Abstract
We define a perturbatively calculable quantity — the on-shell correlator — which furnishes a unified description of particle dynamics in curved spacetime. Specializing to the case of flat and anti-de Sitter space, on-shell correlators coincide precisely with on-shell scattering amplitudes and boundary correlators, respectively. Remarkably, we find that symmetric manifolds admit a generalization of on-shell kinematics in which the corresponding momenta are literally the isometry generators of the spacetime acting on the external kinematic data. These isometric momenta are intrinsically non-commutative but exhibit on-shell conditions that are identical to those of flat space, thus providing a common language for computing and representing on-shell correlators which is agnostic about the underlying geometry. Afterwards, we compute tree-level on-shell correlators for biadjoint scalar (BAS) theory and the nonlinear sigma model (NLSM) and learn that color-kinematics duality is manifested at the level of fields under a mapping of the color algebra to the algebra of gauged isometries on the spacetime manifold. Last but not least, we present a field theoretic derivation of the fundamental BCJ relations for on-shell correlators following from the existence of certain conserved currents in BAS theory and the NLSM.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
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].
Z. Bern and A.G. Morgan, Massive loop amplitudes from unitarity, Nucl. Phys. B 467 (1996) 479 [hep-ph/9511336] [INSPIRE].
Z. Bern, L.J. Dixon and D.A. Kosower, One loop amplitudes for e+e− to four partons, Nucl. Phys. B 513 (1998) 3 [hep-ph/9708239] [INSPIRE].
R. Britto, F. Cachazo and B. Feng, Generalized unitarity and one-loop amplitudes in N = 4 super-Yang-Mills, Nucl. Phys. B 725 (2005) 275 [hep-th/0412103] [INSPIRE].
C.F. Berger et al., An Automated Implementation of On-Shell Methods for One-Loop Amplitudes, Phys. Rev. D 78 (2008) 036003 [arXiv:0803.4180] [INSPIRE].
S. Badger, C. Brønnum-Hansen, H.B. Hartanto and T. Peraro, First look at two-loop five-gluon scattering in QCD, Phys. Rev. Lett. 120 (2018) 092001 [arXiv:1712.02229] [INSPIRE].
S. Abreu, F. Febres Cordero, H. Ita, B. Page and M. Zeng, Planar Two-Loop Five-Gluon Amplitudes from Numerical Unitarity, Phys. Rev. D 97 (2018) 116014 [arXiv:1712.03946] [INSPIRE].
L.F. Alday and J.M. Maldacena, Gluon scattering amplitudes at strong coupling, JHEP 06 (2007) 064 [arXiv:0705.0303] [INSPIRE].
Z. Bern, J.J.M. Carrasco, H. Johansson and D.A. Kosower, Maximally supersymmetric planar Yang-Mills amplitudes at five loops, Phys. Rev. D 76 (2007) 125020 [arXiv:0705.1864] [INSPIRE].
Z. Bern, J.J. Carrasco, L.J. Dixon, M.R. Douglas, M. von Hippel and H. Johansson, D = 5 maximally supersymmetric Yang-Mills theory diverges at six loops, Phys. Rev. D 87 (2013) 025018 [arXiv:1210.7709] [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, Cambridge, U.K. (2016).
N. Arkani-Hamed and J. Trnka, The Amplituhedron, JHEP 10 (2014) 030 [arXiv:1312.2007] [INSPIRE].
J.J.M. Carrasco, A. Edison and H. Johansson, Maximal Super-Yang-Mills at Six Loops via Novel Integrand Bootstrap, arXiv:2112.05178 [INSPIRE].
Z. Bern, J.J.M. Carrasco, L.J. Dixon, H. Johansson and R. Roiban, Simplifying Multiloop Integrands and Ultraviolet Divergences of Gauge Theory and Gravity Amplitudes, Phys. Rev. D 85 (2012) 105014 [arXiv:1201.5366] [INSPIRE].
