Constraints on Vacuum Oscillations from Recent Sol

arXiv:hep-ph/0005053v226Jul2000ConstraintsonVacuumOscillationsfromRecentSolarNeutrinoDataC.E.C.Lima1,∗andH.M.Portella2,†February1,20081CentroBrasileirodePesquisasF´ısicas,RiodeJaneiro,RJ,22290-180,Brazil2UniversidadeFederalFluminense,Niter´oi,RJ,24210-340,BrazilAbstractAdetailedstudyofthesolarneutrinovacuumoscillationismadetakingintoaccountthreeneutrinoflavoursandseasonaleffect.Asetofδm221xsin2(2ω)regionsiscalculatedforarangeoftheparametersin2(2φ)from0to1,withandwithouttheinclusionoftherecoil-electronspectrumintherates.Theaveragedsurvivalprobabilitiesforνeasafunctionoftheenergyareobtained,whatrevealthatsolutionswithvaluesofsin2(2φ)≥0.50andthemaximumexclusionof7Beneutrinosandminimumexclusionofppneutrinosgiveabetterexplanationforthesuppressionratesofalldetectors.∗[e-mail]†[e-mail]11IntroductionForthreedecadesthedeficitontheobservedfluxofsolarneutrinoscomparedtothatoneexpectedfromStandardSolarModels(SSM)haspuzzledthescientistsandoriginatedseveralexplanationsforthisproblem.Thesesolutionsaredividedintoastrophysicalandbeyondthestandardmodel[1].Nowadays,theastrophysicalsolutionshavebeenhighlydisfavouredbythehelioseismologicaldatawhicharewellexplainedbystandardsolarmodels,however,solutionswithnewphysics,inspecialtheneutrinooscillations,thatcanbeinvacuum[2]orinmatter(byMSWeffect)[3],givesagoodfittothesolarneutrinodata.Inthepresentworkwewillfocusattentiononlyonvacuumoscillations,whichoccurintheneutrinowayfromtheSuntoEarth.Theevidenceoftheexistenceofvacuumoscillationscomefromtheobservationoftheseasonaleffect.BecausetheEarthorbitisnotcircular,thevariationsintheEarth-Sundistanceproduceadifferenceintheoscillationprobabilitywhatcanbeveryimportantforneutrinolinesources,inspecial,the7Be(0.861MeV)andistobeconfirmedbytheBorexinodetector[4].Inourstudywewilldealwithtwoandthreeneutrinoflavourstotrytoexplainthesolarneutrinoproblem.Wewillvarytheparametersin2(2φ)thatisconcernedwiththeνe→←ντoscillationsinordertodiscoverwhichscenario(twoorthreegenerations)givesabetterdescriptionfortheexperimentaldata.Thisanalysiswillbemadebythecalculationoftheδm221xsin2(2ω)regionsconsideringfirstonlytheratesofthedetectorswithoutthespectraldistortion,causedbytherecoil-electronspectruminKamiokandeandSuper-Kamiokande,andafterincludingthesespectrumwithinthecalculation.Aninvestigationintheνesurvivalprobabilitywillbemade,whatwillrevealthebehavioroftheneutrinoasafunctionoftheenergy.Insection2wegivetherelationsneededtocalculatethemassregionsandmixinganglesallowedbytheexperiments.Insection3wepresenttheνesurvivalprobabilityaswelltheνμandντconversionprobabilities.Insection4weanalisetheobtaineddatatodescribetheprobabilitiesasafunctionoftheenergyforasetofvaluesofsin2(2φ).Insection5wepresenttheconclusionsofthiswork.2SuppressionRatesThefirstprocedureistocalculatethesuppressionratesoftheexperimentsusingneutrinoos-cillationsandcomparewiththeobservedonestosearchforthevaluesoftheparametersδm221andsin2(2ω)thatfitthefiveexperiments(Homestake[5],GALLEX[6],SAGE[7],Kamiokande[8]andSuper-Kamiokande[9])together.Thesuppressionrate,totheexperimentalcase,isobtainedbydi-vidingthecapturerateofthedetectortothatoneexpectedfromtheSSMandtheoretically,dividingthecaptureratecalculatedusingneutrinooscillationstotheSSMone.WewillmakethiscalculationusingthedatafromtheBP98StandardSolarModel[10]with99%C.L.TheSolarNeutrinoProblemisevidencedwhenwecomparethetheoreticalcapturerateswiththoseobtainedexperimentally.Whenthiscomparisonismadewediscoverthattheseratesnevermatch,givingvaluesvaryingfrom33%forHomestaketo60%forGALLEX.Seetable1.Thesolarneutrinosuppressionrateusingneutrinooscillationisgivenby2Table1:SolarneutrinodataandBP98SSMpredictions[10].Thesecondandthirdcolumnsshowthemeasuredandcalculatedcaptureratesforeachdetector.ThevaluesaregiveninSNUexceptforKamiokandeandSuper-Kamiokande,wheretheparameteristhe8Bneutrinoflux,inunitsof106cm−2s−1.Thesuppressionratesaresetonthefourthcolumnfor1σerrors.ExperimentResultBP98SSMResult/SSMHomestake[5]2.56±0.16±0.167.7+1.2−1.00.33±0.03GALLEX[6]77.5±6.2+4.3−4.7129+8.0−6.00.60±0.06SAGE[7]66.6+7.8−8.1129+8.0−6.00.52±0.06Kamiokande[8]2.80±0.19±0.335.15+1.0−0.70.54±0.07Super-Kamiokande[9]2.44±0.05+0.09−0.075.15+1.0−0.70.47±0.02R=SOSCSSSM(1)whereSOSC=XiΦiZEmaxEminηi(Eν)σ(Eν)Pνeνe(Eν)dEν(2)andSSSMisthesameexpressionwithPνeνe(Eν)=1Theequation(2)isavailableforHomestake,GALLEXandSAGEexperiments,whereirunsforeachneutrinosource,Φiisthetotalneutrinofluxfromthesourcei,ηi(Eν)isitsnormalizedneutrinoenergyspectrum,σ(Eν)isthecrosssectionfortheexperimentconsideredandPνeνe(Eν)istheelectronneutrinosurvivalprobability.Theneutrinosourcesthatweconsiderfortheseexperimentsarepp,7Be,8B,pep,13Nand15O,exceptforHomestakethatisnotsensitivetoppneutrinos.ForSuper-Kamiokandeweconsiderthe8Bneutrinosourceonlyandwehavetotakeintoaccountthefactthatthisexperimentisalsosensitivetotheνμ,eandντ,escattering,soSOSCbecomesSOSC=ΦZEmaxEminη(Eν)nPνeνe(Eν)σνe(Eν)+Pνμνe(Eν)σνμ(Eν)+Pντνe(Eν)σντ(Eν)odEν(3)sinceσνμ(Eν)=σντ(Eν)andPνμνe(Eν)+Pντνe(Eν)=1−Pνeνe(Eν)theequation(3)issimplifiedtoSOSC=ΦZEmaxEminη(Eν)nPνeνe(Eν)