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Galileons as Wess-Zumino terms

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Abstract

We show that the galileons can be thought of as Wess-Zumino terms for the spontaneous breaking of space-time symmetries. Wess-Zumino terms are terms which are not captured by the coset construction for phenomenological Lagrangians with broken symmetries. Rather they are, in d space-time dimensions, d-form potentials for (d + 1)-forms which are non-trivial co-cycles in Lie algebra cohomology of the full symmetry group rela- tive to the unbroken symmetry group. We introduce the galileon algebras and construct the non-trivial (d + 1)-form co-cycles, showing that the presence of galileons and multi-galileons in all dimensions is counted by the dimensions of particular Lie algebra cohomology groups. We also discuss the DBI and conformal galileons from this point of view, showing that they are not Wess-Zumino terms, with one exception in each case.

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References

  1. G. Dvali, G. Gabadadze and M. Porrati, 4 − D gravity on a brane in 5 − D Minkowski space, Phys. Lett. B 485 (2000) 208 [hep-th/0005016] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  2. M.A. Luty, M. Porrati and R. Rattazzi, Strong interactions and stability in the DGP model, JHEP 09 (2003) 029 [hep-th/0303116] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  3. 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].

    MathSciNet  ADS  Google Scholar 

  4. M. Trodden and K. Hinterbichler, Generalizing Galileons, Class. Quant. Grav. 28 (2011) 204003 [arXiv:1104.2088] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  5. A. Padilla, P.M. Saffin and S.-Y. Zhou, Bi-galileon theory I: Motivation and formulation, JHEP 12 (2010) 031 [arXiv:1007.5424] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  6. K. Hinterbichler, M. Trodden and D. Wesley, Multi-field galileons and higher co-dimension branes, Phys. Rev. D 82 (2010) 124018 [arXiv:1008.1305] [INSPIRE].

    ADS  Google Scholar 

  7. A. Nicolis and R. Rattazzi, Classical and quantum consistency of the DGP model, JHEP 06 (2004) 059 [hep-th/0404159] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  8. A. Vainshtein, To the problem of nonvanishing gravitation mass, Phys. Lett. B 39 (1972) 393 [INSPIRE].

    Article  ADS  Google Scholar 

  9. C. Deffayet, G. Dvali, G. Gabadadze and A.I. Vainshtein, Nonperturbative continuity in graviton mass versus perturbative discontinuity, Phys. Rev. D 65 (2002) 044026 [hep-th/0106001] [INSPIRE].

    ADS  Google Scholar 

  10. T. Kobayashi, M. Yamaguchi and J. Yokoyama, G-inflation: Inflation driven by the Galileon field, Phys. Rev. Lett. 105 (2010) 231302 [arXiv:1008.0603] [INSPIRE].

    Article  ADS  Google Scholar 

  11. C. Burrage, C. de Rham, D. Seery and A.J. Tolley, Galileon inflation, JCAP 01 (2011) 014 [arXiv:1009.2497] [INSPIRE].

    Article  ADS  Google Scholar 

  12. P. Creminelli, G. D’Amico, M. Musso, J. Norena and E. Trincherini, Galilean symmetry in the effective theory of inflation: new shapes of non-Gaussianity, JCAP 02 (2011) 006 [arXiv:1011.3004] [INSPIRE].

    Article  ADS  Google Scholar 

  13. P. Creminelli, A. Nicolis and E. Trincherini, Galilean Genesis: An Alternative to inflation, JCAP 11 (2010) 021 [arXiv:1007.0027] [INSPIRE].

    Article  ADS  Google Scholar 

  14. K. Hinterbichler and J. Khoury, The Pseudo-Conformal Universe: Scale Invariance from Spontaneous Breaking of Conformal Symmetry, JCAP 04 (2012) 023 [arXiv:1106.1428] [INSPIRE].

    Article  ADS  Google Scholar 

  15. K. Hinterbichler, A. Joyce and J. Khoury, Non-linear Realizations of Conformal Symmetry and Effective Field Theory for the Pseudo-Conformal Universe, arXiv:1202.6056 [INSPIRE].

  16. N. Chow and J. Khoury, Galileon Cosmology, Phys. Rev. D 80 (2009) 024037 [arXiv:0905.1325] [INSPIRE].

    ADS  Google Scholar 

  17. F.P. Silva and K. Koyama, Self-Accelerating Universe in Galileon Cosmology, Phys. Rev. D 80 (2009) 121301 [arXiv:0909.4538] [INSPIRE].

