Abstract
By following closely Weinberg’s soft theorem, which captures the 1/ω pole contribution to the amplitude for soft graviton emissions (ω < Λ) on top of an arbitrary background hard process, we calculate the expectation value of the graviton’s angular momentum operator for arbitrary collisions dressed with soft radiation. We find that the result becomes independent of the cutoff Λ on the graviton’s frequency, effectively localizing at ω = 0. In this way, our result captures the contribution to the angular momentum that comes from the zero-frequency modes. Like the soft theorem, our formula has an exact dependence on the kinematics of the hard particles and is only a function of their momenta. As an example, we discuss in some detail the case of the 2 → 2 scattering of spinless particles in General Relativity and \( \mathcal{N} \) = 8 supergravity.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
W. D. Goldberger and I. Z. Rothstein, An effective field theory of gravity for extended objects, Phys. Rev. D 73 (2006) 104029 [hep-th/0409156] [INSPIRE].
W. D. Goldberger and A. K. Ridgway, Radiation and the classical double copy for color charges, Phys. Rev. D 95 (2017) 125010 [arXiv:1611.03493] [INSPIRE].
T. Damour, High-energy gravitational scattering and the general relativistic two-body problem, Phys. Rev. D 97 (2018) 044038 [arXiv:1710.10599] [INSPIRE].
A. Luna, I. Nicholson, D. O’Connell and C. D. White, Inelastic black hole scattering from charged scalar amplitudes, JHEP 03 (2018) 044 [arXiv:1711.03901] [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].
A. Cristofoli, N. E. J. Bjerrum-Bohr, P. H. Damgaard and P. Vanhove, Post-Minkowskian Hamiltonians in general relativity, Phys. Rev. D 100 (2019) 084040 [arXiv:1906.01579] [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].
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].
M. Accettulli Huber, A. Brandhuber, S. De Angelis and G. Travaglini, From amplitudes to gravitational radiation with cubic interactions and tidal effects, Phys. Rev. D 103 (2021) 045015 [arXiv:2012.06548] [INSPIRE].
G. U. Jakobsen, G. Mogull, J. Plefka and J. Steinhoff, Classical gravitational Bremsstrahlung from a worldline quantum field theory, Phys. Rev. Lett. 126 (2021) 201103 [arXiv:2101.12688] [INSPIRE].
A. Cristofoli, R. Gonzo, D. A. Kosower and D. O’Connell, Waveforms from amplitudes, arXiv:2107.10193 [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].
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].
Z. Bern et al., Scattering amplitudes and conservative binary dynamics at O(G4), Phys. Rev. Lett. 126 (2021) 171601 [arXiv:2101.07254] [INSPIRE].
Z. Bern et al., Scattering amplitudes, the tail effect, and conservative binary dynamics at O(G4), Phys. Rev. Lett. 128 (2022) 161103 [arXiv:2112.10750] [INSPIRE].
C. Dlapa, G. Kälin, Z. Liu and R. A. Porto, Conservative dynamics of binary systems at fourth post-Minkowskian order in the large-eccentricity expansion, Phys. Rev. Lett. 128 (2022) 161104 [arXiv:2112.11296] [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].
P. Di Vecchia, C. Heissenberg, R. Russo and G. Veneziano, Radiation reaction from soft theorems, Phys. Lett. B 818 (2021) 136379 [arXiv:2101.05772] [INSPIRE].
P. Di Vecchia, C. Heissenberg, R. Russo and G. Veneziano, The eikonal approach to gravitational scattering and radiation at O(G3), JHEP 07 (2021) 169 [arXiv:2104.03256] [INSPIRE].
N. E. J. Bjerrum-Bohr, P. H. Damgaard, L. Planté and P. Vanhove, The amplitude for classical gravitational scattering at third post-Minkowskian order, JHEP 08 (2021) 172 [arXiv:2105.05218] [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].
E. Herrmann, J. Parra-Martinez, M. S. Ruf and M. Zeng, Gravitational Bremsstrahlung from reverse unitarity, Phys. Rev. Lett. 126 (2021) 201602 [arXiv:2101.07255] [INSPIRE].
