Abstract
The eikonal phase which determines the Regge limit of the gravitational scat- tering amplitude of a light particle off a heavy one in Minkowski spacetimes admits an expansion in the ratio of the Schwarzschild radius of the heavy particle to the impact parameter. Such an eikonal phase in AdS spacetimes of any dimensionality has been com- puted to all orders and reduces to the corresponding Minkowski result when both the impact parameter and the Schwarzschild radius are much smaller than the AdS radius. The leading term in the AdS eikonal phase can be reproduced in the dual CFT by a single stress tensor conformal block, but the subleading term is a result of an infinite sum of the double stress tensor contributions. We provide a closed form expression for the OPE coef- ficients of the leading twist double stress tensors in four spacetime dimensions and perform the sum to compute the corresponding lightcone behavior of a heavy-heavy-light-light CFT correlator. The resulting compact expression passes a few nontrivial independent checks. In particular, it agrees with the subleading eikonal phase at large impact parameter.
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References
C.M. Will, The Confrontation between General Relativity and Experiment, Living Rev. Rel.17 (2014) 4 [arXiv:1403.7377] [INSPIRE].
S.B. Giddings, The gravitational S-matrix: Erice lectures, Subnucl. Ser.48 (2013) 93 [arXiv:1105.2036] [INSPIRE].
T. Dray and G. ‘t Hooft, The Gravitational Shock Wave of a Massless Particle, Nucl. Phys.B 253 (1985) 173 [INSPIRE].
G. ‘t Hooft, Graviton Dominance in Ultrahigh-Energy Scattering, Phys. Lett.B 198 (1987) 61 [INSPIRE].
D. Amati, M. Ciafaloni and G. Veneziano, Superstring Collisions at Planckian Energies, Phys. Lett.B 197 (1987) 81 [INSPIRE].
I.J. Muzinich and M. Soldate, High-Energy Unitarity of Gravitation and Strings, Phys. Rev.D 37 (1988) 359 [INSPIRE].
B. Sundborg, High-energy Asymptotics: The One Loop String Amplitude and Resummation, Nucl. Phys.B 306 (1988) 545 [INSPIRE].
D.N. Kabat and M. Ortiz, Eikonal quantum gravity and Planckian scattering, Nucl. Phys.B 388 (1992) 570 [hep-th/9203082] [INSPIRE].
G. D’Appollonio, P. Di Vecchia, R. Russo and G. Veneziano, High-energy string-brane scattering: Leading eikonal and beyond, JHEP11 (2010) 100 [arXiv:1008.4773] [INSPIRE].
D. Neill and I.Z. Rothstein, Classical Space-Times from the S Matrix, Nucl. Phys.B 877 (2013) 177 [arXiv:1304.7263] [INSPIRE].
R. Akhoury, R. Saotome and G. Sterman, High Energy Scattering in Perturbative Quantum Gravity at Next to Leading Power, arXiv:1308.5204 [INSPIRE].
N.E.J. Bjerrum-Bohr, J.F. Donoghue, B.R. Holstein, L. Planté and P. Vanhove, Bending of Light in Quantum Gravity, Phys. Rev. Lett.114 (2015) 061301 [arXiv:1410.7590] [INSPIRE].
N.E.J. Bjerrum-Bohr, J.F. Donoghue, B.R. Holstein, L. Plante and P. Vanhove, Light-like Scattering in Quantum Gravity, JHEP11 (2016) 117 [arXiv:1609.07477] [INSPIRE].
A. Luna, S. Melville, S.G. Naculich and C.D. White, Next-to-soft corrections to high energy scattering in QCD and gravity, JHEP01 (2017) 052 [arXiv:1611.02172] [INSPIRE].
