Abstract
The origin of the graviton from string theory is well understood: it corresponds to a massless state in closed string spectra, whose low-energy effective action, as extracted from string scattering amplitudes, is that of Einstein-Hilbert. In this work, we explore the possibility of such a string-theoretic emergence of ghost-free bimetric theory, a recently proposed theory that involves two dynamical metrics, that around particular backgrounds propagates the graviton and a massive spin-2 field, which has been argued to be a viable dark matter candidate. By choosing to identify the latter with a massive spin-2 state of open string spectra, we compute tree-level three-point string scattering amplitudes that describe interactions of the massive spin-2 with itself and with the graviton. With the mass of the external legs depending on the string scale, we discover that extracting the corresponding low-energy effective actions in four spacetime dimensions is a subtle but consistent process and proceed to appropriately compare them with bimetric theory. Our findings consist in establishing that string and bimetric theory provide to lowest order the same set of two-derivative terms describing the interactions of the massive spin-2 with itself and with the graviton, albeit up to numerical coefficient discrepancies, a fact that we analyze and interpret. We conclude with a mention of future investigations.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
J. Scherk and J.H. Schwarz, Dual Models for Nonhadrons, Nucl. Phys. B 81 (1974) 118 [INSPIRE].
M.B. Green, J.H. Schwarz and L. Brink, N = 4 Yang-Mills and N = 8 Supergravity as Limits of String Theories, Nucl. Phys. B 198 (1982) 474 [INSPIRE].
D.J. Gross and J.H. Sloan, The Quartic Effective Action for the Heterotic String, Nucl. Phys. B 291 (1987) 41 [INSPIRE].
R. Blumenhagen, D. Lüst and S. Theisen, Basic concepts of string theory, Springer-Verlag (2013).
K.S. Stelle, Renormalization of Higher Derivative Quantum Gravity, Phys. Rev. D 16 (1977) 953 [INSPIRE].
K.S. Stelle, Classical Gravity with Higher Derivatives, Gen. Rel. Grav. 9 (1978) 353 [INSPIRE].
D.G. Boulware, G.T. Horowitz and A. Strominger, Zero Energy Theorem for Scale Invariant Gravity, Phys. Rev. Lett. 50 (1983) 1726 [INSPIRE].
F. David and A. Strominger, On the Calculability of Newton’s Constant and the Renormalizability of Scale Invariant Quantum Gravity, Phys. Lett. B 143 (1984) 125 [INSPIRE].
G.T. Horowitz, Quantum Cosmology With a Positive Definite Action, Phys. Rev. D 31 (1985) 1169 [Adv. Ser. Astrophys. Cosmol. 3 (1987) 292] [INSPIRE].
D.J. Gross and E. Witten, Superstring Modifications of Einstein’s Equations, Nucl. Phys. B 277 (1986) 1 [INSPIRE].
S. Deser and B. Tekin, Shortcuts to high symmetry solutions in gravitational theories, Class. Quant. Grav. 20 (2003) 4877 [gr-qc/0306114] [INSPIRE].
S. Deser and B. Tekin, New energy definition for higher curvature gravities, Phys. Rev. D 75 (2007) 084032 [gr-qc/0701140] [INSPIRE].
G. ’t Hooft, A class of elementary particle models without any adjustable real parameters, Found. Phys. 41 (2011) 1829 [arXiv:1104.4543] [INSPIRE].
J. Maldacena, Einstein Gravity from Conformal Gravity, arXiv:1105.5632 [INSPIRE].
H. Lü, C.N. Pope, E. Sezgin and L. Wulff, Critical and Non-Critical Einstein-Weyl Supergravity, JHEP 10 (2011) 131 [arXiv:1107.2480] [INSPIRE].
K. Hinterbichler, Theoretical Aspects of Massive Gravity, Rev. Mod. Phys. 84 (2012) 671 [arXiv:1105.3735] [INSPIRE].
C. de Rham, Massive Gravity, Living Rev. Rel. 17 (2014) 7 [arXiv:1401.4173] [INSPIRE].
G.R. Dvali, G. Gabadadze and M. Porrati, 4D gravity on a brane in 5D Minkowski space, Phys. Lett. B 485 (2000) 208 [hep-th/0005016] [INSPIRE].
