Conserving the lepton number $L_{e}-L_μ-L_τ$ in the exact solution of a 3-3-1 gauge model with right-handed neutrinos

0.11 MB·English

by  Adrian Palcu

Conserving the lepton number L L L in
e µ τ
− −
the exact solution of a 3-3-1 gauge model with
right-handed neutrinos
7
0
0
ADRIAN PALCU
2

n
Departmentof Theoreticaland ComputationalPhysics -West
a
J Universityof Timis¸oara,V. PârvanAve. 4, RO -300223Romania

9

1 Abstract
v
Inthispaperweconsider aplausiblescenariowithconserved leptonnumber
66 L=Le−Lµ−Lτ withintheframeworkoftheexactsolutionofaparticular3-
3-1gaugemodel. Wediscusstheconsequencesofconservingthisgloballeptonic
0
symmetryfromtheviewpointoftheneutrinomassmatrixconstructedviaspecial
1
0 Yukawaterms(involvingtensorproductsamongHiggstriplets).Weprovethatthe
7 actual experimental data can naturally be reproduced by our scenario since soft
0 breakingtermswithrespecttothisleptonsymmetryareproperlyintroduced. Asa
/ consequence,oursolutionpredictsfortheneutrinosectorthecorrectmasssplitting
h
p ratio(∆m212/∆m223 ≃0.033),theinvertedmasshierarchy,thecorrectvaluesfor
- theobservedmixingangles(sin2θ23 ≃0.5andsin2θ12 =0.31)andtheabsolute
p massofthelightestneutrino(m0 ∼ 0.001eV)independentofthebreakingscale
e ofthemodel.
h
PACSnumbers:14.60.St;14.60.Pq;12.60.Fr;12.60.Cn
:
v Keywords:neutrinomasses,3-3-1models,leptonnumber
i
X
r 1 Introduction
a
In a recent series of papers (Refs. [1, 2, 3]), the author has developed an original
methodforcontructingtheneutrinomassmatrixwithintheframeworkofaparticular
3-3-1 gauge model with no other restrictive additional symmetries (such as the lep-
tonnumber). ApropertensorproductamongtheHiggstriplets- designedto recover
(successivelythespontaneoussymmetrybreakdown)thewellknownmass-generating
Yukawatermsintheunitarygauge-isexploitedwithintheexactsolutionofamodel
basedonthegaugegroupSU(3) SU(3) U(1) thatdoesnotcontainparticles
c L X
⊗ ⊗
withexoticelectriccharges.Herewemeanbytheexactsolution(seeforfurtherdetails
thegeneralprescriptionsinRef.[4])aspecificalgebraicalmethodrelyingonanappro-
priateparametrizationin thescalar sectorof themodel. Itprovidesuswith theexact
masseigenstatesandmasseigenvaluesforthegaugebosonsandthecharges(boththe
electric and neutralones) of all the involvedparticles, justby exactly solvingcertain
equations.
1
ThechargedfermionsacquiretheirmassesthroughtraditionalYukawacouplings.
Atthe same time, theyessentially determinethe textureof the neutrinomassmatrix,
since the same coupling coefficient acts for both the charged lepton and its neutrino
