Abstract
We explore color-kinematic duality for tree-level AdS/CFT correlators in momentum space. We start by studying the bi-adjoint scalar in AdS at tree-level as an illustrative example. We follow this by investigating two forms of color-kinematic duality in Yang-Mills theory, the first for the integrated correlator in AdS4 and the second for the integrand in general AdSd+1. For the integrated correlator, we find color-kinematics does not yield additional relations among n-point, color-ordered correlators. To study color-kinematics for the AdSd+1 Yang-Mills integrand, we use a spectral representation of the bulk-to-bulk propagator so that AdS diagrams are similar in structure to their flat space counterparts. Finally, we study color KLT relations for the integrated correlator and double-copy relations for the AdS integrand. We find that double-copy in AdS naturally relates the bi-adjoint theory in AdSd+3 to Yang-Mills in AdSd+1. We also find a double-copy relation at three-points between Yang-Mills in AdSd+1 and gravity in AdSd−1 and comment on the higher-point generalization. By analytic continuation, these results on AdS/CFT correlators can be translated into statements about the wave function of the universe in de Sitter.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
H. Elvang and Y.-t. Huang, Scattering Amplitudes, arXiv:1308.1697 [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. M. Maldacena, The Large N limit of superconformal field theories and supergravity, Int. J. Theor. Phys. 38 (1999) 1113 [hep-th/9711200] [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].
I. Heemskerk, J. Penedones, J. Polchinski and J. Sully, Holography from Conformal Field Theory, JHEP 10 (2009) 079 [arXiv:0907.0151] [INSPIRE].
L. F. Alday, A. Bissi and E. Perlmutter, Holographic Reconstruction of AdS Exchanges from Crossing Symmetry, JHEP 08 (2017) 147 [arXiv:1705.02318] [INSPIRE].
M. S. Costa, V. Gonçalves and J. Penedones, Spinning AdS Propagators, JHEP 09 (2014) 064 [arXiv:1404.5625] [INSPIRE].
C. Sleight and M. Taronna, Spinning Witten Diagrams, JHEP 06 (2017) 100 [arXiv:1702.08619] [INSPIRE].
A. Bzowski, P. McFadden and K. Skenderis, Implications of conformal invariance in momentum space, JHEP 03 (2014) 111 [arXiv:1304.7760] [INSPIRE].
A. Bzowski, P. McFadden and K. Skenderis, Scalar 3-point functions in CFT: renormalisation, β-functions and anomalies, JHEP 03 (2016) 066 [arXiv:1510.08442] [INSPIRE].
A. Bzowski, P. McFadden and K. Skenderis, Renormalised CFT 3-point functions of scalars, currents and stress tensors, JHEP 11 (2018) 159 [arXiv:1805.12100] [INSPIRE].
A. Bzowski, P. McFadden and K. Skenderis, Conformal n-point functions in momentum space, Phys. Rev. Lett. 124 (2020) 131602 [arXiv:1910.10162] [INSPIRE].
A. Bzowski, P. McFadden and K. Skenderis, Conformal correlators as simplex integrals in momentum space, JHEP 01 (2021) 192 [arXiv:2008.07543] [INSPIRE].
H. Isono, T. Noumi and G. Shiu, Momentum space approach to crossing symmetric CFT correlators, JHEP 07 (2018) 136 [arXiv:1805.11107] [INSPIRE].
H. Isono, T. Noumi and G. Shiu, Momentum space approach to crossing symmetric CFT correlators. Part II. General spacetime dimension, JHEP 10 (2019) 183 [arXiv:1908.04572] [INSPIRE].
C. Corianò, L. Delle Rose, E. Mottola and M. Serino, Solving the Conformal Constraints for Scalar Operators in Momentum Space and the Evaluation of Feynman’s Master Integrals, JHEP 07 (2013) 011 [arXiv:1304.6944] [INSPIRE].
C. Corianò and M. M. Maglio, Exact Correlators from Conformal Ward Identities in Momentum Space and the Perturbative T J J Vertex, Nucl. Phys. B 938 (2019) 440 [arXiv:1802.07675] [INSPIRE].
