Abstract
We study the eikonal scattering of two gravitationally interacting bodies, in the regime of large angular momentum and large center of mass energy. We show that eikonal exponentiation of the scattering phase matrix is a direct consequence of the group contraction SU(2) → ISO(2), from rotations to the isometries of the plane, in the large angular momentum limit. We extend it to all orders in the scattering angle, and for all masses and spins. The emergence of the classical limit is understood in terms of the continuous-spin representations admitted by ISO(2). We further investigate the competing classical vs quantum corrections to the leading classical eikonal scattering, and find several interesting examples where quantum corrections are more important than Post-Minkowskian’s. As a case of study, we analyse the scattering of a photon off a massless neutral scalar field, up to next-to-leading order in the Newton constant, and to leading order in the fine structure constant. We investigate the causal structure of the eikonal regime and establish an infinite set of non-linear positivity bounds, of which positivity of time delay is the simplest.
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L. Landau and E.M. Lifshits, Course of theoretical physics. Vol. 3: Quantum mechanics, third edition, Butterworth-Heinemann (1981).
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, Superstring Collisions at Planckian Energies, Phys. Lett. B 197 (1987) 81 [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].
D. Amati, M. Ciafaloni and G. Veneziano, Planckian scattering beyond the semiclassical approximation, Phys. Lett. B 289 (1992) 87 [INSPIRE].
G. ’t Hooft, Graviton Dominance in Ultrahigh-Energy Scattering, Phys. Lett. B 198 (1987) 61 [INSPIRE].
H.L. Verlinde and E.P. Verlinde, Scattering at Planckian energies, Nucl. Phys. B 371 (1992) 246 [hep-th/9110017] [INSPIRE].
D.N. Kabat and M. Ortiz, Eikonal quantum gravity and Planckian scattering, Nucl. Phys. B 388 (1992) 570 [hep-th/9203082] [INSPIRE].
G.F. Giudice, R. Rattazzi and J.D. Wells, Transplanckian collisions at the LHC and beyond, Nucl. Phys. B 630 (2002) 293 [hep-ph/0112161] [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].
G. Kälin and R.A. Porto, Post-Minkowskian Effective Field Theory for Conservative Binary Dynamics, JHEP 11 (2020) 106 [arXiv:2006.01184] [INSPIRE].
G. Mogull, J. Plefka and J. Steinhoff, Classical black hole scattering from a worldline quantum field theory, JHEP 02 (2021) 048 [arXiv:2010.02865] [INSPIRE].
C. Cheung and M.P. Solon, Tidal Effects in the Post-Minkowskian Expansion, Phys. Rev. Lett. 125 (2020) 191601 [arXiv:2006.06665] [INSPIRE].
E. Herrmann, J. Parra-Martinez, M.S. Ruf and M. Zeng, Radiative classical gravitational observables at \( \mathcal{O}\left({G}^3\right) \) from scattering amplitudes, JHEP 10 (2021) 148 [arXiv:2104.03957] [INSPIRE].
P. Di Vecchia, C. Heissenberg, R. Russo and G. Veneziano, The eikonal approach to gravitational scattering and radiation at \( \mathcal{O}\left({G}^3\right) \), JHEP 07 (2021) 169 [arXiv:2104.03256] [INSPIRE].
A. Brandhuber, G. Chen, G. Travaglini and C. Wen, Classical gravitational scattering from a gauge-invariant double copy, JHEP 10 (2021) 118 [arXiv:2108.04216] [INSPIRE].
N.E.J. Bjerrum-Bohr, P.H. Damgaard, L. Planté and P. Vanhove, Classical gravity from loop amplitudes, Phys. Rev. D 104 (2021) 026009 [arXiv:2104.04510] [INSPIRE].
A. Buonanno, M. Khalil, D. O’Connell, R. Roiban, M.P. Solon and M. Zeng, Snowmass White Paper: Gravitational Waves and Scattering Amplitudes, in Snowmass 2021, Seattle, U.S.A. (2022) [arXiv:2204.05194] [INSPIRE].
F. Bastianelli, F. Comberiati and L. de la Cruz, Light bending from eikonal in worldline quantum field theory, JHEP 02 (2022) 209 [arXiv:2112.05013] [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].