C. de Rham, G. Gabadadze and A.J. Tolley, Resummation of Massive Gravity, Phys. Rev. Lett. 106 (2011) 231101 [arXiv:1011.1232] [INSPIRE].
S.F. Hassan and R.A. Rosen, Bimetric Gravity from Ghost-free Massive Gravity, JHEP 02 (2012) 126 [arXiv:1109.3515] [INSPIRE].
S.F. Hassan, R.A. Rosen and A. Schmidt-May, Ghost-free Massive Gravity with a General Reference Metric, JHEP 02 (2012) 026 [arXiv:1109.3230] [INSPIRE].
S.F. Hassan, A. Schmidt-May and M. von Strauss, Higher Derivative Gravity and Conformal Gravity From Bimetric and Partially Massless Bimetric Theory, Universe 1 (2015) 92 [arXiv:1303.6940] [INSPIRE].
S. Deser, E. Joung and A. Waldron, Gravitational- and Self- Coupling of Partially Massless Spin 2, Phys. Rev. D 86 (2012) 104004 [arXiv:1301.4181] [INSPIRE].
F. Del Monte, D. Francia and P.A. Grassi, Multimetric Supergravities, JHEP 09 (2016) 064 [arXiv:1605.06793] [INSPIRE].
A. Schmidt-May and M. von Strauss, Recent developments in bimetric theory, J. Phys. A 49 (2016) 183001 [arXiv:1512.00021] [INSPIRE].
N. Rosen, General Relativity and Flat Space. II, Phys. Rev. 57 (1940) 150 [INSPIRE].
N. Rosen, Bimetric Theory of Gravitation, NATO Sci. Ser. B 27 (1977) 271 [INSPIRE].
C. Aragone and S. Deser, Constraints on gravitationally coupled tensor fields, Nuovo Cim. A 3 (1971) 709 [INSPIRE].
C.J. Isham, A. Salam and J.A. Strathdee, F-dominance of gravity, Phys. Rev. D 3 (1971) 867 [INSPIRE].
L. Bernard, L. Blanchet and L. Heisenberg, Bimetric gravity and dark matter, arXiv:1507.02802 [INSPIRE].
L. Blanchet and L. Heisenberg, Dark Matter via Massive (bi-)Gravity, Phys. Rev. D 91 (2015) 103518 [arXiv:1504.00870] [INSPIRE].
K. Aoki and S. Mukohyama, Massive gravitons as dark matter and gravitational waves, Phys. Rev. D 94 (2016) 024001 [arXiv:1604.06704] [INSPIRE].
E. Babichev et al., Heavy spin-2 Dark Matter, JCAP 09 (2016) 016 [arXiv:1607.03497] [INSPIRE].
E. Babichev et al., Bigravitational origin of dark matter, Phys. Rev. D 94 (2016) 084055 [arXiv:1604.08564] [INSPIRE].
D. Lüst, C. Markou, P. Mazloumi and S. Stieberger, Extracting bigravity from string theory (II), work in progress.
Z. Bern, J.J. Carrasco, M. Chiodaroli, H. Johansson and R. Roiban, The Duality Between Color and Kinematics and its Applications, arXiv:1909.01358 [INSPIRE].
A. Momeni, J. Rumbutis and A.J. Tolley, Massive Gravity from Double Copy, JHEP 12 (2020) 030 [arXiv:2004.07853] [INSPIRE].
L.A. Johnson, C.R.T. Jones and S. Paranjape, Constraints on a Massive Double-Copy and Applications to Massive Gravity, JHEP 02 (2021) 148 [arXiv:2004.12948] [INSPIRE].
S. Ferrara and D. Lüst, Spin-four \( \mathcal{N} \) = 7 W-supergravity: S-fold and double copy construction, JHEP 07 (2018) 114 [arXiv:1805.10022] [INSPIRE].
S. Ferrara, A. Kehagias and D. Lüst, Bimetric, Conformal Supergravity and its Superstring Embedding, JHEP 05 (2019) 100 [arXiv:1810.08147] [INSPIRE].
