Exoti Solutions to the Solar Neutrino Anomaly
M. M.Guzzo
Institutode FsiaGlebWataghin
UniversidadeEstadual deCampinas,UNICAMP
13083-970, Campinas,SP,Brazil
Reeivedon19April,2001
Weanalyzethestatusoftheexotisolutionstothesolarneutrinoproblem,i.e.,thosesolutionsbased
onnewphenomenawhihare notthe usualneutrino osillationsindued by massesand mixing.
Thesesolutions arebasedondierentassumptions: a)resonant spin-avor preessioninduedby
non-vanishing neutrino magnetimoment, b) the existene of non-standard avor-hanging and
non-universalneutrinointerationsand)theviolationoftheequivalenepriniple. Weinvestigate
thequalityofthetprovidedbyeahoneofthesesolutionsnotonlytothetotalratemeasuredby
all solarneutrinoexperimentsbutalso totheday-night andseasonalvariationsofthe eventrate,
aswellasthereoileletronenergyspetrummeasuredbytheSuperKamiokandeollaboration.
I Introdution
Homestake [1℄, GALLEX/GNO [2℄, SAGE [3℄,
Kamiokande[4℄andSuperKamikande[5℄haveobserved
a solar neutrino ux whih is smaller than predited
by the standard solar models (SSM) [6, 7, 8, 9, 10℄.
This disrepany has been alled the solar neutrino
problem [11℄. Better statistis and alibration of
the pioneering experiments, as well as the rst
next-generation experiment SuperKamiokande, measuring
the solar neutrino spetrum and the event rate as a
funtion of the zenith angle with unpreedented
pre-ision, have provided a lot of new information about
thesolarneutrinoproblem[11℄. Onthetheoretialside
several substantial improvements have been made in
the SSM whih now inludes diusion of helium and
heavyelementsand updated low energy nulear ross
setionsrelevantto thesolarneutrinoprodution[12℄.
Furthermore, the SSM has reeived an important
in-dependentonrmationbytheexellentagreement
be-tween itspredited sound speedsand reent
helioseis-mologialobservations[7℄.
Inorder tounderstand the solarneutrinoanomaly
ithasbeensuggestedthatneutrinosareendowedwith
properties whih are notpresent in thestandard
ele-troweak theory [13℄. These new properties allow the
eletronneutrinostobeonvertedalongtheirwayfrom
theenterofthesuntothedetetorsonearthinto
dif-ferent neutrino avors, i.e. into muon, tau, or
possi-bly sterile [14℄ neutrinos. Thefat that theterrestrial
experimentsare lesssensitiveto theseneutrino avors
explainstheobservedlowerountingrates.
Several mehanisms anprovoke the eletron
neu-ourknowledge, themehanismsthat t thesolar
neu-trinodata anbelassiedin fouressentiallydierent
types:
The most famous solutions to the solar neutrino
anomaly assumethat neutrinosare massiveand there
is mixing in the lepton setor. Under this
irun-stane,neutrino avorosillationsan happenin
va-uum[15℄aswellasinmatterwhereitanberesonantly
enhaned [16, 17℄ (the Mikheyev-Smirnov-Wolfenstein
(MSW) eet). In both senarios solar eletron
neu-trinosanbeonvertedinto neutrinoofadierent
a-vor(muonortau)andonsequentlyexplainthedeit
ofobserved solarneutrino deitto the preditionsof
theSSM. Thesesolutionsarealled thestandard
solu-tionsto the solar neutrino problem and are verywell
desribedinreentreferenes[18℄, [19℄and[20℄.
In this review, we will not desribe the solutions
to the solar neutrino anomaly based onmass-indued
osillation phenomenon. We wil onentrateon more
exotisolutionlikethoseonedesribedbelow.
II Resonant spin-avor
phe-nomenon
Assuminganonvanishingtransition magnetimoment
ofneutrinos,ativesolarneutrinosinteratingwiththe
magnetield in theSun anbe spin-avor onverted
intosterilenoneletronneutrinos[21,22℄(ifweare
deal-ingwithDirapartiles)orintoativenoneletron
an-tineutrinos[23℄(iftheinvolvedpartilesareMajorana).
Inbothasestheresultingpartilesinteratwithsolar
nomenonanindueadepletioninthedetetablesolar
neutrinoux. Spin-avorpreessionofneutrinosanbe
resonantlyenhanedinmatter[24,25℄,inloseanalogy
with theMSW eet [16, 17℄. In this asethe
prees-sionstronglydepends ontheneutrinoenergyand
pro-vokesdierentsuppressionsforeahportionofthesolar
neutrinoenergyspetrum. ThereforeRSFP providesa
satisfatory desription [26, 27, 28, 29, 30, 31, 32℄ of
theatualexperimentalpanorama[1,2,3,4,5℄: all
ex-periments detet less than the theoretially predited
solar neutrinouxes [9, 10℄ and dierentsuppressions
are observed in eah experiment, suggesting that the
mehanismtooniliatetheoretialpreditionsand
ob-servationshastodierentiatethedierentpartsofthe
solarneutrinospetrum.
The time evolution of neutrinos interating with
a magneti eld B through a nonvanishing neutrino
magneti moment
in matter is governed by a
Shrodinger-likeequation[24, 25℄;
i d
dt
e
L
R
=
a
e
B
B
m 2
2E +a
e
L
R
;
(1)
where
e and
R
are ative eletron neutrinos and
muonantineutrinos,respetively,m 2
=m 2
m
2
e is
theirsquaredmassdiereneandEistheneutrino
en-ergy, a
e =G
F (2N
e N
n )=
p
2and a
=G
F N
n =
p
2,
with N
e and N
n
being eletron and neutron number
densities, respetively. In eq. (1) it is assumed that
neutrinos are Majorana partiles. Forthe Diraase,
thespin-avorpreessioninvolves
e $
s
,where
s is
asterile neutrinoanda
s =0.
In order to obtain the survival probability one
shouldintegratetheevolutionequations(1)with
vary-ing matter density in the Sun [11℄ for some assumed
proles of the magneti eld whih will be desribed
below. Usingthesolar neutrinoux in ref. [8℄, itwas
omputedinRef.[32℄theexpetedsolarneutrinoevent
rate in eah experiment, taking into aountthe
rele-vant absorption ross setions [11℄ for 71
Ga and 37
Cl
experimentsaswellasthesatteringrosssetionsfor
e
-e and
-e reations inludingalsotheeÆieny
funtion for the SuperKamiokande experiment in the
same way as in ref. [33℄. Note that in this analysis
it was alwaysadopted the solarmodelin ref. [8℄as a
refereneSSM.
It is obvious from theevolutionequations (1) that
the RSFP mehanism ruially depends on the solar
magneti eld prole along the neutrino trajetory.
In the analisys of Ref. [32℄ it was hosen several
dif-ferent proles whih over in general all the
previ-ously [27, 28, 29, 30, 31℄ analysed magneti proles
whihledtoasolutiontothesolarneutrinoanomaly. In
Fig.1,thesemagnetieldsarepresentedintheir
gen-eral aspets. TheonstantmagnetiproleB
1 (r)was
adoptedin referenes[29℄,whilethegeneralaspetsof
theprolesB
3
(r) andB
4
(r)havealreadyappearedin
refs. [28,30,27℄,and [31℄,respetively.
