Lorentz violating kinematics: threshold theorems

  • Valentina Baccetti
  • Kyle Tate
  • Matt Visser


Recent tentative experimental indications, and the subsequent theoretical speculations, regarding possible violations of Lorentz invariance have attracted a vast amount of attention. An important technical issue that considerably complicates detailed calculations in any such scenario, is that once one violates Lorentz invariance the analysis of thresholds in both scattering and decay processes becomes extremely subtle, with many new and naively unexpected effects. In the current article we develop several extremely general threshold theorems that depend only on the existence of some energy momentum relation E(p), eschewing even assumptions of isotropy or monotonicity. We shall argue that there are physically interesting situations where such a level of generality is called for, and that existing (partial) results in the literature make unnecessary technical assumptions. Even in this most general of settings, we show that at threshold all final state particles move with the same 3-velocity, while initial state particles must have 3-velocities parallel/anti-parallel to the final state particles. In contrast the various 3-momenta can behave in a complicatedand counter-intuitive manner.


Space-Time Symmetries Beyond Standard Model Neutrino Physics Global Symmetries 


  1. [1]
    OPERA collaboration, T. Adam et al., Measurement of the neutrino velocity with the OPERA detector in the CNGS beam, arXiv:1109.4897 [INSPIRE].
  2. [2]
    MINOS collaboration, P. Adamson et al., Measurement of neutrino velocity with the MINOS detectors and NuMI neutrino beam, Phys. Rev. D 76 (2007) 072005 [arXiv:0706.0437] [INSPIRE].ADSGoogle Scholar
  3. [3]
    G. Amelino-Camelia et al., OPERA-reassessing data on the energy dependence of the speed of neutrinos, Int. J. Mod. Phys. D 20 (2011) 2623 [arXiv:1109.5172] [INSPIRE].MathSciNetADSGoogle Scholar
  4. [4]
    G.F. Giudice, S. Sibiryakov and A. Strumia, Interpreting OPERA results on superluminal neutrino, Nucl. Phys. B in press, arXiv:1109.5682 [INSPIRE].
  5. [5]
    A.G. Cohen and S.L. Glashow, Pair Creation Constrains Superluminal Neutrino Propagation, Phys. Rev. Lett. 107 (2011) 181803 [arXiv:1109.6562] [INSPIRE].ADSCrossRefGoogle Scholar
  6. [6]
    G. Dvali and A. Vikman, Price for Environmental Neutrino-Superluminality, JHEP 02 (2012)134 [arXiv:1109.5685] [INSPIRE].ADSCrossRefGoogle Scholar
  7. [7]
    J. Alexandre, J. Ellis and N.E. Mavromatos, On the Possibility of Superluminal Neutrino Propagation, Phys. Lett. B 706 (2012) 456 [arXiv:1109.6296] [INSPIRE].ADSGoogle Scholar
  8. [8]
    G. Cacciapaglia, A. Deandrea and L. Panizzi, Superluminal neutrinos in long baseline experiments and SN1987a, JHEP 11 (2011) 137 [arXiv:1109.4980] [INSPIRE].ADSCrossRefGoogle Scholar
  9. [9]
    X.-J. Bi, P.-F. Yin, Z.-H. Yu and Q. Yuan, Constraints and tests of the OPERA superluminal neutrinos, Phys. Rev. Lett. 107 (2011) 241802 [arXiv:1109.6667] [INSPIRE].ADSCrossRefGoogle Scholar
  10. [10]
    F. Klinkhamer, Superluminal muon-neutrino velocity from a Fermi-point-splitting model of Lorentz violation, arXiv:1109.5671 [INSPIRE].
  11. [11]
    S.S. Gubser, Superluminal neutrinos and extra dimensions: Constraints from the null energy condition, Phys. Lett. B 705 (2011) 279 [arXiv:1109.5687] [INSPIRE].ADSGoogle Scholar
