Parametric resonance in neutrino oscillations in m

arXiv:hep-ph/9907435v120Jul1999FISIST/12-99/CFIFhep-ph/9907435Parametricresonanceinneutrinooscillationsinmatter∗E.Kh.Akhmedov†CentrodeF´ısicadasInterac¸c˜oesFundamentais(CFIF)DepartamentodeFisica,InstitutoSuperiorT´ecnicoAv.RoviscoPais,P-1049-001Lisboa,PortugalAbstractNeutrinooscillationsinmattercanexhibitaspecificresonanceenhancement–parametricresonance,whichisdifferentfromtheMSWresonance.Oscillationsofat-mosphericandsolarneutrinosinsidetheearthcanundergoparametricenhancementwhenneutrinotrajectoriescrossthecoreoftheearth.Inthispaperwereviewtheparametricresonanceofneutrinooscillationsinmatter.Inparticular,physicalinter-pretationoftheeffectandtheprospectsofitsexperimentalobservationinoscillationsofsolarandatmosphericneutrinosintheeartharediscussed.∗InvitedreviewarticletobepublishedinthespecialissueofPramanadedicatedtoneutrinophysics†OnleavefromNationalResearchCentreKurchatovInstitute,Moscow123182,Russia.E-mail:[e-mail]1IntroductionItiswellknownthatneutrinooscillationsinmattercandiffersignificantlyfromoscillationsinvacuum,thebeststudiedexamplebeingtheMikheyev-Smirnov-Wolfenstein(MSW)effect[1,2].Itis,however,muchlessknownthattheMSWeffectisnotthesolemechanismbywhichmattercanenhancetransitionsbetweenneutrinosofdifferentflavor.TheMSWeffectenhancestheprobabilitiesofneutrinoflavortransitionsbyamplifyingneutrinomixing:themixingangleinmatterθcanbecomeequalto45◦evenifthevacuummixingangleθ0isverysmall.Itwaspointedoutabout12yearsago[3,4]thattheprobabilitiesofneutrinoflavortransitionscanalsobestronglyenhancediftheoscillationphaseundergoescertainmodificationinmatter.Thiscanhappenifthevariationofthematterdensityalongtheneutrinopathiscorrelatedinacertainwaywiththechangeoftheoscillationphase.Thisamplificationoftheneutrinooscillationprobabilityinmatterduetospecificphaserelationshipshasaninterestingpropertythatitcanaccumulateifthematterdensityprofilealongtheneutrinopathrepeatsitself,i.e.isperiodic.Thephenomenonisanalogoustotheresonanceindynamicalsystemswhoseparametersperiodicallyvarywithtime–parametricresonance.Itwasthereforedubbedparametricresonanceofneutrinooscillations[3,4].Whileperiodicityoftheparametersofthesystemisuseful,itisnotreallynecessary:parametricresonancecanoccureveninstochasticmedia(see,e.g.,[5]).Thestochasticparametricresonanceinneutrinooscillationswasbrieflydiscussedin[6].Theparametricresonancecanleadtolargeprobabilitiesofneutrinoflavortransitioninmatterevenifthemixinganglesbothinvacuumandinmatteraresmall.Thishappensbecauseeachhalf-waveoscillationofthetransitionprobabilityisplacedonthetopofthepreviousone,i.e.thetransitionprobabilitybuildsup(fig.1).Ifmixingangleinmatterisverysmall(matterdensityisfarfromtheMSWresonancedensity),theparametricresonanceenhancementofneutrinooscillationscanmanifestitselfonlyiftheneutrinospassthroughalargenumberofperiodsofdensitymodulation,i.e.travelasufficientlylongdistance.However,ifmatterdensityisnotveryfarfromtheMSWresonanceone,aninterestinginterplaybetweentheMSWeffectandparametriceffectscanoccur.Inparticular,astrongparametricenhancementofneutrinooscillationscantakeplaceeveniftheneutrinospassonlythrough1-2periodsofdensitymodulation[6].Fortheparametricresonancetooccur,theexactshapeofthedensityprofileisnotveryimportant;whatisimportantisthatthechangeinthedensitybesynchronizedwiththechangeoftheoscillationphase.Inparticular,in[3,4]thecaseofthesinusoidaldensityprofilewasconsideredinwhichtheneutrinoevolutionequationreducestotheMathieuequation.In[4]theparametricresonancewasalsoconsideredforneutrinooscillationsinamatterwithaperiodicstepfunction(“castlewall”)densityprofile,whichallowsaverysimpleexactanalyticsolution.Wewilldiscussthissolutioninsections2and4.Althoughtheparametricresonanceinneutrinooscillationsiscertainlyaninterestingphysicalphenomenon,itrequiresthatveryspecialconditionsbesatisfied.Unfortunately,1theseconditionscannotbecreatedinthelaboratorybecausethiswouldrequireeithertoolongabaselineorneutrinopropagationinamatteroftoohighadensity(seesec.5below).Untilrecentlyitwasalsounclearwhetheranaturalobjectexistswheretheseconditionscanbesatisfiedforanyknownsourceofneutrinos.ThissituationhaschangedwithaveryimportantobservationbyLiuandSmirnov[7](seealso[8]),whohaveshownthattheparametricresonanceconditionscanbeapproximatelysatisfiedfortheoscillationsofatmosphericνμintosterileneutrinosνsinsidetheearth.Itisknownthattheearthconsistsoftwomainstructures–themantleandthecore.Withinthemantleandwithinthecorethematterdensitychangesratherslowly(thedensityvariationscaleislargecomparedtothetypicaloscillationlengthsofatmosphericandsolarneutrinos),butattheirborderitjumpssharplybyaboutafactoroftwo.Thereforetoagoodapproximationonecanconsiderthemantleandthecoreasstructuresofconstantdensitiesequaltothecorrespondingaveragedensities(two-layermodel).Neutrinoscom-ingtothedetectorfromthelowerhemisphereatzenithanglesΘintherangedefinedbycosΘ=(−1)÷(−0.837)traversetheearth’smantle,coreandthenagainmantle.Thereforesuchneutrinosexperienceaperiodic“castlewall”potential,andtheiroscillationscanbeparametricallyenhanced.Eventhoughtheneutri