B. Bellazzini, G. Isabella, M. Lewandowski and F. Sgarlata, Gravitational causality and the self-stress of photons, JHEP 05 (2022) 154 [arXiv:2108.05896] [INSPIRE].
B. Bellazzini, J. Elias Miró, R. Rattazzi, M. Riembau and F. Riva, Positive moments for scattering amplitudes, Phys. Rev. D 104 (2021) 036006 [arXiv:2011.00037] [INSPIRE].
S. Caron-Huot, D. Mazac, L. Rastelli and D. Simmons-Duffin, Sharp boundaries for the swampland, JHEP 07 (2021) 110 [arXiv:2102.08951] [INSPIRE].
S. Caron-Huot, Y.-Z. Li, J. Parra-Martinez and D. Simmons-Duffin, Causality constraints on corrections to Einstein gravity, arXiv:2201.06602 [INSPIRE].
D. Amati, M. Ciafaloni and G. Veneziano, Can Space-Time Be Probed Below the String Size?, Phys. Lett. B 216 (1989) 41 [INSPIRE].
G. D’Appollonio, P. Di Vecchia, R. Russo and G. Veneziano, Regge behavior saves String Theory from causality violations, JHEP 05 (2015) 144 [arXiv:1502.01254] [INSPIRE].
D.A. Kosower, B. Maybee and D. O’Connell, Amplitudes, Observables, and Classical Scattering, JHEP 02 (2019) 137 [arXiv:1811.10950] [INSPIRE].
S. Weinberg, The Quantum Theory of Fields. Vol. 1: Foundations, Cambridge University Press (2005).
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].
D. Bai and Y. Huang, More on the Bending of Light in Quantum Gravity, Phys. Rev. D 95 (2017) 064045 [arXiv:1612.07629] [INSPIRE].
M. Abramowitz and I.A. Stegun, Handbook of mathematical functions with formulas, graphs, and mathematical tables, Dover (1964).
G. Szego, Orthogonal polynomials, AMS Press, Providence U.S.A. (1975).
C.L. Frenzen and R. Wong, A uniform asymptotic expansion of the Jacobi polynomials with error bounds, Can. J. Math. 37 (1985) 979.
S.E. Hoffmann, Uniform analytic approximation of Wigner rotation matrices, arXiv:1710.11282.
U. Kol, D. O’connell and O. Telem, The radial action from probe amplitudes to all orders, JHEP 03 (2022) 141 [arXiv:2109.12092] [INSPIRE].
Y.F. Bautista, A. Guevara, C. Kavanagh and J. Vines, Scattering in black hole backgrounds and higher-spin amplitudes. Part I, JHEP 03 (2023) 136 [arXiv:2107.10179] [INSPIRE].
M. Ciafaloni and D. Colferai, Rescattering corrections and self-consistent metric in Planckian scattering, JHEP 10 (2014) 085 [arXiv:1406.6540] [INSPIRE].
N. Arkani-Hamed, T.-C. Huang and Y.-t. Huang, The EFT-Hedron, JHEP 05 (2021) 259 [arXiv:2012.15849] [INSPIRE].
L. Maiani and M. Testa, Unstable systems in relativistic quantum field theory, Annals Phys. 263 (1998) 353 [hep-th/9709110] [INSPIRE].
M.V. Berry and K.E. Mount, Semiclassical approximations in wave mechanics, Rept. Prog. Phys. 35 (1972) 315 [INSPIRE].
P. Morse and H. Feshbach, Methods of Theoretical Physics, McGraw-Hill Book Comp. (1953).
P.H. Damgaard, L. Plante and P. Vanhove, On an exponential representation of the gravitational S-matrix, JHEP 11 (2021) 213 [arXiv:2107.12891] [INSPIRE].
S.B. Giddings, The gravitational S-matrix: Erice lectures, Subnucl. Ser. 48 (2013) 93 [arXiv:1105.2036] [INSPIRE].
S.B. Giddings, M. Schmidt-Sommerfeld and J.R. Andersen, High energy scattering in gravity and supergravity, Phys. Rev. D 82 (2010) 104022 [arXiv:1005.5408] [INSPIRE].
J. Parra-Martinez, M.S. Ruf and M. Zeng, Extremal black hole scattering at \( \mathcal{O}\left({G}^3\right) \): graviton dominance, eikonal exponentiation, and differential equations, JHEP 11 (2020) 023 [arXiv:2005.04236] [INSPIRE].