E. Herrmann, J. Parra-Martinez, M. S. Ruf and M. Zeng, Radiative classical gravitational observables at O(G3) from scattering amplitudes, JHEP 10 (2021) 148 [arXiv:2104.03957] [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].
A. V. Manohar, A. K. Ridgway and C.-H. Shen, Radiated angular momentum and dissipative effects in classical scattering, arXiv:2203.04283 [INSPIRE].
G. Kälin and R. A. Porto, From boundary data to bound states, JHEP 01 (2020) 072 [arXiv:1910.03008] [INSPIRE].
G. Kälin and R. A. Porto, From boundary data to bound states. Part II. Scattering angle to dynamical invariants (with twist), JHEP 02 (2020) 120 [arXiv:1911.09130] [INSPIRE].
G. Cho, G. Kälin and R. A. Porto, From boundary data to bound states. Part III. Radiative effects, JHEP 04 (2022) 154 [arXiv:2112.03976] [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].
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, 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].
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].
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].
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].
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].
Z. Bern, J. Parra-Martinez, R. Roiban, E. Sawyer and C.-H. Shen, Leading nonlinear tidal effects and scattering amplitudes, JHEP 05 (2021) 188 [arXiv:2010.08559] [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].
D. Amati, M. Ciafaloni and G. Veneziano, Superstring collisions at Planckian energies, Phys. Lett. B 197 (1987) 81 [INSPIRE].
G. ’t Hooft, Graviton dominance in ultrahigh-energy scattering, Phys. Lett. B 198 (1987) 61 [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].
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, Higher order gravitational deflection and soft Bremsstrahlung in Planckian energy superstring collisions, Nucl. Phys. B 347 (1990) 550 [INSPIRE].
M. Ciafaloni, D. Colferai, F. Coradeschi and G. Veneziano, Unified limiting form of graviton radiation at extreme energies, Phys. Rev. D 93 (2016) 044052 [arXiv:1512.00281] [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. H. Damgaard, L. Plante and P. Vanhove, On an exponential representation of the gravitational S-matrix, JHEP 11 (2021) 213 [arXiv:2107.12891] [INSPIRE].
A. Cristofoli et al., The uncertainty principle and classical amplitudes, arXiv:2112.07556 [INSPIRE].
A. Laddha and A. Sen, Observational signature of the logarithmic terms in the soft graviton theorem, Phys. Rev. D 100 (2019) 024009 [arXiv:1806.01872] [INSPIRE].
B. Sahoo and A. Sen, Classical and quantum results on logarithmic terms in the soft theorem in four dimensions, JHEP 02 (2019) 086 [arXiv:1808.03288] [INSPIRE].
A. P. Saha, B. Sahoo and A. Sen, Proof of the classical soft graviton theorem in D = 4, JHEP 06 (2020) 153 [arXiv:1912.06413] [INSPIRE].
B. Sahoo and A. Sen, Classical soft graviton theorem rewritten, JHEP 01 (2022) 077 [arXiv:2105.08739] [INSPIRE].
A. Strominger, Lectures on the infrared structure of gravity and gauge theory, Princeton University Press (2018) [arXiv:1703.05448] [INSPIRE].
S. Weinberg, Photons and gravitons in S-matrix theory: derivation of charge conservation and equality of gravitational and inertial mass, Phys. Rev. 135 (1964) B1049 [INSPIRE].
S. Weinberg, Infrared photons and gravitons, Phys. Rev. 140 (1965) B516 [INSPIRE].
P. Di Vecchia, C. Heissenberg, R. Russo and G. Veneziano, The eikonal operator at arbitrary velocities. Part I. The soft-radiation limit, JHEP 07 (2022) 039 [arXiv:2204.02378] [INSPIRE].
G. Veneziano and G. A. Vilkovisky, Angular momentum loss in gravitational scattering, radiation reaction, and the Bondi gauge ambiguity, arXiv:2201.11607 [INSPIRE].
A. Strominger and A. Zhiboedov, Gravitational memory, BMS supertranslations and soft theorems, JHEP 01 (2016) 086 [arXiv:1411.5745] [INSPIRE].
