Abstract
We place model-independent constraints on theories of massive spin-2 particles by considering the positivity of the phase shift in eikonal scattering. The phase shift is an asymptotic S-matrix observable, related to the time delay/advance experienced by a particle during scattering. Demanding the absence of a time advance leads to constraints on the cubic vertices present in the theory. We find that, in theories with massive spin-2 particles, requiring no time advance means that either: (i) the cubic vertices must appear as a particular linear combination of the Einstein-Hilbert cubic vertex and an h 3 μν potential term or (ii) new degrees of freedom or strong coupling must enter at parametrically the mass of the massive spin-2 field. These conclusions have implications for a variety of situations. Applied to theories of large-N QCD, this indicates that any spectrum with an isolated massive spin-2 at the bottom must have these particular cubic self-couplings. Applied to de Rham-Gabadadze-Tolley massive gravity, the constraint is in accord with results obtained from a shockwave calculation: of the two free dimensionless parameters in the theory there is a one parameter line consistent with a subluminal phase shift.
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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].
P. Creminelli, A. Nicolis, M. Papucci and E. Trincherini, Ghosts in massive gravity, JHEP 09 (2005) 003 [hep-th/0505147] [INSPIRE].
G. Goon, K. Hinterbichler, A. Joyce and M. Trodden, Einstein Gravity, Massive Gravity, Multi-Gravity and Nonlinear Realizations, JHEP 07 (2015) 101 [arXiv:1412.6098] [INSPIRE].
M. Torabian, Towards a Lorentz Invariant UV Completion for Massive Gravity: dRGT Theory from Spontaneous Symmetry Breaking, arXiv:1707.04403 [INSPIRE].
M. Porrati, Higgs phenomenon for 4-D gravity in anti-de Sitter space, JHEP 04 (2002) 058 [hep-th/0112166] [INSPIRE].
G. Gabadadze, Scale-up of Λ3 : Massive gravity with a higher strong interaction scale, Phys. Rev. D 96 (2017) 084018 [arXiv:1707.01739] [INSPIRE].
A. Adams, N. Arkani-Hamed, S. Dubovsky, A. Nicolis and R. Rattazzi, Causality, analyticity and an IR obstruction to UV completion, JHEP 10 (2006) 014 [hep-th/0602178] [INSPIRE].
C. de Rham, S. Melville, A.J. Tolley and S.-Y. Zhou, Positivity bounds for scalar field theories, Phys. Rev. D 96 (2017) 081702 [arXiv:1702.06134] [INSPIRE].
C. de Rham, S. Melville, A.J. Tolley and S.-Y. Zhou, UV complete me: Positivity Bounds for Particles with Spin, arXiv:1706.02712 [INSPIRE].
C. Cheung and G.N. Remmen, Positive Signs in Massive Gravity, JHEP 04 (2016) 002 [arXiv:1601.04068] [INSPIRE].
J. Bonifacio, K. Hinterbichler and R.A. Rosen, Positivity constraints for pseudolinear massive spin-2 and vector Galileons, Phys. Rev. D 94 (2016) 104001 [arXiv:1607.06084] [INSPIRE].
G. Velo and D. Zwanziger, Propagation and quantization of Rarita-Schwinger waves in an external electromagnetic potential, Phys. Rev. 186 (1969) 1337 [INSPIRE].
G. Velo and D. Zwanziger, Noncausality and other defects of interaction lagrangians for particles with spin one and higher, Phys. Rev. 188 (1969) 2218 [INSPIRE].
S. Deser and A. Waldron, Acausality of Massive Gravity, Phys. Rev. Lett. 110 (2013) 111101 [arXiv:1212.5835] [INSPIRE].
S. Deser, K. Izumi, Y.C. Ong and A. Waldron, Problems of massive gravities, Mod. Phys. Lett. A 30 (2015) 1540006 [arXiv:1410.2289] [INSPIRE].
S. Deser, M. Sandora, A. Waldron and G. Zahariade, Covariant constraints for generic massive gravity and analysis of its characteristics, Phys. Rev. D 90 (2014) 104043 [arXiv:1408.0561] [INSPIRE].
S. Deser, A. Waldron and G. Zahariade, Propagation peculiarities of mean field massive gravity, Phys. Lett. B 749 (2015) 144 [arXiv:1504.02919] [INSPIRE].
C. Burrage, C. de Rham, L. Heisenberg and A.J. Tolley, Chronology Protection in Galileon Models and Massive Gravity, JCAP 07 (2012) 004 [arXiv:1111.5549] [INSPIRE].
S.F. Hassan and M. Kocic, On the local structure of spacetime in ghost-free bimetric theory and massive gravity, arXiv:1706.07806 [INSPIRE].
G. Goon and K. Hinterbichler, Superluminality, black holes and EFT, JHEP 02 (2017) 134 [arXiv:1609.00723] [INSPIRE].
G. ’t Hooft, Graviton Dominance in Ultrahigh-Energy Scattering, Phys. Lett. B 198 (1987) 61 [INSPIRE].
D.N. Kabat and M. Ortiz, Eikonal quantum gravity and Planckian scattering, Nucl. Phys. B 388 (1992) 570 [hep-th/9203082] [INSPIRE].
