Abstract
We explore the logarithmic terms in the soft theorem in four dimensions by analyzing classical scattering with generic incoming and outgoing states and one loop quantum scattering amplitudes. The classical and quantum results are consistent with each other. Although most of our analysis in quantum theory is carried out for one loop amplitudes in a theory of (charged) scalars interacting via gravitational and electromagnetic interactions, we expect the results to be valid more generally.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
M. Gell-Mann and M.L. Goldberger, Scattering of low-energy photons by particles of spin 1/2, Phys. Rev. 96 (1954) 1433 [INSPIRE].
F.E. Low, Bremsstrahlung of very low-energy quanta in elementary particle collisions, Phys. Rev. 110 (1958) 974 [INSPIRE].
S. Saito, Low-energy theorem for Compton scattering, Phys. Rev. 184 (1969) 1894 [INSPIRE].
T.H. Burnett and N.M. Kroll, Extension of the low soft photon theorem, Phys. Rev. Lett. 20 (1968) 86 [INSPIRE].
J.S. Bell and R. Van Royen, On the low-burnett-kroll theorem for soft-photon emission, Nuovo Cim. A 60 (1969) 62 [INSPIRE].
V. Del Duca, High-energy Bremsstrahlung Theorems for Soft Photons, Nucl. Phys. B 345 (1990) 369 [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].
D.J. Gross and R. Jackiw, Low-Energy Theorem for Graviton Scattering, Phys. Rev. 166 (1968) 1287 [INSPIRE].
R. Jackiw, Low-Energy Theorems for Massless Bosons: Photons and Gravitons, Phys. Rev. 168 (1968) 1623 [INSPIRE].
M. Ademollo et al., Soft Dilations and Scale Renormalization in Dual Theories, Nucl. Phys. B 94 (1975) 221 [INSPIRE].
J.A. Shapiro, On the Renormalization of Dual Models, Phys. Rev. D 11 (1975) 2937 [INSPIRE].
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].
M. Campiglia and A. Laddha, Asymptotic symmetries and subleading soft graviton theorem, Phys. Rev. D 90 (2014) 124028 [arXiv:1408.2228] [INSPIRE].
A. Strominger and A. Zhiboedov, Gravitational Memory, BMS Supertranslations and Soft Theorems, JHEP 01 (2016) 086 [arXiv:1411.5745] [INSPIRE].
M. Campiglia and A. Laddha, New symmetries for the Gravitational S-matrix, JHEP 04 (2015) 076 [arXiv:1502.02318] [INSPIRE].
S. Pasterski, A. Strominger and A. Zhiboedov, New Gravitational Memories, JHEP 12 (2016) 053 [arXiv:1502.06120] [INSPIRE].
D. Kapec, V. Lysov, S. Pasterski and A. Strominger, Higher-Dimensional Supertranslations and Weinberg’s Soft Graviton Theorem, arXiv:1502.07644 [INSPIRE].
M. Campiglia and A. Laddha, Asymptotic symmetries of QED and Weinberg’s soft photon theorem, JHEP 07 (2015) 115 [arXiv:1505.05346] [INSPIRE].
S.G. Avery and B.U.W. Schwab, Burg-Metzner-Sachs symmetry, string theory and soft theorems, Phys. Rev. D 93 (2016) 026003 [arXiv:1506.05789] [INSPIRE].
M. Campiglia and A. Laddha, Asymptotic symmetries of gravity and soft theorems for massive particles, JHEP 12 (2015) 094 [arXiv:1509.01406] [INSPIRE].
M. Campiglia and A. Laddha, Sub-subleading soft gravitons: New symmetries of quantum gravity?, Phys. Lett. B 764 (2017) 218 [arXiv:1605.09094] [INSPIRE].
M. Campiglia and A. Laddha, Subleading soft photons and large gauge transformations, JHEP 11 (2016) 012 [arXiv:1605.09677] [INSPIRE].
M. Campiglia and A. Laddha, Sub-subleading soft gravitons and large diffeomorphisms, JHEP 01 (2017) 036 [arXiv:1608.00685] [INSPIRE].
E. Conde and P. Mao, BMS Supertranslations and Not So Soft Gravitons, JHEP 05 (2017) 060 [arXiv:1612.08294] [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].
M. Asorey, A.P. Balachandran, F. Lizzi and G. Marmo, Equations of Motion as Constraints: Superselection Rules, Ward Identities, JHEP 03 (2017) 136 [arXiv:1612.05886] [INSPIRE].
