Neutrinos as Probes of Lorentz Invariance

Díaz, Jorge S.

doi:https://doi.org/10.1155/2014/962410

Advances in High Energy Physics

On this page

Abstract Introduction Conclusions Acknowledgments References Copyright Related Articles

Special Issue

Through Neutrino Eyes: The Search for New Physics

View this Special Issue

Review Article | Open Access

Volume 2014 | Article ID 962410 | https://doi.org/10.1155/2014/962410

Neutrinos as Probes of Lorentz Invariance

Jorge S. Díaz^1,2

Academic Editor: Michael A. Schmidt

Received07 Feb 2014

Accepted14 Apr 2014

Published16 Jun 2014

Abstract

Neutrinos can be used to search for deviations from exact Lorentz invariance. The worldwide experimental program in neutrino physics makes these particles a remarkable tool to search for a variety of signals that could reveal minute relativity violations. This paper reviews the generic experimental signatures of the breakdown of Lorentz symmetry in the neutrino sector.

1. Introduction

In 1930, Pauli postulated the neutrino as a desperate remedy to save one of the sacred principles of physics, the conservation of energy, which appeared to be violated in beta decays [1]. Today, neutrinos remain as some of the less understood particles of the standard model (SM) and their mysteries are still fascinating. Their ghostly nature makes them barely interact with matter and their interferometric behavior makes them oscillate between different flavors. Neutrino oscillations have led to the remarkable conclusion of massive neutrinos, presenting an established evidence of physics beyond the SM.

In the search for new physics, different candidate theories for quantum gravity involve mechanisms that could trigger the breakdown of one of the most fundamental symmetries in modern physics, Lorentz invariance. In the theoretical front, Lorentz-violating descriptions of neutrino behavior have shown that these fundamental particles can serve as powerful probes of new physics. Experimentally, neutrino oscillations have been used to perform several searches for Lorentz violation. The development of techniques to perform systematic searches for Lorentz violation in many other experimental setups shows a rich phenomenology to be studied, with a large variety of experimental effects that remain unexplored.

This paper summarizes the main experimental signatures of deviations from exact Lorentz invariance in the neutrino sector. The experimental searches for Lorentz violation performed in recent years are presented and future tests of Lorentz symmetry are discussed, ranging from precision measurements of beta decay and double beta decay at low energies to the high energy of astrophysical neutrinos and oscillation experiments using accelerator, atmospheric, and reactor neutrinos.

2. Lorentz Invariance Violation

Deviations from exact Lorentz symmetry have been shown to be possible at very high energies in candidate descriptions of gravity at the quantum level. For instance, mechanisms for the spontaneous breaking of this fundamental symmetry have been identified in string-theory scenarios [2]. Interactions could generate Lorentz-violating terms if a tensor field acquires a nonzero vacuum expectation value , which acts as a background field. In the same fashion as the nonzero vacuum expectation value of the dynamical Higgs field generates mass terms for other fields via interactions, background tensor fields that couple to conventional particles in the SM will generate new terms that break Lorentz invariance. These new terms are Lorentz scalars under coordinate transformations; in fact, the spacetime indices of the background field are all contracted with the indices of the SM operator in the form . This structure guarantees that there is no privileged reference frame because the theory is observer invariant. For instance, consider the case of Lorentz violation generated by a 2-tensor , which will be coupled to some SM operator with the same number of spacetime indices in the form . Under a coordinate transformation both the operator and the tensor background field transform to a new set of coordinates as follows: so that the terms in the Lagrangian remain invariant . The same construction can be applied for a general tensor.

On the other hand, when a Lorentz transformation is performed over the physical system, this is when the experimental apparatus is rotated or boosted rather than the coordinates used to describe it, then the SM operator transforms as shown in (1) but any background field remains unchanged . This so-called particle Lorentz transformation changes the coupling between the background fields and the SM operators, resulting in a physically observable anisotropy of spacetime; this is a violation of Lorentz invariance [3].

Phenomenological approaches to parameterize and experimentally search for particular types of Lorentz violation have been considered since several decades [4–7]. However, effective field theory can be used to incorporate generic operators that break Lorentz invariance for all the particles in the SM. This general framework is known as the standard-model extension (SME) [3, 8, 9], whose action includes general coordinate-invariant terms by contracting operators of conventional fields with controlling coefficients for Lorentz violation and reduces to the SM if all these coefficients vanish. Gravity can also be incorporated by writing the SM in a curved background [9]. The development of the SME has led to a worldwide experimental program searching for violations of Lorentz invariance, whose results are summarized in [10].

Flat spacetime is considered for experiments in particle physics, in which case the coefficients for Lorentz violation that act as background fields can be chosen to be constant and uniform, which guarantees conservation of energy and linear momentum. In this limit, these coefficients represent the vacuum expectation value of the tensor fields of the underlying theory. Excitations of these fields lead to a rich phenomenology; for instance, Nambu-Goldstone modes could play fundamental roles when gravity is included, such as the graviton, the photon in Einstein-Maxwell theory [11–14], and spin-dependent [15] and spin-independent [16] forces.

It should be noted that a subset of operators in the SME also break CPT symmetry. In fact, all the Lorentz-violating terms in the action involving operators with an odd number of spacetime indices are odd under a CPT transformation. In realistic field theories, CPT violation always appears with Lorentz violation [17]. Nonetheless, alternative approaches exist in which CPT violation is implemented with and without Lorentz invariance [18–25].

3. Neutrinos

The general description of three left-handed neutrinos and three right-handed antineutrinos in the presence of Lorentz-violating background fields is given by a effective Hamiltonian of form [28] where the indices indicate the flavors of active neutrinos and antineutrinos . The Lorentz-preserving component is explicitly given by the following: where at leading order the neutrino momentum is given by the energy and the mass-squared matrix is commonly written in terms of the Pontecorvo-Maki-Nakagawa-Sakata (PMNS) matrix [29–31] as , with .

The Lorentz-violating block describing neutrinos in the Hamiltonian (2) is given by the following: The components of the complex matrices and are called coefficients for CPT-odd and CPT-even Lorentz violation, respectively. The spacetime indices encode the nature of the broken symmetry; for instance, isotropic (direction-independent) Lorentz violation appears when only the time components of the coefficients are nonzero; while space anisotropy appears when any of the other components is nonzero, generating direction-dependent effects in the neutrino behavior. The breakdown of invariance under rotations is evident due to the presence of the four-vector that depends on the neutrino direction of propagation .

The block Hamiltonian describing right-handed antineutrinos is obtained as the CP conjugate of the neutrino Hamiltonian , which has the same form as the neutrino Hamiltonian (4) with and . Given the structure of these coefficients in flavor space, it is expected that they will affect neutrino mixing and oscillations. Notice, however, that there exist coefficients that modify the three flavors in the same way, producing no effects on neutrino oscillations because they are proportional to the identity in flavor space. These oscillation-free coefficients and their observable effects are discussed in Sections 5 and 6.

