review article neutrinos as probes of lorentz...

Review ArticleNeutrinos as Probes of Lorentz Invariance

Jorge S Diacuteaz12

1 Institute for Theoretical Physics Karlsruhe Institute of Technology 76128 Karlsruhe Germany2 Physics Department Indiana University Bloomington IN 47405 USA

Correspondence should be addressed to Jorge S Dıaz jorgediazkitedu

Received 7 February 2014 Accepted 14 April 2014 Published 16 June 2014

Academic Editor Michael A Schmidt

Copyright copy 2014 Jorge S Dıaz This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited Thepublication of this article was funded by SCOAP3

Neutrinos can be used to search for deviations from exact Lorentz invariance The worldwide experimental program in neutrinophysics makes these particles a remarkable tool to search for a variety of signals that could reveal minute relativity violations Thispaper 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 tosave one of the sacred principles of physics the conservationof energy which appeared to be violated in beta decays [1]Today neutrinos remain as some of the less understoodparticles of the standard model (SM) and their mysteriesare still fascinating Their ghostly nature makes them barelyinteract withmatter and their interferometric behaviormakesthemoscillate between different flavors Neutrino oscillationshave led to the remarkable conclusion of massive neutrinospresenting an established evidence of physics beyond the SM

In the search for new physics different candidate theoriesfor quantum gravity involve mechanisms that could triggerthe breakdown of one of the most fundamental symmetriesin modern physics Lorentz invariance In the theoreticalfront Lorentz-violating descriptions of neutrino behaviorhave shown that these fundamental particles can serve aspowerful probes of new physics Experimentally neutrinooscillations have been used to perform several searchesfor Lorentz violation The development of techniques toperform systematic searches for Lorentz violation in manyother experimental setups shows a rich phenomenology tobe studied with a large variety of experimental effects thatremain unexplored

This paper summarizes the main experimental signaturesof deviations from exact Lorentz invariance in the neu-trino sector The experimental searches for Lorentz violation

performed in recent years are presented and future testsof Lorentz symmetry are discussed ranging from precisionmeasurements of beta decay and double beta decay at lowenergies to the high energy of astrophysical neutrinos andoscillation experiments using accelerator atmospheric andreactor neutrinos

2 Lorentz Invariance Violation

Deviations from exact Lorentz symmetry have been shown tobe possible at very high energies in candidate descriptions ofgravity at the quantum level For instancemechanisms for thespontaneous breaking of this fundamental symmetry havebeen identified in string-theory scenarios [2] Interactionscould generate Lorentz-violating terms if a tensor fieldacquires a nonzero vacuum expectation value ⟨119879

120572120573120574sdotsdotsdot⟩ =

119905120572120573120574sdotsdotsdot

= 0 which acts as a background field In the samefashion as the nonzero vacuum expectation value of thedynamical Higgs field generates mass terms for other fieldsvia interactions background tensor fields that couple toconventional particles in the SMwill generate new terms thatbreak Lorentz invarianceThese new terms are Lorentz scalarsunder coordinate transformations in fact the spacetimeindices of the background field are all contracted with theindices of the SM operator in the form 119905

120572120573120574sdotsdotsdotO120572120573120574sdotsdotsdot This

structure guarantees that there is no privileged referenceframe because the theory is observer invariant For instanceconsider the case of Lorentz violation generated by a 2-tensor

Hindawi Publishing CorporationAdvances in High Energy PhysicsVolume 2014 Article ID 962410 11 pageshttpdxdoiorg1011552014962410

2 Advances in High Energy Physics

119905120572120573 whichwill be coupled to some SMoperator with the same

number of spacetime indices in the formL = 119905120572120573O120572120573 Under

a coordinate transformation both the operator and the tensorbackground field transform to a new set of coordinates asfollows

O120572120573

997888rarr O12057210158401205731015840

= Λ1205721015840

120572Λ1205731015840

120573O120572120573

119905120572120573997888rarr 11990512057210158401205731015840 = (Λ

minus1

)120582

1205721015840(Λminus1

)120588

1205731015840119905120582120588

(1)

so that the terms in the LagrangianL = 119905sdotO remain invariantL1015840 = 119905

1015840

sdotO1015840 = 119905sdotO = LThe same construction can be appliedfor a general tensor

On the other hand when a Lorentz transformationis performed over the physical system this is when theexperimental apparatus is rotated or boosted rather thanthe coordinates used to describe it then the SM operatortransforms as shown in (1) but any background field remainsunchanged L1015840 = 119905 sdot O1015840 =L This so-called particle Lorentztransformation changes the coupling between the back-ground fields and the SM operators resulting in a physicallyobservable anisotropy of spacetime this is a violation ofLorentz invariance [3]

Phenomenological approaches to parameterize and ex-perimentally search for particular types of Lorentz violationhave been considered since several decades [4ndash7] Howevereffective field theory can be used to incorporate genericoperators that break Lorentz invariance for all the particlesin the SMThis general framework is known as the standard-model extension (SME) [3 8 9] whose action includesgeneral coordinate-invariant terms by contracting operatorsof conventional fieldswith controlling coefficients for Lorentzviolation and reduces to the SM if all these coefficients vanishGravity can also be incorporated by writing the SM in acurved background [9] The development of the SME has ledto a worldwide experimental program searching for viola-tions of Lorentz invariance whose results are summarized in[10]

Flat spacetime is considered for experiments in particlephysics in which case the coefficients for Lorentz violationthat act as background fields can be chosen to be constantand uniform which guarantees conservation of energy andlinear momentum In this limit these coefficients representthe vacuum expectation value of the tensor fields of theunderlying theory Excitations of these fields lead to arich phenomenology for instance Nambu-Goldstonemodescould play fundamental roles when gravity is included suchas the graviton the photon in Einstein-Maxwell theory[11ndash14] and spin-dependent [15] and spin-independent [16]forces

It should be noted that a subset of operators in the SMEalso break CPT symmetry In fact all the Lorentz-violatingterms in the action involving operators with an odd numberof spacetime indices are odd under a CPT transformationIn realistic field theories CPT violation always appears withLorentz violation [17] Nonetheless alternative approachesexist in which CPT violation is implemented with andwithout Lorentz invariance [18ndash25]

3 Neutrinos

The general description of three left-handed neutrinos andthree right-handed antineutrinos in the presence of Lorentz-violating background fields is given by a 6 times 6 effectiveHamiltonian of form [28]

119867 = (

(ℎ0)119886119887

0

0 (ℎ0)lowast

119886119887

) + (

120575ℎ119886119887

120575ℎ119886119887

120575ℎlowast

119886119887

120575ℎ119886119887

) (2)

where the indices indicate the flavors of active neutrinos119886 119887 = 119890 120583 120591 and antineutrinos 119886 119887 = 119890 120583 120591 The Lorentz-preserving component is explicitly given by the following

(ℎ0)119886119887= |p| 120575

119886119887+1198982

119886119887

2 |p| (3)

where at leading order the neutrino momentum is given bythe energy |p| asymp 119864 and themass-squaredmatrix is commonlywritten in terms of the Pontecorvo-Maki-Nakagawa-Sakata(PMNS) matrix [29ndash31] as 1198982 = 119880PMNS(119898

2

119863)119880dagger

PMNS with1198982

119863= diag(1198982

1 1198982

2 1198982

3)

The Lorentz-violating block describing neutrinos in theHamiltonian (2) is given by the following

120575ℎ119886119887= (119886119871)120572

119886119887119901120572+ (119888119871)120572120573

119886119887119901120572119901120573|p| (4)

The components of the 3 times 3 complex matrices (119886119871)120572

119886119887and

(119888119871)120572120573

119886119887are called coefficients for CPT-odd and CPT-even

Lorentz violation respectively The spacetime indices 120572 120573encode the nature of the broken symmetry for instanceisotropic (direction-independent) Lorentz violation appearswhen only the time components of the coefficients arenonzero while space anisotropy appears when any ofthe other components is nonzero generating direction-dependent effects in the neutrino behavior The breakdownof invariance under rotations is evident due to the presenceof the four-vector 119901120572 = (1 119901) that depends on the neutrinodirection of propagation 119901

The block Hamiltonian describing right-handed antineu-trinos is obtained as the CP conjugate of the neutrinoHamiltonian 120575ℎ

119886119887= CP(120575ℎ

119886119887) which has the same form

as the neutrino Hamiltonian (4) with (119886119877)120572

119886119887

= minus(119886119871)120572lowast

119886119887and

(119888119877)120572120573

119886119887

= (119888119871)120572120573lowast

119886119887 Given the structure of these coefficients

in flavor space it is expected that they will affect neutrinomixing and oscillations Notice however that there existcoefficients that modify the three flavors in the same wayproducing no effects on neutrino oscillations because they areproportional to the identity in flavor space These oscillation-free coefficients and their observable effects are discussed inSections 5 and 6

Signals of the breakdown of Lorentz invariance cor-respond to the anisotropy of spacetime due to preferreddirections set by the coefficients for Lorentz violation that actas fixed background fields Taking advantage of the couplingof these background fields with the neutrino direction ofpropagation 119901 we can search for violations of Lorentzinvariance by making measurements with neutrino beams

Advances in High Energy Physics 3

with different orientations which would reveal the presenceof the SME coefficients (119886

119871)120572

119886119887and (119888

119871)120572120573

119886119887 For Earth-based

experiments detectors and source rotate with a well-definedangular frequency 120596

oplus≃ 2120587(23 h 56 min) due to Earthrsquos

rotation which makes the neutrino direction vary withrespect to the fixed background fields This time dependencewill explicitly appear in the relevant observable quantitiesand it can be parameterized as harmonics of the siderealangle 120596

oplus119879oplus Due to the invariance of the theory under

coordinate transformations there is no preferred referenceframe to make the measurements In order to establish aconsistent and systematic search for Lorentz-violating effectsexperimental results are conventionally reported in the Sun-centered equatorial frame described in [10 32] In this framethe sidereal variation of the coupling between the neutrinodirection 119901 and the background fields that break Lorentzsymmetry can be explicitly written in the form

120575ℎ119886119887= (C)