A. Campoleoni, D. Francia and C. Heissenberg, On higher-spin supertranslations and superrotations, JHEP 05 (2017) 120 [arXiv:1703.01351] [INSPIRE].
A. Strominger, Lectures on the Infrared Structure of Gravity and Gauge Theory, arXiv:1703.05448 [INSPIRE].
M. Pate, A.-M. Raclariu and A. Strominger, Color Memory: A Yang-Mills Analog of Gravitational Wave Memory, Phys. Rev. Lett. 119 (2017) 261602 [arXiv:1707.08016] [INSPIRE].
A. Laddha and P. Mitra, Asymptotic Symmetries and Subleading Soft Photon Theorem in Effective Field Theories, JHEP 05 (2018) 132 [arXiv:1709.03850] [INSPIRE].
D. Kapec and P. Mitra, A d-Dimensional Stress Tensor for Mink d+2 Gravity, JHEP 05 (2018) 186 [arXiv:1711.04371] [INSPIRE].
M. Pate, A.-M. Raclariu and A. Strominger, Gravitational Memory in Higher Dimensions, JHEP 06 (2018) 138 [arXiv:1712.01204] [INSPIRE].
A. Campoleoni, D. Francia and C. Heissenberg, Asymptotic Charges at Null Infinity in Any Dimension, Universe 4 (2018) 47 [arXiv:1712.09591] [INSPIRE].
A.H. Anupam, A. Kundu and K. Ray, Double soft graviton theorems and Bondi-Metzner-Sachs symmetries, Phys. Rev. D 97 (2018) 106019 [arXiv:1803.03023] [INSPIRE].
C.D. White, Factorization Properties of Soft Graviton Amplitudes, JHEP 05 (2011) 060 [arXiv:1103.2981] [INSPIRE].
F. Cachazo and A. Strominger, Evidence for a New Soft Graviton Theorem, arXiv:1404.4091 [INSPIRE].
E. Casali, Soft sub-leading divergences in Yang-Mills amplitudes, JHEP 08 (2014) 077 [arXiv:1404.5551] [INSPIRE].
B.U.W. Schwab and A. Volovich, Subleading Soft Theorem in Arbitrary Dimensions from Scattering Equations, Phys. Rev. Lett. 113 (2014) 101601 [arXiv:1404.7749] [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].
S. He, Y.-t. Huang and C. Wen, Loop Corrections to Soft Theorems in Gauge Theories and Gravity, JHEP 12 (2014) 115 [arXiv:1405.1410] [INSPIRE].
A.J. Larkoski, Conformal Invariance of the Subleading Soft Theorem in Gauge Theory, Phys. Rev. D 90 (2014) 087701 [arXiv:1405.2346] [INSPIRE].
F. Cachazo and E.Y. Yuan, Are Soft Theorems Renormalized?, arXiv:1405.3413 [INSPIRE].
N. Afkhami-Jeddi, Soft Graviton Theorem in Arbitrary Dimensions, arXiv:1405.3533 [INSPIRE].
B.U.W. Schwab, Subleading Soft Factor for String Disk Amplitudes, JHEP 08 (2014) 062 [arXiv:1406.4172] [INSPIRE].
M. Bianchi, S. He, Y.-t. Huang and C. Wen, More on Soft Theorems: Trees, Loops and Strings, Phys. Rev. D 92 (2015) 065022 [arXiv:1406.5155] [INSPIRE].
J. Broedel, M. de Leeuw, J. Plefka and M. Rosso, Constraining subleading soft gluon and graviton theorems, Phys. Rev. D 90 (2014) 065024 [arXiv:1406.6574] [INSPIRE].
Z. Bern, S. Davies, P. Di Vecchia and J. Nohle, Low-Energy Behavior of Gluons and Gravitons from Gauge Invariance, Phys. Rev. D 90 (2014) 084035 [arXiv:1406.6987] [INSPIRE].
C.D. White, Diagrammatic insights into next-to-soft corrections, Phys. Lett. B 737 (2014) 216 [arXiv:1406.7184] [INSPIRE].
M. Zlotnikov, Sub-sub-leading soft-graviton theorem in arbitrary dimension, JHEP 10 (2014) 148 [arXiv:1407.5936] [INSPIRE].
C. Kalousios and F. Rojas, Next to subleading soft-graviton theorem in arbitrary dimensions, JHEP 01 (2015) 107 [arXiv:1407.5982] [INSPIRE].
Y.-J. Du, B. Feng, C.-H. Fu and Y. Wang, Note on Soft Graviton theorem by KLT Relation, JHEP 11 (2014) 090 [arXiv:1408.4179] [INSPIRE].