B. Gording and A. Schmidt-May, Ghost-free infinite derivative gravity, JHEP 09 (2018) 044 [Erratum JHEP 10 (2018) 115] [arXiv:1807.05011] [INSPIRE].
M. Porrati, Higgs phenomenon for the graviton in AdS space, Mod. Phys. Lett. A 18 (2003) 1793 [hep-th/0306253] [INSPIRE].
E. Kiritsis, Product CFTs, gravitational cloning, massive gravitons and the space of gravitational duals, JHEP 11 (2006) 049 [hep-th/0608088] [INSPIRE].
O. Aharony, A.B. Clark and A. Karch, The CFT/AdS correspondence, massive gravitons and a connectivity index conjecture, Phys. Rev. D 74 (2006) 086006 [hep-th/0608089] [INSPIRE].
E. Kiritsis and V. Niarchos, Interacting String Multi-verses and Holographic Instabilities of Massive Gravity, Nucl. Phys. B 812 (2009) 488 [arXiv:0808.3410] [INSPIRE].
C. Bachas and I. Lavdas, Quantum Gates to other Universes, Fortsch. Phys. 66 (2018) 1700096 [arXiv:1711.11372] [INSPIRE].
C. Bachas and I. Lavdas, Massive Anti-de Sitter Gravity from String Theory, JHEP 11 (2018) 003 [arXiv:1807.00591] [INSPIRE].
C. Bachas, Massive AdS Supergravitons and Holography, JHEP 06 (2019) 073 [arXiv:1905.05039] [INSPIRE].
S.F. Hassan and R.A. Rosen, Confirmation of the Secondary Constraint and Absence of Ghost in Massive Gravity and Bimetric Gravity, JHEP 04 (2012) 123 [arXiv:1111.2070] [INSPIRE].
S.F. Hassan and R.A. Rosen, On Non-Linear Actions for Massive Gravity, JHEP 07 (2011) 009 [arXiv:1103.6055] [INSPIRE].
N. Arkani-Hamed, H. Georgi and M.D. Schwartz, Effective field theory for massive gravitons and gravity in theory space, Annals Phys. 305 (2003) 96 [hep-th/0210184] [INSPIRE].
Y. Yamashita, A. De Felice and T. Tanaka, Appearance of Boulware-Deser ghost in bigravity with doubly coupled matter, Int. J. Mod. Phys. D 23 (2014) 1443003 [arXiv:1408.0487] [INSPIRE].
C. de Rham, L. Heisenberg and R.H. Ribeiro, On couplings to matter in massive (bi-)gravity, Class. Quant. Grav. 32 (2015) 035022 [arXiv:1408.1678] [INSPIRE].
C. de Rham, A. Matas and A.J. Tolley, New Kinetic Interactions for Massive Gravity?, Class. Quant. Grav. 31 (2014) 165004 [arXiv:1311.6485] [INSPIRE].
C. de Rham, A. Matas and A.J. Tolley, New Kinetic Terms for Massive Gravity and Multi-gravity: A No-Go in Vielbein Form, Class. Quant. Grav. 32 (2015) 215027 [arXiv:1505.00831] [INSPIRE].
A. Matas, Cutoff for Extensions of Massive Gravity and Bi-Gravity, Class. Quant. Grav. 33 (2016) 075004 [arXiv:1506.00666] [INSPIRE].
S.F. Hassan, A. Schmidt-May and M. von Strauss, On Consistent Theories of Massive Spin-2 Fields Coupled to Gravity, JHEP 05 (2013) 086 [arXiv:1208.1515] [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].
I.L. Buchbinder, D.M. Gitman, V.A. Krykhtin and V.D. Pershin, Equations of motion for massive spin-2 field coupled to gravity, Nucl. Phys. B 584 (2000) 615 [hep-th/9910188] [INSPIRE].
V. Nikiforova and T. Damour, Black holes in torsion bigravity, Phys. Rev. D 102 (2020) 084027 [arXiv:2007.08606] [INSPIRE].