119886119887+ (A119904)119886119887sin120596oplus119879oplus

+ (A119888)119886119887cos120596oplus119879oplus

+ (B119904)119886119887sin 2120596

oplus119879oplus+ (B119888)119886119887cos 2120596

oplus119879oplus

(5)

where the amplitude of each sidereal harmonic is a functionof the coefficients for Lorentz violation and experimentalparameters including neutrino energy location of the exper-iment and relative orientation between source and detector[33 34]

The coefficients (119886119871)120572

119886119887and (119888

119871)120572120573

119886119887arise from operators of

dimension three and four respectively Operators of arbitrarydimension 119889 can be incorporated in the theory in which casethe coefficients for Lorentz violation in the Hamiltonian (4)appear as momentum-dependent quantities of the form [3536]

(119886119871)120572

119886119887997888rarr (119886

119871)1205721205821sdotsdotsdot120582119889minus3

1198861198871199011205821

sdot sdot sdot 119901120582119889minus3

119889 odd

(119888119871)120572120573

119886119887997888rarr (119888


1198861198871199011205821


119889 even(6)

The extra derivatives in the Lagrangian appear in the neu-trino Hamiltonian as higher powers of the neutrino energyAlthough the conventional massive-neutrino description ofoscillations accommodates all the established experimentalresults nonnegative powers of the neutrino energy couldhelp to elegantly solve some anomalous results obtained inrecent years in beam experiments [37ndash39] In fact interestingattempts to describe the global data using the SME haveled to the construction of alternative models for neutrinooscillations that can accommodate the results reported bydifferent experiments [40ndash46]

In the following sections we discuss the observablesignatures of these coefficients for Lorentz violation in dif-ferent types of neutrino experiments including oscillationsneutrino velocity and beta decays

4 Neutrino Oscillations

The interferometric nature of neutrino oscillations has beenwidely identified as an ideal experimental setup to search for

new physics in the form of deviations from the conventionaldescription of neutrinosThemixing and oscillation betweenneutrino flavors occur in general due to off-diagonal entriesin the neutrino Hamiltonian leading to eigenstates withdifferent energy

41 Oscillation of Neutrinos and Antineutrinos Neutrino os-cillations are well described by a model of three massive neu-trinos which depends on twomass-squared differencesΔ1198982

21

andΔ119898231controlling the oscillation lengths and threemixing

angles 12057912 12057913 and 120579

23that govern the amplitude of the

oscillation [47] According to this massive-neutrino modelthe oscillation probabilities are proportional to the factorsin2(Δ1198982

1198941198951198714119864) Tests of Lorentz invariance using neutrino

oscillations can be performed by searching for deviationsfrom the conventional behavior For some experiments study-ing neutrinos over a large range of energies and baselinessuch as Super-Kamiokande [48] an exact treatment of thediagonalization of the Hamiltonian is necessary [49] In mostcases the experimental features allow the implementationof approximation methods for determining the oscillationprobabilities as discussed in the following sections

411 Short-Baseline Approximation According to the mas-sive-neutrino model for experiments using neutrinos ofenergy 119864 and baseline 119871 that satisfy Δ1198982

1198941198951198714119864 ≪ 1205872 the

oscillation phase would be too small to impact the neutrinopropagation and no neutrino oscillations should be observedThe effects of themass terms in the conventionalHamiltonianℎ0(3) become negligible and the effectiveHamiltonian can be

approximated by ℎ asymp 120575ℎ Direct calculation of the oscillationprobabilities shows that in this approximation we can writethe appearance probability [33]

119875]119887rarr ]119886

≃ 11987121003816100381610038161003816120575ℎ119886119887

10038161003816100381610038162

119886 = 119887 (7)

with a similar expression for antineutrinos using 120575ℎ119886119887instead

The sidereal decomposition of the Hamiltonian (5) can beused to show that the probability will also exhibit siderealvariations one of the key signatures of Lorentz violationTheHamiltonian contains also a time-independent component(C)119886119887 which can lead to both isotropic and direction-

dependent effects In all cases the energy dependence isdifferent with respect to the conventional case so spectralstudies can be used to study these particular coefficients

Experimental studies using the probability (7) have beenperformed by Double Chooz [50] IceCube [51] LSND [52]MiniBooNE [53 54] andMINOS using its near detector [5556] The absence of a positive signal in all these experimentshas been used to set tight constraints on several coefficientsfor Lorentz violation which are summarized in [10] Itis important to emphasize that since all these searchesuse different oscillation channels they are complementaryaccessing similar coefficients but with different flavor indices

412 Perturbative Approximation Since the phase of theoscillation is given by Δ1198982

1198941198951198714119864 the oscillation probability

can be enhanced by placing a detector at a distance 119871 =


4

3

2

1

00 02 0604 08 1

Local sidereal phase

Even

ts10

18PO

T

Figure 1 Number of events normalized by protons on target (POT)in the MINOS far detector as a function of the sidereal phase Theflat distribution of events is interpreted as the absence of siderealvariations in the oscillation probability Figure adapted from [26]

2120587119864Δ1198982

119894119895from the neutrino source For experiments satis-

fying this condition mass-driven oscillations ℎ0dominate in

the Hamiltonian and we can consider the effects of Lorentzviolation 120575ℎ as a small perturbation [34] In this case theoscillation probability appears as a power series of the form

119875]119887rarr ]119886

= 119875(0)

]119887rarr ]119886

+ 119875(1)

]119887rarr ]119886

+ sdot sdot sdot (8)

where 119875(0)]119887rarr ]119886

is the conventional probability given by themassive-neutrino model and the following terms are given inpowers of the coefficients for Lorentz violationwhose explicitform is given in [34] Once again the sidereal variation ofthe oscillation probability appears as a key signal to search byexperiments For instance the leading-order term119875

(1)

]119887rarr ]119886

canbe generically written as follows

119875(1)

]119887rarr ]119886

2119871= (119875(1)

C )119886119887

+ (119875(1)

A119904

)119886119887

sin120596oplus119879oplus+ (119875(1)

A119888

)119886119887

cos120596oplus119879oplus

+ (119875(1)

B119904

)119886119887

sin 2120596oplus119879oplus+ (119875(1)

B119888

)119886119887

cos 2120596oplus119879oplus

(9)

This is the dominating probability for neutrinomixing as wellas antineutrino oscillations

Since the first-order correction (9) to the oscillationprobability arises from the interference between the con-ventional and the Lorentz-violating effects the sensitivityto the coefficients in 119875

(1)

]119887rarr ]119886

is greater than in the short-baseline approximation presented in the previous sectionFigure 1 shows part of the study performed by the MINOSexperiment which used the expression (9) to search forsidereal variations in the event rate measured at the fardetectorThe sensitivity to different coefficients was improvedby a factor 20ndash510 compared to the previous constraints usingthe near detector [26]

The mixing between neutrinos and antineutrinos is alsopossible due to the block 120575ℎ

119886119887in the Hamiltonian (2) which

is discussed in the following section

42 Neutrino-Antineutrino Mixing The off-diagonal block120575ℎ119886119887in the Hamiltonian (2) can produce the mixing between

neutrinos and antineutrinosThis 3times3matrix is given by [28]

120575ℎ119886119887= 119894radic2(120598

+)120572120572

119886119887

minus 119894radic2(120598+)120572119901120573119892120572120573

119886119887

|p| (10)

where the complex 4-vector (120598+)120572is the neutrino polarization

that can be directly written in terms of the location of theexperiment and the orientation of the neutrino beam [34]Two sets of coefficients for Lorentz violation denoted by 120572

119886119887

and 119892120572120573119886119887

control CPT-even and CPT-odd effects respectivelyThis Hamiltonian can also be decomposed in form (5) withharmonic amplitudes given in terms of the coefficients 120572

119886119887

and 119892120572120573119886119887

[34]Contrary to the neutrino Hamiltonian (4) the neutrino-

antineutrino block always appears with direction-dependenteffects for this reason the search of sidereal variations isan ideal setup to search for these coefficients Followingthe perturbative description presented in Section 412 it hasbeen shown that at first order the oscillation probabilityvanishes in other words neutrino-antineutrino oscillationsappear as a second order effect [34] For this reason thesecond-order probability 119875(2)]

119886rarr ]119887

can be decomposed in form(9) although involving up to four harmonics

The possible oscillation of neutrinos into antineutrinosmodifies for instance the survival probability of muonneutrinos in a beam experiment because now some ]

120583could

disappear into antineutrino states A systematic search of the66 coefficients 120572

119886119887

and 119892120572120573119886119887

producing sidereal variations wasperformed using data from theMINOS experiment [57]Theremaining 15 coefficients producing time-independent effectscould only be explored by a spectral study in a disappearanceexperiment Figure 2 shows a fit to the data from the DoubleChooz experiment searching for the spectral modificationthat could arise in the disappearance of electron antineutrinos[27]

A total of 81 coefficients 120572119886119887

and 119892120572120573119886119887

has been tightly con-strained by these two experimental searches whose resultsare summarized in [10]

5 Neutrino Kinematics

Oscillations are very sensitive to unconventional effects pro-ducing neutrino mixing due to their interferometric natureThere are however terms in the Hamiltonian (2) that areunobservable in oscillations Neutrino oscillations only allowus to measure energy differences between different neutrinostates for this reason the absolute scale of neutrino massescannot be determined from oscillations Similarly somecoefficients for Lorentz violation modify the energy of allflavors in the same way producing no effects in oscillationsNeglectingmixing effects results in a decoupling of the three-flavor system into three copies of a single state One ofthe observable effects of these oscillation-free coefficients isthe modification of the neutrino velocity which producesmeasurable effects in the neutrino time of flight Moreoveras a consequence of the unconventional dispersion relations


50

0

minus50

minus100

minus150

2 4 6 8 10 12

Dat

a pre

dict

ion

05

MeV

Data (stat only error)Best fit wLV params

(MeV)

sin2212057913 = 0109 wo LV params

E

Figure 2 Fit to the disappearance of reactor antineutrinos in theform ]

119890rarr ]119890due to neutrino-antineutrino mixing (red line) and

the conventional oscillation in the absence of Lorentz violation (blueline) in the Double Chooz experiment Figure adapted from [27]

the neutrino phase space and energy-conservation conditionrelevant for decay processes are modified as well

51 Neutrino Velocity The neutrino velocity can be obtainedfrom the Hamiltonian (2) For completeness operators ofarbitrary dimension 119889 can be incorporated in which case theneutrino velocity takes the form [35]

