Abstract
In arXiv:1707.06803 we derived the subleading multiple soft graviton theorem in a generic quantum theory of gravity for arbitrary number of soft external gravitons and arbitrary number of finite energy external states carrying arbitrary mass and spin. In this paper we verify this explicitly using the CHY formula for tree level scattering amplitudes of arbitrary number of gravitons in Einstein gravity. We pay special care to fix the signs of the amplitudes and resolve an apparent discrepancy between our general results in arXiv:1707.06803 and previous results on soft graviton theorem from CHY formula.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
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].
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].
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].
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].
A. Sabio Vera and M.A. Vazquez-Mozo, The Double Copy Structure of Soft Gravitons, JHEP 03 (2015) 070 [arXiv:1412.3699] [INSPIRE].
A.E. Lipstein, Soft Theorems from Conformal Field Theory, JHEP 06 (2015) 166 [arXiv:1504.01364] [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].
J. Rao and B. Feng, Note on Identities Inspired by New Soft Theorems, JHEP 04 (2016) 173 [arXiv:1604.00650] [INSPIRE].
F. Cachazo, P. Cha and S. Mizera, Extensions of Theories from Soft Limits, JHEP 06 (2016) 170 [arXiv:1604.03893] [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. 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].
A. Laddha and A. Sen, Sub-subleading Soft Graviton Theorem in Generic Theories of Quantum Gravity, JHEP 10 (2017) 065 [arXiv:1706.00759] [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].
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].
B.U.W. Schwab, A Note on Soft Factors for Closed String Scattering, JHEP 03 (2015) 140 [arXiv:1411.6661] [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].
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].
A.L. Guerrieri, Soft behavior of string amplitudes with external massive states, Nuovo Cim. C 39 (2016) 221 [arXiv:1507.08829] [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, arXiv:1601.03457 [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].
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].
F. Cachazo, S. He and E.Y. Yuan, New Double Soft Emission Theorems, Phys. Rev. D 92 (2015) 065030 [arXiv:1503.04816] [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].
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].
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].
A.P. Saha, Double Soft Theorem for Perturbative Gravity, JHEP 09 (2016) 165 [arXiv:1607.02700] [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].
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].
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].
G. Georgiou, Multi-soft theorems in Gauge Theory from MHV Diagrams, JHEP 08 (2015) 128 [arXiv:1505.08130] [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].
A. Strominger and A. Zhiboedov, Gravitational Memory, BMS Supertranslations and Soft Theorems, JHEP 01 (2016) 086 [arXiv:1411.5745] [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, 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. Strominger, Lectures on the Infrared Structure of Gravity and Gauge Theory, arXiv:1703.05448 [INSPIRE].
F. Cachazo, S. He and E.Y. Yuan, Scattering equations and Kawai-Lewellen-Tye orthogonality, Phys. Rev. D 90 (2014) 065001 [arXiv:1306.6575] [INSPIRE].
F. Cachazo, S. He and E.Y. Yuan, Scattering of Massless Particles in Arbitrary Dimensions, Phys. Rev. Lett. 113 (2014) 171601 [arXiv:1307.2199] [INSPIRE].
F. Cachazo, S. He and E.Y. Yuan, Scattering of Massless Particles: Scalars, Gluons and Gravitons, JHEP 07 (2014) 033 [arXiv:1309.0885] [INSPIRE].
F. Cachazo, S. He and E.Y. Yuan, Einstein-Yang-Mills Scattering Amplitudes From Scattering Equations, JHEP 01 (2015) 121 [arXiv:1409.8256] [INSPIRE].
F. Cachazo, S. He and E.Y. Yuan, Scattering Equations and Matrices: From Einstein To Yang-Mills, DBI and NLSM, JHEP 07 (2015) 149 [arXiv:1412.3479] [INSPIRE].
Y. Yuan, Scattering equations and S-matrices, https://uwspace.uwaterloo.ca/bitstream/handle/10012/9451/Ye_Yuan.pdf;sequence=3 .
F. Cachazo, S. He and E.Y. Yuan, Detailed Proof of Double Soft Theorems Up to the Sub-Leading Order, https://ellisyeyuan.wordpress.com/2015/03/14/double-soft-theorems-subleading/.
M. Zlotnikov, Leading multi-soft limits from scattering equations, JHEP 10 (2017) 209 [arXiv:1708.05016] [INSPIRE].
Y.-J. Du, B. Feng and F. Teng, Expansion of All Multitrace Tree Level EYM Amplitudes, JHEP 12 (2017) 038 [arXiv:1708.04514] [INSPIRE].
F. Cachazo, S. Mizera and G. Zhang, Scattering Equations: Real Solutions and Particles on a Line, JHEP 03 (2017) 151 [arXiv:1609.00008] [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: 1709.07883
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (https://creativecommons.org/licenses/by/4.0), which permits use, duplication, 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 license, and indicate if changes were made.
About this article
Cite this article
Chakrabarti, S., Kashyap, S.P., Sahoo, B. et al. Testing subleading multiple soft graviton theorem for CHY prescription. J. High Energ. Phys. 2018, 90 (2018). https://doi.org/10.1007/JHEP01(2018)090
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP01(2018)090