Z. Bern, S. Davies, T. Dennen and Y.-t. Huang, Absence of Three-Loop Four-Point Divergences in N = 4 Supergravity, Phys. Rev. Lett. 108 (2012) 201301 [arXiv:1202.3423] [INSPIRE].
Z. Bern, S. Davies, T. Dennen and Y.-t. Huang, Ultraviolet Cancellations in Half-Maximal Supergravity as a Consequence of the Double-Copy Structure, Phys. Rev. D 86 (2012) 105014 [arXiv:1209.2472] [INSPIRE].
Z. Bern, S. Davies and T. Dennen, The Ultraviolet Structure of Half-Maximal Supergravity with Matter Multiplets at Two and Three Loops, Phys. Rev. D 88 (2013) 065007 [arXiv:1305.4876] [INSPIRE].
Z. Bern, S. Davies, T. Dennen, A.V. Smirnov and V.A. Smirnov, Ultraviolet Properties of N = 4 Supergravity at Four Loops, Phys. Rev. Lett. 111 (2013) 231302 [arXiv:1309.2498] [INSPIRE].
Z. Bern, S. Davies and T. Dennen, Enhanced ultraviolet cancellations in \( \mathcal{N} \) = 5 supergravity at four loops, Phys. Rev. D 90 (2014) 105011 [arXiv:1409.3089] [INSPIRE].
Z. Bern, S. Davies and T. Dennen, The Ultraviolet Critical Dimension of Half-Maximal Supergravity at Three Loops, arXiv:1412.2441 [INSPIRE].
Z. Bern, M. Enciso, J. Parra-Martinez and M. Zeng, Manifesting enhanced cancellations in supergravity: integrands versus integrals, JHEP 05 (2017) 137 [arXiv:1703.08927] [INSPIRE].
Z. Bern, J.J.M. Carrasco, W.-M. Chen, H. Johansson, R. Roiban and M. Zeng, Five-loop four-point integrand of N = 8 supergravity as a generalized double copy, Phys. Rev. D 96 (2017) 126012 [arXiv:1708.06807] [INSPIRE].
Z. Bern et al., Ultraviolet Properties of \( \mathcal{N} \) = 8 Supergravity at Five Loops, Phys. Rev. D 98 (2018) 086021 [arXiv:1804.09311] [INSPIRE].
E. Herrmann and J. Trnka, UV cancellations in gravity loop integrands, JHEP 02 (2019) 084 [arXiv:1808.10446] [INSPIRE].
J.L. Bourjaily, E. Herrmann and J. Trnka, Maximally supersymmetric amplitudes at infinite loop momentum, Phys. Rev. D 99 (2019) 066006 [arXiv:1812.11185] [INSPIRE].
A. Edison, E. Herrmann, J. Parra-Martinez and J. Trnka, Gravity loop integrands from the ultraviolet, SciPost Phys. 10 (2021) 016 [arXiv:1909.02003] [INSPIRE].
C. Cheung, I.Z. Rothstein and M.P. Solon, From Scattering Amplitudes to Classical Potentials in the Post-Minkowskian Expansion, Phys. Rev. Lett. 121 (2018) 251101 [arXiv:1808.02489] [INSPIRE].
D.A. Kosower, B. Maybee and D. O’Connell, Amplitudes, Observables, and Classical Scattering, JHEP 02 (2019) 137 [arXiv:1811.10950] [INSPIRE].
Z. Bern, C. Cheung, R. Roiban, C.-H. Shen, M.P. Solon and M. Zeng, Scattering Amplitudes and the Conservative Hamiltonian for Binary Systems at Third Post-Minkowskian Order, Phys. Rev. Lett. 122 (2019) 201603 [arXiv:1901.04424] [INSPIRE].
A. Antonelli, A. Buonanno, J. Steinhoff, M. van de Meent and J. Vines, Energetics of two-body Hamiltonians in post-Minkowskian gravity, Phys. Rev. D 99 (2019) 104004 [arXiv:1901.07102] [INSPIRE].