  12. [12]
    A. Kehagias, Relativistic Superluminal Neutrinos, arXiv:1109.6312 [INSPIRE].
  13. [13]
    P. Wang, H. Wu and H. Yang, Superluminal neutrinos and domain walls, arXiv:1109.6930 [INSPIRE].
  14. [14]
    E.N. Saridakis, Superluminal neutrinos in Hořava-Lifshitz gravity, arXiv:1110.0697 [INSPIRE].
  15. [15]
    W. Winter, Constraints on the interpretation of the superluminal motion of neutrinos at OPERA, Phys. Rev. D 85 (2012) 017301 [arXiv:1110.0424] [INSPIRE].ADSGoogle Scholar
  16. [16]
    J. Alexandre, Lifshitz-type Quantum Field Theories in Particle Physics, Int. J. Mod. Phys. A 26 (2011) 4523 [arXiv:1109.5629] [INSPIRE].ADSGoogle Scholar
  17. [17]
    F. Klinkhamer and G. Volovik, Superluminal neutrino and spontaneous breaking of Lorentz invariance, Pisma Zh. Eksp. Teor. Fiz. 94 (2011) 731 [arXiv:1109.6624] [INSPIRE].Google Scholar
  18. [18]
    R. Cowsik, S. Nussinov and U. Sarkar, Superluminal Neutrinos at OPERA Confront Pion Decay Kinematics, Phys. Rev. Lett. 107 (2011) 251801 [arXiv:1110.0241] [INSPIRE].ADSCrossRefGoogle Scholar
  19. [19]
    L. Maccione, S. Liberati and D.M. Mattingly, Violations of Lorentz invariance in the neutrino sector after OPERA, arXiv:1110.0783 [INSPIRE].
  20. [20]
    N. Dass, OPERA, SN1987a and energy dependence of superluminal neutrino velocity, arXiv:1110.0351 [INSPIRE].
  21. [21]
    J. Carmona and J. Cortes, Constraints from Neutrino Decay on Superluminal Velocities, arXiv:1110.0430 [INSPIRE].
  22. [22]
    S.R. Coleman and S.L. Glashow, Cosmic ray and neutrino tests of special relativity, Phys. Lett. B 405 (1997) 249 [hep-ph/9703240] [INSPIRE].ADSGoogle Scholar
  23. [23]
    S.R. Coleman and S.L. Glashow, High-energy tests of Lorentz invariance, Phys. Rev. D 59 (1999)116008 [hep-ph/9812418] [INSPIRE].ADSGoogle Scholar
  24. [24]
    S. Liberati, T. Jacobson and D. Mattingly, High-energy constraints on Lorentz symmetry violations, hep-ph/0110094 [INSPIRE].
  25. [25]
    T. Jacobson, S. Liberati and D. Mattingly, TeV astrophysics constraints on Planck scale Lorentz violation, Phys. Rev. D 66 (2002) 081302 [hep-ph/0112207] [INSPIRE].ADSGoogle Scholar
  26. [26]
    T. Jacobson, S. Liberati and D. Mattingly, Threshold effects and Planck scale Lorentz violation: Combined constraints from high-energy astrophysics, Phys. Rev. D 67 (2003) 124011 [hep-ph/0209264] [INSPIRE].ADSGoogle Scholar
  27. [27]
    D. Mattingly, T. Jacobson and S. Liberati, Threshold configurations in the presence of Lorentz violating dispersion relations, Phys. Rev. D 67 (2003) 124012 [hep-ph/0211466] [INSPIRE].ADSGoogle Scholar
  28. [28]
    T. Jacobson, S. Liberati and D. Mattingly, A Strong astrophysical constraint on the violation of special relativity by quantum gravity, Nature 424 (2003) 1019 [astro-ph/0212190] [INSPIRE].ADSCrossRefGoogle Scholar
  29. [29]
    T. Jacobson, S. Liberati and D. Mattingly, Comments onImproved limit on quantum space-time modifications of Lorentz symmetry from observations of gamma-ray blazars’, gr-qc/0303001 [INSPIRE].
  30. [30]
    T.A. Jacobson, S. Liberati, D. Mattingly and F. Stecker, New limits on Planck scale Lorentz violation in QED, Phys. Rev. Lett. 93 (2004) 021101 [astro-ph/0309681] [INSPIRE].ADSCrossRefGoogle Scholar
  31. [31]
    T. Jacobson, S. Liberati and D. Mattingly, Quantum gravity phenomenology and Lorentz violation, Springer Proc. Phys. 98 (2005) 83 [gr-qc/0404067] [INSPIRE].CrossRefGoogle Scholar