partner. All these results can be achieved [1] just by tuning a sole free remaining
parameterinthemodel. Regardingtheneutrinosector,thepredictionsclaimedbythis
methodinclude[2]: theinvertedmasshierarchy,thebi-maximalmixingand-forthe
firsttimeintheliterature,asweknow-theminimalabsolutevalueintheneutrinomass
spectrumm 0.0035eV.
0
≃
Thepriceofdealingwithonlyonefreeparameterresidedinaverylargebreaking
scaleofthemodeland,consequently,insomeveryheavynewgaugebosonsthathad
largelyovertakentheir lower experimentalmass limit [5] (arround1TeVfor non-SM
bosons).Inordertoimprovethephenomenologicalconsequencesregardingthebreak-
ingscale,anewapproachwasproposed:acanonicalseesawmechanismcanariseinthe
model,justbyalteringtheparametermatrixofthescalarsectorwithasmallamount(a
fine-tuningparameter)[3]. Thissecondparameterallowsfordecouplingtheneutrino
phenomenologyfrom the breaking scale issue (generatingthus reasonable masses in
thebosonsector), whilekeepingunalteredallthe neutrinomassesandmixingangles
achievedintheone-parameterversion[2].
Hereweintendtoimprovetheoriginalone-parametersolutionofthe3-3-1model
ofinterest(brieflypresentetdinSec. 2)inadifferentway. Thistime,anewsymmetry
istakenintoconsideration,namelythegloballeptonnumberL=L L L (Sec.
e µ τ
− −
3)anditsimplicationsforourmodelarediscussed. Then,somenewsmallparameters
(α,β,γ)areintroducedinthespecialYukawatermsinordertosoftlybreakthisglobal
leptonsymmetry.Whentheµ τ interchangesymmetryisinvokedtheparametersget
−
aparticularratio,namelyγ/β =m(µ)/(τ). Underthesecircumstances,thediagonal-
izationofthemassmatrixleadstopredictionsingoodagreementwiththeexperimental
values for mass splitting ratio and mixing angles. We also consider as a big success
ofourscenariotheminimalabsoluteneutrinomasscomputedindependentofanypa-
rameter,dependingonlyonthepreciseaccountforthemixingangles. Afewremarks
concerningphenomenologicalaspectsofourmethodaresketchedinthefinalsection
(Sec. 4).
2 Preliminaries
Inordertogettotheveryessenceoftheresultingphenomenologyofintroducingthe
globalleptonsymmetryL=L L L intothe3-3-1modelwithoutexoticelec-
e µ τ
− −
triccharges,westartbypresentingtheparticlecontentofthismodelandtheoriginal
mannertogenerateneutrinomassmatrixviaspecialYukawaterms.
2.1 Fermioncontent ofthemodel
Thefemionsectorofthepureleft(subscriptL)3-3-1gaugemodelwithright-handed
neutrinos(namelymodelD inRef. [6])consistsoftwo distinctsectors: leptonfami-
liesandquarkfamilies. Alltheleptongenerationsobeythesamerepresentationwith
respecttothegaugegroup:
2
e
α
f = ν (1,3∗, 1/3) ec (1,1,1) (1)
αL  α  ∼ − αL ∼
νc
α L
 