Signals of the breakdown of Lorentz invariance correspond to the anisotropy of spacetime due to preferred directions set by the coefficients for Lorentz violation that act as fixed background fields. Taking advantage of the coupling of these background fields with the neutrino direction of propagation , we can search for violations of Lorentz invariance by making measurements with neutrino beams with different orientations, which would reveal the presence of the SME coefficients and . For Earth-based experiments, detectors and source rotate with a well-defined angular frequency due to Earth’s rotation, which makes the neutrino direction vary with respect to the fixed background fields. This time dependence will explicitly appear in the relevant observable quantities and it can be parameterized as harmonics of the sidereal angle . Due to the invariance of the theory under coordinate transformations, there is no preferred reference frame to make the measurements. In order to establish a consistent and systematic search for Lorentz-violating effects, experimental results are conventionally reported in the Sun-centered equatorial frame described in [10, 32]. In this frame, the sidereal variation of the coupling between the neutrino direction and the background fields that break Lorentz symmetry can be explicitly written in the form where the amplitude of each sidereal harmonic is a function of the coefficients for Lorentz violation and experimental parameters including neutrino energy, location of the experiment, and relative orientation between source and detector [33, 34].

The coefficients and arise from operators of dimension three and four, respectively. Operators of arbitrary dimension can be incorporated in the theory, in which case the coefficients for Lorentz violation in the Hamiltonian (4) appear as momentum-dependent quantities of the form [35, 36] The extra derivatives in the Lagrangian appear in the neutrino Hamiltonian as higher powers of the neutrino energy. Although the conventional massive-neutrino description of oscillations accommodates all the established experimental results, nonnegative powers of the neutrino energy could help to elegantly solve some anomalous results obtained in recent years in beam experiments [37–39]. In fact, interesting attempts to describe the global data using the SME have led to the construction of alternative models for neutrino oscillations that can accommodate the results reported by different experiments [40–46].

In the following sections we discuss the observable signatures of these coefficients for Lorentz violation in different types of neutrino experiments including oscillations, neutrino velocity, and beta decays.

4. Neutrino Oscillations

The interferometric nature of neutrino oscillations has been widely identified as an ideal experimental setup to search for new physics in the form of deviations from the conventional description of neutrinos. The mixing and oscillation between neutrino flavors occur in general due to off-diagonal entries in the neutrino Hamiltonian leading to eigenstates with different energy.

4.1. Oscillation of Neutrinos and Antineutrinos

Neutrino oscillations are well described by a model of three massive neutrinos, which depends on two mass-squared differences and controlling the oscillation lengths and three mixing angles , and that govern the amplitude of the oscillation [47]. According to this massive-neutrino model, the oscillation probabilities are proportional to the factor . Tests of Lorentz invariance using neutrino oscillations can be performed by searching for deviations from the conventional behavior. For some experiments studying neutrinos over a large range of energies and baselines, such as Super-Kamiokande [48], an exact treatment of the diagonalization of the Hamiltonian is necessary [49]. In most cases, the experimental features allow the implementation of approximation methods for determining the oscillation probabilities, as discussed in the following sections.

4.1.1. Short-Baseline Approximation

According to the massive-neutrino model, for experiments using neutrinos of energy and baseline that satisfy , the oscillation phase would be too small to impact the neutrino propagation and no neutrino oscillations should be observed. The effects of the mass terms in the conventional Hamiltonian (3) become negligible and the effective Hamiltonian can be approximated by . Direct calculation of the oscillation probabilities shows that in this approximation we can write the appearance probability [33] with a similar expression for antineutrinos using instead. The sidereal decomposition of the Hamiltonian (5) can be used to show that the probability will also exhibit sidereal variations, one of the key signatures of Lorentz violation. The Hamiltonian contains also a time-independent component , which can lead to both isotropic and direction-dependent effects. In all cases, the energy dependence is different with respect to the conventional case, so spectral studies can be used to study these particular coefficients.

Experimental studies using the probability (7) have been performed by Double Chooz [50], IceCube [51], LSND [52], MiniBooNE [53, 54], and MINOS using its near detector [55, 56]. The absence of a positive signal in all these experiments has been used to set tight constraints on several coefficients for Lorentz violation, which are summarized in [10]. It is important to emphasize that, since all these searches use different oscillation channels, they are complementary, accessing similar coefficients but with different flavor indices.

4.1.2. Perturbative Approximation

Since the phase of the oscillation is given by , the oscillation probability can be enhanced by placing a detector at a distance from the neutrino source. For experiments satisfying this condition, mass-driven oscillations dominate in the Hamiltonian and we can consider the effects of Lorentz violation as a small perturbation [34]. In this case, the oscillation probability appears as a power series of the form where is the conventional probability given by the massive-neutrino model and the following terms are given in powers of the coefficients for Lorentz violation, whose explicit form is given in [34]. Once again, the sidereal variation of the oscillation probability appears as a key signal to search by experiments. For instance, the leading-order term can be generically written as follows: This is the dominating probability for neutrino mixing as well as antineutrino oscillations.

Since the first-order correction (9) to the oscillation probability arises from the interference between the conventional and the Lorentz-violating effects, the sensitivity to the coefficients in is greater than in the short-baseline approximation presented in the previous section. Figure 1 shows part of the study performed by the MINOS experiment, which used the expression (9) to search for sidereal variations in the event rate measured at the far detector. The sensitivity to different coefficients was improved by a factor 20–510 compared to the previous constraints using the near detector [26].

The mixing between neutrinos and antineutrinos is also possible due to the block in the Hamiltonian (2), which is discussed in the following section.

4.2. Neutrino-Antineutrino Mixing

The off-diagonal block in the Hamiltonian (2) can produce the mixing between neutrinos and antineutrinos. This matrix is given by [28] where the complex 4-vector is the neutrino polarization that can be directly written in terms of the location of the experiment and the orientation of the neutrino beam [34]. Two sets of coefficients for Lorentz violation denoted by and control CPT-even and CPT-odd effects, respectively. This Hamiltonian can also be decomposed in form (5), with harmonic amplitudes given in terms of the coefficients and [34].

Contrary to the neutrino Hamiltonian (4), the neutrino-antineutrino block always appears with direction-dependent effects, for this reason the search of sidereal variations is an ideal setup to search for these coefficients. Following the perturbative description presented in Section 4.1.2, it has been shown that at first order the oscillation probability vanishes, in other words, neutrino-antineutrino oscillations appear as a second order effect [34]. For this reason, the second-order probability can be decomposed in form (9), although involving up to four harmonics.

