The Three Neutrino Scenario

6 Pages·0.18 MB·English

by  Hisakazu Minakata

1
The Three Neutrino Scenario ∗
Hisakazu Minakataa
a Department of Physics, Tokyo Metropolitan University, 1-1 Minami-Osawa,Hachioji, Tokyo
192-0397,Japan, and
1 Research Center for Cosmic Neutrinos, Institute for Cosmic Ray Research, University of Tokyo
0 Kashiwa,Chiba 277-8582,Japan
0
2
I have discussed in my talk several remaining issues in the standard three-flavor mixing scheme of neutrinos,
n in particilar, the sign of ∆m2 and the leptonic CP violating phase. In this report I focus on two topics: (1)
13
a supernova method for determining the former sign, and (2) illuminating how one can detect the signatures for
J bothof themin long-baseline (>10 km)neutrinooscillation experiments. I dothis byformulating perturbative

1 frameworks appropriate for the∼two typical options of such experiments, the high energy and the low energy
2 options with beam energies of 10 GeV and 100 MeV, respectively.
∼ ∼

1
v
1
3 1. INTRODUCTION Let me start by raising the following question;
2 ”Suppose that the standard 3 ν mixing scheme
1 Despite the current trend that many people is what nature exploits and the atmospheric and
0 jumped into the four neutrino scheme (see [1] for
the solar neutrino anomalies are the hints kindly
1 complehensive references), the three-flavor mix-
provided by her to lead us to the scheme. Then,
0
ing scheme of leptons, together with the three-
/ what is left toward our understanding of its full
h flavor mixing scheme of quarks, still constitutes
structure?”
p themostpromisingstandardmodelforthestruc-
It is conceivable that the future atmospheric
-
p ture of the (known to date) most fundamental and solar neutrino observations as well as cur-
e matterinnature. Itisworthtonotethatthetwo
rently planned long baseline experiments will de-
:h of the evidences for the neutrino oscillation, one termine four parameters, ∆m223, ∆m212, θ23, and
v compelling [2] and the other strongly indicative
θ12, to certain accuracies. I then cite four things
i [3], can be nicely fit into the three-flavor mixing
X as in below which will probably be unexploredin
scheme of neutrinos. I want to call the scheme
r the near future:
a the standard 3 ν mixing scheme in my talk. (i) θ13
Iwouldliketoaddressinthistalksomeaspects (ii) the sign of ∆m2 ≡ m2 − m2, the normal
13 3 1
of the three-flavor mixing scheme of neutrinos
versus inverted mass hierarchies
which remain to be explored untill now. While
(iii) the CP violating leptonic Kobayashi-
I have started with a brief remark on robustness
Maskawa phase δ
of the standard 3 ν mixing scheme, I do not re-
(iv) the absolute masses of neutrinos
peatitherebecauseIhavedescribeditelsewhere
I make brief comments on (i) and (iv) one by
[4]. In this manuscript I discuss mainly two key
one before focusing on (ii) and (iii):
issues,(1)thesignof∆m2 ,and(2)howtomea-
13 Measuring θ13 is one of the goals of the cur-
sureleptonicCPviolation. Iwilluse,throughout
rently planned long baseline experiments [6–8],
this manuscript, the MNS matrix [5] in the con-
and therefore I do not discuss it further. Among
vention by Particle Data Group.
themtheexpectedsensitivityinJHF,therecently
approvedexperiment in Japan, is one of the best
∗Talk presented at Europhysics Neutrino Oscillation examined case [9].
Workshop (NOW2000), Otranto, Italy, September 9-16,
Measuring absolute masses of neutrinos is cer-
2000.
2
tainly the ultimate challenge for neutrino exper- We draw in Fig. 1 equal likelihoodcontours as
iments, but it is not clear at this moment how afunctionoftheheavytolightν temperaturera-
one cando it. Presumably,neutrinolessdouble β tio τ ≡Tν¯x/Tν¯e =Tνx/Tν¯e on the space spanned
decay experiments are the most promising. We, by ν¯ temperature and total neutrino luminosity
e
however,donotdiscussitfurther,butjustrecom- bygivingtheneutrinoeventsfromSN1987A.The
mend the interested readersto look into a report data comes from Kamiokande and IMB experi-
at this conference [10]. ments[12]. Inadditiontoitweintroduceanextra
parameterηdefinedbyLνx =Lν¯x =ηLνe =ηLν¯e
2. SIGN OF ∆m2 which describes the departure from equipartition
13
of energies to three neutrino species and exam-
Nunokawa and I recently discussed [4,11] that ine the sensitivity of our conclusion against the
the features of neutrino flavor transformation in change in η.
