Abstract
We derive an on-shell diagram recursion for tree-level scattering amplitudes in \( \mathcal{N} \) = 7 supergravity. The diagrams are evaluated in terms of Grassmannian integrals and momentum twistors, generalising previous results of Hodges in momentum twistor space to non-MHV amplitudes. In particular, we recast five and six-point NMHV amplitudes in terms of \( \mathcal{N} \) = 7 R-invariants analogous to those of \( \mathcal{N} \) = 4 super-Yang-Mills, which makes cancellation of spurious poles more transparent. Above 5-points, this requires defining momentum twistors with respect to different orderings of the external momenta.
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References
N. Arkani-Hamed and J. Trnka, The Amplituhedron, JHEP 10 (2014) 030 [arXiv:1312.2007] [INSPIRE].
N. Arkani-Hamed, H. Thomas and J. Trnka, Unwinding the Amplituhedron in Binary, JHEP 01 (2018) 016 [arXiv:1704.05069] [INSPIRE].
A. Hodges, Eliminating spurious poles from gauge-theoretic amplitudes, JHEP 05 (2013) 135 [arXiv:0905.1473] [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].
N. Arkani-Hamed, J.L. Bourjaily, F. Cachazo, A.B. Goncharov, A. Postnikov and J. Trnka, Grassmannian Geometry of Scattering Amplitudes, Cambridge University Press (2016), https://doi.org/10.1017/CBO9781316091548 [arXiv:1212.5605] [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].
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].
R.H. Boels, On BCFW shifts of integrands and integrals, JHEP 11 (2010) 113 [arXiv:1008.3101] [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].
A.E. Lipstein and L. Mason, From d logs to dilogs the super Yang-Mills MHV amplitude revisited, JHEP 01 (2014) 169 [arXiv:1307.1443] [INSPIRE].
P. Benincasa, On-shell diagrammatics and the perturbative structure of planar gauge theories, arXiv:1510.03642 [INSPIRE].
P. Benincasa and D. Gordo, On-shell diagrams and the geometry of planar \( \mathcal{N} \) < 4 SYM theories, JHEP 11 (2017) 192 [arXiv:1609.01923] [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.L. Bourjaily, S. Franco, D. Galloni and C. Wen, Stratifying On-Shell Cluster Varieties: the Geometry of Non-Planar On-Shell Diagrams, JHEP 10 (2016) 003 [arXiv:1607.01781] [INSPIRE].
L. Ferro and T. Lukowski, Amplituhedra, and Beyond, J. Phys. A 54 (2021) 033001 [arXiv:2007.04342] [INSPIRE].
J. Bedford, A. Brandhuber, B.J. Spence and G. Travaglini, A Recursion relation for gravity amplitudes, Nucl. Phys. B 721 (2005) 98 [hep-th/0502146] [INSPIRE].
F. Cachazo and P. Svrček, Tree level recursion relations in general relativity, hep-th/0502160 [INSPIRE].
P. Heslop and A.E. Lipstein, On-shell diagrams for \( \mathcal{N} \) = 8 supergravity amplitudes, JHEP 06 (2016) 069 [arXiv:1604.03046] [INSPIRE].
E. Herrmann and J. Trnka, Gravity On-shell Diagrams, JHEP 11 (2016) 136 [arXiv:1604.03479] [INSPIRE].
A. Hodges, New expressions for gravitational scattering amplitudes, JHEP 07 (2013) 075 [arXiv:1108.2227] [INSPIRE].
A. Hodges, A simple formula for gravitational MHV amplitudes, arXiv:1204.1930 [INSPIRE].
F.A. Berends, W.T. Giele and H. Kuijf, On relations between multi - gluon and multigraviton scattering, Phys. Lett. B 211 (1988) 91 [INSPIRE].
D. Skinner, Twistor strings for \( \mathcal{N} \) = 8 supergravity, JHEP 04 (2020) 047 [arXiv:1301.0868] [INSPIRE].
Y. Geyer, A.E. Lipstein and L.J. Mason, Ambitwistor Strings in Four Dimensions, Phys. Rev. Lett. 113 (2014) 081602 [arXiv:1404.6219] [INSPIRE].
J.A. Farrow and A.E. Lipstein, From 4d Ambitwistor Strings to On Shell Diagrams and Back, JHEP 07 (2017) 114 [arXiv:1705.07087] [INSPIRE].
N. Arkani-Hamed, F. Cachazo and J. Kaplan, What is the Simplest Quantum Field Theory?, JHEP 09 (2010) 016 [arXiv:0808.1446] [INSPIRE].
J.-Y. Liu and E. Shih, Bonus scaling and BCFW in \( \mathcal{N} \) = 7 supergravity, Phys. Lett. B 740 (2015) 151 [arXiv:1409.1710] [INSPIRE].
J.A. Farrow, Y. Geyer, A.E. Lipstein, R. Monteiro and R. Stark-Muchão, Propagators, BCFW recursion and new scattering equations at one loop, JHEP 10 (2020) 074 [arXiv:2007.00623] [INSPIRE].
S. He, D. Nandan and C. Wen, Note on Bonus Relations for N = 8 Supergravity Tree Amplitudes, JHEP 02 (2011) 005 [arXiv:1011.4287] [INSPIRE].
S. Lal and S. Raju, The Next-to-Simplest Quantum Field Theories, Phys. Rev. D 81 (2010) 105002 [arXiv:0910.0930] [INSPIRE].