V] = 1 minus|119898|2

2|p|2+ sum119889119895119898

(119889 minus 3) |p|119889minus4119890119894119898120596oplus119879oplus0

N119895119898

times ((119886(119889)

of )119895119898minus (119888(119889)

of )119895119898)

(11)

where the factor |119898|2 is a real mass parameter that doesnot participate in oscillations and the Lorentz-violatingcomponent has been written in spherical form The index119889 denotes the effective dimension of the operator and thepair 119895119898 corresponds to angular momentum indices thatlabel the rotational properties of the oscillation-free sphericalcoefficients (119886(119889)of )119895119898 and (119888

(119889)

of )119895119898 controlling CPT-odd (forodd 119889) and CPT-even (for even 119889) effects respectively Thesespherical coefficients can be identified with coefficients inCartesian coordinates used in the previous sections [35] Theexpression for the neutrino velocity (11) has been written inthe Sun-centered frame [10 32] where all the directionalinformation is contained in the angular factors

0N119895119898

andthe sidereal time dependence appears as harmonics functionscontrolled by the index119898

The neutrino velocity (11) exhibits a rich phenomenologyin the form of many physical effects that can affect neu-trino propagation if deviations from Lorentz symmetry arepresent Depending on the dimension 119889 of the operator inthe theory neutrino velocity can be energy dependent for119895 = 0 anisotropic effects appear and the velocity becomes

a function of the direction of propagation for 119898 = 0 timedependence arises in which case the neutrino velocity varieswith sidereal time 119879

oplus and for odd 119889 CPT violation makes

neutrinos and antineutrinos move at different speedBeam experiments are suitable setups to compare the

speed of neutrinos with respect to the speed of photonsFrom the neutrino velocity (11) we clearly find that the massterm makes neutrinos travel slower than light whereas thecoefficients for Lorentz violation can generate subluminalor superluminal velocities depending on the sign of eachcoefficient Different beam experiments have measured thetime for neutrinos to travel a distance 119871 [58ndash64] whichwill experience a delay with respect to photons given by thefollowing

Δ119905 asymp 119871 (1 minus V]) (12)

which can be used to set limits in the oscillation-freecoefficients for Lorentz violation that modify the neutrinovelocity in (11) Since the minute effects of Lorentz violationcan be enhanced by neutrinos travelling a long distance aprecise constraint on the isotropic dimension-four coefficientwas obtained using the few antineutrino events from thesupernova SN1987A [65]

For the particular case of Lorentz invariance violationgenerated by a dimension-four operator the modificationto the neutrino velocity (11) is simply a constant factor Foroperators of dimension 119889 ge 5 low- and high-energy neu-trinos will move at different velocity If a burst of neutrinosof different energies is created at the same time this velocitydifference will generate a spread of neutrinos observable as adelay between high- and low-energy neutrinos at the detector[35] A similar effect has been widely studied for Lorentz-violating photons [66ndash69]

52 Threshold Effects Themodified dispersion relations thatneutrinos satisfy in the presence of Lorentz violation alterthe energy-momentum conservation relation which plays animportant role in meson-decay processes of the form119872

+

rarr

119897+

+ ]119897 It can be shown that above some threshold energy 119864th

these relations can completely block the phase space availablefor the decay [35 70ndash78]The observation of atmospheric andaccelerator neutrinos ]

119897with energy119864

0produced by the decay

of a meson of mass119872119872implies that 119864th gt 119864

0 which can be

used to write the condition

sum119889119895119898

119864119889minus2

0119884119895119898(p) [plusmn(119886(119889)of )119895119898 minus (119888

(119889)

of )119895119898] ≲

1

2(119872119872minus 119898119897plusmn)2

(13)

where the + (minus) sign is for neutrinos (antineutrinos) and119898119897plusmn

is themass of the accompanying charged leptonThis formulahas been used in [35] to constrain several coefficients forLorentz violation including many associated to nonrenor-malizable operators

The sensitivity to the effects of Lorentz violation increaseswith the energy of neutrinos observed as well as the distancethat they travel The observation of very-high-energy neu-trinos reported by the IceCube collaboration [79 80] offers


e+Z0

p

p998400

eminusk

k998400

Figure 3 Electron-positron pair emission as neutrino Cerenkovradiation

a great sensitivity to the effects described in this sectionThe small number of neutrinos observed with energies atthe PeV level [79] allows the study of isotropic effects (119895 =

0) nevertheless a full study of direction-dependent effectswould require several events spread in the sky Althoughthe IceCube results suggest an astrophysical origin for theseenergetic neutrinos tight constraints on different coefficientsfor Lorentz violation can be obtained even in the conservativeinterpretation of these neutrino events having atmosphericorigin [81] The observation of PeV neutrinos created by thedecay of heavy mesons in the upper atmosphere has beenused to implement the threshold condition (13) leading tosensitive limits in several isotropic coefficients of dimension119889 = 4 6 8 and 10 [81]

53 Cerenkov Radiation In same way as some processes canbe forbidden above certain energies the effects of Lorentzviolation can also open particular decay channels that wouldbe otherwise forbidden In particular coefficients leading toV] gt 1 in (11) can produce Cerenkov emission of one or moreparticles [35 82ndash92] Cerenkov radiation makes neutrinoslose energy which distorts the spectrum in long-baselineexperiments using accelerator and atmospheric neutrinosThis feature provides another method to search for Lorentzviolation The observation of high-energy neutrinos afterpropagating a distance 119871 sets a lower value for the character-istic distortion distance119863(119864) = minus119864(119889119864119889119909) in the form 119871 lt

119863(119864)The determination of the characteristic distance for thespectral distortion caused by the isotropic Lorentz-violatingoperator of dimension four (119888

(4)

of )00 is described in [82ndash92] The general calculation including direction-dependenteffects for operators of arbitrary dimension can be found in[35]

Using the PeV neutrinos observed by IceCube [79] thelimits obtained using threshold conditions can be improvedby one order ofmagnitude by determining spectral distortionproduced by Cerenkov radiation [81] For instance theemission of electron-positron pairs in the form ] rarr ]+ 119890minus +119890+ is characterized by a rate of energy loss given by [35 81]

119889119864

119889119909= minus

119862

8int

1205810

12058110158402

(1205812 minus1198722119885)2

12059710038161003816100381610038161003816120581101584010038161003816100381610038161003816

1205971205810

119902 sdot 1198961199021015840

sdot 1198961015840

119902011989601199021015840011989610158400

1198893

1199011015840

119889Ω1205811015840 (14)

where 119862 is a constant the auxiliary 4 vectors 120581 = 119896 +

1198961015840 and 120581

1015840

= 119896 minus 1198961015840 have been defined in terms of the

momentum of the electron and the positron and 1199021199020=

(1 p) 119902101584011990210158400= (1 p1015840) following the conventions in Figure 3

Several orders ofmagnitude in sensitivity can be gainedwhenusing an astrophysical interpretation for the PeV neutrinosin IceCube After travelling astrophysical distances theseneutrinos would rapidly fall below the threshold energyfor Cerenkov emission The observation of these neutrinoswith PeV energies implies that this threshold energy liesabove 1 PeV leading to stringent limits on isotropic Lorentzviolation of dimension 119889 = 4 6 8 and 10 [81] Similar studiesfor the case 119889 = 4 can be found in [93 94]

Direction-dependent effects using high-energy neutrinosrequire several eventsThe recent observation of 26 new ener-getic events in IceCube [80] distributed in the sky allows thesearch of space anisotropy for operators of dimension119889 = 4 6

[81]The simultaneous study of several coefficients producingdirection-dependent effects allows two-sided bounds morerestrictive than the very particular case of isotropic Lorentzviolation considering superluminal velocity that allows one-sided limits only In the future the observation of moreevents should allow a detailed study of operators of higherdimension

6 Beta Decay

The interferometric nature of neutrino oscillations makesthem an ideal type of experiment to search for minutedeviations from exact Lorentz symmetry Nonetheless theeffects thatmodify the kinematics of all neutrino flavors in thesame manner are unobservable in oscillation experimentswhich makes the studies described in Section 5 an importantcomplement to oscillation searches The enhancement ofLorentz-violating effects with the neutrino energy makesalso the study of neutrino velocity and Cerenkov radiationa sensitive probe of Lorentz invariance with high-energyneutrinos Nevertheless it has been shown that low-energyexperiments can also play a key role in the study of Lorentzinvariance In particular signals of oscillation-free operatorsof dimension three (119886

(3)

of )119895119898 are not only unobservable inoscillations but also produce no effects in the neutrinovelocity (11)

The experimental signatures of the coefficients associatedwith these so-called countershaded operators [16 95 96]motivate the study of weak decays The effects of theseoperators are unaffected by the neutrino energy giving low-energy experiments a competitive sensitivity to signals ofLorentz violation It is important to emphasize that Lorentz-violating effects appear as kinematical effects modifying theneutrino phase space nevertheless modifications of thespinor solutions must also be taken into account Beta decayin the context of Lorentz violation in sectors other thanneutrinos has recently been studied theoretically [97 98] andexperimentally [99]

61 Tritium Decay The absolute mass scale of neutrinoscannot be studied in oscillation experiments which only offer


access to mass-squared differences The direct measurementof neutrino masses can be made by searching for a distortionof the electron energy spectrum in tritium decay The mea-surement of beta electrons near the endpoint of the spectrum

119889Γ

119889119879= 119862[(Δ119879)

2

minus1

21198982

]] (15)

allows the study of the effective mass1198982] of electron antineu-trinos where 119862 is approximately constant and Δ119879 = 119879 minus

1198790denotes the kinetic energy of the electron 119879 measured

from the endpoint energy 1198790 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 unde-tected however magnetic fields select the beta electronsemitted in a particular direction to be studied This featurepermits the study of anisotropic effects In the presence ofLorentz-violating neutrinos the spectrum (15) gets correctedby the replacement

Δ119879 997888rarr Δ119879 + (119896(3)

C ) + (119896(3)

A119904

) sin120596oplus119879oplus

+ (119896(3)

A119888

) cos120596oplus119879oplus

(16)

which shows that the distortion near the endpoint canbe shifted and also exhibits a sidereal-time dependenceThe amplitudes in the modification (16) depend on thefour independent coefficients (119886(3)of )00 (119886