    ADS  Google Scholar 

  18. A. De Felice, R. Kase and S. Tsujikawa, Matter perturbations in Galileon cosmology, Phys. Rev. D 83 (2011) 043515 [arXiv:1011.6132] [INSPIRE].

    ADS  Google Scholar 

  19. C. Deffayet, O. Pujolàs, I. Sawicki and A. Vikman, Imperfect Dark Energy from Kinetic Gravity Braiding, JCAP 10 (2010) 026 [arXiv:1008.0048] [INSPIRE].

    Article  ADS  Google Scholar 

  20. C. Deffayet, G. Esposito-Farese and A. Vikman, Covariant Galileon, Phys. Rev. D 79 (2009) 084003 [arXiv:0901.1314] [INSPIRE].

    ADS  Google Scholar 

  21. C. Deffayet, S. Deser and G. Esposito-Farese, Generalized Galileons: All scalar models whose curved background extensions maintain second-order field equations and stress-tensors, Phys. Rev. D 80 (2009) 064015 [arXiv:0906.1967] [INSPIRE].

    ADS  Google Scholar 

  22. C. Deffayet, S. Deser and G. Esposito-Farese, Arbitrary p-form Galileons, Phys. Rev. D 82 (2010) 061501 [arXiv:1007.5278] [INSPIRE].

    ADS  Google Scholar 

  23. J. Khoury, J.-L. Lehners and B.A. Ovrut, Supersymmetric Galileons, Phys. Rev. D 84 (2011) 043521 [arXiv:1103.0003] [INSPIRE].

    ADS  Google Scholar 

  24. S.-Y. Zhou and E.J. Copeland, Galileons with Gauge Symmetries, Phys. Rev. D 85 (2012) 065002 [arXiv:1112.0968] [INSPIRE].

    ADS  Google Scholar 

  25. G. Goon, K. Hinterbichler, A. Joyce and M. Trodden, Gauged Galileons From Branes, arXiv:1201.0015 [INSPIRE].

  26. C. de Rham and G. Gabadadze, Generalization of the Fierz-Pauli Action, Phys. Rev. D 82 (2010) 044020 [arXiv:1007.0443] [INSPIRE].

    ADS  Google Scholar 

  27. C. de Rham, G. Gabadadze and A.J. Tolley, Resummation of Massive Gravity, Phys. Rev. Lett. 106 (2011) 231101 [arXiv:1011.1232] [INSPIRE].

    Article  ADS  Google Scholar 

  28. K. Hinterbichler, Theoretical Aspects of Massive Gravity, Rev. Mod. Phys. 84 (2012) 671 [arXiv:1105.3735] [INSPIRE].

    Article  ADS  Google Scholar 

  29. C. de Rham and A.J. Tolley, DBI and the Galileon reunited, JCAP 05 (2010) 015 [arXiv:1003.5917] [INSPIRE].

    Article  Google Scholar 

  30. D. Lovelock, The Einstein tensor and its generalizations, J. Math. Phys. 12 (1971) 498 [INSPIRE].

    Article  MathSciNet  ADS  MATH  Google Scholar 

  31. G. Goon, K. Hinterbichler and M. Trodden, Symmetries for Galileons and DBI scalars on curved space, JCAP 07 (2011) 017 [arXiv:1103.5745] [INSPIRE].

    Article  ADS  Google Scholar 

  32. G. Goon, K. Hinterbichler and M. Trodden, A New Class of Effective Field Theories from Embedded Branes, Phys. Rev. Lett. 106 (2011) 231102 [arXiv:1103.6029] [INSPIRE].

    Article  ADS  Google Scholar 

  33. C. Burrage, C. de Rham and L. Heisenberg, de Sitter Galileon, JCAP 05 (2011) 025 [arXiv:1104.0155] [INSPIRE].

    Article  ADS  Google Scholar 

  34. G. Goon, K. Hinterbichler and M. Trodden, Galileons on Cosmological Backgrounds, JCAP 12 (2011) 004 [arXiv:1109.3450] [INSPIRE].

