Results of neutrino oscillation experiments have always been presented on the
(sin^2 2theta, Delta m^2) parameter space for the case of two-flavor
oscillations. We point out, however, that this parameterization misses the half
of the parameter space pi/4 < theta <= pi/2 (``the dark side''), which is
physically inequivalent to the region 0 <= theta <= pi/4 (``the light side'')
in the presence of matter effects. The MSW solutions to the solar neutrino
problem can extend to the dark side, especially if we take the conservative
attitude to allow higher confidence levels, ignore some of the experimental
results in the fits, or relax theoretical predictions. Furthermore even the
so-called ``vacuum oscillation'' solution distinguishes the dark and the light
sides. We urge experimental collaborations to present their results on the
entire parameter space.Comment: 4 pages, 3 figures, uses psfig. Fixed typos in Eq. (3). An imprecise
  comment in the footnote remove

Alexander Friedland

André de Gouvêa

Bahcall

Cleveland

de Gouvêa

Fukuda

Guth

Hitoshi Murayama

Physics Letters B

English

arXiv

Crossref

The dark side of the solar neutrino parameter space

Results of neutrino oscillation experiments have always been presented on the(sin^2 2theta, Delta m^2) parameter space for the case of two-flavoroscillations. We point out, however, that this parameterization misses the halfof the parameter space pi/4 < theta <= pi/2 (``the dark side''), which isphysically inequivalent to the region 0 <= theta <= pi/4 (``the light side'')in the presence of matter effects. The MSW solutions to the solar neutrinoproblem can extend to the dark side, especially if we take the conservativeattitude to allow higher confidence levels, ignore some of the experimentalresults in the fits, or relax theoretical predictions. Furthermore even theso-called ``vacuum oscillation'' solution distinguishes the dark and the lightsides. We urge experimental collaborations to present their results on theentire parameter space.Results of neutrino oscillation experiments have always been presented on the (sin^2 2theta, Delta m^2) parameter space for the case of two-flavor oscillations. We point out, however, that this parameterization misses the half of the parameter space pi/4 < theta <= pi/2 (``the dark side''), which is physically inequivalent to the region 0 <= theta <= pi/4 (``the light side'') in the presence of matter effects. The MSW solutions to the solar neutrino problem can extend to the dark side, especially if we take the conservative attitude to allow higher confidence levels, ignore some of the experimental results in the fits, or relax theoretical predictions. Furthermore even the so-called ``vacuum oscillation'' solution distinguishes the dark and the light sides. We urge experimental collaborations to present their results on the entire parameter space.Results of neutrino oscillation experiments have always been presented on the (sin^2 2theta, Delta m^2) parameter space for the case of two-flavor oscillations. We point out, however, that this parameterization misses the half of the parameter space pi/4 < theta <= pi/2 (``the dark side''), which is physically inequivalent to the region 0 <= theta <= pi/4 (``the light side'') in the presence of matter effects. The MSW solutions to the solar neutrino problem can extend to the dark side, especially if we take the conservative attitude to allow higher confidence levels, ignore some of the experimental results in the fits, or relax theoretical predictions. Furthermore even the so-called ``vacuum oscillation'' solution distinguishes the dark and the light sides. We urge experimental collaborations to present their results on the entire parameter space.Results of neutrino oscillation experiments have always been presented on the (sin 2 2 θ , Δm 2 ) parameter space for the case of two-flavor oscillations. We point out, however, that this parameterization misses the half of the parameter space π 4 <θ≤ π 2 (`the dark side'), which is physically inequivalent to the region 0≤θ≤ π 4 (`the light side') in the presence of matter effects. The MSW solutions to the solar neutrino problem can extend to the dark side, especially if we take the conservative attitude to allow higher confidence levels, ignore some of the experimental results in the fits, or relax theoretical predictions. Furthermore, even the so-called `vacuum oscillation' solution distinguishes the dark and the light sides. We urge experimental collaborations to present their results on the entire parameter space