  32. [32]
    T. Jacobson, S. Liberati and D. Mattingly, Astrophysical bounds on Planck suppressed Lorentz violation, Lect. Notes Phys. 669 (2005) 101 [hep-ph/0407370] [INSPIRE].ADSCrossRefGoogle Scholar
  33. [33]
    T. Jacobson, S. Liberati and D. Mattingly, Lorentz violation at high energy: Concepts, phenomena and astrophysical constraints, Annals Phys. 321 (2006) 150 [astro-ph/0505267] [INSPIRE].ADSMATHCrossRefGoogle Scholar
  34. [34]
    D. Mattingly, Modern tests of Lorentz invariance, Living Rev. Rel. 8 (2005) 5 [gr-qc/0502097] [INSPIRE].Google Scholar
  35. [35]
    D. Colladay and V. Kostelecky, Lorentz violating extension of the standard model, Phys. Rev. D 58 (1998) 116002 [hep-ph/9809521] [INSPIRE].ADSGoogle Scholar
  36. [36]
    V. Kostelecky and S. Samuel, Spontaneous Breaking of Lorentz Symmetry in String Theory, Phys. Rev. D 39 (1989) 683 [INSPIRE].ADSGoogle Scholar
  37. [37]
    V. Kostelecky, Gravity, Lorentz violation and the standard model, Phys. Rev. D 69 (2004) 105009 [hep-th/0312310] [INSPIRE].ADSGoogle Scholar
  38. [38]
    V. Kostelecky and R. Lehnert, Stability, causality and Lorentz and CPT violation, Phys. Rev. D 63 (2001) 065008 [hep-th/0012060] [INSPIRE].MathSciNetADSGoogle Scholar
  39. [39]
    V. Kostelecky and M. Mewes, Signals for Lorentz violation in electrodynamics, Phys. Rev. D 66 (2002)056005 [hep-ph/0205211] [INSPIRE].ADSGoogle Scholar
  40. [40]
    V. Kostelecky and M. Mewes, Lorentz and CPT violation in neutrinos, Phys. Rev. D 69 (2004)016005 [hep-ph/0309025] [INSPIRE].ADSGoogle Scholar
  41. [41]
    V. Kostelecky and C.D. Lane, Constraints on Lorentz violation from clock comparison experiments, Phys. Rev. D 60 (1999) 116010 [hep-ph/9908504] [INSPIRE].ADSGoogle Scholar
  42. [42]
    V. Kostelecky and M. Mewes, Cosmological constraints on Lorentz violation in electrodynamics, Phys. Rev. Lett. 87 (2001) 251304 [hep-ph/0111026] [INSPIRE].ADSCrossRefGoogle Scholar
  43. [43]
    D. Bear, R. Stoner, R. Walsworth, V. Kostelecky and C.D. Lane, Limit on Lorentz and CPT violation of the neutron using a two species noble gas maser, Phys. Rev. Lett. 85 (2000) 5038 [Erratum ibid. 89 (2002) 209902] [physics/0007049] [INSPIRE].ADSCrossRefGoogle Scholar
  44. [44]
    D. Anselmi, Renormalization And Lorentz Symmetry Violation, PoS(CLAQG08)010.Google Scholar
  45. [45]
    D. Anselmi and D. Buttazzo, Distance Between Quantum Field Theories As A Measure Of Lorentz Violation, Phys. Rev. D 84 (2011) 036012 [arXiv:1105.4209] [INSPIRE].ADSGoogle Scholar
  46. [46]
    D. Anselmi, Renormalization of Lorentz violating theories, Prepared for 4th Meeting on CPT and Lorentz Symmetry, Bloomington, Indiana, 8-11 Aug 2007 [INSPIRE].
  47. [47]
    D. Anselmi and M. Taiuti, Vacuum Cherenkov Radiation In Quantum Electrodynamics With High-Energy Lorentz Violation, Phys. Rev. D 83 (2011) 056010 [arXiv:1101.2019] [INSPIRE].ADSGoogle Scholar
  48. [48]
    D. Anselmi and E. Ciuffoli, Low-energy Phenomenology Of Scalarless Standard-Model Extensions With High-Energy Lorentz Violation, Phys. Rev. D 83 (2011) 056005 [arXiv:1101.2014] [INSPIRE].ADSGoogle Scholar
  49. [49]
    D. Anselmi and E. Ciuffoli, Renormalization Of High-Energy Lorentz Violating Four Fermion Models, Phys. Rev. D 81 (2010) 085043 [arXiv:1002.2704] [INSPIRE].ADSGoogle Scholar
  50. [50]