X.O. Camanho, J.D. Edelstein, J. Maldacena and A. Zhiboedov, Causality Constraints on Corrections to the Graviton Three-Point Coupling, JHEP 02 (2016) 020 [arXiv:1407.5597] [INSPIRE].
B. Zwiebach, Curvature Squared Terms and String Theories, Phys. Lett. B 156 (1985) 315 [INSPIRE].
G. D’Appollonio, P. Di Vecchia, R. Russo and G. Veneziano, Regge behavior saves String Theory from causality violations, JHEP 05 (2015) 144 [arXiv:1502.01254] [INSPIRE].
J.D. Edelstein, G. Giribet, C. Gomez, E. Kilicarslan, M. Leoni and B. Tekin, Causality in 3D Massive Gravity Theories, Phys. Rev. D 95 (2017) 104016 [arXiv:1602.03376] [INSPIRE].
C. de Rham and G. Gabadadze, Generalization of the Fierz-Pauli Action, Phys. Rev. D 82 (2010) 044020 [arXiv:1007.0443] [INSPIRE].
C. de Rham, G. Gabadadze and A.J. Tolley, Resummation of Massive Gravity, Phys. Rev. Lett. 106 (2011) 231101 [arXiv:1011.1232] [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].
S. Folkerts, A. Pritzel and N. Wintergerst, On ghosts in theories of self-interacting massive spin-2 particles, arXiv:1107.3157 [INSPIRE].
K. Hinterbichler, Ghost-Free Derivative Interactions for a Massive Graviton, JHEP 10 (2013) 102 [arXiv:1305.7227] [INSPIRE].
X.O. Camanho, G. Lucena Gómez and R. Rahman, Causality Constraints on Massive Gravity, Phys. Rev. D 96 (2017) 084007 [arXiv:1610.02033] [INSPIRE].
T. Dray and G. ’t Hooft, The Gravitational Shock Wave of a Massless Particle, Nucl. Phys. B 253 (1985) 173 [INSPIRE].
R. Penrose, On Schwarzschild Causality — A Problem for “Lorentz Covariant” General Relativity, in Essays in General Relativity. A Festschrift for Abraham Taub Academic Press (1980).
K.D. Olum, Superluminal travel requires negative energies, Phys. Rev. Lett. 81 (1998) 3567 [gr-qc/9805003] [INSPIRE].
S. Gao and R.M. Wald, Theorems on gravitational time delay and related issues, Class. Quant. Grav. 17 (2000) 4999 [gr-qc/0007021] [INSPIRE].
E. Babichev, V. Mukhanov and A. Vikman, k-Essence, superluminal propagation, causality and emergent geometry, JHEP 02 (2008) 101 [arXiv:0708.0561] [INSPIRE].
R. Geroch, Faster Than Light?, AMS/IP Stud. Adv. Math. 49 (2011) 59 [arXiv:1005.1614] [INSPIRE].
G. Papallo and H.S. Reall, Graviton time delay and a speed limit for small black holes in Einstein-Gauss-Bonnet theory, JHEP 11 (2015) 109 [arXiv:1508.05303] [INSPIRE].
J.R. Gott, III, Closed timelike curves produced by pairs of moving cosmic strings: Exact solutions, Phys. Rev. Lett. 66 (1991) 1126 [INSPIRE].
S.M. Carroll, E. Farhi and A.H. Guth, An Obstacle to building a time machine, Phys. Rev. Lett. 68 (1992) 263 [Erratum ibid. 68 (1992) 3368] [INSPIRE].
S. Dubovsky, A. Nicolis, E. Trincherini and G. Villadoro, Microcausality in curved space-time, Phys. Rev. D 77 (2008) 084016 [arXiv:0709.1483] [INSPIRE].
S.M. Carroll, Spacetime and geometry: An introduction to general relativity, Addison-Wesley (2004) [INSPIRE].
H. Cheng and T.T. Wu, High-energy elastic scattering in quantum electrodynamics, Phys. Rev. Lett. 22 (1969) 666 [INSPIRE].
M. Levy and J. Sucher, Eikonal approximation in quantum field theory, Phys. Rev. 186 (1969) 1656 [INSPIRE].
H.D.I. Abarbanel and C. Itzykson, Relativistic eikonal expansion, Phys. Rev. Lett. 23 (1969) 53 [INSPIRE].
G.F. Giudice, R. Rattazzi and J.D. Wells, Transplanckian collisions at the LHC and beyond, Nucl. Phys. B 630 (2002) 293 [hep-ph/0112161] [INSPIRE].
S.B. Giddings, The gravitational S-matrix: Erice lectures, Subnucl. Ser. 48 (2013) 93 [arXiv:1105.2036] [INSPIRE].
G. Tiktopoulos and S.B. Treiman, Relativistic eikonal approximation, Phys. Rev. D 3 (1971) 1037 [INSPIRE].
H. Cheng and T.T. Wu, Expanding Protons: Scattering At High-Energies, (1987) [INSPIRE].
D.N. Kabat, Validity of the Eikonal approximation, Comments Nucl. Part. Phys. 20 (1992) 325 [hep-th/9204103] [INSPIRE].