0.0
0.2
0.4
0.6
0.8
1.0
r/R
sun
B
1
(r)
B
2
(r)
B
3
(r)
B
4
(r)
<B>
2<B>
Magnetic Field Profiles
Figure1. Variousmagnetieldprolesusedinthiswork.
ForeaheldhBiisdenedastheaverageoftheeldover
theregionwhereB(r)isnotzero.
Note that lose to the solar surfae(r > 0:95R
)
themagnetieldwasswithedoforalltheproles.
2
isdenedasfollows,
2
= X
i;j (R
th
i R
obs
i )[
2
ij (tot)℄
1
(R th
j R
obs
j
); (2)
where(i;j)runthroughthree experiments,i.e., 71
Ga,
37
ClandSuperKamiokande,andthetotalerrormatrix
2
ij
(tot )andtheexpetedeventratesR
i
areomputed
asfollows. ItwasessentiallyusedtheproedureofRef.
[34℄ for the derivation of the error matrix and to
de-sribetheorrelationsoferrorsweusedin thiswork.
Inludingnowtheexperimentalobservationsonthe
solarneutrinosignalitispossibleto determinethe
re-gion in the m 2
hBi parameter spae whih leads
toaRSFPsolutiontothesolarneutrinoproblemfora
speiedondenelevel. Itispresentedthem 2
hBi
parameter region whih an aount for all the solar
neutrino data, at 90, 95 and 99%C. L. in Figs. 2(a),
(b),()and(d),forthemagnetiprolesB
1 (r),B
2 (r),
B
3
(r)andB
4
(r),respetively.
From Figs. 2 (a) to (d) one observes that a
solu-tionto thesolar neutrinoproblem anbefoundwhen
hBi >
few times10kGandm 2
istheorderof10 8
to10 7
eV 2
foranyofthemagnetiprolesusedinthis
work. Nevertheless,thequalityofthet,measuredby
theminimum 2
riterion,variesalot. Thepoorestt
is obtained when the ontinuously deaying magneti
eld proleB
4
isused, with 2
min
=6:1for one(three
data points-twofreeparameters)degreesoffreedom.
BettertsareobtainedwhentheB
1
(uniform)andB
2
(largetriangle)eldsareemployedshowing 2
min =2:0
and 1.8, respetively. And the best t appears when
thetriangulareld in thesolaronvetivezone, B
3 ,is
employed, witha rathersmall value 2
thisprole,wenotethat,asexpetedfromFig. 3()we
haveseveral loal best t pointsalso indiatedin Fig.
4()bytheopenirleswhoseorresponding 2
min are,
from lefttoright,2.3, 0.29and0.19.
10
100
<B> (kG)
10
-9
10
-8
10
-7
10
-6
∆
m
2
(eV
2
)
90 % C.L.
95 % C.L.
99 % C. L.
(a)
200
10
100
<B> (kG)
10
-9
10
-8
10
-7
10
-6
∆
m
2
(eV
2
)
90 % C.L.
95 % C.L.
99 % C. L.
(b)
200
0
50
100
150
200
<B> (kG)
10
-9
10
-8
10
-7
10
-6
∆
m
2
(eV
2
)
90 % C.L.
95 % C. L.
99 % C. L.
(c)
10
100
<B> (kG)
10
-9
10
-8
10
-7
10
-6
∆
m
2
(eV
2
)
90 % C.L.
95 % C.L.
99 % C.L.
(d)
200
Figure2. TheAllowedRSFPsolutiontothesolarneutrino
problem. The parameter regionallowed at 90, 95 and 99
%C.L.areshownin(a), (b),()and(d)forthemagneti
eld proles,B
1 ,B
2 ,B
3 and B
4
, respetively, skethedin
Fig. 1. We indiatebesttpointsbylled irles. In()
wealsoindiate,bytheopenirles,theloalbesttpoints
insideeahislanddelimitedby90%C.L.urves.
Thereason why averygood t is obtainedfor B
3
is that this prole an provide the required
suppres-sionpatternsofvarious neutrinoux impliedbylatest
the data [35℄asdisussed in ref. [30℄. Firstnote that
lowenergypp neutrinosarenotsosuppressedbeause
where the magneti eld is zero or small. However,
intermediateenergy 7
Beneutrinosanbestrongly
sup-pressed due to the rapid inrease of the eld at the
bottomoftheonvetivezonesinetheirresonane
po-sitionis inaslightlyouterregionthantheppone. On
the other hand, high energy 8
B neutrinos are
moder-atelysuppressed beausetheir resonanepositions are
losertothesolarsurfaethanthe 7
Beones,wherethe
eldisdereasing.
The best tted values of hBi and m 2
as well as
2
min
obtainedfromthesedierentprolesare
summa-rizedin TableI.
Prole hBi(kG) m 2
(10 8
eV 2
)
2
min
1 50.6(40.9) 3.5(2.3) 2.0 (6.2)
2 47.1(40.8) 6.1(4.6) 1.8 (5.7)
3 118(69.4) 1.5(1.2) 0.13(1.3)
4 82.9(81.6) 8.1(6.6) 6.1(11.4)
TABLEI.Thebestttedparametersand 2
min
forthe
Ma-joranaase. Diraase ispresentedintheparentheses.
The sameanalysis forthe Diraneutrino asewas
done. The orresponding plots for the allowed region
arenotshownheresinetheyarerathersimilartowhat
havebeen presentedabove, ifthe samemagnetield
proleis assumed. Instead, for thease ofDira
neu-trinos,weonly presentthe best tted parametersand
2
min
in the parentheses in Table I. We see from this
table that, the Dira ase always leads to a worse t
ifthe same magneti eld prole is assumed. To
un-derstandthis we should note that for the Dira ase,
e
'sareonvertedintotherighthandedmuon(ortau)
neutrino, whih do not ontribute to any of the solar
experimentsinludingthewaterCherenkovexperiment
[36℄. ThismakesitdiÆulttooniliatethedierene
betweentheSuperKamiokandeand 37
Cldata. In
on-trastintheMajoranaase,onvertedrighthanded
neu-trino
's do ontribute to the signalobserved in the
SuperKamiokandedetetor.
Let me now briey omment about the possibility
ofhavingsuhstrongmagnetield intheSun. While
thereisnogenerallyaeptedtheoryofsolarmagneti
eld,itispossibletoboundtheeldstrengthfromvery
generalarguments. Itanbeshown[11℄that the
mag-netieldlessthan10 6
kGinthesolaroreorlessthan
10 4
kGin the solaronvetivezone, will hardlyaet
the thermal struture and nulear reation proesses
well desribed by the standard solar model. These
values ome from the requirement that the magneti
pressureshouldbemuhsmallerthanthegaspressure,
andanberegardedasthemostgenerousupperlimits
of the magneti eld inside the Sun. More stringent
boundsonthemagnetieldintheonvetivezoneare
found in refs. [37, 38℄ where the disussion is based
onthenon-lineareetswhiheventuallypreventsthe
pro-required eld tensionneessary to preventa uid
ele-mentfromsinkingintoamagnetiallystratiedregion,
sothat themagneti ux would notbefurther
ampli-ed. By equating the magneti tensionto theenergy
exessofasinkingelementatthebottomofthe
onve-tive zone, Shmitt andRosner [37℄ obtained 10kG
asan upper bound for the magnetield, whih is of
theorderof themagnitudeweneedto haveagoodt
tothesolardatabyRSFPmehanismforthereferene
valueofmagnetimoment,
=10
11
B .
