Abstract
Soft graviton theorems receive one-loop contributions that are logarithmic in the energy of the soft graviton, and which are closely related to tails of gravitational waveforms. We demonstrate that these logarithmic corrections are encoded in the Ward identity of superrotation symmetries, i.e. they follow from conservation of superrotation charge across spatial infinity i0. Our proof relies on a careful analysis of the radiative phase space admitting such gravitational tails, and the determination of the fluxes through null infinity \( \mathcal{I} \) that act as canonical generators of superrotations on both gravitational and matter fields. All logarithmic terms are derived from the fluxes through correlations of the supertranslation Goldstone mode, provided care is taken in manipulating gravitationally interacting (i.e. dressed) rather than free fields. In cases where massive particles take part in the scattering process, logarithmic corrections also partly arise from the superrotation charge generator at timelike infinity i±.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
A. Strominger, On BMS invariance of gravitational scattering, JHEP 07 (2014) 152 [arXiv:1312.2229] [INSPIRE].
T. He, V. Lysov, P. Mitra and A. Strominger, BMS supertranslations and Weinberg’s soft graviton theorem, JHEP 05 (2015) 151 [arXiv:1401.7026] [INSPIRE].
T. Adamo, E. Casali and D. Skinner, Perturbative gravity at null infinity, Class. Quant. Grav. 31 (2014) 225008 [arXiv:1405.5122] [INSPIRE].
D. Kapec, V. Lysov, S. Pasterski and A. Strominger, Semiclassical Virasoro symmetry of the quantum gravity \( \mathcal{S} \)-matrix, JHEP 08 (2014) 058 [arXiv:1406.3312] [INSPIRE].
S. Weinberg, Infrared photons and gravitons, Phys. Rev. 140 (1965) B516 [INSPIRE].
F. Cachazo and A. Strominger, Evidence for a new soft graviton theorem, arXiv:1404.4091 [INSPIRE].
M. Campiglia and A. Laddha, Asymptotic symmetries of gravity and soft theorems for massive particles, JHEP 12 (2015) 094 [arXiv:1509.01406] [INSPIRE].
H. Bondi, M.G.J. van der Burg and A.W.K. Metzner, Gravitational waves in general relativity. 7. Waves from axisymmetric isolated systems, Proc. Roy. Soc. Lond. A 269 (1962) 21 [INSPIRE].
R.K. Sachs, Gravitational waves in general relativity. 8. Waves in asymptotically flat space-times, Proc. Roy. Soc. Lond. A 270 (1962) 103 [INSPIRE].
R. Sachs, Asymptotic symmetries in gravitational theory, Phys. Rev. 128 (1962) 2851 [INSPIRE].
G. Barnich and C. Troessaert, Symmetries of asymptotically flat 4 dimensional spacetimes at null infinity revisited, Phys. Rev. Lett. 105 (2010) 111103 [arXiv:0909.2617] [INSPIRE].
G. Barnich and C. Troessaert, Aspects of the BMS/CFT correspondence, JHEP 05 (2010) 062 [arXiv:1001.1541] [INSPIRE].
Z. Bern, S. Davies and J. Nohle, On loop corrections to subleading soft behavior of gluons and gravitons, Phys. Rev. D 90 (2014) 085015 [arXiv:1405.1015] [INSPIRE].
A. Laddha and A. Sen, Logarithmic terms in the soft expansion in four dimensions, JHEP 10 (2018) 056 [arXiv:1804.09193] [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].
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, The gravitational eikonal: from particle, string and brane collisions to black-hole encounters, arXiv:2306.16488 [INSPIRE].
L. Blanchet and T. Damour, Tail transported temporal correlations in the dynamics of a gravitating system, Phys. Rev. D 37 (1988) 1410 [INSPIRE].
L. Blanchet and G. Schaefer, Gravitational wave tails and binary star systems, Class. Quant. Grav. 10 (1993) 2699 [INSPIRE].
L. Donnay, K. Nguyen and R. Ruzziconi, Loop-corrected subleading soft theorem and the celestial stress tensor, JHEP 09 (2022) 063 [arXiv:2205.11477] [INSPIRE].