Z. Bern, C. Cheung, R. Roiban, C.-H. Shen, M.P. Solon and M. Zeng, Black Hole Binary Dynamics from the Double Copy and Effective Theory, JHEP 10 (2019) 206 [arXiv:1908.01493] [INSPIRE].
Z. Bern et al., Scattering Amplitudes and Conservative Binary Dynamics at \( \mathcal{O} \)(G4), Phys. Rev. Lett. 126 (2021) 171601 [arXiv:2101.07254] [INSPIRE].
E. Herrmann, J. Parra-Martinez, M.S. Ruf and M. Zeng, Gravitational Bremsstrahlung from Reverse Unitarity, Phys. Rev. Lett. 126 (2021) 201602 [arXiv:2101.07255] [INSPIRE].
E. Herrmann, J. Parra-Martinez, M.S. Ruf and M. Zeng, Radiative classical gravitational observables at \( \mathcal{O} \)(G3) from scattering amplitudes, JHEP 10 (2021) 148 [arXiv:2104.03957] [INSPIRE].
Z. Bern et al., Scattering Amplitudes, the Tail Effect, and Conservative Binary Dynamics at O(G4), arXiv:2112.10750 [INSPIRE].
Z. Bern, J.J.M. Carrasco and H. Johansson, New Relations for Gauge-Theory Amplitudes, Phys. Rev. D 78 (2008) 085011 [arXiv:0805.3993] [INSPIRE].
H. Kawai, D.C. Lewellen and S.H.H. Tye, A Relation Between Tree Amplitudes of Closed and Open Strings, Nucl. Phys. B 269 (1986) 1 [INSPIRE].
Z. Bern, J.J.M. Carrasco and H. Johansson, Perturbative Quantum Gravity as a Double Copy of Gauge Theory, Phys. Rev. Lett. 105 (2010) 061602 [arXiv:1004.0476] [INSPIRE].
Z. Bern, J.J. Carrasco, M. Chiodaroli, H. Johansson and R. Roiban, The Duality Between Color and Kinematics and its Applications, arXiv:1909.01358 [INSPIRE].
J. Penedones, Writing CFT correlation functions as AdS scattering amplitudes, JHEP 03 (2011) 025 [arXiv:1011.1485] [INSPIRE].
A.L. Fitzpatrick, J. Kaplan, J. Penedones, S. Raju and B.C. van Rees, A Natural Language for AdS/CFT Correlators, JHEP 11 (2011) 095 [arXiv:1107.1499] [INSPIRE].
L. Rastelli and X. Zhou, Mellin amplitudes for AdS5 × S5, Phys. Rev. Lett. 118 (2017) 091602 [arXiv:1608.06624] [INSPIRE].
L. Rastelli and X. Zhou, How to Succeed at Holographic Correlators Without Really Trying, JHEP 04 (2018) 014 [arXiv:1710.05923] [INSPIRE].
C. Sleight, A Mellin Space Approach to Cosmological Correlators, JHEP 01 (2020) 090 [arXiv:1906.12302] [INSPIRE].
C. Sleight and M. Taronna, Bootstrapping Inflationary Correlators in Mellin Space, JHEP 02 (2020) 098 [arXiv:1907.01143] [INSPIRE].
C. Sleight and M. Taronna, From AdS to dS exchanges: Spectral representation, Mellin amplitudes, and crossing, Phys. Rev. D 104 (2021) L081902 [arXiv:2007.09993] [INSPIRE].
C. Sleight and M. Taronna, From dS to AdS and back, JHEP 12 (2021) 074 [arXiv:2109.02725] [INSPIRE].
S.B. Giddings, The Boundary S matrix and the AdS to CFT dictionary, Phys. Rev. Lett. 83 (1999) 2707 [hep-th/9903048] [INSPIRE].
V. Balasubramanian, S.B. Giddings and A.E. Lawrence, What do CFTs tell us about Anti-de Sitter space-times?, JHEP 03 (1999) 001 [hep-th/9902052] [INSPIRE].