Finally, we briey disuss how the reoil eletron
energy spetra in the SuperKamiokande detetor will
beaeted bytheRSFPmehanism[39℄.
0.0
5.0
10.0
15.0
Neutrino Energy (MeV)
0.0
0.2
0.4
0.6
0.8
1.0
Electron Neutrino Survival Probability
Profile 1
Profile 2
Profile 3
Profile 4
(a)
5.0
10.0
15.0
Electron Energy (MeV)
0.0
0.2
0.4
0.6
0.8
1.0
Prediction/SSM
profile 1
profile 2
profile 3
profile 4
(b)
Figure 3. We plot in(a) eletronneutrino survival
prob-ability asafuntionof energywiththebest tted
param-eters for various eld proles. In (b) we plot reoil
ele-tronenergyspetraexpetedfromRSFPsenariousingour
best t parameters, divided by the SSM predition. The
SuperKamiokandedataare alsoshownbythe lledirles
witherrorbars. Thelast datapointinludesthe
ontribu-tionfromtheeletronswithenergylarger than14MeV.
In Fig. 3(a)weplot the eletron neutrinosurvival
probabilitiesasafuntionofneutrinoenergyusingthe
besttparameters. InFig.3(b)weplotthereoil
ele-expeted to beobserved in the SuperKamiokande
de-tetor,usingalsothebesttparametersasinFig.3(a).
In Fig. 3(b) we also plot the latest data from
Su-perKamiokande [5℄. As we an see from the plot the
observed data indiate some distortion mainly due to
thelastthreedatapointsinthehigherenergybins.We,
however, note from this plotthat it seemsdiÆult to
exlude, at thismoment, anyof ourpreditedspetra
expetedfromdierenteldproles,beauseofthe
ex-perimentalerrors. We haveto waitfor morestatistis
andmorearefulanalysisfrom theexperimentalgroup
beforedrawinganydeniteonlusion.
In onlusion, reanalysing the RSFP mehanism
as a solution to the solar neutrino problem in the
light of the latest experimental data as well as the
theoretial preditions, one nds that the quality of
the RSFPsolutionto the solarneutrino anomaly
ru-ially depends on the solar magneti eld
ongura-tionalongtheneutrinotrajetoryinsidetheSun. One
nds that the best t to the observed solar neutrino
data, whih seems to be even better than the usual
MSW solution asfarasthe totalrates areonerned,
is obtained if intensive magneti els in the
onve-tive zone is assumed, in agreement with the
onlu-sion found in ref. [30℄, whereas the linearly deaying
magneti eld gives the worst t. Note that the
re-quired magnitude of the free parameters involved in
the proess, i.e., the magneti eld strength
multi-pliedbytheneutrinomagnetimoment
hBiandthe
squared mass dierene m 2
, points to the same
or-der,
hBi fewtimes10 11
B
10 kGand m 2
fewtimes10 8
eV 2
,foranyoftheeldprolesassumed
in thiswork.
Ourignoraneabouttheproleaswellasthe
mag-nitude ofthesolarmagneti eldmakesthisapproah
to the solar neutrino observation less preditive than
itsalternativeapproahes[15,19,40℄. Neverthelessthe
preseneofthismehanismopenssomeinteresting
pos-sibilities.
Onepossibility is tolook foranytime variationof
thesolarneutrinosignal[41℄whihannotbeexpeted
inotheralternativesolutionsfoundinrefs. [15,19,40℄.
Any time variation of the solar neutrino signal whih
an be attributed to some time variation of the solar
magneti eld an be a good signature of this
meh-anism. Although SuperKamiokande has not yet
on-rmedanysignianttimevariationuptoexperimental
unertaintythispossibilityremains.
Anotherpossibility is to look for thesolar
e ux,
whih annot beproduedin the usualMSW or
va-uum osillation ase but an be produed in RSFP
mehanism if the avor mixing is inluded.
pro-dued by RSFP mehanism an be onverted into
e
by theusual vauumosillation. Ref. [42℄ suggeststo
observe(or to put upper bound of)
e
Su-energy solar neutrino experiment suh as Borexinoor
Hellaz.
We nally stress that RSFP mehanism an still
provideagoodsolutiontothesolarneutrinoproblem,
omparableinqualitytoMSWorJustSosolution,and
isnotexludedbythepresentsolarneutrinodata.
III Non-standard neutrino
in-terations
InhisseminalpaperWolfenstein[16℄observedthat
non-standardneutrinointerations(NSNI)withmatteran
also generate neutrino osillations. In partiular this
mehanism ould be relevant to solar neutrinos
inter-atingwiththedensesolarmatteralongtheirpathfrom
the ore of the sun to its surfae [44, 45, 46, 47, 48,
49, 50℄. In this ase the avor hanging neutrino
in-terations (FCNI) are responsible for the o-diagonal
elements in the neutrino propagation matrix (similar
to the m 2
sin 2
2 term induedby vauum mixing).
Formasslessneutrinosresonantlyenhanedonversions
an our due to an interplay between the standard
eletroweakneutrinointerationsandnon-universal
a-vor diagonal neutrino interations (FDNI) with
mat-ter [44,51℄.
Whilemanyextensionsofthestandardmodelallow
formassiveneutrinos,itisimportanttostressthatalso
manyNewPhysismodelspreditnewneutrino
inter-ations. Theminimalsupersymmetristandardmodel
withoutR -parityhasbeenevokedasanexpliitmodel
thatouldprovidetheFCNIandFDNIneededforthis
mehanism. Systemati studies of the data
demon-strated that resonantly enhaned osillations indued
by FCNI and FDNIformasslessneutrinos [45,49℄, or
FCNI in ombination with massive neutrinos [45, 50℄
ansolvethesolarneutrinoproblem.
Here we present the status of the solution to the
solar neutrino problem based on NSNI follwoing the
same steps of Ref. [40℄. We present a omprehensive
statistial analysis of this solution. Ouranalysis
om-prises boththemeasuredtotalratesofHomestake[1℄,
GALLEX[2℄,SAGE[3℄andSuperKamiokande[5℄and
the full SuperKamiokande data set (orresponding to
825eetivedaysofoperation)inludingthereoil
ele-tronspetrumand theday-nightasymmetry. Wehave
notinluded in our 2
analysis the seasonalvariation
butwewillommentonthiseet. Forthesolarinput
we take the solar neutrinouxes and their
unertain-tiesaspreditedinthestandardsolarmodelbyBahall
andPinsonneault(hereafterBP98SSM)[9℄. TheBP98
SSM inludeshelium and heavyelementsdiusion, as
wellasthenewreommendedvalue[12℄forthelow
en-ergy S-fator, S
17 = 19
+4
2
eV b. We also study the
dependeneoftheallowedparameterspaeonthehigh
energy 8
Bneutrinoux,byvaryingtheux
normaliza-tionasafreeparameter.
