Does the Sun Appear Brighter at Night in Neutrinos

J.N.BahcallandP.I.Krastev

SchoolofNaturalSciences,InstituteforAdvancedStudy

Princeton,NJ080

Wecalculateaccuratelythenumberofsolarneutrinoeventsexpectedas

afunctionofsolarzenithangle,withandwithoutneutrinooscillations,fordetectorsatthelocationsofSuper-Kamiokande,SNO,andtheGranSassoNationalLaboratory.Usingdifferentearthmodelstoestimategeophysicaluncertainties,anddifferentsolarmodelstoestimatesolaruncertainties,weevaluatedistortionspredictedbytheMSWeffectinthezenithangledistribu-tionsofsolarneutrinoevents.Thedistortionsarecausedbyoscillationsandbyν−einteractionsintheearththatregenerateνefromνµorντ.Weshowthatthefirsttwomomentsofthezenith-angledistributionaremoresensitivetothesmallmixingangleMSWsolutionthantheconventionallystudiedday-nightasymmetry.Wepresentiso-sigmacontoursthatillustratethepotentialofSuper-Kamiokande,SNO,BOREXINO,ICARUSandHERON/HELLAZfordetectingtheearthregenerationeffectattheiractuallocations(andattheequator).MSWsolutionsfavoredbythefourpioneeringsolarneutrinoexperimentspredictcharacteristicdistortionsforSuper-Kamiokande,SNO,BOREXINO,andICARUSthatrangefrombeingunmeasurablysmallto>5σ(stat)afteronlyafewyearsofobservations.

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I.INTRODUCTION

Fouroperatingsolarneutrinoexperiments(Chlorine[1],Kamiokande[2],GALLEX[3])andSAGE[4]havedetectedneutrinosfromnuclearfusionintheinteriorofthesunwithap-proximatelythenumbersandenergiesexpectedfromstandardsolarmodels[5,6].Moreover,soundspeedscalculatedfromthestandardsolarmodelsagreewiththehelioseismologicallydeterminedsoundspeedstoarmsaccuracyofbetterthan0.2%throughoutessentiallytheentiresun[7].

Nevertheless,quantitativediscrepancieshavepersistedforalmostthreedecadesbetweenthepredictionsofthestandardsolarmodelsandtheobservationsofsolarneutrinoexper-iments[8–10].Severalsuggestedmodificationsofneutrinopropertiesprovideexcellentfitstotheexistingsolarneutrinodata[11].

Aretherepotential“smokinggun”indicationsofnewphysics?Yes,themostpopularneutrinophysicssolution,theMikheyev-Smirnov-Wolfenstein(MSW)effect[12],predictsseveralcharacteristicanduniquephenomena.TheMSWeffectexplainssolarneutrinoob-servationsastheresultofconversionsinthesolarinteriorofνeproducedinnuclearreactionstothemoredifficulttodetectνµorντ.

PotentiallydecisivesignaturesofnewphysicsthataresuggestedbytheMSWeffectincludeobservingthatthesunisbrighterinneutrinosatnight(the‘earthregenerationeffect’)[13–15],detectingdistortionsintheincidentsolarneutrinoenergyspectrum[16],andobservingthatthefluxofalltypesofneutrinosexceedsthefluxofjustelectrontypeneutrinos[17,18].Ademonstrationthatanyofthesephenomenaexistswouldprovideevidenceforphysicsbeyondtheminimalstandardelectroweakmodel.

Theregenerationeffectisanespeciallypowerfuldiagnosticofnewphysicssincenodiffer-enceispredictedbetweenthecountingratesobservedduringthedayandatnight(or,moregenerally,anydependenceofthecountingrateonthesolarzeithangle)bysuchpopularal-ternativestotheMSWeffectasvacuumoscillations[19],magneticmomenttransitions[20],orviolationsoftheequivalenceprinciple[21].

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Inthispaper,weinvestigatethesensitivityofnewsolarneutrinoexperiments,Super-Kamiokande[22],SNO[23],ICARUS[24],BOREXINO[25],HERON[26]andHELLAZ[27],totheearthregenerationeffect.TheMSWeffectpredictsthat,forcertainvaluesoftheneutrinomassesandmixingangles,ν−einteractionsintheearth(atnight)mayconvertνµorντfromthesunbackintothemoreeasilydetectableνe.

AnaccurateevaluationofthesystematicsignificanceofexperimentalresultswillrequirethedetailedMonteCarlosimulationsthatwillbecarriedoutbytheexperimentalcollab-orations;thecollaborationswilldeterminethebestestimatesanduncertaintiesforallthequantitiesthataffecttheexperimentalresult.Theseresultswillthenbeanalyzedusingcomputercodesthatincludetheexperimentaldetailsandwhichmakeuseofoptimalsta-tisticaltechniquessuchasmaximumlikelihoodanalysis.IntheabsenceofdetailedMonteCarlosimulationsoftheexperimentalcharacteristics(yettobedetermined),weestimateinthispaperthestatisticalsignificanceofexpectedresultsbycomparingthepredictionsofvariousMSWscenarioswithrespecttotheno-oscillationscenarioincludingonlystatisticalerrorsandanalyzingtheresultswithaχ2statistic.

ThereaderwhowantstoseetheapproximatepowerofthenewexperimentscanturnimmediatelytoFig.9,whichshowsthesignificancelevel(statisticalerrorsonly)withwhichnewsolarexperimentscoulddetecttheregenerationeffect.Experiencewiththeoperatingexperimentsoveraperiodofyearsmaybenecessarytodeterminethesizeofthesystematicerrors.

Thispaperisorganizedasfollows.InSec.IIwesummarizethegeneralfeaturesofνeregenerationintheearth.WethendescribeinSec.IIIthedifferentmodelsoftheearthusedforestimatingtheuncertaintiesinthenumericalcalculationsduetouncertaintiesinthedensityprofileandthechemicalcompositionoftheearth’sinterior.Inthedetailedcalculationsthatfollow,weusethedifferencesbetweentheresultsobtainedfromthedifferentmodelsoftheearthtodeterminethegeophysicaluncertaintiesintheMSWpredictions.Afterthesepreliminaryconsiderations,wedetermineinSec.IVtheregionsinthemixingangleandmassdifferenceplane,sin22θ-∆m2,thatareallowedbythelatestsolarneutrino

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data,takingintoaccounttheinfluenceofearthregenerationonthepredictedcountingratesinthechlorine,Kamiokande,GALLEX,andSAGEexperiments.InadditiontothefamiliarlargemixingangleandsmallmixingangleMSWsolutions,wefindanadditionalLOWsolution(largemixingangle,smallerneutrinomassdifference)thathasanacceptableconfidencelevelonlywhenearthregenerationisincludedinthecalculations.

Wetheninvestigatethesensitivityofnext-generationsolarneutrinoexperimentstotheearthregenerationeffect.Webeginbydefiningandcalculatingthezenith-angleexposurefunctioninSec.V.Thisfunctiondependsonlyonthelocationofaneutrinodetectoronthesurfaceoftheearth;itisindependentofthecharacteristicsofthedetector.WealsocalculateinSec.Vthedistortedzenithangledistributionthattakesaccountofregenerationintheearth.

TheresultsforMSWregenerationgivenpreviouslyintheliteratureinvolvemakingapproximationseitherinthemodeldescriptionoftheearthorinthecalculationoftheaveragesurvivalprobabilityafterregeneration,orboth.Instead,weintegratenumericallythedifferentialequationsdescribingtheevolutionoftheneutrinostatesintraversinganaccuratemodeloftheearth,therebyavoidingthenecessityofarguingthatanapproximationschemeissufficientlyaccurate.Inseveraltablesinthispaper,wepresentnumericalresultstoaprecisionof0.01%,anaccuracymuchhigherthancanbemeasuredexperimentally.Theseprecisenumericalpredictionsaregiveninordertoillustratethesmalleffectonmeasurablequantitiesofsomeofthesystematicdifferences.

WeintroduceinSec.VIthefirsttwomomentsofthezenith-angledistributionofneu-trinoeventsandcalculatethedependenceofthemomentsonneutrinoparametersforthenewsolarneutrinoexperiments:Super-Kamiokande,SNO,ICARUS,BOREXINOandHERON/HELLAZ.Forcomparison,wecalculateaccuratelyinSec.VIItheconventionalday-nightasymmetry;wepresentvaluesoftheday-nightasymmetryforthenewsolarneu-trinodetectorsmentionedabove.

Whichcharacterizationismoresensitivetonewphysics:themomentsofthezenith-angledistributionortheday-nightasymmetry?WeshowinSec.VIIIthatthemomentsaremore

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sensitivetothesmallmixingangleMSWsolutionandtheday-nightasymmetryismoresensitivetothelargemixinganglesolution.AlthoughlessstatisticallypowerfulthanthefullMonteCarlosimulationsthatwillbecarriedoutbytheexperimentalcollaborations,theanalysesusing,e..g,momentsortheday-nightasymmetrycanbecarriedoutquicklybytheoreticiansinterestedindeterminingwhethernewparticlephysicsscenarioscanbetestedbytheexperimentsorwhethertheyarealreadyinconsistentwithdatathathavebeenpublished.Forcompleteness,wepresentinSec.IXthemomentsoftheelectronrecoilenergyspectrumforSuper-KamiokandeandSNOthatwerecomputedincludingtheearthregenerationeffect.

WethenshowinSec.XthattheMSWpredictionsforregenerationintheearthdependonlyslightlyontheadopteddensityprofileoftheearth,thechemicalcompositionoftheearth,andthedetailsofthesolarmodel.

FollowingthesuggestionofGelb,Kwong,andRosen[28],wecalculateinSec.XItheincreaseinthesensitivitytotheregenerationeffectthatcouldbeachievedbybuildingdetectorsattheequator.WediscussandsummarizeourmainresultsinSec.XII.

II.THEEARTHREGENERATIONEFFECT

Weworkinatwo-neutrinomixingschemeinvolvingνe(producedinthesun)andeitherνµorντ(producedbyoscillations).SoonafterMikheyevandSmirnovsuggestedtheMSWeffect[12]asapossiblesolutionofthesolarneutrinoproblems,severalauthorspointedout[13]thatday-nightvariationsoftheeventratesinsolarneutrinodetectorscouldprovidespectacularconfirmationoftheMSWeffectandthusofnewphysics.