(3)

of )10 (119886(3)

of )11 and(119886(3)

of )1minus1 and experimental quantities such as location of thelaboratory orientation of the apparatus and intensity of themagnetic fields used to select the beta electrons for theiranalysis [96]

An interesting feature appears when the effective coeffi-cients (119888(2)of )1119898 are considered which arise as a consequenceof neutrino mass and CPT-even Lorenz violation [35] Thesecoefficients can mimic the effects of a mass parameter inwhich case the spectrum (15) gets modified in the form

1198982

] 997888rarr 1198982

= 1198982

] + (119896(2)

C ) + (119896(2)

A119904


+ (119896(2)

A119888


(17)

We find that the experimental mass-squared parameter 1198982involves the actual neutrino mass119898] however the mass canbe screened by the effects of the three coefficients (119888(2)of )1119898(119898 = 0 plusmn1) varying with sidereal time and depending on theorientation of the apparatus Notice also that the sign of theexperimental mass-squared parameter1198982 is not restricted tobe positive so the coefficients (119888(2)of )1119898 could even mimic atachyonic neutrino [103]

Alternative approaches have been considered to searchfor isotropic Lorentz violation in tritium decay for otheroperators in [104 105]

62 Neutron Decay Neutrons are fascinating laboratories tostudy the validity of fundamental symmetries The effects ofdeviations from exact Lorentz invariance would affect the

spectrum of the beta electrons as well as the measurementsof particular experimental asymmetries Contrary to tritiumdecay experiments the study of neutron decay covers thewhole energy spectrumwhich takes the formof the spectrum(15) neglecting the neutrino mass that plays no role far fromthe endpoint and the factor119862 can no longer be approximatedby a constant so it becomes a function of the electron energyFor experiments only counting the number of beta electronsper energy range all the anisotropic effects disappear afterintegrating over all the neutrino orientations The net effectis a distortion of the whole spectrum that can be studied bysearching for deviations from the conventional spectrumTheresidual spectrum is proportional to the coefficient (119886(3)of )00[96]

Anisotropic effects can be studied by constructing asym-metries Aexp in experiments that can determine the direc-tionality of some of the decay products For experimentsusing unpolarized neutrons an asymmetry counting elec-trons emitted in the same direction as the antineutrino 119873

+

compared to events in which the two leptons are emitted inopposite directions 119873

minuscan be constructed for the measure-

ment of the electron-antineutrino correlation 119886 in the formof

119886exp =119873+minus 119873minus

119873++ 119873minus

(18)

Similarly experiments using polarized neutrons that are ableto measure the electron and the recoiling proton can be usedto search for electron-proton coincidence events useful forthe measurement of the neutrino asymmetry 119861 As shown inFigure 4 the experimental asymmetry counts events inwhichproton and electron are emitted against119873

minusminusor along119873

++ the

direction of the neutron polarization n can be written in theform

119861exp =119873minusminusminus 119873++

119873minusminus+ 119873++

(19)

Effects of Lorentz violation arise from themodified spinorsolutions that affect the matrix element of the decay as well asthe unconventional form of the neutrino phase space due tothe modified dispersion relations making the asymmetries(18) and (19) have the general form [96]

Aexp = (AC) + (AA119904

) sin120596oplus119879oplus+ (AA

119888

) cos120596oplus119879oplus (20)

where the amplitudes depend on the coefficients for Lorentzviolation (119886

(3)

of )10 (119886(3)

of )11 and (119886(3)

of )1minus1 and experimentalquantities including the orientation and location of theapparatus

63 Double Beta Decay The same coefficients modifying thespectrum for beta decay can also introduce observable effectsin double beta decay experiments Since the antineutrinosescape unobserved the simplest test of Lorentz invarianceis an alteration of the two-electron spectrum for the two-neutrino mode of double beta decay produced by thecoefficient (119886(3)of )00 [96 106] Similar to the neutron-decay


e

e

nn

N++Nminusminus

p

p

nn

Figure 4 Electron-proton coincidence events for the asymmetry 119861exp The polarization of the neutron is denoted by n

spectrum the resulting effect is a distortion of the wholespectrum that can be studied by searching for deviations fromthe conventional spectrumThe energy at which this effect ismaximal has been identified for several isotopes which willguide these types of studies [106]

The neutrinoless mode of double beta decay offers accessto other type of coefficient one that modifies the neutrinopropagatorThisMajorana coupling in the SMEdenoted |119892120582120588

120573120573|

is a combination of other coefficients in the SME that cantrigger neutrinoless double beta decay even if the Majoranamass is negligible In terms of this effective coefficient forCPT-odd Lorentz violation the half-life of an isotope ofradius 119877 is given by [106]

(1198790]12)minus1

= 1198660]100381610038161003816100381610038161198720]10038161003816100381610038161003816

2

100381610038161003816100381610038161003816119892120582120588

120573120573

100381610038161003816100381610038161003816

2

41198772

(21)

where1198660] is the phase-space factor regarding the two emittedelectrons and 119872

0] is the relevant nuclear matrix elementLimits on the Majorana mass parameter |119898

120573120573| can be used

to constrain the coefficient |119892120582120588120573120573| Since the Lorentz-violating

neutrinoless double beta decay depends on the nuclear size 119877of the isotope used a future observation of this decay modecan be distinguished because theMajorana-mass mechanismdepends on the isotope only through the nuclear matrixelements

7 Conclusions

In this paper we have presented a general overview ofthe effects of deviations from exact Lorentz invariance inneutrinos in the context of the Standard-Model Extension Ingeneral the signatures of the breakdown of Lorentz symme-try are direction and time dependence of the relevant observ-ables for Earth-based experiments as well as unconventionaldependence on the neutrino energy Neutrino oscillations are

sensitive probes of new physics which makes this type ofexperiment an ideal setup to search for violations of Lorentzinvariance In oscillations some effects of Lorentz violationinclude direction and time dependence of the oscillationprobability oscillation phases that grow with the neutrinoenergy CPT violation and mixing between neutrinos andantineutrinos

Some effects are unobservable in neutrino oscillationsin which case kinematical effects become a complementarytechnique Effects of Lorentz violation appear as modifi-cations to the neutrino velocity as well as unconventionalbehavior in decay processes In particular some decays withneutrinos in the final state can become forbidden abovecertain threshold energy similarly some forbidden processescan become allowed including Cerenkov radiation of oneor more particles Most of these effects are enhanced bythe neutrino energy which makes high-energy neutrinos ofparticular interest for future tests of Lorentz invariance

Finally there are operators in the theory whose exper-imental signatures are independent of the neutrino energyIn this case the high precision of low-energy experimentscan play a fundamental role in the test of Lorentz symme-try for some particular operators that are unobservable inoscillations and that leave the neutrino velocity unchangedFor these countershaded operators beta decay is the idealexperimental setup Depending on the properties of theexperiment themain features have been identified for studiesof tritium decay neutron decay and double beta decay

To date there is no compelling evidence of Lorentz viola-tion nevertheless only a few of the experimental signatureshave been studied [10] Neutrinos offer great sensitivity andnumerous ways to test the validity of the cornerstone ofmodern physics Many of the different techniques presentedin this review are currently being implemented by a variety ofexperimental collaborations Interesting new tests of Lorentzsymmetry will be performed in the near future in whichlow- and high-energy neutrinos will play a key role in ourunderstanding of the nature of spacetime


Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work was supported in part by the Department ofEnergy under Grant DE-FG02-13ER42002 the Indiana Uni-versity Center for Spacetime Symmetries and the HelmholtzAlliance for Astroparticle Physics (HAP) under Grant noHA-301

References

[1] W Pauli Offener Brief an die Gruppe der Radioaktiven bei derGauvereins-Tagung zu Tubingen 1930

[2] V A Kostelecky and S Samuel ldquoSpontaneous breaking ofLorentz symmetry in string theoryrdquo Physical Review D vol 39article 683 1989

[3] D Colladay and V A Kostelecky ldquoCPT violation and thestandard modelrdquo Physical Review D vol 55 no 11 pp 6760ndash6774 1997

[4] H B Nielsen and I Picek ldquoThe Redei-like model and testingLorentz invariancerdquo Physics Letters B vol 114 no 2-3 pp 141ndash146 1982

[5] H B Nielsen and I Picek ldquoLorentz non-invariancerdquo NuclearPhysics B vol 211 no 2 pp 269ndash296 1983

[6] R Huerta and J L Lucio M ldquoTesting Lorentz invariance inbaryon decayrdquo Physics Letters B vol 131 no 4ndash6 pp 471ndash4731983

[7] R Huerta-Quintanilla and J L Lucio M ldquoConstraint onLorentz non-invariance from the Michel parameterrdquo PhysicsLetters B vol 124 no 5 pp 369ndash370 1983

[8] D Colladay and V A Kostelecky ldquoLorentz-violating extensionof the standardmodelrdquo Physical ReviewD vol 58 no 11 ArticleID 116002 1998

[9] V A Kostelecky ldquoGravity Lorentz violation and the standardmodelrdquo Physical Review D vol 69 Article ID 105009 2004

[10] V A Kostelecky and N Russell ldquoData tables for Lorentz andCPT violationrdquo Reviews of Modern Physics vol 83 article 112011

[11] V A Kostelecky and S Samuel ldquoPhenomenological gravita-tional constraints on strings and higher-dimensional theoriesrdquoPhysical Review Letters vol 63 article 224 1989

[12] V A Kostelecky and R Potting ldquoGravity from local Lorentzviolationrdquo General Relativity and Gravitation vol 37 no 10 pp1675ndash1679 2005

[13] R Bluhm andV A Kostelecky ldquoSpontaneous Lorentz violationNambu-Goldstone modes and gravityrdquo Physical Review D vol71 Article ID 065008 2005

[14] O Bertolami and J Paramos ldquoVacuum solutions of a gravitymodel with vector-induced spontaneous Lorentz symmetrybreakingrdquo Physical Review D vol 72 no 4 Article ID 0440012005

[15] N Arkani-HamedH C ChengM Luty and JThaler ldquoUniver-sal dynamics of spontaneous Lorentz violation and a new spin-dependent inverse-square law forcerdquo Journal of High EnergyPhysics vol 7 article 029 2005