N. Anand, Z. U. Khandker and M. T. Walters, Momentum space CFT correlators for Hamiltonian truncation, JHEP 10 (2020) 095 [arXiv:1911.02573] [INSPIRE].
M. Gillioz, Conformal 3-point functions and the Lorentzian OPE in momentum space, Commun. Math. Phys. 379 (2020) 227 [arXiv:1909.00878] [INSPIRE].
J. A. Farrow, A. E. Lipstein and P. McFadden, Double copy structure of CFT correlators, JHEP 02 (2019) 130 [arXiv:1812.11129] [INSPIRE].
B. Nagaraj and D. Ponomarev, Spinor-helicity formalism for massless fields in AdS4. Part II. Potentials, JHEP 06 (2020) 068 [arXiv:1912.07494] [INSPIRE].
B. Nagaraj and D. Ponomarev, Spinor-helicity formalism for massless fields in AdS4 III: contact four-point amplitudes, JHEP 08 (2020) 012 [arXiv:2004.07989] [INSPIRE].
S. Jain, R. R. John and V. Malvimat, Momentum space spinning correlators and higher spin equations in three dimensions, JHEP 11 (2020) 049 [arXiv:2005.07212] [INSPIRE].
S. Raju, BCFW for Witten Diagrams, Phys. Rev. Lett. 106 (2011) 091601 [arXiv:1011.0780] [INSPIRE].
S. Raju, Recursion Relations for AdS/CFT Correlators, Phys. Rev. D 83 (2011) 126002 [arXiv:1102.4724] [INSPIRE].
S. Raju, New Recursion Relations and a Flat Space Limit for AdS/CFT Correlators, Phys. Rev. D 85 (2012) 126009 [arXiv:1201.6449] [INSPIRE].
S. Raju, Four Point Functions of the Stress Tensor and Conserved Currents in AdS4/CFT3, Phys. Rev. D 85 (2012) 126008 [arXiv:1201.6452] [INSPIRE].
D. Meltzer and A. Sivaramakrishnan, CFT unitarity and the AdS Cutkosky rules, JHEP 11 (2020) 073 [arXiv:2008.11730] [INSPIRE].
S. Albayrak and S. Kharel, Towards the higher point holographic momentum space amplitudes, JHEP 02 (2019) 040 [arXiv:1810.12459] [INSPIRE].
S. Albayrak, C. Chowdhury and S. Kharel, New relation for Witten diagrams, JHEP 10 (2019) 274 [arXiv:1904.10043] [INSPIRE].
S. Albayrak and S. Kharel, Towards the higher point holographic momentum space amplitudes. Part II. Gravitons, JHEP 12 (2019) 135 [arXiv:1908.01835] [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. Albayrak and S. Kharel, Spinning loop amplitudes in anti-de Sitter space, Phys. Rev. D 103 (2021) 026004 [arXiv:2006.12540] [INSPIRE].
J. M. Maldacena, Non-Gaussian features of primordial fluctuations in single field inflationary models, JHEP 05 (2003) 013 [astro-ph/0210603] [INSPIRE].
D. Harlow and D. Stanford, Operator Dictionaries and Wave Functions in AdS/CFT and dS/CFT, arXiv:1104.2621 [INSPIRE].
A. Ghosh, N. Kundu, S. Raju and S. P. Trivedi, Conformal Invariance and the Four Point Scalar Correlator in Slow-Roll Inflation, JHEP 07 (2014) 011 [arXiv:1401.1426] [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, arXiv:2005.04234 [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, arXiv:2007.09993 [INSPIRE].
E. Pajer, Building a Boostless Bootstrap for the Bispectrum, JCAP 01 (2021) 023 [arXiv:2010.12818] [INSPIRE].
E. Pajer, D. Stefanyszyn and J. Supeł, The Boostless Bootstrap: Amplitudes without Lorentz boosts, JHEP 12 (2020) 198 [arXiv:2007.00027] [INSPIRE].
