Abstract
In this paper we consider how the strong-coupling scale, or perturbative cutoff, in a multi-gravity theory depends upon the presence and structure of interactions between the different fields. This can elegantly be rephrased in terms of the size and structure of the ‘theory graph’ which depicts the interactions in a given theory. We show that the question can be answered in terms of the properties of various graph-theoretical matrices, affording an efficient way to estimate and place bounds on the strong-coupling scale of a given theory. In light of this we also consider the problem of relating a given theory graph to a discretised higher dimensional theory, à la dimensional deconstruction.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
N. Boulanger, T. Damour, L. Gualtieri and M. Henneaux, Inconsistency of interacting, multigraviton theories, Nucl. Phys. B 597 (2001) 127 [hep-th/0007220] [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].
D.G. Boulware and S. Deser, Can gravitation have a finite range?, Phys. Rev. D 6 (1972) 3368 [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].
S.F. Hassan and R.A. Rosen, On non-linear actions for massive gravity, JHEP 07 (2011) 009 [arXiv:1103.6055] [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, 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 and R.A. Rosen, Bimetric gravity from ghost-free massive gravity, JHEP 02 (2012) 126 [arXiv:1109.3515] [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].
K. Hinterbichler and R.A. Rosen, Interacting spin-2 fields, JHEP 07 (2012) 047 [arXiv:1203.5783] [INSPIRE].
U. Aydemir, M.M. Anber and J.F. Donoghue, Self-healing of unitarity in effective field theories and the onset of new physics, Phys. Rev. D 86 (2012) 014025 [arXiv:1203.5153] [INSPIRE].
C. Burrage, N. Kaloper and A. Padilla, Strong coupling and bounds on the spin-2 mass in massive gravity, Phys. Rev. Lett. 111 (2013) 021802 [arXiv:1211.6001] [INSPIRE].
N. Arkani-Hamed, A.G. Cohen and H. Georgi, (De)constructing dimensions, Phys. Rev. Lett. 86 (2001) 4757 [hep-th/0104005] [INSPIRE].
C. de Rham, A. Matas and A.J. Tolley, Deconstructing dimensions and massive gravity, Class. Quant. Grav. 31 (2014) 025004 [arXiv:1308.4136] [INSPIRE].
C. de Rham, G. Gabadadze and A.J. Tolley, Ghost free massive gravity in the Stückelberg language, Phys. Lett. B 711 (2012) 190 [arXiv:1107.3820] [INSPIRE].
M. Fasiello and A.J. Tolley, Cosmological stability bound in massive gravity and bigravity, JCAP 12 (2013) 002 [arXiv:1308.1647] [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].
C. de Rham, M. Fasiello and A.J. Tolley, Galileon duality, Phys. Lett. B 733 (2014) 46 [arXiv:1308.2702] [INSPIRE].
J. Noller and J.H.C. Scargill, The decoupling limit of multi-gravity: multi-Galileons, dualities and more, JHEP 05 (2015) 034 [arXiv:1503.02700] [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].
T. Hanada, K. Kobayashi, K. Shinoda and K. Shiraishi, Classical and quantum cosmology of multigravity, Class. Quant. Grav. 27 (2010) 225010 [arXiv:1004.5435] [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. Yan, Properties of spectra of graphs and line graphs, Appl. Math. J. Chin. Univ. B 17 (2002) 371.
M. Doob, An interrelation between line graphs, eigenvalues, and matroids, J. Combinat. Theor. B 15 (1973) 40.
R. Brualdi, The mutually beneficial relationship of graphs and matrices, in Regional conference series in mathematics, American Mathematical Soc., U.S.A. (2011).
N. Arkani-Hamed and M.D. Schwartz, Discrete gravitational dimensions, Phys. Rev. D 69 (2004) 104001 [hep-th/0302110] [INSPIRE].
M.D. Schwartz, Constructing gravitational dimensions, Phys. Rev. D 68 (2003) 024029 [hep-th/0303114] [INSPIRE].
C. Deffayet and J. Mourad, Multigravity from a discrete extra dimension, Phys. Lett. B 589 (2004) 48 [hep-th/0311124] [INSPIRE].
N. Kan and K. Shiraishi, Multigraviton theory: a latticized dimension and the cosmological constant, Class. Quant. Grav. 20 (2003) 4965 [gr-qc/0212113] [INSPIRE].
R. Grone and R. Merris, The Laplacian spectrum of a graph II, SIAM J. Discrete Math. 7 (1994) 221.
D. Stevanović, Bounding the largest eigenvalue of trees in terms of the largest vertex degree, Lin. Alg. Appl. 360 (2003) 35.
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].
J. Noller and S. Melville, The coupling to matter in massive, bi- and multi-gravity, JCAP 01 (2015) 003 [arXiv:1408.5131] [INSPIRE].
S. Melville and J. Noller, Generalised matter couplings in massive bigravity, arXiv:1511.01485 [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: 1511.02877
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
Scargill, J.H.C., Noller, J. Strong-coupling scales and the graph structure of multi-gravity theories. J. High Energ. Phys. 2016, 29 (2016). https://doi.org/10.1007/JHEP01(2016)029
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP01(2016)029