CPT Violating Decoherence and LSND: a possible window to Planck scale Physics

0.34 MB·English

by  Gabriela Barenboim

CPT Violating Decoherence and LSND: a possible window to Planck scale Physics
Gabriela Barenboima and Nick E. Mavromatosb
aDepartamento de F´ısica Te´orica and IFIC, Centro Mixto,
Universidad de Valencia-CSIC, E-46100, Burjassot, Valencia, Spain.
bKing’s College London, University of London, Department of Physics, Strand WC2R 2LS, London, U.K.
(Dated: February 1, 2008)
DecoherencehasthepotentialtoexplainallexistingneutrinodataincludingLSNDresults,with-
outenlargingtheneutrinosector. ThisparticularformofCPTviolationcanpreservetheequalityof
massesandmixinganglesbetweenparticleandantiparticle sectors,andstillprovideseizablediffer-
encesintheoscillationpatterns. Asimplifiedminimalmodelofdecoherencecanexplaintheexisting
neutrinodata as well as thestandard three oscillation scenario, while making dramatic predictions
for the upcoming experiments. Such a model can easily accomodate the LSND result but cannot
fit the spectral distortions seen by KamLAND. Some comments on the order of the decoherence
5
parameters in connection with theoretically expected values from some models of quantum-gravity
0
aregiven. Inparticular, thequantumgravity decoherenceasaprimary origin of theneutrinomass
0
differences scenario is explored, and even a speculative link between the neutrino mass-difference
2
scale and thedark energy density component of theUniversetoday is drawn.

n
a
J INTRODUCTION: THE LSND PUZZLE AND ogy of QG” may be easily proven to be wishful think-

4 ABANDONING OF SYMMETRIES ing, if all the symmetries and properties that character-
2 ize the current version of what is called particle physics

phenomenology are still valid in a full quantum the-
4 Although the Special Theoryof Relativity, a theoryof
ory of gravity. This is mainly due to the fact that the
v flat space time physics based on Lorentz symmetry, is
4 dimension-full coupling constant of gravity, the Newton
verywelltested,andinfactnextyearitcelebratesacen-
1 constant G = 1/M2, appears as a very strong sup-
tury of enormous success, having passed very stringent N P
0 pression factor of any physical observable that could be
4 experimental precision tests, a quantum theory of Grav-
associatedwithpredictions ofquantumgravity,ina the-
0 ity, that is a consistent quantized version of Einstein’s
oryinwhichLorentzinvarianceandthelawsofquantum
4 General Relativity, a dynamical model for curved space
0 mechanics (or better quantum field theory), such as uni-
times, still eludes us. The main reason for this is the
/ tary temporal evolution and locality of interactions, still
h lack of any concrete observationalevidence on the struc-
hold.
p ture of space time at the characteristicscale of quantum
-
p gravity (QG), the (four-dimensional) Planck mass scale In recent years, however, more and more physicists
e M = 1019 GeV. The situation concerning QG is to be contemplate the idea that such laws, which are charac-
P
h contrasted with that characterizing its classical counter- teristic of a flat space time quantum field theory, may
:
v part,GeneralRelativity,andthequantumversionofSpe- not be valid in a full theory of curved space times. For
i cialRelativisticfieldtheories,bothofwhichcanbetested instance,theCPTtheorem,oneofthemostprofoundre-
X
to enormous precision to date. For instance, the perihe- sults of quantum field theory [1], which is a consequence
r
a lion precessionof planets such as mercury,as well as the of Lorentz invariance, locality, as well as quantum me-
deflection of light by the sun, made the General Theory chanics (specifically unitary evolution of a system), may
ofRelativityaninstantsuccess,almostimmediatelyafter notcharacterizeaquantumgravitytheory. The possibil-
it was proposed in 1915. Similarly, the modern versions ity of a violation of CPT invariance by quantum gravity
of relativisticquantum field theories,including the foun- has been raisedin a number of theoreticalmodels ofQG
dations of the standard model of particle physics, which thatgobeyondconventionallocalquantumfieldtheoretic
have been tested extremely well up to now, leave little treatments of gravity [2, 4, 5, 6].
doubt on the validity of Special Relativity as a quantum
In a phenomenological, rather model-independent set-
theory of flat space time, that is in the absence of gravi-
ting, CPT Violation has been invoked recently [8] in
tational effects.
an attempt to explain within a three generation sce-
One would hope that, like any other successful physi- nario, without the introduction of sterile neutrinos, a
caltheory,aphysicallyrelevantquantumtheoryofgrav- puzzling experimentalresultin neutrino physics, namely
ity should lead to experimental predictions that should the claims from LSND Collaboration [7] on evidence of
be testable in the foreseeable future. In the QG case, oscillations in the antineutrino sector ν ν , due to
µ e
→
however, such predictions may be subtle, due to the observed excess of ν events, but lack of evidence (de-
e
extremely weak nature of the gravitational interaction pending though on interpretation) for oscillations in the
as compared with the rest of the known interactions corresponding neutrino sector ν ν . In order to ex-
µ e
→
in nature. Indeed, any hope for a true “phenomenol- plaintheseresultswithintheconventionaloscillationsce-
2
nario, one should invoke different neutrino mass differ- ticles.
encesintheparticleandantiparticlesectorsofthemodel. It is the point of this work to claim that decoherence
This would signify CPT violation. Specifically, from so- scenaria combined with conventional oscillations, led as
larandatmosphericneutrinodata,themasssquareddif- usual by the mass differences between the mass eigen-
ferences that account for the observed oscillations are of states, can account for all the available neutrino data,
order∆m2 10 5eV2 and10 3 eV2,respectively,while including the LSND result, within a three generation
− −
∼
the LSND result would require ∆m2 10 1 eV2, which model. The LSND result would then evidence CPT vi-
−
∼
would explain the suppressed signal in the neutrino sec- olation in the sense of different decohering interactions
tor and the strong signal in the antineutrino one. To betweenparticle andantiparticle sectors,while the mass
reconcile, therefore, the LSND result with the rest of differencesbetweenthetwosectorsremainthesame. For
the available neutrino data, in conventional oscillation detailedreviewsofdecoherenceintwo-levelsystems,with
scenaria, one would have to invoke CPT violating mass emphasis on phenomenology including neutrinos, we re-
differences between neutrinos and antineutrinos of order fer the reader to the literature [11]. In what follows we
m2 m2 10 1 eV2. Although such a CPT viola- shall only expose the very basic features, which we shall
| ν¯ − ν| ∼ −
tion may look rather drastic, the most stringent limit make use of in this work in order to perform our phe-
we have on the CPT symmetry today, the one com- nomenological analysis. It is important to stress that in
ing from the neutral kaon system, if reinterpreted as a decoherence scenaria CPT symmetry is violated in its
limit on the possible differences in mass squared, gives: strong form, in the sense of the CPT operator not being
m2(K0) m2(K¯0) <0.25eV2 showingthatsuchdiffer- well defined. This is an important issue which we now
| − |
ences are just barely explored even in the kaon system. come to discuss briefly.