Wendthatnon-standardneutrinointerationsan
provideagoodttothesolarneutrinodataifthereare
ratherlargenon-universalFDNI(oforder 0:5G
F )and
smallFCNI(oforderafewtimes10 3
G
F
).
Neverthe-lessitisshowninRef.[40℄thatphenomenologial
on-traintsindiate thatFCNIouldonly belargeenough
to provide
e !
transitions, while
e !
transi-tionsarenotrelevantforthesolutionofthesolar
neu-trinoproblem,beauseofstrongexperimental bounds.
Large FDNI an only be indued by an intermediate
doubletofSU(2)
L
(a salaroravetorboson)orbya
neutralvetorsinglet. We onludethat the minimal
supersymmetrimodelwithbrokenR -parity[55℄isthe
favoritemodelforthissenario.
Anymodelbeyondthestandardeletroweaktheory
thatgivesrisetotheproesses
e
f !
`
f; (3)
f !
f; (4)
where (here and below) f = u;d;e and ` = ; and
=e;;, is potentially relevant forneutrino
osilla-tionsinthesun,sinetheseproessesmodifythe
ee-tivemassofneutrinospropagatingindensematter.
Theevolutionequationsformasslessneutrinosthat
interatwithmatterviathestandardweakinterations
andthenon-standardinterationsin(3)and(4)isgiven
by[44,45℄:
i d
dr
A
e (r)
A
` (r)
= p
2G
F
n
e
(r)
f
` n
f (r)
f
` n
f (r)
0 f
` n
f (r)
A
e (r)
A
` (r)
; (5)
d
whereA
e
(r)andA
`
(r)are,respetively,theprobability
amplitudestodeteta
e and
`
atpositionr. For
neu-trinosthathavebeenoherentlyproduedas
e in the
solaroreatpositionr ,theequationsin(5)aresubjet
totheboundaryonditionsA
e (r
0
)=1andA
` (r
0 )=0.
WhileW-exhangeof
e
withthebakgroundeletrons
am-plitude p 2G F n e
(r), the FCNI in (3) indue a avor
hanging forward sattering amplitude p 2G F f ` n f (r)
andthenon-universalFDNIareresponsibleforthe
a-vordiagonalentry p 2G F 0 f ` n f
(r)ineq.(5). Here
n f (r)= n n
(r)+2n
p
(r) f =u
2n
n (r)+n
p
(r) f =d
(6)
isthe respetivefermion numberdensity atposition r
in termsoftheproton [neutron℄numberdensity n
p (r) [n n (r)℄and "= f ` G f e` G F ;" 0 = 0 f ` G f `` G f ee G F ; (7)
desribe,respetively,therelativestrengthoftheFCNI
in (3), and thenew avordiagonal, but non-universal
interations in (4). G f
(; =e;;) denotesthe
eetiveouplingofthefour-fermionoperator
O f ( )(
ff): (8)
thatgivesrisetosuhinterations. TheLorentz
stru-ture of O f
depends on the New Physisthat indues
thisoperator.Operatorswhihinvolveonlyleft-handed
neutrinos(andwhihonservetotalleptonnumberL)
an be deomposed into a (V A)(V A) and
a (V A)(V +A) omponent. (Any single New
Physisontribution that isindued byhiral
intera-tions yieldsonly one of these twoomponents.) It is,
however, important to note that only the vetor part
ofthebakgroundfermionurrentaetstheneutrino
propagationforanunpolarizedmediumatrest[16,52℄.
Hene only the (V A)(V) part of O f
is relevant
forneutrinoosillationsinnormalmatter. One
meha-nismtoinduesuhoperatorsisduetotheexhangeof
heavybosons that appear in various extensions ofthe
standardmodel. Analternativemehanismariseswhen
extending the fermioni setorof the standard model
and is due to Z-indued avor-hanging neutral
ur-rents(FCNCs). Fora disussion of Z-indued FCNC
eetsonsolarneutrinos,seeRefs.[53,54℄.
Aresonaneourswhenthediagonalentriesofthe
evolutionmatrixineq.(5) oinideat somepointr
res
alongthetrajetoryoftheneutrino,leadingtothe
res-onaneondition 0 f ` n f (r res )=n
e (r
res
): (9)
AnimmediateonsequeneisthatnewFDNIforf =e
alone annot indue resonant neutrino avor
onver-sions.
As it was observed in Ref. [40℄ that only
e !
onversions are ompatible with the existing
phe-nomenologial onstraints on f ` and 0 f `
. We note
that in the minimal supersymmetri standard model
with broken R -parity[55℄ the relevantparametersare
givenby d = 0 331 0 131 4M 2 ~ b p 2G F ; 0 d = j 0 331 j 2 j 0 131 j 2 4M 2 ~ b p 2G F ; (10)
intermsofthetrilinearouplings 0
ijk
andthebottom
squarkmassM
~
b .
The neutrino evolution matrix in eq. (5) vanishes
in vauumand isnegligibly small for thematter
den-sities of the earth's atmosphere. Therefore the
prob-ability of nding an eletron neutrino arriving at the
detetorduringdaytimeiseasilyobtainedbyevolving
theequationsin(5)fromtheneutrinoprodutionpoint
to the solar surfae. Furthermore, typially there are
manyosillationsbetweentheneutrinoprodutionand
detetion point and aresonane. Therefore the phase
information before and after the resonaneis usually
lostafterintegrationovertheprodutionanddetetion
regionandonemayuselassialsurvivalprobabilities.
Thenatdaytimewehave[45℄
P day e!e =jA e (r s )j 2 ' 1 2 +( 1 2 P )os2
p m os2 s m ; (11) where r s
is the solar surfae position and in the
an-alyti expressionin eq. (11)wedenote by p m and s m ,
respetively,theeetive,matter-induedmixingatthe
neutrinoprodutionpointand atthesolarsurfae. In
terms of the New Physis parameters ", " 0
and the
fermiondensitiestheeetivemixingisgivenby[44,45℄
tan2 m = 2 f ` n f 0 f ` n f n e : (12)
Note that tan2
m = 2 e ` =( 0 e `
1) is onstant for
f = e. P
is the level rossing probability. The
ap-proximateLandau-Zenerexpressionis [44,45℄
P
=exp[ =2℄ (13)
with =4 p 2G F ( f ` = 0 f ` ) 2 0 f ` n e d dx n f ne res : (14)
Whenneutrinosarriveatthedetetorduringthenight,
amodiationofthesurvivalprobabilityhastobe
in-trodued sinethe non-standardneutrino interations
withtheterrestrialmattermayregenerateeletron
neu-trinosthat havebeentransformedin thesun.
Assum-ingthattheneutrinosreahtheEarthasaninoherent
survival probability during night-time an be written
as [56℄:
P night
e!e =
P day
e!e sin
2
s
m +P
2e (1 2P
day
e!e )
os2 s
m
:
(15)
HereP
2e
istheprobabilityofatransitionfromthestate
2
to theavor eigenstate
e
along the neutrino path
in theEarth.