S. Raju, BCFW for Witten Diagrams, Phys. Rev. Lett. 106 (2011) 091601 [arXiv:1011.0780] [INSPIRE].
M.S. Costa, V. Gonçalves and J. Penedones, Spinning AdS Propagators, JHEP 09 (2014) 064 [arXiv:1404.5625] [INSPIRE].
J. Liu, E. Perlmutter, V. Rosenhaus and D. Simmons-Duffin, d-dimensional SYK, AdS Loops, and 6j Symbols, JHEP 03 (2019) 052 [arXiv:1808.00612] [INSPIRE].
S. Giombi, C. Sleight and M. Taronna, Spinning AdS Loop Diagrams: Two Point Functions, JHEP 06 (2018) 030 [arXiv:1708.08404] [INSPIRE].
L. Di Pietro, V. Gorbenko and S. Komatsu, Analyticity and unitarity for cosmological correlators, JHEP 03 (2022) 023 [arXiv:2108.01695] [INSPIRE].
M. Hogervorst, J. Penedones and K.S. Vaziri, Towards the non-perturbative cosmological bootstrap, arXiv:2107.13871 [INSPIRE].
A.L. Fitzpatrick and J. Kaplan, Unitarity and the Holographic S-matrix, JHEP 10 (2012) 032 [arXiv:1112.4845] [INSPIRE].
L.F. Alday and S. Caron-Huot, Gravitational S-matrix from CFT dispersion relations, JHEP 12 (2018) 017 [arXiv:1711.02031] [INSPIRE].
D. Meltzer, E. Perlmutter and A. Sivaramakrishnan, Unitarity Methods in AdS/CFT, JHEP 03 (2020) 061 [arXiv:1912.09521] [INSPIRE].
D. Ponomarev, From bulk loops to boundary large-N expansion, JHEP 01 (2020) 154 [arXiv:1908.03974] [INSPIRE].
D. Meltzer and A. Sivaramakrishnan, CFT unitarity and the AdS Cutkosky rules, JHEP 11 (2020) 073 [arXiv:2008.11730] [INSPIRE].
D. Meltzer, The inflationary wavefunction from analyticity and factorization, JCAP 12 (2021) 018 [arXiv:2107.10266] [INSPIRE].
D. Meltzer, Dispersion Formulas in QFTs, CFTs, and Holography, JHEP 05 (2021) 098 [arXiv:2103.15839] [INSPIRE].
D. Baumann, W.-M. Chen, C. Duaso Pueyo, A. Joyce, H. Lee and G.L. Pimentel, Linking the Singularities of Cosmological Correlators, arXiv:2106.05294 [INSPIRE].
H. Goodhew, S. Jazayeri and E. Pajer, The Cosmological Optical Theorem, JCAP 04 (2021) 021 [arXiv:2009.02898] [INSPIRE].
S. Jazayeri, E. Pajer and D. Stefanyszyn, From locality and unitarity to cosmological correlators, JHEP 10 (2021) 065 [arXiv:2103.08649] [INSPIRE].
S. Melville and E. Pajer, Cosmological Cutting Rules, JHEP 05 (2021) 249 [arXiv:2103.09832] [INSPIRE].
H. Goodhew, S. Jazayeri, M.H. Gordon Lee and E. Pajer, Cutting cosmological correlators, JCAP 08 (2021) 003 [arXiv:2104.06587] [INSPIRE].
P. Diwakar, A. Herderschee, R. Roiban and F. Teng, BCJ amplitude relations for Anti-de Sitter boundary correlators in embedding space, JHEP 10 (2021) 141 [arXiv:2106.10822] [INSPIRE].
L. Eberhardt, S. Komatsu and S. Mizera, Scattering equations in AdS: scalar correlators in arbitrary dimensions, JHEP 11 (2020) 158 [arXiv:2007.06574] [INSPIRE].