[16] V A Kostelecky and J D Tasson ldquoProspects for large relativityviolations in matter-gravity couplingsrdquo Physical Review Lettersvol 102 no 1 Article ID 010402 2009

[17] O W Greenberg ldquoCPT violation implies violation of Lorentzinvariancerdquo Physical Review Letters vol 89 no 23 Article ID231602 2002

[18] J Ellis and N E Mavromatos ldquoComments on CP T and CPTviolation in neutral kaon decaysrdquo Physics Report vol 320 no1ndash6 pp 341ndash354 1999

[19] F R Klinkhamer ldquoA CPT anomalyrdquo Nuclear Physics B vol 578no 1-2 pp 277ndash289 2000

[20] C Adam and F R Klinkhamer ldquoCausality and CPT violationfrom an Abelian Chern-Simons-like termrdquo Nuclear Physics Bvol 607 no 1-2 pp 247ndash267 2001

[21] G Barenboim L Borissov J Lykken and A Y SmirnovldquoNeutrinos as the messengers of CPT violationrdquo Journal of HighEnergy Physics vol 10 article 001 2002

[22] G Barenboim J F Beacom L Borissov and B Kayser ldquoCPTviolation and the nature of neutrinosrdquo Physics Letters B vol 537no 3-4 pp 227ndash232 2002

[23] F R Klinkhamer and C Rupp ldquoSpacetime foam CPT anomalyand photon propagationrdquo Physical Review D vol 70 no 4Article ID 045020 2004

[24] N EMavromatos ldquoCPT violation and decoherence in quantumgravityrdquo in Planck Scale Effects in Astrophysics and Cosmologyvol 669 of Planck Scale Effects in Astrophysics and Cosmologypp 245ndash320 2005

[25] M Chaichian A D Dolgov V A Novikov and A TureanuldquoCPT violation does not lead to violation of Lorentz invarianceand vice versardquo Physics Letters B vol 699 no 3 pp 177ndash1802011

[26] P Adamson D J Auty MINOS Collaboration et al ldquoSearchfor Lorentz invariance and CPT violation with the MINOS fardetector rdquo Physical Review Letters vol 105 Article ID 1516012010

[27] J S Dıaz T Katori J Spitz and J M Conrad ldquoSearch forneutrinondashantineutrino oscillations with a reactor experimentrdquoPhysics Letters B vol 727 no 4-5 pp 412ndash416 2013

[28] V A Kostelecky and M Mewes ldquoLorentz and CPT violation inneutrinosrdquo Physical Review D vol 69 Article ID 016005 2004

[29] Z Maki M Nakagawa and S Sakata ldquoRemarks on the unifiedmodel of elementary particlesrdquo Progress of Theoretical Physicsvol 28 no 5 pp 870ndash880 1962

[30] B Pontecorvo ldquoNeutrino experiments and the problem ofconservation of leptonic chargerdquo Journal of Experimental andTheoretical Physics vol 53 pp 1717ndash1725 1967

[31] B Pontecorvo ldquoNeutrino experiments and the question ofleptonic-charge conservationrdquo Soviet PhysicsmdashJETP vol 26 pp984ndash988 1968

[32] R Bluhm V A Kostelecky C D Lane and N Russell ldquoClock-comparison tests of Lorentz and CPT symmetry in spacerdquoPhysical Review Letters vol 88 Article ID 090801 2002

[33] V A Kostelecky and M Mewes ldquoLorentz violation and short-baseline neutrino experimentsrdquo Physical Review D vol 70Article ID 076002 2004

[34] J S Dıaz V A Kostelecky and M Mewes ldquoPerturbativeLorentz and CPT violation for neutrino and antineutrinooscillationsrdquo Physical ReviewD vol 80 Article ID 076007 2009

[35] V A Kostelecky and M Mewes ldquoNeutrinos with Lorentz-violating operators of arbitrary dimensionrdquo Physical Review Dvol 85 Article ID 096005 2012


[36] V A Kostelecky and M Mewes ldquoFermions with Lorentz-violating operators of arbitrary dimensionrdquo Physical Review Dvol 88 Article ID 096006 2013

[37] A Aguilar-Arevalo L B Auerbach and LSND CollaborationldquoEvidence for neutrino oscillations from the observation of 120592

119890

appearance in a 120592120583beamrdquo Physical Review D vol 64 Article ID

112007 2001[38] A A Aguilar-Arevalo A O Bazarko MiniBooNE Collabora-

tion et al ldquoSearch for Electron Neutrino Appearance at the9987791198982

sim 11198901198812 Scalerdquo Physical Review Letters vol 98 Article ID

231801 2007[39] A A Aguilar-Arevalo C E Anderson MiniBooNE Collabo-

ration et al ldquoUnexplained excess of electronlike events from a1-GeV neutrino beamrdquo Physical Review Letters vol 102 ArticleID 101802 2009

[40] V A Kostelecky and M Mewes ldquoLorentz and CPT violationin the neutrino sectorrdquo Physical Review D vol 70 Article ID031902 2004

[41] T Katori V A Kostelecky and R Tayloe ldquoGlobal three-parameter model for neutrino oscillations using Lorentz viola-tionrdquo Physical Review D vol 74 Article ID 105009 2006

[42] V Barger D Marfatia and K Whisnant ldquoChallenging Lorentznoninvariant neutrino oscillations without neutrino massesrdquoPhysics Letters B vol 653 no 2ndash4 pp 267ndash277 2007

[43] V Barger J Liao D Marfatia and K Whisnant ldquoLorentznoninvariant oscillations of massless neutrinos are excludedrdquoPhysical Review D vol 84 no 5 Article ID 056014 2011

[44] J S Dıaz and V A Kostelecky ldquoThree-parameter Lorentz-violating texture for neutrino mixingrdquo Physics Letters B vol700 no 1 pp 25ndash28 2011

[45] J S Dıaz and V A Kostelecky ldquoLorentz- and CPT-violatingmodels for neutrino oscillationsrdquo Physical Review D vol 85Article ID 016013 2012

[46] S J Rong and Q Y Liu ldquoThe perturbed puma modelrdquo ChinesePhysics Letters vol 29 no 4 Article ID 041402 2012

[47] J Beringer J F Arguin Particle Data Group et al ldquoReview ofParticle Physicsrdquo Physical Review D vol 86 Article ID 0100012012

[48] T Akiri and Super-Kamiokande Collaboration ldquoSensitivity ofatmospheric neutrinos in Super-Kamiokande to Lorentz vio-lationrdquo High Energy Physics-Phenomenology httparxivorgabs13082210

[49] J S Dıaz IUHET-585[50] Y Abe C Aberle Double Chooz Collaboration et al ldquoFirst

test of Lorentz violation with a reactor-based antineutrinoexperimentrdquo Physical Review D vol 86 Article ID 112009 2012

[51] R Abbasi Y Abdou IceCube Collaboration et al ldquoSearch for aLorentz-violating sidereal signal with atmospheric neutrinos inIceCuberdquo Physical Review D vol 82 Article ID 112003 2010

[52] L B Auerbach R L Burman LSNDCollaboration et al ldquoTestsof Lorentz violation in V

120583rarr V119890oscillationsrdquo Physical ReviewD

vol 72 Article ID 076004 2005[53] A A Aguilar-Arevalo C E Anderson MiniBooNE Collab-

oration et al ldquoTest of Lorentz and CPT violation with shortbaseline neutrino oscillation excessesrdquo Physics Letters B vol718 no 4ndash5 pp 1303ndash1308 2013

[54] T Katori ldquoTests of lorentz and cpt violation with MiniBooNEneutrino oscillation excessesrdquoModern Physics Letters A vol 27no 25 Article ID 1230024 2012

[55] P Adamson et al ldquoTesting Lorentz invariance and CPT con-servation with NuMI neutrinos in the MINOS near detectorrdquoPhysical Review Letters vol 101 Article ID 151601 2008

[56] P Adamson D S Ayres The MINOS Collaboration et alldquoSearch for Lorentz invariance and CPT violation with muonantineutrinos in the MINOS near detectorrdquo Physical Review Dvol 85 Article ID 031101 2012

[57] B Rebel and S Mufson ldquoThe search for neutrinondashantineutrinomixing resulting from Lorentz invariance violation using neu-trino interactions in MINOSrdquo Astroparticle Physics vol 48 pp78ndash81 2013

[58] J Alspector G R Kalbfleisch N Baggett et al ldquoExperimentalcomparison of neutrino and muon velocitiesrdquo Physical ReviewLetters vol 36 no 15 pp 837ndash840 1976

[59] G R Kalbfleisch N Baggett E C Fowler and J Alspec-tor ldquoExperimental comparison of neutrino antineutrino andmuon velocitiesrdquo Physical Review Letters vol 43 no 19 pp1361ndash1364 1979

[60] P Adamson C Andreopoulos MINOS Collaboration et alldquoMeasurement of neutrino velocity with the MINOS detectorsand NuMI neutrino beamrdquo Physical Review D vol 76 ArticleID 072005 2007

[61] T AdamN Agafonova OPERACollaboration et al ldquoMeasure-ment of the neutrino velocity with the OPERA detector in theCNGS beamrdquo Journal of High Energy Physics vol 2012 p 932012

[62] M Antonello P Aprili ICARUSCollaboration et al ldquoMeasure-ment of the neutrino velocity with the ICARUS detector at theCNGS beamrdquo Physics Letters B vol 713 no 1 pp 17ndash22 2012

[63] P Alvarez Sanchez R Barzaghi Borexino Collaboration et alldquoMeasurement of CNGS muon neutrino speed with BorexinordquoPhysics Letters B vol 716 pp 401ndash405 2012

[64] N Y Agafonova M Aglietta LVD Collaboration et al ldquoMea-surement of the velocity of neutrinos from the CNGS beamwith the large volume detectorrdquo Physical Review Letters vol 109Article ID 070801 2012

[65] M J Longo ldquoTests of relativity from SN1987Ardquo Physical ReviewD vol 36 no 10 pp 3276ndash3277 1987

[66] V A Kostelecky and M Mewes ldquoLorentz-violating electro-dynamics and the cosmic microwave backgroundrdquo PhysicalReview Letters vol 99 Article ID 011601 2007