For ouranalysis weassume a stepfuntion prole
for the Earth matter density, whih has been shown
to beagood approximationin otherontexts(seee.g.
Ref. [57℄ for areentanalysis of matter eets for
at-mospherineutrinos). Thentheearthmattereetson
the neutrino propagation orrespond to a parametri
resonaneandanbealulatedanalytially[25℄,
P
2e =sin
2
s
m +W
2
1 os2
s
m +W
1 W
3 sin2
s
m
; (16)
wheretheparametersW
1 andW
3
ontainallthe
infor-mationoftheEarthdensityandaredenedinRef.[25℄.
(The only dierene is that in our ase also the
o-diagonalelementoftheneutrinoevolutionmatrixvaries
when the neutrinopropagatesthrough theearth
mat-ter.)
Itisthisinterationwiththeterrestrialmatterthat
anprodueaday-nightvariationofthesolarneutrino
ux and, onsequently, a seasonal modulation of the
data. (Notethatthisseasonalvariationisofadierent
nature than the oneexpeted for vauum osillations
from thehangeofthebaselineduetotheeentriity
oftheearth'sorbitaroundthesun.)
Ourmaingoalistodeterminethevaluesof"and" 0
thatanexplaintheexperimentalobservationswithout
modifyingthestandardsolarmodelpreditions.
First we onsider the data on the total event rate
measuredbytheChlorine(Cl)experiment[1℄,the
Gal-lium (Ga) detetors GALLEX [2℄ and SAGE [3℄ and
the water Cherenkov experiment SuperKamiokande
(SK) [5℄. We ompute the allowed regions in
param-eterspaeaordingtotheBP98SSM[9℄andompare
the results with the regions obtained for an arbitrary
normalization f
B
of thehigh energy neutrino 8
B
neu-trinouxes.
Weusetheminimal 2
statistial treatmentof the
0.40
0.45
0.50
0.55
0.60
0.65
0.70
0.75
ε
’
10
-3
10
-2
10
-1
10
0
ε
Ga + Cl + SK (Rates Only)
90 % C.L.
95 % C.L.
99 % C.L.
d-quark
(a)
8
B flux fixed to BP98 SSM
0.40 0.45
0.50
0.55 0.60
0.65
0.70
0.75
ε
’
10
-3
10
-2
10
-1
10
0
ε
Ga + Cl + SK (Rates Only)
90 % C.L.
95 % C.L.
99 % C.L.
d-quark
(b)
8
B flux free
Figure 4. Region of " = d
and "
0
= 0
d
whih an
ex-plainthetotalratesmeasuredbytheHomestake,GALLEX,
SAGEandSuperKamiokandesolarneutrinoexperimentsin
termsofnon-standardneutrinointerations withd-quarks.
(a)Thebest t (indiatedby the openirle) isobtained
for(";" 0
)=(0:0032;0:610)with 2
min
=2:44for 4 2=2
DOF.Aseond(loal) 2
minimum(indiatedbythesolid
square)is foundat (";" 0
) =(0:034;0:610) with 2
=2:63.
(b) Allowing for an arbitrary 8
B ux normalization fB,
the best t (indiated by the open irle) is obtained for
(";" 0
)=(0:022;0:590)and fB =1:36 with 2
min
=0:91for
4 3=1DOF.
InFig.4theallowedregionsintheparameterspae
of d
and
0 d
for neutrino sattering o d-quarks are
shown at 90, 95 and 99% ondene level (CL). In
Fig. 4a, the 8
B ux is xed by the BP98 SSM
pre-dition(f
B
=1). Thebest tpointofthis analysis is
foundat
d
=3:210 3
and
0 d
=0:61; (17)
with 2
min
= 2:44 for 4 2 = 2 degrees of freedom
(DOF). Allowing an arbitrary 8
B ux normalization,
a dierent best t point is obtained for ( d
;
0 d
) =
(2:210 2
;0:59) and f
B
= 1:36 with 2
min
= 0:91
for4 3=1DOF.Theresultofthisanalysisisshown
in Fig.4b. (Eetsdue to deviations of thehep
neu-trinouxfromthestandardsolarmodelpreditionare
0.36
0.38
0.40
0.42
0.44
0.46
0.48
ε
’
10
-4
10
-3
10
-2
10
-1
ε
Ga + Cl + SK (Rates Only)
90 % C.L.
95 % C.L.
99 % C.L.
u-quark
(a)
8
B flux fixed to BP98 SSM
0.36
0.38
0.40
0.42
0.44
0.46
0.48
ε
’
10
-4
10
-3
10
-2
10
-1
ε
Ga + Cl + SK (Rates Only)
90 % C.L.
95 % C.L.
99 % C.L.
u-quark
(b)
8
B flux free
Figure 5. Same as in Fig. 4 but for u-quarks. (a)
The bestt (indiatedby the openirle) is obtained for
(";" 0
) = (0:0013;0:430) with 2
min
= 2:75 for 4 2 = 2
DOF.Aseond(loal) 2
minimum(indiatedbythesolid
square)isfoundat(";" 0
)=(0:0083;0:425)with 2
=2:70.
(b) Allowing for an arbitrary 8
B ux normalization f
B ,
the best t (indiated by the open irle) is obtained for
(";" 0
)=(0:0058;0:425)andf
B
=1:34with 2
min
=0:96for
4 3=1DOF.
InFig.5theallowedregionsintheparameterspae
of u
and
0 u
for neutrino sattering o u-quarks are
shownat90,95and99% CL.InFig.5a,the 8
Buxis
xed bytheBP98 SSMpredition. Thebestt point
ofthisanalysisisfoundat
u
=1:3210 3
and
0 u
=0:43; (18)
with 2
min
=2:64fortwoDOF. Allowinganarbitrary
8
Buxnormalizationf
B
,adierentbesttpointis
ob-tainedfor( u
;
0 u
)=(5:810 3
;0:425)andf
B =1:34
with 2
min
=0:96foroneDOF.Theresultofthis
anal-ysisisshownin Fig.5b.
It isremarkablethat theneutrinoavoronversion
mehanism based on NSNI provides quite a good t
to the total ratesdespite the fat that theonversion
probabilities(11)and (15)do notdepend on the
neu-trino energy. This is unlike the ase of the vauum
and the MSW onversion mehanisms whih provide
theappropriateenergydependene toyield agoodt.