wheree =e,µ,τarethewellknownchargedleptonsfromSM,whileν =ν ,ν ,ν
α α e µ τ
are their neutrinopartners. We mean by superscriptc the chargeconjugationcarried
outbytheoperatorC =iγ2γ0(formoredetailswerefferthereadertoAppendixBin
Ref. [4]).
Thequarkscome(accordingtotheanomalycancellationrequirementthatpreserves
therenormalizability)inthreedistinctgenerationsasfollows:
u d
i
Q = d (3,3,0) Q= u (3,3∗,1/3) (2)
iL i
  ∼   ∼
D U
i L L
   
(d )c,(d )c (3∗,1,1/3) (u )c,(u )c (3∗,1, 2/3) (3)
L iL L iL
∼ ∼ −
(U )c (3∗,1, 2/3) (D )c (3∗,1,1/3) (4)
L iL
∼ − ∼
with i = 1,2 and capital letters denoting exotic quarks. The numbers in brackets
followingthe left-handedfermionfelds (Eqs. (1) - (4)) labelthe representationsand
theircharacterswithrespecttothegaugegroupSU(3) SU(3) U(1) . Notethat
c L X
⊗ ⊗
thethirdquarkfamilyhastoobeyadifferentrepresentationcomparedtotheothertwo
families.
ThemainphenomenologyoftheexactsolutionofthismodelcanbefoundinRef.
[1]whereaspecialgeneralizedWeinbergtransformationisalsogiveninordertosep-
aratetheneutralbosons(A -theelectromagneticone,Z - theWeinbergbosonfrom
µ µ
Standard Model, and Z′- the new neutral boson of the present theory) that get their
µ
exacteigenstates.Hereweareconcernedonlywiththeneutrinomassissuesincesome
specialcouplingtermsareintroducedinthe Yukawasectoranda globalsymmetryis
addedaswell.
2.2 The resulting neutrino massmatrix
ThemassesofthefermionsinthemodelaregeneratedbytheYukawaLagrangian. In
ourapproach[1]itlookslike:
G f¯ (φ(ρ)ec +Sfc )+h.c. (5)
αβ αL βL βL
with S = φ−1 φ(η) φ(χ)+φ(χ) φ(η) (1,6, 2/3) and G as the cou-
αβ
⊗ ⊗ ∼ −
plingcoefficientsofthe leptonsector. The Yukawasectorreliesonthe scalartriplets
φ(ρ) (1,3∗,2(cid:0)/3),φ(χ),φ(η) (1,3∗, 1(cid:1)/3) (see for details of constructingthe
∼ ∼ −
Higgssectorin3-3-1modelswithnoexoticelectricchargesRef. [1]). Werecallthat
(cid:8) (cid:9)
theexactsolutionforthe3-3-1modelofinteresthereleadstotheone-parameterma-
trix η = 1 η2 diag a/2cos2θ ,1 a,a 1 tan2θ /2 which determines
− 0 W − − W
(cid:0) (cid:1) (cid:2) (cid:0) (cid:1) (cid:3)
3
theVEVsalignmentintheHiggssector φ(i) = ηi φ ,i = 1,2,3(successivelythe
h i
SSB),accordingtothegeneralprescriptionsofthemethodshowninSec. 4.1ofRef.
(cid:10) (cid:11)
[4]. Obviously,θ istheWeinberganglefromSM.
W
NotethattheHiggsfieldφplaystheroleofthe”norm”forthegeometrizedscalar
sector and it is therefore a main factor in all the VEVs. It is appropriate for one to
consider it as the homologueof the neutral scalar field of the Standard Model, since
their vacuumexpectationvaluessupplythe massesfor the particle of the modelthey
actin.
ThechargedleptonsgettheirmassesviatraditionalYukawacouplingsasonecan
easily observe in Eq.(5): m(e) = A φ(ρ) , m(µ) = B φ(ρ) , m(τ) = C φ(ρ) .
Obviously, G = A, G = B, G = C, but we specify in advance that our
ee µµ ττ
(cid:10) (cid:11) (cid:10) (cid:11) (cid:10) (cid:11)
notationsinclude: G = D, G = E, G = F forthe off-diagonalterms in the
eµ eτ µτ
flavorbasis.
Neutrino mixingis expressedby ν (x) = 3 U ν (x), where α = e,µ,ν
αL i=1 αi iL
labelthe flavor space (flavor eigenstates)while i = 1,2,3denotethe massive physi-
P
caleigenstates. WeconsiderthroughoutthepaperthephysicalneutrinosasMajorana
fields,i.e. νc (x)=ν (x). TheneutrinomasstermintheYukawasectoryieldsthen:
iL iL
1
= ν¯ Mνc +H.c (6)
−LY 2 L L
T
withν = ν ν ν wherethe superscriptT standsfor”transposed”. The
L e µ τ L
mixing matrix U that diagonalizesthe mass matrix in the mannerU+MU = m δ
ij j
(cid:0) (cid:1)
hasinthestandardparametrizationtheform:
c c s c s e−iδ
2 3 2 3 3
U = s c c s s eiδ c c s s s eiδ c s (7)
2 1 2 1 3 1 2 2 3 1 3 1
 − − − 
s s c c s eiδ s c s s c eiδ c c
2 1 2 1 3 1 2 2 3 1 3 1
− − −
 
with natural substitutions: sinθ = s , sinθ = s , sinθ = s , cosθ = c ,
23 1 12 2 13 3 23 1
cosθ =c ,cosθ =c forthemixingangles,andδfortheCPphase.
12 2 13 3
Ourprocedureprovides(aftertheSSB)thefollowingnewsymmetricmassmatrix
fortheneutrinosinvolvedinthemodel:
A D E
φ(η) φ(χ)
M =4 D B F (8)
  φ
E F C (cid:10) h(cid:11)(cid:10)i (cid:11)
 
