Abstract
The AdS boundary correlators and their dual correlation functions of boundary operators have been the main dynamic observables of the holographic duality relating a bulk AdS theory and a boundary conformal field theory. We show that tree-level AdS boundary correlators for generic states can be expressed as nonlocal differential operators of a certain structure acting on contact Witten diagrams. We further write the boundary correlators in a form that is very similar to flat space amplitudes, with Mandelstam variables replaced by certain combinations of single-state conformal generators, prove that all tree-level AdS boundary correlators have a differential representation, and detail the conversion of such differential expressions to position space. We illustrate the construction through the computation of the boundary correlators of scalars coupled to gluons and gravitons; when converted to position space, they reproduce known results. Color-kinematics duality and BCJ relations can be defined in analogy with their flat space counterparts, and are respected by the scalar correlators with a gluon exchange. We also discuss potential approaches to the double copy and find that its direct generalization may require nontrivial extensions.
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References
L. Susskind, The World as a hologram, J. Math. Phys. 36 (1995) 6377 [hep-th/9409089] [INSPIRE].
G. ’t Hooft, Dimensional reduction in quantum gravity, Conf. Proc. C 930308 (1993) 284 [gr-qc/9310026] [INSPIRE].
J.M. Maldacena, The Large N limit of superconformal field theories and supergravity, Adv. Theor. Math. Phys. 2 (1998) 231 [hep-th/9711200] [INSPIRE].
S.S. Gubser, I.R. Klebanov and A.M. Polyakov, Gauge theory correlators from noncritical string theory, Phys. Lett. B 428 (1998) 105 [hep-th/9802109] [INSPIRE].
E. Witten, Anti-de Sitter space and holography, Adv. Theor. Math. Phys. 2 (1998) 253 [hep-th/9802150] [INSPIRE].
G. Arutyunov and S. Frolov, Four point functions of lowest weight CPOs in N = 4 SYM(4) in supergravity approximation, Phys. Rev. D 62 (2000) 064016 [hep-th/0002170] [INSPIRE].
G. Arutyunov and E. Sokatchev, Implications of superconformal symmetry for interacting (2,0) tensor multiplets, Nucl. Phys. B 635 (2002) 3 [hep-th/0201145] [INSPIRE].
G. Arutyunov, F.A. Dolan, H. Osborn and E. Sokatchev, Correlation functions and massive Kaluza-Klein modes in the AdS/CFT correspondence, Nucl. Phys. B 665 (2003) 273 [hep-th/0212116] [INSPIRE].
E. D’Hoker, D.Z. Freedman and L. Rastelli, AdS/CFT four point functions: How to succeed at z integrals without really trying, Nucl. Phys. B 562 (1999) 395 [hep-th/9905049] [INSPIRE].
E. D’Hoker, D.Z. Freedman, S.D. Mathur, A. Matusis and L. Rastelli, Graviton exchange and complete four point functions in the AdS/CFT correspondence, Nucl. Phys. B 562 (1999) 353 [hep-th/9903196] [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].
A. Bzowski, P. McFadden and K. Skenderis, Implications of conformal invariance in momentum space, JHEP 03 (2014) 111 [arXiv:1304.7760] [INSPIRE].
J.A. Farrow, A.E. Lipstein and P. McFadden, Double copy structure of CFT correlators, JHEP 02 (2019) 130 [arXiv:1812.11129] [INSPIRE].
S. Albayrak and S. Kharel, Towards the higher point holographic momentum space amplitudes, JHEP 02 (2019) 040 [arXiv:1810.12459] [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].
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.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].
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].
C. Armstrong, A.E. Lipstein and J. Mei, Color/kinematics duality in AdS4, JHEP 02 (2021) 194 [arXiv:2012.02059] [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].
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].
S. Jain, R.R. John, A. Mehta, A.A. Nizami and A. Suresh, Higher spin 3-point functions in 3d CFT using spinor-helicity variables, JHEP 09 (2021) 041 [arXiv:2106.00016] [INSPIRE].
G. Mack, D-dimensional Conformal Field Theories with anomalous dimensions as Dual Resonance Models, Bulg. J. Phys. 36 (2009) 214 [arXiv:0909.1024] [INSPIRE].
G. Mack, D-independent representation of Conformal Field Theories in D dimensions via transformation to auxiliary Dual Resonance Models. Scalar amplitudes, arXiv:0907.2407 [INSPIRE].
J. Penedones, Writing CFT correlation functions as AdS scattering amplitudes, JHEP 03 (2011) 025 [arXiv:1011.1485] [INSPIRE].
M.F. Paulos, Towards Feynman rules for Mellin amplitudes, JHEP 10 (2011) 074 [arXiv:1107.1504] [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].
S. Kharel and G. Siopsis, Tree-level Correlators of scalar and vector fields in AdS/CFT, JHEP 11 (2013) 159 [arXiv:1308.2515] [INSPIRE].
