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Lorentz-violating neutrino oscillations
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Lorentz-violating neutrino oscillations
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{{Short description|Quantum phenomenon}}Lorentz-violating neutrino oscillation refers to the quantum phenomenon of neutrino oscillations described in a framework that allows the breakdown of Lorentz invariance. Today, neutrino oscillation or change of one type of neutrino into another is an experimentally verified fact; however, the details of the underlying theory responsible for these processes remain an open issue and an active field of study. The conventional model of neutrino oscillations assumes that neutrinos are massive, which provides a successful description of a wide variety of experiments; however, there are a few oscillation signals that cannot be accommodated within this model, which motivates the study of other descriptions. In a theory with Lorentz violation, neutrinos can oscillate with and without masses and many other novel effects described below appear. The generalization of the theory by incorporating Lorentz violation has shown to provide alternative scenarios to explain all the established experimental data through the construction of global models.- the content below is remote from Wikipedia
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Introduction
Conventional Lorentz-preserving descriptions of neutrinos explain the phenomenon of oscillations by endowing these particles with mass. However, if Lorentz violation occurs, oscillations could be due to other mechanisms. The general framework for Lorentz violation is called the Standard-Model Extension (SME).D. Colladay and V.A. Kostelecky, CPT Violation and the Standard Model, Phys. Rev. D 55, 6760 (1997). arXiv:hep-ph/9703464D. Colladay and V.A. Kostelecky, Lorentz-Violating Extension of the Standard Model, Phys. Rev. D 58, 116002 (1998). arXiv:hep-ph/9809521V.A Kostelecky, Gravity, Lorentz Violation, and the Standard Model, Phys. Rev. D 69, 105009 (2004). arXiv:hep-th/0312310 The neutrino sector of the SME provides a description of how Lorentz and CPT violation would affect neutrino propagation, interactions, and oscillations. This neutrino framework first appeared in1997 as part of the general SME for Lorentz violation in particle physics, which is built from the operators of the Standard Model. An isotropic limit of the SME,including a discussion on Lorentz-violating neutrino oscillations, was presented in a 1999 publication.S. Coleman and S.L. Glashow, High-energy tests of Lorentz invariance, Phys. Rev. D 59, 116008 (1999). arXiv:hep-ph/9812418 Full details of the general formalism for Lorentz and CPT symmetry in the neutrino sector appeared in a 2004 publication.V.A. Kostelecky and M. Mewes, Lorentz and CPT violation in neutrinos, Phys. Rev. D 69, 016005 (2004). arxiv=hep-ph/0309025 This work presented the minimal SME (mSME) for the neutrino sector, which involves only renormalizable terms. The incorporation of operators of arbitrary dimension in the neutrino sector was presented in 2011.V.A. Kostelecky and M. Mewes, Neutrinos with Lorentz-Violating Operators of Arbitrary Dimension (2011). arXiv:1112.6395The Lorentz-violating contributions to the Lagrangian are built as observer Lorentz scalars by contracting standard field operators with controlling quantities called coefficients for Lorentz violation. These coefficients, arising from the spontaneous breaking of Lorentz symmetry, lead to non-standard effects that could be observed in current experiments. Tests of Lorentz symmetry attempt to measure these coefficients. A nonzero result would indicate Lorentz violation.The construction of the neutrino sector of the SME includes the Lorentz-invariant terms of the standard neutrino massive model, Lorentz-violating terms that are even under CPT, and ones that are odd under CPT.Since in field theory the breaking of CPT symmetry is accompanied by the breaking of Lorentz symmetry,O.W. Greenberg, CPT Violation Implies Violation of Lorentz Invariance, Phys. Rev. Lett. 89, 231602 (2002). arXiv:hep-ph/0201258 the CPT-breaking terms are necessarily Lorentz breaking. It is reasonable to expect that Lorentz and CPT violation are suppressed at the Planck scale, so the coefficients for Lorentz violation are likely to be small. The interferometric nature of neutrino oscillation experiments, and also of neutral-meson systems, gives them exceptional sensitivity to such tiny effects. This holds promise for oscillation-based experiments to probe new physics and access regions of the SME coefficient space that are still untested.General predictions
Current experimental results indicate that neutrinos do indeed oscillate. These oscillations have a variety of possible implications, including the existence of neutrino masses, and the presence of several types of Lorentz violation. In the following, each category of Lorentz breaking is outlined.Spectral anomalies