Thenexttaskistodiagonalizethematrix(8)inordertogetthephysicaleigenstates
ofthemassiveneutrinos.Thisprocedurewilllead(withintheCase1inRef.[1],which
istheonlyacceptableonefromthephenomenologicalpointofview)tothefollowing
genericsolution:
2a 1 2sin2θ
m =f [θ ,θ ,θ ,m(e),m(µ),m(τ)] − W (9)
i i 12 23 13 (cid:18)√1−a(cid:19)p cos2θW
with i = 1,2,3. In these expressions f s are analytical functions depending on the
i
mixinganglesandthechargedleptonmassesinaparticularwaytobedetermined.
4
=
3 Conserving the lepton number L L L L
e µ τ
− −
InthefollowingweassumethattheleptonnumberL = L L L isconserved
e µ τ
− −
(orapproximatelyconserved,assuggestedbyalotofpapersconcerningthisissue[7]
-[21])inthe3-3-1modelpresentedabove. Wefirstanalyzetheconsequencesofthis
globalsymmetryfor the neutrinosector andthenthe modificationsprovidedby soft-
breaking terms with respect to this symmetry. The modifications occur in the mass
matrixduetosomenewsmallparametersintroducedintheYukawasector.
3.1 Neutrino massspectrum
Atthispoint,ifoneimposesfortheleptonsectoranadditionalglobalsymmetrygiven
bytheleptonnumberL=L L L ,certaincouplingcoefficientsintheYukawa
e µ τ
− −
Lagrangiangetsuppresed,namelyA=B =C =F =0,sincethefollowingassigne-
mentfortheleptonnumberholds:L(f )=L(e )=1,L(f )=L(µ )= 1and
eL R µL R
−
L(f )=L(τ )= 1,whileallthescalarfieldscarryzeroleptonnumber.
τL R
−
Hence,themassmatrixwiththeexactglobalLsymmetrybecomes:
0 D E
2a 1 2sin2θ
M = D 0 0 − W φ (10)
 E 0 0 (cid:18)√1−a(cid:19)p cos2θW h i
 
Theconcreteformsoff s remaintobe computedbysolvingthefollowingsetof
i
equations:
f = 2c c s D+2s c s E
1 1 2 2 1 2 2
−
0= c s2D+c s2D s c2E+s s2E
 − 1 2 1 2 − 1 2 1 2
 0=cf1c222=D002==−c1sccc1122ssss22212DDDD+++−scc2121sscs11222cEEE2s−2Es1c22E (11)
0=s c D+c c E
obtainedstraightforwardlyfromE0q=. (s1110sf)223vD=ia+U0c+11Ms22UE =diag(m1,m2,m3). Thelines
3, 6, 7 and 8 in Eqs. (11) express the same condition, namely: D = Ecotθ .
23
−
Thelines2and4inthesetofequations(11)arefulfiledsimultaneouslyifandonlyif
cos2θ =sin2θ (maximalsolarmixingangle).
12 12
Under these circumstances, taking into consideration the maximal atmospheric
mixingangletoo,thesolutionreads:
2a 1 2sin2θ
m = m =√2D − W φ (12)
1 | 2| (cid:18)√1−a(cid:19)p cos2θW h i
m =0 (13)
3
5
givingrisetoaµ τ interchangesymmetry[22]-[30].
−
3.2 Introducing softbreaking terms
We exploit in the following the consequencesof assuming some soft breaking terms
withrespecttothegloballeptonnumberL,byintroducingthefreeparametersα,β,γ
intheYukawaLangrangian:
G f¯ (φ(ρ)ec +αSfc )+h.c.
ee eL L eL
G f¯ (φ(ρ)µc +βSfc )+h.c. (14)
 µµ µL L µL
 Gττf¯τL(φ(ρ)τLc +γSfτcL)+h.c.
Obviously,α, β, γ [0,1). A small nonzeroF = βGµτ couplingcoefficientis also
∈
allowed.Now,themassmatrixoftheneutrinosectorstands:
αA D D
2a 1 2sin2θ
M = D βB βF − W φ (15)
 D βF γC (cid:18)√1−a(cid:19)p cos2θW h i
 
