Abstract
We consider the eikonal phase associated with the gravitational scattering of a highly energetic light particle off a very heavy object in AdS spacetime. A simple expression for this phase follows from the WKB approximation to the scattering amplitude and has been computed to all orders in the ratio of the impact parameter to the Schwarzschild radius of the heavy particle. The eikonal phase is related to the deflection angle by the usual stationary phase relation. We consider the flat space limit and observe that for sufficiently small impact parameters (or angular momenta) the eikonal phase develops a large imaginary part; the inelastic cross-section is exactly the classical absorption cross-section of the black hole. We also consider a double scaling limit where the momentum becomes null simultaneously with the asymptotically AdS black hole becoming very large. In the dual CFT this limit retains contributions from all leading twist multi stress tensor operators, which are universal with respect to the addition of higher derivative terms to the gravitational lagrangian. We compute the eikonal phase and the associated Lyapunov exponent in the double scaling limit.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
S.B. Giddings, The gravitational S-matrix: Erice lectures, Subnucl. Ser. 48 (2013) 93 [arXiv:1105.2036] [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. Amati, M. Ciafaloni and G. Veneziano, Classical and Quantum Gravity Effects from Planckian Energy Superstring Collisions, Int. J. Mod. Phys. A 3 (1988) 1615 [INSPIRE].
D. Amati, M. Ciafaloni and G. Veneziano, Higher Order Gravitational Deflection and Soft Bremsstrahlung in Planckian Energy Superstring Collisions, Nucl. Phys. B 347 (1990) 550 [INSPIRE].
H.L. Verlinde and E.P. Verlinde, Scattering at Planckian energies, Nucl. Phys. B 371 (1992) 246 [hep-th/9110017] [INSPIRE].
S.B. Giddings and M. Srednicki, High-energy gravitational scattering and black hole resonances, Phys. Rev. D 77 (2008) 085025 [arXiv:0711.5012] [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, JHEP 11 (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, Phys. Rev. D 103 (2021) 064036 [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, JHEP 11 (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, JHEP 01 (2017) 052 [arXiv:1611.02172] [INSPIRE].
F. Cachazo and A. Guevara, Leading Singularities and Classical Gravitational Scattering, JHEP 02 (2020) 181 [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, 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. 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].
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].
N.E.J. Bjerrum-Bohr, A. Cristofoli and P.H. Damgaard, Post-Minkowskian Scattering Angle in Einstein Gravity, JHEP 08 (2020) 038 [arXiv:1910.09366] [INSPIRE].
T. Damour, Classical and quantum scattering in post-Minkowskian gravity, Phys. Rev. D 102 (2020) 024060 [arXiv:1912.02139] [INSPIRE].
Z. Bern, H. Ita, J. Parra-Martinez and M.S. Ruf, Universality in the classical limit of massless gravitational scattering, Phys. Rev. Lett. 125 (2020) 031601 [arXiv:2002.02459] [INSPIRE].
J. Blümlein, A. Maier, P. Marquard and G. Schäfer, Testing binary dynamics in gravity at the sixth post-Newtonian level, Phys. Lett. B 807 (2020) 135496 [arXiv:2003.07145] [INSPIRE].
C. Cheung and M.P. Solon, Classical gravitational scattering at \( \mathcal{O} \)(G3) from Feynman diagrams, JHEP 06 (2020) 144 [arXiv:2003.08351] [INSPIRE].
A. Cristofoli, P.H. Damgaard, P. Di Vecchia and C. Heissenberg, Second-order Post-Minkowskian scattering in arbitrary dimensions, JHEP 07 (2020) 122 [arXiv:2003.10274] [INSPIRE].
D. Bini, T. Damour and A. Geralico, Binary dynamics at the fifth and fifth-and-a-half post-Newtonian orders, Phys. Rev. D 102 (2020) 024062 [arXiv:2003.11891] [INSPIRE].
Z. Bern, A. Luna, R. Roiban, C.-H. Shen and M. Zeng, Spinning Black Hole Binary Dynamics, Scattering Amplitudes and Effective Field Theory, arXiv:2005.03071 [INSPIRE].
J. Parra-Martinez, M.S. Ruf and M. Zeng, Extremal black hole scattering at O(G3): graviton dominance, eikonal exponentiation, and differential equations, JHEP 11 (2020) 023 [arXiv:2005.04236] [INSPIRE].