TheMSWsolutionofthesolarneutrinoproblemsrequiresthatelectronneutrinospro-ducedinnuclearreactionsinthecenterofthesunareconvertedtomuonortauneutrinosbyinteractionswithsolarelectronsontheirwayfromtheinteriorofthesuntothedetectoronearth.Theconversioninthesunisprimarilyaresonancephenomenon,whichoccursataspecificdensitythatcorrespondstoadefiniteneutrinoenergy(foraspecifiedneutrinomass

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difference).

Duringday-time,thehigher-energyneutrinosarrivingateartharemostlyνµ(orντ)withsomeadmixtureofνe.Atnight-time,neutrinosmustpassthroughtheearthinordertoreachthedetector.Asaresultoftraversingtheearth,thefractionofthemoreeasilydetectedνeincreasesbecauseoftheconversionofνµ(orντ)toνebyneutrinooscillations.ForthesmallmixingangleMSWsolution,interactionswithelectronsintheearthincreasetheeffectivemixingangleandenhancetheconversionprocess.ForthelargemixingangleMSWsolution,theconversionofνµ(orντ)toνeoccursbyoscillationsthatareonlyslightlyenhancedovervacuummixing.Thisprocessofincreasingintheearththefractionoftheneutrinosthatareνeiscalledthe“regenerationeffect”andhastheoppositeeffecttotheconversionofνetoνµ(orντ)inthesun.

Becauseofthechangeofneutrinotypeintheearth,theMSWmechanismpredictsthatsolarneutrinodetectorsshouldgenerallymeasurehighereventratesatnightthanduringday-time.

Figure1illustratesaschematicviewofasolarneutrinodetectoratthegeographiclatitude,φ.SincetheearthissphericallysymmetrictoO(10−2.5),itissufficienttoconsiderthecross-sectionsliceshowninthefigure.∗Twolinesdeterminethegeometry:onelinedefinesthezenithdirection,andtheotherlineisthetrajectoryoftheneutrino.Thezenithangleα(00<α<1800)betweenthesetwolinesspecifiestheneutrinotrajectoryintheearth.Thesurvivalprobabilitydependsontheneutrinooscillationparameters,∆m2andsin22θ,ontheneutrinoenergy,E,andonthepath(i.e.,α)theneutrinotravelsthroughtheearth.Sinceαchangesduetotheapparentmotionofthesun,theneutrinosurvivalprobabilityshouldchangewithtimeaswell,resultinginanasymmetricdistortionoftheangulardistributionofevents.

Real-timedetectors,whichrecordthetimesatwhichneutrinosinteractwithinthedetec-tor,arebestsuitedforstudyingtheearthregenerationeffect.Inradiochemicalexperimentsthetimeofdetectionispoorlyknown,sinceatypicalrunusuallylastsbetweenseveralweeks(gallium)andseveralmonths(chlorine).

†

Kamiokande,areal-timeneutrinoelectronscatteringdetector,didnotseeanysignalfortheearthregenerationeffect.TheKamiokandecollaborationusedthisnon-observationtoruleoutanimportantregioninparameterspaceforwhichthepredictedday-nightasymme-try,orzenith-angledependence,islarge[30].However,thesensitivityoftheKamiokandedetectorwasinsufficienttoprobethefull∆m2-sin22θparameterspaceforwhichtheremightbeanappreciableday-nighteffect,measurablebySuper-KamiokandeorSNO.Thereareseveralcalculationsintheliterature[31–35]oftheexpectedmagnitudeoftheregenerationeffectinfutureexperiments.Differentgroupsofresearchershaveuseddifferentmodelsoftheearthintheircalculations.Noquantitativeestimatehasbeenmadepreviouslyofthesensitivityofthemeasurablequantitiestotheadopteddensityprofileoftheearth;eachgrouphastypicallypresentedresultsusingaspecificdensityprofile,oftennotthebestavailableprofile.Inthesubsequentsections,wedescribedirectnumericalcomputationsoftheearthregenerationeffectforsixdifferentmodelsoftheearth.Thuswequantifythedependenceofthecalculatedcharacteristicsoftheearthregenerationeffectonthemodeloftheearthandexhibitthecorrespondinguncertaintieswhichturnouttoberathersmall.ThedensityprofilesinthesixmodelsoftheeartharedescribedinSec.III.

III.EARTHMODELS

TheMSWeffectintheearthdependsupontheelectronnumberdensityasafunctionofradius.Inthissection,wedeterminebest-estimatesandarangeofuncertaintiesforthe

totalmassdensityandforthechemicalcomposition.Weusethebestavailableearthmodelsandchemicalcompositionformostofthecalculationsperformedinthispaper,butwealsocarryoutcalculationsforfiveoldermassmodelsinordertodeterminetheuncertaintiesinthepredictedMSWeffectsthatarisefromuncertaintiesinthemodeloftheearth’sdensityprofile.Weuseabest-estimatechemicalcompositionforthecorethatisinferredfromtheseismologicalmeasurements.TotestthesensitivityoftheMSWpredictionstotheassumedchemicalcompositionoftheearth,wemakeextremeassumptionsthatmaximizeorminimizetheaveragechargetomassratioandcarryoutcalculationsalsofortheseextremecases.

A.DensityProfiles

Thedensitydistributioninsidetheearthisknownwithaprecisionofafewpercent[36].Alargesetofseismicmeasurementshasbeenusedtoobtainthemostaccuratemodel,PREM[37](thePreliminaryReferenceEarthModel),fortheearth’sdensitydistribution.WewillusethePREMmodelforallofourbest-estimatecalculations.ThismodelhasalsobeenusedbyLisiandMontanino[34]asthebasisoftheirrecentanalyticstudyofearthregeneration.OthermodelsaredescribedinRefs.[38]and[39].Todeterminethesensitivitytotheassumeddensityprofile,wehaveperformedcalculationswitharepresentativesetofsixdifferentearthmodels,allsphericallysymmetricandwiththesameradius,R⊕=6371km.

Figure2showsthedensityprofilesofthesixearthmodels.Thedensitydistributionsinthesemodelsaredividedintofivezones:a)acrustwithathicknessofafewtensofkilometers,b)anuppermantleextendingdownfromthecrusttoabout1000km,c)alowermantledowntoabout2900km,d)anoutercorebetween1250kmand3480kmfromthecenter,ande)aninnercoreofradius≈1220km.Thedensitychangesabruptlybetweentheinnerandoutercore,andalsoattheborderbetweenthelowermantleandtheoutercore.Thepositionsoftheseabruptchangesareknownwithanaccuracyofbetterthanapercentfromseismologicaldata.

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TableIcomparesthemassandmomentofinertiathatwehavecomputedforeachofthesixearthmodelswiththemeasuredvalues.Therecentmodelsgivenin[37]and[39]reproducethetotalmassandmomentofinertiawithexcellentprecision.Theoldermodels[40–43]giveslightlyworsefitstothemassandmomentofinertia.ThemodelslistedinthethirdtosixthrowinTableIareinconflictwithseismologicalmeasurements.

Thesetofsixmodelsrepresentsasamplethatallowsforvariationsofthedensitydistri-butionlargerthantheuncertaintiesinthePREMmodel.AsshowninSec.VI(andthelastfourcolumnsofTableI),thelargedifferencesbetweenthesixdensitymodelsproducerela-tivelysmall(butnotalwaysnegligible,seeSec.VI)changesinthepredictedcharacteristicsoftheearthregenerationeffectfortheSuper-Kamiokande,SNO,BOREXINO,ICARUS,andHERON/HELLAZexperiments.

Figure2showsthatthelargestdifferencesbetweenthesixmodelsoftheearthareinthecore,below2000km.Theoperatingsolarneutrinoexperiments,andthosecurrentlyunderdevelopment,arelocatedatrelativelyhighnorthernlatitudes.Solarneutrinosthataredetectedintheseexperimentsnevercrosstheinnercoreoftheearth.Amongthereal-timeexperimentsthatarecurrentlyoperatingorareunderconstruction,Super-Kamiokandeismostsensitivetothecoredistribution.Nevertheless,thefractionofayearduringwhichtheneutrinoscrosstheoutercoreattheKamiokasiteissmall(≃7%).

B.ChemicalCompositions

Measurementsofthepropagationofseismologicalwavesintheearth’sinteriorandstudiesofthepropertiesofmineralsunderhighpressure,havebeencombinedtodeterminethechemicalcompositionoftheearth’sinteriorwithrelativelyhighaccuracy[44,45].Usingtheresultsofreferences[44,45],weadoptabest-estimatechargetomassratio,Z/A,of0.468forthecore(83%Fe,9%Ni,and8%lightelementswithZ/A=0.5)and0.497forthemantle(41.2%SiO2,52.7%MgOand6.1%FeO).

Alowerlimitforthechargetomassratiointhecoreis0.465,whichcorrespondsto

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assumingacompositionof100%iron.Fromtheseismicandmineraldata,geophysicistshaveconcluded[44]thattheminimumamountofironinthecoreis80%.WedetermineamaximumvalueofZ/A=0.472inthecorebyassumingacompositionof80%ironand20%lightelements.Thetotalrangeoftheelectronnumberdensityduetotheimperfectlyknowncompositionofthecoreisabout1.5%.

Thechemicalcompositioninthemantleisbelievedknowntoabout1%(seeref.[36]).Weconsiderherevariationsof−1%and−2%.ThevalueofZ/Xinthemantlecannotbeincreasedsignificantlyabovethestandardvalueof0.496becausethatwouldrequirethepresenceofalargeamountofhydrogeninthemantle.

IV.AVERAGEEVENTRATESANDMSWSOLUTIONS

Includingtheearthregenerationeffect,wehavecalculatedtheexpectedone-yearaver-ageeventratesasfunctionsoftheneutrinooscillationparameters∆m2andsin22θforallfouroperatingexperimentswhichhavepublishedresultsfromtheirmeasurementsofsolarneutrinoeventrates.Theseincludethechlorineexperiment,Kamiokande,GALLEXandSAGE.ThusweupdateourpreviousresultsgiveninRef.[11],inwhichtheeartheffectwasneglected.Wetakeintoaccount,asbefore,theknownthresholdandcross-sectionforeachdetector.InthecaseofKamiokande,wealsotakeintoaccounttheknownenergyresolution(20%1σatelectronenergy10MeV)andtriggerefficiencyfunction[46].