H. Elvang, Y.-t. Huang and C. Peng, On-shell superamplitudes in N < 4 SYM, JHEP 09 (2011) 031 [arXiv:1102.4843] [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].
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].
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].
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].
Y.-t. Huang and A.E. Lipstein, Dual Superconformal Symmetry of N = 6 Chern-Simons Theory, JHEP 11 (2010) 076 [arXiv:1008.0041] [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].
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].
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, C. Cheung and J. Kaplan, A Duality For The S Matrix, JHEP 03 (2010) 020 [arXiv:0907.5418] [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].
D. Nandan, A. Volovich and C. Wen, A Grassmannian Etude in NMHV Minors, JHEP 07 (2010) 061 [arXiv:0912.3705] [INSPIRE].
J.M. Drummond, M. Spradlin, A. Volovich and C. Wen, Tree-Level Amplitudes in N = 8 Supergravity, Phys. Rev. D 79 (2009) 105018 [arXiv:0901.2363] [INSPIRE].
P. Benincasa and M. Parisi, Positive geometries and differential forms with non-logarithmic singularities. Part I, JHEP 08 (2020) 023 [arXiv:2005.03612] [INSPIRE].
J. Trnka, private communication.
L.F. Alday and J.M. Maldacena, Gluon scattering amplitudes at strong coupling, JHEP 06 (2007) 064 [arXiv:0705.0303] [INSPIRE].
A. Brandhuber, P. Heslop and G. Travaglini, MHV amplitudes in N = 4 super Yang-Mills and Wilson loops, Nucl. Phys. B 794 (2008) 231 [arXiv:0707.1153] [INSPIRE].
J.M. Drummond, J. Henn, G.P. Korchemsky and E. Sokatchev, On planar gluon amplitudes/Wilson loops duality, Nucl. Phys. B 795 (2008) 52 [arXiv:0709.2368] [INSPIRE].
L.J. Mason and D. Skinner, The Complete Planar S-matrix of N = 4 SYM as a Wilson Loop in Twistor Space, JHEP 12 (2010) 018 [arXiv:1009.2225] [INSPIRE].
S. Caron-Huot, Notes on the scattering amplitude/Wilson loop duality, JHEP 07 (2011) 058 [arXiv:1010.1167] [INSPIRE].
A. Brandhuber, P. Heslop, A. Nasti, B. Spence and G. Travaglini, Four-point Amplitudes in N = 8 Supergravity and Wilson Loops, Nucl. Phys. B 807 (2009) 290 [arXiv:0805.2763] [INSPIRE].
E. Herrmann, C. Langer, J. Trnka and M. Zheng, Positive geometry, local triangulations, and the dual of the Amplituhedron, JHEP 01 (2021) 035 [arXiv:2009.05607] [INSPIRE].
J.L. Bourjaily, E. Herrmann, C. Langer, A.J. McLeod and J. Trnka, Prescriptive Unitarity for Non-Planar Six-Particle Amplitudes at Two Loops, JHEP 12 (2019) 073 [arXiv:1909.09131] [INSPIRE].
N. Berkovits and E. Witten, Conformal supergravity in twistor-string theory, JHEP 08 (2004) 009 [hep-th/0406051] [INSPIRE].
T. Adamo and L. Mason, Einstein supergravity amplitudes from twistor-string theory, Class. Quant. Grav. 29 (2012) 145010 [arXiv:1203.1026] [INSPIRE].
L. Dolan and J.N. Ihry, Conformal Supergravity Tree Amplitudes from Open Twistor String Theory, Nucl. Phys. B 819 (2009) 375 [arXiv:0811.1341] [INSPIRE].
T. Adamo and L. Mason, Conformal and Einstein gravity from twistor actions, Class. Quant. Grav. 31 (2014) 045014 [arXiv:1307.5043] [INSPIRE].
J.A. Farrow and A.E. Lipstein, New Worldsheet Formulae for Conformal Supergravity Amplitudes, JHEP 07 (2018) 074 [arXiv:1805.04504] [INSPIRE].
Z. Bern, D. Kosower and J. Parra-Martinez, Two-loop n-point anomalous amplitudes in N = 4 supergravity, Proc. Roy. Soc. Lond. A 476 (2020) 20190722 [arXiv:1905.05151] [INSPIRE].
H. Johansson and J. Nohle, Conformal Gravity from Gauge Theory, arXiv:1707.02965 [INSPIRE].
H. Johansson, G. Mogull and F. Teng, Unraveling conformal gravity amplitudes, JHEP 09 (2018) 080 [arXiv:1806.05124] [INSPIRE].
J.A. Farrow, A Monte Carlo Approach to the 4D Scattering Equations, JHEP 08 (2018) 085 [arXiv:1806.02732] [INSPIRE].
J.L. Bourjaily, Positroids, Plabic Graphs, and Scattering Amplitudes in Mathematica, arXiv:1212.6974 [INSPIRE].
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Armstrong, C., Farrow, J.A. & Lipstein, A.E. \( \mathcal{N} \) = 7 On-shell diagrams and supergravity amplitudes in momentum twistor space. J. High Energ. Phys. 2021, 181 (2021). https://doi.org/10.1007/JHEP01(2021)181
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DOI: https://doi.org/10.1007/JHEP01(2021)181