InordertoremaininthespiritoftheleptonsymmetryL = L L L which
e µ τ
− −
alsofavorstheinterchangesymmetryµ τ,onecanapproximate:
−
βB =γC (16)
∼
Hence,oneofthethreenewparametershasdisappeared. Thetworemainingones
finally will have to fulfil a particular ratio in order to be compatible with the phe-
nomenologicaldata, so one remains with only one parameter to be tuned apart from
themainone(ainourmodel).
Thenewresultingmassmatrix
αA D D
2a 1 2sin2θ
M(ν)= D βB βF − W φ (17)
 D βF βB (cid:18)√1−a(cid:19)p cos2θW h i
 
isnowtobediagonalized.ThistaskwasalreadycarriedoutinRef. [2]forthegeneral
case,nowwejusthavetoinsertthenewdiagonalentriesinthesolutionobtainedthere
(Eqs. (13)inRef. [2])andtoimposethemaximalatmosphericmixingangleatleast
inthenumerators,whilededenominatorswillbecomputedinthefinalstageofsolv-
ingthemassissuebyinsertingasuitableapproximationforthemaximalatmospheric
mixingangle.Hence,aninterestingmasssplitisobtained:
sin2θ 1 sin2θ
f = βB 12 +αA − 12 ,
1 − 1 2sin2θ 1 2sin2θ 1 2sin2θ
23 12 (cid:0) 12(cid:1)
− − −
(cid:0) (cid:1)(cid:0) (cid:1) (cid:0) (cid:1)
sin2θ sin2θ
12 12
f =βB +αA ,
2 1 2sin2θ 1 2sin2θ 1 2sin2θ
23 12 12
− − −
(cid:0) (cid:1)(cid:0) (cid:1) (cid:0) (cid:1)
f =βB. (18)
3
6
Evidently,itisrequiredaγ5transformationperformedonthefirstneutrinofieldin
ordertogetthesignchangeforitsmass(m ).
1
Themassspectrumintheneutrinosectorbecomes:
βm(µ)sin2θ αm(e) 1 sin2θ 2a 1 2sin2θ
m = 12 − 12 − W,
| 1| " 1−2sin2θ23 1−2sin2θ12 − 1−(cid:0)2sin2θ12 (cid:1)#(cid:18)√1−a(cid:19)p cos2θW
(cid:0) (cid:1)(cid:0) (cid:1) (cid:0) (cid:1)
βm(µ)sin2θ αm(e)sin2θ 2a 1 2sin2θ
12 12 W
m = + − ,
2 " 1−2sin2θ23 1−2sin2θ12 1−2sin2θ12 #(cid:18)√1−a(cid:19)p cos2θW
(cid:0) (cid:1)(cid:0) (cid:1) (cid:0) (cid:1)
2a 1 2sin2θ
m =βm(µ) − W (19)
3 (cid:18)√1−a(cid:19)p cos2θW
3.3 Phenomenological predictions
The physical relevant magnitudes for the neutrino oscillations are the mass squared
differencesfor solar and atmosphericneutrinos, defined as: ∆m2 = m2 m2 and
12 2 − 1
∆m2 =m2 m2. Theyresultfromtheaboveexpressions(Eqs. (19)):
23 3− 2
m(e)m(µ)sin2θ 4a2 1 2sin2θ
∆m2 =2αβ 12 − W (20)
12 ∼ 1−2sin2θ12 2 1−2sin2θ23 (cid:18)1−a(cid:19)(cid:18) cos4θW (cid:19)
(cid:0) (cid:1) (cid:0) (cid:1)
m2(µ)sin4θ 4a2 1 2sin2θ
∆m2 =β2 12 − W (21)
23 ∼ 1−2sin2θ12 2 1−2sin2θ23 2 (cid:18)1−a(cid:19)(cid:18) cos4θW (cid:19)
Themasssp(cid:0)littingratiode(cid:1)fin(cid:0)edasr =∆m(cid:1)2 /∆m2 yieldsinourscenario:
∆ 12 23
α m(e) 1 2sin2θ
r =2 − 23 (22)
∆ β m(µ) sin2θ
(cid:18) (cid:19)(cid:18) (cid:19)(cid:18) 12 (cid:19)
As inallthe caseswhereourmethodofgeneratingneutrinomassesis employed,
the mass splitting ratio comes out independently of the breaking scale in the theory.
The latter is determined by the parameter a which is not involved in formula (22).
Once again, we have got an important conclusion: the breaking scale of the model