In the standard Lorentz-invariant description of massive-neutrinos, the oscillation phase is proportional to the baseline L and inversely proportional to the neutrino energy E. The mSME introduces dimension-three operators that lead to oscillation phases with no energy dependence. It also introduces dimension-four operators generating oscillation phases proportional to the energy. Standard oscillation amplitudes are controlled by three mixing anglesand one phase, all of which are constant. In the SME framework, Lorentz violation can lead to energy-dependent mixing parameters.When the whole SME is considered and nonrenormalizable terms in the theory are not neglected, the energy dependence of the effective hamiltonian takes the form of an infinite series in powers of neutrino energy. The fast growth of elements in the hamiltonian could produce oscillation signals in short-baseline experiment, as in the puma model.The unconventional energy dependence in the theory leads to other novel effects, including corrections to the dispersion relations that would make neutrinos move at velocities other than the speed of light. By this mechanism neutrinos could become faster-than-light particles. The most general form of the neutrino sector of the SME has been constructed by including operators of arbitrary dimension. In this formalism, the speed of propagation of neutrinos is obtained. Some of the interesting new features introduced by the violation of Lorentz invariance include dependence of this velocity on neutrino energy and direction of propagation. Moreover, different neutrino flavors could also have different speeds.L â E conflicts
The L â E conflicts refer to null or positive oscillation signals for values of L and E that are not consistent with the Lorentz-invariant explanation. For example, KamLAND and SNO observationsJOURNAL, KamLAND Collaboration
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, require a mass-squared difference Delta m^2_odotsimeq8times10^{-5},mbox{eV}^2 to be consistent with the Lorentz-invariant phase proportional to L/E. Similarly, Super-Kamiokande, K2K, and MINOS observations
JOURNAL
, SNO Collaboration
, Electron energy spectra, fluxes, and day-night asymmetries of 8B solar neutrinos from measurements with NaCl dissolved in the heavy-water detector at the Sudbury Neutrino Observatory
, Physical Review C
, 72, 5, 055502
, nucl-ex/0502021
, 2005PhRvC..72e5502A
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, Ahmed
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, A.
, Beier
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, Bellerive
, A.
, Bergevin
, M.
, Biller
, S.
, Boger
, J.
, Boulay
, M., Bowler
, M.
, Bullard
, T.
, Chan
, Y.
, Chen
, M.
, Chen
, X.
, Cleveland
, B.
, Cox
, G.
, Currat
, C.
, Dai
, X.
, Dalnoki-Veress
, F.
, Deng
, H.
, Doe
, P.
, Dosanjh
, R.
, Doucas
, G.
, Duba
, C.
, Duncan
, F.
, Dunford
, M.
, Dunmore
, J.
, Earle
, E.
, Elliott
, S.
, Evans
, H.
, 119350768
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, require a mass-squared difference Delta m^2_odotsimeq8times10^{-5},mbox{eV}^2 to be consistent with the Lorentz-invariant phase proportional to L/E. Similarly, Super-Kamiokande, K2K, and MINOS observations
, Super-Kamiokande Collaboration
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, Measurement of atmospheric neutrino oscillation parameters by Super-Kamiokande I
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, of atmospheric-neutrino oscillations require a mass-squared difference Delta m^2_text{atm}simeq2.5times10^{-3},mbox{eV}^2. Any neutrino-oscillation experiment must be consistent with either of
these two mass-squared differences for Lorentz invariance to hold. To date, this is the only class of signal for which there is positive evidence. The LSND experiment observedJOURNAL
, 2006
, Observation of Muon Neutrino Disappearance with the MINOS Detectors in the NuMI Neutrino Beam
, Physical Review Letters
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, of atmospheric-neutrino oscillations require a mass-squared difference Delta m^2_text{atm}simeq2.5times10^{-3},mbox{eV}^2. Any neutrino-oscillation experiment must be consistent with either of
, LSND Collaboration
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, Evidence for neutrino oscillations from the observation of {{Subatomic particle, Electron antineutrino, appearance in a {{Subatomic particle|Muon antineutrino}} beam
|journal=Physical Review D
|volume=64 |issue= 11|pages=112007
|arxiv=hep-ex/0104049
|bibcode = 2001PhRvD..64k2007A
|doi = 10.1103/PhysRevD.64.112007
|last2=Auerbach
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|display-authors=8
}} oscillations leading to a mass-squared difference that is inconsistent with results from solar- and atmospheric-neutrino observations. The oscillation phase requires Delta m^2_text{LSND}simeq 1,mbox{eV}^2. This anomaly can be understood in the presence of Lorentz violation.