S. Pasterski, A comment on loop corrections to the celestial stress tensor, JHEP 01 (2023) 025 [arXiv:2205.10901] [INSPIRE].
T. He, D. Kapec, A.-M. Raclariu and A. Strominger, Loop-corrected Virasoro symmetry of 4D quantum gravity, JHEP 08 (2017) 050 [arXiv:1701.00496] [INSPIRE].
S. He, P. Mao and X.-C. Mao, Loop corrections versus marginal deformation in celestial holography, arXiv:2307.02743 [INSPIRE].
H. Krishna and B. Sahoo, Universality of loop corrected soft theorems in 4d, JHEP 11 (2023) 233 [arXiv:2308.16807] [INSPIRE].
D. Kapec, P. Mitra, A.-M. Raclariu and A. Strominger, 2D stress tensor for 4D gravity, Phys. Rev. Lett. 119 (2017) 121601 [arXiv:1609.00282] [INSPIRE].
L. Donnay, A. Fiorucci, Y. Herfray and R. Ruzziconi, Carrollian perspective on celestial holography, Phys. Rev. Lett. 129 (2022) 071602 [arXiv:2202.04702] [INSPIRE].
L. Donnay, A. Fiorucci, Y. Herfray and R. Ruzziconi, Bridging Carrollian and celestial holography, Phys. Rev. D 107 (2023) 126027 [arXiv:2212.12553] [INSPIRE].
E. Himwich et al., The soft \( \mathcal{S} \)-matrix in gravity, JHEP 09 (2020) 129 [arXiv:2005.13433] [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].
K. Nguyen and J. Salzer, Celestial IR divergences and the effective action of supertranslation modes, JHEP 09 (2021) 144 [arXiv:2105.10526] [INSPIRE].
L. Donnay and R. Ruzziconi, BMS flux algebra in celestial holography, JHEP 11 (2021) 040 [arXiv:2108.11969] [INSPIRE].
A. Ashtekar and M. Streubel, Symplectic geometry of radiative modes and conserved quantities at null infinity, Proc. Roy. Soc. Lond. A 376 (1981) 585 [INSPIRE].
M. Campiglia and A. Laddha, BMS algebra, double soft theorems, and all that, arXiv:2106.14717 [INSPIRE].
M. Campiglia and A. Laddha, Loop corrected soft photon theorem as a Ward identity, JHEP 10 (2019) 287 [arXiv:1903.09133] [INSPIRE].
S. Atul Bhatkar, Ward identity for loop level soft photon theorem for massless QED coupled to gravity, JHEP 10 (2020) 110 [arXiv:1912.10229] [INSPIRE].
S. Weinberg, The quantum theory of fields. Volume 1: foundations, Cambridge University Press, Cambridge, U.K. (2005) [https://doi.org/10.1017/CBO9781139644167] [INSPIRE].
S. Pasterski, A. Puhm and E. Trevisani, Revisiting the conformally soft sector with celestial diamonds, JHEP 11 (2021) 143 [arXiv:2105.09792] [INSPIRE].
K. Nguyen, A. Rios Fukelman and C.D. White, Celestial soft dressings from generalised Wilson lines, PoS CORFU2022 (2023) 289 [arXiv:2304.01250] [INSPIRE].
M. Campiglia, Null to time-like infinity Green’s functions for asymptotic symmetries in Minkowski spacetime, JHEP 11 (2015) 160 [arXiv:1509.01408] [INSPIRE].
G. Compère, A. Fiorucci and R. Ruzziconi, Superboost transitions, refraction memory and super-Lorentz charge algebra, JHEP 11 (2018) 200 [Erratum ibid. 04 (2020) 172] [arXiv:1810.00377] [INSPIRE].
G. Compère, A. Fiorucci and R. Ruzziconi, The Λ-BMS4 charge algebra, JHEP 10 (2020) 205 [arXiv:2004.10769] [INSPIRE].
L. Blanchet et al., Multipole expansion of gravitational waves: from harmonic to Bondi coordinates, JHEP 02 (2021) 029 [arXiv:2011.10000] [INSPIRE].
A. Strominger and A. Zhiboedov, Gravitational memory, BMS supertranslations and soft theorems, JHEP 01 (2016) 086 [arXiv:1411.5745] [INSPIRE].