Subsequent global analyses of neutrino data [9], how-
ever, seem to disfavor two and three generationCPT vi-
olating scenaria [6, 8] thereby leaving the CPT violat- QUANTUM GRAVITY AND DECOHERENCE:
ing scenaria involving sterile neutrinos as the only sur- BRIEF REVIEW
viving possibility for the explanation of the LSND re-
sults. Specifically, it has been argued in [10], that al- A characteristic example of such a violation occurs in
lowing for CPT violation, the mixing matrix elements quantumgravitymodelsthatinvolvesingularspace-time
with the forth generationbetween neutrino and antineu- configurations,integratedoverin a path integralformal-
trino sectors need no longer be the same, thereby evad- ism,whicharesuchthattheaxiomsofquantumfieldthe-
ing stringent experimental constraints that killed three- ory, as well as conventional quantum mechanical behav-
generation models. ior,cannotbemaintained[3]. Suchconfigurationsconsist
Is, however, CPT violation evidenced only as an in- of wormholes,microscopic (Planck size) black holes, and
equality between neutrino-antineutrino masses the only other topologically non-trivial solitonic objects, such as
way a violation of this symmetry can manifest itself in geons [12], etc. Collectively, we may call such config-
nature? Such a question becomes extremely relevant for urations space time foam, a terminology given by J.A.
thecaseofLSND,becauseitispossiblethatothermech- Wheeler [13] who first conceived the idea that the struc-
anismsleadingtoCPTviolationexist,unrelated,inprin- ture of the quantum space time at Planck scales (10 35
−
ciple,tomassdifferencesbetweenparticlesandantiparti- m) may indeed be fuzzy.
cles. SuchadditionalmechanismsforCPTviolationmay It has been argued that, as result, a mixed state de-
well be capable of explaining the LSND results within a scription must be used (QG-induced decoherence) [2, 3],
three generationscenario without invoking a sterile neu- given that such objects cannot be accessible to low-
trino (a scenario which, on the other hand, is getting energy observers, and as such must be traced over in
totally excludedas new experimentaldata become avail- an effective field theory context. For the case of micro-
able). It is therefore necessary to explore whether alter- scopic black holes this can be readily understood by the
nativewaysexisttoaccountfortheLSNDresultwithout loss of information across microscopic event horizons, as
invoking extra (sterile) neutrino states. acorollaryofwhichithasbeenargued[14],alreadyback
Asweshallargueinthisarticle,quantumdecoherence in 1978, that CPT invariance in its strong form must be
maybethekeytoanswerthisquestion. Indeed,quantum abandoned in a foamy quantum gravity theory. Such a
decoherenceinmatterpropagationoccurswhenthemat- breakdownofCPTsymmetryisafundamentalone,and,
tersubsysteminteractswithan‘environment’,according in particular, implies that a proper CPT operator may
to the rules of open-system quantum mechanics. At a be ill defined in such QG decoherence cases.
fundamental level, such a decoherence may be the result The main reason behind such a failure is the non fac-
of propagation of matter in quantum gravity space-time torisability of the super-scattering matrix, $, connecting
backgrounds with ‘fuzzy’ properties, which may be re- asymptotic“in” and“out”density matrices[3]. The lat-
sponsible for violation of CPT in a way not necessarily ter describe mixed quantum states due to the existence
relatedtomassdifferencesbetweenparticlesandantipar- ofaquantum-gravitational“environment”ofmicroscopic
3
singular space-time configurations. The presence of the chanical evolution of a matter system under considera-
environment leads to decoherence, like in any other case tion, as a result of the interaction with the QG environ-
ofanopenquantum-mechanicalsystem,implyinganevo- ment [2]. Such an evolution will result in many observ-
lution from an initial pure quantum mechanical state able consequences,whichmay be testable inthe nearfu-
φ > to a mixed state ρ = Trψ >< ψ (the trace is turewithaprecisionthatinsomecasescanreachPlanck
ψ
| | |
overunobserveddegreesoffreedom,‘lost’intheenviron- scalesensitivity. Historically,thefirstprobeofsuchnon-
ment). This in turn implies [14] the non invertibility of quantummechanicaldecoherenceandCPTviolatingbe-
the $-matrix, as a quantum mechanical operator acting havior (in the sense of the strong form violation of [14])
ondensitymatrices. The associatedCPTnoninvariance is the neutral kaon system [2, 4], in both single-particle
is then linkedto the factthatif CPTwere awelldefined experiments,suchastheCPLEAR[17]andKTeVexper-
operator, stemming from the invariance of the system iments [18], as well as multi-particle situations, such as
under its action, then $ would have been invertible, in kaon (φ) factories [19]. Other neutral mesons, such as B
obvious contradiction with the information loss and de- mesons [20], can also be used as sensitive probes of QG
coherence inherent to the problem. decoherence.
As emphasized recently in [15] this would also imply Insingle-particleneutralmesonexperiments onelooks
that the concept of antiparticle, may be ill defined, with at time profiles of asymmetries [4]. The mass difference
interesting testable consequences in multiparticle situa- betweenthe energyeigenstatesK ,K isresponsiblefor
L S
tions,suchasthose encounteredin(EPR-likeentangled) the ‘bulk’ behavior of the asymmetry, for instance lead-
meson factories. Such CPT violation is distinct from ingto anoscillatorypatterninsomeofthe asymmetries,
other violations of this symmetry as a result of Lorentz such as A , decoherence on the other hand can be re-
2π
symmetry breakdown in the so-called Standard Model sponsible for a slight distortion of such a behavior [4].
extension(SME)[5],orlocalityviolations[6],whichhave The non observation of such distortions results in strin-
been considered in the recent literature as other exper- gentbounds onthe decoherenceparameters. Mathemat-
imentally testable approaches to quantum gravity. It ically, one does not have to have a detailed knowledge
shouldbenotedthattheselatterviolationsofsymmetries of the dynamics of quantum gravity in order to perform
mayormaynotcoexistwithQGdecoherence,giventhat experimental low-energy tests of decoherence. This can
Lorentzinvarianceisnotnecessarilyinconsistentwithde- be achieved following the so-called Lindblad or mathe-
coherence [16]. maticalsemi-groupsapproachtodecoherence[21],which
Some caution should be paid regarding CPT Viola- is a very efficient way of studying open systems in quan-
tionthroughdecoherence. As emphasizedin [14], froma tum mechanics. The time irreversibility in the evolution
formal view point, the non-invertibility of the $-matrix, of such semigroups, which is linked to decoherence, is
which implies a strong violation of CPT in the sense de- inherent in the mathematical property of the lack of an
scribed above, does not preclude a softer form of CPT inverse in the semigroup. This approach has been fol-
invariance,inthesensethatanystrongformofCPTvio- lowed for the study of quantum-gravity decoherence in
lationdoes not necessarilyhave to show up in any single the case of neutral kaons in [2, 4].