, 2001
, Evidence for neutrino oscillations from the observation of {{Subatomic particle, Electron antineutrino, appearance in a {{Subatomic particle|Muon antineutrino}} beam
|journal=Physical Review D
|volume=64 |issue= 11|pages=112007
|arxiv=hep-ex/0104049
|bibcode = 2001PhRvD..64k2007A
|doi = 10.1103/PhysRevD.64.112007
|last2=Auerbach
|first2=L.
|last3=Burman
|first3=R.
|last4=Caldwell
|first4=D.
|last5=Church
|first5=E.
|last6=Cochran
|first6=A.
|last7=Donahue
|first7=J.
|last8=Fazely
|first8=A.
|last9=Garvey
|first9=G.|last10=Gunasingha
|first10=R.
|last11=Imlay
|first11=R.
|last12=Louis
|first12=W.
|last13=Majkic
|first13=R.
|last14=Malik
|first14=A.
|last15=Metcalf
|first15=W.
|last16=Mills
|first16=G.
|last17=Sandberg
|first17=V.
|last18=Smith
|first18=D.
|last19=Stancu
|first19=I.
|last20=Sung
|first20=M.
|last21=Tayloe
|first21=R.
|last22=Vandalen
|first22=G.
|last23=Vernon
|first23=W.
|last24=Wadia
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|last25=White
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|s2cid=118686517
|display-authors=8
}} oscillations leading to a mass-squared difference that is inconsistent with results from solar- and atmospheric-neutrino observations. The oscillation phase requires Delta m^2_text{LSND}simeq 1,mbox{eV}^2. This anomaly can be understood in the presence of Lorentz violation.
Periodic variations
Laboratory experiments follow complicated trajectories as the Earth rotates on its axis and revolves around the Sun. Since the fixed SME background fields are coupled with the particle fields, periodic variations associated with these motions would be one of the signatures of Lorentz violation.There are two categories of periodic variations:- Sidereal variations: As the Earth rotates, the source and detector for any neutrino experiment will rotate along with it at a sidereal frequency of omega_oplussim2pi/23,mbox{h}, 56 ,mbox{min}. Since the 3-momentum of the neutrino beam is coupled to the SME background fields, this can lead to sidereal variations in the observed oscillation probability data. Sidereal variations are among the most commonly sought signals in Lorentz tests in other sectors of the SME.
- Annual variations: Variations with a period of one year can arise due to the motion of the Earth around the Sun. The mechanism is the same as for sidereal variations, arising because the particle fields couple to the fixed SME background fields. These effects, however, are challenging to resolve because they require the experiment to provide data for a comparable length of time. There are also boost effects that arise because the earth moves around the Sun at more than 30 kilometers per second. However, this is one ten thousandth of the speed of light, and means the boost effects are suppressed by four orders of magnitude relative to purely rotational effects.