C. Cheung and J. Mangan, Covariant color-kinematics duality, JHEP 11 (2021) 069 [arXiv:2108.02276] [INSPIRE].
S. Jain, R.R. John, A. Mehta, A.A. Nizami and A. Suresh, Double copy structure of parity-violating CFT correlators, JHEP 07 (2021) 033 [arXiv:2104.12803] [INSPIRE].
A. Bzowski, P. McFadden and K. Skenderis, Renormalised 3-point functions of stress tensors and conserved currents in CFT, JHEP 11 (2018) 153 [arXiv:1711.09105] [INSPIRE].
J.A. Farrow, A.E. Lipstein and P. McFadden, Double copy structure of CFT correlators, JHEP 02 (2019) 130 [arXiv:1812.11129] [INSPIRE].
A.E. Lipstein and P. McFadden, Double copy structure and the flat space limit of conformal correlators in even dimensions, Phys. Rev. D 101 (2020) 125006 [arXiv:1912.10046] [INSPIRE].
X. Zhou, Double Copy Relation in AdS Space, Phys. Rev. Lett. 127 (2021) 141601 [arXiv:2106.07651] [INSPIRE].
L.F. Alday, C. Behan, P. Ferrero and X. Zhou, Gluon Scattering in AdS from CFT, JHEP 06 (2021) 020 [arXiv:2103.15830] [INSPIRE].
S. Albayrak, S. Kharel and D. Meltzer, On duality of color and kinematics in (A)dS momentum space, JHEP 03 (2021) 249 [arXiv:2012.10460] [INSPIRE].
A. Sivaramakrishnan, Towards color-kinematics duality in generic spacetimes, JHEP 04 (2022) 036 [arXiv:2110.15356] [INSPIRE].
C. Armstrong, A.E. Lipstein and J. Mei, Color/kinematics duality in AdS4, JHEP 02 (2021) 194 [arXiv:2012.02059] [INSPIRE].
L.F. Alday, V. Gonçalves and X. Zhou, Super Gluon Five-Point Amplitudes in AdS Space, arXiv:2201.04422 [INSPIRE].
A. Herderschee, R. Roiban and F. Teng, On the Differential Representation and Color-Kinematics Duality of AdS Boundary Correlators, arXiv:2201.05067 [INSPIRE].
D.G. Boulware and L.S. Brown, Symmetric space scalar field theory, Annals Phys. 138 (1982) 392 [INSPIRE].
R. Camporesi, Harmonic analysis and propagators on homogeneous spaces, Phys. Rept. 196 (1990) 1 [INSPIRE].
D.G. Boulware and L.S. Brown, Tree Graphs and Classical Fields, Phys. Rev. 172 (1968) 1628 [INSPIRE].
P. Ramond, Frontiers in Physics. Vol. 74: Field Theory: A Modern Primer, Avalon Publishing, New York, U.S.A. (1997).
C. Cheung and J. Mangan, Scattering Amplitudes and the Navier-Stokes Equation, arXiv:2010.15970 [INSPIRE].
C. Cheung and Z. Moss, Symmetry and Unification from Soft Theorems and Unitarity, JHEP 05 (2021) 161 [arXiv:2012.13076] [INSPIRE].
C. Cheung, A. Helset and J. Parra-Martinez, Geometric soft theorems, JHEP 04 (2022) 011 [arXiv:2111.03045] [INSPIRE].
C. Cheung, K. Kampf, J. Novotny and J. Trnka, Effective Field Theories from Soft Limits of Scattering Amplitudes, Phys. Rev. Lett. 114 (2015) 221602 [arXiv:1412.4095] [INSPIRE].
K. Hinterbichler and A. Joyce, Hidden symmetry of the Galileon, Phys. Rev. D 92 (2015) 023503 [arXiv:1501.07600] [INSPIRE].
J. Bonifacio, K. Hinterbichler, A. Joyce and R.A. Rosen, Shift Symmetries in (Anti) de Sitter Space, JHEP 02 (2019) 178 [arXiv:1812.08167] [INSPIRE].