C. Anastasiou and K. Melnikov, Higgs boson production at hadron colliders in NNLO QCD, Nucl. Phys. B 646 (2002) 220 [hep-ph/0207004] [INSPIRE].
C. Anastasiou, L.J. Dixon and K. Melnikov, NLO Higgs boson rapidity distributions at hadron colliders, Nucl. Phys. B Proc. Suppl. 116 (2003) 193 [hep-ph/0211141] [INSPIRE].
C. Anastasiou, L.J. Dixon, K. Melnikov and F. Petriello, Dilepton rapidity distribution in the Drell-Yan process at NNLO in QCD, Phys. Rev. Lett. 91 (2003) 182002 [hep-ph/0306192] [INSPIRE].
C. Anastasiou, C. Duhr, F. Dulat, E. Furlan, F. Herzog and B. Mistlberger, Soft expansion of double-real-virtual corrections to Higgs production at N3LO, JHEP 08 (2015) 051 [arXiv:1505.04110] [INSPIRE].
F.V. Tkachov, A Theorem on Analytical Calculability of Four Loop Renormalization Group Functions, Phys. Lett. B 100 (1981) 65 [INSPIRE].
K.G. Chetyrkin and F.V. Tkachov, Integration by Parts: The Algorithm to Calculate beta Functions in 4 Loops, Nucl. Phys. B 192 (1981) 159 [INSPIRE].
V.A. Smirnov, Analytic tools for Feynman integrals, Springer Tracts in Modern Physics. Vol. 250, Springer (2012), https://doi.org/10.1007/978-3-642-34886-0 [INSPIRE].
R.N. Lee, Presenting LiteRed: a tool for the Loop InTEgrals REDuction, arXiv:1212.2685 [INSPIRE].
R.N. Lee, LiteRed 1.4: a powerful tool for reduction of multiloop integrals, J. Phys. Conf. Ser. 523 (2014) 012059 [arXiv:1310.1145] [INSPIRE].
A.V. Kotikov, Differential equations method: New technique for massive Feynman diagrams calculation, Phys. Lett. B 254 (1991) 158 [INSPIRE].
A.V. Kotikov, Differential equation method: The Calculation of N point Feynman diagrams, Phys. Lett. B 267 (1991) 123 [Erratum ibid. 295 (1992) 409] [INSPIRE].
Z. Bern, L.J. Dixon and D.A. Kosower, Dimensionally regulated one loop integrals, Phys. Lett. B 302 (1993) 299 [hep-ph/9212308] [Erratum ibid. 318 (1993) 649] [INSPIRE].
T. Gehrmann and E. Remiddi, Differential equations for two loop four point functions, Nucl. Phys. B 580 (2000) 485 [hep-ph/9912329] [INSPIRE].
J.M. Henn, Multiloop integrals in dimensional regularization made simple, Phys. Rev. Lett. 110 (2013) 251601 [arXiv:1304.1806] [INSPIRE].
S. Caron-Huot and J.M. Henn, Iterative structure of finite loop integrals, JHEP 06 (2014) 114 [arXiv:1404.2922] [INSPIRE].
O. Gituliar and V. Magerya, Fuchsia and master integrals for splitting functions from differential equations in QCD, PoS LL2016 (2016) 030 [arXiv:1607.00759] [INSPIRE].
O. Gituliar and V. Magerya, Fuchsia: a tool for reducing differential equations for Feynman master integrals to epsilon form, Comput. Phys. Commun. 219 (2017) 329 [arXiv:1701.04269] [INSPIRE].
J.M. Henn, Lectures on differential equations for Feynman integrals, J. Phys. A 48 (2015) 153001 [arXiv:1412.2296] [INSPIRE].
V.A. Smirnov, Analytical result for dimensionally regularized massless on shell double box, Phys. Lett. B 460 (1999) 397 [hep-ph/9905323] [INSPIRE].
J.B. Tausk, Nonplanar massless two loop Feynman diagrams with four on-shell legs, Phys. Lett. B 469 (1999) 225 [hep-ph/9909506] [INSPIRE].
S. Dubovsky, A. Nicolis, E. Trincherini and G. Villadoro, Microcausality in curved space-time, Phys. Rev. D 77 (2008) 084016 [arXiv:0709.1483] [INSPIRE].
