Neutrino oscillations in a model with a source and

arXiv:hep-ph/9710289v18Oct1997NeutrinooscillationsinamodelwithasourceanddetectorKenKiers∗DepartmentofPhysicsBrookhavenNationalLaboratory,Upton,NY11973-5000,USANathanWeiss†SchoolofPhysicsandAstronomy,TelAvivUniversity,TelAviv,IsraelDepartmentofParticlePhysics,WeizmannInstitute,Rehovot,IsraelDepartmentofPhysicsandAstronomy,UniversityofBritishColumbiaVancouver,B.C.,V6T1Z1CanadaAbstractWestudytheoscillationsofneutrinosinamodelinwhichtheneutrinoiscoupledtoalocalized,idealizedsourceanddetector.Byvaryingthespatialandtemporalresolutionofthesourceanddetectorweareabletomodelthefullrangeofsourceanddetectortypesrangingfromcoherenttoincoherent.WefindthatthisapproachisusefulinunderstandingtheinterfacebetweentheQuantumMechanicalnatureofneutrinooscillationsontheonehandandtheproductionanddetectionsystemsontheotherhand.ThismethodcaneasilybeextendedtostudytheoscillationsofotherparticlessuchastheneutralKandBmesons.Wefindthatthisapproachgivesareliablewaytotreatthevariousambiguitieswhicharisewhenoneexaminestheoscillationsfromawavepacketpointofview.Wedemonstratethattheconventionaloscillationformulaiscorrectintherelativisticlimitandthatseveralrecentclaimsofanextrafactorof2intheoscillationlengthareincorrect.Wealsodemonstrateexplicitlythattheoscillationsofneutrinoswhichhaveseparatedspatiallymaybe“revived”byalongcoherentmeasurement.∗Email:[e-mail]†Email:[e-mail]flavouroscillationsofparticlesareafascinatingdemonstrationofquantummechan-icsinthemacroscopicworld.Flavouroscillationscangenericallyoccurwhenthestateswhichareproducedanddetectedinagivenexperimentaresuperpositionsoftwoormoreeigenstateswhichhavedifferentmasses.TheoscillationsofKandBmesonshavebeenobservedexperimentally[1]andhavebeenusedtoplacestringentconstraintsonphysicsbeyondtheStandardModel.Ifneutrinosaremassive,theytoomayoscillate,andthiscouldleadtotheresolutionofthewell-knownsolarneutrinoproblem[2–4].Morerecently,thediscussionofparticleoscillationshasbeenextendedtoincludesupersymmetricparticlesinsupersymmetricextensionsoftheStandardModel[5].Thephenomenonofparticleoscillationshasbeenstudiedextensivelyandisgenerallythoughttobeverywellunderstood.Thereneverthelessremainseveralsubtleissueswhichcontinuetocausesomeconfusion.Thekeytoacompleteunderstandingofanysuchissuelaysintreatingcorrectlythenecessaryinterplaybetweenthe“classical”and“quantum”naturesoftheparticleswhichareinterferingtoproducetheoscillations.Thus,forexample,theinterferenceeffectitselfispurely“quantum”innature(itrequiresthattheparticlesbedescribedbywaves),andyettheresultingoscillationsinspaceareonlyobservableiftheparticlesaresufficientlylocalizedinspace[6].Thisexamplehighlightsthefactthatanydiscussionofparticleoscillationsimplicitlyassumesthatthemasseigenstateswhichareinterferingtoproducetheoscillationsaredescribedbysomesortofwavepackets.Despitethesuccessofthewavepacketapproachinclarifyingmanyaspectsofthephe-nomenonofparticleoscillations[6–9],theapproachisnotwithoutitsowndifficulties.Theresultsofagivencalculationwilldepend,forexample,onthedetailsoftheinitialmasseigenstatewavepackets(includingtheirshape,spectrumandrelativenormalization).Oneparticularlydifficultproblemwhicharisesistheconversionofthefinaltime-evolvedwavepacketsintoanexperimentallyobservablequantity:sinceitisgenerallythefluxofparticleswhichismeasuredinanexperiment,oneisrequiredtocalculateacurrentdensityratherthanaprobabilitydensity.1Thedifferencebetweenacurrentdensityandaprobabilityden-sity,atleastnaively,involvesafactorofthevelocityv,whichisverysignificantifthemasseigenstateshavequitedifferentmasses.Thus,ifonewouldcalculatetheprobabilitydensityatthedetectorandintegrateitovertime,theresultingexpressionwouldhavefactorsof1/vpre-multiplyingthevariousterms,leadingtoanenhancementofthetermscorrespondingtoheaviermasseigenstates.Inthecaseofneutrinos,aswasnotedin[8–10],thiswouldskewtheusualoscillationformulaquitedramaticallyifoneofthemasseigenstateneutrinoswasnon-relativistic.Effortstoconstructanappropriatecurrentdensitywhichretainsthenec-essarywavepacketfeatureshavehadmixedsuccess.Acalculationinthekaoncaseappearstogivereliableresults[11],butitcanbeshownthatunphysicaleffectsariseifoneattemptstodefineasuitablecurrentwhenthemasseigenstateshaveverydifferentmasses[12].Thereisanotherverystrikingapparent“ambiguity”whicharisesifonedoesnottreat1Wavepacketcalculationsleadnaturallytoexpressionsfortheprobabilitydensity,whichareappropriatelyintegratedoverspace,nottime.Foroscillationsinspaceonewantsaquantitywhichisappropriatelyintegratedovertime,i.e.,acurrent.2thedelicateinterplaybetweentheclassicalandquantumnaturesoftheparticlescorrectly.Inthiscasethe“ambiguity”leadstoanallegederrorbyafactorof“2”inthecalculationoftheoscillationlength[13].Inordertounderstandthesourceoftheambiguity,wefollowthediscussiongivenbyLipkin[14].SupposeweconsidertheoscillationsintimeofasystemforwhichtheinitialandfinalstatesarenoteigenstatesofthefreeHamiltonian.Thephaseoftheinterferencetermwillthenbegivenbyφ(t)=(E1−E2)t,whereEi≡(p2+m2i)1/2.Detectorsdonotmeasureoscillationsdirectlyasafuncti