D. Bonocore, E. Laenen, L. Magnea, L. Vernazza and C.D. White, The method of regions and next-to-soft corrections in Drell-Yan production, Phys. Lett. B 742 (2015) 375 [arXiv:1410.6406] [INSPIRE].
B.U.W. Schwab, A Note on Soft Factors for Closed String Scattering, JHEP 03 (2015) 140 [arXiv:1411.6661] [INSPIRE].
A. Sabio Vera and M.A. Vazquez-Mozo, The Double Copy Structure of Soft Gravitons, JHEP 03 (2015) 070 [arXiv:1412.3699] [INSPIRE].
P. Di Vecchia, R. Marotta and M. Mojaza, Soft theorem for the graviton, dilaton and the Kalb-Ramond field in the bosonic string, JHEP 05 (2015) 137 [arXiv:1502.05258] [INSPIRE].
F. Cachazo, S. He and E.Y. Yuan, New Double Soft Emission Theorems, Phys. Rev. D 92 (2015) 065030 [arXiv:1503.04816] [INSPIRE].
A.E. Lipstein, Soft Theorems from Conformal Field Theory, JHEP 06 (2015) 166 [arXiv:1504.01364] [INSPIRE].
T. Klose, T. McLoughlin, D. Nandan, J. Plefka and G. Travaglini, Double-Soft Limits of Gluons and Gravitons, JHEP 07 (2015) 135 [arXiv:1504.05558] [INSPIRE].
A. Volovich, C. Wen and M. Zlotnikov, Double Soft Theorems in Gauge and String Theories, JHEP 07 (2015) 095 [arXiv:1504.05559] [INSPIRE].
M. Bianchi and A.L. Guerrieri, On the soft limit of open string disk amplitudes with massive states, JHEP 09 (2015) 164 [arXiv:1505.05854] [INSPIRE].
P. Di Vecchia, R. Marotta and M. Mojaza, Double-soft behavior for scalars and gluons from string theory, JHEP 12 (2015) 150 [arXiv:1507.00938] [INSPIRE].
A.L. Guerrieri, Soft behavior of string amplitudes with external massive states, Nuovo Cim. C 39 (2016) 221 [arXiv:1507.08829] [INSPIRE].
S.D. Alston, D.C. Dunbar and W.B. Perkins, n-point amplitudes with a single negative-helicity graviton, Phys. Rev. D 92 (2015) 065024 [arXiv:1507.08882] [INSPIRE].
Y.-t. Huang and C. Wen, Soft theorems from anomalous symmetries, JHEP 12 (2015) 143 [arXiv:1509.07840] [INSPIRE].
P. Di Vecchia, R. Marotta and M. Mojaza, Soft Theorems from String Theory, Fortsch. Phys. 64 (2016) 389 [arXiv:1511.04921] [INSPIRE].
M. Bianchi and A.L. Guerrieri, On the soft limit of closed string amplitudes with massive states, Nucl. Phys. B 905 (2016) 188 [arXiv:1512.00803] [INSPIRE].
M. Bianchi and A.L. Guerrieri, On the soft limit of tree-level string amplitudes, in Proceedings, 14th Marcel Grossmann Meeting on Recent Developments in Theoretical and Experimental General Relativity, Astrophysics and Relativistic Field Theories (MG14) (In 4 Volumes), Rome, Italy, July 12–18, 2015, vol. 4, pp. 4157–4163 (2017) [https://doi.org/10.1142/9789813226609_0555] [arXiv:1601.03457] [INSPIRE].
J. Rao and B. Feng, Note on Identities Inspired by New Soft Theorems, JHEP 04 (2016) 173 [arXiv:1604.00650] [INSPIRE].
P. Di Vecchia, R. Marotta and M. Mojaza, Subsubleading soft theorems of gravitons and dilatons in the bosonic string, JHEP 06 (2016) 054 [arXiv:1604.03355] [INSPIRE].
S. He, Z. Liu and J.-B. Wu, Scattering Equations, Twistor-string Formulas and Double-soft Limits in Four Dimensions, JHEP 07 (2016) 060 [arXiv:1604.02834] [INSPIRE].
F. Cachazo, P. Cha and S. Mizera, Extensions of Theories from Soft Limits, JHEP 06 (2016) 170 [arXiv:1604.03893] [INSPIRE].
A.P. Saha, Double Soft Theorem for Perturbative Gravity, JHEP 09 (2016) 165 [arXiv:1607.02700] [INSPIRE].
