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arXiv:hep-ph/0303173v28Apr2003QuantumFieldTheoryTreatmentofNeutrinoOscillationsinVacuumandinMatterDiegoPallin∗andH˚akanSnellman†DivisionofMathematicalPhysics,DepartmentofPhysics,RoyalInstituteofTechnology,KTH-SCFAB,SE-10961Stockholm,Sweden(February1,2008)AbstractWestudyneutrinooscillationsinvacuumandinmatterusingfieldtheoreticalmethodsandwave-packets.Inparticular,wecalculatetheneutrinopropaga-torinthepresenceofmatterwithconstantdensityforthecaseoftwoflavors,obtainingtheresonanceformula.Intheextremerelativisticlimit,theresultoftheusualquantummechanicaltreatmentisrecoveredwithinteresting,butsmallmodifications.I.INTRODUCTIONEvidenceforneutrinoshavingsmall,butfinite,massesisaccumulatingfast.Themainparadigmforthisistheidea,firstproposedbyPontecorvo[1],thatiftheneutrinofla-vorswereasuperpositionofmassiveeigenstates,thentheneutrinoscouldoscillatebetweenthesedifferentflavors.Therearemanyexperimentslookingforneutrinooscillations:solarneutrinomeasurements,atmosphericneutrinomeasurements,reactorexperiments,andac-celeratorexperiments.Severalofthemhaveclaimedevidenceforneutrinooscillations.ThefinalbreakthroughcameinJune1998whentheSuper-KamiokandecollaborationreportedstrongevidenceforneutrinooscillationsfromtheatmosphericneutrinoUP-DOWNasymme-try.Themeasurementsonthedepletionofatmosphericmuonneutrinofluxfitwelltoatwoflavorνμ↔ντoscillationmodel.ThissummertheSNOcollaboration[2]showedthatthesolarneutrinodeficit,pioneeredbyDavies[3],wasduetoconversionofelectronneutrinostomu-andtauneutrinos.Recently,alsotheKamLANDexperiment[4]showsevidenceforneutrinooscillations.Inthestandardquantummechanicaltreatmentofneutrinooscillations,themasseigen-statesareassumedtoberelativisticandtohavethesamemomentum,andthus,differentenergies.Thefamiliarquantummechanicalmodeldescribingtheflavormixingprocesshas∗E-mail:diego@theophys.kth.se†E-mail:snell@theophys.kth.se1severalconceptualdifficultiescomparedtoquantumfieldtheorymodels,seeRefs.[5–9].Forexample,energymomentumconservationintheproductionanddetectionprocessesthatforcesneutrinostobeinamasseigenstateisincompatiblewithneutrinooscillations.Theneutrinooscillationprobabilityisindependentofthedetailsconcerningtheproductionanddetectionprocessesonlyintheextremelyrelativisticlimit.Hence,inthecasethatsomeofthemasseigenstatescannotbeconsideredtobeextremelyrelativistic,onehastousequantumfieldtheory.Ofcourse,thequantumfieldtheoryexpressionmustreproducethequantummechanicsoscillationprobabilityintheultra-relativisticlimit.AverydetailedreviewregardingdifferentaspectsandquestionsofneutrinooscillationsinquantumfieldtheorycanbefoundinRef.[5].AnotherinterestingquestionwhichisimportantinbothaquantummechanicalorquantumfieldtheorytreatmentisifthereexistsaFockspacefortheflavoreigenstates,sincethereexistsaFockspaceforthemasseigenstates.ThisquestionhasbeendiscussedbyGiuntietal.[10].Fujietal.[11]giveargumentsthataFockspaceofflavorneutrinosdoesnotexist.Whenneutrinospropagatethroughmattertheirbehaviormaybeaffectedsignificantly,aswaspointedoutbyWolfenstein[12]in1978andemphasizedbyMikheyevandSmirnov[13]asbeingimportantforneutrinooscillations.Thisisduetothefactthatinthepres-enceofmattertheeffectivemassinducedbytheforwardscatteringofneutrinosbythebackgroundchangestheflavoroscillatingparameters.Aquantummechanicaltreatmentofneutrinosinteractingwithmattercanbefoundinalmostallbookstreatingneutrinos.PeltoniemiandSipil¨ainen[14]havestudiedneutrinopropagationinmatterusingawavepacketapproach.CardallandChung[15]treatsneutrinooscillationsinastaticuniformbackgroundbyquantumfieldtheoryandshowthattheyrecoverthequantummechanicaloscillationamplitudeintherelativisticlimit.AlsoFujietal.[11]treatsneutrinooscillationsinastaticmatterbackgroundusingBogoliubovtransformations.Inthepresentpaper,wegiveashortpresentationoftheuseoffieldtheorymethodstodescribeneutrinooscillationsinvacuumandinmatter.Forneutrinooscillationsinmatterwecarryoutthecalculationsforthecaseoftwoflavorsindetail,whichdisplaysthemainfeaturesofourapproach.OuraimisinparticulartocalculateexplicitlythesingularitystructureoftheneutrinoGreen’sfunctioninthepresenceofmattertostudytheresonaceformula.InSec.II,wereviewthebasicformalismtobeused.SectionIIIcontainsanelaborationoftheuseofwave-packetsforflavorstates.WethencalculatetheoscillationamplitudeforthecaseofvacuumoscillationsinSec.IV,andthen,treatexplicitlythecaseoftwoflavorsinthepresenceofmatterwithconstantdensityinSec.V.II.BASICFORMALISMTheequationofmotionfortheneutrinofieldνj,withmassmj,canbewritteninthefollowingform(i∂/−mj)νj(x)=χj(x),(1)wherewehaveintroducedthesourcetermχj(x),whichcanbeobtainedfromthecompleteLagrangiandensityofthestandardelectroweaktheoryintheunitarygauge.Theinteractionpartofprimaryinteresttousisgivenby2Lint(x)=Xβ=e,μ,τ−g√2νβL(x)γαψβL(x)Wα(x)−g√2ψβL(x)γανβL(x)W†α(x)−g2cosθWνβL(x)γανβL(x)Zα(x)−3Xj=11vmνjνj(x)νj(x)σ,(2)wheregisadimensionlesscouplingconstantandWα(x)andZαarethefieldsthatdescribestheWandZbosonswithspin1.ThelastpartcontainstheHiggsfieldσandtheelectroweakvacuumexpectationvaluev.Weobservethattheinteractionpartcontainstheauxiliaryflavorfieldsνβ.Thes