J. M. Maldacena and G. L. Pimentel, On graviton non-Gaussianities during inflation, JHEP 09 (2011) 045 [arXiv:1104.2846] [INSPIRE].
T. Adamo, E. Casali, L. Mason and S. Nekovar, Plane wave backgrounds and colour-kinematics duality, JHEP 02 (2019) 198 [arXiv:1810.05115] [INSPIRE].
E. Casali and A. Puhm, Double Copy for Celestial Amplitudes, Phys. Rev. Lett. 126 (2021) 101602 [arXiv:2007.15027] [INSPIRE].
C. Armstrong, A. E. Lipstein and J. Mei, Color/kinematics duality in AdS4, JHEP 02 (2021) 194 [arXiv:2012.02059] [INSPIRE].
Y.-J. Du, B. Feng and C.-H. Fu, BCJ Relation of Color Scalar Theory and KLT Relation of Gauge Theory, JHEP 08 (2011) 129 [arXiv:1105.3503] [INSPIRE].
F. Cachazo, S. He and E. Y. Yuan, Scattering of Massless Particles: Scalars, Gluons and Gravitons, JHEP 07 (2014) 033 [arXiv:1309.0885] [INSPIRE].
C. D. White, Exact solutions for the biadjoint scalar field, Phys. Lett. B 763 (2016) 365 [arXiv:1606.04724] [INSPIRE].
N. Arkani-Hamed, P. Benincasa and A. Postnikov, Cosmological Polytopes and the Wavefunction of the Universe, arXiv:1709.02813 [INSPIRE].
A. Momeni, J. Rumbutis and A. J. Tolley, Massive Gravity from Double Copy, JHEP 12 (2020) 030 [arXiv:2004.07853] [INSPIRE].
L. A. Johnson, C. R. T. Jones and S. Paranjape, Constraints on a Massive Double-Copy and Applications to Massive Gravity, JHEP 02 (2021) 148 [arXiv:2004.12948] [INSPIRE].
Y.-t. Huang, H. Johansson and S. Lee, On Three-Algebra and Bi-Fundamental Matter Amplitudes and Integrability of Supergravity, JHEP 11 (2013) 050 [arXiv:1307.2222] [INSPIRE].
A. Sivaramakrishnan, Color-Kinematic Duality in ABJM Theory Without Amplitude Relations, Int. J. Mod. Phys. A 32 (2017) 1750002 [arXiv:1402.1821] [INSPIRE].
N. Arkani-Hamed and J. Maldacena, Cosmological Collider Physics, arXiv:1503.08043 [INSPIRE].
A. Hillman, Symbol Recursion for the dS Wave Function, arXiv:1912.09450 [INSPIRE].
S. Rychkov, EPFL Lectures on Conformal Field Theory in D ≥ 3 Dimensions, SpringerBriefs in Physics (2016) [DOI] [arXiv:1601.05000] [INSPIRE].
H. Liu and A. A. Tseytlin, On four point functions in the CFT/AdS correspondence, Phys. Rev. D 59 (1999) 086002 [hep-th/9807097] [INSPIRE].
Z. Bern, A. De Freitas and H. L. Wong, On the coupling of gravitons to matter, Phys. Rev. Lett. 84 (2000) 3531 [hep-th/9912033] [INSPIRE].
S. Y. Li, Y. Wang and S. Zhou, KLT-Like Behaviour of Inflationary Graviton Correlators, JCAP 12 (2018) 023 [arXiv:1806.06242] [INSPIRE].
N. Arkani-Hamed and P. Benincasa, On the Emergence of Lorentz Invariance and Unitarity from the Scattering Facet of Cosmological Polytopes, arXiv:1811.01125 [INSPIRE].
P. Benincasa, From the flat-space S-matrix to the Wavefunction of the Universe, arXiv:1811.02515 [INSPIRE].
P. Benincasa, Cosmological Polytopes and the Wavefuncton of the Universe for Light States, arXiv:1909.02517 [INSPIRE].
P. Benincasa, A. J. McLeod and C. Vergu, Steinmann Relations and the Wavefunction of the Universe, Phys. Rev. D 102 (2020) 125004 [arXiv:2009.03047] [INSPIRE].