The possible oscillation of neutrinos into antineutrinos modifies, for instance, the survival probability of muon neutrinos in a beam experiment because now some could disappear into antineutrino states. A systematic search of the 66 coefficients and producing sidereal variations was performed using data from the MINOS experiment [57]. The remaining 15 coefficients producing time-independent effects could only be explored by a spectral study in a disappearance experiment. Figure 2 shows a fit to the data from the Double Chooz experiment, searching for the spectral modification that could arise in the disappearance of electron antineutrinos [27].

A total of 81 coefficients and has been tightly constrained by these two experimental searches, whose results are summarized in [10].

5. Neutrino Kinematics

Oscillations are very sensitive to unconventional effects producing neutrino mixing due to their interferometric nature. There are, however, terms in the Hamiltonian (2) that are unobservable in oscillations. Neutrino oscillations only allow us to measure energy differences between different neutrino states; for this reason, the absolute scale of neutrino masses cannot be determined from oscillations. Similarly, some coefficients for Lorentz violation modify the energy of all flavors in the same way producing no effects in oscillations. Neglecting mixing effects results in a decoupling of the three-flavor system into three copies of a single state. One of the observable effects of these oscillation-free coefficients is the modification of the neutrino velocity, which produces measurable effects in the neutrino time of flight. Moreover, as a consequence of the unconventional dispersion relations, the neutrino phase space and energy-conservation condition relevant for decay processes are modified as well.

5.1. Neutrino Velocity

The neutrino velocity can be obtained from the Hamiltonian (2). For completeness, operators of arbitrary dimension can be incorporated, in which case the neutrino velocity takes the form [35] where the factor is a real mass parameter that does not participate in oscillations, and the Lorentz-violating component has been written in spherical form. The index denotes the effective dimension of the operator and the pair corresponds to angular momentum indices that label the rotational properties of the oscillation-free spherical coefficients and , controlling CPT-odd (for odd ) and CPT-even (for even ) effects, respectively. These spherical coefficients can be identified with coefficients in Cartesian coordinates used in the previous sections [35]. The expression for the neutrino velocity (11) has been written in the Sun-centered frame [10, 32], where all the directional information is contained in the angular factors and the sidereal time dependence appears as harmonics functions controlled by the index .

The neutrino velocity (11) exhibits a rich phenomenology in the form of many physical effects that can affect neutrino propagation if deviations from Lorentz symmetry are present. Depending on the dimension of the operator in the theory, neutrino velocity can be energy dependent; for , anisotropic effects appear and the velocity becomes a function of the direction of propagation; for , time dependence arises, in which case the neutrino velocity varies with sidereal time ; and for odd , CPT violation makes neutrinos and antineutrinos move at different speed.

Beam experiments are suitable setups to compare the speed of neutrinos with respect to the speed of photons. From the neutrino velocity (11), we clearly find that the mass term makes neutrinos travel slower than light, whereas the coefficients for Lorentz violation can generate subluminal or superluminal velocities depending on the sign of each coefficient. Different beam experiments have measured the time for neutrinos to travel a distance [58–64], which will experience a delay with respect to photons given by the following: which can be used to set limits in the oscillation-free coefficients for Lorentz violation that modify the neutrino velocity in (11). Since the minute effects of Lorentz violation can be enhanced by neutrinos travelling a long distance, a precise constraint on the isotropic dimension-four coefficient was obtained using the few antineutrino events from the supernova SN1987A [65].

For the particular case of Lorentz invariance violation generated by a dimension-four operator, the modification to the neutrino velocity (11) is simply a constant factor. For operators of dimension , low- and high-energy neutrinos will move at different velocity. If a burst of neutrinos of different energies is created at the same time, this velocity difference will generate a spread of neutrinos, observable as a delay between high- and low-energy neutrinos at the detector [35]. A similar effect has been widely studied for Lorentz-violating photons [66–69].

5.2. Threshold Effects

The modified dispersion relations that neutrinos satisfy in the presence of Lorentz violation alter the energy-momentum conservation relation, which plays an important role in meson-decay processes of the form . It can be shown that above some threshold energy these relations can completely block the phase space available for the decay [35, 70–78]. The observation of atmospheric and accelerator neutrinos with energy produced by the decay of a meson of mass implies that , which can be used to write the condition where the + (−) sign is for neutrinos (antineutrinos) and is the mass of the accompanying charged lepton. This formula has been used in [35] to constrain several coefficients for Lorentz violation, including many associated to nonrenormalizable operators.

The sensitivity to the effects of Lorentz violation increases with the energy of neutrinos observed as well as the distance that they travel. The observation of very-high-energy neutrinos reported by the IceCube collaboration [79, 80] offers a great sensitivity to the effects described in this section. The small number of neutrinos observed with energies at the PeV level [79] allows the study of isotropic effects (); nevertheless, a full study of direction-dependent effects would require several events spread in the sky. Although the IceCube results suggest an astrophysical origin for these energetic neutrinos, tight constraints on different coefficients for Lorentz violation can be obtained even in the conservative interpretation of these neutrino events having atmospheric origin [81]. The observation of PeV neutrinos created by the decay of heavy mesons in the upper atmosphere has been used to implement the threshold condition (13), leading to sensitive limits in several isotropic coefficients of dimension and [81].

5.3. erenkov Radiation

In same way as some processes can be forbidden above certain energies, the effects of Lorentz violation can also open particular decay channels that would be otherwise forbidden. In particular, coefficients leading to in (11) can produce erenkov emission of one or more particles [35, 82–92]. erenkov radiation makes neutrinos lose energy, which distorts the spectrum in long-baseline experiments using accelerator and atmospheric neutrinos. This feature provides another method to search for Lorentz violation. The observation of high-energy neutrinos after propagating a distance sets a lower value for the characteristic distortion distance in the form . The determination of the characteristic distance for the spectral distortion caused by the isotropic Lorentz-violating operator of dimension four is described in [82–92]. The general calculation including direction-dependent effects for operators of arbitrary dimension can be found in [35].

Using the PeV neutrinos observed by IceCube [79], the limits obtained using threshold conditions can be improved by one order of magnitude by determining spectral distortion produced by erenkov radiation [81]. For instance, the emission of electron-positron pairs in the form is characterized by a rate of energy loss given by [35, 81] where is a constant, the auxiliary 4 vectors and have been defined in terms of the momentum of the electron and the positron, and , , following the conventions in Figure 3. Several orders of magnitude in sensitivity can be gained when using an astrophysical interpretation for the PeV neutrinos in IceCube. After travelling astrophysical distances these neutrinos would rapidly fall below the threshold energy for erenkov emission. The observation of these neutrinos with PeV energies implies that this threshold energy lies above 1 PeV, leading to stringent limits on isotropic Lorentz violation of dimension , and [81]. Similar studies for the case can be found in [93, 94].