¯SE,foralargenumberofWefirstcalculatetheoneyearaveragesurvivalprobability,P

valuesof∆m2andsin22θusingthemethoddescribedinAppendixA.Thenwecomputethecorrespondingoneyearaverageeventratesineachdetector.Weperformaχ2analysistakingintoaccounttheoreticaluncertaintiesandexperimentalerrorsasdescribedin[47].TableIIsummarizesthereportedmeaneventratesfromeachdetector.Weobtainallowedregionsin∆m2-sin22θparameterspacebyfindingtheminimumχ2andplotting

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contoursofconstantχ2=χ2min+∆χwhere∆χ=5.99for95%C.L.and9.21for99%

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C.L.‡.

Thebestfitisobtainedforthesmallmixingangle(SMA)solution:

∆m2=5.0×10−6eV2,(1a)sin22θ=8.7×10−3,

(1b)

whichhasaχ2min=0.25.Therearetwomorelocalminimaofχ2

.Thebestfitforthewell

knownlargemixingangle(LMA)solutionoccursat

∆m2=1.3×10−5eV2,(2a)sin22θ=0.63,

(2b)

withχ2min=1.1.Thereisalsoalessprobablesolution,whichwerefertoastheLOWsolution(lowprobability,lowmass),at[48,31]

∆m2=1.1×10−7eV2,(3a)sin22θ=0.83.

(3b)

withχ2min=6.9.TheLOWsolutionisacceptableonlyat96.5%C.L.

Figure3showstheallowedregionsintheplanedefinedby∆m2andsin22θ.TheC.L.is95%fortheallowedregionsoftheSMAandLMAsolutionsand99%fortheLOWsolution.Theblackdotswithineachallowedregionindicatethepositionofthelocalbest-fitpointinparameterspace.TheresultsshowninFig.3werecalculatedusingthepredictionsofthe1995standardsolarmodelofBahcallandPinsonneault[6],whichincludesheliumandheavyelementdiffusion;theshapeoftheallowedcontoursdependsonlyslightlyupontheassumedsolarmodel(seeFig.1ofref.[11]).

Theresultsgivenherediffersomewhatfromthosegivenearlierinref.[11],bothbecausewearenowincludingtheregenerationeffectandalsobecauseweareusingmorerecentexperimentaldataforthepioneeringsolarneutrinoexperiments.Comparingtheresults

giveninEqs.(1)–(3)andFigs.3withthecorrespondingallowedregionsobtainedforthesameinputneutrinoexperimentaldatabutwithoutincludingtheeartheffectshowsthatterrestrialregenerationchangesonlyslightlythebest-fitsolutionsfortheSMAsolution<5%in∆m2andsin22θ)andtheLMAsolution(<10%in∆m2andsin22θ).The(∼∼regenerationeffectisincluded;otherwise,theLOWsolutionisruledoutat99.9%C.L.Figure4comparesthecomputedsurvivalprobabilitiesfortheday(noregeneration),thenight(withregeneration),andtheannualaverage.Theseresultsareusefulinunderstandingtheday-nightshiftsintheenergyspectrumthatarecomputedanddiscussedinSec.IX.TheresultsinthefigurerefertoadetectoratthelocationofSuper-Kamiokande,butthediffer-encesareverysmallbetweenthesurvivalprobabilitiesatthepositionsofSuper-Kamiokande,SNO,andtheGranSassoUndergroundLaboratory.

valuesofχ2minarealsonotsignificantlyaffected.TheLOWsolutionisacceptableonlyifthe

V.THEZENITH-ANGLEEXPOSUREFUNCTIONANDTHEZENITH-ANGLE

DISTRIBUTIONFUNCTION

Inthissection,wedefineandcalculatethezenith-angleexposurefunctionandshowhowtheexposurefunctioncanbedistortedbyoscillationsandbyneutrino-electroninteractionsintheearthintothezenith-angledistributionfunction.

Thefractionalnumberofneutrinoeventsobservedasafunctionofsolarzenithangle,α(seeFig.1),is,forstandardneutrinophysics,determinedonlybythelatitudeoftheneutrinodetector.Thenumberofeventsislargestforzenithanglesatwhichthesunspendsthemosttimeinitsapparentmotionaroundtheearth.Inwhatfollows,weshallrefertothenormalizednumberdistributionofevents,Y(α),asthe“zenith-angleexposurefunction”;thisfunctiondescribestherelativeamountoftimethedetectorisexposedtothesunatafixedzenithangle.Wepresentinthissectionthezenith-angleexposurefunctioncalculatedassumingmasslessneutrinos(nooscillations)anddetectorsplacedatKamioka,Japan(KamiokandeandSuper-Kamiokande),Sudbury,Canada(SNO),andtheGranSasso

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UndergroundLaboratory(GALLEX,ICARUS,BOREXINOandHERON/HELLAZ).TheexposurefunctionfortheHomestakedetectorhasbeencalculatedbyCherryandLande‘[13].

Thezenith-angledistributionofeventscanbedistortedbytheearthregenerationeffectimpliedbyMSWsolutions,sinceatsolarzenithangleslargerthan900neutrinospassthroughtheearthontheirwaytothedetectorandthereforeνe’scanberegeneratedbyinteractionswithelectronsintheearth’sinterior.Thedistorteddistributionfunction,f(α),willbereferredtoasthe“zenith-angledistribution.”Themaingoalofthispaperistocalculateandanalyzetheshapeoff(α)predictedbydifferentMSWsolutionsandtoestimatethesensitivityofdetectorstothedifferencebetweenf(α)andthezenith-angleexposurefunction,Y(α).

A.Zenith-AngleExposureFunction

Inordertorepresentaccuratelythezenith-angleexposurefunction,weconsider360angles,αi,separatedby0.5◦intervalsbetween0and180◦.Thefractionoftimeinayearduringwhichthezenithangle,α,(seeFig.1)ofthesunisclosetoαkisproportionalto:

Y(αk)=

′

N󰀃i=1

θ(α(ti)−αk)θ(αk+1−α(ti))[1AU/R(ti)]−2.

(4)

Hereα(ti)isthesolarzenithangleattimeti,N=T/∆t,Tisthedurationofonecalendaryear,and∆tisthetime-step.Thefunctionθisthewellknownstep-functionθ(x)=0,x<0andθ(x)=1,x≥0.ThefunctionR(t)istheinstantaneousearth-sundistance,and1AUistheoneyearaverageearth-sundistanceforwhichsolarneutrinofluxesareroutinelycalculatedinstandardsolarmodels(seeAppendixBforanexplicitrepresentationofthetime-dependentearth-sundistance).

ThesuminEq.(4)wascomputedbysimulatingthemotionofthesunonthecelestialsphereduringonecalendaryearusingtheformulaeinAppendixB.Computationswithdifferenttimestepsbetweenafewsecondstoafewminutesarepracticallyequivalentforcalculatingeventratesinthevarioussolarneutrinodetectors.

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Thenormalizedzenith-angleexposurefunction,Y(αk),isobtaineddirectlyfromEq.(2),

Y(αk)=Y(αk)/

′

N󰀃i=1

Y(αi).

′

(5)

ThusY(αk)isthefractionofthetimeduringonecalendaryearthatthesun’szenithangleiswithintheinterval(αk−0.250,αk+0.250).Ifthesolarneutrinofluxisconstantintime(noneutrinooscillationsoccur),thenthefunctionY(α)willbethenormalizedangulardistributionofeventsinthedetector.WeusetheanalyticalexpressionsforR(t)giveninRef.[49](seeEq.B2).Thezenithangleexposurefunctionwascalculatedanalyticallyinref.[34],withoutincludingthevariationduetothechangingearth-sundistance.

Figures5and6showtheundistortedzenith-angleexposurefunctionsforSuper-Kamiokande,SNO,and,thedetectorsassumedtooperateatGranSasso(ICARUS,BOREX-INO,andHERON/HELLAZ).ConvenientnumericaltablesfortheY(αk)areavailableathttp://www.sns.ias.edu/∼jnb.TableIIIliststhelatitudes(allnorthern)ofeachofthesolarneutrinodetectors.

ThepositionsofthesharppeaksinFigs.5and6aredeterminedbythelocationofthedetectorandbytheobliquity,ǫ,oftheearth’sorbit(approximately23.40).Theabsolutevalueofthedifferencebetweenthemaximum(orminimum)possiblezenithangleforagivenlocationandthepositionoftheclosestpeakisequaltoφ+ǫ.Atsummersolstice,thesun’szenithanglechangesfromtheminimumpossibletotheanglecorrespondingtothesecondpeak(atanangle>900).Atwintersolsticethesungoesbetweenthepeakatangle<900andthemaximumpossiblezenithangle.Thusduringwintersolarneutrinospassclosertotheearth’scenter,whereasduringsummertheygothroughlowerdensitylayersofthemantle.

B.Zenith-AngleDistributionFunction

Wecalculatetheeventrates,Qi,alongeachdirection,αi,withtheaidofasetofsurvival

i

probabilities,PSE,computed(justonceforeachdirection,mixingangle,andE/∆m2)along

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afixedsetoftrajectoriesthroughtheearth.ThenumericalvalueofeachQiisobtainedfromEq.C1ofAppendixC.Thenormalizedzenith-angledistributionfunctionis

f(αi)=Q(αi)Y(αi)/

󰀃

k

Q(αk)Y(αk).(6)

Intheabsenceofoscillations,thedetectoreventrate,Qi,isindependentofdirectionanddisappearsfromtheright-handsideoftheequation.Inthiscasef(α)≡Y(α).

InFigs.5and6,wepresenttheexpecteddistortedangulardistributionsfortheSMA,LMA,andLOWsolutionsinSuper-Kamiokande,SNO,ICARUS,BOREXINOandHERON/HELLAZ.InthepanelsforSuper-Kamiokande,SNOandICARUS,weshowonlythedistributionscorrespondingtothebest-fitpointsintheSMAandLMAsolutions;thecurvescorrespondingtotheLOWsolutionarevirtuallyundistorted.Correspondingly,inthepanelsforBOREXINOandHERON/HELLAZ,weshowonlytheexpectedangulardis-tributionfortheLOWsolution(seeEq.3b),sincetheSMAandLMAsolutionsimplyonlyanegligibledistortionofthezenith-angledistributionfortheselowenergydetectors.

