Lorentz-Breaking Field Theory

Dear Colleagues,

The possibility of Lorentz and/or CPT symmetry breaking began to be intensively discussed about thirty years ago. There are numerous factors that have called attention to such studies. First, any physical theory, including special and general relativity, should have some limits of applicability, which clearly establish the question of their determination. In particular, such an interest has been supported by studies of elementary particles, where the violation of various symmetries, for example, CP-symmetry, for certain processes has been proved. Second, the possible breaking of symmetries, including Lorentz-CPT symmetry, is supported by cosmological studies: indeed, most cosmological models claim that the expansion of the universe is irreversible, thus, there is a privileged direction, that is, the positive direction of the time axis, and, moreover, it is well known that our universe is characterized by matter–antimatter asymmetry. Third, Lorentz-breaking models could be interesting effective models for describing condensed matter, where, for example, one can encounter privileged directions generated by crystal symmetry axes. Fourth, string theory and loop quantum gravity establish the idea of minimal length, thus, it is natural to expect that at the distances of the order of this length the Lorentz symmetry can be broken.

The first field theory model involving the small Lorentz-breaking correction was formulated in [1]. In this paper, the four-dimensional generalization of the Chern–Simons term, called the Carroll–Field–Jackiw (CFJ) term was proposed, as a result, the four-dimensional gauge invariant massive theory was established. As a consequence, many aspects of this new theory, both classical and quantum ones, began to be discussed. The theory became a natural laboratory for studying the impacts of the Lorentz symmetry breaking, displaying many new effects, both at the classical level (for example, the birefringence of plane wave solutions [2]) and at the quantum level (ambiguity of the CFJ term generated as a quantum correction [3]). The success of the Lorentz-violating QED with the CFJ term implied an introduction of the Lorentz-breaking standard model extension [2] where many other Lorentz-breaking additive terms involving not only gauge but scalar and spinor fields as well were introduced.

The real breakthrough in studies of Lorentz-CPT breaking theories began after publishing the paper [4] where the CFJ term was shown to arise as a quantum correction, being finite.

Essentially, the main result of this paper consisted of establishing the idea that the Lorentz-breaking effects can naturally arise as a consequence of quantum corrections, in the low-energy limit of some fundamental theory, as a result of integration over matter fields. Many quantum aspects of perturbative generation of the CFJ term were considered, the most important among them is the relation of this term to the ABJ anomaly, implied in the ambiguity of the result [3], which, in its part, called attention to various schemes for generating this term (besides [3, 4], see, for example, [5] and the references therein). Further, the methodology of perturbative generating of Lorentz-breaking contributions has been generated to other terms not only in the gauge sector but in other sectors as well (for a review, see, for example, [6]).

A very important problem within the context of the Lorentz and/or CPT symmetry breaking is the problem of a consistent implementation of the breaking of these symmetries within the gravity context. The first example this was performed in [7] where the four-dimensional Chern–Simons modified gravity was formulated. This model displays CPT symmetry breaking, and, for the Chern–Simons coefficient, also Lorentz symmetry breaking. Further, the 4D gravitational Chern–Simons term has been generated as a one-loop correction [8] that is finite and ambiguous. As in the case of the CFJ term, this ambiguity is related to the anomaly, in this case, the gravitational one. Further, other examples of CPT and/or Lorentz breaking terms in gravity have been proposed, a review of these is presented in [9].

While the 4D gravitational Chern–Simons term represents itself as one of the most successful attempts to introduce Lorentz-CPT breaking within a gravitational context, the general study of Lorentz symmetry breaking in curved space-time encounters several fundamental problems. First, Lorentz symmetry in curved space-time is extended to diffeomorphism symmetry, which, at the same time, plays the role of the gauge symmetry. Therefore, the possible Lorentz-breaking terms in gravity would break the gauge symmetry at the same time (for a description of the problem of the diffeomorphism symmetry breaking, see [10]). Second, it is well known that in space the Lorentz symmetry is usually broken through the introduction of a constant vector or tensor. In curved space-time, a constant tensor cannot be introduced in a consistent way; indeed, the requirement of the simple derivative of a tensor to be zero is not covariant, while the requirement of a covariant derivative of a tensor to vanish imposes strong, and, in many cases, inconsistent conditions on the background metric. Otherwise, one can start with the usual Lorentz-breaking extension of any theory (for example, QED) and extend it to a curved space-time, promoting the constant vectors or tensors to some background tensors, but, in this case, within perturbative calculations, terms proportional to the covariant derivatives of these tensors will arise leading to results with a very complicated structure, see, for example, [11]. As a consequence, explicit Lorentz symmetry breaking based on a definition of a special vector or tensor that introduces privileged space-time dimensions from the very beginning appears to be less advantageous than spontaneous Lorentz symmetry breaking where the privileged space-time direction emerges as a result of the choice of a minimum for some potential for a vector or tensor. The most successful manner to extend a theory with spontaneous Lorentz symmetry breaking to curved space-time is the bumblebee theory [12] where the dynamics of the vector field with a nontrivial potential are embedded into curved space-time. However, until now, only the first steps in the study of gravity coupled with a bumblebee model have been undertaken, see, for example, [13].

