Neutrino oscillations in structured matter

arXiv:hep-ph/0004075v21Jun20001NeutrinoOscillationsinStructuredMatterPaulM.FishbanePhysicsDept.andInstituteforNuclearandParticlePhysics,Univ.ofVirginia,Charlottesville,VA22903AbstractAlayeredmaterialstructureinamonochromaticneutrinobeamproducesinterferenceeffectsthatcouldbeusedforthemeasurementoffeaturesoftheneutrinomassmatrix.Thephenomenonwouldbemostusefulathighenergies.2TheMSWeffect[1,2]describeshowelectronneutrinosinmatterpropagatedifferentlyfromotherneutrinos,andfromelectronneutrinosinvacuum.Thiseffectisanelementintheinterpretationofrecentexperiments[3-6]thathaveexploredtheneutrinomassspectrum.Thephenomenonalsodescribestheeffectsofthepresenceofboundariesbetweendifferentmediaonneutrinopropagationandoscillation.Asrecentwork[7-9]hasshown,theboundariesintroducethepossibilityofinterferencebetweendifferentamplitudesforneutrinopropagation.Indeed,thepresenceofboundarieswithintheearthhasimplications[7,8]fortheinterpretationofthedataofreference3.Herewepointoutthatanarrangementoflayersofmaterials,containingmanyboundaries,canprovideanotherangleontheinterferencebetweenthepropagationmodes.Inadditiontodescribingthebasicmechanismforatwo-familyneutrinostructure,webrieflyaddressthequestionofwherethiseffectcouldbemostprofitablyemployed.I.ReviewofMSWandBoundaryInterferenceeffectsNeutrinooscillationoccursbecausetheelectroweakeigenstatesarenotthemasseigenstates.Letusconsidertwofamiliesofneutrinos,withelectroweaklabelseandμ(genericallyGreekletters)andmasseigenstatelabels1and2,with1labelingthelightestneutrino.ThematrixU(theleptonicCKMmatrix)connectsthesestatesaccordingto1,2iiiUαανν==∑,(1.1)whereUtakesthegenericformcossinsincosUθθθθ⎡⎤=⎢⎥-⎣⎦.(1.2)withθsuchthatcos(2θ)ispositive.Startingwithapurebeamof,say,νβ,withdefinitemomentum(assumedthroughout),thetimeevolutionisgovernedbythemasseigenstatesandgivesaftertheneutrinobeamhastraveledadistancex≅tabeamcontainingamixtureofeachtypeofneutrinointheusualfashion.Inparticulartheprobabilityfortheconversionβ→αisPβα(t)=sin22θsin2φ,(1.3)wheretheangleφisdeterminedbytheenergydifference,()2211,24mEExxEφΔ=-≅(1.4)withΔm2≡m22-m12.TheapproximationreferstothelimitinwhichthemassdifferenceΔmismuchlessthanthemomentumofthebeam,alimitthatwillbeofinteresttous.Theoscillationlengthø(E)isthedistancethatcorrespondstoachangeinφbyπ;thatisø(E)=4πE/Δm2.Finallynotethattheprobabilityfornonconversion,i.e.thatβ→β,isPββ=cos2φ+cos22θsin2φ=1-Pβα.(1.5)Propagationinmatter.Inthepresenceofmatter,eachneutrinospeciemayhaveadifferentpotential.(Innormalmatteritisonlytheelectronneutrinosforwhichthereisapotentialassociatedwithrescatteringfromelectronsinthematerial.)Inparticular,supposethatthepotentialdifferenceofthetwoneutrinospeciesisVk,whereklabelsthematerial.Thentheeffectofthematteronthepropagationparametersisdescribedby3()()2222222222cos2sin24sin2sin2cos2sin2kkkkkkkkmmmmxEθξθφφθθθθθξθΔ→Δ=Δ-+Δ→=→∋=-+(1.6)where22kkEVmξ≡Δ(1.7)ThevalueofEforwhichξi=cos2θisreferredtoastheMSWresonance;forthatvaluetheangle2θigoesthroughπ/2.forgeneralvaluesofθ,totalconversioninasinglethicknessofanymedium,includingvacuum,isnotpossible,butitcanoccurforanappropriatethicknessofmediumatMSWresonance.TheexpressionforθiaboveisequivalenttotheformgiveninRef.[2],namely20tan2tan241sec2iEmθθπθ=-ΔA(1.8)withthereplacementø0=2π/Vi.Letususethelabelkforthematerial(carryinginformationnotonlyonthecompositionthroughtheneutrinopotentialsbutonthelayerthicknessaswell).ThenthegenericamplitudeAρσ{k}thataρ-neutrinoentersandaσ-neutrinoleaveslayer{k}isgivenforthetwoneutrinotypesαandβby{}{}{}{}coscos2sin,coscos2sin,sin2sin.kkkkkkkkkkkkAiAiAAiααββαββαφθφφθφθφ=+=-==-(1.9)SinceweshallmainlybeconcernedwithtotalconversionitisusefulforlatercomparisontogiveherethelengthXkformaximumconversioninasinglelayerofmaterialk,immediatelyfoundfromthelastoftheseequations.ThemaximumpossibleconversionisrealizedforalengthXksuchthatthefactorsinφk=1,i.e.,222farbelowMSW4cot2atMSW2faraboveMSWkkkkkkkkkVEmXmmVVVπξπππξθπ⎧⎪⎪⎪Δ⎪===⎨ΔΔ⎪⎪⎪⎪⎩(1.10)Themaximumconversionprobabilityissin22θk,whichissin22θfarbelowMSW,unityatMSW,andsin22θ/ξk2,asymptoticallysmall,faraboveMSW.4ThesmallestvalueofenergyandhenceofξkleadstothesmallestlengthXk.Atthesametime,itiseasiesttodetectenergeticneutrinos.Itisthereforehelpfultohavesomeideaoftheenergiesthatareinvolvedforneutrinos.WenotethatwhenweusethenumberVk=6×10-9cmappropriateforearth[2],thenξk=(E/(Δmc2)2)×2.5×10-13eVForΔmc2=O(10-3)eV,ξkisontheorderof1forE=O(4MeV).OnecanthinkofthisenergyasroughlythedividinglineforwhetheroneisbeloworaboveMSW,althoughofcoursethepreciseMSWenergydependsnotonlyonθbutonthecorrectvalueofΔm2aswell,andthelatternumberisnotyetfullyunderstood[10].Propagationthroughlayers.Whentherearelayersofmatterwithdifferingdensities,theninterferenceispossible.Thehistoryoftheeffectsofpassagethroughrepeatedlayersonmixingisinfactextensive,beginningperhapswithadiscussionofneutron-antineutronoscillationsinanonuniformmagneticfield[11],thencontinuingwithworkonneutrinosbyErmilovaetal.[12]andlaterbyAkhmedov[13]inalongseriesofpapers.Theselatterreferencesdescribewhatthephenomenonofparametri