experimental measurement. This implies that, despite In the parameterization of [2] for the decoherence ef-
thegeneralevolutionofpuretomixedstates,itmaystill fects, using three decoherence parameters with dimen-
be possible in the laboratory to ensure that the system sions of energy, α,β,γ, one finds the following bounds
evolvesfromaninitialpure stateψ to a singlefinalstate from the CPLEAR experiment [17]:
φ, and that the weak form of CPT invariance is mani-
α<4 10 17 GeV, β <2.3 10 19 GeV,
fested through the equality of probabilities between the × − | | × −
states ψ, φ: γ <3.7 10−21 GeV (2)
×
P(ψ φ)=P(θ 1φ θψ) , (1) where the positivity of ρ, required by the fact that its
−
→ → diagonal elements express probability densities, implies
with θ a CPT operator acting on the subspace of pure α,γ 0, and αγ β2.
≥ ≥
state vectorsinthe laboratory,suchthatfordensity ma- The Lindblad approach to decoherence does not re-
trices Θρ = θρθ†, θ† = θ−1, and $† = Θ−1$Θ−1, but quire any detailed knowledge of the environment, apart
−
$ † =$ −1, unless full CPT invariance holds, which is fromenergyconservation,entropyincreaseandcomplete
6
the case of no decoherence. If this is the case, then the positivityofthe(reduced)densitymatrixρ(t)ofthesub-
decoherence-induced CPT violation will not show up in system under consideration. The basic evolution equa-
any experimental measurement. It is therefore impor- tionforthe(reduced)densitymatrixofthesubsystemin
tanttocheckwhetheritispossibletotestexperimentally the Lindblad approach is linear in ρ(t) and reads:
which case is realized in nature. Fortunately this can be
∂ρ 1
done in a clear way. ∂t =−i[Heff,ρ]+ 2 [bj,ρ(t)b†j]+[bjρ(t),b†j] ,
Indeed, the main consequence of the quantum-gravity
Xj (cid:16) (cid:17)
induced decoherence is the modified non-quantum me- (3)
4
whereH istheeffectiveHamiltonianofthe subsystem, onal, with only one non vanishing entry occupied by the
eff
and the operators b represent the interaction with the decoherence parameter γ >0 [25].
j
environment,andareassumedbounded. Noticethatthe
Lindblad part cannot be written as a commutator (of a
Hamiltonian function) with ρ. Environmental contribu- TWO GENERATION NEUTRINO MODELS AND
tionsthatcanbecastinHamiltonianevolution(commu- DECOHERENCE: BRIEF REVIEW
tator form) are absorbed in H .
eff
It must be noted at this stage that the requirement If the above formalismis applied to two-levelneutrino
of complete positivity, which essentially pertains to the oscillationphysics,inanattempttodiscusstheinfluence
positivity of the map ρ(t) as the time evolvesin the case of a quantum-gravity environment on neutrino oscilla-
ofmanyparticlesituations,suchasmesonfactories(two- tions, then there are two possible forms of the matrix :
L
kaonstates(φ-factory),ortwo-B-mesonstatesetc.),may =Diag(0,0, γ,0)inthecasewhereenergyandlepton
L −
not be an exact property of quantum gravity, whose in- number are conserved, and = Diag(0,0,0, γ) in the
L −
teractionswiththeenvironmentcouldbenon linear[22]. casethatenergyandleptonnumberareviolatedbyquan-
Nevertheless, complete positivity leads to a convenient tum gravity, but flavor is conserved. In [23] the energy
and simple parametrization, and it has been assumed conservingcasehasbeenconsidered,withthe conclusion
so far in many phenomenological analyses of quantum thatpuredecoherence,i.e. inducedoscillationsintheab-
gravity decoherence in generic two state systems, such senceofneutrinomassdifferences,wasincompatiblewith
as two-flavor neutrino systems [23, 24]. In fact, in the the observed phenomenology of neutrino oscillations.
parametrization of [2], the imposition of complete posi- Thisisinagreementwiththekaoncase[4],whereagain
tivity leads to α=β =0, leaving only γ >0 as the only theobservedbulkbehaviorofthetimeprofilesofthevar-
decoherence parameter in the two-state system [25]. iouskaonasymmetries,aswellasthe observedCPviola-
Formally, the bounded Lindblad operators of an N- tion,cannotbeexplainedsolelybyquantumdecoherence,
level quantum mechanical system can be expanded in a the latter providing only distortions to the ‘bulk’ behav-
basis of matrices satisfying standard commutation rela- ior. Including the mass-induced oscillations, combined
tions of Lie groups. For a two-level system [2, 4] such with decoherence,andcomparing with atmosphericneu-
matrices are the SU(2) generators (Pauli matrices) plus trinodata,theauthorsof[23]managedtoobtainbounds
the 2 2 identity operator, while for a three level sys- for the decoherence parameter γ in the presence of mass
×
tem [26], which will be relevant for our purposes in this differences (oscillations); as in the kaon case [4], such
article,thebasiscomprisesoftheeightGell-MannSU(3) bounds are obtained merely by the non-observation of
matrices Λ , i = 1,...8 plus the 3 3 identity matrix the slight distortions to the mass-inducedneutrino oscil-
i
×
I . lations that the presence of decoherence would cause.
