Abstract
We analyze the single subleading soft graviton theorem in (d + 1) dimensions under compactification on S1. This produces the single soft theorems for the graviton, vector and scalar fields in d dimension. For the compactification of 11-dimensional supergravity theory, this gives the soft factorization properties of the single graviton, dilaton and RR 1-form fields in type IIA string theory in ten dimensions. For the case of the soft vector field, we also explicitly check the result obtained from compactification by computing the amplitudes with external massive spin two and massless finite energy states interacting with soft vector field. The former are the Kaluza-Klein excitations of the d + 1 dimensional metric. Describing the interaction of the KK-modes with the vector field at each level by the minimally coupled Fierz-Pauli Lagrangian, we find agreement with the results obtained from the compactification if the gyromagnetic ratio in the minimally coupled Fierz-Pauli Lagrangian is taken to be g = 1.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
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].
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].
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].
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 et al., 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].
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].
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 et al., 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 et al., Testing subleading multiple soft graviton theorem for CHY prescription, JHEP 01 (2018) 090 [arXiv:1709.07883] [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, A classical proof of the classical soft graviton theorem in D > 4, arXiv:1906.08288 [INSPIRE].
P. Vecchia, R. Marotta and M. Mojaza, Multiloop soft theorem for gravitons and dilatons in the bosonic string, JHEP 01 (2019) 038 [arXiv:1808.04845] [INSPIRE].
P. Di Vecchia, R. Marotta and M. Mojaza, Multiloop soft theorem of the dilaton in the bosonic string, Phys. Rev. D 100 (2019) 041902 [arXiv:1907.01036] [INSPIRE].
C.D. White, Factorization properties of soft graviton amplitudes, JHEP 05 (2011) 060 [arXiv:1103.2981] [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].
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].
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].
S. Chakrabarti et al., Subleading soft theorem for multiple soft gravitons, JHEP 12 (2017) 150 [arXiv:1707.06803] [INSPIRE].
S. Atul Bhatkar and B. Sahoo, Subleading soft theorem for arbitrary number of external soft photons and gravitons, JHEP 01 (2019) 153 [arXiv:1809.01675] [INSPIRE].
D.R. Yennie, S.C. Frautschi and H. Suura, The infrared divergence phenomena and high-energy processes, Annals Phys. 13 (1961) 379 [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].
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].
F.E. Low, Bremsstrahlung of very low-energy quanta in elementary particle collisions, Phys. Rev. 110 (1958) 974 [INSPIRE].
S. Weinberg, Photons and gravitons in s matrix theory: derivation of charge conservation and equality of gravitational and inertial mass, Phys. Rev. B 135 (1964) 1049.
P. Di Vecchia, R. Marotta, M. Mojaza and J. Nohle, New soft theorems for the gravity dilaton and the Nambu-Goldstone dilaton at subsubleading order, Phys. Rev. D 93 (2016) 085015 [arXiv:1512.03316] [INSPIRE].
M. Campiglia and L. Coito, Asymptotic charges from soft scalars in even dimensions, Phys. Rev. D 97 (2018) 066009 [arXiv:1711.05773] [INSPIRE].
D. Francia and C. Heissenberg, Two-form asymptotic symmetries and scalar soft theorems, Phys. Rev. D 98 (2018) 105003 [arXiv:1810.05634] [INSPIRE].
E. Witten, String theory dynamics in various dimensions, Nucl. Phys. B 443 (1995) 85 [hep-th/9503124] [INSPIRE].
A. Sen, A note on marginally stable bound states in type-II string theory, Phys. Rev. D 54 (1996) 2964 [hep-th/9510229] [INSPIRE].
M. Bianchi et al., Exploring soft constraints on effective actions, JHEP 10 (2016) 036 [arXiv:1605.08697] [INSPIRE].
A.L. Guerrieri, Y.-t. Huang, Z. Li and C. Wen, On the exactness of soft theorems, JHEP 12 (2017) 052 [arXiv:1705.10078] [INSPIRE].
F. Loebbert, M. Mojaza and J. Plefka, Hidden conformal symmetry in tree-level graviton scattering, JHEP 05 (2018) 208 [arXiv:1802.05999] [INSPIRE].
R.H. Boels and W. Wormsbecher, Spontaneously broken conformal invariance in observables, arXiv:1507.08162 [INSPIRE].
M. Fierz and W. Pauli, On relativistic wave equations for particles of arbitrary spin in an electromagnetic field, Proc. Roy. Soc. Lond. A 173 (1939) 211 [INSPIRE].
Y.M. Cho and S.W. Zoh, Virasoro invariance and theory of internal string, Phys. Rev. D 46 (1992) 3483 [INSPIRE].
S. Deser and A. Waldron, Inconsistencies of massive charged gravitating higher spins, Nucl. Phys. B 631 (2002) 369 [hep-th/0112182] [INSPIRE].
A. Hosoya, K. Ishikawa, Y. Ohkuwa and K. Yamagishi, Gyromagnetic ratio of heavy particles in the Kaluza-Klein theory, Phys. lett. B 134 (1984) 44.
M.J. Duff, J.T. Liu and J. Rahmfeld, g = 1 for Dirichlet 0-branes, Nucl. Phys. B 524 (1998) 129 [hep-th/9801072] [INSPIRE].
M.J. Duff, R.R. Khuri and J.X. Lu, String solitons, Phys. Rept. 259 (1995) 213 [hep-th/9412184] [INSPIRE].
C. Csáki, TASI lectures on extra dimensions and branes, in From fields to strings. Volume 2, H. Shifman ed., World Scientific, Singapore (2004), hep-ph/0404096 [INSPIRE].
Y.M. Cho and S.W. Zoh, Explicit construction of massive spin two fields in Kaluza-Klein theory, Phys. Rev. D 46 (1992) R2290 [INSPIRE].
J. Bonifacio and K. Hinterbichler, Unitarization from geometry, JHEP 12 (2019) 165 [arXiv:1910.04767] [INSPIRE].
R. Rahman, Higher spin theory — Part I, PoS(ModaveVIII)004 [arXiv:1307.3199] [INSPIRE].
M. Porrati and R. Rahman, Intrinsic cutoff and acausality for massive spin 2 fields coupled to electromagnetism, Nucl. Phys. B 801 (2008) 174 [arXiv:0801.2581] [INSPIRE].
M.J. Duff, Kaluza-Klein theory in perspective, in The Oskar Klein centenary proceedings, U. Lindstroem ed., World Scientific, Singapore (1995), hep-th/9410046.
C.R. Nappi and L. Witten, Interacting Lagrangian for massive spin two field, Phys. Rev. D 40 (1989) 1095 [INSPIRE].
L. Dolan and M.J. Duff, Kac-Moody symmetries of Kaluza-Klein theories, Phys. Rev. D 52 (1984) 14.
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: 1911.05099
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
Marotta, R., Verma, M. Soft theorems from compactification. J. High Energ. Phys. 2020, 8 (2020). https://doi.org/10.1007/JHEP02(2020)008
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP02(2020)008