J. Penedones, J.A. Silva and A. Zhiboedov, Nonperturbative Mellin Amplitudes: Existence, Properties, Applications, JHEP 08 (2020) 031 [arXiv:1912.11100] [INSPIRE].
L. Rastelli and X. Zhou, Mellin amplitudes for AdS5 × S5, Phys. Rev. Lett. 118 (2017) 091602 [arXiv:1608.06624] [INSPIRE].
X. Zhou, On Superconformal Four-Point Mellin Amplitudes in Dimension d > 2, JHEP 08 (2018) 187 [arXiv:1712.02800] [INSPIRE].
L. Rastelli and X. Zhou, How to Succeed at Holographic Correlators Without Really Trying, JHEP 04 (2018) 014 [arXiv:1710.05923] [INSPIRE].
L.F. Alday and X. Zhou, All Holographic Four-Point Functions in All Maximally Supersymmetric CFTs, Phys. Rev. X 11 (2021) 011056 [arXiv:2006.12505] [INSPIRE].
L.F. Alday and X. Zhou, All Tree-Level Correlators for M-theory on AdS7 × S4, Phys. Rev. Lett. 125 (2020) 131604 [arXiv:2006.06653] [INSPIRE].
X. Zhou, Double Copy Relation in AdS Space, Phys. Rev. Lett. 127 (2021) 141601 [arXiv:2106.07651] [INSPIRE].
L.F. Alday, A. Bissi and X. Zhou, One-loop gluon amplitudes in AdS, JHEP 02 (2022) 105 [arXiv:2110.09861] [INSPIRE].
L.F. Alday, V. Gonçalves and X. Zhou, Supersymmetric Five-Point Gluon Amplitudes in AdS Space, Phys. Rev. Lett. 128 (2022) 161601 [arXiv:2201.04422] [INSPIRE].
C. Sleight and M. Taronna, On the consistency of (partially-)massless matter couplings in de Sitter space, JHEP 10 (2021) 156 [arXiv:2106.00366] [INSPIRE].
C. Sleight and M. Taronna, Bootstrapping Inflationary Correlators in Mellin Space, JHEP 02 (2020) 098 [arXiv:1907.01143] [INSPIRE].
V. Gonçalves, J. Penedones and E. Trevisani, Factorization of Mellin amplitudes, JHEP 10 (2015) 040 [arXiv:1410.4185] [INSPIRE].
C. Sleight and M. Taronna, Spinning Mellin Bootstrap: Conformal Partial Waves, Crossing Kernels and Applications, Fortsch. Phys. 66 (2018) 1800038 [arXiv:1804.09334] [INSPIRE].
V. Gonçalves, R. Pereira and X. Zhou, 20′ Five-Point Function from AdS5 × S5 Supergravity, JHEP 10 (2019) 247 [arXiv:1906.05305] [INSPIRE].
I. Heemskerk, J. Penedones, J. Polchinski and J. Sully, Holography from Conformal Field Theory, JHEP 10 (2009) 079 [arXiv:0907.0151] [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].
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].
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].
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, JHEP 02 (2022) 073 [arXiv:2007.07234] [INSPIRE].
H. Gomez, R.L. Jusinskas and A. Lipstein, Cosmological Scattering Equations, Phys. Rev. Lett. 127 (2021) 251604 [arXiv:2106.11903] [INSPIRE].
A. Sivaramakrishnan, Towards color-kinematics duality in generic spacetimes, JHEP 04 (2022) 036 [arXiv:2110.15356] [INSPIRE].
F.A. Berends and W.T. Giele, Recursive Calculations for Processes with n Gluons, Nucl. Phys. B 306 (1988) 759 [INSPIRE].
C.R. Mafra and O. Schlotterer, Berends-Giele recursions and the BCJ duality in superspace and components, JHEP 03 (2016) 097 [arXiv:1510.08846] [INSPIRE].
S. Mizera and B. Skrzypek, Perturbiner Methods for Effective Field Theories and the Double Copy, JHEP 10 (2018) 018 [arXiv:1809.02096] [INSPIRE].
L.M. Garozzo, L. Queimada and O. Schlotterer, Berends-Giele currents in Bern-Carrasco-Johansson gauge for F3- and F4-deformed Yang-Mills amplitudes, JHEP 02 (2019) 078 [arXiv:1809.08103] [INSPIRE].
E. Bridges and C.R. Mafra, Algorithmic construction of SYM multiparticle superfields in the BCJ gauge, JHEP 10 (2019) 022 [arXiv:1906.12252] [INSPIRE].
C. Cheung and J. Mangan, Covariant color-kinematics duality, JHEP 11 (2021) 069 [arXiv:2108.02276] [INSPIRE].
C. Cheung, J. Parra-Martinez and A. Sivaramakrishnan, On-shell Correlators and Color-Kinematics Duality in Curved Symmetric Spacetimes, arXiv:2201.05147 [INSPIRE].
