Coherence and oscillations of cosmic neutrinos

arXiv:0803.0495v2[hep-ph]1Sep2008September1,2008CoherenceandoscillationsofcosmicneutrinosYasamanFarzana1andAlexeiYu.Smirnovb,c2aInstituteforresearchinfundamentalsciences(IPM),P.O.Box19395-5531,Tehran,IranbInternationalCentreforTheoreticalPhysics,StradaCostiera11,34014Trieste,Italy,cInstituteforNuclearResearch,RussianAcademyofSciences,Moscow,RussiaAbstractForcosmicneutrinoswestudytheconditionsandtheeffectsofthecoherencelossaswellascoherentbroadeningofthespectrum.Weevaluatethewidthoftheneutrinowavepacketproducedbychargedparticlesundervariouscircumstances:inaninteraction-freeenviron-ment,inaradiation-dominatedmedium(typicalofthesourcesofthegammaraybursts)andinthepresenceofamagneticfield.Theef-fectofthemagneticfieldonthewavepacketsizeappearstobemoreimportantthanthescattering.Ifthemagneticfieldatthesourceislargerthan∼10Gauss,thecoherenceofneutrinoswillbelostwhiletravelingovercosmologicaldistances.Variousapplicationsoftheseresultshavebeenconsidered.Wefindthatforlargemagneticfields(B109Gauss)andhighenergies(EνPeV),“coherentbroad-ening”canmodifytheenergyspectrumofneutrinos.Inthecoher-entcase,averagingouttheoscillatorytermsoftheprobabilitiesdoesnotinduceanystatisticaluncertaintybeyondwhatexpectedintheabsenceoftheseterms.Adeviationfromthestandardquantumme-chanicsthatpreservesaverageenergyandunitaritycannotalterthepicture.PACS:14.60.PqKEYWORDS:Cosmicneutrinos,Coherence,[e-mail][e-mail]11IntroductionConstructionandoperationoftheneutrinotelescopesopennewwindowstowardsexploringtheUniverse.Bystudyingtheneutrinosarrivingatthesetelescopesfromsourceslocatedatcosmologicaldistances,wecanderiveun-precedentedinformationonthesourcesofsuchneutrinosaswellasonthepropagationandpropertiesoftheneutrinos,themselves.Operatingdetectorshavenotsofardetectedcosmicneutrinos.Thedataaccumulatedduring2000-2004bytheAMANDAexperimentshowsnoindi-cationofneutrinofluxfrombeyondtheatmosphereoftheEarth[1]whichleadstoanupperboundonthepointsourcefluxE2νdΦνdEν∼10−8GeVcm−2sec−1sr−1,(1)intheenergyrangebetween1.6TeVand2.5PeV.Thisboundimpliesthatthetotalnumberofeventsperyearata1km3scaledetectorsuchasICE-CUBE[2]cannotbeonaveragelargerthan104.Ahostoftheoreticalmodelspredictcosmicneutrinofluxesinthe(1-100)TeVenergyrangewhichcanbedetectedbya1km3scaledetector:e.g.,thefireballmodelsforGammaRayBurst(GRB)sources[3,4],modelsforsupernovaetypeIb/c[5,6]andmod-elsfortheActiveGalacticNuclei(AGN)[7].TheneutrinofluxfromGRBinthe(1-10)TeVrangecansaturatethepresentboundfromAMANDA[4].Weshouldprepareforsuchgenerosityofnatureandanticipatewhatcanbelearnedabouttheneutrinopropertiesaswellasthemechanismbehindtheproductionofneutrinosatthesource.Becauseofthisprospect,wemainlyconcentrateonneutrinoswithenergiesinthe(1-100)TeVrangethroughoutthepresentpaper.Accordingtothemodels,neutrinosareproducedthroughthefollowingchainofreactions[8]p+γ→Δ→π++X,(2)π+→νμ+μ+,μ+→νe+¯νμ+e+.(3)Alongwiththeprocess(2),bothπ+andπ−canbeproducedviatheppcolli-sionsorviathecollisionoftheprotonswiththenucleipresentinthemedium.2Iftheinteractionratesofπandμarenegligible,theflavorcompositionoftheneutrinofluxatthesourcewillbeasthefollowing(F0e+F0¯e):(F0μ+F0¯μ):(F0τ+F0¯τ)=1:2:0,(4)whereF0αandF0¯αrespectivelydenotethefluxesofναand¯να.Neutrinooscillationsaltertheflavorcompositionoftheneutrinofluxdur-ingpropagationfromthesourcetothedetector.Iftheflavorcompositionatthesourceisknown,bystudyingtheflavorcompositionatthedetector,onecanextractinformationontheoscillationparameters[9].Inparticularin[10,11],asthekeypartoftheprogramofreconstructingtheunitaritytriangleandderivingtheleptonsectorJarlskoginvariant,itwassuggestedtostudythecosmicneutrinostoextracttheUμ1elementofthePontecorvo-Maki-Nakagawa-Sakatamatrix.ThereisalsorichliteratureonthepossibilityofdirectlydeterminingtheDiracCP-violatingphasefromtheneutrinotele-scopedata[12].Oneoftheimportanteffectsofneutrinopropagationwhichcaninprin-cipleinfluenceobservationsisthelossofcoherence.Thedifferentneutrinomasseigenstateshavingthesameenergyhavedifferentvelocities.Thus,thewavepacketsofthemasseigenstatescomposinganeutrinostatewillcomeapartastheypropagate.Ifthetraveleddistanceissolargethatthesecom-ponentscompletelyseparatefromeachother,theywillceasetointerfereatthedetector.Inthecaseofcoherenceloss,theoscillatorytermsoftheoscil-lationprobabilitiesdisappear.Conversely,inthecoherentcase,thedifferentmasseigenstateskeepinterferingandasaresult,theoscillatorytermsarestillpresent.Ofcourse,eveninthelattercase,becauseofthefiniteenergyresolutionofthedetectorandthefactthatthesourcesofdifferenteventsarelocatedatvariousdistances,theeffectsoftheoscillatorytermsaveragetozero.Asdiscussedin[13,14],thetwocasesareinpracticeindistinguishable.Inthispaper,wediscussindetailunderwhichconditionsthecoherenceislost.Wealsoevaluatethestatisticaluncertaintyinthepresenceoftheoscil-latorytermsforthecoherentcase.Coherentbroadeningofthewavepacketcanleadtodeformationoftheenergyspectrumwhichhasdirectobservationalconsequences.Inthispaper,wediscussunderwhatcircumstances,theeffectwillbenoticeable.Thepaperisorganizedasfoll