ForNSNI theonly way to distinguishbetween
neutri-nos of dierentenergies is viathe position of the
res-onane r
res
. Note that aording to eq. (9), r
res is
a funtion of " 0
only. In the Sun, n
e =n
f
(f = d;u)
is a smooth and monotoni funtion of the distane
fromthesolarenterr,allowingtouniquelydetermine
r for a given value of " 0
. Consequently, it follows
that aresonaneanonly ourif 0
d
2[0:50;0:77℄for
NSNI withd-quarks or 0
u
2[0:40;0:46℄for NSNIwith
u-quarks. Forbothasesthemajorpartofthese
inter-vals orrespondsto r
res <
0:2R
(R
beingthesolar
radius). For 0
d;u
withinthe90%CLregions(indiated
in Fig.4andFig.5) wendr
res
0:1 R
. Sine the
nulear reations that produe neutrinos with higher
energies in general take plae loser to the solar
en-ter (see hapter 6 of Ref. [11℄ for the various spatial
distributions of the neutrino prodution reations), a
resonanepositionlosetothesolarenterimpliesthat
predominantlythehighenergyneutrinosareonverted
byaresonanttransition. Forr
res
0:1R
pratially
all 8
B-neutrinos ross the resonanelayer, fewer 7
Be-neutrinospassthroughtheresonane,whilemostofthe
pp-neutrinosarenotbeaetedbytheresonanesine
their prodution region extends well beyond the
reso-nane layer. Therefore for most of theallowedregion
inFig.4andFig.5theorrespondingaveragesurvival
probabilityfulllthefollowingrelations:
hP( 8
B)i<hP( 7
Be)i<hP(pp)i: (19)
Wenotethattheaboverelationisstillvalidwhen
tak-ing into aount that a signiant fration of the pp
neutrinosrossestheresonanelayertwie, iftheyare
produed just outside resonane. This is { roughly
speaking { beause a
e
whih undergoes a resonant
avortransitionwhenenteringthesolarinterioratr
res
is reonverted into a
e
at the seondresonanewhen
itemergesagainfrom thesolarore. Inournumerial
alulationsweproperlytakeintoaounttheeetsof
suhdoubleresonanes.
Animmediateonsequeneoftherelationineq.(19)
is that as longas f
B
= 1 the NSNI solution predits
thatR
SK <R
Cl <R
Ga
,whihisinonsistentwiththe
observed hierarhy of the rates, R
Cl < R
SK < R
Ga ,
leading to a somewhat worse t than the standard
MSW solutions. However when treating f
B
as a free
parameter,forf
B
1:3 1:4theSKrateissuÆiently
enhanedtogivetheorretrelationbetweentherates.
Inthis asealsotheneutralurrentontributionfrom
;
e satteringisinreasedduetoalarger
; ux,
whihis onsistent withtheSuper-Kamiokande
obser-vations. Wendthatforthebesttpointsfor(";" 0
)in
Figs.4band5bandf
B
1:35thesurvivalprobability
for 8
B, 7
Be and pp-neutrinosare 0:24, 0:4 and 0:7,
respetively.
Next, we onsider the zenith angle dependene of
thesolarneutrinodataoftheSuperKamiokande
exper-iment. As mentioned above, NSNI with matter may
aet the neutrino propagation throughthe earth
re-sulting in a dierene between theevent rates during
day and night time. The data obtained by the
Su-perKamiokandeollaborationaredividedintovebins
ontainingtheeventsobservedatnightandonebinfor
averaged overthe period of SuperKamiokande
opera-tion: 403.2eetivedaysforthe dayeventsand421.5
eetive days for the night events. The experimental
results suggest an asymmetry between the total data
olleted during the day (D) and the total data
ob-servedduring thenight(N)[60℄:
A=2
N D
N+D
=0:0650:031(stat:)0:013(syst:):
(20)
Inordertotakeintoaounttheearthmattereet
wedene thea 2
-funtionthat haraterizesthe
de-viations ofthesixmeasured(Z obs
i
)from thepredited
(Z th
i
) values of the rate as a funtion of zenith angle
(seeRef. [40℄).
0.40
0.45
0.50
0.55
0.60
0.65
0.70
0.75
ε
’
10
-3
10
-2
10
-1
10
0
ε
SK Zenith Angle Only
excluded at 99 % C.L.
95 % C.L.
90 % C.L.
0.40
0.45
0.50
0.55
0.60
0.65
0.70
0.75
ε
’
10
-3
10
-2
10
-1
10
0
ε
a) d-quark
0.36
0.38
0.40
0.42
0.44
0.46
ε
’
10
-3
10
-2
10
-1
10
0
ε
SK Zenith Angle Only
excluded at 99 % C.L.
95 % C.L.
90 % C.L.
0.36
0.38
0.40
0.42
0.44
0.46
ε
’
10
-3
10
-2
10
-1
10
0
ε
SK Zenith Angle Only
b) u-quark
Figure 6. Regionof"= d
and "
0
= 0d
whihisexluded
bydayandnightdata(ontainedin1+5bins)asmeasured
bytheSuperKamiokandesolarneutrinoexperimentinterms
ofnon-standardneutrinointerationswith(a)d-quarksand
(b) u-quarks. For d-quarks, thebest t (indiatedby the
open irle) is obtained for (";" 0
) = (0:251;0:620) and
Z
=0:819with 2
min
=1:10for6 3=3DOF.Aseond
(loal) 2
minimum(indiatedbythesolidsquared)isfound
at (";" 0
) =(0:0316;0:570)and
Z
=1:02 with 2
=5:20.
For u-quarks,the bestt(indiatedbythe openirle)is
obtained for (";" 0
) = (0:229;0:690) and
Z
= 0:685 with
2
min
=1:44for6 3=3DOF.
InFig.6weshowtheallowedregionsin the(";" 0
)
parameter spae for neutrino sattering o d- and
u-quarks, respetively. The ontours in Fig. 6
orre-spond to the allowed regions at 90, 95 and 99% CL.
The best t (indiatedby theopen irle) isobtained
for ( d
;
0 d
) = (0:251;0:62) and
Z
= 0:819 with
2
min
= 1:10 for neutrino sattering o d-quarks and
at ( u
; 0
u
) = (0:229;0:690) and
Z
= 0:685 with
2
min
=1:44forneutrinosattering ou-quarks
(hav-ing6 3=3DOFinbothases).
0.4
0.5
0.6
R/R
SSM
-1.0
cos(
Φ
Z
)
0.0
0.2
0.4
0.6
0.8
1.0
(
ε
,
ε
’)=(0.251,0.620)
(
ε,ε
’)=(0.018,0.585)
D
N1
N2
N3
N4
N5
a) d-quark
0.4
0.5
0.6
R/R
SSM
-1.0
cos(
Φ
Z
)
0.0
0.2
0.4
0.6
0.8
1.0
b) u-quark
(
ε
,
ε
’)=(0.091,0.430)
(
ε,ε
’)=(6.3x10
-3
,0.425)
D
N1
N2
N3
N4
N5
Figure7. Expetedzenith angle dependenewiththe our
besttvaluesof(";" 0
)determinedbytheSKZenithangle
onlyas wellas theombinedanalysisfor (a)d-quarks and
(b)u-quarks.
Finally,inFig.7weshowtheexpetedzenithangle
distributionsforSuperKamiokandeusing thevaluesof
(; 0
) determined by thebest t. Foromparison, we
alsopresentinthisguretheexpetedzenithangle
dis-tributionsfor thebest t valuesof (; 0
)found in the
ombinedanalysis (thatwillbedisussedlater).
We also onsider the measurements of the reoil
eletronspetrumbySuperKamiokande[60℄. Dataare
dividedinto 18bins. 17 ofthese bins haveawidth of
0.5MeVandaregroupedintotwobinsforasuperlow
energyanalysiswithenergiesbetween5.5MeVand6.5
MeVand15 binswithenergiesranging from 6.5MeV
(thelowenergylimit)to14MeV.Thelastbininludes
alltheeventswithenergieslargerthan14MeV.