supernova(SN)issensitivetothesignof∆m2 ≡
13
m2 − m2, making contrast between the normal
3 1
(∆m2 > 0) vs. inverted (∆m2 < 0) mass hi- Likelihood Contours for Inverted Mass Heirachy
13 13 1155..00
erarchies. Therefore, one can obtain insight on
the sign of ∆m213 by analyzing neutrino events η = 1.0
from supernova. With use of the unique data η = 0.7
η = 1.3
at hand from SN1987A [12], we have obtained a
strongindicationthatthenormalmasshierarchy, 1100..00
]]
m3 ≫m1 ∼m2, is favoredoverthe invertedone, rgrg τ=2 τ=1
m1 ∼m2 ≫m3. e e
33
The point is that there are always two MSW 00 5 5
resonance points in SN for neutrinos with cos- 11
[[
mologicallyinterestingmassrange,mν <∼100eV. EE b b 55..00
The higher density point, which I denote the
H resonance, plays a deterministic role. If the
H resonance is adiabatic the feature of ν fla-
vor transformationin SN is best characterizedas
00..00
νe−νheavy exchange, as first pointed out in Ref. 11..00 22..00 33..00 44..00 55..00 66..00
[13]. Here, νheavy collectively denotes νµ and ντ, TT
νν
which are physically indistinguishable in SN. See ee
also Ref. [14] for a recent complehensive treat-
ment of 3 ν flavor conversionin SN.
Nowthequestionis: howdoesthesignof∆m2 Fig.1: Contours of constant likelihood which cor-
13 respondto95.4%confidenceregionsfortheinverted
make difference? The answer is: if ∆m2 is posi-
13 mass hierarchy under the assumption of adiabatic
tive(negative)theneutrino(antineutrino)under-
goes the resonance. Then, if the inverted mass HTνxr/eTsoν¯nea=nc2e,.1.8F,r1o.m6,1l.e4ft,1t.2oarnigdht1,.0τw≡herTeν¯xx/=Tν¯µe,τ=.
hierarchyis the caseandassumingthe adiabatic- Best-fit points for Tν¯e and Eb are also shown by
ity of H resonance, the ν¯ which to be observed the open circles. The parameter η parametrizes the
e
interrestrialdetectorscomesfromoriginalν¯heavy departure from the equipartition of energy, Lνx =
in neutrinosphere. It is widely recognized that, Lν¯x = ηLνe = ηLν¯e (x = µ,τ), and the dotted
duetotheirweakerinteractionswithsurrounding lines (with best fit indicated by open squares) and
thedashedlines(withbestfitindicatedbystars)are
matter,ν andν¯ aremoreenergeticthan
heavy heavy
for the cases η = 0.7 and 1.3, respectively. Theoret-
ν andν¯ . Sincetheν¯ inducedCCreactionisthe
e e e ical predictions from supernova models are indicated
dominant reaction channel in water Cherenkov
by theshadowed box.
detectors,the effectofsuchflavortransformation
would be sensitively probed by them.
3
Atτ =1, that is at equal ν¯e and νe temperatures, e−iλ5θ13Γ+δe−iλ7θ23 αβνβ, into theform
the95%likelihood contourmarginally overlapswith
the theoretical expectation [15] represented by the (cid:2)d (cid:3)
i ν˜α=(H)αβ,ν˜β (1)
shadowed box in Fig. 1. When the temperature ra- dx
tio τ is varied from unity to 2 the likelihood contour
where Hamiltonian H contains the following three
movestotheleft, indicatingless andless consistency
terms:
betweenthestandardtheoreticalexpectationandthe
observed feature of the neutrino events. This is sim- 0 0 0 c213 0 c13s13
pbleyinbteecrapurseetetdheasobtsheartveodf tehneerogryigsinpaelctornuemooffν¯νh¯eeamvyuisnt H =  00 00 ∆20E13 +a(x)" c130s13 00 s0213 #
ptbhylieeasptrfheasacettnotcrheeoofofrτtihgtiehnMaanlSν¯Wtehteeemffobepcseterraivnteudν¯reochnmaeu,nsnlteeabl.deiInltogwimteor-  +∆2E12  c1s221s212 c1c221s212 00 (2).