P. Di Vecchia, R. Marotta and M. Mojaza, Soft behavior of a closed massless state in superstring and universality in the soft behavior of the dilaton, JHEP 12 (2016) 020 [arXiv:1610.03481] [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].
H. Elvang, C.R.T. Jones and S.G. Naculich, Soft Photon and Graviton Theorems in Effective Field Theory, Phys. Rev. Lett. 118 (2017) 231601 [arXiv:1611.07534] [INSPIRE].
C. Cheung, K. Kampf, J. Novotny, C.-H. Shen and J. Trnka, A Periodic Table of Effective Field Theories, JHEP 02 (2017) 020 [arXiv:1611.03137] [INSPIRE].
A.P. Saha, Double soft limit of the graviton amplitude from the Cachazo-He-Yuan formalism, Phys. Rev. D 96 (2017) 045002 [arXiv:1702.02350] [INSPIRE].
A. Sen, Soft Theorems in Superstring Theory, JHEP 06 (2017) 113 [arXiv:1702.03934] [INSPIRE].
A. Sen, Subleading Soft Graviton Theorem for Loop Amplitudes, JHEP 11 (2017) 123 [arXiv:1703.00024] [INSPIRE].
P. Di Vecchia, R. Marotta and M. Mojaza, Double-soft behavior of the dilaton of spontaneously broken conformal invariance, JHEP 09 (2017) 001 [arXiv:1705.06175] [INSPIRE].
A. Laddha and A. Sen, Sub-subleading Soft Graviton Theorem in Generic Theories of Quantum Gravity, JHEP 10 (2017) 065 [arXiv:1706.00759] [INSPIRE].
S. Chakrabarti, S.P. Kashyap, B. Sahoo, A. Sen and M. Verma, Subleading Soft Theorem for Multiple Soft Gravitons, JHEP 12 (2017) 150 [arXiv:1707.06803] [INSPIRE].
Y. Hamada and G. Shiu, Infinite Set of Soft Theorems in Gauge-Gravity Theories as Ward-Takahashi Identities, Phys. Rev. Lett. 120 (2018) 201601 [arXiv:1801.05528] [INSPIRE].
Z.-Z. Li, H.-H. Lin and S.-Q. Zhang, Infinite Soft Theorems from Gauge Symmetry, Phys. Rev. D 98 (2018) 045004 [arXiv:1802.03148] [INSPIRE].
S. Higuchi and H. Kawai, Universality of soft theorem from locality of soft vertex operators, Nucl. Phys. B 936 (2018) 400 [arXiv:1805.11079] [INSPIRE].
A. Laddha and A. Sen, Gravity Waves from Soft Theorem in General Dimensions, JHEP 09 (2018) 105 [arXiv:1801.07719] [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, arXiv:1806.01872 [INSPIRE].
P.C. Peters, Relativistic gravitational bremsstrahlung, Phys. Rev. D 1 (1970) 1559 [INSPIRE].
L. Blanchet and T. Damour, Hereditary effects in gravitational radiation, Phys. Rev. D 46 (1992) 4304 [INSPIRE].
W.D. Goldberger and A. Ross, Gravitational radiative corrections from effective field theory, Phys. Rev. D 81 (2010) 124015 [arXiv:0912.4254] [INSPIRE].
W.D. Goldberger, A. Ross and I.Z. Rothstein, Black hole mass dynamics and renormalization group evolution, Phys. Rev. D 89 (2014) 124033 [arXiv:1211.6095] [INSPIRE].
A. Gruzinov and G. Veneziano, Gravitational Radiation from Massless Particle Collisions, Class. Quant. Grav. 33 (2016) 125012 [arXiv:1409.4555] [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].
A. Addazi, M. Bianchi and G. Veneziano, Soft gravitational radiation from ultra-relativistic collisions at sub- and sub-sub-leading order, arXiv:1901.10986 [INSPIRE].
M. Ciafaloni, D. Colferai and G. Veneziano, Infrared features of gravitational scattering and radiation in the eikonal approach, arXiv:1812.08137 [INSPIRE].
G. Grammer Jr. and D.R. Yennie, Improved treatment for the infrared divergence problem in quantum electrodynamics, Phys. Rev. D 8 (1973) 4332 [INSPIRE].
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.
Author information
Authors and Affiliations
Corresponding author
Additional information
ArXiv ePrint: 1808.03288
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made.
The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder.
To view a copy of this licence, visit https://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
Sahoo, B., Sen, A. Classical and quantum results on logarithmic terms in the soft theorem in four dimensions. J. High Energ. Phys. 2019, 86 (2019). https://doi.org/10.1007/JHEP02(2019)086
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP02(2019)086