3x3
Let ,µ=0,...8(3)beasetofSU(3)(SU(2))gener- The so-obtained limits for the decoherence Lindblad
µ
J
atorsforathree(two)-levelsystem;then,onemayexpand parameterγ in[23]canbesummarizedasfollows: adopt-
the various terms in (3) in terms of to arrive at the ing the parametrization γ = γ (E/GeV)n, they consid-
µ 0
J
generic form: eredthreecases: (a)n=0,inwhichtheexperimentaldata
imply γ < 10 23 GeV, (b) n=2, which is the case ex-
0 −
∂ρµ pected in some rather optimistic (from the prospect of
= h ρ f + ρ ,
i j ijµ µν µ
∂t L experimental detection) theoretical models of quantum
ij ν
X X gravity [4, 22], for which the most stringent bound de-
µ,ν =0,...N2 1, i,j =1,...N2 1 (4)
− − rived is γ0 < 0.9 10−27, and (c) n=-1, which is the
×
case in which the decoherence parameter exhibits an en-
with N = 3(2) for three(two) level systems, and f
ijk ergydependencethatmimicstheconventionaloscillation
the structure constants of the SU(N) group. The re-
scenario, for which γ <2 10 21 GeV2.
quirement for entropy increase implies the hermiticity 0 × −
Itisworthnoticing,thatasfarascase(b)isconcerned,
of the Lindblad operators b , as well as the fact that
i
atleastbydimensionalconsiderations,oneistemptedto
the matrix of the the non-Hamiltonian part of the
L assume theoretical values of the decoherence parameter
evolution has the properties that = = 0,
= 1 b(n)b(n)f f , withL0tµhe notLatµi0on b which are of the form E2/MQG, with MQG the effec-
Lij 2 k,ℓ,m m k imk ℓkj j ≡ tive quantum gravity scale ‘felt’ by the neutrinos. The
µbµ(j)JPµ. above bounds indicate a value MQG ∼ 1027 GeV for
Inthe two-levelcaseof[2]thedecoherencematrix E = (few GeV) which are typical in atmospheric neu-
µν
P L O
is parametrized by a 4 4 matrix, whose non vanishing trino experiments. This is much higher than the theo-
×
entriesareoccupiedbythethreeparameterswiththedi- reticallyexpected(onnaturalnessgrounds)Planckvalue
mensionsofenergyα,β,γ withthe propertiesmentioned M 1019 GeV. Thisis to be contrastedwiththe case
QG
∼
above. If the requirement of a completely positive map of kaons [2], where, as mentioned above, the bound on
ρ(t) is imposed, then the 4 4 matrix becomes diag- thedecoherenceparameterγ <10 21GeVobtainedfrom
−
× L
5
CPLEAR[17],wasfoundtobemuchclosertothePlanck [ ]=Diag(0, γ , γ , γ , γ , γ , γ , γ , γ )
µν 1 2 3 4 5 6 7 8
L − − − − − − − −
scale,the naturalQGscale. Onthe otherhand, the case indirectanalogywiththetwo-levelcaseofcompletepos-
(c), with aninverse energy E dependence, is purely phe- itivity[23,25]. Aswehavementionedalready,thereisno
nomenologicalanddoesnotstemdirectly fromaspecific strong physical motivation behind such restricted forms
modelofdecoherence. Theboundonγ <10 23 (GeV)2 of decoherence. This assumption, however, leads to the
0 −
obtained in that case is of the order of the quantity simplest possible decoherence models, and, for our phe-
∆m2/E for atmospheric neutrino energies, and square nomenological purposes in this work, we will assume the
mass differences of order 10 5 eV2 which is assumed to aboveform,whichwewillusetofitalltheavailableneu-
−
be the right order for solar neutrino oscillations in the trino data. It must be clear to the reader though, that
conventional oscillation scenario. suchasimplification,ifproventobesuccessful(which,as
Unfortunately, the analysis of the data presented in weshallarguebelow,is the casehere),justaddsmorein
[23] within a two-generation scenario did not show any favor of decoherence models, given the restricted num-
concrete evidence for such a type-(c) decoherence, even ber of available parameters for the fit in this case. In
after inclusion [27] of the first K2K data [28], in com- fact, any other non-minimal scenario will have it easier
bination with Super Kamiokande (SK) data [29]. This to accommodate data because it will have more degrees
prompted these authors to draw the pessimistic conclu- of freedom available for such a purpose.
sion that the extension of the neutrino oscillation plus Specifically we shall look at transition probabili-
decoherence scenario to the fully-fledged case of three ties [26]:
neutrinogenerationswasnotworthy. Addingtothispes-
P(ν ν )=Tr[ρα(t)ρβ]=
simism,there werealsosometheoreticalestimatesofthe α β
→
gdreacvoihteyreinncdeucpeadradmeectoehrerienncsepecfoifircgemnoedrieclstwofo-qleuvaenltusyms-- 31 + 21 eλktDikDk−j1ραj(0)ρβi (5)
i,k,j
tems [30], accordingto whichγ (∆m2)2/E2M , with X
0 P
∼
∆m2 the mass-squared difference between, say, the two where α,β = e,µ,τ stand for the three neutrino flavors,
neutrinos(orkaonsetc). ForMP 1019GeV,thePlanck andLatin indices run over1,...8. The quantities λk are
∼
scale, this is beyond any prospects for experimental de- the eigenvalues of the matrix appearing in the evolu-
M
tection in oscillation experiments in the foreseeable fu- tion (4), after taking into account probability conserva-
ture. However, as remarked in [22], such estimates are tion, which decouples ρ (t)= 2/3, leaving the remain-
0
specifictothe modelconsideredin[30],andindeedthere ing equations in the form: ∂ρ /∂t = ρ . The
pk jMkj j
are concrete examples of theoretical (stringy) models of matrices are the matrices that diagonalize [21].
ij
D P M
quantum-gravity-induceddecoherencewheretheyarenot Explicit forms of these matrices, the eigenvalues λ , and
k
applicable. We shall discuss such issues later on in our consequentlythetransitionprobabilities(5),aregivenin
article. [26].