" 0 0 0 #
stronger inconsistency at larger τ.
ThesolidlinesinFig. 1areforthecaseofequipar- We first note the order of magnitude of a relevant
titionofenergyintothreeflavors,η=1,whereasthe quantity to observe the hierarchies of various terms
dotted and the dashed lines are for η = 0.7 and 1.3, in theHamiltonian:
respectively. We observe that our result is very in-
sensitive against thechange in η. ∆m2 =10−13 ∆m2 E −1eV. (3)
We conclude that if the temperature ratio τ is in E 10−3eV2 10GeV
(cid:18) (cid:19)(cid:16) (cid:17)
the range 1.4-2.0 as the SN simulations indicate, the
Itmaybecomparedwiththematterpotentiala(x)=
invertedhierarchyofneutrinomassesisdisfavoredby
theneutrinodataofSN1987AunlesstheHresonance √2GFNe(x)where Ne denoteselectron numberden-
isnonadiabatic,i.e.,unlesss2 < afew 10−4 [4,11]. sity in the earth;
13
×
∼
a(x)=1.04 10−13 ρ Ye eV, (4)
3. HANOWDCTPOVMIOELAASTUIORNE ISNIGNNEUOTFR∆INmO213 × (cid:18)2.7g/cm3(cid:19)(cid:16)0.5(cid:17)
OSCILLATION EXPERIMENTS ? where Ye ≡Np/(Np+Nn) is theelectron fraction.
Inviewoftheseresultsonecanidentifytwotypical
The possibility that SN can tell about the sign of cases, the high and low energy options with ν beam
∆m2 is, I think, interesting and in fact it is the energies 10GeVand 100MeV,respectively,each
13 ∼ ∼
uniqueavailablehintonthequestionatthismoment. with a hierarchy of energy scales:
We, the authors of Ref. [11], feel that our argument (1) High energy option
andtheanalysis donewith theSN1987A data isrea- ∆m2 ∆m2
sonably robust. But, of course, it would be much 13 a(x) 12 (5)
E ∼ ≫ E
nicerifwecanhaveindependentconfirmationbyter-
restrialexperiments. WithregardtotheCPviolating (2) Low energy option
effectmentionedin(iii)itappears,tomyunderstand-
∆m2 ∆m2
ing, that the best place for its measurement is long 13 a(x) 12 (6)
(> 10km)baselineneutrinooscillationexperiments. E ≫ ∼ E
∼We develop an analytic method by which we can Nowletusdiscussthehighandlowenergyoptions
explorevariousregionsofexperimentallyvariablepa- onebyone. Thefocuswillbeonthesignof∆m2 in
13
rameters to illuminate at where CP violating effects the former and theCP violation in thelatter.
are large and how one can avoid serious matter ef-
fect contamination. Actually we formulate below a 3.1. High energy option: matter enhanced
perturbativeframework tohaveabird-eyeviewofat θ13 mechanism
where the sign of ∆m2 is clearly displayed and the In the high energy option one can formulate per-
13
CP violating phase manifests itself. Some of theear- turbation theory by regarding the 1st and the 2nd
lier attempts to formulate perturbative treatment to terms in the Hamiltonian in (2) as unperturbedpart
explorethe verious regions may befound in [16]. andthe3rdtermasperturbation;solar ∆m2 pertur-
We rewrite the Schr¨odinger equation by using the bation theory. Theunperturbedsystemisessentially
basis ν˜ defined by (Γ is a CP phase matrix) ν˜α = thetwo-flavorMSWsystemanditiswellknownthat
4
itleadstothematterenhancedθ13mechanisminneu- If the measurement is done in both neutrino and
trino (if ∆m2 > 0) or antineutrino (if ∆m2 < 0) antineutrino channel, onemay obtain thedifference
13 13
channels. Therefore,thehighenergyoptionisadvan-
tagious if θ13 is extremely small. ∆P ≡ P(νµ→νe)−P(ν¯µ →ν¯e)
In leading order one can easily compute the oscil- 8 Ea ∆m2 L 2
lation probability in matter under the adiabatic ap- ≃ 3 ∆m2 L 4E13 Pvac (12)
proximation. It reads (cid:18) 13 (cid:19)(cid:18) (cid:19)
P(νµ →νe)=s223sin22θ1M3sin2 ξHE∆m4E213L (7) bIfasIeluinsee ∆∼m1021300=km3×an1d0−en3eergVy2,∼∆1P0 G∼eV0.1sPinvcaectfhoer
(cid:18) (cid:19) first parencesis is of order unity in the high energy
sin2θM = sin2θ13, (8) option. Thus, the sign of ∆m213 is determined as the
13 ξHE sign of ∆P [20].