(andconsequently,thebosonmassspectrum)andtheneutrinophenomenologydonot
influenceoneanother(thesamehappensinRef. [3],butadifferentstrategybasedon
a seesaw mechanismwas employedtherein). Therefore,as soon as new data regard-
ingtheprecisionmeasurementsforthemassofthenewneutralboson(otherthanthe
WeinbergbosonfromSM)areavailable,onecanestablishtheparametera. Asfaras
we know [5], the lower limit is m(Z′) 1.5TeV, that claims in our solution for the
≥
3-3-1mode:la 0.06and φ 1TeV[1,3].
≤ h i≥
The next step is to implement the specific values for the atmospheric and solar
mixinganglesintoformula(22)andestimatehowtheycaninfluencethemasssplitting
7
ratio. For instance, if the plausible sin2θ = 0.499 and sin2θ = 0.31 are taken
23 12
into account, and assuming the charged lepton masses [5] m(e) = 0.511 MeV and
m(µ)= 106MeV,oneobtainesr 0.033(veryclosetothevaluesuppliedbydata)
∆
≃
if
α
=530.5 (23)
β
Thisestimate suggeststhat the addedmatrix responsableforthe softviolationof the
initialconservedLsymmetrycanbeactuallyconsideredasasmallperturbationofthe
form:δM =εdiag(1,1,1).
Themethodpresentedaboveallowsonetoestimatethesumoftheabsolutemasses
intheneutrinosector:
3 2βm(µ)sin2θ 2a 1 2sin2θ
m 12 − W (24)
Xi=1 i ≃ 1−2sin2θ12 1−2sin2θ23 (cid:18)√1−a(cid:19)p cos2θW
Thisisexpe(cid:0)rimentallyrest(cid:1)ri(cid:0)ctedto: 3 m(cid:1) 1eV,ifwetakeintoconsideration
i=1 i ∼
theTroitsk[31]andMainz[32,33]experiments. Ontheotherhand,combiningEqs.
P
(19)and(23)oneobtaines:
3 2sin2θ
m = 12 m (25)
i 1 2sin2θ 1 2sin2θ 0
Xi=1 − 12 − 23
withminimalneutrinomassm(cid:0)0 = m3 . This(cid:1)le(cid:0)ads(withthea(cid:1)boveconsideredvalues
formixingangles)tom 0.001eV.
0
≃
4 Concluding remarks
Inthispaperwehavedevelopedastrategythatcombinestheexactsolutionofapartic-
ular3-3-1gaugemodelwithpossiblegloballeptonicsymmetryL = L L L .
e µ τ
− −
Whenthisadditionalsymmetryisrigourouslyconserved,theneutrinomassspectrum
exhibitsan invertedhierarchywith two degeneratemasses and the third one equalto
zero.Atthesametimethissymmetryrestrictsthemixinganglestothebi-maximalset-
tingonly. Thisstateofaffairscanbenaturallyovertakenifonedealswiththeapprox-
imate globalleptonicsymmetry,by introducingsomesmall termsthatsoftly violates
thissymmetryintheYukawasector. Theresultsareamazing: thecorrectpredictions
regardingthemasssplittingratio∆m2 /∆m2 0.033andtheobservedvaluesfor
themixinganglessin2θ 0.5ands1i2n2θ 2=3 ≃0.31ariseindependentlyofthemain
23 12
≃
parameter a in the 3-3-1 model of interest. At the same time, the minimal absolute
mass in the neutrino sector can be computed based on an exact formula depending
onlyontheaccurateaccountforthemixingangles.Inourapproximation,thescenario
leadstotheminimalmassintheneutrinospectrum: m(ν ) = 0.001eV,butthisorder
3
ofmagnitudecandecreaseiftheatmosphericanglegetscloser-to-maximalvalues.
All the results regardingthe mass squared differences and the mixing angles are
