Softly broken $muleftrightarrowtau$ symmetry in the minimal

Neutrino oscillations data indicates that neutrino mixings are consistent with an apparent $\nu_\mu - \nu_\tau$ exchange symmetry in neutrino mass matrix. We observe that in the mininimally extended standard model with the see-saw mechanism, one can impose

Softlybrokenµ τsymmetryintheminimalsee-sawmodel

JuanCarlosG´omez-Izquierdo andAbdelP´erez-Lorenzana

DepartamentodeF´ sica,CentrodeInvestigaci´onydeEstudiosAvanzadosdelI.P.N.

Apdo.Post.14-740,07000,M´exico,D.F.,M´exico

(Dated:September,2007)Neutrinooscillationsdataindicatesthatneutrinomixingsareconsistentwithanapparentνµ ντexchangesymmetryinneutrinomassmatrix.Weobservethatinthemininimallyextendedstandardmodelwiththesee-sawmechanism,onecanimposeµ τsymmetryatthetreelevelonallLagrangianterms,butforthemassdi erenceamongµandτleptons.Intheabsenceofanynewextraphysics,thismassdi erencebecomestheonlysourceforthebreakingofsuchasymmetry,whichinduces,viaquantumcorrections,smallbutpredictablevaluesforθ13,andforthedeviationofθATMfrommaximallity.IntheCPconservingcase,thepredictionsonlydependonneutrinomasshierarchyandmayprovideauniquewaytotestfornewphysicswithneutrinoexperiments.

PACSnumbers:14.60.Pq,12.60.-i,11.30.Fs

arXiv:0711.0045v2 [hep-ph] 27 Jun 2008

I.INTRODUCTION

Convincingevidencethatneutrinoshavemassandos-cillatehasbeenprovidedalongrecentyearsbyneutrinooscillationexperiments[1].Inthestandardframework,onlythreeweakneutrinospecies,νe;νµandντ,areneededtoconsistentlydescribetheexperimentalresults,withtheadditionofneutrinomassesandmixingsasnewparameterstothestandardmodel.Centralideaisthatneutrinomasseigenstates,ν1,2,3,andweakeigenstatesaredi erent,buttheycanbewrittenaslinearcombina-tionsofeach otherbyusingacomplexunitarymatrix,U,asν =iU iνi,for =e,µ,τandi=1,2,3,wherewereferonlytolefthandedstates.Acommonparameter-izationforMajorananeutrinosoftheUmatrixisgivenintermsofthreeanglesandthreeCPphases,suchthatU=VK,whereK=diag{1,eiφ1,eiφ2},withφ1,φ2thephysicalCP-oddMajoranaphases,andtheelementsoftheVmixingmatrixparameterizedas[2]

c12c13s12c13z

V= s12c23 c12s23zc12c23 s12s23zs23c13 ;

s12s23 c12c23z c12s23 s12c23zc23c13

wherecijandsijstandforcosθijandsinθijrespectivelyandz=s13ei .Thekinematicalscalesfortheoscillation,ontheotherhand,aregivenbythetwomasssquareddif-2

ferences:thesolar/KamLANDscale m2⊙= m12;and

22

theatmosphericscale m2ATM=| m23|≈| m13|.Combinedanalysisofalldata[1]indicatesthatattwo

+0.5 5

eV2; m2sigmalevel m2⊙=7.6 0.3×10ATM=

.05

2.4±0.3×10 3eV2,whereassin2θ12=0.32+0 0.04,

.132

sin2θATM=sin2θ23=0.5+0 0.12,andsinθ13≤0.033.Thus,dataisconsistentwithθ13≈0,andθATM≈π/4,whichmakestheDiracCPphase, ,hardtobemea-sured.Currentandnewexperimentsonneutrinophysics

Neutrino oscillations data indicates that neutrino mixings are consistent with an apparent $\nu_\mu - \nu_\tau$ exchange symmetry in neutrino mass matrix. We observe that in the mininimally extended standard model with the see-saw mechanism, one can impose

proposetotreatµ τasasoftlybrokensymmetryoftheminimalsee-sawmodel.Therefore,attreelevel,allphysicsnotdirectlyrelatedtomµ,τwouldbedescribedbythesymmetriclimit,allowingusto xthefreeparam-etersofthemodelatlowenergy.Nevertheless,quantumcorrectionsshallcommunicate

the

symmetry

breakingtotheneutrinosector[8].Inparticular,oneloopcorrectionswillalreadyproducesmalldeviationstoθATMfromπ/4,andanonzeroθ13.Becausethemodelhasnotextraunknowningredients,onecanmakede nitepredictionsforthesephysicalobservablesintermsofsymmetriclevelresults.Thosearethemainpointswewanttodiscussinwhatfollows.

II.THEMINIMALµ τMODEL

Themodelwewillexploreconsiders, rst,themini-malsee-sawextensionthatincludesthreerighthanded

neutrinos,withalladditionalLagrangiantermsthatareconsistentwiththestandardmodelsymmetries,h L¯ H R+y ′L¯ HN ′+(h.c.)+(MR) ′N¯ cN ′,(1)wheresumoverindicesshouldbeunderstood.Here,L

standsforthestandardleptondoubletsandHfortheHiggs eld.Inordertoimplementµ τsymmetryinameaningfulway,wehavechosentoworkinthebasiswherethechargeleptonYukawacouplings,andsotheirmasses,arediagonalandreal.Also,wehavechosenrighthandedneutrinostocarryleptonnumber,andproperlyidenti edtheindex.ItisworthmentioningthatifNi= werenotsubjectedtoµ τsymmetry,asde nedbe-low,then,neutrinoYukawacouplingswouldbecamesuchthatyµi=yτi,underµ τsymmetry,regardlessofthechosenbasisforNi.FollowingthisimplieddegeneracyofsecondandthirdrowsontheDiracmassmatrix,thelefthandedmasslessneutrinostateν′=(νµ ντ)/

√2

θ12m0eµ;

m2=m0ee+

√

2tan8m0eµ

m2

2| 13

+ m2course,extraparametersyetexistforthewhole23|.the-Of

ory[seeEq.(1)].Theybelongtothehighenergyrighthandedneutrinosectorandcannotbe xedfromtheseresults.However,aswewillshowbelow,thoseparame-terswillnotberequiredtomakefurtherpredictionsforthelowenergyphysics.

III.

SOFTBREAKINGOFµ τSYMMETRY

Exactµ τsymmetrywouldalsoimplythathµ=hτ,whichgivesthewrongresultmµ=mτ.Thisistheonlyplacewherethesymmetryisbeingexplicitelybroken.Henceforth,wewilltakehµ=hτ,achoicethatofcourserespectsallgaugesymmetries,andbyde nitionitisex-pectedtobevalidatanyenergy.Weshallnotassumeanydynamicaloriginforsuchadi erenceontheYukawacouplingsinordertokeepthemodeltrulyminimal.No-ticethat,asamatterofinthestandardmodel,iL

¯fact,allleptonic γµDµLL +i ¯kineticµterms

RγDµR RwithDµL,Rthecorrespondingcovariantderivatives,areinvari-antunderµ τexchangeduetotheuniversalityofgaugeinteractions.Incontrast,outoftheLagrangiantermsgiveninEq.(1),thechargedleptonYukawacou-plingsnowhavetheform

heL

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