Describing neutrino oscillations in matter with Ma

arXiv:0803.1967v1[hep-ph]13Mar2008DescribingneutrinooscillationsinmatterwithMagnusexpansionA.N.Ioannisiana,b,A.Yu.Smirnovc,daYerevanPhysicsInstitute,AlikhanianBr.2,375036Yerevan,ArmeniabInstituteforTheoreticalPhysicsandModeling,375036Yerevan,ArmeniacInternationalCentreforTheoreticalPhysics,StradaCostiera11,34014Trieste,ItalydInstituteforNuclearResearch,RussianAcademyofSciences,Moscow,RussiaWepresentnewformalismfordescriptionofneutrinooscillationsinmatterwithvaryingdensity.TheformalismisbasedontheMagnusexpansionandhasavirtuethattheunitarityofS-matrixismaintainedineachorderofperturbationtheory.WeshowthattheMagnusserieshasbetterconvergence:restorationofunitarityleadstosmallerdeviationfromtheexactresultsespeciallyintheregionoflargetransitionprobabilities.VariousexpansionsareobtaineddependingonbasisofneutrinostatesandawaytosplittheHamiltonianintotheself-commutingandnon-commutingparts.WeapplytheformalismtoneutrinooscillationsinmatteroftheEarthandshowthatthesecondorderoftheexpansionhasbetterthan3%accuracyuptoE∼90MeVforsolarΔm2andupto2.5GeVforatmosphericΔm2.PACSnumbers:14.60.Pq,95.85.Ry,14.60.Lm,26.65.+tI.INTRODUCTIONNeutrinophysicsenterstheeraofprecisionmeasurements,studiesofthesub-leadingoscillationeffectsandsearchesfornewphysicsbeyondthestandardneutrinoscenario.NeutrinoflavorconversionsbecomeatoolofexplorationofotherparticlesandobjectssuchasinteriorsoftheEarthandstars.Oneofthekeyelementsofthesestudiesisneutrinooscillationsinmatterwithvaryingdensity,andinparticular,oscillationsinsidetheEarth.Thelatterisrelevantforthesolar,supernovaandatmosphericneutrinos,aswellasforthecosmicandacceleratorneutrinos.Inthisconnectionitisimportanttohavepreciseanalyticalorsemi-analyticalexpressionsforoscillationprobabilitiesvalidinwideenergyranges.Theseexpressionswillallowusnotonlytofurthersimplifynumericalcomputationsbutalsotogivedeeperinsightintophysicsinvolved.Theresultscanbeofspecialinterestinviewofdiscussionsoffutureexperimentswiththemegaton-scalefinestructuredunderwater/undericedetectors.Severalanalyticapproacheshavebeendevelopedrecentlywhichusevariousperturbationtheories[1],[2],[3],[4],[5],[6],[7],[8],[9],[10],[11].Inthepreviouspublications[2],[3],wehaveproposedaformalismwhichdescribesneutrinooscillationsinmatterwithlowdensity.ItmakesuseofsmallnessofthematterpotentialVincomparisonwiththekineticterm:V≪Δm2/2E,whereΔm2isthemasssquareddifferenceandEistheenergyofneutrino.Essentially,theexpansionparameterisgivenbytheintegralalongthetrajectoryI=ZdxV(x)cosφ(x),whereφ(x)istheadiabaticphase.ThefirstapproximationworksverywellatlowenergiesE20MeV[2].Validityoftheresultscanbeextendedtohigherenergieswhenthesecondorderterm,∼I2,istakenintoaccount[3].Itcanbefurtherimprovedincertainenergyrangesifexpansionisperformedwithrespecttothedeviationofthepotentialfromsomeaveragevalue.Theproblemofthisandsomeothersimilarapproachesisthattheunitarityofoscillationamplitudesisnotguar-anteed,andinfact,isviolatedathighenergies[3].Thisviolationcanproducecertainproblemsinnumericalcomputations.Inthispaperweproposethenewtypeofperturbationtheorieswhichmaintaintheunitarityexplicitlyineachorderofexpansion,andthereforeatanytruncationoftheseries.TheapproachisbasedontheMagnusexpansion[12],[13]whichwaspreviouslyusedfordescriptionofthenonadiabaticneutrinoconversioninmediumwithmonotonouslyvaryingdensity[14],[15][16].RecentlythefirstorderMagnusexpansionhasbeenappliedtothelowenergyneutrinooscillationsinmatteroftheEarth[10].TheformulafortheregenerationfactorintheEarthhasbeenobtainedwhichgeneralizesourresultin[3].InthispaperwedevelopvariousperturbationtheoriesusingexplicitlytwoordersoftheMagnusexpansion.Asapartofthepresentstudywereproducetheformulafrom[10].Essentially,arestorationofunitarityintheMagnusexpansionisachievedbyaneffectivere-summationofcertaincontributionstooscillationamplitudes.Thisleadstobetteraccuracyofthesemi-analyticresultsandallowsustofurtherextendtherangeofapplicationsoftheapproach.Furthermore,itgivesbetterunderstandingofthepreviouslyobtainedresultsandtheirlimitsofvalidity.Thepaperisorganizedasfollows.Insec.2wepresenttheformalismofMagnusexpansionandobtaingeneralexpressionsfortheS-matrix.Wecalculatetheoscillationprobabilitiesusingvariousperturbationapproachesbased2ontheMagnusexpansioninsec.3.Wecompareresultsofdifferentsemi-analyticapproachesinsec.4.Conclusionsfollowinsec.5.II.MAGNUSEXPANSIONA.S-matrixandMagnusexpansionInwhatfollowswewillmainlystudythecaseof2ν−mixing.Theevolutionmatrixofneutrinosinmatter,S(x,x0),obeysthefirstorder(operator)differentialequation,idS(x,x0)dx=H(x)S(x,x0),(1)wheretheHamiltonianH(x)isgivenintheflavorbasisbyH=MM†2E+V=12EU(θ)M2ΔU(θ)†+V.(2)HereV≡diag(V,0)isthematrixofpotentials,U(θ)≡cosθsinθ−sinθcosθ.(3)isthemixingmatrixandM2Δ≡diag(0,Δm2)isthediagonalmatrixofthemasssquareddifferences.Formally,thesolutionoftheequation(1)canbewrittenasthechronologicalproductS(xf,x0)=Te−iRxfx0H(x)dx≡limn→∞e−iH(xn)Δx·e−iH(xn−1)Δx···e−iH(x1)Δx,(4)Δx=xf−x0n.Inourpreviouspapers,[2],[3],weperformedexpansionofeachexponentialfactorsineq.(4)andthentooklimitn→∞.Suchaproceduredo