VI.MOMENTSOFTHEZENITH-ANGLEDISTRIBUTION

Inthissection,weevaluatetheexpectedMSWdistortionsoftheangulardistributionandcomparethefirsttwomomentsofthepredictedangulardistributionofeventswiththecalculatedmomentsexpectedintheabsenceofoscillations.Thiscomparisonconstitutesanewand,forthesmallmixinganglesolution,amorepowerfulwayofanalyzingthetimedependenceoftheobservedneutrinoevents.ThepredicteddistortionsoftherecoilelectronenergyspectrainSuper-KamiokandeandinSNOwereinvestigatedinRef.[50,34]intermsoftheanalogousmomentsoftheenergydistribution.

InSec.VIA,wedefinethefirstandsecondmomentsofthezenith-angledistributionandthencalculateinSec.VIBthepredictedMSWchangesinthezenith-angledistributions.Weplottherelativeshiftofthefirstmomentintheplaneoftheneutrinooscillationparameters∆m2andsin22θ,illustratingtherangeofpossiblevaluesoftheshiftofthefirstmoment

14

withinthe95%C.L.allowedbytheChlorine[1],Kamiokande[2],GALLEX[3],andSAGE[4]experiments.

HowsensitivewilltheSuper-Kamiokande,SNO,ICARUS,BOREXINOandHEL-LAZ/HERONexperimentsbetotheregenerationeffect?

Weanswerthisquestionin

Sec.VIBbyplottingthenumberofstandarddeviationseachMSWsolutionisseparatedfromtheno-oscillationsolutionintheplanedefinedbythevaluesofthefirsttwomoments.

A.MomentDefinitions

Thefirsttwomomentsofthezenith-angledistributionaredefinedby

󰀁α󰀂=

and

σ=

2

󰀄

αf(α)dα,(7a)

󰀄

(α−󰀁α󰀂)2f(α)dα.

(7b)

Thefractionalshiftsinthesetwomomentsaredefinedas

∆󰀁α󰀂

2σ0

22

=(σ2−σ0)/σ0,

(8b)

2

whereα0andσ0arethemomentsofthezenith-angleexposurefunctiondistribution(no

oscillations)and󰀁α󰀂andσ2arethemomentsofthedistorted(byoscillations)zenith-angledistribution.BecauseofthesymmetryofY(α)aboutα=90deg,α0=π/2forallthedetectorlocations.Thevaluesoftheundistortedsecondmoment,σ0,are:37.90(Super-Kamiokande),32.90(SNO),34.90(GranSasso),33.90(Homestake),34.50(Baksan),and47.50(Equator).

TheKamiokandecollaborationexcludedasignificantregionin∆m2-sin22θparameterspacebygroupingtheeventsfrombelowthehorizonintofivebins,correspondingtodifferent

15

zenithangles[2].LisiandMontaninoinRef.[34]calculatedbinnedangulardistributionsforSuper-KamiokandeandSNOfordifferentvaluesof∆m2andθ2.

Thecalculationofthefirsttwomomentsofthezenith-angledistributionissubjecttofewercomplicationsthanthemoretraditionalmethodusedinreferences[2,34]ofbinningthedataanddoingaχ2analysis.Forbinneddata,theaprioriunknownnormalizationandtheangulardependenceofthedetectorsensitivityare,forexample,twooftheaspectsofneutrinoexperimentsthatintroducebin-to-bincorrelationswhichareoftendifficulttoestimatefromtheobservationsortoevaluateincludingalloftherelevantdetectorcharacteristics.Ontheotherhand,thecalculationofthefirstandsecondmomentsdirectlyfromthedataisstraightforward.

B.PredictedMSWMoments

Figures7and8showtheexpectedshiftsinthefirsttwomomentsasafunctionoftheneutrinooscillationparameters,∆m2andsin22θ.WeplotinFig.7(Fig.8)contoursofcon-2

stantfractionalshift,∆α/α0(∆σ2/σ0),ofthefirstmoment(secondmoment)inthe∆m2-

sin22θplane.Thefivepanelsare,forbothfigures,forSuper-Kamiokande,SNO,ICARUS,BOREXINOandHERON/HELLAZ.ForSuper-Kamiokande,∆α/α0variesbetween−0.1%and3%intheSMAregionandbetween0.5%and7%intheLMAregion.ThecorrespondingrangesforSNOare(−0.1,4.5)%(SMA)and(0.3,15)%(LMA).SNOissomewhatmoresen-sitivethanSuper-Kamiokandetotheshiftofthefirstmomentbecauseνµs(andντs)donotcontributetothecharged-currentsignalinSNO.ICARUSandSNOhavesimilarsensitivi-tiestotheregenerationeffectsincebothobservechargedcurrentreactionswithhigh-energythresholdsof10.9MeVand6MeV,respectively.Thelow-energyexperiments,BOREXINOandHERON/HELLAZ,areinsensitivetoboththeSMAandLMAallowedregions,butwillbeabletotesttheLOWsolutiontowhichneitherSNOnorSuper-Kamiokandearesensitive.Theresultsforthesecondmoment,showninFig.8,exhibitsimilartrends.

Figure9summarizesthepotentialofthesecondgenerationofsolarneutrinoexperiments

16

fordiscoveringnewphysicsviatheearthregenerationeffect.Thefiguredisplaysiso-sigmaellipses,statisticalerrorsonly,intheplaneofthefractionalpercentageshiftsofthefirst

twomoments,∆α/α0and∆σ2/σ2

0.Assumingatotalnumberofeventsof30000,(which

correspondsto∼5yearsofstandardoperationforSuper-Kamiokandeand∼10yearsforSNO),wehavecomputedthesamplingerrorsonthefirsttwomomentsaswellasthe

󰀅

correlationoftheerrorsusingthefollowingwellknownformulae[51]:σ(m1)=

󰀅

N

,σ(m2)=

N

,andρ(m1,m2)=

µ√

µ3

2

Weevaluatenumericallytheday-nightasymmetryforSuper-Kamiokande,SNO,ICARUS,BOREXINO,andHERON/HELLAZ.Wepresentquantitativeestimatesofthesensitivityoffutureexperimentstothepredicteddifferenceinnight-timeandday-timeeventrates.

Theday-nightasymmetryisdefinedas:

AQn−d=

n−Qd

ThedifferencesbetweenSNOandICARUSaremainlyduetothedifferentassumedneu-trinothresholds(6.4MeVand10.9MeV,respectively)andtothelocationsofthedetectors.Forlargemixingangles,forwhichtheregenerationeffecttakesplacemainlyintheman-tle,thesensitivitiesofthetwodetectorsareparticularlysimilar.Atsmallmixingangles(sin22θ<0.3)and5×10−6<∆m2/eV2<10−5,ICARUSisslightlymoresensitivetotheeartheffectbecauseitislocatedatalowerlatitudethanSNO.

Thethreefuturedetectorsthatwillmeasurelow-energyneutrinos,BOREXINO(7Be),HERON(pp),andHELLAZ(pp),willbesensitivetothelargeday-nightasymmetriespre-dictedintheLOWsolution,butinsensitivetotheasymmetriespredictedbytheSMAandLMAsolutions.Therangeofday-nightasymmetriesexpectedfortheLOWsolutionare(16±5%)forHERON/HELLAZand(16±8%)forBOREXINO.

VIII.WHICHSTATISTICALTESTISBEST?

Whichstatisticaltestsaremostpowerfulindetectingnewphysics?Whattypeofanalysiswillmostclearlyshowdeparturesfromthezenith-angleexposurefunctionduetotheregen-erationeffect?Byanalyzingsimulateddatainthissection,weshallseethatthepreferredstatisticalanalysisdependsuponwhichsolutionNaturehaschosen.

TableVIcomparesthesensitivityofSuper-KamiokandeandSNOtotheearthregener-ationeffectforthreedifferentstatisticaltests.Wehavecomputedthenumberofstandarddeviationsbywhichthebest-fitMSWsolutions(describedinSec.IV)differfromtheundis-tortedzenith-angleexposurefunction.Weconsiderthefirstandsecondmomentsofthezenith-angledistribution(seeSectionV),theday-nightasymmetry(An−d)(seeSec.VI),andtheKolmogorov-Smirnovtestofthedistortedzenith-angledistribution.Weassume30000eventsaredetectedinthecaseoftheSMAsolution.Thecomparisonismadeafteronly5000eventsareobservedforthemore-easilyrecognizedLMAsolution.

FortheSMAsolution,themomentsanalysisismostsensitive.ThedifferenceforSNO,between4.9σ(day-nightanalysis)and6.5σ(momentsdistribution),correspondsto20000

19

events,orapproximately7yearsofdatataking.Allthreestatisticaltestscaneasilyrevealthebest-fitLMAsolution,althoughtheday-nightasymmetryisthemostefficientchar-acterizationinthiscase.TheKolmogorov-SmirnovtestistheleastsensitivetotheSMAsolution,butperformsbetterthanthemomentsmethodfortheLMAsolution.

WecanunderstandphysicallywhytheSMAdistortionismosteasilydetectedbymeasur-ingthemomentswhiletheLMAdistortionismostprominentintheday-nightasymmetry.Figure11showsforSuper-Kamiokandethefractionaldistortion,[f(α)−Y(α)]/Y(α),ofthezenith-angledistributionforthebest-fitSMAandLMAsolutions.Onecancrudelyapproximatethedistortionsby,fortheSMAsolution,adelta-functionnearthemaximumallowedzenithangleand,fortheLMAsolution,astepfunctionnearπ/2.Withthesesimpleapproximations,onecanshowanalyticallythatthefirstmomentandtheday-nightasymmetryhavesimilarstatisticalpowerfortheSMAsolution,andthesecondmomentismorediscriminatorythaneitherthefirstmomentortheday-nightasymmetry.ThereasonthatthesecondmomentissousefulfortheSMAsolutionisthatinthiscasethedistortionmostlyariseswhentheneutrinospassthroughthecoreatlargezenithangles.Becausethevacuummixingangleissmall,theenhancedmixing[ρres=7gcm−3(E/10MeV)]duetotheearthmattereffectisparticularlysignificantwhentheneutrinostraversethecore.SincethevacuummixingangleislargefortheLMA,thematterenhancementisnotespeciallysignificantinthiscaseandthemainregenerationfortheLMAsolutionisduetooscillationsthatoccurinthemantle,i.e.,wheneverα>π/2.Theday-nightasymmetryiswell-tunedtothisdistortionsinceAn−dcomparestheaverageeventrateforα>π/2withtheeventrateforα<π/2.TheLOWsolutionproducesarelativedistortion,[f(α)−Y(α)]/Y(α),thathasashapesimilartotheLMAsolutionandisthereforemosteasilydetectedbytheday-nightasymmetry.TheKolmogorov-SmirnovtestisnotoptimallytunedtoanyofthethreebestMSWsolutionsandisthereforenotaspowerfulasthemomentsortheday-nightasymmetry.