F. Cachazo and A. Guevara, Leading Singularities and Classical Gravitational Scattering, arXiv:1705.10262 [INSPIRE].
N.E.J. Bjerrum-Bohr, P.H. Damgaard, G. Festuccia, L. Planté and P. Vanhove, General Relativity from Scattering Amplitudes, Phys. Rev. Lett.121 (2018) 171601 [arXiv:1806.04920] [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, JHEP02 (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. Koemans Collado, P. Di Vecchia and R. Russo, Revisiting the second post-Minkowskian eikonal and the dynamics of binary black holes, Phys. Rev.D 100 (2019) 066028 [arXiv:1904.02667] [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].
E. Witten, Anti-de Sitter space and holography, Adv. Theor. Math. Phys.2 (1998) 253 [hep-th/9802150] [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].
L. Cornalba, M.S. Costa, J. Penedones and R. Schiappa, Eikonal Approximation in AdS/CFT: From Shock Waves to Four-Point Functions, JHEP08 (2007) 019 [hep-th/0611122] [INSPIRE].
L. Cornalba, M.S. Costa, J. Penedones and R. Schiappa, Eikonal Approximation in AdS/CFT: Conformal Partial Waves and Finite N Four-Point Functions, Nucl. Phys.B 767 (2007) 327 [hep-th/0611123] [INSPIRE].
L. Cornalba, M.S. Costa and J. Penedones, Eikonal approximation in AdS/CFT: Resumming the gravitational loop expansion, JHEP09 (2007) 037 [arXiv:0707.0120] [INSPIRE].
L. Cornalba, Eikonal methods in AdS/CFT: Regge theory and multi-reggeon exchange, arXiv:0710.5480 [INSPIRE].
M.S. Costa, V. Goncalves and J. Penedones, Conformal Regge theory, JHEP12 (2012) 091 [arXiv:1209.4355] [INSPIRE].
X.O. Camanho, J.D. Edelstein, J. Maldacena and A. Zhiboedov, Causality Constraints on Corrections to the Graviton Three-Point Coupling, JHEP02 (2016) 020 [arXiv:1407.5597] [INSPIRE].
M. Kulaxizi, A. Parnachev and A. Zhiboedov, Bulk Phase Shift, CFT Regge Limit and Einstein Gravity, JHEP06 (2018) 121 [arXiv:1705.02934] [INSPIRE].
M.S. Costa, T. Hansen and J. Penedones, Bounds for OPE coefficients on the Regge trajectory, JHEP10 (2017) 197 [arXiv:1707.07689] [INSPIRE].
N. Afkhami-Jeddi, T. Hartman, S. Kundu and A. Tajdini, Einstein gravity 3-point functions from conformal field theory, JHEP12 (2017) 049 [arXiv:1610.09378] [INSPIRE].
D. Li, D. Meltzer and D. Poland, Conformal Bootstrap in the Regge Limit, JHEP12 (2017) 013 [arXiv:1705.03453] [INSPIRE].
N. Afkhami-Jeddi, T. Hartman, S. Kundu and A. Tajdini, Shockwaves from the Operator Product Expansion, JHEP03 (2019) 201 [arXiv:1709.03597] [INSPIRE].
D. Meltzer and E. Perlmutter, Beyond a = c: gravitational couplings to matter and the stress tensor OPE, JHEP07 (2018) 157 [arXiv:1712.04861] [INSPIRE].
D. Meltzer, Higher Spin ANEC and the Space of CFTs, JHEP07 (2019) 001 [arXiv:1811.01913] [INSPIRE].
N. Afkhami-Jeddi, S. Kundu and A. Tajdini, A Bound on Massive Higher Spin Particles, JHEP04 (2019) 056 [arXiv:1811.01952] [INSPIRE].
A. Belin, D.M. Hofman and G. Mathys, Einstein gravity from ANEC correlators, JHEP08 (2019) 032 [arXiv:1904.05892] [INSPIRE].
M. Kologlu, P. Kravchuk, D. Simmons-Duffin and A. Zhiboedov, Shocks, Superconvergence and a Stringy Equivalence Principle, arXiv:1904.05905 [INSPIRE].