Compass asymmetries
The breaking of rotation invariance can also lead to time-independent signals arising in the form of directional asymmetries at the location of the detector. This type of signal can cause differences in observed neutrino properties for neutrinos originating from different directions.Neutrino-antineutrino mixing
Some of the mSME coefficients lead to mixing between neutrinos and antineutrinos. These processes violate lepton-number conservation, but can readily be accommodated in the Lorentz-breaking SME framework. The breaking of invariance under rotations leads to the non-conservation of angular momentum, which allows a spin flip of the propagating neutrino that can oscillate into an antineutrino. Because of the loss of rotational symmetry, coefficients responsible for this type of mixing always introduce direction dependence.Classic CPT tests
Since CPT violation implies Lorentz violation, traditional tests of CPT symmetry can also be used to search for deviations from Lorentz invariance. This test seeks evidence of P_{nu_arightarrownu_b}neq P_{barnu_brightarrowbarnu_a}. Some subtle features arise. For example, although CPT invariance implies P_{nu_arightarrownu_b}=P_{barnu_brightarrowbarnu_a}, this relation can be satisfied even in the presence of CPT violation.Global models of neutrino oscillations with Lorentz violation
Global models are descriptions of neutrino oscillations that are consistent with all the established experimental data: solar, reactor, accelerator, and atmospheric neutrinos. The general SME theory of Lorentz-violating neutrinos has shown to be very successful as an alternative description of all observed neutrino data. These global models are based on the SME and exhibit some of the key signals of Lorentz violation described in the previous section.Bicycle model
The first phenomenological model using Lorentz-violating neutrinos was proposed by Kostelecky and Mewes in a 2004 paper.V.A. Kostelecky and M. Mewes, Lorentz and CPT violation in the neutrino sector, Phys. Rev. D 70, 031902 (2004).arXiv:hep-ph/0308300 This so-called bicycle model exhibits direction dependence and only two parameters (two non-zero SME coefficients), instead of the six of the conventional massive model. One of the main characteristics of this model is that neutrinos are assumed to be massless. This simple model is compatible with solar, atmospheric, and long-baseline neutrino oscillation data. A novel feature of the bicycle model occurs at high energies, where the two SME coefficients combine to create a direction-dependent pseudomass. This leads to maximal mixing and an oscillation phase proportional to L/E, as in the massive case.Generalized bicycle model
The bicycle model is an example of a very simple and realistic model that can accommodate most of the observed data using massless neutrinos in the presence of Lorentz violation. In 2007, Barger, Marfatia, and Whisnant constructed a more general version of this model by including more parameters.V. Barger, D. Marfatia, and K. Whisnant, Challenging Lorentz noninvariant neutrino oscillations without neutrino masses, Phys. Lett. B 653, 267 (2007) arXiv:0706.1085 In this paper, it is shown that a combined analysis of solar, reactor, and long-baseline experiments excluded the bicycle model and its generalization. Despite this, the bicycle served as starting point for more elaborate models.Tandem model
The tandem modelT. Katori, V.A. Kostelecky, and R. Tayloe Global three-parameter model for neutrino oscillations using Lorentz violation, Phys. Rev. D 74, 105009 (2006). arXiv:hep-ph/0606154 is an extended version of the bicycle presented in 2006 by Katori, Kostelecky, and Tayloe. It is a hybrid model that includes Lorentz violation and also mass terms for a subset of neutrino flavors. It attempts to construct a realistic model by applying a number of desirable criteria. In particular, acceptable models for neutrino violation should:- be based on quantum field theory,
- involve only renormalizable terms,
- offer an acceptable description of the basic features of neutrino-oscillation data,
- have a mass scale lesssim0.1,text{eV} for seesaw compatibility,
- involve fewer parameters than the four used in the standard picture,
- have coefficients for Lorentz violation consistent with a Planck-scale suppression lesssim10^{-17}, and
- accommodate the LSND signal.