20

IX.MOMENTSOFTHEENERGYSPECTRUM

Werefineinthissectionourpreviouscalculationsofthefirsttwomomentsoftheenergyspectrumfromelectronrecoilsproducedbyinteractionswith8Bneutrinos.WeuseheretheslightlyimprovedMSWsolutions,describedinSec.IV,thatincluderegenerationintheearth.Thereaderisreferredtoourearlierpaper[50]fortherelevantdefinitionsandnotation(seealsoref.[34]forasimilarcalculation).

TableVIIpresents,foranassumedthresholdof5MeV,thefirstandsecondmomentsoftheelectronrecoilenergyspectrum,andthepercentageshiftswithrespecttotheaverageelectronkineticenergy,T0,andthedispersioninthekineticenergy,σ0,intheabsenceofoscillations.TheresultsforT0andσ0differbysmallamounts(<1%)fromourearlierresultsgiveninref.[50];thepresentresultsarenumericallymoreprecise.TheresultsinTableVIIaregivenforthebest-estimateMSWsolutions(SMA,LMA,andLOW)describedinSec.IV.Forcompleteness,welisttheoneyearaveragemomentsoftheenergyspectrumforday-time,night-time,andthetotalyear.ThecalculatedmomentsfortheSuper-KamiokandeexperimentaregivenintheupperpartofthetableandthemomentsforSNOarelistedinthelowerpartofthetable.TableVIIIpresentsthesameresultsforanassumedthresholdof6MeV.

WedidnotincludeinourcalculationtheunknowntriggerefficienciesofSuper-KamiokandeorSNO.TheinclusionofthesetriggerfunctionscanchangethepredictedfirstandsecondmomentsoftheenergydistributionbyafewpercentandwillcertainlybeincludedinthecarefulMonteCarlocalculationsthatwillbeperformedultimatelybytheSuper-KamiokandeandSNOexperimentalgroups.

Comparingthecalculateddayandnightrates,TableVIIandTableVIIIshowthatregenerationintheearthslightlydecreases,forboththeSMAandtheLOWsolutions,theaveragekineticenergyoftherecoilelectronsinbothSuper-KamiokandeandSNO.Thisdecreaseoccursbecauseinthesunthesetwosolutionspreferentiallytransformlowenergyneutrinosfromνetoνµ(orντ)andthereforethereisarelativelylargerchanceatlowenergy

21

ofregeneratingνefromνµ(orντ)intheearth.FortheLMAsolution,regenerationincreasestheaveragekineticenergysinceinthiscasethehigh-energypartofthe8Bneutrinoenergyspectrumispreferentiallydepletedofνeinthesun.

TheshiftbetweendayandnightofthemomentsismostsignificantfortheLMAsolution.Infact,ifthenaturehaschosentheLMAsolution,thenthespectraldistortionmaybehighlightedbycomparingtheday-timeandnight-timemoments.

X.SENSITIVITYTOEARTHMODELSANDSOLARMODELS

WecalculateinSec.XAthesensitivityoftheMSWpredictionstotheassumeddensityprofileandchemicalcompositionoftheearthmodelandinSec.XBthedependenceupontheassumedmodelofthesun.

A.UncertaintiesDuetoEarthModels

TableIpresentsthecalculatedpercentageshiftsofthefirsttwomomentsofthezenith-angleeventdistributionforallsixmodelsoftheearthdiscussedinSec.III;thecalculationsweremadeassumingeithertheSMAortheLMAsolutions.Thefractionalchangesofthefirstmomentvarybyonly∼0.02%fortheSMAsolutionand∼0.2%fortheLMAsolution,althoughthedensityprofilesinsomeofthesemodelsaresignificantlydifferentfromtherangeallowedbycurrentseismologicaldata.Weconcludethattheshapeofthezenith-angledistributioncanbecalculatedwithacceptableaccuracyforanyoftherecentlypublisheddensityprofilesoftheearth.

TableIXillustratestheuncertaintiesintheMSWpredictionsoftheshiftsofthefirstandsecondmomentsofthezenith-angledistributionduetouncertaintiesintheelectronnumberdensityinthemantleandinthecore.TherangesofZ/Aincludedinthetable(±2%inthecoreand−1%,−2%inthemantle)arelargerthanthecurrentestimatesofthegeophysicaluncertainties(seethediscussioninSec.IIIandreference[44,45]).Weconclude

22

fromTableIXthatuncertaintiesinthechemicalcompositionaffectthepredictedmomentshiftsduetoregenerationbyatmostafewpercentoftheirvalues.

AsimplifiedmodelwithauniformcompositionofZ/A=0.5hasbeenusedinrefer-ence[35](andinmanyormostoftheearlycalculationsrelatedtotheregenerationeffect,see[13]).Thepredictionsfromthisconstant-compositionmodelarealsogiveninTableIX;thiscrudemodelleadstoimprecise,butnotgrosslyerroneous,predictionsofthemomentsofthezenith-angledistribution.

B.UncertaintiesDuetoSolarModels

Tothebestofourknowledge,allpreviousdiscussionsoftheearthregenerationeffecthavedescribedthisphenomenonasifitwerecompletelyindependentofsolarmodels.Thisimplicitassumptionisnotexactlycorrectsincethesizeoftheearthregenerationeffectdependsupontheflavorcontentoftheincidentneutrinobeam,whichmustbecalculatedbyusingasolarmodeltodescribe(forspecifiedMSWparameters)theproductionandconversionprobabilitiesof8Bsolarneutrinosasafunctionofthepositioninthesunatwhichtheneutrinosarecreatedandtheneutrinoenergy.Theslightlydifferentdensitydistributionsindifferentsolarmodelshavethelargesteffect,whichisstillquitesmallasweshallseebelow,ontheinferredflavorcontentoftheincidentsolarneutrinoflux.

Inordertoquantifythedependenceofthepredictedearthregenerationeffectuponthecharacteristicsofthesolarmodel,wehavecalculatedthefractionalshiftsofthefirstandsecondmomentsofthezenith-angleeventdistributionusingthreedifferentsolarmodels.Asourstandardsolarmodel,weadoptthemodelwithheliumandheavyelementdiffusionofBahcallandPinsonneault[6].Forcomparison,weusethe1992modelofBahcallandPinsonneault[52],whichincludesheliumdiffusion(butnotheavyelementdiffusion)andsomewhatlessaccurateinputphysics.Finally,weusethe1988modelofBahcallandUl-rich[53],whichdoesnotincludeanydiffusionandhaslesspreciseopacities,equationofstate,andotherinputdata.

23

TableIVshowsthattheMSWpredictionsareessentiallyidenticalforthe1992solarmodelwithheliumdiffusionandthe1995solarmodelwithheliumandheavyelementdiffu-sionplusimprovedinputdata.The1988solarmodelleadstopredictionsthatcandifferbyasmuchas10%fortheSMAmomentsthatwillbemeasuredbySNO.However,this1988modelisinconsistentwithrecenthelioseismologicalmeasurementssincethe1988modeldoesnotincludediffusion[7].

Weconcludethatpredictionsoftheearthregenerationeffectarepracticallyindependentofsolarmodelsaslongasthemodelsincludediffusion(i.e.,areconsistentwithhelioseismol-ogy).

XI.FUTUREEXPERIMENTSATTHEEQUATOR

RecentlyGelb,KwongandRosen[28]suggestedbuildinganewdetectorsimilartoSNOclosetotheequatorinordertoincreasethesensitivityoftheexperimenttotheearthregenerationeffect.Anequatoriallocationmaximizesthetimeneutrinospassthroughthecoreoftheearthduringonecalendaryear.

Inthissection,wecalculatethesizeoftheregenerationeffectforhypotheticalequa-torialdetectorsandcomparewiththesensitivityofthedetectorsintheiractualposi-tions.WeconsiderequatorialanaloguesoftheSuper-Kamiokande,SNO,BOREXINO,andHERON/HELLAZdetectors.

Figure12showsthepredictedzenith-angledistributionfordetectorsattheequator.Thecurveintheupperleftpanelisthezenith-angleexposurefunction.TheotherfivepanelsshowthedistortionduetoregenerationforequatorialanaloguesofSuper-Kamiokande,SNO,ICARUS,BOREXINO,andHERON/HELLAZ.Foreachdetector,onlythepredictedangu-lardistributionfunctionsareshownforthebest-fitsolutionstowhichtherelevantdetectorissensitive:SMAandLMAforthehigh-energyboronneutrinodetectors(Super-Kamiokande,SNOandICARUS)andLOWforthelow-energyneutrinodetectors(BOREXINOandHERON/HELLAZ).

24

Figure13showstheiso-sigmaellipsesforthefourequatorialdetectors.BOREXINOandHERON/HELLAXwouldbemoresensitivetotheLOWsolutionifthesedetectorswerebuiltattheequator.However,thehigh-energyneutrinodetectors(Super-Kamiokande,SNO,andICARUS)wouldremaininsensitivetotheLOWsolutioneveniftheyweremovedtotheequator.Low-energyneutrinodetectors(BOREXINOandHERON/HELLAZ)alsoremaininsensitivetotheSMAandLMAsolutionsevenattheequator.