Theimportantpointtostressisthat,ingenericmodels
ofoscillationplusdecoherence,theeigenvaluesλ depend
k
EXTENSION TO THREE GENERATIONS: on both the decoherence parameters γi and the mass
COMBINING OSCILLATIONS WITH differences ∆m2. For instance, λ = 1[ (γ + γ )
DECOHERENCE ij 1 2 − 1 2 −
(γ γ )2 4∆2 ],withthe notation∆ ∆m2/2p,
2− 1 − 12 ij ≡ ij
i,j = 1,2,3. Note that, to leading order in the (small)
p
Pessimistic thoughts have also appeared in the work squared-massdifferences, one may replace p by the total
of [26], which considered the extension of the com- neutrino energy E, and this will be understood in what
pletely positive decoherence scenario to the standard follows. For detailed expressions of the rest of the pa-
three-generation neutrino oscillations case. The above- rameters we refer the reader to [26]. However, we note,
describedthree-stateLindbladproblemhasbeenadopted for future use, that it is a generic feature of the λ to
k
for the discussion of this case. The relativistic neutrino depend on the quantities Ω which are given by
ij
Hamiltonian H p2 + m2/2p, with m the neutrino
eff
∼
mass, has been used as the Hamiltonian of the subsys- Ω = (γ γ )2 4∆2
tem in the evolution of eq.(3). 12 2− 1 − 12
q
In terms of the generators Jµ, µ = 0,...8 of Ω13 = (γ5−γ4)2−4∆213 (6)
the SU(3) group, H can be expanded as [26]:
eff q
Heff = 21p 2/3 6p2+ 3i=1m2i J0 + 21p(∆m212)J3 + Ω23 = (γ7−γ6)2−4∆223
2√13p ∆m21p3+∆m(cid:16)223 J8P, with (cid:17)the obvious notation Fromthe aboveexpresqsionsforthe eigenvaluesλk, itbe-
∆m2 =m2 m2, i,j =1,2,3. comes clear that, when decoherence and oscillations are
ij(cid:0) i − j (cid:1)
The analysis of [26] assumed ad hoc a diagonal presentsimultaneously,oneshoulddistinguishtwocases,
form for the 9 9 decoherence matrix in (4): according to the relative magnitudes of ∆ and ∆γ
ij kl
× L ≡
6
γ γ: (i) 2∆ ∆γ , and (ii) 2∆ < ∆γ . In probabilitiesinbothsectorsareequal,i.e. P(ν ν )=
k l ij kℓ ij kℓ α α
− | |≥| | | | | | →
theformercase,theprobabilities(5)containtrigonomet- P(ν ν ). From (5) we have in this case [26]:
α α
→
ric (sine and cosine) functions, whilst in the latter they
1 1 1
exhibit hyperbolic sin and cosine dependence. P =P + e γ3t+ e γ8t (7)
νe→νe νµ→νµ ≃ 3 2 − 6 −
Assuming mixing between the flavors, amounts to ex-
pressing neutrino flavor eigenstates ν >, α = e,µ,τ in From the CHOOZ experiment [31], for which L/E
α
t(eurnmitsaroyf)mmaastsriexigUen:stνates>|=νi >3, i|U= 1ν,2,>3.thTrhouisghima- 1K023K/3exmp/eMrimeVen,tw[e28h]awvieththLa/tEhPν¯e2→50ν¯e/i1.3≃ k1m, /wGheilVe htha∼es
| α i=1 α∗i| i ∼
pexlipersetshseadt itnhetedremnssitoyf mmaastsrixeigoPefnastflaatevsorass:taρtαe ρ=α νcan>b<e ocobnsetrravdedicetivnegnttshceomthpeaotriebtliecawlipthrehdPicνµti→onνµsi(≃7)0o.f7,ptuhreeredbey-
α
|
ν = U U ν >< ν . From this we can deter- coherence.
α| i,j α∗i αj| i j|
mineρα =2Tr(ρα ),aquantityneededtocalculatethe However, this conclusion is based on the fact that in
transitµiPon probabiJlitµies (5). the antineutrino sector the decoherence matrix is the
The important comment we would like to raise at this same as that in the neutrino sector. In generalthis need
pointisthat,whenconsideringtheaboveprobabilitiesin not be the case, in view of CPT violation, which could
the antineutrino sector, the respective decoherence pa- imply a different interaction of the antiparticle with the
rameters γ¯ in general may be different from the corre- gravitationalenvironmentascomparedwiththeparticle.
i
sponding ones in the neutrino sector, as a result of the In our tests we take into account this possibility, but
strongformofCPTviolation. Infact,asweshalldiscuss as can be seen from the figure, and will be evident later
next, this will be crucial for accommodating the LSND on, once the analysis will be explained in some depth,
result without conflicting with the rest of the available pure decoherence can be excluded also in this case, as it
neutrino data. This feature is totally unrelated to mass is clearly incompatible with the totality of the available
differences between flavors. data.
In order to check our model, we have performed a χ2
comparison(asopposedtoaχ2 fit)toSuperKamiokande
COMBINING DECOHERENCE, OSCILLATIONS sub-GeV and multi GeV data (the forty data points
AND THE LSND RESULT that are shown in the figures), CHOOZ data (15 data
points) LSND (1 datum) and KamLAND spectral infor-
In[26]apessimisticconclusionwasdrawnonthe“clear mation [32] (sampled in 13 bins), for a sample point in
incompatibility between neutrino data and theoretical thevastparameterspaceofourextremelysimplifiedver-
expectations”, as followed by their qualitative tests for sion of decoherence models. Let us emphasize that we
decoherence. It is a key feature of our present work to have not performed a χ2-fit and therefore the point we
point out that we do not share at all this point of view. are selecting (by “eye” and not by χ) is not optimized
In fact, as we shall demonstrate below, if one takes into to give the best fit to the existing data. Instead, it must
account all the available neutrino data, including the fi- beregardedasoneamongthemanyequallygoodsonsin
nal LSND results [7], which the authors of [26] did not this family of solutions, being extremely possible to find
do,andallowsfortheabovementionedCPTviolationin a better fitting one through a complete (and highly time
thedecoherencesector,thenonewillarriveatexactlythe consuming) scan over the whole parameter space.
opposite conclusion, namely that three-generation deco- Cutting the longstoryshort,andto makethe analysis
herence and oscillations can fit the data successfully! easier, we have set all the γi in the neutrino sector to
As we shall argue here, compatibility of all avail- zero, restricting this way, all the decoherence effects to
able data, including CHOOZ [31] and LSND, can be theantineutrinoonewhere,wehaveassumedforthesake
achieved through a set of decoherence parameters γ of simplicity,
j
with energy dependences γ0E and γ0/E, with γ0
j j j ∼ γ¯ =γ¯ for i=1,4,6,7 and γ¯ =γ¯ ,γ¯ =γ¯ (8)
10 18,10 24 (GeV)2, respectively, for some j’s. As it i i+1 1 4 3 8
− −
willbe clearlater,the qualityofthe fitdiminishes tothe Furthermore, we have also set the CP violating phase of
level of the standard three-oscillation case once Kam- the NMS matrix to zero, so that all the mixing matrix
LAND spectral distortion data [32] are included but is elements become real.