Direction-dependent effects using high-energy neutrinos require several events. The recent observation of 26 new energetic events in IceCube [80] distributed in the sky allows the search of space anisotropy for operators of dimension [81]. The simultaneous study of several coefficients producing direction-dependent effects allows two-sided bounds, more restrictive than the very particular case of isotropic Lorentz violation considering superluminal velocity that allows one-sided limits only. In the future, the observation of more events should allow a detailed study of operators of higher dimension.

6. Beta Decay

The interferometric nature of neutrino oscillations makes them an ideal type of experiment to search for minute deviations from exact Lorentz symmetry. Nonetheless, the effects that modify the kinematics of all neutrino flavors in the same manner are unobservable in oscillation experiments, which makes the studies described in Section 5 an important complement to oscillation searches. The enhancement of Lorentz-violating effects with the neutrino energy makes also the study of neutrino velocity and erenkov radiation a sensitive probe of Lorentz invariance with high-energy neutrinos. Nevertheless, it has been shown that low-energy experiments can also play a key role in the study of Lorentz invariance. In particular, signals of oscillation-free operators of dimension three are not only unobservable in oscillations but also produce no effects in the neutrino velocity (11).

The experimental signatures of the coefficients associated with these so-called countershaded operators [16, 95, 96] motivate the study of weak decays. The effects of these operators are unaffected by the neutrino energy, giving low-energy experiments a competitive sensitivity to signals of Lorentz violation. It is important to emphasize that Lorentz-violating effects appear as kinematical effects modifying the neutrino phase space; nevertheless, modifications of the spinor solutions must also be taken into account. Beta decay in the context of Lorentz violation in sectors other than neutrinos has recently been studied theoretically [97, 98] and experimentally [99].

6.1. Tritium Decay

The absolute mass scale of neutrinos cannot be studied in oscillation experiments, which only offer access to mass-squared differences. The direct measurement of neutrino masses can be made by searching for a distortion of the electron energy spectrum in tritium decay. The measurement of beta electrons near the endpoint of the spectrum allows the study of the effective mass of electron antineutrinos, where is approximately constant and denotes the kinetic energy of the electron measured from the endpoint energy . This type of experimental measurements has been made by Troitsk [100] and Mainz [101], and high precision will be achieved by KATRIN [102].

In these experiments the antineutrino escapes undetected; however, magnetic fields select the beta electrons emitted in a particular direction to be studied. This feature permits the study of anisotropic effects. In the presence of Lorentz-violating neutrinos, the spectrum (15) gets corrected by the replacement which shows that the distortion near the endpoint can be shifted and also exhibits a sidereal-time dependence. The amplitudes in the modification (16) depend on the four independent coefficients , , , and and experimental quantities such as location of the laboratory, orientation of the apparatus, and intensity of the magnetic fields used to select the beta electrons for their analysis [96].

An interesting feature appears when the effective coefficients are considered, which arise as a consequence of neutrino mass and CPT-even Lorenz violation [35]. These coefficients can mimic the effects of a mass parameter, in which case the spectrum (15) gets modified in the form We find that the experimental mass-squared parameter involves the actual neutrino mass ; however, the mass can be screened by the effects of the three coefficients (), varying with sidereal time and depending on the orientation of the apparatus. Notice also that the sign of the experimental mass-squared parameter is not restricted to be positive, so the coefficients could even mimic a tachyonic neutrino [103].

Alternative approaches have been considered to search for isotropic Lorentz violation in tritium decay for other operators in [104, 105].

6.2. Neutron Decay

Neutrons are fascinating laboratories to study the validity of fundamental symmetries. The effects of deviations from exact Lorentz invariance would affect the spectrum of the beta electrons as well as the measurements of particular experimental asymmetries. Contrary to tritium decay experiments, the study of neutron decay covers the whole energy spectrum, which takes the form of the spectrum (15) neglecting the neutrino mass that plays no role far from the endpoint and the factor can no longer be approximated by a constant so it becomes a function of the electron energy. For experiments only counting the number of beta electrons per energy range, all the anisotropic effects disappear after integrating over all the neutrino orientations. The net effect is a distortion of the whole spectrum that can be studied by searching for deviations from the conventional spectrum. The residual spectrum is proportional to the coefficient [96].

Anisotropic effects can be studied by constructing asymmetries in experiments that can determine the directionality of some of the decay products. For experiments using unpolarized neutrons, an asymmetry counting electrons emitted in the same direction as the antineutrino compared to events in which the two leptons are emitted in opposite directions can be constructed for the measurement of the electron-antineutrino correlation in the form of Similarly, experiments using polarized neutrons that are able to measure the electron and the recoiling proton can be used to search for electron-proton coincidence events, useful for the measurement of the neutrino asymmetry . As shown in Figure 4, the experimental asymmetry counts events in which proton and electron are emitted against or along ; the direction of the neutron polarization can be written in the form

Effects of Lorentz violation arise from the modified spinor solutions that affect the matrix element of the decay as well as the unconventional form of the neutrino phase space due to the modified dispersion relations, making the asymmetries (18) and (19) have the general form [96] where the amplitudes depend on the coefficients for Lorentz violation , , and and experimental quantities including the orientation and location of the apparatus.

6.3. Double Beta Decay

The same coefficients modifying the spectrum for beta decay can also introduce observable effects in double beta decay experiments. Since the antineutrinos escape unobserved, the simplest test of Lorentz invariance is an alteration of the two-electron spectrum for the two-neutrino mode of double beta decay produced by the coefficient [96, 106]. Similar to the neutron-decay spectrum, the resulting effect is a distortion of the whole spectrum that can be studied by searching for deviations from the conventional spectrum. The energy at which this effect is maximal has been identified for several isotopes, which will guide these types of studies [106].

The neutrinoless mode of double beta decay offers access to other type of coefficient, one that modifies the neutrino propagator. This Majorana coupling in the SME denoted is a combination of other coefficients in the SME that can trigger neutrinoless double beta decay even if the Majorana mass is negligible. In terms of this effective coefficient for CPT-odd Lorentz violation, the half-life of an isotope of radius is given by [106] where is the phase-space factor regarding the two emitted electrons and is the relevant nuclear matrix element. Limits on the Majorana mass parameter can be used to constrain the coefficient . Since the Lorentz-violating neutrinoless double beta decay depends on the nuclear size of the isotope used, a future observation of this decay mode can be distinguished because the Majorana-mass mechanism depends on the isotope only through the nuclear matrix elements.