G. Compère and J. Long, Vacua of the gravitational field, JHEP 07 (2016) 137 [arXiv:1601.04958] [INSPIRE].
R. Geroch, Asymptotic structure of space-time, in the proceedings of the Symposium on asymptotic structure of space-time, Springer, Boston, MA, U.S.A. (1977), p. 1 [https://doi.org/10.1007/978-1-4684-2343-3_1] [INSPIRE].
M. Campiglia and J. Peraza, Generalized BMS charge algebra, Phys. Rev. D 101 (2020) 104039 [arXiv:2002.06691] [INSPIRE].
K. Nguyen, Schwarzian transformations at null infinity, PoS CORFU2021 (2022) 133 [arXiv:2201.09640] [INSPIRE].
G. Barnich and R. Ruzziconi, Coadjoint representation of the BMS group on celestial Riemann surfaces, JHEP 06 (2021) 079 [arXiv:2103.11253] [INSPIRE].
L. Freidel, D. Pranzetti and A.-M. Raclariu, Higher spin dynamics in gravity and w1+∞ celestial symmetries, Phys. Rev. D 106 (2022) 086013 [arXiv:2112.15573] [INSPIRE].
G. Compère, S.E. Gralla and H. Wei, An asymptotic framework for gravitational scattering, Class. Quant. Grav. 40 (2023) 205018 [arXiv:2303.17124] [INSPIRE].
G. Compère and F. Dehouck, Relaxing the parity conditions of asymptotically flat gravity, Class. Quant. Grav. 28 (2011) 245016 [Erratum ibid. 30 (2013) 039501] [arXiv:1106.4045] [INSPIRE].
H. Friedrich, Spin two fields on Minkowski space near space-like and null infinity, Class. Quant. Grav. 20 (2003) 101 [gr-qc/0209034] [INSPIRE].
C. Troessaert, The BMS4 algebra at spatial infinity, Class. Quant. Grav. 35 (2018) 074003 [arXiv:1704.06223] [INSPIRE].
M. Henneaux and C. Troessaert, BMS group at spatial infinity: the Hamiltonian (ADM) approach, JHEP 03 (2018) 147 [arXiv:1801.03718] [INSPIRE].
M.M.A. Mohamed and J.A.V. Kroon, Asymptotic charges for spin-1 and spin-2 fields at the critical sets of null infinity, J. Math. Phys. 63 (2022) 052502 [arXiv:2112.03890] [INSPIRE].
K. Prabhu, Conservation of asymptotic charges from past to future null infinity: supermomentum in general relativity, JHEP 03 (2019) 148 [arXiv:1902.08200] [INSPIRE].
K. Prabhu and I. Shehzad, Conservation of asymptotic charges from past to future null infinity: Lorentz charges in general relativity, JHEP 08 (2022) 029 [arXiv:2110.04900] [INSPIRE].
F. Capone, K. Nguyen and E. Parisini, Charge and antipodal matching across spatial infinity, SciPost Phys. 14 (2023) 014 [arXiv:2204.06571] [INSPIRE].
A. Ashtekar, M. Campiglia and A. Laddha, Null infinity, the BMS group and infrared issues, Gen. Rel. Grav. 50 (2018) 140 [arXiv:1808.07093] [INSPIRE].
Acknowledgments
We thank Francesco Alessio, Shamik Banerjee, Miguel Campiglia, Geoffrey Compère, Laurent Freidel, Alok Laddha, Lorenzo Magnea, Prahar Mitra, Andrea Puhm, Ali Seraj and Beniamino Valsesia for stimulating discussions. S.A. and L.D. are partially supported by INFN Iniziativa Specifica ST&FI. K.N. is supported by the STFC grants ST/P000258/1 and ST/T000759/1. R.R. is supported by the Titchmarsh Research Fellowship in Mathematical Physics at the University of Oxford.
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: 2309.11220
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
Agrawal, S., Donnay, L., Nguyen, K. et al. Logarithmic soft graviton theorems from superrotation Ward identities. J. High Energ. Phys. 2024, 120 (2024). https://doi.org/10.1007/JHEP02(2024)120
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP02(2024)120