E. Sezgin and P. van Nieuwenhuizen, New Ghost Free Gravity Lagrangians with Propagating Torsion, Phys. Rev. D 21 (1980) 3269 [INSPIRE].
E. Sezgin, Class of Ghost Free Gravity Lagrangians With Massive or Massless Propagating Torsion, Phys. Rev. D 24 (1981) 1677 [INSPIRE].
K. Hayashi and T. Shirafuji, Gravity from Poincaré Gauge Theory of the Fundamental Particles. 1. Linear and Quadratic Lagrangians, Prog. Theor. Phys. 64 (1980) 866 [Erratum ibid. 65 (1981) 2079] [INSPIRE].
K. Hayashi and T. Shirafuji, Gravity From Poincaré Gauge Theory of the Fundamental Particles. 2. Equations of motion for test bodies and various limits, Prog. Theor. Phys. 64 (1980) 883 [Erratum ibid. 65 (1981) 2079] [INSPIRE].
K. Hayashi and T. Shirafuji, Gravity From Poincaré Gauge Theory of the Fundamental Particles. 3. Weak Field Approximation, Prog. Theor. Phys. 64 (1980) 1435 [Erratum ibid. 66 (1981) 741] [INSPIRE].
K. Hayashi and T. Shirafuji, Gravity From Poincaré Gauge Theory of the Fundamental Particles. 4. Mass and Energy of Particle Spectrum, Prog. Theor. Phys. 64 (1980) 2222 [INSPIRE].
V.P. Nair, S. Randjbar-Daemi and V. Rubakov, Massive Spin-2 fields of Geometric Origin in Curved Spacetimes, Phys. Rev. D 80 (2009) 104031 [arXiv:0811.3781] [INSPIRE].
V. Nikiforova, S. Randjbar-Daemi and V. Rubakov, Infrared Modified Gravity with Dynamical Torsion, Phys. Rev. D 80 (2009) 124050 [arXiv:0905.3732] [INSPIRE].
C. Deffayet and S. Randjbar-Daemi, Non linear Fierz-Pauli theory from torsion and bigravity, Phys. Rev. D 84 (2011) 044053 [arXiv:1103.2671] [INSPIRE].
T. Damour and V. Nikiforova, Spherically symmetric solutions in torsion bigravity, Phys. Rev. D 100 (2019) 024065 [arXiv:1906.11859] [INSPIRE].
N. Boulanger, T. Damour, L. Gualtieri and M. Henneaux, Inconsistency of interacting, multigraviton theories, Nucl. Phys. B 597 (2001) 127 [hep-th/0007220] [INSPIRE].
S. Stieberger, Open & Closed vs. Pure Open String Disk Amplitudes, arXiv:0907.2211 [INSPIRE].
S. Stieberger and T.R. Taylor, Disk Scattering of Open and Closed Strings (I), Nucl. Phys. B 903 (2016) 104 [arXiv:1510.01774] [INSPIRE].
I.R. Klebanov and L. Thorlacius, The Size of p-branes, Phys. Lett. B 371 (1996) 51 [hep-th/9510200] [INSPIRE].
S.S. Gubser, A. Hashimoto, I.R. Klebanov and J.M. Maldacena, Gravitational lensing by p-branes, Nucl. Phys. B 472 (1996) 231 [hep-th/9601057] [INSPIRE].
M.R. Garousi and R.C. Myers, Superstring scattering from D-branes, Nucl. Phys. B 475 (1996) 193 [hep-th/9603194] [INSPIRE].
A. Hashimoto and I.R. Klebanov, Decay of excited D-branes, Phys. Lett. B 381 (1996) 437 [hep-th/9604065] [INSPIRE].
A. Hashimoto and I.R. Klebanov, Scattering of strings from D-branes, Nucl. Phys. B Proc. Suppl. 55 (1997) 118 [hep-th/9611214] [INSPIRE].
D. Lüst, P. Mayr, R. Richter and S. Stieberger, Scattering of gauge, matter, and moduli fields from intersecting branes, Nucl. Phys. B 696 (2004) 205 [hep-th/0404134] [INSPIRE].