    D. Anselmi and M. Taiuti, Renormalization Of High-Energy Lorentz Violating QED, Phys. Rev. D 81 (2010) 085042 [arXiv:0912.0113] [INSPIRE].
  51. [51]
    D. Anselmi, Standard Model Without Elementary Scalars And High Energy Lorentz Violation, Eur. Phys. J. C 65 (2010) 523 [arXiv:0904.1849] [INSPIRE].ADSCrossRefGoogle Scholar
  52. [52]
    D. Anselmi, Weighted power counting, neutrino masses and Lorentz violating extensions of the Standard Model, Phys. Rev. D 79 (2009) 025017 [arXiv:0808.3475] [INSPIRE].ADSGoogle Scholar
  53. [53]
    D. Anselmi, Weighted power counting and Lorentz violating gauge theories. II. Classification, Annals Phys. 324 (2009) 1058 [arXiv:0808.3474] [INSPIRE].MathSciNetADSMATHCrossRefGoogle Scholar
  54. [54]
    D. Anselmi, Weighted power counting and Lorentz violating gauge theories. I. General properties, Annals Phys. 324 (2009) 874 [arXiv:0808.3470] [INSPIRE].MathSciNetADSMATHCrossRefGoogle Scholar
  55. [55]
    D. Anselmi, Weighted scale invariant quantum field theories, JHEP 02 (2008) 051 [arXiv:0801.1216] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  56. [56]
    D. Anselmi and M. Halat, Renormalization of Lorentz violating theories, Phys. Rev. D 76 (2007)125011 [arXiv:0707.2480] [INSPIRE].ADSGoogle Scholar
  57. [57]
    P. Hořava, Quantum Gravity at a Lifshitz Point, Phys. Rev. D 79 (2009) 084008 [arXiv:0901.3775] [INSPIRE].ADSGoogle Scholar
  58. [58]
    M. Visser, Lorentz symmetry breaking as a quantum field theory regulator, Phys. Rev. D 80 (2009)025011 [arXiv:0902.0590] [INSPIRE].MathSciNetADSGoogle Scholar
  59. [59]
    M. Visser, Power-counting renormalizability of generalized Hořava gravity, arXiv:0912.4757 [INSPIRE].
  60. [60]
    T.P. Sotiriou, M. Visser and S. Weinfurtner, Quantum gravity without Lorentz invariance, JHEP 10 (2009) 033 [arXiv:0905.2798] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  61. [61]
    T.P. Sotiriou, M. Visser and S. Weinfurtner, Phenomenologically viable Lorentz-violating quantum gravity, Phys. Rev. Lett. 102 (2009) 251601 [arXiv:0904.4464] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  62. [62]
    S. Weinfurtner, T.P. Sotiriou and M. Visser, Projectable Hořava-Lifshitz gravity in a nutshell, J. Phys. Conf. Ser. 222 (2010) 012054 [arXiv:1002.0308] [INSPIRE].ADSCrossRefGoogle Scholar
  63. [63]
    M. Visser, Status of Hořava gravity: A personal perspective, J. Phys. Conf. Ser. 314 (2011) 012002 [arXiv:1103.5587] [INSPIRE].ADSCrossRefGoogle Scholar
  64. [64]
    S. Judes and M. Visser, Conservation laws inDoubly special relativity’, Phys. Rev. D 68 (2003)045001 [gr-qc/0205067] [INSPIRE].MathSciNetADSGoogle Scholar
  65. [65]
    S. Liberati, S. Sonego and M. Visser, Interpreting doubly special relativity as a modified theory of measurement, Phys. Rev. D 71 (2005) 045001 [gr-qc/0410113] [INSPIRE].MathSciNetADSGoogle Scholar
  66. [66]
    C. Barcelo, S. Liberati and M. Visser, Analogue gravity, Living Rev. Rel. 8 (2005) 12 [gr-qc/0505065] [INSPIRE].Google Scholar
  67. [67]
    M. Visser, Acoustic black holes: Horizons, ergospheres and Hawking radiation, Class. Quant. Grav. 15 (1998) 1767 [gr-qc/9712010] [INSPIRE].MathSciNetADSMATHCrossRefGoogle Scholar

Copyright information

© SISSA, Trieste, Italy 2012

Authors and Affiliations

  1. 1.School of Mathematics, Statistics, and Operations ResearchVictoria University of WellingtonWellingtonNew Zealand

Personalised recommendations