TableXshowsthegaininsensitivitythatwouldoccurifdetectorslikeSuper-KamiokandeandSNOwerebuiltattheequator.Theenhancementisrepresentedinthetablebytheshiftinthefirstandsecondmomentofthezenith-angledistributionandbytheday-nightasymmetry.Theenhancementswouldbeimportantforthebest-fitSMAsolution,butlesssignificantfortheLMAsolution.However,regionsofparameterspaceintheLMAsolutionforwhichthepredictedshiftsinthefirstandsecondmomentsaresmallcouldbeprobedmorepreciselywithdetectorsattheequator.

XII.DISCUSSIONANDCONCLUSIONS

Theconversionintheearthofνµ(orντ)tothemoreeasilydetectedνeisadistinctivepredictionoftheMSWeffectthatoffersthepossibilityofunambiguouslyestablishingtheexistenceofphysicsbeyondthestandardelectroweakmodel.Becauseoftheimportanceofthissubject,wehavecarriedoutprecisenumericalcalculationsofthesizeoftheregenerationeffectpredictedbydifferentMSWparametersthatareconsistentwiththeexperimentalresultsfromthechlorine,Kamiokande,GALLEX,andSAGEexperiments.Ourresultsshowthepotentialofthenewexperiments,Super-Kamiokande,SNO,ICARUS,BOREXINO,HERON,andHELLAZ,fordiscoveringtheregenerationeffect.

Ourresultsprovidethemostprecisepredictionsavailableoftheexpectedzenith-angledistributionofthesolarneutrinoeventsintheabsenceofnewphysicsandinthepresenceofMSWdistortions.TheresultsareobtainedbynumericalcalculationsthatarediscussedinSec.VandillustratedinFig.5(forSuper-KamiokandeandSNO)andFig.6(fortheGran

25

SassoexperimentsICARUS,BOREXINO,andHERON/HELLAZ).

Wepresentthepredictionsforthesmallmixingangle(SMA),largemixingangle(LMA),andlowmass(lowprobability,LOW)MSWsolutionsofthesolarneutrinoproblems.TheparametersoftheseMSWsolutions,whichareconsistentwiththeresultsofthechlorine,Kamiokande,GALLEX,andSAGEexperiments,aregiveninSec.IV.Oursolutionsincludeself-consistentlytheeffectsofearthregeneration.

Figure3showstheallowedregionsofthethreeMSWsolutionsinthe∆m2-sin22θplane.Figure4presentsthesurvivalprobabilitiesasafunctionofenergyforνecreatedinthesun.Thisfigurecomparessurvivalprobabilitiescomputedfortheday(withoutregeneration)withsurvivalprobabilitiesforthenight(withregeneration)andwiththeaverageannualsurvivalprobabilities.

WedescribethepredictedMSWdistortionsintermsofthefirsttwomomentsofthezenithangledistributionofneutrinoevents(seeSec.VI),aswellasintermsofthetraditionalday-nightasymmetry(seeSec.VII).WeanalyzesimulateddatainSec.VIIIandshowthatthemomentsofthezenith-angledistributionaremoresensitivetotheharder-to-detectSMAsolution.ThepredictedlargeeffectoftheLMAsolutionismoreeasilydiscoveredwiththeconventionalday-nightasymmetry.

The“bottomline”isillustratedsuccinctlyinFig.9.Thisfigureshowsthatthecurrentbest-estimateMSWsolutionspredictstatisticallysignificantdeviationsfromtheundistortedzenith-anglemomentsfortheSuper-Kamiokande,SNO,andICARUSexperiments(whicharesensitivetotheSMAandLMAsolutions)andtheBOREXINOandHERON/HELLAZexperiments(whicharesensitivetotheLOWsolution).

Wehaveconsideredanumberofeffectsthathavenotbeenpreviouslyinvestigatedinconnectionwiththeearthregenerationeffect.WehavecalculatedthesensitivityoftheMSWpredictionstoawiderangeofdensityprofilesoftheearthandalsotoasetofextremechemicalcompositions.ThesecalculationsarediscussedinSec.IIIandSec.XA.Wealsoevaluatetheslightdependenceofthepredictedearthregenerationeffectupontheassumedsolarmodelusedtocalculatetheflavorcontentoftheincidentneutrinobeam(seeSec.XB).

26

Ourresultsshowthattheseusually-neglectedeffectsassociatedwiththeearthandsolarmodelsarerathersmall.

Forcompleteness,wehavecarriedoutcalculationsforhypotheticalnewdetectorsthatmightbebuiltneartheequator.ThesecalculationsaredescribedinSec.XIandshowquan-titativelytheenhancedsensitivitytotheearthregenerationeffectofequatorialdetectors,asemphasizedbyGelbetal.[28].

Usingthebest-fitMSWsolutionscalculatedherethatincludetheearthregenerationeffect,wehaveevaluatedthefirstandsecondmomentsoftheelectronrecoilenergyspectrumfor8BneutrinosdetectedinSuper-KamiokandeandSNO.Thesecalculations,summarizedinSec.VIandinTableVIIandTableVIII,refineourearlierresults[50](seealsoref.[34])forthemomentsoftheelectronrecoilspectrum.Perhapsmostimportantly,theyshowthatfortheLMAsolutionthecomparisonoftherecoilelectronenergyspectrumbetweendayandnightmayrevealadistortionthatisnotapparentinthetemporalaverageoftheenergyspectrum.

Whatwouldwelearnfromanobservationwhichshowedthattheneutrinocountingratedependeduponsolardirection?TheexperimentaldemonstrationofadependenceofsolarneutrinoeventrateuponthedirectionofthesunwouldnotonlyconstituteadirectproofofnewphysicsbutwouldatthesametimeeliminateanumberofthepopularalternativestotheMSWeffect.ManyofthealternativestotheMSWeffect,suchasvacuumoscillations,magneticmomenttransitions,andviolationsoftheequivalenceprinciplepredictthatthecountingrateisindependentofthezenithanglepositionofthesun.

ACKNOWLEDGMENTS

ThisworkhasbeensupportedbyNSFgrant#PHY-9513835.WeareindebtedtoM.Fukugita,E.Lisi,andA.Smirnovforvaluablecommentsonthedraftmanuscript.WearegratefultoE.Lisi,W.Press,P.Rosen,andA.SmirnovforstimulatingdiscussionsandtoD.L.Anderson,P.Goldreich,F.Press,andA.Rubinforvaluablecommunicationsregarding

27

seismologicalmodelsoftheearth.

APPENDIXA:NEUTRINOSURVIVALPROBABILITIES

Inordertocalculatetheeventratesasafunctionoftimewefirstcompute,followingtheprescriptioninRef.[],theelectronneutrinosurvivalprobabilities,PSE,aftertraversingtheearth.WebeginbyusingtheanalyticalapproximationdevelopedinRef.[55]ofthesurvivalprobabilities,PS,foranelectronneutrinopassingthroughthesun;thesesolarsurvivalprobabilitiesareaveragedovertherelevantneutrinoproductionregionsfor8B,7Be,pp,pep,andCNOneutrinos.Assumingthattheneutrinosarrivingattheearthrepresentanincoherentsuperpositionofmass-eigenstates[56],wecalculatetheelectronneutrinosurvivalprobabilityafterpassingthroughtheearth,PSE,fromtheexpression:

PSE=

PS−sin2θ+P2e(1−2PS)

and

10−4≤sin22θ≤1.

(A2b)

Thenumericalprecisionofthecalculatedsurvivalprobabilitiesalongeachtrajectoryisbetterthan0.1%.

Theone-yearaveragedsurvivalprobabilityisgivenby:

¯SE=P

N󰀃i=1

i

PSEY(αi),

(A3)

wherethesumisoverzenithanglesfrom0to180degrees.(N=360inourcase).Thezenith-angleexposurefunctionisdefinedinSec.V.Correspondingly,theone-yearaveragednight-timesurvivalprobabilityisgivenby

¯nPSE

=

N󰀃

iPSEY(αi),

(A4)

i=N/2

wherethesumnowrunsoveranglesfrom90to180degrees.Sincethenight-timeandday-timeintervalswithinoneyearareequal,theday-timeeventrateissimply

¯d=0.5PSE.PSE

(A5)

Withthecalculatedsurvivalprobabilities,itisstraightforwardtodeterminethecorrespond-ingtotaleventrates,aswellastheday-timeandnight-timeone-yearaveragedeventratesinanysolarneutrinodetector.

APPENDIXB:TIMEDEPENDENCEOFTHEZENITHANGLE

Thedependenceofthesolarzenithangleonthetimeoftheyearandonthegeographiclocationofthedetectorsisgivenbythefollowingsetofformulae[49]:

cosα=sinδsinφ+cosδcosφcosH,

sinδ=sinǫsinλ,L=2800.461+00.9856003n,

29

(B1a)(B1b)(B1c)

n=−1462.5+D+H,(B1d)g=3570.528+00.9856003n,

(B1e)

and

λ=L+10.915sing+00.020sin2g.

(B1f)

TheprecisionintheapparentcoordinatesoftheSunis00.01andtheprecisionoftheequationoftimeis6secondsbetweentheyears1950and2050.HereHisthefractionofdayfrom0hUT,Disdayoftheyear(countingfromJanuary1),nisthenumberofdaysfromJulianyear2000.0,λistheeclipticlongitude,Listhemeanlongitudeofthesun(correctedforaberration),ǫ=230.439−0.0000004nistheobliquityoftheecliptic,δisthesun’sdeclination,andgisthemeananomaly§.

Thedistanceofthesunfromtheearthinastronomicalunits(1AU=1.495978706(2)1011m)isgivenbytheformula:

R=1.0014−0.01671cosg−0.00014cos2g.

(B2)

Equation(B2)hasbeenusedinthecalculationoftheday-nightasymmetriesandtheshiftsofthefirsttwomomentsofthezenith-angledistribution.

APPENDIXC:CHARACTERISTICSOFFUTUREDETECTORS

WedescribeinthisAppendixthecharacteristicswehaveassumedforSuper-Kamiokande[22],SNO[23],ICARUS[24],BOREXINO[25],HERON[26]andHELLAZ[27].TheSuper-Kamiokande,SNOandICARUSdetectorsaresensitiveonlytohigh-energy8Bneutrinos,whileBOREXINOissensitiveprimarilyto7Beneutrinos(Eν=0.862MeV)andtheHERONandHELLAZdetectorsarebeingdesignedtodetectlowenergy(Eν<0.44MeV)ppneu-trinos.