    Article  ADS  Google Scholar 

  35. S.R. Coleman, J. Wess and B. Zumino, Structure of phenomenological Lagrangians. 1., Phys. Rev. 177 (1969) 2239 [INSPIRE].

    Article  ADS  Google Scholar 

  36. C.G. Callan Jr., S.R. Coleman, J. Wess and B. Zumino, Structure of phenomenological Lagrangians. 2., Phys. Rev. 177 (1969) 2247 [INSPIRE].

    Article  ADS  Google Scholar 

  37. D.V. Volkov, Phenomenological Lagrangians, Sov. J. Part. Nucl. 4 (1973) 3.

    Google Scholar 

  38. J. Wess and B. Zumino, Consequences of anomalous Ward identities, Phys. Lett. B 37 (1971) 95 [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  39. E. Witten, Global Aspects of Current Algebra, Nucl. Phys. B 223 (1983) 422 [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  40. C. Chevalley and S. Eilenberg, Cohomology Theory of Lie Groups and Lie Algebras, Trans. Am. Math. Soc. 63 (1948) 85 [INSPIRE].

    Article  MathSciNet  MATH  Google Scholar 

  41. J. de Azcarraga, J. Izquierdo and J. 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].

    MATH  Google Scholar 

  42. J. de Azcarraga, A. Macfarlane and J. Perez Bueno, Effective actions, relative cohomology and Chern Simons forms, Phys. Lett. B 419 (1998) 186 [hep-th/9711064] [INSPIRE].

    Article  ADS  Google Scholar 

  43. E. D’Hoker and S. Weinberg, General effective actions, Phys. Rev. D 50 (1994) 6050 [hep-ph/9409402] [INSPIRE].

    MathSciNet  ADS  Google Scholar 

  44. A. Nicolis, Galilean currents and charges, Phys. Rev. D 85 (2012) 085026 [arXiv:1011.3057] [INSPIRE].

    ADS  Google Scholar 

  45. J.A. de Azcarraga and J.M. Izquierdo, Lie groups, Lie algebras, cohomology and some applications in physics, Cambridge University Press, Cambridge U.K. (1999).

    Google Scholar 

  46. J. Brugues, T. Curtright, J. Gomis and L. Mezincescu, Non-relativistic strings and branes as non-linear realizations of Galilei groups, Phys. Lett. B 594 (2004) 227 [hep-th/0404175] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  47. S. Weinberg, Nonlinear realizations of chiral symmetry, Phys. Rev. 166 (1968) 1568 [INSPIRE].

    Article  ADS  Google Scholar 

  48. V.I. Ogievetsky, Nonlinear Realizations of Internal and Space-time Symmetries, Proceedings of X-th Winter School of Theoretical Physics in Karpacz. Vol. 1, Wroclaw Poland (1974), pg. 227.

  49. B. Zumino, Effective Lagrangians and broken symmetries, in Lectures On Elementary Particles And Quantum Field Theory. Vol. 2, M.I.T. Press, Cambridge U.S.A. (1970), pg. 437.

  50. R. Camporesi, Harmonic analysis and propagators on homogeneous spaces, Phys. Rept. 196 (1990) 1 [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  51. S. Kobayashi and K. Nomizu, Foundations of Differential Geometry. I, J. Wiley, New York U.S.A. (1963).

    MATH  Google Scholar 

  52. H.B. Nielsen and S. Chadha, On How to Count Goldstone Bosons, Nucl. Phys. B 105 (1976) 445 [INSPIRE].

    Article  ADS  Google Scholar 

  53. 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].

    Article  ADS  Google Scholar 

  54. H. Watanabe and H. Murayama, Unified Description of Non-Relativistic Nambu-Goldstone bosons, arXiv:1203.0609 [INSPIRE].

  55. Y. Hidaka, Counting rule for Nambu-Goldstone modes in nonrelativistic systems, arXiv:1203.1494 [INSPIRE].

  56. E. Ivanov and V. Ogievetsky, The Inverse Higgs Phenomenon in Nonlinear Realizations, Teor. Mat. Fiz. 25 (1975) 164 [INSPIRE].

    Article  Google Scholar 

  57. I. McArthur, Nonlinear realizations of symmetries and unphysical Goldstone bosons, JHEP 11 (2010) 140 [arXiv:1009.3696] [INSPIRE].

