Abstract
The existence of a ghost free theory of massive gravity begs for an interpre-tation as a Higgs phase of General Relativity. We revisit the study of massive gravity as a Higgs phase. Absent a compelling microphysical model of spontaneous symmetry breaking in gravity, we approach this problem from the viewpoint of nonlinear realizations. We employ the coset construction to search for the most restrictive symmetry breaking pattern whose low energy theory will both admit the de Rham-Gabadadze-Tolley (dRGT) potentials and nonlinearly realize every symmetry of General Relativity, thereby providing a new perspective from which to build theories of massive gravity. In addition to the known ghost-free terms, we find a novel parity violating interaction which preserves the constraint structure of the theory, but which vanishes on the normal branch of the theory. Finally, the procedure is extended to the cases of bi-gravity and multi-vielbein theories. Analogous parity violating interactions exist here, too, and may be non-trivial for certain classes of multi-metric theories.
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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].
N. Arkani-Hamed, H.-C. Cheng, M.A. Luty and S. Mukohyama, Ghost condensation and a consistent infrared modification of gravity, JHEP 05 (2004) 074 [hep-th/0312099] [INSPIRE].
D. Blas and S. Sibiryakov, Completing Lorentz violating massive gravity at high energies, JETP 120 (2015) 509 [Zh. Eksp. Teor. Fiz. 147 (2015) 578] [arXiv:1410.2408] [INSPIRE].
M. Porrati, Higgs phenomenon for 4 − D gravity in anti-de Sitter space, JHEP 04 (2002) 058 [hep-th/0112166] [INSPIRE].
M. Porrati, Higgs phenomenon for the graviton in AdS space, Mod. Phys. Lett. A 18 (2003) 1793 [hep-th/0306253] [INSPIRE].
V.I. Ogievetsky, Infinite-dimensional algebra of general covariance group as the closure of finite-dimensional algebras of conformal and linear groups, Lett. Nuovo Cim. 8 (1973) 988 [INSPIRE].
A.B. Borisov and V.I. Ogievetsky, Theory of Dynamical Affine and Conformal Symmetries as Gravity Theory, Theor. Math. Phys. 21 (1975) 1179 [INSPIRE].
E.A. Ivanov and J. Niederle, Gauge Formulation of Gravitation Theories. 1. The Poincaré, de Sitter and Conformal Cases, Phys. Rev. D 25 (1982) 976 [INSPIRE].
A. Pashnev, Nonlinear realizations of the (super)diffeomorphism groups, geometrical objects and integral invariants in the superspace, hep-th/9704203 [INSPIRE].
F. Riccioni and P. West, Local E11, JHEP 04 (2009) 051 [arXiv:0902.4678] [INSPIRE].
L.V. Delacrétaz, S. Endlich, A. Monin, R. Penco and F. Riva, (Re-)Inventing the relativisticwheel: gravity, cosets and spinning objects, JHEP 11 (2014) 008 [arXiv:1405.7384] [INSPIRE].
E.A. Ivanov and V.I. Ogievetsky, Gauge theories as theories of spontaneous breakdown, JETP Lett. 23 (1976) 606 [INSPIRE].
G. Goon, A. Joyce and M. Trodden, Spontaneously broken gauge theories and the coset construction, Phys. Rev. D 90 (2014) 025022 [arXiv:1405.5532] [INSPIRE].
D.G. Boulware and S. Deser, Can gravitation have a finite range?, Phys. Rev. D 6 (1972) 3368 [INSPIRE].
C. de Rham, L. Heisenberg and R.H. Ribeiro, Quantum Corrections in Massive Gravity, Phys. Rev. D 88 (2013) 084058 [arXiv:1307.7169] [INSPIRE].
G. Goon, K. Hinterbichler, A. Joyce and M. Trodden, Galileons as Wess-Zumino Terms, JHEP 06 (2012) 004 [arXiv:1203.3191] [INSPIRE].
K. Hinterbichler and R.A. Rosen, Interacting Spin-2 Fields, JHEP 07 (2012) 047 [arXiv:1203.5783] [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].
C. de Rham, Massive Gravity, Living Rev. Rel. 17 (2014) 7 [arXiv:1401.4173] [INSPIRE].
K. Hinterbichler, Theoretical Aspects of Massive Gravity, Rev. Mod. Phys. 84 (2012) 671 [arXiv:1105.3735] [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].
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].
S.F. Hassan and R.A. Rosen, Resolving the ghost problem in non-linear massive gravity, Phys. Rev. Lett. 108 (2012) 041101 [arXiv:1106.3344] [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].
B. Zumino, Effective Lagrangians and broken symmetries, Brandeis University Lectures on Elementary Particles and Quantum Field Theory 2 (1970) 437.
S. Groot Nibbelink, M. Peloso and M. Sexton, Nonlinear Properties of Vielbein Massive Gravity, Eur. Phys. J. C 51 (2007) 741 [hep-th/0610169] [INSPIRE].
D. Ivanenko and G. Sardanashvily, The gauge treatment of gravity, Phys. Rept. 94 (1983) 1 [INSPIRE].
I. Kirsch, A Higgs mechanism for gravity, Phys. Rev. D 72 (2005) 024001 [hep-th/0503024] [INSPIRE].
N. Boulanger and I. Kirsch, A Higgs mechanism for gravity. Part II. Higher spin connections, Phys. Rev. D 73 (2006) 124023 [hep-th/0602225] [INSPIRE].
G. ’t Hooft, Unitarity in the Brout-Englert-Higgs Mechanism for Gravity, arXiv:0708.3184 [INSPIRE].
A.H. Chamseddine and V. Mukhanov, Higgs for Graviton: Simple and Elegant Solution, JHEP 08 (2010) 011 [arXiv:1002.3877] [INSPIRE].