FortheSuper-Kamiokandedetector,weadoptathresholdof5MeVandatriggeref-ficiencyof50%atthisenergy[22].TheenergyresolutionfunctionisassumedtohaveagaussianshapewithFWHMof1.6MeVatelectronkineticenergy10MeV.Wehaveper-formedcalculations,whichshowthatthesensitivityofourresultstotheassumedenergyresolutionandtriggerefficiencyofthedetector.

FortheSNOdetector,wecalculateonlytherateoftheCCreaction,namelyνe+d→p+p+e−.Weadopt[58]athresholdof5MeVandanenergyresolutionfunctionwitha1σuncertaintyof1.1MeVat10MeVelectronkineticenergy.TheCCcross-sectionforSNOwastakenfromRef.[18].Thetriggerefficiencyfunctionhasbeenapproximatedwithastepfunctionatthethresholdofthedetector.

Reference[50]givesfurtherdetailsregardingourcharacterizationofSNOandSuper-Kamiokande.

ForICARUS,wehaveconsideredonlythesuperallowedtransitionandhaveusedtheneutrinoabsorptioncrosssectionsgiveninref.[5].Wehaveassumedaneutrinothresholdfordetectionthatcorrespondstoelectronsbeingproducedwithatleast5MeVofenergy,whichrequiresaminimumneutrinoenergyof10.9MeV.

TheBOREXINOdetectorisbeingdevelopedasaneutrino-electronscatteringexper-imentthatwillmeasurethefluxof7Beneutrinos,usingthestepintheenergydistribu-tionofneutrino-electronscatteringeventsatthemaximumrecoil-electronkineticenergyofTe=0.62MeV.Thedetectorcharacteristicsthatareknowableapriori(whichdoesnotincludethecrucialbackgroundrateversusenergy)arenotasimportantasforSNOandSuper-Kamiokandeandweneedtocomputeonlytheνesurvivalprobability.Weincluderadiativecorrectionstotheneutrino-electroncross-section,calculatedin[59],whichforre-coilelectronenergiesbelow0.62MeVarelessthan1%.Theratioofthe(νµe)to(νee)totalcross-sectionsforneutrinoenergy0.862MeVisσνµ,e/σνe,e=0.221.

HERONandHELLAZarealsoneutrino-electronscatteringexperimentswithverydif-ferentpreliminarydesigns,butwiththesametargetmaterial,helium.Theywillmeasurethefluxandspectralshapeofppneutrinos.Wehaveassumedathresholdof0.1MeV

31

andaperfectenergyresolution(deltafunction).Thetriggerefficiencyisrepresentedbyastep-functionatthethresholdofthedetector.Weagainuseneutrino-electroncross-sectionsincludingradiativecorrections[59].

Theeventrate,Q,averagedovercertaintimeinterval,τ,inaneutrino-electron-scatteringexperiment,suchasSuper-Kamiokande,BOREXINO,HERONorHELLAZisgivenby

󰀁Q󰀂τ=

󰀄

Emax

0

Φ(Eν)Zνe(Eν)󰀁P(Eν)󰀂τ+Zνµ(Eν)(1−󰀁P(Eν)󰀂τ)dEν

󰀁󰀂

(C1)

HereZνe(νµ)aretheresponsefunctionsofthedetectortoeitherνeorνµ,Φisthesolarneutrinofluxtowhichthedetectorissensitive,Eνistheneutrinoenergy,Emaxistheendpointoftheneutrinoenergyspectrumand󰀁P󰀂τistheaveragesurvivalprobabilityforthechosentimeintervalτ.Theintervalcanbe,e.g.,thetotalday-timeornight-timeduringonecalendaryear,orawholecalendaryearincludingbothdaysandnights.ThecalculationoftheaveragesurvivalprobabilitiesisdescribedinAppendixA(Eqs.A1-A5).Theresponsefunctionsrepresenttheconvolutionoftheabsorptioncrosssectionswiththedetectorcharacteristics(seeRef.[50]fordetails).

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34

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Venice,Italy,1993,editedbyM.Baldo-Ceolin(PaduaUniv.,Padua,Italy,1994),p.161;G.Bonvicini,Nucl.Phys.B35,438(1994).

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ref.[13].

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andS.T.Petcov,hep-ph/9703207.

35

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Jeanholz,J.Geophys.Res.96,16169(1991);R.Jeanholz,Annu.Rev.EarthPlanet.Sci.18,357(1990).

[45]Y.ZhaoandD.L.Anderson,Phys.EarthPlanet.Interior,85,273(1994).[46]K.S.Hirataetal.,Phys.Rev.D44,2241(1991);seealsoRef.[2].[47]G.L.FogliandE.Lisi,Astropart.Phys.3,185(1995).

[48]P.I.KrastevandS.T.Petcov,Phys.Lett.B299,99(1993);G.L.Fogli,E.Lisi,andD.

Montanino,Phys.Rev.D,2048(1996).

[49]TheAstronomicalAlmanac,1996(U.S.GovernmentPrintingOffice,Washington).[50]J.N.Bahcall,P.I.Krastev,andE.Lisi,Phys.Rev.C55,494(1997).

[51]M.G.KendallandA.Stuart,TheAdvancedTheoryofStatistics(Hafner,NewYork,1969),

Vol.I.

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36

[53]J.N.BahcallandR.K.Ulrich,Rev.Mod.Phys.60,297(1988).

[]S.P.MikheyevandA.Yu.Smirnov,inNewandExoticPhenomena,proceedingsoftheSeventh

MoriondWorkshop,editedbyO.FacklerandY.TrˆanThanhVˆan(EditionsFronti`eres,Gif-sur-Yvette,1987),p.403.

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(1987)];Prog.Part.Nucl.Phys.23,41(1988).

[57]S.P.MikheyevandA.Yu.Smirnov,NuovoCimento9C,17(1986).[58]E.W.Beier(privatecommunication).

[59]J.N.Bahcall,M.Kamionkowski,andA.Sirlin,Phys.Rev.D51,6146(1995).[60]D.L.Anderson,TheoryoftheEarth,(BlackwellScientificPublications,19).

TABLEI.Sensitivitytothemodeloftheearth.ThetableillustratestheweakdependenceonthemodeloftheearthofthecalculatedchangesinthefirstandsecondmomentsoftheangulardistributionofeventsintheSuper-KamiokandeandSNOdetectors.Thedensitydistributionsinthesixmodelsoftheearthlistedinthetablespanarangeofpossibilitiesthatismuchlargerthansuggestedbycurrentgeophysicalknowledge.Thesecondandthirdcolumnsgivethetotalmass,M⊕(in1027g),andthemomentofinertia,I(in1045gcm2),foreachmodel.Themassoftheearthis5.97370±0.00076,thepolarvalueofthemomentofinertiais0.804,andtheequatorialvalueis0.801[60].]Thelastfourcolumnsgivethefractionalshiftinpercentofthefirsttwomoments

2)ofthezenith-angledistributionofeventsintheSuper-KamiokandeandSNO(∆α/α0and∆σ2/σ0

detectorsfortheSMA(upperentry)andfortheLMA(lowerentry)solutions.

∆󰀁α󰀂/α0

Model

M⊕

I

Super-Kamiokande(%)

2∆σ2/σ0

∆󰀁α󰀂/α0SNO(%)

2∆σ2/σ0

Super-Kamiokande(%)SNO(%)

19813.23−0.335.28−2.20

B4971973

5.0670.8001.013.22

2.03−0.26

1.035.26

2.41−2.05

A′′1967

5.9460.7981.003.29

2.01−0.51

1.015.35

2.39−2.55

HOMESTAKEKAMIOKANDEGALLEXSAGE

2.56±0.16±0.142.80±0.19±0.3569.7±6.7±

72

+12+5−10−7

+3.9−4.5

.2

9.5+1−1.4

SNU106cm−2s−1

SNUSNU

0.27±0.0220.42±0.0600.51±0.0580.53±0.095

[1][2][3][4]

6.62

+0.93

−1.12

136.8+8−7136.8+8−7

38

TABLEIII.Northernlatitudes(indegrees)forseveralsolarneutrinodetectors.TheseanglescorrespondtoφinFig.1.Homestake

Kamioka

GranSasso

Baksan

Sudbury

1995

Super-Kamiokande

19921988

1.021.020.96

2.032.031.94

3.223.223.22

−0.32−0.32−0.32

TABLEV.Day-nightasymmetryinSuper-KamiokandeandSNO.Thetablegivesthemagnitudeoftheexpectedday-nightasymmetry(An−d,seeEq.(9))(inpercent)inSuper-KamiokandeandSNOforvaluesoftheneutrinooscillationparameters∆m2andsin22θcorrespondingtothebest-fitSMAandLMAsolutions(seeEqs.1a,1band2a,2b).Theindicateduncertaintiesdescribetheexpectedlimitsat95%C.L.Solution

Super-Kamiokande

SNO

Super-KamiokandeSMALMA

55.5

4.49.2

3.76.3

TABLEVII.MomentsoftheEnergySpectrum.Themomentsintheabsenceofoscillationsare

2=3.391MeV2forSuper-KamiokandeandT=7.6MeVandσ2=3.032T0=7.293MeVandσ000

MeV2forSNO.MSWSolution

󰀁T󰀂(MeV)

(T−T0)/T0

(%)

σ2(MeV2)

2)/σ2(σ2−σ00

(%)

Day

SMA

NightAverageDay

LMA

NightAverageDay

LOW

NightAverage

7.4087.4037.4057.2757.3107.2947.2987.2907.294

1.581.501.−0.250.230.0080.06−0.040.008

3.5913.5743.5823.3683.4393.4083.4113.3983.404

5.885.385.−0.671.420.480.580.200.38

Day

SMA

NightAverageDay

LMA

NightAverageDay

LOW

NightAverage

7.8697.8467.8587.47.7197.6877.6827.6687.675

2.912.622.76−0.030.950.530.470.280.37

3.1613.1443.1523.0323.1133.0843.0693.0603.0

4.253.683.96−0.032.661.721.220.901.06

41

TABLEVIII.MomentsoftheEnergySpectrum.Themomentsintheabsenceofoscillationsare

2=2.825MeV2forSuper-KamiokandeandT=8.178MeVandσ2=2.348T0=8.057MeVandσ000

MeV2forSNO.MSWSolution

󰀁T󰀂(MeV)