F. Serra, J. Serra, E. Trincherini and L.G. Trombetta, Causality constraints on black holes beyond GR, JHEP 08 (2022) 157 [arXiv:2205.08551] [INSPIRE].
A. Adams, N. Arkani-Hamed, S. Dubovsky, A. Nicolis and R. Rattazzi, Causality, analyticity and an IR obstruction to UV completion, JHEP 10 (2006) 014 [hep-th/0602178] [INSPIRE].
Z. Bern, D. Kosmopoulos and A. Zhiboedov, Gravitational effective field theory islands, low-spin dominance, and the four-graviton amplitude, J. Phys. A 54 (2021) 344002 [arXiv:2103.12728] [INSPIRE].
D.J. Gross and R. Jackiw, Low-Energy Theorem for Graviton Scattering, Phys. Rev. 166 (1968) 1287 [INSPIRE].
H.D.I. Abarbanel and M.L. Goldberger, low-energy theorems, dispersion relations and superconvergence sum rules for compton scattering, Phys. Rev. 165 (1968) 1594 [INSPIRE].
K. Häring and A. Zhiboedov, Gravitational Regge bounds, arXiv:2202.08280 [INSPIRE].
M. Ruhdorfer, J. Serra and A. Weiler, Effective Field Theory of Gravity to All Orders, JHEP 05 (2020) 083 [arXiv:1908.08050] [INSPIRE].
M. Accettulli Huber, A. Brandhuber, S. De Angelis and G. Travaglini, Eikonal phase matrix, deflection angle and time delay in effective field theories of gravity, Phys. Rev. D 102 (2020) 046014 [arXiv:2006.02375] [INSPIRE].
B. Bellazzini, C. Cheung and G.N. Remmen, Quantum Gravity Constraints from Unitarity and Analyticity, Phys. Rev. D 93 (2016) 064076 [arXiv:1509.00851] [INSPIRE].
G. Sommer, Present state of rigorous analytic properties of scattering amplitudes, Fortsch. Phys. 18 (1970) 577 [INSPIRE].
C.Y.R. Chen, C. de Rham, A. Margalit and A.J. Tolley, A cautionary case of casual causality, JHEP 03 (2022) 025 [arXiv:2112.05031] [INSPIRE].
B. Bellazzini, M. Riembau and F. Riva, IR side of positivity bounds, Phys. Rev. D 106 (2022) 105008 [arXiv:2112.12561] [INSPIRE].
J. Henriksson, B. McPeak, F. Russo and A. Vichi, Bounding violations of the weak gravity conjecture, JHEP 08 (2022) 184 [arXiv:2203.08164] [INSPIRE].
S. Caron-Huot, Y.-Z. Li, J. Parra-Martinez and D. Simmons-Duffin, Graviton partial waves and causality in higher dimensions, arXiv:2205.01495 [INSPIRE].
S. Weinberg, Infrared photons and gravitons, Phys. Rev. 140 (1965) B516 [INSPIRE].
R. Akhoury, R. Saotome and G. Sterman, Collinear and Soft Divergences in Perturbative Quantum Gravity, Phys. Rev. D 84 (2011) 104040 [arXiv:1109.0270] [INSPIRE].
M. Ciafaloni, D. Colferai and G. Veneziano, Infrared features of gravitational scattering and radiation in the eikonal approach, Phys. Rev. D 99 (2019) 066008 [arXiv:1812.08137] [INSPIRE].
P. Di Vecchia, C. Heissenberg, R. Russo and G. Veneziano, Classical Gravitational Observables from the Eikonal Operator, arXiv:2210.12118 [INSPIRE].
M. Jacob and G.C. Wick, On the General Theory of Collisions for Particles with Spin, Annals Phys. 7 (1959) 404 [INSPIRE].
M.M. Riva and F. Vernizzi, Radiated momentum in the post-Minkowskian worldline approach via reverse unitarity, JHEP 11 (2021) 228 [arXiv:2110.10140] [INSPIRE].
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Bellazzini, B., Isabella, G. & Riva, M.M. Classical vs quantum eikonal scattering and its causal structure. J. High Energ. Phys. 2023, 23 (2023). https://doi.org/10.1007/JHEP04(2023)023
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DOI: https://doi.org/10.1007/JHEP04(2023)023