J. Bonifacio, K. Hinterbichler, A. Joyce and D. Roest, Exceptional scalar theories in de Sitter space, arXiv:2112.12151 [INSPIRE].
R.W. Brown and S.G. Naculich, Color-factor symmetry and BCJ relations for QCD amplitudes, JHEP 11 (2016) 060 [arXiv:1608.05291] [INSPIRE].
V. Del Duca, L.J. Dixon and F. Maltoni, New color decompositions for gauge amplitudes at tree and loop level, Nucl. Phys. B 571 (2000) 51 [hep-ph/9910563] [INSPIRE].
A. Herderschee, A New Framework for Higher Loop Witten Diagrams, arXiv:2112.08226 [INSPIRE].
H. Gomez, R.L. Jusinskas and A. Lipstein, Cosmological Scattering Equations at Tree-level and One-loop, arXiv:2112.12695 [INSPIRE].
M.F. Paulos, Towards Feynman rules for Mellin amplitudes, JHEP 10 (2011) 074 [arXiv:1107.1504] [INSPIRE].
S. Parikh, Holographic dual of the five-point conformal block, JHEP 05 (2019) 051 [arXiv:1901.01267] [INSPIRE].
C.B. Jepsen and S. Parikh, Propagator identities, holographic conformal blocks, and higher-point AdS diagrams, JHEP 10 (2019) 268 [arXiv:1906.08405] [INSPIRE].
S. Albayrak, C. Chowdhury and S. Kharel, Study of momentum space scalar amplitudes in AdS spacetime, Phys. Rev. D 101 (2020) 124043 [arXiv:2001.06777] [INSPIRE].
S. Parikh, A multipoint conformal block chain in d dimensions, JHEP 05 (2020) 120 [arXiv:1911.09190] [INSPIRE].
D. Karateev, P. Kravchuk and D. Simmons-Duffin, Weight Shifting Operators and Conformal Blocks, JHEP 02 (2018) 081 [arXiv:1706.07813] [INSPIRE].
M.S. Costa and T. Hansen, AdS Weight Shifting Operators, JHEP 09 (2018) 040 [arXiv:1805.01492] [INSPIRE].
N. Arkani-Hamed, D. Baumann, H. Lee and G.L. Pimentel, The Cosmological Bootstrap: Inflationary Correlators from Symmetries and Singularities, JHEP 04 (2020) 105 [arXiv:1811.00024] [INSPIRE].
D. Baumann, C. Duaso Pueyo, A. Joyce, H. Lee and G.L. Pimentel, The cosmological bootstrap: weight-shifting operators and scalar seeds, JHEP 12 (2020) 204 [arXiv:1910.14051] [INSPIRE].
D. Baumann, C. Duaso Pueyo, A. Joyce, H. Lee and G.L. Pimentel, The Cosmological Bootstrap: Spinning Correlators from Symmetries and Factorization, SciPost Phys. 11 (2021) 071 [arXiv:2005.04234] [INSPIRE].
A. Hillman and E. Pajer, A differential representation of cosmological wavefunctions, JHEP 04 (2022) 012 [arXiv:2112.01619] [INSPIRE].
K. Roehrig and D. Skinner, Ambitwistor strings and the scattering equations on AdS3 × S3, JHEP 02 (2022) 073 [arXiv:2007.07234] [INSPIRE].
J. Penedones, TASI lectures on AdS/CFT, in Theoretical Advanced Study Institute in Elementary Particle Physics: New Frontiers in Fields and Strings, Boulder, U.S.A. (2017), pg. 75 [arXiv:1608.04948] [INSPIRE].
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
ArXiv ePrint: 2201.05147
Rights and permissions
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.
About this article
Cite this article
Cheung, C., Parra-Martinez, J. & Sivaramakrishnan, A. On-shell correlators and color-kinematics duality in curved symmetric spacetimes. J. High Energ. Phys. 2022, 27 (2022). https://doi.org/10.1007/JHEP05(2022)027
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP05(2022)027