[67] V A Kostelecky and M Mewes ldquoAstrophysical tests of Lorentzand CPT violation with photonsrdquo The Astrophysical JournalLetters vol 689 no 1 article L1 2008

[68] V A Kostelecky and M Mewes ldquoElectrodynamics withLorentz-violating operators of arbitrary dimension rdquo PhysicalReview D vol 80 Article ID 015020 2009

[69] V Vasileiou A Jacholkowska F Piron et al ldquoConstraints onLorentz invariance violation from Fermi-Large Area Telescopeobservations of gamma-ray burstsrdquo Physical Review D vol 87no 12 Article ID 122001 2013

[70] B Altschul ldquoConsequences of neutrino Lorentz violation forleptonic meson decaysrdquo Physical Review D vol 84 Article ID091902 2011

[71] A Chodos V A Kostelecky R Potting and E Gates ldquoNullexperiments for neutrinomassesrdquoModern Physics Letters A vol7 no 6 article 467 1992

[72] A Chodos andVA Kostelecky ldquoNuclear null tests for spacelikeneutrinosrdquo Physics Letters B vol 336 no 3-4 pp 295ndash302 1994

[73] V A Kostelecky ldquoMass bounds for spacelike neutrinosrdquo inTop-ics in Quantum Gravity and Beyond Essays in Honor of LouisWitten F Mansouri and J J Scanio Eds p 369 WorldScientific Singapore 1993


[74] L Gonzalez-Mestres ldquoAstrophysical consequences of theOPERA superluminal neutrinordquo General Physics httparxivorgabs11096630

[75] X J Bi P F Yin Z H Yu and Q Yuan ldquoConstraints and testsof the OPERA superluminal neutrinosrdquo Physical Review Lettersvol 107 Article ID 241802 2011

[76] R Cowsik S Nussinov and U Sarkar ldquoSuperluminal nutrinosat OPERA confront pion decay kinematicsrdquo Physical ReviewLetters vol 107 no 25 Article ID 251801 2011

[77] M Mannarelli M Mitra F L Villante and F Vissani ldquoNon-standard neutrino propagation and pion decayrdquo Journal of HighEnergy Physics vol 2012 no 1 article 136 2012

[78] V Baccetti K Tate and M Visser ldquoLorentz violating kinemat-ics threshold theoremsrdquo Journal of High Energy Physics vol2012 article 87 2012

[79] M G Aartsen et al ldquoFirst observation of PeV-energy neutrinoswith iceCuberdquoPhysical ReviewLetters vol 111 Article ID0211032013

[80] M G Aartsen et al ldquoEvidence for high-energy extraterrestrialneutrinos at the icecube detectorrdquo Science vol 342 no 6161Article ID 1242856 2013

[81] J S Dıaz V A Kostelecky and M Mewes ldquoTesting relativitywith high-energy astrophysical neutrinosrdquo Physical Review Dvol 89 Article ID 043005 2014

[82] S R Coleman and S L Glashow ldquoHigh-energy tests of Lorentzinvariancerdquo Physical Review D vol 59 Article ID 116008 1999

[83] DMattingly T Jacobson and S Liberati ldquoThreshold configura-tions in the presence of Lorentz violating dispersion relationsrdquoPhysical Review D vol 67 no 12 Article ID 124012 2003

[84] D M Mattingly L MacCione M Galaverni S Liberati andG Sigl ldquoPossible cosmogenic neutrino constraints on Planck-scale Lorentz violationrdquo Journal of Cosmology and AstroparticlePhysics vol 2010 no 2 article 007 2010

[85] J Alexandre J Ellis and N E Mavromatos ldquoOn the possibilityof superluminal neutrino propagationrdquo Physics Letters B vol706 no 4-5 pp 456ndash461 2012

[86] A G Cohen and S L Glashow ldquoPair creation constrainssuperluminal neutrino propagationrdquo Physical Review Lettersvol 107 no 18 Article ID 181803 2011

[87] J M Carmona and J L Cortes ldquoConstraints from neu-trino decay on superluminal velocitiesrdquo High Energy Physicshttparxivorgabs11100430

[88] L Maccione S Liberati and D M Mattingly ldquoViolations ofLorentz invariance in the neutrino sector an improved analysisof anomalous threshold constraintsrdquo Journal of Cosmology andAstroparticle Physics vol 2013 no 3 article 039 2013

[89] S Mohanty and S Rao ldquoConstraint on super-luminal neutri-nos from vacuum Cerenkov processesrdquo High Energy Physicshttparxivorgabs11112725

[90] M Li D Liu J Meng T Wang and L Zhou ldquoReplayingneutrino bremsstrahlung with general dispersion relationsrdquoHigh Energy Physics httparxivorgabs11113294

[91] Y Huo T Li Y Liao D V Nanopoulos and Y Qi ldquoConstraintson neutrino velocities revisitedrdquo Physical Review D vol 85 no3 Article ID 034022 2012

[92] S J Brodsky and S Gardner ldquoPair production constraints onsuperluminal neutrinos revisitedrdquo High Energy Physics httparxivorgabs11121090

[93] E Borriello S Chakraborty A Mirizzi and P D SerpicoldquoStringent constraint on neutrino Lorentz invariance violation

from the two IceCube PeV neutrinosrdquo Physical Review D vol87 no 11 Article ID 116009 2013

[94] F W Stecker ldquoConstraining superluminal electron and neu-trino velocities using the 2010Crab nebula flare and the iceCubePeV neutrino eventsrdquoHigh Energy Physics httparxivorgabs13066095

[95] V A Kostelecky and J D Tasson ldquoMatter-gravity couplings andLorentz violationrdquo Physical Review D vol 83 Article ID 0160132011

[96] J S Dıaz et al ldquoRelativity violations and beta decayrdquo PhysicalReview D vol 88 Article ID 071902 2013

[97] J P Noordmans H W Wilschut and R G E TimmermansldquoLorentz violation in neutron decay and allowed nuclear betadecayrdquo Physical Review C vol 87 Article ID 055502 2013

[98] B Altschul ldquoContributions to pion decay from Lorentz viola-tion in the weak sectorrdquo Physical Review D vol 88 Article ID076015 2013

[99] S EMMuller E ADijckH Bekker et al ldquoFirst test of Lorentzinvariance in the weak decay of polarized nucleirdquo PhysicalReview D vol 88 Article ID 071901 2013

[100] V N Aseev A I Belesev A I Berlev et al ldquoUpper limit onthe electron antineutrino mass from the Troitsk experimentrdquoPhysical Review D vol 84 Article ID 112003 2011

[101] C Kraus B Bornschein L Bornschein et al ldquoFinal results fromphase II of the Mainz neutrino mass searchin tritium 120573 decayrdquoThe European Physical Journal C vol 40 no 4 pp 447ndash4682005

[102] J Angrik and KATRIN Collaboration KATRIN design reportFZKA Scientific Report 7090 2005

[103] A Chodos A I Hauser and V A Kostelecky ldquoThe neutrino asa tachyonrdquo Physics Letters B vol 150 no 6 pp 431ndash435 1985

[104] J M Carmona and J L Cortes ldquoTesting Lorentz invarianceviolations in the tritium beta-decay anomalyrdquo Physics Letters Bvol 494 no 1-2 pp 75ndash80 2000

[105] A E Bernardini andO Bertolami ldquoLorentz violating extensionof the standard model and the 120573-decay endpointrdquo PhysicalReview D vol 77 Article ID 085032 2008

[106] J S Dıaz ldquoLimits on Lorentz and CPT violation from doublebeta decayrdquo Physical Review D vol 89 Article ID 036002 2014

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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119905120572120573 whichwill be coupled to some SMoperator with the same

number of spacetime indices in the formL = 119905120572120573O120572120573 Under

a coordinate transformation both the operator and the tensorbackground field transform to a new set of coordinates asfollows

O120572120573

997888rarr O12057210158401205731015840

= Λ1205721015840

120572Λ1205731015840

120573O120572120573

119905120572120573997888rarr 11990512057210158401205731015840 = (Λ

minus1

)120582

1205721015840(Λminus1

)120588

1205731015840119905120582120588

(1)

so that the terms in the LagrangianL = 119905sdotO remain invariantL1015840 = 119905

1015840

sdotO1015840 = 119905sdotO = LThe same construction can be appliedfor a general tensor

On the other hand when a Lorentz transformationis performed over the physical system this is when theexperimental apparatus is rotated or boosted rather thanthe coordinates used to describe it then the SM operatortransforms as shown in (1) but any background field remainsunchanged L1015840 = 119905 sdot O1015840 =L This so-called particle Lorentztransformation changes the coupling between the back-ground fields and the SM operators resulting in a physicallyobservable anisotropy of spacetime this is a violation ofLorentz invariance [3]

Phenomenological approaches to parameterize and ex-perimentally search for particular types of Lorentz violationhave been considered since several decades [4ndash7] Howevereffective field theory can be used to incorporate genericoperators that break Lorentz invariance for all the particlesin the SMThis general framework is known as the standard-model extension (SME) [3 8 9] whose action includesgeneral coordinate-invariant terms by contracting operatorsof conventional fieldswith controlling coefficients for Lorentzviolation and reduces to the SM if all these coefficients vanishGravity can also be incorporated by writing the SM in acurved background [9] The development of the SME has ledto a worldwide experimental program searching for viola-tions of Lorentz invariance whose results are summarized in[10]

Flat spacetime is considered for experiments in particlephysics in which case the coefficients for Lorentz violationthat act as background fields can be chosen to be constantand uniform which guarantees conservation of energy andlinear momentum In this limit these coefficients representthe vacuum expectation value of the tensor fields of theunderlying theory Excitations of these fields lead to arich phenomenology for instance Nambu-Goldstonemodescould play fundamental roles when gravity is included suchas the graviton the photon in Einstein-Maxwell theory[11ndash14] and spin-dependent [15] and spin-independent [16]forces

It should be noted that a subset of operators in the SMEalso break CPT symmetry In fact all the Lorentz-violatingterms in the action involving operators with an odd numberof spacetime indices are odd under a CPT transformationIn realistic field theories CPT violation always appears withLorentz violation [17] Nonetheless alternative approachesexist in which CPT violation is implemented with andwithout Lorentz invariance [18ndash25]

