Neutrino Oscillations and the MSW effect in Random

arXiv:hep-ph/9602398v25Mar1996BUTP-96/8NeutrinoOscillationsandtheMSWeffectinRandomSolarMatterE.TorrenteLujan.Inst.furTheoretischePhysik,UniversitatBernSidlerstrasse5,3012Bern,Switzerland.e-mail:[e-mail]heeffectsofrandomdensityfluctuationsonneutrinooscillationsintheSunenvironment.WeshowhowtheaverageofcertainquantitieswhichcanbeusedtodescribetheMSWeffectcanbecomputedanalytically.Weexaminealsothehypothesiscommonlyacceptedthatonlyperturbationsinsidetheresonancelayercanhaverelevance.Theaverageamplitude,whichgivesthe”coherentprobability”,iscomputedinananalyticalandexactwayforthecaseofcoloredδ-correlatedGaus-siannoise:therandomperturbationinducesarenormalizationofthematterdensitywhichacquiresanimaginarypartproportionaltothefluctuationmagnitudeintheresonanceregion.Integralequationsaregivenforthedensitymatrixofthesysteminthe”optical”approximation.PACS:96.60.Kx,02.50.Ey,14.60.Pq,95.30.Cq,96.60.Hv,14.60.Gh.11IntroductionInthisworkweinvestigatematter-enhancedneutrinoflavortransformations,theMSWeffect[1,2],forthecaseofaexponentiallydecayingmatterdensitywithacoloredGaussiannoise.Thereareseveralexamplesofinterestingtransportandscatteringprocessesinducedormodifiedbythepresenceofadisorderedorrandommedium:dislocationsincrystalsandthesolid-liquidtransition;randomimpuritypotentialsproducethelocalizationofquantumwavefunctionswhichenablesonetounderstandthetransitionbetweeninsulatorsandconductors;resistanceanomaliesatlowtemperatureandinpresenceofmagneticfieldsof”weaklocalized”electronsystemssubjecttoarandompotential([3,4]).Thesamebasiclocalizationphenomenaexplainalsoinopticsthebackscatteringenhancementinelectromagneticwavescatteringfromarandomlyroughsurface([5]).Usually,astrongdependenceonthedimensionalityoftheproblemisobserved.Insomeothercasesasinnuclearphysicstheconsiderationofrandommatrixhamiltoniansallowsthesimplificationofotherwiseunmanageablesystems.Forneutrinooscillationsinpresenceofrandomorrapidlytwistingmagneticfieldsconsiderableworkhasbeendonealready[6,7,8,9].Randomdensityfluctuationshavereceivedsomeattentiononlyrecently.In[6],adifferentialequationfortheaveragedsurvivalprobabilitywasderivedforthecaseinwhichtherandomnoisewastakentobeadelta-correlatedwhiteGaussiandistribution.Thedifferentialequationwassolvednumericallyandtheneutrinoevolutionobtained.Argumentsweregivenwhichindicatethatifthecorrelationofthematterdensityfluctuationsissmallcomparedtotheneutrinooscillationlengthatresonance,oneobtainsthesameresultasforthecaseofadelta-correlatednoise.In[10]themorerealisticcaseofcolorednoisewasconsideredalsoinanumericalwayandappliedtoSupernovadynamics.Anapproximatedifferentialequationfortheaveragedsurvivalprobabilitieswasobtainedusingthehypothesisthatthefluctuationsshouldnotaffecttheevolutionfarfromtheresonance.In[11],theimplicationsofrandomperturbationsupontheSolarneutrinodeficitareconsiderednumerically.ItisfoundthattheMSWeffectisratherstableunderthesefluctuationsspeciallyinthesmallmixingcasebutinanycasetheexperimental(Δm2,cosθ)exclusioncurvesgetmodifiedinanappreciableway.Inthisworkwetrytodevelopanalyticalresultsforneutrinoflavoroscillationsinducedbyarandommatterdensity.WeareinterestedmainlyinthepersistenceoftheMSWef-fect.Theobtentionofconcretevaluesforthesurvivalprobabilitiesandphenomenologicalconsequencesforthesolarneutrinoproblemisleftforasubsequentwork.Theobten-tionofthesephenomenologicalconsequencesisnotaneasytask,atleast,becausetheamplitudeofpossibledensityfluctuationsintheSunispoorlyknownexperimentallyandtheoretically.Differentargumentscangiveeasilyvaluesforitdifferingbytwoordersofmagnitude([11]):between0.1%−10%ofthelocaldensity.Strongerlocalinhomogeneities,forexampleatthenear-surfacedarkspotsshouldnotbediscarded.Ourbasicstartingpointwillbetheexactanalyticalsolutionobtainedin[12]fortheneutrinooscillationamplitudesinpresenceofanexponentiallydecayingmatterdensity.2Therandomcomponentisconsiderasaperturbationtothissolution.Inspiredbytheelectromagneticwavescatteringinrandommedia,coherentandincoherenttransitionprobabilitiesaredefined.Thebasicresultofthisworkisthatthecoherentprobability,whichcomesessentiallyfromanaveragedamplitude,canbecomputedexactlyinsomecases.Differentintegralequationsaregivenfortheincoherentprobabilityusingan”opti-cal”approximation.Approximatesolutionsvalidinalimitedrangeareobtainedforthem.Bytheverynatureoftheproceduretheresultsarevalidoreasilygeneralizabletoanynumberofneutrinospecies.Theoutlineofthisworkisasfollows.Thefirstsection,aftertheintroductionofsomeknownresults,isdedicatedtoexplorediversequantitieswhosestochasticaveragecanbecomputedexactlyoratleastbyaneasyapproximation.Oneofthesequantitiesisthedeterminantoftheevolutionoperatorofthesystem.Weusea”naive”argumenttoshowtheplausibilitythattheinfluenceoftherandomperturbationontheneutrinooscillationscanbedescribedbyacomplexredefinitionofthematterdensity.Thissuggestionallowstodefinesomeansatzprobability.Anotherquantityisthe”totalcrosssection”ofthesystem,wesuggesttheinterestinfurtherstudyingthisquantitywhichcancontaininformationonthepresenceandlocalizationoft