My next and the last message about the high en-
where
ergy option is that the CP violating effect is small.
2Ea It is obvious in leading order that no CP violating
ξHE = (cos2θ13± ∆m2 c213)2+sin22θ13 (9) effectisinduced;itisatwo-flavorproblemandhence
r 13
there is no room for CP violation even in matter.
where referstoantineutrinoandneutrinochannels,
± Therefore, wehavetogobeyondtheleadingorderto
respectively.
haveCPviolation. Then,theCP andT violating ef-
Let us expand the oscillation probability by the
fect is always accompanied by the suppression factor
mpaorsatmoefttehre∆pm4rEa213cLti.caIlncfaascets,;it is a small parameter in ∆∆mm212123 ≃ 0.1−0.01 which comes from the energy de-
nominator. Therefore, CP-odd effect is small in the
∆m213L = 0.127 high energy option.3
4E
∆m2 L E −1 3.2. Low energy option: matter enhanced
× 10−3e1V32 1000km 10GeV (10). θ12 mechanism
(cid:18) (cid:19)(cid:16) (cid:17)(cid:16) (cid:17) The high energy option is certainly advantagious
Then,theoscillationprobabilityreadstonexttolead- for the determination of the sign of ∆m2 thanks to
13
ing order as larger matter effects by available longer baseline due
∆m2 L 2 tobetterfocusingoftheν beam. Ontheotherhand,
P(νµ →νe) = s223sin22θ13 4E13 I will show that the low energy option is the natural
(cid:18) (cid:19) place to look for genuine CP violation.
1 ∆m2 L 2 Because of the hierarchy in the energy scale (6),
× 1− 3ξH2E 4E13 (11) thefirstterm in theHamiltonian in (2)is theunper-
" (cid:18) (cid:19) # turbedtermandthematterandthe∆m2 termsare
12
smallperturbations. Itisimportanttorecognizethat
Thefirsttermin(11) isidenticalwiththevacuum
oscillation probability Pvac under the small ∆m4E213L idteigsenaedraecgyenienrattheepuenrtpuerrbtautriboendthHeaomryiltboenciaauns.eTofhtehne,
approximation. It is the simplest version of the vac-
onemustfirstdiagonalizethedegeneratesubspaceto
uum mimicking mechanism discussed in Ref. [17]
obtain zeroth order wave function and the first order
where a much more extensive version including the
correctiontotheenergyeigenvalues. Then,thezeroth
CP (or T) violating piece is uncovered.2
order wave function contains the CP violating phase
2 InpassingIhaveafewcomments onthevacuum mim- effect. This is the reason why the low energy option
icking mechanism. It might be curious that it works at
theMSWresonancepointbecausethemixingangleiscer- 3Itappearsthatthisstatementmadesomeoftheneutrino
tainlyexhanced. Butitworksinsuchawaythatthereisa factoryworkersunhappy;probablytheyfeltitdifficultto
prolongationofoscillationlengthwhichexactlycancelsthe reconsilethisstatementwiththereportedenomoussensi-
exhacementofthemixingangle[17]. Butthephenonenon tivitiesthatextends towardaverysmallvalueofsin2θ13
of vacuum mimicking is more general which occurs not whichwillbeachievedbyneutrinofactory[21]. However,
only off resonance but alsoinnonresonant channel as far it appears that they are actually consistent because the
asneutrinopathlengthisshorterthanthevacuumoscilla- sensitivity is in fact achieved by the CP conserving cosδ
tionlength. Thismechanismhastriggeredsomeinterests termnotbytheCPviolatingtermintheoscillationprob-
quiterecently[18,19]. abilityatleastforL<∼1000km[22].