W.-Z. Feng, D. Lüst, O. Schlotterer, S. Stieberger and T.R. Taylor, Direct Production of Lightest Regge Resonances, Nucl. Phys. B 843 (2011) 570 [arXiv:1007.5254] [INSPIRE].
M. Guillen, H. Johansson, R.L. Jusinskas and O. Schlotterer, Scattering Massive String Resonances through Field-Theory Methods, Phys. Rev. Lett. 127 (2021) 051601 [arXiv:2104.03314] [INSPIRE].
D. Friedan, E.J. Martinec and S.H. Shenker, Conformal Invariance, Supersymmetry and String Theory, Nucl. Phys. B 271 (1986) 93 [INSPIRE].
P. Mayr and S. Stieberger, Dilaton, antisymmetric tensor and gauge fields in string effective theories at the one loop level, Nucl. Phys. B 412 (1994) 502 [hep-th/9304055] [INSPIRE].
I.G. Koh, W. Troost and A. Van Proeyen, Covariant Higher Spin Vertex Operators in the Ramond Sector, Nucl. Phys. B 292 (1987) 201 [INSPIRE].
W.-Z. Feng, D. Lüst and O. Schlotterer, Massive Supermultiplets in Four-Dimensional Superstring Theory, Nucl. Phys. B 861 (2012) 175 [arXiv:1202.4466] [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].
R. Blumenhagen, B. Körs, D. Lüst and S. Stieberger, Four-dimensional String Compactifications with D-branes, Orientifolds and Fluxes, Phys. Rept. 445 (2007) 1 [hep-th/0610327] [INSPIRE].
S. Ferrara, A. Kehagias and D. Lüst, Aspects of Weyl Supergravity, JHEP 08 (2018) 197 [arXiv:1806.10016] [INSPIRE].
J. Polchinski, String theory. Volume 1. An introduction to the bosonic string, in Cambridge Monographs on Mathematical Physics, Cambridge University Press, Cambridge U.K. (1998).
D. Lüst, S. Theisen and G. Zoupanos, Four-dimensional Heterotic Strings and Conformal Field Theory, Nucl. Phys. B 296 (1988) 800 [INSPIRE].
S. Stieberger and T.R. Taylor, Graviton as a Pair of Collinear Gauge Bosons, Phys. Lett. B 739 (2014) 457 [arXiv:1409.4771] [INSPIRE].
S. Stieberger and T.R. Taylor, Graviton Amplitudes from Collinear Limits of Gauge Amplitudes, Phys. Lett. B 744 (2015) 160 [arXiv:1502.00655] [INSPIRE].
S. Stieberger and T.R. Taylor, Subleading terms in the collinear limit of Yang-Mills amplitudes, Phys. Lett. B 750 (2015) 587 [arXiv:1508.01116] [INSPIRE].
S. Stieberger and T.R. Taylor, New relations for Einstein-Yang-Mills amplitudes, Nucl. Phys. B 913 (2016) 151 [arXiv:1606.09616] [INSPIRE].
C. de Rham and G. Gabadadze, Selftuned Massive Spin-2, Phys. Lett. B 693 (2010) 334 [arXiv:1006.4367] [INSPIRE].
C. de Rham and G. Gabadadze, Generalization of the Fierz-Pauli Action, Phys. Rev. D 82 (2010) 044020 [arXiv:1007.0443] [INSPIRE].
H. van Dam and M.J.G. Veltman, Massive and massless Yang-Mills and gravitational fields, Nucl. Phys. B 22 (1970) 397 [INSPIRE].
V.I. Zakharov, Linearized gravitation theory and the graviton mass, JETP Lett. 12 (1970) 312 [Pisma Zh. Eksp. Teor. Fiz. 12 (1970) 447] [INSPIRE].
K. Aoki, Nonlinearly ghost-free higher curvature gravity, Phys. Rev. D 102 (2020) 124049 [arXiv:2009.11739] [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: 2106.04614
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
Lüst, D., Markou, C., Mazloumi, P. et al. Extracting bigravity from string theory. J. High Energ. Phys. 2021, 220 (2021). https://doi.org/10.1007/JHEP12(2021)220
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP12(2021)220