T. Adamo and L. Mason, Einstein supergravity amplitudes from twistor-string theory, Class. Quant. Grav. 29 (2012) 145010 [arXiv:1203.1026] [INSPIRE].
D. Skinner, Twistor strings for \( \mathcal{N} \) = 8 supergravity, JHEP 04 (2020) 047 [arXiv:1301.0868] [INSPIRE].
T. Adamo and L. Mason, Conformal and Einstein gravity from twistor actions, Class. Quant. Grav. 31 (2014) 045014 [arXiv:1307.5043] [INSPIRE].
T. Adamo, Gravity with a cosmological constant from rational curves, JHEP 11 (2015) 098 [arXiv:1508.02554] [INSPIRE].
T. Adamo, D. Skinner and J. Williams, Twistor methods for AdS5, JHEP 08 (2016) 167 [arXiv:1607.03763] [INSPIRE].
B. Nagaraj and D. Ponomarev, Spinor-Helicity Formalism for Massless Fields in AdS4, Phys. Rev. Lett. 122 (2019) 101602 [arXiv:1811.08438] [INSPIRE].
F. Cachazo, S. He and E. Y. Yuan, Scattering equations and Kawai-Lewellen-Tye orthogonality, Phys. Rev. D 90 (2014) 065001 [arXiv:1306.6575] [INSPIRE].
F. Cachazo, S. He and E. Y. Yuan, Scattering of Massless Particles in Arbitrary Dimensions, Phys. Rev. Lett. 113 (2014) 171601 [arXiv:1307.2199] [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].
K. Roehrig and D. Skinner, Ambitwistor Strings and the Scattering Equations on AdS3 × S3, arXiv:2007.07234 [INSPIRE].
G. Chen and Y.-J. Du, Amplitude Relations in Non-linear Sigma Model, JHEP 01 (2014) 061 [arXiv:1311.1133] [INSPIRE].
J. J. M. Carrasco, C. R. Mafra and O. Schlotterer, Semi-abelian Z-theory: N LSM + ϕ3 from the open string, JHEP 08 (2017) 135 [arXiv:1612.06446] [INSPIRE].
C. Cheung and C.-H. Shen, Symmetry for Flavor-Kinematics Duality from an Action, Phys. Rev. Lett. 118 (2017) 121601 [arXiv:1612.00868] [INSPIRE].
A. L. Fitzpatrick and J. Kaplan, Unitarity and the Holographic S-matrix, JHEP 10 (2012) 032 [arXiv:1112.4845] [INSPIRE].
O. Aharony, L. F. Alday, A. Bissi and E. Perlmutter, Loops in AdS from Conformal Field Theory, JHEP 07 (2017) 036 [arXiv:1612.03891] [INSPIRE].
S. Caron-Huot, Analyticity in Spin in Conformal Theories, JHEP 09 (2017) 078 [arXiv:1703.00278] [INSPIRE].
D. Ponomarev, From bulk loops to boundary large-N expansion, JHEP 01 (2020) 154 [arXiv:1908.03974] [INSPIRE].
D. Meltzer, E. Perlmutter and A. Sivaramakrishnan, Unitarity Methods in AdS/CFT, JHEP 03 (2020) 061 [arXiv:1912.09521] [INSPIRE].
A. Costantino and S. Fichet, Opacity from Loops in AdS, JHEP 02 (2021) 089 [arXiv:2011.06603] [INSPIRE].
H. Goodhew, S. Jazayeri and E. Pajer, The Cosmological Optical Theorem, arXiv:2009.02898 [INSPIRE].
S. Céspedes, A.-C. Davis and S. Melville, On the time evolution of cosmological correlators, JHEP 02 (2021) 012 [arXiv:2009.07874] [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].
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: 2012.10460
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
Albayrak, S., Kharel, S. & Meltzer, D. On duality of color and kinematics in (A)dS momentum space. J. High Energ. Phys. 2021, 249 (2021). https://doi.org/10.1007/JHEP03(2021)249
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP03(2021)249