R. Akhoury, R. Saotome and G. Sterman, High Energy Scattering in Perturbative Quantum Gravity at Next to Leading Power, arXiv:1308.5204 [INSPIRE].
N.E.J. Bjerrum-Bohr, J.F. Donoghue, B.R. Holstein, L. Plante and P. Vanhove, Light-like Scattering in Quantum Gravity, JHEP 11 (2016) 117 [arXiv:1609.07477] [INSPIRE].
R. Saotome and R. Akhoury, Relationship Between Gravity and Gauge Scattering in the High Energy Limit, JHEP 01 (2013) 123 [arXiv:1210.8111] [INSPIRE].
P.C. Aichelburg and R.U. Sexl, On the Gravitational field of a massless particle, Gen. Rel. Grav. 2 (1971) 303 [INSPIRE].
R. Britto, F. Cachazo, B. Feng and E. Witten, Direct proof of tree-level recursion relation in Yang-Mills theory, Phys. Rev. Lett. 94 (2005) 181602 [hep-th/0501052] [INSPIRE].
P. Benincasa and F. Cachazo, Consistency Conditions on the S-matrix of Massless Particles, arXiv:0705.4305 [INSPIRE].
P.C. Schuster and N. Toro, Constructing the Tree-Level Yang-Mills S-matrix Using Complex Factorization, JHEP 06 (2009) 079 [arXiv:0811.3207] [INSPIRE].
K. Benakli, S. Chapman, L. Darmé and Y. Oz, Superluminal graviton propagation, Phys. Rev. D 94 (2016) 084026 [arXiv:1512.07245] [INSPIRE].
M.S. Costa, J. Penedones, D. Poland and S. Rychkov, Spinning Conformal Correlators, JHEP 11 (2011) 071 [arXiv:1107.3554] [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].
K. Hinterbichler and R.A. Rosen, Note on ghost-free matter couplings in massive gravity and multigravity, Phys. Rev. D 92 (2015) 024030 [arXiv:1503.06796] [INSPIRE].
M. Mohseni, Exact plane gravitational waves in the de Rham-Gabadadze-Tolley model of massive gravity, Phys. Rev. D 84 (2011) 064026 [arXiv:1109.4713] [INSPIRE].
K. Hinterbichler, Cosmology of Massive Gravity and its Extensions, in Proceedings, 51st Rencontres de Moriond, Cosmology session: La Thuile, Italy, March 19–26, 2016, pp. 223-232 [arXiv:1701.02873] [INSPIRE].
S.F. Hassan and R.A. Rosen, Bimetric Gravity from Ghost-free Massive Gravity, JHEP 02 (2012) 126 [arXiv:1109.3515] [INSPIRE].
K. Hinterbichler and R.A. Rosen, Interacting Spin-2 Fields, JHEP 07 (2012) 047 [arXiv:1203.5783] [INSPIRE].
G. D’Amico, G. Gabadadze, L. Hui and D. Pirtskhalava, Quasidilaton: Theory and cosmology, Phys. Rev. D 87 (2013) 064037 [arXiv:1206.4253] [INSPIRE].
G. Gabadadze, K. Hinterbichler, J. Khoury, D. Pirtskhalava and M. Trodden, A Covariant Master Theory for Novel Galilean Invariant Models and Massive Gravity, Phys. Rev. D 86 (2012) 124004 [arXiv:1208.5773] [INSPIRE].
T. Hartman, S. Jain and S. Kundu, Causality Constraints in Conformal Field Theory, JHEP 05 (2016) 099 [arXiv:1509.00014] [INSPIRE].
T. Hartman, S. Jain and S. Kundu, A New Spin on Causality Constraints, JHEP 10 (2016) 141 [arXiv:1601.07904] [INSPIRE].
A. Nicolis, R. Rattazzi and E. Trincherini, The Galileon as a local modification of gravity, Phys. Rev. D 79 (2009) 064036 [arXiv:0811.2197] [INSPIRE].
P. Creminelli, M. Serone, G. Trevisan and E. Trincherini, Inequivalence of Coset Constructions for Spacetime Symmetries, JHEP 02 (2015) 037 [arXiv:1403.3095] [INSPIRE].
A. Sagnotti and M. Taronna, String Lessons for Higher-Spin Interactions, Nucl. Phys. B 842 (2011) 299 [arXiv:1006.5242] [INSPIRE].
S. Caron-Huot, Z. Komargodski, A. Sever and A. Zhiboedov, Strings from Massive Higher Spins: The Asymptotic Uniqueness of the Veneziano Amplitude, JHEP 10 (2017) 026 [arXiv:1607.04253] [INSPIRE].
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Hinterbichler, K., Joyce, A. & Rosen, R.A. Massive spin-2 scattering and asymptotic superluminality. J. High Energ. Phys. 2018, 51 (2018). https://doi.org/10.1007/JHEP03(2018)051
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DOI: https://doi.org/10.1007/JHEP03(2018)051