G. Kälin and R.A. Porto, Post-Minkowskian Effective Field Theory for Conservative Binary Dynamics, JHEP 11 (2020) 106 [arXiv:2006.01184] [INSPIRE].
P. Di Vecchia, C. Heissenberg, R. Russo and G. Veneziano, Universality of ultra-relativistic gravitational scattering, Phys. Lett. B 811 (2020) 135924 [arXiv:2008.12743] [INSPIRE].
T. Damour, Radiative contribution to classical gravitational scattering at the third order in G, Phys. Rev. D 102 (2020) 124008 [arXiv:2010.01641] [INSPIRE].
C. Cheung, N. Shah and M.P. Solon, Mining the Geodesic Equation for Scattering Data, Phys. Rev. D 103 (2021) 024030 [arXiv:2010.08568] [INSPIRE].
L. Cornalba, M.S. Costa, J. Penedones and R. Schiappa, Eikonal Approximation in AdS/CFT: From Shock Waves to Four-Point Functions, JHEP 08 (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, JHEP 09 (2007) 037 [arXiv:0707.0120] [INSPIRE].
R.C. Brower, M.J. Strassler and C.-I. Tan, On the eikonal approximation in AdS space, JHEP 03 (2009) 050 [arXiv:0707.2408] [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, JHEP 12 (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, JHEP 02 (2016) 020 [arXiv:1407.5597] [INSPIRE].
M. Kulaxizi, A. Parnachev and A. Zhiboedov, Bulk Phase Shift, CFT Regge Limit and Einstein Gravity, JHEP 06 (2018) 121 [arXiv:1705.02934] [INSPIRE].
D. Li, D. Meltzer and D. Poland, Conformal Bootstrap in the Regge Limit, JHEP 12 (2017) 013 [arXiv:1705.03453] [INSPIRE].
M.S. Costa, T. Hansen and J. Penedones, Bounds for OPE coefficients on the Regge trajectory, JHEP 10 (2017) 197 [arXiv:1707.07689] [INSPIRE].
D. Meltzer, AdS/CFT Unitarity at Higher Loops: High-Energy String Scattering, JHEP 05 (2020) 133 [arXiv:1912.05580] [INSPIRE].
S. Giusto, M.R.R. Hughes and R. Russo, The Regge limit of AdS3 holographic correlators, JHEP 11 (2020) 018 [arXiv:2007.12118] [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].
A.L. Fitzpatrick and K.-W. Huang, Universal Lowest-Twist in CFTs from Holography, JHEP 08 (2019) 138 [arXiv:1903.05306] [INSPIRE].
R. Karlsson, M. Kulaxizi, A. Parnachev and P. Tadić, Black Holes and Conformal Regge Bootstrap, JHEP 10 (2019) 046 [arXiv:1904.00060] [INSPIRE].
Y.-Z. Li, Z.-F. Mai and H. Lü, Holographic OPE Coefficients from AdS Black Holes with Matters, JHEP 09 (2019) 001 [arXiv:1905.09302] [INSPIRE].
M. Kulaxizi, G.S. Ng and A. Parnachev, Subleading Eikonal, AdS/CFT and Double Stress Tensors, JHEP 10 (2019) 107 [arXiv:1907.00867] [INSPIRE].
A.L. Fitzpatrick, K.-W. Huang and D. Li, Probing universalities in d > 2 CFTs: from black holes to shockwaves, JHEP 11 (2019) 139 [arXiv:1907.10810] [INSPIRE].
R. Karlsson, M. Kulaxizi, A. Parnachev and P. Tadić, Leading Multi-Stress Tensors and Conformal Bootstrap, JHEP 01 (2020) 076 [arXiv:1909.05775] [INSPIRE].
Y.-Z. Li, Heavy-light Bootstrap from Lorentzian Inversion Formula, JHEP 07 (2020) 046 [arXiv:1910.06357] [INSPIRE].
R. Karlsson, Multi-stress tensors and next-to-leading singularities in the Regge limit, JHEP 08 (2020) 037 [arXiv:1912.01577] [INSPIRE].