5
allows large CP violation unsuppressed by the hier- σE =10 MeV. Wepresent our results in Fig. 2.
archical mass ratio ∆m212, which is to my knowledge
theuniquecase. ∆m213 105
In this setting one can derive the oscillation prob- (a) Neutrino
104
ability P(νµ νe) as follows:
→ )
νe103
→ No CPV, No Matter
νµ102 CPV + Matter
P (νµ νe) N(
→ ∆m2 L 101
= 4s2 c2 s2 sin2 13
23 13 13 (cid:18) 4E (cid:19) 100
+ c2 sin2θM[(c2 s2 s2 )sin2θM (b) Anti-Neutrino
13 12 23− 23 13 12 104
∆m2 L
+ 2c23s23s13cosδcos2θ1M2]sin2(cid:18)ξLE 4E12 (cid:19) → ν)e103 No CPV, No Matter
− 2JM(θ1M2,δ)sin ξLE∆m2E212L (13) νN(µ102 CPV + Matter
(cid:18) (cid:19) 101
100
where ξLE = ξHE(θ13 → θ12,∆m213 → ∆m212), and 1.4 (c) Ratio
JM is the matter enhanced Jarlskog factor. The 1.2
probability(13)represents,apartfromthecosδterm 1.0
wickhiinchgmisescmhaalnlidsmueitnoittshemfoascttoerxtse1n3s,itvheefovramcuiunmclumdiimng- atio 0.8 No CPV, No Matter
R 0.6 CPV + Matter
theCPviolatingJarlskogterm. Tocheckhowwellthe
0.4 Only CPV
system mimics vacuum oscillations see Ref [17].
Only Matter
0.2
The number of appearance events in water
Cherenkovdetectorfora beam energyE =100 MeV 0.0
1 10 100 1000
isestimatedbyassuming10timesstrongerνµ fluxat
L=250 km than theK2K design flux (despitelower Distance from the Source [km]
energy!) and 100% conversion of νµ to νe as
Fig.2: Expected number of events for (a) neutri-
nos, N(νµ νe), (b) anti-neutrinos, N(ν¯µ ν¯e),
N 6300 L −2 V F250 .(14) and(c)their→ratioR≡N(νµ →νe)/N(ν¯µ →ν¯→e)with
≃ 100 km 1 Mton 10FK2K a Gaussian type neutrino energy beam with hEνi =
(cid:16) (cid:17) (cid:16) (cid:17)(cid:16) (cid:17) 100 MeV with σ = 10 MeV are plotted asa function
of distance from the source. Neutrino fluxes are as-
sumedtovaryas 1/L2 inallthedistancerangewe
wtehcetorer Fat25L0 a=nd25F0K2kKmaarnedthteheasdseusmigendnfleuuxtraintoaflduex- consider. sin22θ13∼istakenas0.1, a”maximalvalue”
allowed bytheCHOOZlimit. The remaining mixing
at SK in K2K experiment, respectively. The latter
is approximately, 3×106 P10O2T0 cm−2 where POT p[1a7r]a.mTehteeresrruosredbaarrseaorfetohnelyLMstaAtisMtiScaWl. solution; see
standsfor proton on targe(cid:16)t. (cid:17)
To estimate the optimal distance we compute the
Whilethisparticularproposalmayhaveseveralex-
expected number of events in neutrino and anti-
perimental problems it is sufficiantly illuminative of
neutrino channels as well as their ratios as a func-
the fact that the low energy option is in principle
tion of distance by taking into account of neutrino
more appropriate for experimental search for CP vi-
beamenergyspread. Fordefiniteness,weassumethat
olating effect. There is a large CP violation and the
theaverageenergyofneutrinobeam E =100MeV
h i matter effect is small or controllable. The remain-
andbeamenergyspreadofGaussiantypewithwidth
6
ing question is of course how to develop a feasible 6. JHF Neutrino Working Group, Y. Itow et
experimental proposal. A possibility which employs al., Letter of Intent: A Long Baseline
medium energy ( 1 2 GeV) conventional ν beam Neutrino Oscillation Experiment Using the
∼ −
israisedbyaneminentexperimentalistandtriggered JHF 50 GeV Proton-Synchrotron and the
much interests [23]. Super-Kamiokande Detector, February 3, 2000,
There were many debates between supporters of http://neutrino.kek.jp/jhfnu
high and low energy options in the workshop. I have 7. The MINOS Collaboration, P. Adamson et al.,
concludedwithmypersonalbestthreeflavorscenario; MINOSDetectors Technical Design Report, Ver-
we measure CP violation by low energy superbeam sion 1.0, NuMI-L-337, October1998.
in Japan, and you measure δ by neutrino factory in 8. OPERACollaboration, M.Guleretal.,OPERA:
Europe, and then let us compare theresults! AnAppearanceExperimenttoSearchforNu/Mu
Nu/Tau Oscillations in the CNGS Beam.