significanlty higher otherwise, showing that either addi- With these assumptions, the otherwise cumbersome
tional energy dependences are needed or non-diagonal expression for the transition probability for the antineu-
elements arerequiredto getbetter agreementwith data. trino sector takes the form,
Some important remarks are in order. First of all,
1 1 Ω t
itnhrtehe-egeannearlaytsiiosnosfce[2n6a]ripau,raesdinectowhoergeennceeraitsioenxcolnuedse,dduine Pν¯α→ν¯β = 3 + 2 ρα1ρβ1 cos | 212| e−γ¯1t
(cid:26) (cid:18) (cid:19)
to the fact that the transition probabilities in the case Ω t
∆m2 = 0 (pure decoherence) are such that the survival + ρα4ρβ4cos | 213| e−γ¯4t
ij (cid:18) (cid:19)
7
+ ρα6ρβ6 cos |Ω223|t e−γ¯6t edffenecctess. inInthoeur(ncoanse,cownistthanttw)oddeicffoehreernetnceenecrogeyffidceiepnetns-,
(cid:18) (cid:19)
there seems to be no master condition that guarantees
+ e−γ¯3t ρα3ρβ3 +ρα8ρβ8 . (9) thecompletepositivityinawayindependentoftheener-
) gies/momentaofthe neutrinos. As we shall discuss later
(cid:16) (cid:17)
on in the article, we attribute the 1/E-dependent deco-
where the Ω were defined in the previous section and
ij
herent coefficients to conventional matter effects, while
are the same in both sectors (due to our choice of γ ’s)
i
theE-dependentonesareassociatedwithnoveleffectsof
and
quantum gravity, which increase with the energy of the
ρα1 =2Re(Uα∗1Uα2) probe. The negative probabilities that may occur out-
ρα = U 2 U 2 side the regime of parameters to be specified below are
3 | α1| −| α2| then interpreted as implying simply that our linear (and
ρα =2Re(U U ) (10)
4 α∗1 α3 simplified) parametrization of quantum gravity decoher-
ρα =2Re(U U )
6 α∗2 α3 ence (11) ceases to be valid in such regimes, and more
ρα = 1 (U 2+ U 2 2U 2) complicated, probably non linear, entanglement may be
8 √3 | α1| | α2| − | α3| in operation, as expected in a quantum theory of grav-
ity [22].
wherethemixingmatricesarethesameasintheneutrino
Withtheseinmind,wenowobservethat,inoursimpli-
sector. For the neutrino sector, as there are no decoher-
fiedmodelofdecoherence,parametrizedbyadiagonalde-
ence effects, the standard expression for the transition
coherence matrix, positivity of the relevant probabilities
probability is valid.
seems to be guaranteed for γ L > γ L, which, with our
It is obvious now that, since the neutrino sector does 1 3
parametrization(11),implies: L/E 1024 1025GeV 2,
not suffer from decoherence, there is no need to include ≤ − −
and E > 1 MeV. These sufficient conditions are met by
the solardatainto the fit. We areguaranteedto havean
all the current and planned terrestrial(anti)neutrino ex-
excellent agreement with solar data, as long as we keep
periments. Outsidethisregime,ourparametrizationsim-
the relevantmass difference and mixing angle within the
plyfails,andoneneedstoresorttomorecomplicatedsit-
LMA region, something which we shall certainly do.
uations,whichfallbeyondthescopeofthepresentpaper,
As mentioned previously, CPT violation is driven by,
and will be presented elsewhere.
andrestrictedto,thedecoherenceparameters,andhence
At this point it is important to stress that the inclu-
masses and mixing angles are the same in both sectors,
sion of two new degrees of freedom is not sufficient to
and selected to be
∆m2 =∆m 2 =7 10 5 eV2, guarantee that one will indeed be able to account for all
∆m212=∆m 122 =2.5· 10− 3 eV2, the experimentalobservations. We have to keep in mind
23 23 · − that,inno-decoherencesituations,theadditionofaster-
θ =θ =π/4, θ =θ =.45,
23 23 12 12
ileneutrino (whichcomesalongwithfour newdegreesof
θ =θ =.05,
13 13
freedom -excluding again the possibility of CP violating
as indicated by the state of the art analysis.
phases)didnotseemtobe sufficientformatchingallthe
For the decoherence parameters we have chosen (c.f.
available experimental data, at least in CPT conserving
(8))
situations.
γ1 =γ2 =γ4 = γ5 =2 10−18 E In order to test our model with these two decoher-
· ·
and ence parameters in the antineutrino sector, we have cal-
γ =γ =γ = γ =1 10 24/E , (11) culated the zenith angle dependence of the ratio “ob-
3 6 7 8 −
· served/(expected in the no oscillation case)”, for muon
where E is the neutrino energy, and barred quantities andelectronatmosphericneutrinos,forthesub-GeVand
refer to the antineutrinos, given that decoherence takes multi-GeV energy ranges, when mixing is taken into ac-
place only in this sector in our model. All the other count. Sincemattereffectsareimportantforatmospheric
parameters are assumed to be zero. All in all, we have neutrinos, we have implemented them through a two-
introduced only two new parameters, two new degrees shell model, where the density in the mantle (core) is
of freedom, γ and γ , and we shall try to explain with taken to be roughly 3.35 (8.44) gr/cm3, and the core ra-
1 3
them all the available experimental data. diusistakentobe2887km. Weshouldnoteatthisstage
It can be checked straightforwardly, by means of that a “fake” CPT Violation appears due to matter ef-
(9), that for the regime of the parameters relevant to fects,arisingfromarelativesigndifference ofthe matter
the experiments we are considering in this work, this potentialbetweentherespectiveinteractionsofneutrinos
parametrizationguaranteesthe positivity ofthe relevant and antineutrinos with ordinary matter. This, however,
probabilities. This is an important issue, because usu- is easily disentangled from our genuine (due to quantum
ally negative probabilities are viewed as a signal of in- gravity) CPT Violation, used here to parametrize our
consistency in parameterizing the pertinent decoherence model fit to LSND results; indeed, a systematic study of
8
sucheffects[33]hasshownthat“fake”CPTViolationin-
creaseswiththeoscillationlength,butdecreaseswiththe sub−GeV e sub−GeV µ multi−GeV e multi−GeV µ
neutrinoenergy,E,vanishinginthelimitE ;more-
→∞
over, no independent information regarding such effects
canbe obtainedbylookingatthe antineutrinosector,as
comparedwithdata fromthe neutrinosector,due to the
fact that in the presence of “fake” CPT Violation, but
in the absence of any genuine CPT breaking, the perti- cosθ cosθ cosθ cosθ
nent CPT probability differences between neutrinos and
antineutrinos are related, ∆PCPT = ∆PCPT, where sub−GeV e sub−GeV µ multi−GeV e multi−GeV µ
αβ − βα
∆PCPT =P P , andthe Greek indices denote neu-
αβ αβ− βα
trino flavors. These features are to be contrasted with
our dominant decoherence effects γ (11), proportional
1
to the antineutrino energy, E, which are dominant only
in the antineutrino sector. For the same reason, our ef-
fectscanbedisentangledfrom“fake”decoherenceeffects cosθ cosθ cosθ cosθ
arising from Gaussian averages of the oscillation proba-
sub−GeV e sub−GeV µ multi−GeV e multi−GeV µ
bility due to, say, uncertainties in the energy of the neu-
trino beams [34], which are the same for both neutrinos
and antineutrinos. We, therefore, claim that the com-
plexenergydependence in(11),withbothL E andL/E
·
terms being present in the antineutrino sector, may be a
characteristicfeatureofnewphysics,withtheL E terms
·
beingrelatedtoquantum-gravityinduced(genuine)CPT cosθ cosθ cosθ cosθ
Violating decoherence.