Sinetheeletronneutrinosurvivalprobabilitydoes
not depend on the neutrino energy in the NSNI
se-nario,thespetraldistortionofthereoileletronsfrom
8
B neutrino due to the presene of a
;
omponent
in the neutrino ux is expeted to be very small [61℄
and therefore, even a relatively small spetral
distor-tion (suh as the one expeted in small mixing angle
MSWsolution)ouldruleoutthissolution.
The 2
-funtion that haraterizes the deviations
of the measured (S obs
i
) from the predited (S th
i )
val-uesforthe eletronreoilspetrumtherefore provides
present data to our senario we obtain 2
min
= 20:0
for 18 1=17 DOF, whih is still aeptable at the
27% CL.
Theearth matter eets on neutrino avor
transi-tions indue aseasonal variation of the data (beyond
theexpetedvariationofthesolarneutrinouxdueto
the eentriity of the earth's orbit) due to the
varia-tion ofthe day andnight timeduring the year. Sine
these variationsan be relevant to other neutrino
os-illation senarios [62℄, a positive signal ouldhelp to
distinguish the various solutions and it is worthwhile
to analyze theeets of suh avariation in the NSNI
senario.
0.40
0.45
0.50
0.55
0.60
0.65
0.70
0.75
ε
’
10
−3
10
−2
10
−1
10
0
ε
Ga + Cl + SK (Rates + Zenith + Spectrum)
90 % C.L.
95 % C.L.
99 % C.L.
d−quark
(a)
8
B flux fixed to BP98 SSM
0.40
0.45
0.50
0.55
0.60
0.65
0.70
0.75
ε
’
10
−3
10
−2
10
−1
10
0
ε
Ga + Cl + SK (Rates + Zenith + Spectrum)
90 % C.L.
95 % C.L.
99 % C.L.
d−quark
(b)
8
B flux free
Figure 8. Theallowedregion for " = d
and "
0
= 0d
ob-tained by the ombined analysis using 4 rates +6 zenith
angle bins+ 18 spetrum bins for non-standard neutrino
interations with d-quarks. (a) Fixing fB = 1 the best
t (indiated by the open irle) is obtained for (";" 0
) =
(0:028;0:585) with 2
min
= 29:05 for 28 4 = 24 DOF.
There are two additional (loal) 2
minima at (";" 0
) =
(0:0033;0:610) with 2
= 29:40 (indiated by the solid
square)and (";" 0
)=(0:21;0:61)with 2
=33:1(indiated
by the ross). (b)Same as in(a) butallowing afree f
B .
The bestt (indiatedby the openirle) is obtained for
(";" 0
)=(0:018;0:585)andf
B
=1:38with 2
min
=26:62for
28 5=23DOF.
ThepresentSKsolarneutrinodatadonotprovide
anyonlusiveevidenein favorofsuhavariation. It
onlyindiatesthatthevariationmightbelargerfor
re-oileletronenergiesabove11.5MeV.Inoursenario,
however,wedonotexpetanyorrelationbetweenthe
seasonalvariationandthereoileletronenergies,sine
theeletronneutrinosurvivalprobabilitydoesnot
de-parametersthatleadstoaonsiderableseasonal
modu-lationforenergiesabove11.5MeVisdisfavoredbythe
data forlowerenergies. However,forthe rangeof
pa-rameters(";" 0
)that ansolvethesolarneutrino
prob-lem,earthregenerationeetsareneverstrongenough
to indue a signiant seasonalvariation. Hene
tak-ingintoaountthedataonseasonalvariationsneither
hanges the shapeof theallowed region, northe best
tpoints.
0.36
0.38
0.40
0.42
0.44
0.46
0.48
ε’
10
-4
10
-3
10
-2
10
-1
ε
Ga + Cl + SK (Rates+Zenith+Spectrum)
90 % C.L.
95 % C.L.
99 % C.L.
u-quark
a)
8
B flux fixed to BP98 SSM
0.36
0.38
0.40
0.42
0.44
0.46
0.48
ε
’
10
-4
10
-3
10
-2
10
-1
ε
90 % C.L.
95 % C.L.
99 % C.L.
b)
8
B flux free
u-quark
Ga+Cl+SK (Rates+Zenith+Spectrum)
Figure 9. Same as inFig. ?? butfor u-quarks. (a)
Fix-ing f
B
= 1 the best t (indiated by the open irle) is
obtainedfor (";" 0
)=(0:0083;0:425)with 2
min
=28:45for
28 4=24 DOF.A seond(loal) 2
minimumis found
at (";" 0
) = (0:0013;0:430) with 2
= 30:27 (indiated by
thesolidsquare)(b)Sameasin(a)butallowingafreefB.
The bestt (indiated by the openirle) is obtained for
(";" 0
)=(0:0063;0:426)and fB =1:34 with 2
min
=26:59
for28 5=23DOF.
Our nal result is the t derived from the
om-bined analysis of all presentlyavailable solar neutrino
data. In Fig. 8 and Fig. 9 we show the allowed
re-gionsfor( d
;
0 d
)and( u
;
0 u
),respetively,usingboth
the results from the total rates from the Chlorine,
GALLEX,SAGEandSuperKamiokandesolarneutrino
experiments together with the 6 bins from the
Su-perKamiokandezenithangledatadisussedpreviously.
Although addingthespetralinformationto our
anal-ysis does nothange the shape of allowed regions nor
thebestt points,itisinludedin ordertodetermine
the quality ofthe globalt. However, wedo nottake
Forneutrinosatteringo d-quarksthebestt for
theombineddataisobtainedfor
d
=0:028 and
0 d
=0:585 (21)
with 2
min
=29:05for28 4=24DOF,orresponding
to a solution at the 22% CL (see Fig. 8a). Allowing
f
B
6=1,thebesttisfoundat( d
;
0 d
)=(0:018;0:585)
and f
B
= 1:38 with 2
min
= 26:62 for 28 5 = 23
DOF, orrespondingto asolutionat the27% CL(see
Fig.8b). Forneutrino satteringou-quarksthe best
tfortheombineddataisobtainedfor
u
=0:0083 and
0 u
=0:425 (22)
with 2
min
=28:45 for 28 4 =24 DOF
orrespond-ing to a solution at the 24% CL (see Fig. 9a).
Al-lowingf
B
6=1, the best t is obtained for ( u
;
0 u
) =
(0:0063;0:426) and f
B
= 1:34with 2
min
= 26:59 for
28 5= 23 DOF, orresponding to a solution at the
27% CL (see Fig.9b). These resultshaveto be
om-pared with the t for standardmodel neutrinos, that
do not osillate(where theCL is smaller than 10 7
),
as well as to the standard solutionsof the solar
neu-trino problem in terms of usual neutrino osillations
(36%CL)[18,19℄.
Finally, in Fig.7 weshowthe expeted zenith
an-gle distributions for SuperKamiokande using the best
ttedvaluesof(; 0
)from theombinedanalysis.
Aordingtoour 2
analysisnon-standardneutrino
interations (NSNI) an provide a good t to the
so-lar neutrinodata provided that there are rather large
non-universalFDNI(oforder0:5G
F
)and smallFCNI
(oforder10 2
10 3
G
F
). Thetto theobserved
to-tal rate, day-nightasymmetry, seasonalvariationand
spetrum distortion of the reoil eletron spetrum is
omparableinqualitytotheoneforstandardneutrino
osillations.
