSPIRE].
M. Bullimore, L.J. Mason and D. Skinner, MHV Diagrams in Momentum Twistor Space, JHEP 12 (2010) 032 [arXiv:1009.1854] [INSPIRE].
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Broedel, J., de Leeuw, M. & Rosso, M. A dictionary between R-operators, on-shell graphs and Yangian algebras. J. High Energ. Phys. 2014, 170 (2014). https://doi.org/10.1007/JHEP06(2014)170
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DOI: https://doi.org/10.1007/JHEP06(2014)170