7. Conclusions

In this paper we have presented a general overview of the effects of deviations from exact Lorentz invariance in neutrinos in the context of the Standard-Model Extension. In general, the signatures of the breakdown of Lorentz symmetry are direction and time dependence of the relevant observables for Earth-based experiments as well as unconventional dependence on the neutrino energy. Neutrino oscillations are sensitive probes of new physics, which makes this type of experiment an ideal setup to search for violations of Lorentz invariance. In oscillations, some effects of Lorentz violation include direction and time dependence of the oscillation probability, oscillation phases that grow with the neutrino energy, CPT violation, and mixing between neutrinos and antineutrinos.

Some effects are unobservable in neutrino oscillations, in which case kinematical effects become a complementary technique. Effects of Lorentz violation appear as modifications to the neutrino velocity as well as unconventional behavior in decay processes. In particular, some decays with neutrinos in the final state can become forbidden above certain threshold energy; similarly, some forbidden processes can become allowed, including erenkov radiation of one or more particles. Most of these effects are enhanced by the neutrino energy, which makes high-energy neutrinos of particular interest for future tests of Lorentz invariance.

Finally, there are operators in the theory whose experimental signatures are independent of the neutrino energy. In this case, the high precision of low-energy experiments can play a fundamental role in the test of Lorentz symmetry for some particular operators that are unobservable in oscillations and that leave the neutrino velocity unchanged. For these countershaded operators, beta decay is the ideal experimental setup. Depending on the properties of the experiment, the main features have been identified for studies of tritium decay, neutron decay, and double beta decay.

To date, there is no compelling evidence of Lorentz violation; nevertheless, only a few of the experimental signatures have been studied [10]. Neutrinos offer great sensitivity and numerous ways to test the validity of the cornerstone of modern physics. Many of the different techniques presented in this review are currently being implemented by a variety of experimental collaborations. Interesting new tests of Lorentz symmetry will be performed in the near future, in which low- and high-energy neutrinos will play a key role in our understanding of the nature of spacetime.

Conflict of Interests

The author declares that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

This work was supported in part by the Department of Energy under Grant DE-FG02-13ER42002, the Indiana University Center for Spacetime Symmetries, and the Helmholtz Alliance for Astroparticle Physics (HAP) under Grant no. HA-301.