The results are shown in Fig. 1 (c), where, for the sub−GeV e sub−GeV µ multi−GeV e multi−GeV µ
sake of comparison, we have also included the experi-
mental data. We also present in that figure the pure
decoherence scenario in the antineutrino sector (a), as
well as in both sectors (b). For completeness, we also
present a scenario with neutrino mixing but with deco-
herence operative in both sectors (d). The conclusion is cosθ cosθ cosθ cosθ
straightforward: while pure decoherence appears to be
excluded,decoherence plus mixing provides an astonish- FIG.1: Decoherencefits,from toptobottom: (a)puredeco-
ing agreement with experiment[45]. herence in antineutrino sector, (b) pure decoherence in both
sectors,(c)mixingplusdecoherenceintheantineutrinosector
As bare eye comparisons can be misleading, we have
only, (d) mixing plus decoherence in both sectors. The dots
alsocalculatedtheχ2 valueforeachofthecases,defining
correspond to SK data.
the atmospheric χ2 as
10 (Rexp Rth )2
χ2 = α,i − α,i . (12) present the χ2 comparison for the following cases: (a)
atm σ2
M,Sα=e,µi=1 αi puredecoherenceinthe antineutrinosector,(b)purede-
X X X
coherenceinbothsectors,(c)mixingplusdecoherencein
Here σ are the statistical errors, the ratios R be-
α,i α,i the antineutrino sector, (d) mixing plus decoherence in
tween the observed and predicted signal can be written
bothsectors,and(e)mixingonly-thestandardscenario
as
forallexperimentaldatawith(Table2)andwithout(Ta-
ble 1) KamLAND spectral distortion .
Rexp =Nexp/NMC (13)
α,i α,i α,i From the tables it becomes clear that the mixing plus
(with α indicating the lepton flavor and i counting the decoherencescenariointheantineutrinosectorcaneasily
differentbins,tenintotal)andM,S standforthemulti- accountforalltheavailableexperimentalinformation,in-
GeV and sub-GeV data respectively. For the CHOOZ cluding LSND when KamLAND spectraldistortiondata
experiment we used the 15 data points with their statis- are excluded but does not significantly improve over the
ticalerrors,whereineachbinweaveragedtheprobability standardcaseoncetheyareincluded,showingclearlythe
overenergyandforLSNDonedatumhasbeenincluded. limitations of our simplified scenario.
The results with which we hope all our claims become Suchachangecanbeeasilyundestoodbynoticingthat
crystal clear are summarized in Table 1 and 2 , were we scenario(c)improvesthe fitoverthestandardcase(case
9
model χ2 without LSND χ2 including LSND almost totally by the oscillations driven by the mixing.
This was somehow expected, since decoherenceis known
(a) 1097.6 1104.3
not to be the leading force behind the atmospheric neu-
(b) 1037.8 1044.4 trinos. In fact we are taking advantage of the fact that
(c) 45.7 45.9 forourparticularsetofparameters,nosignificanteffects
(d) 52.5 52.7 are expected in the (2,3) channels.
(e) 53.9 60.7 One might wonder then, whether decohering effects,
which affect the antineutrino sector sufficiently to ac-
TABLE I: χ2 obtained for (a) pure decoherence in antineu- countfortheLSNDresult,haveanyimpactonthesolar-
trinosector, (b)puredecoherenceinbothsectors, (c)mixing neutrino related parameters,measured through antineu-
plus decoherence in the antineutrino sector only, (d) mixing trinos in the KamLAND experiment [32, 35]. In order
plus decoherence in both sectors, (e) standard scenario with to answer this question, it will be sufficient to calculate
and without the LSND result for all experimental data but the electron survival probability for KamLAND in our
KamLAND spectral distortions.
model, which turns out to be P .57, in
ν¯α→ν¯β |KamLAND≃
perfect agreement with observations. As is well known,
model χ2 without LSND χ2 including LSND KamLAND is sensitive to a bunch of different reactors
withdistancesspanningfrom80to800km. However,the
(a) 1135.8 1142.3
bulk of the signal comes from just two of those, whose
(b) 1095.6 1102.3 distances are 160 and 179 km. These parameters have
(c) 83.9 84.2 been used to compute the survival probability. Never-
(d) 90.7 90.9 theless, decohering effects, at least of the simplified type
(e) 78.6 85.4 discussed in this work, are not able to account for any
spectral distortion; they rather provide an overall sup-
TABLE II: χ2 obtained for (a) pure decoherence in antineu- pression. Thus,thespectraldistortionthatisapparently
being observed by KamLAND [32], tends to favor the
trinosector, (b)puredecoherenceinbothsectors, (c)mixing
plus decoherence in the antineutrino sector only, (d) mixing standard three-generation scenario (without KamLAND
plus decoherence in both sectors, (e) standard scenario with spectralinformationbutincludingLSNDtheχ2formod-
and without theLSND result, for all data. els (c) and (e) would be 45.9 and 60.7, respectively).
Therefore,ifKamLANDevidence getsstrongerasimpli-
fied decoherence model of the class discussed here would
(e)) by∆χ2 =14.8(seeTable 1)but the priceto payfor be ruled-out.However, this should by no means be re-
itistogiveupspectraldistortionsinKamLAND.Infact, garded as implying that decoherence models in general
for KamLAND data alone we find a χ2 = 24.7 for case will not survive such a case. On the contrary, it is our
(e) and χ2 = 38.4 for an energy independent suppres- beliefthatCPTViolatingdecoherencemodelswithcom-
sion (case (c)), implying a ∆χ2 = 13.7, which essentialy plex energy dependences of the decoherence parameters
meansthattheneteffectistoputourdecoherencemodel as the one discussed here, but probably of more com-
back to the standad case level. plicated, even non linear form, stand a good chance of
It is important to stress once more that our sample describing nature.
point was not obtained through a scan over all the pa- It is also interesting to notice that in our model, the
rameter space, but by an educated guess, and therefore LSND effect is not given by the phase inside the oscilla-
plenty of room is left for improvements. On the other tion term ( which is proportionalto the mass difference)
hand, for the mixing-only/no-decoherence scenario, we but rather by the decoherence factor multiplying the os-
have taken the best fit values of the state of the art cillationterm. Thereforethe tensionbetweenLSNDand
analysis and therefore no significant improvements are KARMEN [36] data is naturally eliminated, because the
expected. At this point a word of warning is in order: difference in length leads to an exponential suppression.
although superficially it seems that scenario (d), deco- Thus, while we predict a 0.24 % anti-electron neutrino
herence plus mixing in both sectors, provides an equally appearanceprobabilityforLSND,thecorrespondingone
good fit, one should remember that including decoher- for KARMEN gets only to 0.14%. And although this
ence effects in the neutrino sector can have undesirable number is already below their experimental sensitivity,
effects in solar neutrinos, especially due to the fact that oneshouldnoticethatKARMENcombinesneutrinoand
decoherenceeffectsareweightedbythedistancetraveled antineutrino channels in their analysis, so that the ac-
bytheneutrino,somethingthatmayleadtoseizable(not tual appearanceprobability is even smaller,as our effect
observed!) effects in the solar case. shows up only in the antineutrino sector.