M. Kulaxizi, G.S. Ng and A. Parnachev, Black Holes, Heavy States, Phase Shift and Anomalous Dimensions, SciPost Phys.6 (2019) 065 [arXiv:1812.03120] [INSPIRE].
R. Karlsson, M. Kulaxizi, A. Parnachev and P. Tadíc, Black Holes and Conformal Regge Bootstrap, arXiv:1904.00060 [INSPIRE].
A.L. Fitzpatrick, J. Kaplan and M.T. Walters, Universality of Long-Distance AdS Physics from the CFT Bootstrap, JHEP08 (2014) 145 [arXiv:1403.6829] [INSPIRE].
A.L. Fitzpatrick, J. Kaplan and M.T. Walters, Virasoro Conformal Blocks and Thermality from Classical Background Fields, JHEP11 (2015) 200 [arXiv:1501.05315] [INSPIRE].
S. Collier, Y. Gobeil, H. Maxfield and E. Perlmutter, Quantum Regge Trajectories and the Virasoro Analytic Bootstrap, JHEP05 (2019) 212 [arXiv:1811.05710] [INSPIRE].
A.L. Fitzpatrick and K.-W. Huang, Universal Lowest-Twist in CFTs from Holography, JHEP08 (2019) 138 [arXiv:1903.05306] [INSPIRE].
Y.-Z. Li, Z.-F. Mai and H. Lü, Holographic OPE Coefficients from AdS Black Holes with Matters, JHEP09 (2019) 001 [arXiv:1905.09302] [INSPIRE].
Z. Komargodski and A. Zhiboedov, Convexity and Liberation at Large Spin, JHEP11 (2013) 140 [arXiv:1212.4103] [INSPIRE].
A.L. Fitzpatrick, J. Kaplan, D. Poland and D. Simmons-Duffin, The Analytic Bootstrap and AdS Superhorizon Locality, JHEP12 (2013) 004 [arXiv:1212.3616] [INSPIRE].
A.L. Fitzpatrick and J. Kaplan, Unitarity and the Holographic S-matrix, JHEP10 (2012) 032 [arXiv:1112.4845] [INSPIRE].
A.L. Fitzpatrick, J. Kaplan, M.T. Walters and J. Wang, Eikonalization of Conformal Blocks, JHEP09 (2015) 019 [arXiv:1504.01737] [INSPIRE].
H. Maxfield, A view of the bulk from the worldline, arXiv:1712.00885 [INSPIRE].
A.L. Fitzpatrick, J. Kaplan, M.T. Walters and J. Wang, Hawking from Catalan, JHEP05 (2016) 069 [arXiv:1510.00014] [INSPIRE].
J. Penedones, Writing CFT correlation functions as AdS scattering amplitudes, JHEP03 (2011) 025 [arXiv:1011.1485] [INSPIRE].
J.A. Minahan and R. Pereira, Three-point correlators from string amplitudes: Mixing and Regge spins, JHEP04 (2015) 134 [arXiv:1410.4746] [INSPIRE].
G. Arutyunov, S. Frolov and A.C. Petkou, Operator product expansion of the lowest weight CPOs in \( \mathcal{N} \) = 4 SYM4 at strong coupling, Nucl. Phys.B 586 (2000) 547 [Erratum ibid.B 609 (2001) 539] [hep-th/0005182] [INSPIRE].
V.V. Bazhanov, S.L. Lukyanov and A.B. Zamolodchikov, Integrable structure of conformal field theory, quantum KdV theory and thermodynamic Bethe ansatz, Commun. Math. Phys.177 (1996) 381 [hep-th/9412229] [INSPIRE].
J. de Boer and D. Engelhardt, Remarks on thermalization in 2D CFT, Phys. Rev.D 94 (2016) 126019 [arXiv:1604.05327] [INSPIRE].
A. Maloney, G.S. Ng, S.F. Ross and I. Tsiares, Thermal Correlation Functions of KdV Charges in 2D CFT, JHEP02 (2019) 044 [arXiv:1810.11053] [INSPIRE].