References

W. Pauli, Offener Brief an die Gruppe der Radioaktiven bei der Gauvereins-Tagung zu Tübingen, 1930.
V. A. Kostelecký and S. Samuel, “Spontaneous breaking of Lorentz symmetry in string theory,” Physical Review D, vol. 39, article 683, 1989.
View at: Publisher Site | Google Scholar
D. Colladay and V. A. Kostelecký, “CPT violation and the standard model,” Physical Review D, vol. 55, no. 11, pp. 6760–6774, 1997.
View at: Google Scholar
H. B. Nielsen and I. Picek, “The Rédei-like model and testing Lorentz invariance,” Physics Letters B, vol. 114, no. 2-3, pp. 141–146, 1982.
View at: Google Scholar
H. B. Nielsen and I. Picek, “Lorentz non-invariance,” Nuclear Physics B, vol. 211, no. 2, pp. 269–296, 1983.
View at: Publisher Site | Google Scholar
R. Huerta and J. L. Lucio M., “Testing Lorentz invariance in baryon decay,” Physics Letters B, vol. 131, no. 4–6, pp. 471–473, 1983.
View at: Google Scholar
R. Huerta-Quintanilla and J. L. Lucio M., “Constraint on Lorentz non-invariance from the Michel parameter,” Physics Letters B, vol. 124, no. 5, pp. 369–370, 1983.
View at: Google Scholar
D. Colladay and V. A. Kostelecký, “Lorentz-violating extension of the standard model,” Physical Review D, vol. 58, no. 11, Article ID 116002, 1998.
View at: Google Scholar
V. A. Kostelecký, “Gravity, Lorentz violation, and the standard model,” Physical Review D, vol. 69, Article ID 105009, 2004.
View at: Publisher Site | Google Scholar
V. A. Kostelecký and N. Russell, “Data tables for Lorentz and CPT violation,” Reviews of Modern Physics, vol. 83, article 11, 2011.
View at: Publisher Site | Google Scholar
V. A. Kostelecký and S. Samuel, “Phenomenological gravitational constraints on strings and higher-dimensional theories,” Physical Review Letters, vol. 63, article 224, 1989.
View at: Publisher Site | Google Scholar
V. A. Kostelecký and R. Potting, “Gravity from local Lorentz violation,” General Relativity and Gravitation, vol. 37, no. 10, pp. 1675–1679, 2005.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
R. Bluhm and V. A. Kostelecký, “Spontaneous Lorentz violation, Nambu-Goldstone modes, and gravity,” Physical Review D, vol. 71, Article ID 065008, 2005.
View at: Publisher Site | Google Scholar
O. Bertolami and J. Páramos, “Vacuum solutions of a gravity model with vector-induced spontaneous Lorentz symmetry breaking,” Physical Review D, vol. 72, no. 4, Article ID 044001, 2005.
View at: Publisher Site | Google Scholar | MathSciNet
N. Arkani-Hamed, H. C. Cheng, M. Luty, and J. Thaler, “Universal dynamics of spontaneous Lorentz violation and a new spin-dependent inverse-square law force,” Journal of High Energy Physics, vol. 7, article 029, 2005.
View at: Publisher Site | Google Scholar | MathSciNet
V. A. Kostelecký and J. D. Tasson, “Prospects for large relativity violations in matter-gravity couplings,” Physical Review Letters, vol. 102, no. 1, Article ID 010402, 2009.
View at: Publisher Site | Google Scholar | MathSciNet
O. W. Greenberg, “CPT violation implies violation of Lorentz invariance,” Physical Review Letters, vol. 89, no. 23, Article ID 231602, 2002.
View at: Publisher Site | Google Scholar
J. Ellis and N. E. Mavromatos, “Comments on CP, T and CPT violation in neutral kaon decays,” Physics Report, vol. 320, no. 1–6, pp. 341–354, 1999.
View at: Google Scholar
F. R. Klinkhamer, “A CPT anomaly,” Nuclear Physics B, vol. 578, no. 1-2, pp. 277–289, 2000.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
C. Adam and F. R. Klinkhamer, “Causality and CPT violation from an Abelian Chern-Simons-like term,” Nuclear Physics B, vol. 607, no. 1-2, pp. 247–267, 2001.
View at: Publisher Site | Google Scholar
G. Barenboim, L. Borissov, J. Lykken, and A. Y. Smirnov, “Neutrinos as the messengers of CPT violation,” Journal of High Energy Physics, vol. 10, article 001, 2002.
View at: Publisher Site | Google Scholar
G. Barenboim, J. F. Beacom, L. Borissov, and B. Kayser, “CPT violation and the nature of neutrinos,” Physics Letters B, vol. 537, no. 3-4, pp. 227–232, 2002.
View at: Publisher Site | Google Scholar
F. R. Klinkhamer and C. Rupp, “Spacetime foam, CPT anomaly, and photon propagation,” Physical Review D, vol. 70, no. 4, Article ID 045020, 2004.
View at: Publisher Site | Google Scholar | MathSciNet
N. E. Mavromatos, “CPT violation and decoherence in quantum gravity,” in Planck Scale Effects in Astrophysics and Cosmology, vol. 669 of Planck Scale Effects in Astrophysics and Cosmology, pp. 245–320, 2005.
View at: Publisher Site | Google Scholar
M. Chaichian, A. D. Dolgov, V. A. Novikov, and A. Tureanu, “CPT violation does not lead to violation of Lorentz invariance and vice versa,” Physics Letters B, vol. 699, no. 3, pp. 177–180, 2011.
View at: Publisher Site | Google Scholar
P. Adamson, D. J. Auty, MINOS Collaboration et al., “Search for Lorentz invariance and CPT violation with the MINOS far detector,” Physical Review Letters, vol. 105, Article ID 151601, 2010.
View at: Publisher Site | Google Scholar
J. S. Díaz, T. Katori, J. Spitz, and J. M. Conrad, “Search for neutrino–antineutrino oscillations with a reactor experiment,” Physics Letters B, vol. 727, no. 4-5, pp. 412–416, 2013.
View at: Publisher Site | Google Scholar
V. A. Kostelecký and M. Mewes, “Lorentz and CPT violation in neutrinos,” Physical Review D, vol. 69, Article ID 016005, 2004.
View at: Publisher Site | Google Scholar
Z. Maki, M. Nakagawa, and S. Sakata, “Remarks on the unified model of elementary particles,” Progress of Theoretical Physics, vol. 28, no. 5, pp. 870–880, 1962.
View at: Publisher Site | Google Scholar
B. Pontecorvo, “Neutrino experiments and the problem of conservation of leptonic charge,” Journal of Experimental and Theoretical Physics, vol. 53, pp. 1717–1725, 1967.
View at: Google Scholar
B. Pontecorvo, “Neutrino experiments and the question of leptonic-charge conservation,” Soviet Physics—JETP, vol. 26, pp. 984–988, 1968.
View at: Google Scholar
R. Bluhm, V. A. Kostelecký, C. D. Lane, and N. Russell, “Clock-comparison tests of Lorentz and CPT symmetry in space,” Physical Review Letters, vol. 88, Article ID 090801, 2002.
View at: Publisher Site | Google Scholar
V. A. Kostelecký and M. Mewes, “Lorentz violation and short-baseline neutrino experiments,” Physical Review D, vol. 70, Article ID 076002, 2004.
View at: Publisher Site | Google Scholar
J. S. Díaz, V. A. Kostelecký, and M. Mewes, “Perturbative Lorentz and CPT violation for neutrino and antineutrino oscillations,” Physical Review D, vol. 80, Article ID 076007, 2009.
View at: Publisher Site | Google Scholar
V. A. Kostelecký and M. Mewes, “Neutrinos with Lorentz-violating operators of arbitrary dimension,” Physical Review D, vol. 85, Article ID 096005, 2012.
View at: Publisher Site | Google Scholar
V. A. Kostelecký and M. Mewes, “Fermions with Lorentz-violating operators of arbitrary dimension,” Physical Review D, vol. 88, Article ID 096006, 2013.
View at: Publisher Site | Google Scholar
A. Aguilar-Arevalo, L. B. Auerbach, and LSND Collaboration, “Evidence for neutrino oscillations from the observation of ${\bar{υ}}_{e}$ appearance in a ${\bar{υ}}_{μ}$ beam,” Physical Review D, vol. 64, Article ID 112007, 2001.
View at: Publisher Site | Google Scholar