3 Neutrinos

The general description of three left-handed neutrinos andthree right-handed antineutrinos in the presence of Lorentz-violating background fields is given by a 6 times 6 effectiveHamiltonian of form [28]

119867 = (

(ℎ0)119886119887

0

0 (ℎ0)lowast

119886119887

) + (

120575ℎ119886119887

120575ℎ119886119887

120575ℎlowast

119886119887

120575ℎ119886119887

) (2)

where the indices indicate the flavors of active neutrinos119886 119887 = 119890 120583 120591 and antineutrinos 119886 119887 = 119890 120583 120591 The Lorentz-preserving component is explicitly given by the following

(ℎ0)119886119887= |p| 120575

119886119887+1198982

119886119887

2 |p| (3)

where at leading order the neutrino momentum is given bythe energy |p| asymp 119864 and themass-squaredmatrix is commonlywritten in terms of the Pontecorvo-Maki-Nakagawa-Sakata(PMNS) matrix [29ndash31] as 1198982 = 119880PMNS(119898

2

119863)119880dagger

PMNS with1198982

119863= diag(1198982

1 1198982

2 1198982

3)

The Lorentz-violating block describing neutrinos in theHamiltonian (2) is given by the following

120575ℎ119886119887= (119886119871)120572

119886119887119901120572+ (119888119871)120572120573

119886119887119901120572119901120573|p| (4)

The components of the 3 times 3 complex matrices (119886119871)120572

119886119887and

(119888119871)120572120573

119886119887are called coefficients for CPT-odd and CPT-even

Lorentz violation respectively The spacetime indices 120572 120573encode the nature of the broken symmetry for instanceisotropic (direction-independent) Lorentz violation appearswhen only the time components of the coefficients arenonzero while space anisotropy appears when any ofthe other components is nonzero generating direction-dependent effects in the neutrino behavior The breakdownof invariance under rotations is evident due to the presenceof the four-vector 119901120572 = (1 119901) that depends on the neutrinodirection of propagation 119901

The block Hamiltonian describing right-handed antineu-trinos is obtained as the CP conjugate of the neutrinoHamiltonian 120575ℎ

119886119887= CP(120575ℎ

119886119887) which has the same form

as the neutrino Hamiltonian (4) with (119886119877)120572

119886119887

= minus(119886119871)120572lowast

119886119887and

(119888119877)120572120573

119886119887

= (119888119871)120572120573lowast

119886119887 Given the structure of these coefficients

in flavor space it is expected that they will affect neutrinomixing and oscillations Notice however that there existcoefficients that modify the three flavors in the same wayproducing no effects on neutrino oscillations because they areproportional to the identity in flavor space These oscillation-free coefficients and their observable effects are discussed inSections 5 and 6

Signals of the breakdown of Lorentz invariance cor-respond to the anisotropy of spacetime due to preferreddirections set by the coefficients for Lorentz violation that actas fixed background fields Taking advantage of the couplingof these background fields with the neutrino direction ofpropagation 119901 we can search for violations of Lorentzinvariance by making measurements with neutrino beams



119871)120572

119886119887and (119888

119871)120572120573







120575ℎ119886119887= (C)

119886119887+ (A119904)119886119887sin120596oplus119879oplus

+ (A119888)119886119887cos120596oplus119879oplus

+ (B119904)119886119887sin 2120596

oplus119879oplus+ (B119888)119886119887cos 2120596

oplus119879oplus

(5)



119886119887and (119888

119871)120572120573



(119886119871)120572

119886119887997888rarr (119886


1198861198871199011205821


119889 odd

(119888119871)120572120573

119886119887997888rarr (119888


1198861198871199011205821


119889 even(6)







21


angles 12057912 12057913 and 120579






1198941198951198714119864 ≪ 1205872 the



119875]119887rarr ]119886

≃ 11987121003816100381610038161003816120575ℎ119886119887

10038161003816100381610038162

119886 = 119887 (7)









4

3

2

1

00 02 0604 08 1


Even

ts10

18PO

T


2120587119864Δ1198982




119875]119887rarr ]119886

= 119875(0)

]119887rarr ]119886

+ 119875(1)

]119887rarr ]119886


where 119875(0)]119887rarr ]119886


(1)

]119887rarr ]119886


119875(1)

]119887rarr ]119886

2119871= (119875(1)

C )119886119887

+ (119875(1)

A119904

)119886119887

sin120596oplus119879oplus+ (119875(1)

A119888

)119886119887


+ (119875(1)

B119904

)119886119887

sin 2120596oplus119879oplus+ (119875(1)

B119888

)119886119887


(9)



(1)

]119887rarr ]119886







120575ℎ119886119887= 119894radic2(120598

+)120572120572

119886119887

minus 119894radic2(120598+)120572119901120573119892120572120573

119886119887

|p| (10)



119886119887

and 119892120572120573119886119887


119886119887

and 119892120572120573119886119887



119886rarr ]119887



120583could


119886119887

and 119892120572120573119886119887



and 119892120572120573119886119887





50

0

minus50

minus100

minus150

2 4 6 8 10 12

Dat

a pre

dict

ion

05

MeV


(MeV)

sin2212057913 = 0109 wo LV params

E






V] = 1 minus|119898|2

2|p|2+ sum119889119895119898


N119895119898

times ((119886(119889)

of )119895119898minus (119888(119889)

of )119895119898)

(11)


(119889)


0N119895119898







Δ119905 asymp 119871 (1 minus V]) (12)




+

rarr

119897+






0 which can be


sum119889119895119898

119864119889minus2

0119884119895119898(p) [plusmn(119886(119889)of )119895119898 minus (119888

(119889)

of )119895119898] ≲

1

2(119872119872minus 119898119897plusmn)2

(13)





e+Z0

p

p998400

eminusk

k998400






(4)



119889119864

119889119909= minus

119862

8int

1205810

12058110158402

(1205812 minus1198722119885)2

12059710038161003816100381610038161003816120581101584010038161003816100381610038161003816

1205971205810

119902 sdot 1198961199021015840

sdot 1198961015840

119902011989601199021015840011989610158400

1198893

1199011015840

119889Ω1205811015840 (14)


1198961015840 and 120581

1015840







6 Beta Decay


(3)






119889Γ

119889119879= 119862[(Δ119879)

2

minus1

21198982

]] (15)






Δ119879 997888rarr Δ119879 + (119896(3)

C ) + (119896(3)

A119904


+ (119896(3)

A119888


(16)


(3)

of )10 (119886(3)

of )11 and(119886(3)



1198982

] 997888rarr 1198982

= 1198982

] + (119896(2)

C ) + (119896(2)

A119904


+ (119896(2)

A119888


(17)






+




119886exp =119873+minus 119873minus

119873++ 119873minus

(18)



++ the



119873minusminus+ 119873++

(19)


Aexp = (AC) + (AA119904


119888



(3)

of )10 (119886(3)

of )11 and (119886(3)




e

e

nn

N++Nminusminus

p

p

nn




120573120573|


(1198790]12)minus1

= 1198660]100381610038161003816100381610038161198720]10038161003816100381610038161003816

2

100381610038161003816100381610038161003816119892120582120588

120573120573

100381610038161003816100381610038161003816

2

41198772

(21)



120573120573| can be used



7 Conclusions









Acknowledgments


References







































119890



















































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics





119871)120572

119886119887and (119888

119871)120572120573







120575ℎ119886119887= (C)

119886119887+ (A119904)119886119887sin120596oplus119879oplus

+ (A119888)119886119887cos120596oplus119879oplus

+ (B119904)119886119887sin 2120596

oplus119879oplus+ (B119888)119886119887cos 2120596

oplus119879oplus

(5)



119886119887and (119888

119871)120572120573



(119886119871)120572

119886119887997888rarr (119886


1198861198871199011205821


119889 odd

(119888119871)120572120573

119886119887997888rarr (119888


1198861198871199011205821


119889 even(6)







21


angles 12057912 12057913 and 120579






1198941198951198714119864 ≪ 1205872 the



119875]119887rarr ]119886

≃ 11987121003816100381610038161003816120575ℎ119886119887

10038161003816100381610038162

119886 = 119887 (7)









4

3

2

1

00 02 0604 08 1


Even

ts10

18PO

T


2120587119864Δ1198982




119875]119887rarr ]119886

= 119875(0)

]119887rarr ]119886

+ 119875(1)

]119887rarr ]119886


where 119875(0)]119887rarr ]119886


(1)

]119887rarr ]119886


119875(1)

]119887rarr ]119886

2119871= (119875(1)

C )119886119887

+ (119875(1)

A119904

)119886119887

sin120596oplus119879oplus+ (119875(1)

A119888

)119886119887


+ (119875(1)

B119904

)119886119887

sin 2120596oplus119879oplus+ (119875(1)

B119888

)119886119887


(9)



(1)

]119887rarr ]119886







120575ℎ119886119887= 119894radic2(120598

+)120572120572

119886119887

minus 119894radic2(120598+)120572119901120573119892120572120573

119886119887

|p| (10)



119886119887

and 119892120572120573119886119887


119886119887

and 119892120572120573119886119887



119886rarr ]119887



120583could


119886119887

and 119892120572120573119886119887



and 119892120572120573119886119887





50

0

minus50

minus100

minus150

2 4 6 8 10 12

Dat

a pre

dict

ion

05

MeV


(MeV)

sin2212057913 = 0109 wo LV params

E






V] = 1 minus|119898|2

2|p|2+ sum119889119895119898


N119895119898

times ((119886(119889)

of )119895119898minus (119888(119889)

of )119895119898)

(11)


(119889)


0N119895119898







Δ119905 asymp 119871 (1 minus V]) (12)




+

rarr

119897+






0 which can be


sum119889119895119898

119864119889minus2

0119884119895119898(p) [plusmn(119886(119889)of )119895119898 minus (119888

(119889)

of )119895119898] ≲

1

2(119872119872minus 119898119897plusmn)2

(13)





e+Z0

p

p998400

eminusk

k998400






(4)



119889119864

119889119909= minus

119862

8int

1205810

12058110158402

(1205812 minus1198722119885)2

12059710038161003816100381610038161003816120581101584010038161003816100381610038161003816

