Abstract
Scattering amplitudes in 4d \( \mathcal{N}=4 \) super Yang-Mills theory (SYM) can be described by Grassmannian contour integrals whose form depends on whether the external data is encoded in momentum space, twistor space, or momentum twistor space. After a pedagogical review, we present a new, streamlined proof of the equivalence of the three integral formulations. A similar strategy allows us to derive a new Grassmannian integral for 3d \( \mathcal{N}=6 \) ABJM theory amplitudes in momentum twistor space: it is a contour integral in an orthogonal Grassmannian with the novel property that the internal metric depends on the external data. The result can be viewed as a central step towards developing an amplituhedron formulation for ABJM amplitudes. Various properties of Grassmannian integrals are examined, including boundary properties, pole structure, and a homological interpretation of the global residue theorems for \( \mathcal{N}=4 \) SYM.
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References
N. Arkani-Hamed, F. Cachazo, C. Cheung and J. Kaplan, A duality for the S matrix, JHEP 03 (2010) 020 [arXiv:0907.5418] [INSPIRE].
L.J. Mason and D. Skinner, Dual superconformal invariance, momentum twistors and Grassmannians, JHEP 11 (2009) 045 [arXiv:0909.0250] [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 et al., Scattering amplitudes and the positive grassmannian, arXiv:1212.5605 [INSPIRE].
N. Arkani-Hamed and J. Trnka, The amplituhedron, JHEP 1410 (2014) 30 [arXiv:1312.2007] [INSPIRE].
N. Arkani-Hamed and J. Trnka, Into the amplituhedron, arXiv:1312.7878 [INSPIRE].
N. Arkani-Hamed, J.L. Bourjaily, F. Cachazo, A. Hodges and J. Trnka, A note on polytopes for scattering amplitudes, JHEP 04 (2012) 081 [arXiv:1012.6030] [INSPIRE].
O. Aharony, O. Bergman, D.L. Jafferis and J. Maldacena, N = 6 superconformal Chern-Simons-matter theories, M2-branes and their gravity duals, JHEP 10 (2008) 091 [arXiv:0806.1218] [INSPIRE].
K. Hosomichi, K.-M. Lee, S. Lee, S. Lee and J. Park, N = 5, 6 superconformal Chern-Simons theories and M2-branes on orbifolds, JHEP 09 (2008) 002 [arXiv:0806.4977] [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].
S. Lee, Yangian invariant scattering amplitudes in supersymmetric Chern-Simons theory, Phys. Rev. Lett. 105 (2010) 151603 [arXiv:1007.4772] [INSPIRE].
Y.-T. Huang and C. Wen, ABJM amplitudes and the positive orthogonal Grassmannian, JHEP 02 (2014) 104 [arXiv:1309.3252] [INSPIRE].
Y.-t. Huang, C. Wen and D. Xie, The positive orthogonal Grassmannian and loop amplitudes of ABJM, arXiv:1402.1479 [INSPIRE].
J. Kim and S. Lee, Positroid stratification of orthogonal Grassmannian and ABJM Amplitudes, JHEP 09 (2014) 085 [arXiv:1402.1119] [INSPIRE].
A. Henriques and D. Speyer, The multidimensional cube recurrence, Adv. Math. 223 (2010) 1107.
T. Lam, Electroid varieties and a compactification of the space of electrical networks, arXiv:1402.6261.
H. Elvang and Y.-t. Huang, Scattering Amplitudes, to be published as a textbook with Cambridge University Press, arXiv:1308.1697 [INSPIRE].
S.J. Parke and T.R. Taylor, An amplitude for n gluon scattering, Phys. Rev. Lett. 56 (1986) 2459 [INSPIRE].
V.P. Nair, A current algebra for some gauge theory amplitudes, Phys. Lett. B 214 (1988) 215 [INSPIRE].
I.M. Gel’fand and G.E. Shilov, Generalized functions. Volume 1. Properties and operations, Academic Press, New York U.S.A. (1977).
L.J. Mason and D. Skinner, Scattering amplitudes and BCFW recursion in twistor space, JHEP 01 (2010) 064 [arXiv:0903.2083] [INSPIRE].