M.S. Costa, V. Gonçalves and J. Penedones, Spinning AdS Propagators, JHEP 09 (2014) 064 [arXiv:1404.5625] [INSPIRE].
M.S. Costa, J. Penedones, D. Poland and S. Rychkov, Spinning Conformal Correlators, JHEP 11 (2011) 071 [arXiv:1107.3554] [INSPIRE].
M. Günaydin and D. Minic, Singletons, doubletons and M-theory, Nucl. Phys. B 523 (1998) 145 [hep-th/9802047] [INSPIRE].
M. Günaydin, D. Minic and M. Zagermann, 4D doubleton conformal theories, CPT and IIB string on AdS5 × S5 , Nucl. Phys. B 534 (1998) 96 [Erratum ibid. 538 (1999) 531] [hep-th/9806042] [INSPIRE].
R. Monteiro and D. O’Connell, The Kinematic Algebra From the Self-Dual Sector, JHEP 07 (2011) 007 [arXiv:1105.2565] [INSPIRE].
C. Fronsdal, Massless Fields with Integer Spin, Phys. Rev. D 18 (1978) 3624 [INSPIRE].
D.Z. Freedman, S.D. Mathur, A. Matusis and L. Rastelli, Correlation functions in the CFT(d)/AdS(d+1) correspondence, Nucl. Phys. B 546 (1999) 96 [hep-th/9804058] [INSPIRE].
W. Mueck and K.S. Viswanathan, Conformal field theory correlators from classical scalar field theory on AdS(d+1), Phys. Rev. D 58 (1998) 041901 [hep-th/9804035] [INSPIRE].
X. Zhou, Recursion Relations in Witten Diagrams and Conformal Partial Waves, JHEP 05 (2019) 006 [arXiv:1812.01006] [INSPIRE].
L. Rastelli, K. Roumpedakis and X. Zhou, AdS3 × S3 Tree-Level Correlators: Hidden Six-Dimensional Conformal Symmetry, JHEP 10 (2019) 140 [arXiv:1905.11983] [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].
A. Hillman and E. Pajer, A differential representation of cosmological wavefunctions, JHEP 04 (2022) 012 [arXiv:2112.01619] [INSPIRE].
N. Arkani-Hamed and J. Maldacena, Cosmological Collider Physics, arXiv:1503.08043 [INSPIRE].
N. Arkani-Hamed, P. Benincasa and A. Postnikov, Cosmological Polytopes and the Wavefunction of the Universe, arXiv:1709.02813 [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].
C. Sleight, A Mellin Space Approach to Cosmological Correlators, JHEP 01 (2020) 090 [arXiv:1906.12302] [INSPIRE].
A. Hillman, Symbol Recursion for the dS Wave Function, arXiv:1912.09450 [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].
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].
D.J. Binder, D.Z. Freedman and S.S. Pufu, A bispinor formalism for spinning Witten diagrams, JHEP 02 (2022) 040 [arXiv:2003.07448] [INSPIRE].
S. Raju, Recursion Relations for AdS/CFT Correlators, Phys. Rev. D 83 (2011) 126002 [arXiv:1102.4724] [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].
A.L. Fitzpatrick and J. Kaplan, Unitarity and the Holographic S-matrix, JHEP 10 (2012) 032 [arXiv:1112.4845] [INSPIRE].
S. Caron-Huot, Analyticity in Spin in Conformal Theories, JHEP 09 (2017) 078 [arXiv:1703.00278] [INSPIRE].
D. Simmons-Duffin, D. Stanford and E. Witten, A spacetime derivation of the Lorentzian OPE inversion formula, JHEP 07 (2018) 085 [arXiv:1711.03816] [INSPIRE].
D. Meltzer, E. Perlmutter and A. Sivaramakrishnan, Unitarity Methods in AdS/CFT, JHEP 03 (2020) 061 [arXiv:1912.09521] [INSPIRE].
D. Meltzer and A. Sivaramakrishnan, CFT unitarity and the AdS Cutkosky rules, JHEP 11 (2020) 073 [arXiv:2008.11730] [INSPIRE].
D. Meltzer, Dispersion Formulas in QFTs, CFTs, and Holography, JHEP 05 (2021) 098 [arXiv:2103.15839] [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 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].
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 N = 4 super-Yang-Mills, Nucl. Phys. B 725 (2005) 275 [hep-th/0412103] [INSPIRE].
C. Sleight, Interactions in Higher-Spin Gravity: a Holographic Perspective, J. Phys. A 50 (2017) 383001 [arXiv:1610.01318] [INSPIRE].
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Herderschee, A., Roiban, R. & Teng, F. On the differential representation and color-kinematics duality of AdS boundary correlators. J. High Energ. Phys. 2022, 26 (2022). https://doi.org/10.1007/JHEP05(2022)026
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DOI: https://doi.org/10.1007/JHEP05(2022)026