Abstract
We take up the study of integrable structures behind non-planar contributions to scattering amplitudes in \( \mathcal{N}=4 \) super Yang-Mills theory. Focusing on leading singularities, we derive the action of the Yangian generators on color-ordered subsets of the external states. Each subset corresponds to a single boundary of the non-planar on-shell diagram. While Yangian invariance is broken, we find that higher-level Yangian generators still annihilate the non-planar on-shell diagram. For a given diagram, the number of these generators is governed by the degree of non-planarity. Furthermore, we present additional identities involving integrable transfer matrices. In particular, for diagrams on a cylinder we obtain a conservation rule similar to the Yangian invariance condition of planar on-shell diagrams. To exemplify our results, we consider a five-point MHV on-shell function on a cylinder.
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References
J.M. Drummond, J. Henn, G.P. Korchemsky and E. Sokatchev, Dual superconformal symmetry of scattering amplitudes in \( \mathcal{N}=4 \) 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 \( \mathcal{N}=4 \) super Yang-Mills theory, JHEP 05 (2009) 046 [arXiv:0902.2987] [INSPIRE].
N. Beisert et al., Review of AdS/CFT Integrability: An overview, Lett. Math. Phys. 99 (2012) 3 [arXiv:1012.3982] [INSPIRE].
N. Beisert, On Yangian Symmetry in Planar N = 4 SYM, in proceedings of Quantum chromodynamics and beyond: Gribov-80 memorial volume, Memorial Workshop devoted to the 80th birthday of V.N. Gribov, Trieste, Italy, 26–28 May 2010, World Scientific, arXiv:1004.5423 [INSPIRE].
Z. Bern, E. Herrmann, S. Litsey, J. Stankowicz and J. Trnka, Logarithmic Singularities and Maximally Supersymmetric Amplitudes, JHEP 06 (2015) 202 [arXiv:1412.8584] [INSPIRE].
N. Arkani-Hamed, J.L. Bourjaily, F. Cachazo and J. Trnka, Singularity Structure of Maximally Supersymmetric Scattering Amplitudes, Phys. Rev. Lett. 113 (2014) 261603 [arXiv:1410.0354] [INSPIRE].
Z. Bern, E. Herrmann, S. Litsey, J. Stankowicz and J. Trnka, Evidence for a Nonplanar Amplituhedron, arXiv:1512.08591 [INSPIRE].
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].
R. Britto, F. Cachazo and B. Feng, Generalized unitarity and one-loop amplitudes in \( \mathcal{N}=4 \) super-Yang-Mills, Nucl. Phys. B 725 (2005) 275 [hep-th/0412103] [INSPIRE].
N. Arkani-Hamed, J.L. Bourjaily, F. Cachazo, A.B. Goncharov, A. Postnikov and J. Trnka, Scattering Amplitudes and the Positive Grassmannian, Cambridge University Press (2012).
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].
S. Caron-Huot, Loops and trees, JHEP 05 (2011) 080 [arXiv:1007.3224] [INSPIRE].
N. Arkani-Hamed, J.L. Bourjaily, F. Cachazo, S. Caron-Huot and J. Trnka, The All-Loop Integrand For Scattering Amplitudes in Planar \( \mathcal{N}=4 \) SYM, JHEP 01 (2011) 041 [arXiv:1008.2958] [INSPIRE].
E. Witten, Perturbative gauge theory as a string theory in twistor space, Commun. Math. Phys. 252 (2004) 189 [hep-th/0312171] [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, J.L. Bourjaily, F. Cachazo, A. Postnikov and J. Trnka, On-Shell Structures of MHV Amplitudes Beyond the Planar Limit, JHEP 06 (2015) 179 [arXiv:1412.8475] [INSPIRE].
S. Franco, D. Galloni, B. Penante and C. Wen, Non-Planar On-Shell Diagrams, JHEP 06 (2015) 199 [arXiv:1502.02034] [INSPIRE].
J.M. Drummond and L. Ferro, The Yangian origin of the Grassmannian integral, JHEP 12 (2010) 010 [arXiv:1002.4622] [INSPIRE].
J.M. Drummond and L. Ferro, Yangians, Grassmannians and T-duality, JHEP 07 (2010) 027 [arXiv:1001.3348] [INSPIRE].
M. Gekhtman, M. Shapiro and A. Vainshtein, Poisson Geometry of Directed Networks in an Annulus, J. Eur. Math. Soc. 14 (2012) 541 [arXiv:0901.0020].
S. Franco, D. Galloni and A. Mariotti, The Geometry of On-Shell Diagrams, JHEP 08 (2014) 038 [arXiv:1310.3820] [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].
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].
A. Molev, M. Nazarov and G. Olshansky, Yangians and classical Lie algebras, Russ. Math. Surveys 51 (1996) 205 [hep-th/9409025] [INSPIRE].
L.D. Faddeev, How algebraic Bethe ansatz works for integrable model, in proceedings of Relativistic gravitation and gravitational radiation, School of Physics, Les Houches, France, 26 September – 6 October 1995, hep-th/9605187 [INSPIRE].
R. Frassek, D. Meidinger, D. Nandan and M. Wilhelm, On-shell diagrams, Graßmannians and integrability for form factors, JHEP 01 (2016) 182 [arXiv:1506.08192] [INSPIRE].
C. Kristjansen, Review of AdS/CFT Integrability, Chapter IV.1: Aspects of Non-Planarity, Lett. Math. Phys. 99 (2012) 349 [arXiv:1012.3997] [INSPIRE].
B. Basso, J. Caetano, L. Cordova, A. Sever and P. Vieira, OPE for all Helicity Amplitudes, JHEP 08 (2015) 018 [arXiv:1412.1132] [INSPIRE].
B. Basso, S. Komatsu and P. Vieira, Structure Constants and Integrable Bootstrap in Planar N = 4 SYM Theory, arXiv:1505.06745 [INSPIRE].
N. Arkani-Hamed and J. Trnka, The Amplituhedron, JHEP 10 (2014) 030 [arXiv:1312.2007] [INSPIRE].
Z. Bajnok and R.A. Janik, String field theory vertex from integrability, JHEP 04 (2015) 042 [arXiv:1501.04533] [INSPIRE].
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Frassek, R., Meidinger, D. Yangian-type symmetries of non-planar leading singularities. J. High Energ. Phys. 2016, 110 (2016). https://doi.org/10.1007/JHEP05(2016)110
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DOI: https://doi.org/10.1007/JHEP05(2016)110