From the model-independent analysis of Ref. [40℄
weleranthatNSNI induedbytheexhangeof heavy
bosons annot provide large enough
e
!
transi-tions, while
e
FCNI in priniple ould be
suÆ-iently strong. However, the urrent bounds will be
improved by the up-oming B-fatories, providing an
independent test of the NSNI solution. The required
largenon-universalFDNI(for
e
transitionsintoboth
and
) anbe ruled out by the upper bounds on
leptonuniversality,unlesstheyareinduedbyan
inter-mediatedoubletofSU(2)
L
(asalaroravetorboson)
or bya neutralvetorsinglet. For
e !
there
ex-ists abound due tothe limitonompositeness inthis
ase, butfor
e !
thereis nosigniantonstraint
at present.
Generiallyonlyveryfewmodels anfulll the
re-quirements needed for the solution disussed in this
paper: massless neutrinos, small FCNI and relatively
large non-universal FDNI. As for the vetor bosons
themost attrativesenario is to evokean additional
U(1)
B 3L
gauge symmetry (where B is the baryon
numberandL
denotesthetauleptonnumber),whih
would introdue anadditionalvetorsinglet that only
ouplestothethirdgenerationleptonsandquarks[63℄.
Among the attrative theories beyond the standard
modelwhereneutrinosarenaturallymasslessasaresult
of a proteting symmetry, are supersymmetri SU(5)
models[64℄ thatonserveB L,and theorieswithan
extendedgaugestruturesuhasSU(3)
C
SU(3)
L
U(1)
N
models [65℄,where a hiral symmetryprevents
the neutrino from getting a mass. These partiular
models,however,donotontributesigniantlytothe
spei interations we are interested in this paper.
SU(5)modelshavenegligibleNSNI sinetheyare
me-diatedbyvetorbosonswhihhavemassesattheGUT
sale. SU(3)
C
SU(3)
L
U(1)
N
models an provide
large
e and
0
e
, but thesemodelsdo notindue NSNI
withquarks. Fromeq. (9) it follows that noresonant
onversionanourinthis ase.
Therefore we onludethat the best andidate for
the senario we studied are supersymmetri models
withbrokenR -parity,wheretherelevantNSNIare
me-diated by a salar doublet, namely the \left-handed"
bottom squark. Although in this model neutrino
masses are not naturally proteted from aquiring a
mass, one may either evoke an additional symmetry
orassume that non-zeroneutrino masses are not in a
rangethatwouldspoilthesolutionintermsofthe
non-standardneutrinoosillations.
IV Violation of the equivalene
priniple
Letme brey mentionthat adierent solutionto the
solar neutrino anomaly based on the violation of the
equivalenepriniplewasreentlyrevisited[72℄. Inthis
ontext,neutrinomixing andavorosillationsare
in-duedby gravitationalfores. It isassumed that
neu-trinosofdiferentspeieswill inurdierenttimedelay
totheweak,statigravitationalledintheintervening
spae on their way from the sun to the earth. Their
motioninthisgravitationaleldisdesribedassuming
adierentneutrinogravitationalouplingforeah
neu-trinotype. In this way, weak intereating eigenstates
andgravitationalinteratingeigenstateswillberelated
byaunitarytransformationthat anbeparametrized,
assumingonlytwoneutrinoavors,byasingle
param-eter, the mixing angle
G
whih an leadto neutrino
osillations. Theevolutionequationsfor these avors,
whihareassumedtobedegenerateinmass,
propagat-ingthroughthegravitationalpotential(r)inabsene
i d
dt
e
x
=2E(r)
os2
G
sin2
G
sin2
G
os2
G
e
x
; (23)
d
where E istheneutrinoenergy,
x =
;
or
S
;
is the quantity whih measures the magnitude of the
violationof theequivalene prinipleand itis the
dif-ferene ofthe gravitationalouplingsbetweenthetwo
neutrinosinvolvednormalizedbythesum.
Case sin 2
2
G
jj f
B
2
min
Rates 1.0 1.71 x=1 1.49
Rates 1.0 1.70 0.81 0.32
Spetrum 0.98 1.00 0.80 15.8
Combined 1.0 1.65 0.82 22.0
TABLEII.Thebestttedparametersand 2
min forthe
violationoftheequivalenepriniplesenario.
Applying the same statistial treatment shown in
Ref. [34℄, the allowed region in the parameter spae
jjsin 2
2
G
determinedonlybytheratesisshown
inFig.10(a),whihwastakendiretlyfromRef. [72℄.
Itwasfoundthat,onsideringafreenormalization
fa-torf
B forthe
8
B neutrino ux, 2
mim
=0:32 for3-3
degrees of
10
−24
10
−23
10
−22
|
φ∆γ|
0.0
0.2
0.4
0.6
0.8
1.0
sin
2
2
θ
G
10
−26
10
−25
10
−24
10
−23
10
−22
|
φ∆γ|
0.0
0.2
0.4
0.6
0.8
1.0
sin
2
2
θ
G
10
−24
10
−23
|
φ∆γ|
90 % C.L.
95 % C.L.
99 %.C.L.
(b) SK Spectrum Only
Cl+Ga+SK
8
B flux free
8
B flux free
(c) Rates
8
B flux free
+ SK Spectrum
(a) Rates Only
Figure 10. Allowed parameter region for (a) the rates,
(b) SuperKamiokande spetrum, and () rates plus
Su-perkamiokandespetrumombined. Thebesttpointsare
indiatedbytherossesandtheloalbesttpointsinother
90%C.L.islandsareindiatedbytheopenirles.
freedom. Consideringaspetralshapeanalysis, 2
mim =
Fig.10(b). Finally, theombinedresult (rates
+spe-trum) is shown in Fig. 10(), whih provides 2
mim =
22.0 for 21-3 degrees of freedom. Details of this
pi-ture andthe valuesof thebest t points arefound in
TableII.
In onlusion, a solution to the solar neutrino
anomaly whih is oomparable in quality of the t to
theothersuggestedones,anbefoundbymeansofthe
violationoftheequivalenepriniple.
V Conlusion
Even though the onventional osillation mehanisms
an be onsidered the most plausible solutions to the
solarneutrinoproblem,itisimportanttorealizethatin
generalNewPhysisintheneutrinosetorinlude
neu-trinomassesandmixing,aswellasnewneutrino
inter-ationswhihangenerate,inpriniple,largeneutrino
magneti moment, neutrino avor-hangingand
non-universal proesses, violation of the equivalene
prin-iple. While it is diÆult to explain the atmospheri
neutrino problem [74℄ and the LSND anomalies [75℄
by these alternative mehanisms, we have shown in
this review that asolutionof thesolar neutrino
prob-lembasedonphenomenawhiharenottheonvention
mass-indued neutrino osillationsis still viable. The
ultimate goal is of ourse a diret experimental test
ofthesesolution. Theupomingsolarneutrino
experi-mentswillprovidealotofnewinformationwhih
hope-fully will reveal the true nature of the solar neutrino
problem.
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