achieved just by tuning a second small parameter, let it be α or β. There are many
8
advantagesofthepresentmethodoverthepreviousonesfollowedbytheauthorinRefs.
[2]and[3]respectively.Intheone-parametercase[2]withnoleptonnumberconserved
andnoµ τ interchangesymmetry,theneutrinophenomenologyrequiredaverylarge
−
breakingscale and a radiativemechanismwas set to supplypossible deviationsfrom
the bi-maximalmixing. In Ref. [3] a second parameterwas introduced(that time in
theHiggssector)accompaniedbyseekforacanonicalseesawinterpretationinorderto
getaconsiderableautonomyfortheneutrinophenomenologyfromthebreakingscale
issue. Here, althougha second parameteris introducedtoo, there is no need for any
seesaw mechanism. Instead, finally a µ τ interchange symmetry remains. Along
−
with all the old valuablepredictionssuppliedby the exactsolutionof ourmodel, the
approachbasedonthissecondparameterexplainestheneutrinomassphenomenology
and, in addition, successfully passes the difficult challenge of predicting the correct
solarandatmosphericmixingangles.
References
[1] A.Palcu,Mod.Phys.Lett.A21,1203(2006).
[2] A.Palcu,Mod.Phys.Lett.A21,2027(2006).
[3] A.Palcu,Mod.Phys.Lett.A21,2591(2006).
[4] I.I.Cota˘escu,Int.J.Mod.Phys.A12,1483(1997).
[5] ParticleDataGroup(S.Eidelmanet.al.),Phys.Lett.B592,1(2004).
[6] W.A.Ponce,J.B.FlorezandL.A.Sanchez,Int.J.Mod.Phys.A17,643(2002).
[7] S.T.Petcov,Phys.Lett.B110,245(1982).
[8] R.Barbieri,L.Hall,D.Smith,A.StrumiaandN.WeinerJHEP12,017(1998).
[9] A.JoshipuraandS.Rindani,Eur.Phys.J.C14,85(2000).
[10] R. N. Mohapatra, A. Perez-Lorenzanaand C. A. de S. Pires, Phys. Lett. B 474,
355(2000).
[11] Q.ShafiandZ.Tavartkiladze,Phys.Lett.B482,145(2000).
[12] L.Lavoura,Phys.Rev.D62,093011(2000).
[13] W.GrimusandL.Lavoura,Phys.Rev.D62,093012(2000).
[14] T.KitabayashiandM.Yassue,Phys.Rev.D63,095002(2001).
[15] R.N.Mohapatra,Phys.Rev.D64,091301(2001).
[16] S.BabuandR.N.Mohapatra,Phys.Lett.B532,77(2002).
[17] H.S.Goh,R.N.MohapatraandS.P.Ng,Phys.Lett.B542,116(2002).
9
[18] D.A.DicusandH.-J.HeandJ.N.Ng,Phys.Lett.B536,83(2002).
[19] G.K.Leontaris,J.RizosandA.Psallidas,Phys.Lett.B597,182(2004).
[20] W.GrimusandL.Lavoura,J.Phys.G31,683(2005).
[21] G.AltarelliandR.Franceschini,JHEP03,047(2006).
[22] T.FukuyamaandH.Nishiura,hep-ph/9702253.
[23] R.N.MohapatraandS.Nussinov,Phys.Rev.D60,013002(1999).
[24] E.MaandM.Raidal,Phys.Rev.Lett.87,011802(2001).
[25] C.S.Lam,Phys.Lett.B507,214(2001).
[26] W.GrimusandL.Lavoura,Phys.Lett.B572,189(2003).
[27] T.KitabayashiandM.Yasue,Phys.Rev.D67,015006(2003).
[28] W.GrimusandL.Lavoura,J.Phys.G30,73(2004).
[29] Y.Koide,Phys.Rev.D69,093001(2004).
[30] A.Ghosal,Mod.Phys.Lett.A19,2579(2004).
[31] V.M.Lobashevetal.,Prog.Part.Nucl.Phys.48,123(2002).
[32] C.Weinheimeretal.,Nucl.Phys.Proc.Suppl.118,279(2003).
[33] Ch.Kraussetal.,Eur.Phys.J.C40,447(2005).
10