V.A. Kostelecky and S. Samuel, Gravitational Phenomenology in Higher Dimensional Theories and Strings, Phys. Rev. D 40 (1989) 1886 [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].
S.R. Coleman, J. Wess and B. Zumino, Structure of phenomenological Lagrangians. 1., Phys. Rev. 177 (1969) 2239 [INSPIRE].
C.G. Callan Jr., S.R. Coleman, J. Wess and B. Zumino, Structure of phenomenological Lagrangians. 2., Phys. Rev. 177 (1969) 2247 [INSPIRE].
D.V. Volkov, Phenomenological Lagrangians, Sov. J. Part. Nucl. 4 (1973) 3.
S. Weinberg, Nonlinear realizations of chiral symmetry, Phys. Rev. 166 (1968) 1568 [INSPIRE].
D.V. Volkov, Phenomenological Lagrangians, Fiz. Elem. Chast. Atom. Yadra 4 (1973) 3 [INSPIRE].
V. Ogievetsky, Nonlinear Realizations of Internal and Space-time Symmetries, in proceedings of the X-th Winter School of Theoretical Physics in Karpacz 1 (1974).
E. Ivanov and V. Ogievetsky, The inverse Higgs phenomenon in nonlinear realizations, Teor. Mat. Fiz. 25 (1975) 164 [INSPIRE].
I. Low and A.V. Manohar, Spontaneously broken space-time symmetries and Goldstone’s theorem, Phys. Rev. Lett. 88 (2002) 101602 [hep-th/0110285] [INSPIRE].
I.N. McArthur, Nonlinear realizations of symmetries and unphysical Goldstone bosons, JHEP 11 (2010) 140 [arXiv:1009.3696] [INSPIRE].
A. Nicolis, R. Penco, F. Piazza and R.A. Rosen, More on gapped Goldstones at finite density: more gapped Goldstones, JHEP 11 (2013) 055 [arXiv:1306.1240] [INSPIRE].
T. Brauner and H. Watanabe, Spontaneous breaking of spacetime symmetries and the inverse Higgs effect, Phys. Rev. D 89 (2014) 085004 [arXiv:1401.5596] [INSPIRE].
D. Lovelock, The Einstein tensor and its generalizations, J. Math. Phys. 12 (1971) 498 [INSPIRE].
A. Mardones and J. Zanelli, Lovelock-Cartan theory of gravity, Class. Quant. Grav. 8 (1991) 1545 [INSPIRE].
J. Zanelli, Chern-Simons Forms in Gravitation Theories, Class. Quant. Grav. 29 (2012) 133001 [arXiv:1208.3353] [INSPIRE].
G. Gabadadze, K. Hinterbichler, D. Pirtskhalava and Y. Shang, Potential for general relativity and its geometry, Phys. Rev. D 88 (2013) 084003 [arXiv:1307.2245] [INSPIRE].
N.A. Ondo and A.J. Tolley, Complete Decoupling Limit of Ghost-free Massive Gravity, JHEP 11 (2013) 059 [arXiv:1307.4769] [INSPIRE].
M. Andrews, G. Goon, K. Hinterbichler, J. Stokes and M. Trodden, Massive Gravity Coupled to Galileons is Ghost-Free, Phys. Rev. Lett. 111 (2013) 061107 [arXiv:1303.1177] [INSPIRE].
C. Deffayet, J. Mourad and G. Zahariade, A note on ‘symmetric’ vielbeins in bimetric, massive, perturbative and non perturbative gravities, JHEP 03 (2013) 086 [arXiv:1208.4493] [INSPIRE].
S.F. Hassan, A. Schmidt-May and M. von Strauss, Metric Formulation of Ghost-Free Multivielbein Theory, arXiv:1204.5202 [INSPIRE].
J. Noller, J.H.C. Scargill and P.G. Ferreira, Interacting spin-2 fields in the Stückelberg picture, JCAP 02 (2014) 007 [arXiv:1311.7009] [INSPIRE].
H.R. Afshar, E.A. Bergshoeff and W. Merbis, Interacting spin-2 fields in three dimensions, JHEP 01 (2015) 040 [arXiv:1410.6164] [INSPIRE].
J.H.C. Scargill, J. Noller and P.G. Ferreira, Cycles of interactions in multi-gravity theories, JHEP 12 (2014) 160 [arXiv:1410.7774] [INSPIRE].
C.J. Isham, A. Salam and J.A. Strathdee, Nonlinear realizations of space-time symmetries. Scalar and tensor gravity, Annals Phys. 62 (1971) 98 [INSPIRE].
J.A. de Azcarraga, J.M. Izquierdo and J.C. Perez Bueno, An Introduction to some novel applications of Lie algebra cohomology in mathematics and physics, Rev. R. Acad. Cien. Exactas Fis. Nat. Ser. A Mat. 95 (2001) 225 [physics/9803046] [INSPIRE].
E. D’Hoker and S. Weinberg, General effective actions, Phys. Rev. D 50 (1994) 6050 [hep-ph/9409402] [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].
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].
R. Kimura and D. Yamauchi, Derivative interactions in de Rham-Gabadadze-Tolley massive gravity, Phys. Rev. D 88 (2013) 084025 [arXiv:1308.0523] [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, arXiv:1505.00831 [INSPIRE].
J. Noller, On consistent kinetic and derivative interactions for gravitons, JCAP 04 (2015) 025 [arXiv:1409.7692] [INSPIRE].
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Goon, G., Hinterbichler, K., Joyce, A. et al. Einstein gravity, massive gravity, multi-gravity and nonlinear realizations. J. High Energ. Phys. 2015, 101 (2015). https://doi.org/10.1007/JHEP07(2015)101
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DOI: https://doi.org/10.1007/JHEP07(2015)101