Dissipative neutrino oscillations in randomly fluc

arXiv:hep-ph/0412311v121Dec2004DISSIPATIVENEUTRINOOSCILLATIONSINRANDOMLYFLUCTUATINGMATTERF.BenattiDipartimentodiFisicaTeorica,Universit`adiTriesteStradaCostiera11,34014Trieste,ItalyandIstitutoNazionalediFisicaNucleare,SezionediTriesteR.FloreaniniIstitutoNazionalediFisicaNucleare,SezionediTriesteDipartimentodiFisicaTeorica,Universit`adiTriesteStradaCostiera11,34014Trieste,ItalyAbstractThegeneralizeddynamicsdescribingthepropagationofneutrinosinrandomlyfluctuatingmediaisanalyzed:ittakesintoaccountmatter-induced,decoherencephenomenathatgobeyondthestandardMSWeffect.Awidelyadopteddensityfluctuationpatternisfoundtobephysicallyuntenable:amoregeneralmodelneedstobeinsteadconsid-ered,leadingtoflavorchangingeffectiveneutrino-matterinteractions.Theyinducenew,dissipativeeffectsthatmodifytheneutrinooscillationpatterninawayamenabletoadirectexperimentalanalysis.11.INTRODUCTIONWhenaneutrinopropagatesinaconstantdistributionofmatter,coherentforwardscatteringphenomenacanaffectitstimeevolution.Despitethesmallnessofthecross-sectioninducedbytheneutrinointeractionwiththemedium,thesemattereffectscansignificantlymodifytheoscillationpattern,throughtheso-calledMSWmechanism.[1,2]However,forwardscatteringphenomenaarejustthesimplestmatterinducedeffectsthatcanoccurtoaneutrinowhenthemediumisallowedtofluctuate.Inthiscase,theneutrinocanbeviewedasanopensystem,i.e.asubsystemimmersedinanexternalenvi-ronment(themedium);[3-7]itstimeevolution,obtainedfromthetotalneutrino+matterdynamicsbyeliminating(i.e.integratingover)thematterdegreesoffreedom,isnolongerunitary:ittakesintoaccountpossibleexchangesofentropyandenergybetweentheneu-trinoandthefluctuatingmedium.Inmanyphysicalsituations,onecansafelyignorethedetailsofthematterdynamicsanduseaneffectivedescriptionofthemediumasaclassical,randomexternalfield.Quiteingeneral,anyenvironmentcanbemodeledinthisway,providedthecharacteristicdecaytimeoftheassociatedcorrelationsissufficientlysmallwithrespecttothetypicalevolutiontimeofthesubsystem.Inthecaseofrelativisticneutrinos,thistimescalecanberoughlyidentifiedwiththevacuumoscillationlength:weshallthereforeconsidermediathatfluc-tuateontimescalesshorterthanthis.Ithasbeenrecentlypointedoutthattheinteriorofthesuncouldindeedsatisfysuchacondition,[8]aslikelyastheearthmantel.Thus,aneutrinocreatedinthesunormovingthroughtheearthwouldeffectivelyseearandomfluctuatingdistributionofscatteringcentersandthereforebesubjectedtostochastic,in-coherentinteractionwiththemedium.Inthissituation,correlationsinthemediumplayafundamentalrole:theyareresponsibleforthegenerationofnewmattereffects,beyondtheMSWones,leadingtoirreversibilityandlossofquantumcoherence.Theeffectsoffluctuatingmatteronneutrinopropagationhavebeenfirstdiscussedin[9-11],andrecentlyreconsideredin[12-14].However,alltheseanalysisdealwithasimpledensityfluctuationpattern,naturallysuggestedbythestandardMSWtreatment.Further,thesefluctuationsareassumedtobeexactlyδ-correlated;thisisahighlyidealizeddescriptionoftheenvironment,thate.g.forheatbathscanbeattainedonlyinthelimitofinfinitetemperature.Instead,inthefollowingamorerealisticexponentiallydampedformforthecorrela-tionfunctionsinthemediumwillbeadopted.Limitingforsimplicitythediscussiontotheanalysisoftheoscillationsoftwospeciesofneutrinos,weshallseethattheeffectsinducedbymatterfluctuationscanbefullydescribedintermsofalimitednumberofphenomeno-logicalparameters.Theyaffecttheoscillationpatterninaverydistinctiveway,thatisamenabletoadirectexperimentalstudy.Ontheotherhand,whenthesimplifieddensityfluctuationhypothesisconsideredin[9-14]isadopted,asingleconstantissufficienttoparametrizethenewmattereffects.However,thisapproximationappearsphysicallyuntenable,sincebyadoptingitcertaintransitionprobabilitiestakeunacceptablenegativevalues;thisseriousinconsistencycanbecuredonlybyallowingmoregeneralmatterfluctuations,pointingtowardsthepresence2offlavorchangingneutrino-matterinteractions.†Asafinalremark,itisinterestingtopointoutthatthedissipativeeffectsinducedbyarandomlyfluctuatingmediumonneutrinooscillationsinvolveingeneraltheCP-violatingphasethatispresentinthemixingmatrixforMajorananeutrinos.Therefore,contrarytothevacuumcase,matteroscillationexperimentscanprovide,atleastinprinciple,awaytodistinguishbetweenDiracandMajorananeutrinos.2.MASTEREQUATIONIndiscussingthemixingoftwoneutrinospecies,weshalladoptthefamiliareffectivedescriptionintermsofatwo-dimensionalHilbertspace;[18-22]theflavorstates,thatweshallconventionallycall|νeiand|νμi,willbechosenasbasisstates.Withrespecttothisbasis,thephysicalneutrinostatesarethenrepresentedbydensitymatricesR,i.e.byhermitian2×2matrices,withnon-negativeeigenvaluesandunittrace.TheirtimeevolutionequationcanbecastinastandardLiouville–vonNeumannform:[23,24]∂R(t)∂t=−iH0,R(t)+LtR(t).(2.1)Thefirstpieceonther.h.s.describesthepropagationoftheneutrinosinvacuum;inthechosenbasis,theeffectivehamiltonianH0takesthestandardform:H0=ω~n·~σ,(2.2)whereω=Δm2/4E,Δm2beingthesquaremassdifferenceofthetwomasseigenstatesandEtheaverageneutrinoenergy,whiletheunitvector~n=(sin2θ,0,−cos2θ)containsthedependenceonthemixingangle,~σ=(σ1,σ2,σ3)be