Discussing Lorentz symmetry breaking, it is very important to consider its possible applications within studies of condensed matter. As we already noted, the reasons for these applications are, first, the natural anisotropy of crystals, and, second, the fact that in a vacuum the invariant velocity is the speed of light, while, in condensed matter it is the Fermi velocity, which is essentially anisotropic. This allows us to formulate the Lorentz-breaking effective theory for various condensed matter systems such as Weyl semi-metals [14] and topological insulators [15]. In these studies, this effective theory has been represented in a form similar to the known Lorentz-CPT breaking QED. Therefore, it is natural to expect that other Lorentz-breaking extensions of QED can also be applied for other condensed matter systems.

All this allows us to establish the following lines of study in Lorentz symmetry breaking.

The most important direction involves the consideration of various Lorentz-breaking extensions of gravity, both on a classical level (impacts of Lorentz-breaking modifications for exact solutions and gravitational waves propagation, new cosmological effects, PPN approach, etc.) and on a quantum level (perturbative generation, especially the search for cases where the corresponding quantum correction turns out to be finite, the study of possible modifications of gravitational anomalies, consideration of quantum aspects of bumblebee theory on a curved background, etc.). It should be noted that most researches of the Lorentz-breaking aspects of gravity have been performed within the framework of Riemannian geometry. At the same time, such researches within other geometrical methodologies, for example, those involving torsion and nonmetricity were not carried out, hence, it can be important to perform these studies.

The second important direction is the detailed discussion of tree-level and perturbative impacts of new additive Lorentz-breaking terms in the fermionic sector, both of higher-dimension quadratic additive terms [16] and of higher-dimension interaction terms [17]. The third important direction is the development of more applications of Lorentz-breaking theories within the condensed matter context.

These directions will be presented in various papers of our Special Issue.

[1] S. M. Carroll, G. B. Field and R. Jackiw, Phys. Rev. D 41, 1231 (1990).

[2] D. Colladay and V. A. Kostelecky, Phys. Rev. D 55, 6760 (1997) [hep-ph/9703464]; D. Colladay and V. A. Kostelecky, Phys. Rev. D 58, 116002 (1998) [hep-ph/9809521].

[3] R. Jackiw, Int. J. Mod. Phys. B 14, 2011 (2000) [hep-th/9903044].

[4] R. Jackiw and V. A. Kostelecky, Phys. Rev. Lett. 82, 3572 (1999) [hep-ph/9901358].

[5] T. Mariz, J. R. Nascimento, E. Passos, R. F. Ribeiro and F. A. Brito, JHEP 0510, 019 (2005) [hep-th/0509008].

[6] A. F. Ferrari, J. R. Nascimento and A. Y. Petrov, arXiv:1812.01702 [hep-th].

[7] R. Jackiw and S. Y. Pi, Phys. Rev. D 68, 104012 (2003) [gr-qc/0308071].

[8] T. Mariz, J. R. Nascimento, E. Passos and R. F. Ribeiro, Phys. Rev. D 70, 024014 (2004) [hep-th/0403205].

[9] V. A. Kostelecky, Phys. Rev. D 69, 105009 (2004) [hep-th/0312310].

[10] V. A. Kostelecky and M. Mewes, Phys. Lett. B 779, 136 (2018) [arXiv:1712.10268 [gr-qc]].

[11] G. de Berredo-Peixoto and I. L. Shapiro, Phys. Lett. B 642, 153 (2006) [hep-th/0607109].

[12] O. Bertolami and J. Paramos, Phys. Rev. D 72, 044001 (2005) [hep-th/0504215].

[13] M. D. Seifert, Phys. Rev. D 81, 065010 (2010) [arXiv:0909.3118 [hep-ph]].

[14] A. G. Grushin, Phys. Rev. D 86, 045001 (2012) [arXiv:1205.3722 [hep-th]].

[15] A. Martín-Ruiz, M. Cambiaso and L. F. Urrutia, Phys. Rev. D 92, 125015 (2015) [arXiv:1511.01170 [cond-mat.other]].

[16] A. Kostelecky and M. Mewes, Phys. Rev. D 88, 096006 (2013) [arXiv:1308.4973 [hep-ph]].

[17] A. Kostelecky and Z. Li, arXiv:1812.11672 [hep-ph].

Prof. Dr. Albert Petrov
Guest Editor

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• Lorentz and Poincare symmetries
• CPT symmetry