R. Karlsson, M. Kulaxizi, A. Parnachev and P. Tadić, Stress tensor sector of conformal correlators operators in the Regge limit, JHEP 07 (2020) 019 [arXiv:2002.12254] [INSPIRE].
Y.-Z. Li and H.-Y. Zhang, More on heavy-light bootstrap up to double-stress-tensor, JHEP 10 (2020) 055 [arXiv:2004.04758] [INSPIRE].
A. Parnachev, Near Lightcone Thermal Conformal Correlators and Holography, J. Phys. A 54 (2021) 155401 [arXiv:2005.06877] [INSPIRE].
A.L. Fitzpatrick, K.-W. Huang, D. Meltzer, E. Perlmutter and D. Simmons-Duffin, Model-dependence of minimal-twist OPEs in d > 2 holographic CFTs, JHEP 11 (2020) 060 [arXiv:2007.07382] [INSPIRE].
M. Soldate, Partial Wave Unitarity and Closed String Amplitudes, Phys. Lett. B 186 (1987) 321 [INSPIRE].
G. Festuccia and H. Liu, A Bohr-Sommerfeld quantization formula for quasinormal frequencies of AdS black holes, Adv. Sci. Lett. 2 (2009) 221 [arXiv:0811.1033] [INSPIRE].
V. Balasubramanian, B. Craps, M. De Clerck and K. Nguyen, Superluminal chaos after a quantum quench, JHEP 12 (2019) 132 [arXiv:1908.08955] [INSPIRE].
B. Craps, M. De Clerck, P. Hacker, K. Nguyen and C. Rabideau, Slow scrambling in extremal BTZ and microstate geometries, JHEP 03 (2021) 020 [arXiv:2009.08518] [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].
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].
V.E. Hubeny, H. Liu and M. Rangamani, Bulk-cone singularities & signatures of horizon formation in AdS/CFT, JHEP 01 (2007) 009 [hep-th/0610041] [INSPIRE].
V. Cardoso, A.S. Miranda, E. Berti, H. Witek and V.T. Zanchin, Geodesic stability, Lyapunov exponents and quasinormal modes, Phys. Rev. D 79 (2009) 064016 [arXiv:0812.1806] [INSPIRE].
M. Bianchi, A. Grillo and J.F. Morales, Chaos at the rim of black hole and fuzzball shadows, JHEP 05 (2020) 078 [arXiv:2002.05574] [INSPIRE].
D. Berenstein, Z. Li and J. Simon, ISCOs in AdS/CFT, Class. Quant. Grav. 38 (2021) 045009 [arXiv:2009.04500] [INSPIRE].
L.N. Lipatov, High-energy scattering in QCD and in quantum gravity and two-dimensional field theories, Nucl. Phys. B 365 (1991) 614 [INSPIRE].
D. Amati, M. Ciafaloni and G. Veneziano, Effective action and all order gravitational eikonal at Planckian energies, Nucl. Phys. B 403 (1993) 707 [INSPIRE].
L. Fidkowski, V. Hubeny, M. Kleban and S. Shenker, The Black hole singularity in AdS/CFT, JHEP 02 (2004) 014 [hep-th/0306170] [INSPIRE].
M. Grinberg and J. Maldacena, Proper time to the black hole singularity from thermal one-point functions, JHEP 03 (2021) 131 [arXiv:2011.01004] [INSPIRE].
V. Gorbenko, S. Rychkov and B. Zan, Walking, Weak first-order transitions, and Complex CFTs, JHEP 10 (2018) 108 [arXiv:1807.11512] [INSPIRE].
J. Maldacena, S.H. Shenker and D. Stanford, A bound on chaos, JHEP 08 (2016) 106 [arXiv:1503.01409] [INSPIRE].
K.-W. Huang, Stress-tensor commutators in conformal field theories near the lightcone, Phys. Rev. D 100 (2019) 061701 [arXiv:1907.00599] [INSPIRE].
K.-W. Huang, Lightcone Commutator and Stress-Tensor Exchange in d > 2 CFTs, Phys. Rev. D 102 (2020) 021701 [arXiv:2002.00110] [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: 2011.06920
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
Parnachev, A., Sen, K. Notes on AdS-Schwarzschild eikonal phase. J. High Energ. Phys. 2021, 289 (2021). https://doi.org/10.1007/JHEP03(2021)289
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP03(2021)289