←→
Experimental Proposal, CERN-SPSC-2000-028,
ACKNOWLEDGMENTS
CERN-SPSC-P-318, LNGS-P25-00, Jul 2000.
I express deep gratitude to G. L. Fogli for cordial 9. T. Kobayashi, Talk at Neutrino Oscillations and
invitation to such a focused and the well-organized TheirOrigin,NOON2000,Tokyo,Japan,Decem-
workshop, in which I was able to enjoy the stimu- ber6-8, 2000.
lating atmosphere of the Europian neutrino physics 10. H.V.Klapdor-Kleingr¨othaus, Talk at NOW2000,
community. I thank Hiroshi Nunokawa for collab- in these Proceedings.
oration and his kind help in dealing with the else- 11. H.MinakataandH.Nunokawa,hep-ph/0010240.
vier latex format. This work was supported by the 12. K. S. Hirata et al., Phys. Rev. Lett. 58 (1987)
Grant-in-AidforScientificResearchinPriorityAreas 1490; Phys. Rev. D38 (1988) 448; R. M. Bionta
No. 11127213,JapanMinistryofEducation,Culture, et al., Phys. Rev.Lett. 58 (1987) 1494.
Sports, Science and Technology. 13. H. Minakata and H. Nunokawa, Phys. Rev. D41
(1990) 2976.
14. A.S.DigheandA.Yu.Smirnov,Phys.Rev.D62
REFERENCES
(2000) 033007.
1. C. Giunti, Talk at NOW2000, in these Proceed- 15. H.-T. Janka, in Vulcano Workshop 1992; Fron-
ings. tierObjects inAstrophysics andParticlePhysics,
2. Y. Fukuda et al. (Kamiokande collaboration), Proceedings of the Workshop Vulcano, Italy,
Phys. Lett. B335 (1994) 237; Y. Fukuda et 1992, edited by F. Giovannelli and G. Mannochi,
al.(SuperKamiokandecollaboration), Phys.Rev. IPS Conf. Proc. No. 40 (Italian Physical Society,
Lett. 81 (1998) 1562; T. Kajita, in Neutrino Vulcano,1993).
Physics and Astrophysics, Proceedings of the 16. J. Arafune and J. Sato, Phys. Rev. D55 (1997)
XVIIIth International Conference on Neutrino 1653; J. Arafune, M. Koike and J. Sato, Phys.
PhysicsandAstrophysics(Neutrino’98), June4- Rev. D56 (1997) 3093; Erratum ibid., D 60
9, 1998, Takayama, Japan, edited by Y. Suzuki (1999) 119905; H. Minakata and H. Nunokawa,
and Y. Totsuka, (Elsevier Science B.V., Amster- Phys. Rev. D57 (1998) 4403; Phys. Lett. B413
dam, 1999) page 123. (1997)369;M.KoikeandJ.Sato,Phys.Rev.D61
3. HomestakeCollaboration, K.Landeetal.,Astro- (2000) 073012; O. Yasuda, Acta. Phys. Polon. B
phys.J.496,505(1998);SAGECollaboration,J. 30 (1999) 3089.
N. Abdurashitov et al., Phys. Rev.C 60, 055801 17. H.MinakataandH.Nunokawa,Phys.Lett.B495
(1999); GALLEX Collaboration, W. Hampel et (2000) 369, hep-ph/0004114.
al., Phys. Lett. B 447, 127 (1999); Kamiokande 18. P.Lipari, privatecommunications.
Collaboration, Y. Fukudaet al. Phys. Rev. Lett. 19. S.J. Parkeand T. J. Weiler, hep-ph/0011247.
77, 1683 (1996); SuperKamiokande Collabora- 20. This point has first been explicitly discussed by
tion, Y. Fukuda et al., Phys. Rev. Lett. 81, P.Lipari, in Phys.Rev. D61 (2000) 113004.
1158 (1998); ibid.81, 4279 (1998); ibid.82, 2430 21. A.Cervera,A.Donini,M.B.Gavela,J.J.Gomez
(1999); ibid. 82, 1810 (1999). C´adenas,P.Hern´andez,O.Mena,andS.Rigolin,
4. H.Minakata,TalkatDark2000,hep-ph/0101148. hep-ph/0002108.
5. Z. Maki, M. Nakagawa, and S. Sakata, Prog. 22. Private communications with Osamu Yasuda.
Theor. Phys. 28 (1962) 870. 23. B. Richter,hep-ph/0008222.