A. A. Aguilar-Arevalo, A. O. Bazarko, MiniBooNE Collaboration et al., “Search for Electron Neutrino Appearance at the ${∆ m}^{2} ~ 1 {e V}^{2}$ Scale,” Physical Review Letters, vol. 98, Article ID 231801, 2007.
View at: Publisher Site | Google Scholar
A. A. Aguilar-Arevalo, C. E. Anderson, MiniBooNE Collaboration et al., “Unexplained excess of electronlike events from a 1-GeV neutrino beam,” Physical Review Letters, vol. 102, Article ID 101802, 2009.
View at: Publisher Site | Google Scholar
V. A. Kostelecký and M. Mewes, “Lorentz and CPT violation in the neutrino sector,” Physical Review D, vol. 70, Article ID 031902, 2004.
View at: Publisher Site | Google Scholar
T. Katori, V. A. Kostelecký, and R. Tayloe, “Global three-parameter model for neutrino oscillations using Lorentz violation,” Physical Review D, vol. 74, Article ID 105009, 2006.
View at: Publisher Site | Google Scholar
V. Barger, D. Marfatia, and K. Whisnant, “Challenging Lorentz noninvariant neutrino oscillations without neutrino masses,” Physics Letters B, vol. 653, no. 2–4, pp. 267–277, 2007.
View at: Publisher Site | Google Scholar
V. Barger, J. Liao, D. Marfatia, and K. Whisnant, “Lorentz noninvariant oscillations of massless neutrinos are excluded,” Physical Review D, vol. 84, no. 5, Article ID 056014, 2011.
View at: Publisher Site | Google Scholar
J. S. Díaz and V. A. Kostelecký, “Three-parameter Lorentz-violating texture for neutrino mixing,” Physics Letters B, vol. 700, no. 1, pp. 25–28, 2011.
View at: Publisher Site | Google Scholar
J. S. Díaz and V. A. Kostelecký, “Lorentz- and CPT-violating models for neutrino oscillations,” Physical Review D, vol. 85, Article ID 016013, 2012.
View at: Publisher Site | Google Scholar
S. J. Rong and Q. Y. Liu, “The perturbed puma model,” Chinese Physics Letters, vol. 29, no. 4, Article ID 041402, 2012.
View at: Publisher Site | Google Scholar
J. Beringer, J. F. Arguin, Particle Data Group et al., “Review of Particle Physics,” Physical Review D, vol. 86, Article ID 010001, 2012.
View at: Google Scholar
T. Akiri and Super-Kamiokande Collaboration, “Sensitivity of atmospheric neutrinos in Super-Kamiokande to Lorentz violation,” High Energy Physics-Phenomenology, http://arxiv.org/abs/1308.2210.
View at: Google Scholar
J. S. Díaz, IUHET-585.
Y. Abe, C. Aberle, Double Chooz Collaboration et al., “First test of Lorentz violation with a reactor-based antineutrino experiment,” Physical Review D, vol. 86, Article ID 112009, 2012.
View at: Publisher Site | Google Scholar
R. Abbasi, Y. Abdou, IceCube Collaboration et al., “Search for a Lorentz-violating sidereal signal with atmospheric neutrinos in IceCube,” Physical Review D, vol. 82, Article ID 112003, 2010.
View at: Publisher Site | Google Scholar
L. B. Auerbach, R. L. Burman, LSND Collaboration et al., “Tests of Lorentz violation in ${\bar{v}}_{μ} \to {\bar{v}}_{e}$ oscillations,” Physical Review D, vol. 72, Article ID 076004, 2005.
View at: Publisher Site | Google Scholar
A. A. Aguilar-Arevalo, C. E. Anderson, MiniBooNE Collaboration et al., “Test of Lorentz and CPT violation with short baseline neutrino oscillation excesses,” Physics Letters B, vol. 718, no. 4–5, pp. 1303–1308, 2013.
View at: Publisher Site | Google Scholar
T. Katori, “Tests of lorentz and cpt violation with MiniBooNE neutrino oscillation excesses,” Modern Physics Letters A, vol. 27, no. 25, Article ID 1230024, 2012.
View at: Publisher Site | Google Scholar
P. Adamson et al., “Testing Lorentz invariance and CPT conservation with NuMI neutrinos in the MINOS near detector,” Physical Review Letters, vol. 101, Article ID 151601, 2008.
View at: Publisher Site | Google Scholar
P. Adamson, D. S. Ayres, The MINOS Collaboration et al., “Search for Lorentz invariance and CPT violation with muon antineutrinos in the MINOS near detector,” Physical Review D, vol. 85, Article ID 031101, 2012.
View at: Publisher Site | Google Scholar
B. Rebel and S. Mufson, “The search for neutrino–antineutrino mixing resulting from Lorentz invariance violation using neutrino interactions in MINOS,” Astroparticle Physics, vol. 48, pp. 78–81, 2013.
View at: Publisher Site | Google Scholar
J. Alspector, G. R. Kalbfleisch, N. Baggett et al., “Experimental comparison of neutrino and muon velocities,” Physical Review Letters, vol. 36, no. 15, pp. 837–840, 1976.
View at: Publisher Site | Google Scholar
G. R. Kalbfleisch, N. Baggett, E. C. Fowler, and J. Alspector, “Experimental comparison of neutrino, antineutrino, and muon velocities,” Physical Review Letters, vol. 43, no. 19, pp. 1361–1364, 1979.
View at: Publisher Site | Google Scholar
P. Adamson, C. Andreopoulos, MINOS Collaboration et al., “Measurement of neutrino velocity with the MINOS detectors and NuMI neutrino beam,” Physical Review D, vol. 76, Article ID 072005, 2007.
View at: Publisher Site | Google Scholar
T. Adam, N. Agafonova, OPERA Collaboration et al., “Measurement of the neutrino velocity with the OPERA detector in the CNGS beam,” Journal of High Energy Physics, vol. 2012, p. 93, 2012.
View at: Publisher Site | Google Scholar
M. Antonello, P. Aprili, ICARUS Collaboration et al., “Measurement of the neutrino velocity with the ICARUS detector at the CNGS beam,” Physics Letters B, vol. 713, no. 1, pp. 17–22, 2012.
View at: Google Scholar
P. Alvarez Sanchez, R. Barzaghi, Borexino Collaboration et al., “Measurement of CNGS muon neutrino speed with Borexino,” Physics Letters B, vol. 716, pp. 401–405, 2012.
View at: Publisher Site | Google Scholar
N. Y. Agafonova, M. Aglietta, LVD Collaboration et al., “Measurement of the velocity of neutrinos from the CNGS beam with the large volume detector,” Physical Review Letters, vol. 109, Article ID 070801, 2012.
View at: Publisher Site | Google Scholar
M. J. Longo, “Tests of relativity from SN1987A,” Physical Review D, vol. 36, no. 10, pp. 3276–3277, 1987.
View at: Publisher Site | Google Scholar
V. A. Kostelecký and M. Mewes, “Lorentz-violating electrodynamics and the cosmic microwave background,” Physical Review Letters, vol. 99, Article ID 011601, 2007.
View at: Publisher Site | Google Scholar
V. A. Kostelecký and M. Mewes, “Astrophysical tests of Lorentz and CPT violation with photons,” The Astrophysical Journal Letters, vol. 689, no. 1, article L1, 2008.
View at: Publisher Site | Google Scholar
V. A. Kostelecký and M. Mewes, “Electrodynamics with Lorentz-violating operators of arbitrary dimension,” Physical Review D, vol. 80, Article ID 015020, 2009.
View at: Publisher Site | Google Scholar
V. Vasileiou, A. Jacholkowska, F. Piron et al., “Constraints on Lorentz invariance violation from Fermi-Large Area Telescope observations of gamma-ray bursts,” Physical Review D, vol. 87, no. 12, Article ID 122001, 2013.
View at: Publisher Site | Google Scholar
B. Altschul, “Consequences of neutrino Lorentz violation for leptonic meson decays,” Physical Review D, vol. 84, Article ID 091902, 2011.
View at: Publisher Site | Google Scholar
A. Chodos, V. A. Kostelecký, R. Potting, and E. Gates, “Null experiments for neutrino masses,” Modern Physics Letters A, vol. 7, no. 6, article 467, 1992.
View at: Publisher Site | Google Scholar
