The matter fluctuation effect to T violation at a

arXiv:hep-ph/0106086v18Jun2001June8,2001OU-HET389hep-ph/0106086ThematterfluctuationeffecttoTviolationataneutrinofactoryTakahiroMiura∗,TetsuoShindou†,EiichiTakasugi‡,andMasakiYoshimura§DepartmentofPhysics,OsakaUniversityToyonaka,Osaka560-0043,JapanAbstractWederivedananalyticformulaforTviolationbyusingtheperturbationtheoryforsmallquantities,Δm221L/2Eandδa(x)L/2E,whereδa(x)representssymmetricandasymmetricmatterfluctuations,i.e.,deviationsfromtheaveragedensity.WeanalyzedtheeffectofmatterfluctuationstoTviolation,byassumingPREMprofileofearthmatterdensity.WefoundthatmatterfluctuationsdonotgiveanyviablecontributionforL6000km,whilethefluctuationeffectbecomeslargeduetoresonancesforL7000km.For7000kmL8000km,matterfluctuationscontributedestructivelytotheaveragedensitytermandthenetresultissmall,whileforL8000km,thecontributionfrommatterfluctuationsbecomeslargebutcontributesconstructively.∗e-mailaddress:[e-mail]†e-mailaddress:[e-mail]‡e-mailaddress:[e-mail]§e-mailaddress:[e-mail]1IntroductionInourpreviouspaper[1],weanalyzedthematterfluctuationeffecttoTviolation,P(να→νβ)−P(νβ→να)(α6=β)ataneutrinofactory[2]byusingtheperturbationmethoddevelopedbyKoikeandSato[3]andOtaandSato[4].Theperturbationismadewithrespecttosmallquantities,Δm221L/2Eandδa(x)L/2E,whereδa(x)representsmatterfluctuations,i.e.,deviationsfromtheaveragedensity.WeexaminedTviolationuptothe2ndorderandfoundthatitarisesfromtheaveragedensity,the1stordertermwhichisproportionaltoΔm221L/2Eandtermfrommatterfluctuations,the2ndordertermproportionalto(Δm221L/2E)(δa(x)L/2E).Thezerothordertermandtermsproportionalto(Δm221L/2E)2and(δa(x)L/2E)n(n=1,2,3)donotcontributetoTviolation.The1stordertermandthe2ndordertermfromsymmetricmatterfluctuationswhichwedenoteδascontributetosinδtermandthe2ndordertermfromasymmetricmatterfluctuations,δaadoestothefakecosδterm,whereδistheCPviolationphaseinMNSneutrinomixingmatrix[5].Byusingthepreliminaryreferenceearthmodel(PREM)[6]forsymmetricmatterfluc-tuationsandassumingthatasymmetricmatterfluctuationsaremuchlessthansymmetricmatterfluctuationsgivenbyPREM,wecomputedTviolationandfoundthatthe2ndordertermfromsymmetricandasymmetricmatterfluctuationsgivesonlynegligiblecon-tributionstoTviolation,andthustheconstant(average)matterapproximationisvalidforL=3000km.Ontheotherhand,forL=7332km,wefoundthatthecontributionfromsymmetricmatterfluctuationsbecomesaslargeasthe1stordertermandmoreovertheycontributedestructivelysothatTviolationbecomesverysmall.Thismeansthattheconstant(average)matterapproximationisnotvalidforL7000kmandalsothevalidityofour2ndorderformulashouldbeexamined.Inthispaper,wediscussthefollowingthreequestions:(1)Isthe2ndorderformulavalid?(2)Whatisthelengthwheretheconstant(average)matterapproximationfailsforTviolation?Thatis,atwhatlength,thematterfluctuationeffectbecomesimportant.(3)WhatisthesizeofTviolationforL7000km?1Toanswerthesequestions,wecomputedthenextorder(3rdorder)contributiontoTviolation,i.e.,thetermproportionalto(Δm221L/2E)(δa(x)sL/2E)2.Ourresultisasfollows:ThecontributionfrommatterfluctuationcanbesafelyneglectedforL6000km.WhenwediscussTviolationwithlengthlargerthan6000km,thematterfluctuationeffectshouldbetakenintoaccount.The3rdordercontributionisnegligibleincomparisonwiththe1standthe2ndordertermforalldistances.Therefore,Tviolationcanbereliablyestimatedforalldistancesanditbecomesverysmallfor7000kmL8000km.ForL8000km,the1standthe2ndtermscontributeconstructivelyandTviolationbecomeslarge.Thispaperisorganizedasfollows:InSec.2,theanalyticformulaforTviolationisgiven.ThenumericalanalysisforTviolationbyusingPREMprofileisgiveninSec.3andthemechanismhowthecancellationoccursfor7000kmL8000kmisexplainedanalytically.ThesummaryisgiveninSec.4.ThederivationoftheanalyticformulaisgiveninAppendix.2TviolationformulaForcompleteness,wegivethe1standthe2ndordercontributionstoTviolationandthedefinitionofparametersintheformula.Wealsogivethe3rdordercontributionfromsymmetricfluctuations.(a)NotationWebeginwithdefiningtheneutrinomixingmatrixasU=eiθyλ7diag(1,1,eiδ)eiθzλ5eiθxλ2=cxczsxczsz−sxcy−cxsyszeiδcxcy−sxsyszeiδsyczeiδsxsy−cxcyszeiδ−cxsy−sxcyszeiδcyczeiδ,(1)whereλj(j=2,5,7)areGell-Mannmatricesandca=cosθaandsa=sinθa.SincetheMajoranaCP-violationphasesareirrelevanttotheneutrinooscillations(flavoros-cillations)[7],weneglectedthem.Therelationbetweentheflavoreigenstates,|ναi(α=2e,μ,τ),andthemasseigenstates,|νii(i=1,2,3),isgivenby|ναi=Uαi|νii.(2)TheevolutionoftheflavoreigenstatesinmatterwithenergyEisgivenbyiddx|νβ(x)i=H(x)βα|να(x)i,(3)whereHamiltonianH(x)βαisgivenbyH(x)βα=12EUβi0Δm221Δm231iiU†iα+a(x)00βα.(4)HereΔm2ij≡m2i−m2jwithmibeingthemassof|νii,GFistheFermicouplingconstantanda(x)≡2√2GFne(x)E=7.56×10−5ρ(x)g/cm3!Ye0.5EGeVeV2,(5)wherene(x),Yeandρ(x)aretheelectronnumberdensity,theelectronfractionandthematterdensity,respectively.Fortheelectronfraction,weuseYe=0.5.Weseparatethematterdensityfluctuationfromitsaverage¯a,δa(x)≡a(x)−¯a,(6)andconsiderthedevia