A. Hodges, Eliminating spurious poles from gauge-theoretic amplitudes, JHEP 05 (2013) 135 [arXiv:0905.1473] [INSPIRE].
P. Orlik and H. Terao, Arrangements of hyperplanes, Springer, Germany (1992).
A. Brandhuber, P. Heslop and G. Travaglini, One-loop amplitudes in N = 4 super Yang-Mills and anomalous dual conformal symmetry, JHEP 08 (2009) 095 [arXiv:0905.4377] [INSPIRE].
H. Elvang, D.Z. Freedman and M. Kiermaier, Dual conformal symmetry of 1-loop NMHV amplitudes in N = 4 SYM theory, JHEP 03 (2010) 075 [arXiv:0905.4379] [INSPIRE].
J.M. Drummond and J.M. Henn, All tree-level amplitudes in N = 4 SYM, JHEP 04 (2009) 018 [arXiv:0808.2475] [INSPIRE].
M. Bullimore, L.J. Mason and D. Skinner, Twistor-strings, grassmannians and leading singularities, JHEP 03 (2010) 070 [arXiv:0912.0539] [INSPIRE].
J. Brodel and S. He, Dual conformal constraints and infrared equations from global residue theorems in N = 4 SYM theory, JHEP 06 (2010) 054 [arXiv:1004.2400] [INSPIRE].
Y. Bai and S. He, The amplituhedron from momentum twistor diagrams, arXiv:1408.2459 [INSPIRE].
T. Bargheer, F. Loebbert and C. Meneghelli, Symmetries of tree-level scattering amplitudes in N = 6 superconformal Chern-Simons theory, Phys. Rev. D 82 (2010) 045016 [arXiv:1003.6120] [INSPIRE].
D. Gang, Y.-t. Huang, E. Koh, S. Lee and A.E. Lipstein, Tree-level recursion relation and dual superconformal symmetry of the ABJM theory, JHEP 03 (2011) 116 [arXiv:1012.5032] [INSPIRE].
A.E. Lipstein and L. Mason, Amplitudes of 3D Yang-Mills Theory, JHEP 01 (2013) 009 [arXiv:1207.6176] [INSPIRE].
P.A.M. Dirac, Wave equations in conformal space, Annals Math. 37 (1936) 429.
G. Mack and A. Salam, Finite component field representations of the conformal group, Annals Phys. 53 (1969) 174 [INSPIRE].
S.L. Adler, Massless, Euclidean quantum electrodynamics on the five-dimensional unit hypersphere, Phys. Rev. D 6 (1972) 3445 [Erratum ibid. D 7 (1973) 3821] [INSPIRE].
R. Marnelius and B.E.W. Nilsson, Manifestly conformally covariant field equations and a possible origin of the Higgs mechanism, Phys. Rev. D 22 (1980) 830 [INSPIRE].
T. Goddard, P. Heslop and V.V. Khoze, Uplifting amplitudes in special kinematics, JHEP 10 (2012) 041 [arXiv:1205.3448] [INSPIRE].
S. Caron-Huot and S. He, Three-loop octagons and n-gons in maximally supersymmetric Yang-Mills theory, JHEP 08 (2013) 101 [arXiv:1305.2781] [INSPIRE].
D. Gang, Y. -t. Huang, E. Koh, S. Lee and A.E. Lipstein, unpublished notes.
Y.-t. Huang and S. Lee, A new integral formula for supersymmetric scattering amplitudes in three dimensions, Phys. Rev. Lett. 109 (2012) 191601 [arXiv:1207.4851] [INSPIRE].
O.T. Engelund and R. Roiban, A twistor string for the ABJ(M) theory, JHEP 06 (2014) 088 [arXiv:1401.6242] [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].
T.M. Olson, Orientations of BCFW charts on the Grassmannian, in preparation.
P.A. Griffiths and J. Harris, Principles of algebraic geometry, John Wiley & Sons, U.S.A. (1978).
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Elvang, H., Huang, Yt., Keeler, C. et al. Grassmannians for scattering amplitudes in 4d \( \mathcal{N}=4 \) SYM and 3d ABJM. J. High Energ. Phys. 2014, 181 (2014). https://doi.org/10.1007/JHEP12(2014)181
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DOI: https://doi.org/10.1007/JHEP12(2014)181