    Article  ADS  Google Scholar 

  58. S. Bellucci, E. Ivanov and S. Krivonos, AdS/CFT equivalence transformation, Phys. Rev. D 66 (2002) 086001 [Erratum ibid. D 67 (2003) 049901] [hep-th/0206126] [INSPIRE].

  59. E. Inonu and E.P. Wigner, On the Contraction of groups and their represenations, Proc. Nat. Acad. Sci. 39 (1953) 510 [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  60. J.P. Gauntlett, J. Gomis and P. Townsend, Particle actions as Wess-Zumino terms for space-time (super)symmetry groups, Phys. Lett. B 249 (1990) 255 [INSPIRE].

    Article  ADS  Google Scholar 

  61. P.C. West, Automorphisms, nonlinear realizations and branes, JHEP 02 (2000) 024 [hep-th/0001216] [INSPIRE].

    Article  ADS  Google Scholar 

  62. C. Chryssomalakos, J. de Azcarraga, J. Izquierdo and J. Perez Bueno, The Geometry of branes and extended superspaces, Nucl. Phys. B 567 (2000) 293 [hep-th/9904137] [INSPIRE].

    Article  ADS  Google Scholar 

  63. J. Gomis, K. Kamimura and P.C. West, The Construction of brane and superbrane actions using non-linear realisations, Class. Quant. Grav. 23 (2006) 7369 [hep-th/0607057] [INSPIRE].

    Article  MathSciNet  ADS  MATH  Google Scholar 

  64. T.Y. Thomas, On Conformal Geometry, Proc. Nat. Acad. Sci. 12 (1926) 352.

    Article  ADS  MATH  Google Scholar 

  65. T.Y. Thomas, Conformal Tensors, Proc. Nat. Acad. Sci. 18 (1932) 103.

    Article  ADS  MATH  Google Scholar 

  66. T.N. Bailey, M.G. Eastwood and A.R. Gover, Thomas’s Structure Bundle for Conformal, Projective and Related Structures, Rocky Mountain J. Math. 24 (1994) 1191.

    Article  MathSciNet  MATH  Google Scholar 

  67. M. Eastwood, Notes on Conformal Differential Geometry, Suppl. Rendi. Circ. Mat. Palermo 43 (1996) 57.

    MathSciNet  Google Scholar 

  68. A. Gover, A. Shaukat and A. Waldron, Weyl Invariance and the Origins of Mass, Phys. Lett. B 675 (2009) 93 [arXiv:0812.3364] [INSPIRE].

    Article  ADS  Google Scholar 

  69. A. Gover, A. Shaukat and A. Waldron, Tractors, Mass and Weyl Invariance, Nucl. Phys. B 812 (2009) 424 [arXiv:0810.2867] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  70. R. Bonezzi, E. Latini and A. Waldron, Gravity, Two Times, Tractors, Weyl Invariance and Six Dimensional Quantum Mechanics, Phys. Rev. D 82 (2010) 064037 [arXiv:1007.1724] [INSPIRE].

    ADS  Google Scholar 

  71. Z. Komargodski and A. Schwimmer, On renormalization group flows in four dimensions, JHEP 12 (2011) 099 [arXiv:1107.3987] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  72. Z. Komargodski, The Constraints of Conformal Symmetry on RG Flows, arXiv:1112.4538 [INSPIRE].

  73. G.L. Goon, K. Hinterbichler and M. Trodden, Stability and superluminality of spherical DBI galileon solutions, Phys. Rev. D 83 (2011) 085015 [arXiv:1008.4580] [INSPIRE].

    ADS  Google Scholar 

  74. S. Mizuno and K. Koyama, Primordial non-Gaussianity from the DBI Galileons, Phys. Rev. D 82 (2010) 103518 [arXiv:1009.0677] [INSPIRE].

    ADS  Google Scholar 

  75. V. 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].

    Article  MathSciNet  Google Scholar 

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  1. Center for Particle Cosmology, Department of Physics and Astronomy, University of Pennsylvania, Philadelphia, PA, 19104, U.S.A.

    Garrett Goon, Kurt Hinterbichler, Austin Joyce & Mark Trodden

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  1. Garrett Goon
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Goon, G., Hinterbichler, K., Joyce, A. et al. Galileons as Wess-Zumino terms. J. High Energ. Phys. 2012, 4 (2012). https://doi.org/10.1007/JHEP06(2012)004

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