S. Weinberg, The quantum theory of fields. Volume 1: foundations, Cambridge University Press (2005).
Y. F. Bautista and A. Laddha, Soft constraints on KMOC formalism, arXiv:2111.11642 [INSPIRE].
Y. F. Bautista and A. Guevara, From scattering amplitudes to classical physics: universality, double copy and soft theorems, arXiv:1903.12419 [INSPIRE].
S. Mougiakakos, M. M. Riva and F. Vernizzi, Gravitational Bremsstrahlung in the post-Minkowskian effective field theory, Phys. Rev. D 104 (2021) 024041 [arXiv:2102.08339] [INSPIRE].
S. E. Gralla and K. Lobo, Self-force effects in post-Minkowskian scattering, Class. Quant. Grav. 39 (2022) 095001 [arXiv:2110.08681] [INSPIRE].
F. Alessio and P. Di Vecchia, Radiation reaction for spinning black-hole scattering, Phys. Lett. B 832 (2022) 137258 [arXiv:2203.13272] [INSPIRE].
S. G. Naculich, H. Nastase and H. J. Schnitzer, Two-loop graviton scattering relation and IR behavior in N = 8 supergravity, Nucl. Phys. B 805 (2008) 40 [arXiv:0805.2347] [INSPIRE].
S. G. Naculich and H. J. Schnitzer, Eikonal methods applied to gravitational scattering amplitudes, JHEP 05 (2011) 087 [arXiv:1101.1524] [INSPIRE].
C. D. White, Factorization properties of soft graviton amplitudes, JHEP 05 (2011) 060 [arXiv:1103.2981] [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].
S. Melville, S. G. Naculich, H. J. Schnitzer and C. D. White, Wilson line approach to gravity in the high energy limit, Phys. Rev. D 89 (2014) 025009 [arXiv:1306.6019] [INSPIRE].
P. Di Vecchia, A. Luna, S. G. Naculich, R. Russo, G. Veneziano and C. D. White, A tale of two exponentiations in N = 8 supergravity, Phys. Lett. B 798 (2019) 134927 [arXiv:1908.05603] [INSPIRE].
P. Di Vecchia, S. G. Naculich, R. Russo, G. Veneziano and C. D. White, A tale of two exponentiations in N = 8 supergravity at subleading level, JHEP 03 (2020) 173 [arXiv:1911.11716] [INSPIRE].
D. Bonocore, Asymptotic dynamics on the worldline for spinning particles, JHEP 02 (2021) 007 [arXiv:2009.07863] [INSPIRE].
D. Bonocore, A. Kulesza and J. Pirsch, Classical and quantum gravitational scattering with generalized Wilson lines, JHEP 03 (2022) 147 [arXiv:2112.02009] [INSPIRE].
R. Soldati, Field theory 1. Introduction to quantum field theory, http://www.robertosoldati.com/archivio/news/107/QFT1.pdf.
R. Gonzo and A. Pokraka, Light-ray operators, detectors and gravitational event shapes, JHEP 05 (2021) 015 [arXiv:2012.01406] [INSPIRE].
D. Christodoulou, Nonlinear nature of gravitation and gravitational wave experiments, Phys. Rev. Lett. 67 (1991) 1486 [INSPIRE].
A. G. Wiseman and C. M. Will, Christodoulou’s nonlinear gravitational wave memory: evaluation in the quadrupole approximation, Phys. Rev. D 44 (1991) R2945 [INSPIRE].
L. Blanchet and T. Damour, Hereditary effects in gravitational radiation, Phys. Rev. D 46 (1992) 4304 [INSPIRE].
F. Bloch and A. Nordsieck, Note on the radiation field of the electron, Phys. Rev. 52 (1937) 54 [INSPIRE].
W. Thirring and B. Touschek, A covariant formulation of the Block-Nordsieck method, Phil. Mag. Ser. 7 42 (1951) 244.
M. Mirbabayi and M. Porrati, Dressed hard states and black hole soft hair, Phys. Rev. Lett. 117 (2016) 211301 [arXiv:1607.03120] [INSPIRE].
S. Choi and R. Akhoury, BMS supertranslation symmetry implies Faddeev-Kulish amplitudes, JHEP 02 (2018) 171 [arXiv:1712.04551] [INSPIRE].