Puma model
The puma model was proposed by Diaz and Kostelecky in 2010 as a three-parameter modelJ.S. Diaz and V.A. Kostelecky, Three-parameter Lorentz-violating texture for neutrino mixing, Phys. Lett. B 700, 25 (2011). arXiv:1012.5985.J.S. Diaz and V.A. Kostelecky, Lorentz- and CPT-violating models for neutrino oscillations, arXiv:1108.1799. that exhibits consistency with all the established neutrino data (accelerator, atmospheric, reactor, and solar) and naturally describes the anomalous low-energy excess observed in MiniBooNE that is inconsistent with the conventional massive model. This is a hybrid model that includes Lorentz violation and neutrino masses. One of the main differences between this model and the bicycle and tandem models described above is the incorporation of nonrenormalizable terms in the theory, which lead to powers of the energy greater than one. Nonetheless, all these models share the characteristic of having a mixed energy dependence that leads to energy-dependent mixing angles, a feature absent in the conventional massive model. At low energies, the mass term dominates and the mixing takes the tribimaximal form, a widely used matrix postulated to describe neutrino mixing. This mixing added to the 1/E dependence of the mass term guarantees agreement with solar and KamLAND data. At high energies, Lorentz-violating contributions take over making the contribution of neutrino masses negligible. A seesaw mechanism is triggered, similar to that in the bicycle model, making one of the eigenvalues proportional to 1/E, which usually come with neutrino masses. This feature lets the model mimic the effects of a mass term at high energies despite the fact that there are only non-negative powers of the energy. The energy dependence of the Lorentz-violating terms produce maximal nu_muleftrightarrownu_tau mixing, which makes the model consistent with atmospheric and accelerator data. The oscillation signal in MiniBooNE appears because the oscillation phase responsible for the oscillation channel nu_murightarrownu_e grows rapidly with energy and the oscillation amplitude is large only for energies below 500 MeV. The combination of these two effects produces an oscillation signal in MiniBooNE at low energies, in agreement with the data. Additionally, since the model includes a term associated to a CPT-odd Lorentz-violating operator, different probabilities appear for neutrinos and antineutrinos. Moreover, since the amplitude for nu_murightarrownu_e decreases for energies above 500 MeV, long-baseline experiments searching for nonzero theta_{13} should measure different values depending on the energy; more precisely, the MINOS experiment should measure a value smaller than the T2K experiment according to the puma model, which agrees with current measurements.T2K Collaboration (K. Abe et al.), Indication of Electron Neutrino Appearance from an Accelerator-produced Off-axis Muon Neutrino Beam, Phys. Rev. Lett. 107, 041801 (2011). arXiv:1106.2822,MINOS Collaboration (P. Adamson et al.), Improved search for muon-neutrino to electron-neutrino oscillations in MINOS, arXiv:1108.0015.Isotropic bicycle model
In 2011, Barger, Liao, Marfatia, and Whisnant studied general bicycle-type models (without neutrino masses) that can be constructed using the minimal SME that are isotropic (direction independent).V. Barger, J. Liao, D. Marfatia, and K. Whisnant, Lorentz noninvariant oscillations of massless neutrinos are excluded, arXiv:1106.6023. Results show that long-baseline accelerator and atmospheric data can be described by these models in virtue of the Lorentz-violating seesaw mechanism; nevertheless, there is a tension between solar and KamLAND data. Given this incompatibility, the authors concluded that renormalizable models with massless neutrinos are excluded by the data.Mathematical theory
From a general model-independent point of view, neutrinos oscillate because the effective hamiltonian describing their propagation is not diagonal in flavor space and has a non-degenerate spectrum, in other words, the eigenstates of the hamiltonian are linear superpositions of the flavor eigenstates of the weak interaction and there are at least two different eigenvalues. If we find a transformation U_{a'a} that puts the effective hamiltonian in flavor basis (heff)ab in the diagonal form
E_{a'b'}=mathrm{diag}(lambda_1,lambda_2,lambda_3)
(where the indices a, b = e, μ, Ï and a′, b′ =1, 2, 3 denote the flavor and diagonal basis, respectively), then we can write the oscillation probability from a flavor state |nu_brangle to |nu_arangle as
P_{nu_brightarrownu_a}=left|leftlangle nu_a|nu_b(L)rightrangle right|^{2}=left|sum_{a'}U_{a'a}^{*}U_{a'b}, e^{ -i lambda_{a'} L }right|^{2},
where lambda_{a'}frac{}{} are the eigenvalues. For the conventional massive model lambda_{a'}=m^2_{a'}/2E.In the SME formalism, the neutrino sector is described by a 6-component vector with three active left-handed neutrinos and three right-handed antineutrinos. The effective Lorentz-violating Hamiltonian is a 6 Ã 6 matrix that takes the explicit form
-isqrt2p_alpha(epsilon_+)_beta[(g^{alphabetagamma}p_gamma-H^{alphabeta})]_{abar b}\
isqrt2p_alpha(epsilon_+)_beta^*[(g^{alphabetagamma}p_gamma-H^{alphabeta})]_{bar ab}^*&
[(a_R)^alpha p_alpha-(c_R)^{alphabeta} p_alpha p_beta]_{bar abar b}end{pmatrix} .
