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Lorentz Breaking Effective Field Theory Models for Matter and Gravity: Theory and Observational Constraints

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Gravity: Where Do We Stand?

Abstract

A number of different approaches to quantum gravity are at least partly phenomenologically characterized by their treatment of Lorentz symmetry, in particular whether the symmetry is exact or modified/broken at the smallest scales. For example, string theory generally preserves Lorentz symmetry while analog gravity and Lifshitz models break it at microscopic scales. In models with broken Lorentz symmetry, there are a vast number of constraints on departures from Lorentz invariance that can be established with low-energy experiments by employing the techniques of effective field theory in both the matter and gravitational sectors. We shall review here the low-energy effective field theory approach to Lorentz breaking in these sectors, and present various constraints provided by available observations.

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Notes

  1. 1.

    Anisotropic scaling [35–37] techniques were recently recognized to be the most appropriate way of handling higher-order operators in Lorentz breaking theories and in this case the highest-order operators are indeed crucial in making the theory power counting renormalizable. This is why we shall adopt sometimes the expression “naively non-renormalizable.”

  2. 2.

    A notable exception to this assumption is the SME and associated tests. Rotational invariance is not assumed in the program of the SME as it considers all terms at each mass dimension.

  3. 3.

    We disregard here the possible appearance of dissipative terms [74] in the dispersion relation, as this would correspond to a theory with unitarity loss and to a more radical departure from standard physics than that envisaged in the framework discussed herein (albeit a priori such dissipative scenarios are logically consistent and even plausible within some quantum/emergent gravity frameworks).

  4. 4.

    We consider here only \(\kappa = 3,4\), for which these relationships have been demonstrated.

  5. 5.

    Actually these criteria allow the addition of other (CPT even) terms, but these would not lead to modified dispersion relations (they can be thought of as extra, Planck suppressed, interaction terms) [39].

  6. 6.

    Note that for an object located at cosmological distance (let \(z\) be its redshift), the distance \(d\) becomes

    $$ d(z) = \frac{1}{H_{0}}\int^{z}_0 \frac{1+z'}{\sqrt{\Omega_{\Lambda} + \Omega_{m}(1+z')^{3}}}\,dz'\;, $$
    (23)

    where \(d(z)\) is not exactly the distance of the object as it includes a \((1+z)^{2}\) factor in the integrand to take into account the redshift acting on the photon energies.

  7. 7.

    Faraday rotation is negligible at these energies.

  8. 8.

    The same paper also claims a strong constraint on the parameter \(\xi ^{(4)}\). Unfortunately, such a claim is based on the erroneous assumption that the EFT order six operators responsible for this term imply opposite signs for opposite helicities of the photon. We have instead seen that the CPT evenness of the relevant dimension-six operators imply a helicity independent dispersion relation for the photon (see Eq. (8)).

  9. 9.

    One could of course consider any lepton/antilepton pair as produced, for example, the reaction \(e^- \rightarrow e^- \nu \overline {\nu }\). While standard particle physics arguments imply that the rate will be roughly equivalent to the \(e^-e^+\) pair production case [119] given the same order of coefficients, and the threshold will be slightly lower, the constraints are on a higher-dimensional parameter space and so less useful.

  10. 10.

    Note that the bounds presented here are weaker by a factor of \(10^{-3}\), as we have used the CMB frame as the rest frame rather than the Sun-centered frame and therefore the strengths of the bounds are weakened by the \(v/c\) of the Sun with respect to the CMB.

  11. 11.

    Note the index symmetry \({}{}{Z}{^{\beta \alpha }_{\delta \gamma }} = {}{}{Z}{^{\alpha \beta }_{\gamma \delta }}\).

  12. 12.

    Note that one could have fixed the norm to be different than \(-1\), however, one can simply scale \(u^\alpha \) to have norm \(-1\) and absorb the scaling into the coefficients.

  13. 13.

    Note however, that some knowledge of DSR phenomenology can be obtained by considering that, as in special relativity, any phenomenon that implies the existence of a preferred reference frame is forbidden. Thus, the detection of such a phenomenon would imply the falsification of both special and DSR. An example of such a process is the decay of a massless particle.

  14. 14.

    This is a somewhat harsh statement given that it was shown in [174] that a substantial (albeit reduced) high-energy gamma-ray flux is still expected also in the case of mixed composition, so that in principle the previously discussed line of reasoning based on the absence of upper threshold for UHE gamma rays might still work.

  15. 15.

    UHE nuclei suffer mainly from photodisintegration losses as they propagate in the intergalactic medium. Because photodisintegration is indeed a threshold process, it can be strongly affected by LV. According to [176], and in the same way as for the proton case, the mean free paths of UHE nuclei are modified by LV in such a way that the final UHECR spectra after propagation can show distinctive LV features. However, a quantitative evaluation of the propagated spectra has not been performed yet.

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Acknowledgements

We wish to thank Luca Maccione for useful insights, discussions, and feedback on the manuscript preparation.

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  1. SISSA and INFN, Via Bonomea 265, Trieste, Italy

    Stefano Liberati

  2. University of New Hampshire, 03824, Durham, NH, USA

    David Mattingly

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  2. Dipartimento di Fisica G. Occhialini, Università degli Studi di Milano-Bicocca, Milano, Italy

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  3. Dipartimento di Scienza e Alta Tecnologia, Università degli Studi dell’Insubria, Como, Italy

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    Ugo Moschella

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Liberati, S., Mattingly, D. (2016). Lorentz Breaking Effective Field Theory Models for Matter and Gravity: Theory and Observational Constraints. In: Peron, R., Colpi, M., Gorini, V., Moschella, U. (eds) Gravity: Where Do We Stand?. Springer, Cham. https://doi.org/10.1007/978-3-319-20224-2_11

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