(T−T0)/T0

(%)

σ2(MeV2)

2)/σ2(σ2−σ00

(%)

Day

SMA

NightAverageDay

LMA

NightAverageDay

LOW

NightAverage

8.1498.1438.1468.0468.0748.0618.08.0588.061

1.141.061.10−0.140.220.050.080.010.046

2.9682.92.9622.8042.8612.8332.8362.8272.832

5.074.5.84−0.761.250.280.400.0480.24

Day

SMA

NightAverageDay

LMA

NightAverageDay

LOW

NightAverage

8.3328.3158.3238.1778.2398.2128.2068.1968.201

1.881.671.77−0.0432.631.840.340.220.28

2.4672.4522.4592.3472.4102.3922.3792.3702.374

5.0.414.72−0.0432.631.841.280.941.11

42

TABLEIX.Dependenceofmomentsonassumedchemicalcomposition.Thetablegivestherelativeshifts(inpercent)ofthefirstandsecondmoments(∆µi/µi)ofthezenith-angledistributioninSuper-KamiokandeandSNOasafunctionoftheassumedchemicalcomposition.TheratioZ/Ainthecorehasbeenvariedby±0.5%,±1%,±2%(seesecondcolumn)fromthecentralvalue[(Z/A)core=0.465]adoptedintherestofthepaper.Theratiointhemantlehasbeenvariedby−1%,−2%fromthestandardvalueof[(Z/A)mantle=0.496].ThelastrowforeachdetectorcorrespondstoasimplifiedmodelwithZ/A=0.5bothinthemantleandinthecore.ThefourcolumnsforeachdetectorcorrespondtothefirstandsecondmomentintheSMAandLMAbest-fitsolutionsrespectively.

SMA

∆(Z/A)/(Z/A)

Detector

(%)

∆󰀁α󰀂/α0

(%)

2∆σ2/σ0

LMA

∆󰀁α󰀂/α0

(%)

2∆σ2/σ0

(%)(%)

−1

mantle

−2

1.011.00

2.032.04

3.193.16

−0.336−0.358

−2−1−0.50.0

SNO

+0.5

1.051.041.041.041.04

2.462.442.432.422.41

5.275.275.275.285.28

−2.23−2.22−2.21−2.20−2.20

43

+1+2

1.031.03

2.402.38

5.285.29

−2.19−2.17

Z/A=0.501.012.285.33−2.11

TABLEX.Equatorialenhancement.Themagnitudeoftheregenerationeffectiscomparedfordetectorslocatedattheiractualpositionsandattheequator.Thesizeoftheeffectisrepresentedbythenumberofstandarddeviationsthefirstandsecondmomentsoftheangulardistributiondifferfromtheundistortedexposurefunction(and,inparentheses,theday-nightasymmetryinpercent).Thevaluesoftheneutrinooscillationparameters∆m2andsin22θcorrespondtothebest-fitSMAandLMAsolutions(seeEqs.1a,1band2a,2b).Forcomparison,thebest-fitSMAandLMAsolutionsproduce5σand13σeffectsattheactuallocationofSuper-Kamiokandeand6.5σand25σattheactuallocationofSNO.Location

Solution

Super-Kamiokande

SNO

FIG.1.Schematicviewofdetector’slocationandsun’sdirection.ThezenithisdefinedasthelinefromthecenteroftheEarththroughthecenterofthedetector.Thezenithangle,α,andthelatitudeofthedetector’slocation,φ,arealsoshowninthefigure.

FIG.2.Densityprofilesforsixdifferentmodelsoftheearth.Themodelsare:1)PREM[37],2)Stacey’smodel[39],3)modelHA[43],4)modelHB1[40],5)modelB497[41],and6)modelB1[42].Theradiusisgiveninkilometersfromthecenteroftheearth.Allmodelsaresphericallysymmetric.

FIG.3.AllowedMSWsolutionswithregeneration.Theallowedregionsareshownfortheneutrinooscillationparameters∆m2andsin22θ.TheC.L.fortheouterregionsis99%andtheC.Lfortheinnerregionsis99%(onlyappliestotheLMAandSMAsolutions).Thedatausedherearefromthechlorine[1],Kamiokande[2],GALLEX[3],andSAGE[4]experiments.ThesolarmodelusedisthebeststandardmodelofBahcallandPinsonneault(1995)withheliumandheavyelementdiffusion[6].Thepointswhereχ2hasalocalminimumareindicatedbyacircle.

′

FIG.4.SurvivalprobabilitiesforMSWsolutions.ThefigurepresentsthesurvivalprobabilitiesforaνecreatedinthesuntoremainaνeuponarrivalattheSuper-Kamiokandedetector.Thebest-fitMSWsolutionsincludingregenerationintheeartharedescribedinSec.IV.Thefulllinereferstotheaveragesurvivalprobabilitiescomputedtakingintoaccountregenerationintheearthandthedottedlinereferstocalculationsfortheday-timethatdonotincluderegeneration.Thedashedlineincludesregenerationatnight.ThereareonlyslightdifferencesbetweenthecomputedregenerationprobabilitiesforthedetectorslocatedatthepositionsofSuper-Kamiokande,SNOandtheGranSassoUndergroundLaboratory.

45

FIG.5.Super-KamiokandeandSNOzenith-angledistributions.Thefigureshowstheexpectedzenith-angledistributionofneutrinoeventsduringonecalendaryearintheSuper-KamiokandeandtheSNOdetectors.Theangle,α,representstheangularseparationbetweenthedirectiontothesunandthedirectionofthelocalzenith(seeFig.1).Thetwoleftpanelsdisplaythezenith-angleexposurefunctions,theundistortedangulardistributionsintheabsenceofoscillations.Theex-posurefunctionsaredeterminedbythelocationofthetwodetectorsat,respectively,Kamioka,Japan,andSudbury,Canada.Thedistortedzenith-angledistributionsduetotheregenerationeffectintheEarthareshowninthetworightpanels;theneutrinosolutionsareindicatedby:SMA(solidline)andLMA(dottedline).

FIG.6.GranSassozenith-angledistributions.Thefigureshowstheexpectedzenith-angledistributionofeventsduringonecalendaryearindetectorslocatedattheGranSassoLaboratoryinItaly:ICARUS,BOREXINO,HERONandHELLAZ.Theupperleftpanelshowsthezenith-angleexposurefunction,whichdoesnotdependondetectorcharacteristics.Thethreeadditionalpanelsdisplaythedistortedzenith-angledistributionsduetotheregenerationeffectintheEarth;thesolutionsareindicatedby:SMA(solidline),LMA(dottedline)andLOW(dashedline).

FIG.7.Contoursofconstantrelativeshift(inpercent)oftheaveragezenithangle,(󰀁α󰀂−α0)/α0,duetoνeregenerationintheearthasafunctionoftheneutrinooscillationparameters,sin22θand∆m2.Hereα0=900istheaverageangleoftheundistortedangulardistributionwithnooscillations.TheshadedregionsinthepanelsforSuper-KamiokandeandSNOareallowedbythelatestsolarneutrinodataat95%C.L.andrepresenttheSMAandLMAsolutions.Inthelowertwopanels(BOREXINOandHERON/HELLAZ)thethreeshadedregionsareallowedat99%C.L.,thelow-massregionrepresentingtheLOWsolution(seetextfordetails).Theblackcirclewithineachallowedregionrepresentsthepointwhichcorresponds(locally)tothebest-fittothedata.

46

FIG.8.Contoursofconstantrelativeshift(inpercent)ofthedispersionofthezenithangle,

2)/σ2,duetoνregenerationintheearthasafunctionoftheneutrinooscillationparameters,(σ2−σ0e02aregiveninthetextforeachoftheexperiments.Thedefinitionsin22θand∆m2.Thevaluesofσ0

oftheshadedregionsisthesameasinFig.7.FIG.9.Howmanysigmas?

ThefigureshowsthesensitivityofSuper-Kamiokande,SNO,

ICARUS,BOREXINOandHERON/HELLAZtotheregenerationeffect.Iso-sigmacontours,sta-tisticalerrorsonly,delineatethefractionalpercentageshiftsofthefirsttwomomentsoftheangulardistributionofeventsforanassumed30000observedevents.ForallbuttheICARUSexperiment,thebest-fitMSWsolutionsareindicatedbyblackcircles(SMA),squares(LMA),andtriangles(LOW);thebest-fitsolutionsarepresentedinSec.IV.Theerrorbarsonthepredictedmomentscorrespondto∆m2andsin22θwithinallowedsolutionspaceat95%C.L.(forSuper-Kamiokande,SNO,andICARUS)or99%C.L.(BOREXINOandHERON/HELLAZ).ForICARUS,wehaveindicatedthebest-fitsolutionsbyatransparentcircle,square,ortriangle.Thebest-fitSMAandLOWsolutionsforICARUSandtheLOWsolutionforSNOareallthreeclosetogetheratabout3σfromthenooscillationsolution.Inordertoavoidtoomuchcrowdinginthefigure,wehavenotshownthetheoreticaluncertaintiesforICARUS.

FIG.10.Contoursofconstantday-nightasymmetry,An−d(seeEq.(9)),inSuper-Kamiokande,SNO,ICARUS,BOREXINOandHERON/HELLAZ.TheshadedregionsarethesameasinFig.7.

FIG.11.RelativedistortionforSuper-Kamiokande.Thefigureshowsthefractionaldistortion,[f(α)−Y(α)]/Y(α),ofthezenith-angledistributionforthebest-estimateSMAandLMAMSWsolutions.

FIG.12.Thezenith-angledistributionforequatorialdetectorswiththecharacteristicsofSu-per-Kamiokande,SNO,ICARUS,BOREXINOandHERON/HELLAZ.NotationisthesameasinFigs.5and6.

47

FIG.13.Howmanysigmasattheequator?Thefigureshowsthesensitivitytotheregenera-tioneffectofequatorialdetectorswiththecharacteristicsofSuper-Kamiokande,SNO,ICARUS,BOREXINOandHERON/HELLAZ.NotationisthesameasinFig.9.

48

ZenithαDetectorφEqtouarEarthνSunFigure󰀀149

50

51

52

53

55

56

57

58

59

60

61