A. Chodos and V. A. Kostelecký, “Nuclear null tests for spacelike neutrinos,” Physics Letters B, vol. 336, no. 3-4, pp. 295–302, 1994.
View at: Publisher Site | Google Scholar
V. A. Kostelecký, “Mass bounds for spacelike neutrinos,” in Topics in Quantum Gravity and Beyond: Essays in Honor of Louis Witten, F. Mansouri and J. J. Scanio, Eds., p. 369, World Scientific, Singapore, 1993.
View at: Google Scholar
L. Gonzalez-Mestres, “Astrophysical consequences of the OPERA superluminal neutrino,” General Physics, http://arxiv.org/abs/1109.6630.
View at: Google Scholar
X. J. Bi, P. F. Yin, Z. H. Yu, and Q. Yuan, “Constraints and tests of the OPERA superluminal neutrinos,” Physical Review Letters, vol. 107, Article ID 241802, 2011.
View at: Publisher Site | Google Scholar
R. Cowsik, S. Nussinov, and U. Sarkar, “Superluminal nutrinos at OPERA confront pion decay kinematics,” Physical Review Letters, vol. 107, no. 25, Article ID 251801, 2011.
View at: Publisher Site | Google Scholar
M. Mannarelli, M. Mitra, F. L. Villante, and F. Vissani, “Non-standard neutrino propagation and pion decay,” Journal of High Energy Physics, vol. 2012, no. 1, article 136, 2012.
View at: Publisher Site | Google Scholar
V. Baccetti, K. Tate, and M. Visser, “Lorentz violating kinematics: threshold theorems,” Journal of High Energy Physics, vol. 2012, article 87, 2012.
View at: Publisher Site | Google Scholar
M. G. Aartsen et al., “First observation of PeV-energy neutrinos with iceCube,” Physical Review Letters, vol. 111, Article ID 021103, 2013.
View at: Publisher Site | Google Scholar
M. G. Aartsen et al., “Evidence for high-energy extraterrestrial neutrinos at the icecube detector,” Science, vol. 342, no. 6161, Article ID 1242856, 2013.
View at: Publisher Site | Google Scholar
J. S. Díaz, V. A. Kostelecký, and M. Mewes, “Testing relativity with high-energy astrophysical neutrinos,” Physical Review D, vol. 89, Article ID 043005, 2014.
View at: Publisher Site | Google Scholar
S. R. Coleman and S. L. Glashow, “High-energy tests of Lorentz invariance,” Physical Review D, vol. 59, Article ID 116008, 1999.
View at: Publisher Site | Google Scholar
D. Mattingly, T. Jacobson, and S. Liberati, “Threshold configurations in the presence of Lorentz violating dispersion relations,” Physical Review D, vol. 67, no. 12, Article ID 124012, 2003.
View at: Publisher Site | Google Scholar
D. M. Mattingly, L. MacCione, M. Galaverni, S. Liberati, and G. Sigl, “Possible cosmogenic neutrino constraints on Planck-scale Lorentz violation,” Journal of Cosmology and Astroparticle Physics, vol. 2010, no. 2, article 007, 2010.
View at: Publisher Site | Google Scholar
J. Alexandre, J. Ellis, and N. E. Mavromatos, “On the possibility of superluminal neutrino propagation,” Physics Letters B, vol. 706, no. 4-5, pp. 456–461, 2012.
View at: Publisher Site | Google Scholar
A. G. Cohen and S. L. Glashow, “Pair creation constrains superluminal neutrino propagation,” Physical Review Letters, vol. 107, no. 18, Article ID 181803, 2011.
View at: Publisher Site | Google Scholar
J. M. Carmona and J. L. Cortes, “Constraints from neutrino decay on superluminal velocities,” High Energy Physics, http://arxiv.org/abs/1110.0430.
View at: Google Scholar
L. Maccione, S. Liberati, and D. M. Mattingly, “Violations of Lorentz invariance in the neutrino sector: an improved analysis of anomalous threshold constraints,” Journal of Cosmology and Astroparticle Physics, vol. 2013, no. 3, article 039, 2013.
View at: Publisher Site | Google Scholar
S. Mohanty and S. Rao, “Constraint on super-luminal neutrinos from vacuum Cerenkov processes,” High Energy Physics, http://arxiv.org/abs/1111.2725.
View at: Google Scholar
M. Li, D. Liu, J. Meng, T. Wang, and L. Zhou, “Replaying neutrino bremsstrahlung with general dispersion relations,” High Energy Physics, http://arxiv.org/abs/1111.3294.
View at: Google Scholar
Y. Huo, T. Li, Y. Liao, D. V. Nanopoulos, and Y. Qi, “Constraints on neutrino velocities revisited,” Physical Review D, vol. 85, no. 3, Article ID 034022, 2012.
View at: Publisher Site | Google Scholar
S. J. Brodsky and S. Gardner, “Pair production constraints on superluminal neutrinos revisited,” High Energy Physics, http://arxiv.org/abs/1112.1090.
View at: Google Scholar
E. Borriello, S. Chakraborty, A. Mirizzi, and P. D. Serpico, “Stringent constraint on neutrino Lorentz invariance violation from the two IceCube PeV neutrinos,” Physical Review D, vol. 87, no. 11, Article ID 116009, 2013.
View at: Publisher Site | Google Scholar
F. W. Stecker, “Constraining superluminal electron and neutrino velocities using the 2010 Crab nebula flare and the iceCube PeV neutrino events,” High Energy Physics, http://arxiv.org/abs/1306.6095.
View at: Google Scholar
V. A. Kostelecký and J. D. Tasson, “Matter-gravity couplings and Lorentz violation,” Physical Review D, vol. 83, Article ID 016013, 2011.
View at: Publisher Site | Google Scholar
J. S. Díaz et al., “Relativity violations and beta decay,” Physical Review D, vol. 88, Article ID 071902, 2013.
View at: Publisher Site | Google Scholar
J. P. Noordmans, H. W. Wilschut, and R. G. E. Timmermans, “Lorentz violation in neutron decay and allowed nuclear beta decay,” Physical Review C, vol. 87, Article ID 055502, 2013.
View at: Google Scholar
B. Altschul, “Contributions to pion decay from Lorentz violation in the weak sector,” Physical Review D, vol. 88, Article ID 076015, 2013.
View at: Publisher Site | Google Scholar
S. E. M. Müller, E. A. Dijck, H. Bekker et al., “First test of Lorentz invariance in the weak decay of polarized nuclei,” Physical Review D, vol. 88, Article ID 071901, 2013.
View at: Publisher Site | Google Scholar
V. N. Aseev, A. I. Belesev, A. I. Berlev et al., “Upper limit on the electron antineutrino mass from the Troitsk experiment,” Physical Review D, vol. 84, Article ID 112003, 2011.
View at: Publisher Site | Google Scholar
C. Kraus, B. Bornschein, L. Bornschein et al., “Final results from phase II of the Mainz neutrino mass searchin tritium β decay,” The European Physical Journal C, vol. 40, no. 4, pp. 447–468, 2005.
View at: Publisher Site | Google Scholar
J. Angrik and KATRIN Collaboration, KATRIN design report, FZKA Scientific Report 7090, 2005.
A. Chodos, A. I. Hauser, and V. A. Kostelecký, “The neutrino as a tachyon,” Physics Letters B, vol. 150, no. 6, pp. 431–435, 1985.
View at: Publisher Site | Google Scholar | MathSciNet
J. M. Carmona and J. L. Cortes, “Testing Lorentz invariance violations in the tritium beta-decay anomaly,” Physics Letters B, vol. 494, no. 1-2, pp. 75–80, 2000.
View at: Publisher Site | Google Scholar
A. E. Bernardini and O. Bertolami, “Lorentz violating extension of the standard model and the β-decay endpoint,” Physical Review D, vol. 77, Article ID 085032, 2008.
View at: Publisher Site | Google Scholar
J. S. Díaz, “Limits on Lorentz and CPT violation from double beta decay,” Physical Review D, vol. 89, Article ID 036002, 2014.
View at: Publisher Site | Google Scholar

Copyright

Copyright © 2014 Jorge S. Díaz. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The publication of this article was funded by SCOAP³.

PDF Download Citation

Download other formats

Order printed copies

Views

1659

Downloads

1068

Citations