N. Arkani-Hamed, M. Pate, A.-M. Raclariu and A. Strominger, Celestial amplitudes from UV to IR, JHEP 08 (2021) 062 [arXiv:2012.04208] [INSPIRE].
S. Weinberg, Gravitation and cosmology: principles and applications of the general theory of relativity, John Wiley and Sons (1972).
D. Garfinkle, S. Hollands, A. Ishibashi, A. Tolish and R. M. Wald, The memory effect for particle scattering in even spacetime dimensions, Class. Quant. Grav. 34 (2017) 145015 [arXiv:1702.00095] [INSPIRE].
A. Campoleoni, D. Francia and C. Heissenberg, Electromagnetic and color memory in even dimensions, Phys. Rev. D 100 (2019) 085015 [arXiv:1907.05187] [INSPIRE].
S. Caron-Huot and Z. Zahraee, Integrability of black hole orbits in maximal supergravity, JHEP 07 (2019) 179 [arXiv:1810.04694] [INSPIRE].
D. Bini and T. Damour, Gravitational radiation reaction along general orbits in the effective one-body formalism, Phys. Rev. D 86 (2012) 124012 [arXiv:1210.2834] [INSPIRE].
D. Bini, T. Damour and A. Geralico, Radiative contributions to gravitational scattering, Phys. Rev. D 104 (2021) 084031 [arXiv:2107.08896] [INSPIRE].
P. C. Peters, Gravitational radiation and the motion of two point masses, Phys. Rev. 136 (1964) B1224 [INSPIRE].
B. DeWitt, Bryce DeWitt’s lectures on gravitation, Lect. Notes Phys. 826 (2011) 1 [INSPIRE].
K. S. Thorne, Multipole expansions of gravitational radiation, Rev. Mod. Phys. 52 (1980) 299 [INSPIRE].
B. Bonga and E. Poisson, Coulombic contribution to angular momentum flux in general relativity, Phys. Rev. D 99 (2019) 064024 [arXiv:1808.01288] [INSPIRE].
L. Blanchet and G. Faye, Flux-balance equations for linear momentum and center-of-mass position of self-gravitating post-Newtonian systems, Class. Quant. Grav. 36 (2019) 085003 [arXiv:1811.08966] [INSPIRE].
M. Maggiore, Gravitational waves. Volume 1: theory and experiments, Oxford University Press (2007).
A. Ashtekar and B. Bonga, On a basic conceptual confusion in gravitational radiation theory, Class. Quant. Grav. 34 (2017) 20LT01 [arXiv:1707.07729] [INSPIRE].
A. Ashtekar and B. Bonga, On the ambiguity in the notion of transverse traceless modes of gravitational waves, Gen. Rel. Grav. 49 (2017) 122 [arXiv:1707.09914] [INSPIRE].
G. Compère, R. Oliveri and A. Seraj, The Poincaré and BMS flux-balance laws with application to binary systems, JHEP 10 (2020) 116 [arXiv:1912.03164] [INSPIRE].
M. E. Peskin and D. V. Schroeder, An introduction to quantum field theory, Addison-Wesley (1995).
C. Heissenberg, Infrared divergences and the eikonal exponentiation, Phys. Rev. D 104 (2021) 046016 [arXiv:2105.04594] [INSPIRE].
D. Kosmopoulos, Simplifying D-dimensional physical-state sums in gauge theory and gravity, Phys. Rev. D 105 (2022) 056025 [arXiv:2009.00141] [INSPIRE].
S. J. Kovacs and K. S. Thorne, The generation of gravitational waves. 3. Derivation of Bremsstrahlung formulas, Astrophys. J. 217 (1977) 252 [INSPIRE].
S. J. Kovacs and K. S. Thorne, The generation of gravitational waves. 4. Bremsstrahlung, Astrophys. J. 224 (1978) 62 [INSPIRE].
S. E. Gralla and K. Lobo, Electromagnetic scoot, Phys. Rev. D 105 (2022) 084053 [arXiv:2112.01729] [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: 2203.11915
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
Di Vecchia, P., Heissenberg, C. & Russo, R. Angular momentum of zero-frequency gravitons. J. High Energ. Phys. 2022, 172 (2022). https://doi.org/10.1007/JHEP08(2022)172
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP08(2022)172