end{align}The indices of this effective Hamiltonian take the six values A, B = e, μ, Ï, {{overline|e}}, {{overline|μ}}, {{overline|Ï}}, for neutrinos and antineutrinos. The lowercase indices indicate neutrinos (a, b = e, μ, Ï), and the barred lowercase indices indicate antineutrinos ({{overline|a}}, {{overline|b}} = {{overline|e}}, {{overline|μ}}, {{overline|Ï}}). Notice that the ultrarelativistic approximation Esimeq|vec p| has been used.The first term is diagonal and can be removed because it does not contribute to oscillations; however, it can play an important role in the stability of the theory.V.A. Kostelecky and R. Lehnert, "Stability, Causality, and Lorentz and CPT Violation", Phys. Rev. D 63, 065008 (2001). arXiv:hep-th/0012060 The second term is the standard massive-neutrino Hamiltonian. The third term is the Lorentz-violating contribution. It involves four types of coefficients for Lorentz violation. The coefficients (a_L)^alpha_{ab} and (c_L)^{alphabeta}_{ab} are of dimension one and zero, respectively. These coefficients are responsible for the mixing of left-handed neutrinos, leading to Lorentz-violating neutrinoâneutrino oscillations. Similarly, the coefficients (a_R)^alpha_{bar abar b} and (c_R)^{alphabeta}_{bar abar b} mix right-handed antineutrinos, leading to Lorentz-violating antineutrinoâantineutrino oscillations. Notice that these coefficients are 3 à 3 matrices having both spacetime (Greek) and flavor indices (Roman). The off-diagonal block involves the dimension-zero coefficients, g^{alphabetagamma}_{abar b}, and the dimension-one coefficients, H^{alphabeta}_{abar b}. These lead to neutrinoâantineutrino oscillations. All spacetime indices are properly contracted forming observer Lorentz scalars. The four-momentum shows explicitly that the direction of propagation couples to the mSME coefficients, generating the periodic variations and compass asymmetries described in the previous section. Finally, note that coefficients with an odd number of spacetime indices are contracted with operators that break CPT. It follows that the a- and g-type coefficients are CPT-odd. By similar reasoning, the c- and H-type coefficients are CPT-even.isqrt2p_alpha(epsilon_+)_beta^*[(g^{alphabetagamma}p_gamma-H^{alphabeta})]_{bar ab}^*&
[(a_R)^alpha p_alpha-(c_R)^{alphabeta} p_alpha p_beta]_{bar abar b}end{pmatrix} .
Applying the theory to experiments
Negligible-mass description
For most short baseline neutrino experiments, the ratio of experimental baseline to neutrino energy, L/E, is small, and neutrino masses can be neglected because they are not responsible for oscillations. In these cases, the possibility exists of attributing observed oscillations to Lorentz violation, even if the neutrinos are massive. This limit of the theory is sometimes called the short-baseline approximation. Caution is necessary in this point, because, in short-baseline experiments, masses can become relevant if the energies are sufficiently low.An analysis of this limit, presenting experimentally accessible coefficients for Lorentz violation, first appeared in a 2004 publication.V.A. Kostelecky and M. Mewes, "Lorentz violation and short-baseline neutrino experiments", Phys. Rev. D 70, 076002 (2004). arXiv:hep-ph/0406255 Neglecting neutrino masses, the neutrino Hamiltonian becomesPerturbative Lorentz-violating description
For experiments where L/E is not small, neutrino masses dominate the oscillation effects. In these cases, Lorentz violation can be introduced as a perturbative effect in the form, V.A., Kostelecky, N., Russell
, Data Tables for Lorentz and CPT Violation
, 2010
, Reviews of Modern Physics
, 83, 1, 11â31
, 0801.0287
, 2011RvMP...83...11K
, 10.1103/RevModPhys.83.11, 3236027,
, Data Tables for Lorentz and CPT Violation
, 2010
, Reviews of Modern Physics
, 83, 1, 11â31
, 0801.0287
, 2011RvMP...83...11K
, 10.1103/RevModPhys.83.11, 3236027,
See also
- Standard-Model Extension
- Lorentz-violating electrodynamics
- Antimatter Tests of Lorentz Violation
- Bumblebee models
- Neutrino oscillation
- Tests of special relativity
- Test theories of special relativity
External links
References
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