1205971205810

119902 sdot 1198961199021015840

sdot 1198961015840

119902011989601199021015840011989610158400

1198893

1199011015840

119889Ω1205811015840 (14)


1198961015840 and 120581

1015840







6 Beta Decay


(3)






119889Γ

119889119879= 119862[(Δ119879)

2

minus1

21198982

]] (15)






Δ119879 997888rarr Δ119879 + (119896(3)

C ) + (119896(3)

A119904


+ (119896(3)

A119888


(16)


(3)

of )10 (119886(3)

of )11 and(119886(3)



1198982

] 997888rarr 1198982

= 1198982

] + (119896(2)

C ) + (119896(2)

A119904


+ (119896(2)

A119888


(17)






+




119886exp =119873+minus 119873minus

119873++ 119873minus

(18)



++ the



119873minusminus+ 119873++

(19)


Aexp = (AC) + (AA119904


119888



(3)

of )10 (119886(3)

of )11 and (119886(3)




e

e

nn

N++Nminusminus

p

p

nn




120573120573|


(1198790]12)minus1

= 1198660]100381610038161003816100381610038161198720]10038161003816100381610038161003816

2

100381610038161003816100381610038161003816119892120582120588

120573120573

100381610038161003816100381610038161003816

2

41198772

(21)



120573120573| can be used



7 Conclusions









Acknowledgments


References







































119890



















































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics




4

3

2

1

00 02 0604 08 1


Even

ts10

18PO

T


2120587119864Δ1198982




119875]119887rarr ]119886

= 119875(0)

]119887rarr ]119886

+ 119875(1)

]119887rarr ]119886


where 119875(0)]119887rarr ]119886


(1)

]119887rarr ]119886


119875(1)

]119887rarr ]119886

2119871= (119875(1)

C )119886119887

+ (119875(1)

A119904

)119886119887

sin120596oplus119879oplus+ (119875(1)

A119888

)119886119887


+ (119875(1)

B119904

)119886119887

sin 2120596oplus119879oplus+ (119875(1)

B119888

)119886119887


(9)



(1)

]119887rarr ]119886







120575ℎ119886119887= 119894radic2(120598

+)120572120572

119886119887

minus 119894radic2(120598+)120572119901120573119892120572120573

119886119887

|p| (10)



119886119887

and 119892120572120573119886119887


119886119887

and 119892120572120573119886119887



119886rarr ]119887



120583could


119886119887

and 119892120572120573119886119887



and 119892120572120573119886119887





50

0

minus50

minus100

minus150

2 4 6 8 10 12

Dat

a pre

dict

ion

05

MeV


(MeV)

sin2212057913 = 0109 wo LV params

E






V] = 1 minus|119898|2

2|p|2+ sum119889119895119898


N119895119898

times ((119886(119889)

of )119895119898minus (119888(119889)

of )119895119898)

(11)


(119889)


0N119895119898







Δ119905 asymp 119871 (1 minus V]) (12)




+

rarr

119897+






0 which can be


sum119889119895119898

119864119889minus2

0119884119895119898(p) [plusmn(119886(119889)of )119895119898 minus (119888

(119889)

of )119895119898] ≲

1

2(119872119872minus 119898119897plusmn)2

(13)





e+Z0

p

p998400

eminusk

k998400






(4)



119889119864

119889119909= minus

119862

8int

1205810

12058110158402

(1205812 minus1198722119885)2

12059710038161003816100381610038161003816120581101584010038161003816100381610038161003816

1205971205810

119902 sdot 1198961199021015840

sdot 1198961015840

119902011989601199021015840011989610158400

1198893

1199011015840

119889Ω1205811015840 (14)


1198961015840 and 120581

1015840







6 Beta Decay


(3)






119889Γ

119889119879= 119862[(Δ119879)

2

minus1

21198982

]] (15)






Δ119879 997888rarr Δ119879 + (119896(3)

C ) + (119896(3)

A119904


+ (119896(3)

A119888


(16)


(3)

of )10 (119886(3)

of )11 and(119886(3)



1198982

] 997888rarr 1198982

= 1198982

] + (119896(2)

C ) + (119896(2)

A119904


+ (119896(2)

A119888


(17)






+




119886exp =119873+minus 119873minus

119873++ 119873minus

(18)



++ the



119873minusminus+ 119873++

(19)


Aexp = (AC) + (AA119904


119888



(3)

of )10 (119886(3)

of )11 and (119886(3)




e

e

nn

N++Nminusminus

p

p

nn




120573120573|


(1198790]12)minus1

= 1198660]100381610038161003816100381610038161198720]10038161003816100381610038161003816

2

100381610038161003816100381610038161003816119892120582120588

120573120573

100381610038161003816100381610038161003816

2

41198772

(21)



120573120573| can be used



7 Conclusions









Acknowledgments


References







































119890



















































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics




50

0

minus50

minus100

minus150

2 4 6 8 10 12

Dat

a pre

dict

ion

05

MeV


(MeV)

sin2212057913 = 0109 wo LV params

E






V] = 1 minus|119898|2

2|p|2+ sum119889119895119898


N119895119898

times ((119886(119889)

of )119895119898minus (119888(119889)

of )119895119898)

(11)


(119889)


0N119895119898







Δ119905 asymp 119871 (1 minus V]) (12)




+

rarr

119897+






0 which can be


sum119889119895119898

119864119889minus2

0119884119895119898(p) [plusmn(119886(119889)of )119895119898 minus (119888

(119889)

of )119895119898] ≲

1

2(119872119872minus 119898119897plusmn)2

(13)





e+Z0

p

p998400

eminusk

k998400






(4)



119889119864

119889119909= minus

119862

8int

1205810

12058110158402

(1205812 minus1198722119885)2

12059710038161003816100381610038161003816120581101584010038161003816100381610038161003816

1205971205810

119902 sdot 1198961199021015840

sdot 1198961015840

119902011989601199021015840011989610158400

1198893

1199011015840

119889Ω1205811015840 (14)


1198961015840 and 120581

1015840







6 Beta Decay


(3)






119889Γ

119889119879= 119862[(Δ119879)

2

minus1

21198982

]] (15)






Δ119879 997888rarr Δ119879 + (119896(3)

C ) + (119896(3)

A119904


+ (119896(3)

A119888


(16)


(3)

of )10 (119886(3)

of )11 and(119886(3)



1198982

] 997888rarr 1198982

= 1198982

] + (119896(2)

C ) + (119896(2)

A119904


+ (119896(2)

A119888


(17)






+




119886exp =119873+minus 119873minus

119873++ 119873minus

(18)



++ the



119873minusminus+ 119873++

(19)


Aexp = (AC) + (AA119904


119888



(3)

of )10 (119886(3)

of )11 and (119886(3)




e

e

nn

N++Nminusminus

p

p

nn




120573120573|


(1198790]12)minus1

= 1198660]100381610038161003816100381610038161198720]10038161003816100381610038161003816

2

100381610038161003816100381610038161003816119892120582120588

120573120573

100381610038161003816100381610038161003816

2

41198772

(21)



120573120573| can be used



7 Conclusions









Acknowledgments


References







































119890



















































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics




e+Z0

p

p998400

eminusk

k998400






(4)



119889119864

119889119909= minus

119862

8int

1205810

12058110158402

(1205812 minus1198722119885)2

12059710038161003816100381610038161003816120581101584010038161003816100381610038161003816

1205971205810

119902 sdot 1198961199021015840

sdot 1198961015840

119902011989601199021015840011989610158400

1198893

1199011015840

119889Ω1205811015840 (14)


1198961015840 and 120581

1015840







6 Beta Decay


(3)






119889Γ

119889119879= 119862[(Δ119879)

2

minus1

21198982

]] (15)






Δ119879 997888rarr Δ119879 + (119896(3)

C ) + (119896(3)

A119904


+ (119896(3)

A119888


(16)


(3)

of )10 (119886(3)

of )11 and(119886(3)



1198982

] 997888rarr 1198982

= 1198982

] + (119896(2)

C ) + (119896(2)

A119904


+ (119896(2)

A119888


(17)






+




119886exp =119873+minus 119873minus

119873++ 119873minus

(18)



++ the



119873minusminus+ 119873++

(19)


Aexp = (AC) + (AA119904


119888



(3)

of )10 (119886(3)

of )11 and (119886(3)




e

e

nn

N++Nminusminus

p

p

nn




120573120573|


(1198790]12)minus1

= 1198660]100381610038161003816100381610038161198720]10038161003816100381610038161003816

2

100381610038161003816100381610038161003816119892120582120588

120573120573

100381610038161003816100381610038161003816

2

41198772

(21)



120573120573| can be used



7 Conclusions









Acknowledgments


References







































119890



















































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics





119889Γ

119889119879= 119862[(Δ119879)

2

minus1

21198982

]] (15)






Δ119879 997888rarr Δ119879 + (119896(3)

C ) + (119896(3)

A119904


+ (119896(3)

A119888


(16)


(3)

of )10 (119886(3)

of )11 and(119886(3)



1198982

] 997888rarr 1198982

= 1198982

] + (119896(2)

C ) + (119896(2)

A119904


+ (119896(2)

A119888


(17)






+




119886exp =119873+minus 119873minus

119873++ 119873minus

(18)



++ the



119873minusminus+ 119873++

(19)


Aexp = (AC) + (AA119904


119888



(3)

of )10 (119886(3)

of )11 and (119886(3)




e

e

nn

N++Nminusminus

p

p

nn




120573120573|


(1198790]12)minus1

= 1198660]100381610038161003816100381610038161198720]10038161003816100381610038161003816

2

100381610038161003816100381610038161003816119892120582120588

120573120573

100381610038161003816100381610038161003816

2

41198772

(21)



120573120573| can be used



7 Conclusions









Acknowledgments


References







































119890



















































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics




e

e

nn

N++Nminusminus

p

p

nn




120573120573|


(1198790]12)minus1

= 1198660]100381610038161003816100381610038161198720]10038161003816100381610038161003816

2

100381610038161003816100381610038161003816119892120582120588

120573120573

100381610038161003816100381610038161003816

2

41198772

(21)



120573120573| can be used



7 Conclusions









Acknowledgments


References







































119890



















































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics






Acknowledgments


References







































119890



















































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics






119890



















































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics











































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics



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