A. Maloney, G.S. Ng, S.F. Ross and I. Tsiares, Generalized Gibbs Ensemble and the Statistics of KdV Charges in 2D CFT, JHEP03 (2019) 075 [arXiv:1810.11054] [INSPIRE].
A. Dymarsky and K. Pavlenko, Generalized Gibbs Ensemble of 2d CFTs at large central charge in the thermodynamic limit, JHEP01 (2019) 098 [arXiv:1810.11025] [INSPIRE].
A. Dymarsky and K. Pavlenko, Exact generalized partition function of 2D CFTs at large central charge, JHEP05 (2019) 077 [arXiv:1812.05108] [INSPIRE].
A. Dymarsky and K. Pavlenko, Generalized Eigenstate Thermalization Hypothesis in 2D Conformal Field Theories, Phys. Rev. Lett.123 (2019) 111602 [arXiv:1903.03559] [INSPIRE].
S. Datta, P. Kraus and B. Michel, Typicality and thermality in 2d CFT, JHEP07 (2019) 143 [arXiv:1904.00668] [INSPIRE].
E.M. Brehm and D. Das, On KdV characters in large c CFTs, arXiv:1901.10354 [INSPIRE].
R.G. Leigh, A.C. Petkou, P.M. Petropoulos and P.K. Tripathy, The Geroch group in Einstein spaces, Class. Quant. Grav.31 (2014) 225006 [arXiv:1403.6511] [INSPIRE].
P.M. Petropoulos and K. Siampos, Integrability, Einstein spaces and holographic fluids, arXiv:1510.06456 [INSPIRE].
A.C. Petkou, P.M. Petropoulos and K. Siampos, Geroch group for Einstein spaces and holographic integrability, PoS(PLANCK2015)104 (2015) [arXiv:1512.04970] [INSPIRE].
D. Klemm, M. Nozawa and M. Rabbiosi, On the integrability of Einstein-Maxwell-(A)dS gravity in the presence of Killing vectors, Class. Quant. Grav.32 (2015) 205008 [arXiv:1506.09017] [INSPIRE].
J. Gath, A. Mukhopadhyay, A.C. Petkou, P.M. Petropoulos and K. Siampos, Petrov Classification and holographic reconstruction of spacetime, JHEP09 (2015) 005 [arXiv:1506.04813] [INSPIRE].
Y. Chervonyi and O. Lunin, Killing(-Yano) Tensors in String Theory, JHEP09 (2015) 182 [arXiv:1505.06154] [INSPIRE].
V. Frolov, P. Krtous and D. Kubiznak, Black holes, hidden symmetries and complete integrability, Living Rev. Rel.20 (2017) 6 [arXiv:1705.05482] [INSPIRE].
I. Bena, D. Turton, R. Walker and N.P. Warner, Integrability and Black-Hole Microstate Geometries, JHEP11 (2017) 021 [arXiv:1709.01107] [INSPIRE].
J.L. Burchnall and T.W. Chaundy, The Hypergeometric Identities of Cayley, Orr and Bailey, Proc. Lond. Math. Soc.s2-50 (1948) 56.
A.A. Belavin, A.M. Polyakov and A.B. Zamolodchikov, Infinite Conformal Symmetry in Two-Dimensional Quantum Field Theory, Nucl. Phys.B 241 (1984) 333 [INSPIRE].
P. Di Francesco, P. Mathieu and D. Senechal, Conformal Field Theory, Graduate Texts in Contemporary Physics, Springer-Verlag, New York (1997).
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Kulaxizi, M., Ng, G.S. & Parnachev, A. Subleading eikonal, AdS/CFT and double stress tensors. J. High Energ. Phys. 2019, 107 (2019). https://doi.org/10.1007/JHEP10(2019)107
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DOI: https://doi.org/10.1007/JHEP10(2019)107