Another point to stress is that, as can be easily seen Another potential source of concern for the present
in the figure, decoherence plays no role for atmospheric model of decoherence might be accelerator neutrino ex-
neutrino energies and baselines, where the effect is given periments, which involve high energies and long base-
10
lines, and where the decoherence L E scaling can po- periment [38] confirms previous LSND claims, then this
·
tentially be probed. This, however, is not the case. Ac- may be a significant result. One would be tempted to
celerator experiments typically join their neutrino and conclude that if the above estimate holds, this would
antineutrino data, with the antineutrino statistics being probably mean that the neutrino mass differences might
alwayssmallerthantheneutrinoone. Thisfact,together be due to quantum gravity decoherence. Theoretically
with the smaller antineutrino cross section, renders our it is still unknown how the neutrinos acquire a mass,
potential signal consistent with the backgroundcontam- or what kind of mass (Majorana or Dirac) they possess.
ination. Even more, in order to constrain decoherence Therearescenariainwhichthe massofneutrino maybe
effectsofthekindweareproposingherethroughacceler- duetosomepeculiarbackgroundsofstringtheoryforin-
atorexperiments, excellent controlandknowledgeof the stance. Iftheabovemodelturnsouttoberightwemight
beam background are mandatory. The new KTeV data thenhave,for the firsttime inlowenergyphysics,anin-
[37] on kaon decay branching ratios, for example, will dicationofadirectdetectionofaquantumgravityeffect,
change the ν background enough to make any conclu- which disguised itself as an induced decohering neutrino
e
sionontheviabilityofdecoherencemodelsuseless. After massdifference. Noticethatinoursamplepointonlyan-
all, the predicted signal in our decoherence scenario will tineutrinos have non-trivial decoherence parameters γ ,
i
be at the level of the electron neutrino contamination, for i = 1 and 3, while the corresponding quantities in
and therefore one would need to disentangle one from the neutrino sector vanish. This implies that there is a
the other. single cause for mass differences, the decoherence in an-
Having said that, it is clear that althougth this very tineutrinosector,whichiscompatiblewithcommonmass
simplified decoherence model (once neutrino mixing is differencesinbothsectors. Ifitturnsouttobetrue,this
takeninto account)does not accountfor all the observa- would be truly amazing.
tionsincludingtheLSNDresultbetterthanthestandard
three-neutrino model, it certainly suggests that less sim-
plifiedmodelsareworthexploring. Thisscenario,which SPECULATIONS ON THEORETICAL MODELS
makes dramatic predictions for the upcoming neutrino OF DECOHERENCE
experiments, expresses a strong observable form of CPT
violationinthelaboratory,and,inthissense,ourfitdoes Moreover, if the neutrino masses are actually related
notseemtogive(yet)aconclusiveanswertothequestion to decoherence as a result of quantum gravity, this may
askedinthe introductionastowhetherthe weakformof have far reaching consequences for our understanding of
CPTinvariance(1)isviolatedinNature,butitcertainly the Early stages of our Universe, and even the issue of
encourages further studies. Dark Energy that came up recently as a result of as-
This CPT violating pattern, with equal mass spec- trophysical observations on a current acceleration of the
tra for neutrinos and antineutrinos, will have dramatic Universefromeitherdistantsupernovaedata[40]ormea-
signatures in future neutrino oscillation experiments. surements on Cosmic Microwave Background tempera-
The most striking consequence will be seen in Mini- ture fluctuations from the WMAP satellite [41]. Indeed,
BooNE [38], According to our picture, MiniBooNE will asdiscussedin [4,22],decoherenceimpliesanabsenceof
be able to confirm LSND only when running in the an- a well-defined scattering S-matrix, which in turn would
tineutrinomodeandnotintheneutrinoone,asdecoher- implyCPTviolationinthestrongform,accordingtothe
ence effects live only in the former. Smaller but exper- theorem of [14]. A positive cosmological constant Λ > 0
imentally accessible signatures will be seen also in MI- willalsoleadto anilldefinitionofanS-matrix,precisely
NOS [39], by comparing conjugated channels (most no- due to the existence, in such a case, of an asymptotic-
ticeably, the muon survival probability). futuredeSitter(inflationary)phaseoftheuniverse,with
Wenextremarkthatfitswithdecoherenceparameters Hubbleparameter √Λ,implyingtheexistenceofacos-
∼
withenergydependences ofthe form(11)imply thatthe mic(Hubble) horizon. This inturnwillpreventaproper
exponential factors eλkt in (5) due to decoherence will definition of pure asymptotic states.
modify the amplitudes of the oscillatory terms due to We would like to point out at this stage that the
mass differences, and while one term depends on L/E claimed value of the dark energy density component of
theotheroneisdrivenbyL E,wherewehavesett=L, the (four-dimensional) Universe today, Λ 10 122M4,
· ∼ − P
withLtheoscillationlength(weareworkingwithnatural with M 1019 GeV (the Planck mass scale), can actu-
P
∼
units where c=1). ally be accounted for (in an amusing coincidence?) by
The order of the coefficients of these quantities, γ0 the scale of the neutrino mass differences used in or-
j ∼
10 18,10 24 (GeV)2, found in our sample point, im- der to explain the oscillation experiments. Indeed, Λ
− −
∼
plies that for energies of a few GeV, which are typical [(∆m2)2/M4]M4 10 122M4 for ∆m2 10 5 eV2,
P P ∼ − P ∼ −
of the pertinent experiments, such values are not far the order of magnitude of the solar neutrino mass differ-
from γ0 ∆m2 . If our conclusions survive the next enceassumedinoscillationexperiments(whichistheone
j ∼ ij
round of experiments, and therefore if MiniBOONE ex- thatencompassesthe decoherenceeffects, as canbe seen