de Gouvea, Andre

Friedland, Alexander

Murayama, Hitoshi

CERN Document Server

arXiv:hep-ph/0002064   7 Feb 2000TheDarkSideoftheSolarNeutrinoParameterSpaceAndredeGouve^aCERN-TheoryDivision,CH-1211Geneva23,SwitzerlandAlexanderFriedlandandHitoshiMurayamaDepartmentofPhysics,UniversityofCalifornia,Berkeley,CA94720,USATheoryGroup,LawrenceBerkeleyNationalLaboratory,Berkeley,CA94720,USA(February7,2000)Resultsofneutrinooscillationexperimentshavealwaysbeenpresentedonthe(sin22;m2)param-eterspaceforthecaseoftwo-avoroscillations.Wepointout,however,thatthisparameterizationmissesthehalfoftheparameterspace4<2(\thedarkside"),whichisphysicallyinequivalenttotheregion04(\thelightside")inthepresenceofmattereects.TheMSWsolutionstothesolarneutrinoproblemcanextendtothedarkside,especiallyifwetaketheconservativeattitudetoallowhighercondencelevels,ignoresomeoftheexperimentalresultsinthets,orrelaxtheoreticalpredictions.Furthermore,eventheso-called\vacuumoscillation"solutiondistinguishesthedarkandthelightsides.Weurgeexperimentalcollaborationstopresenttheirresultsontheentireparameterspace.IntheStandardModelofparticlephysics,neutrinosarestrictlymassless.Recently,however,theSuper-Kamiokandecollaborationstudiedatmosphericneutrinosandreportedastrongevidenceforneutrinooscillations[1],andhenceaniteneutrinomass.Themostlikelyinterpretationoftheirdataistheoscillationbetweenand.Thismadeitalsonaturaltointerpretanotherlong-standingissueinneutrinophysics,thedecitofthesolareux[2],intermsofneutrinooscillations.How-ever,thesolarneutrinodecithasnotbeenregardedasconvincingevidenceforneutrinooscillationsinthecom-munity.Thereasonisprobablymultifoldbuttwomainobjectionsarethefollowing.(1)Neutrinoexperimentsaresodicultthatitispossiblethatsomeofthedataarenotentirelycorrect.(2)ThephysicsoftheSunissocomplexthattheneutrinouxcalculationsintheStan-dardSolarModel(SSM)mayhaveunderestimatedthetheoreticaluncertainties.Toresolvethissituation,anewgenerationofsolarneutrinoexperiments,suchasSuper-Kamiokande,SNO,Borexino,GNO,KamLAND,etc,islookingforanevi-denceforsolarneutrinooscillationswithoutrelyingontheSSMinwell-understoodexperimentalenvironments.Theyaimnotonlyatestablishingoscillationsbutalsoatoverdeterminingthesolutioninthenextfewyears.Suchdatawilleventuallysupersededatafromthepastexper-iments.Itisthereforeimportanttoanalyzethefuturedatawithouttoomuchprejudicebasedonthepastdata.Inthisletter,wepointoutthatthestudyofneutrinooscillationsonthe(m2;sin22)parameterspacedonetraditionallyisincomplete,sinceitcoversonlytherange04(\thelightside").Indeed,someofthesolu-tionstothesolarneutrinopuzzleextendtotheotherhalfoftheparameterspace4<2,whichwecall\thedarkside,"andhenceitisphenomenologicallynecessarytoincludebothhalvesoftheparameterspace.Thisisespeciallytrueonceoneemploysamoreconservativeatti-tudewhicheitherallowshighercondencelevels,ignoressomeoftheexperimentaldata(especiallyHomestake[3]),orrelaxesthetheoreticalpredictiononthe8Bux.Neutrinooscillationsoccurifneutrinomasseigenstatesaredierentfromneutrinoweakeigenstates.Assumingthatonlytwoneutrinostatesmix,therelationbetweenmasseigenstates(1and2)andavoreigenstates(forexampleeand)issimplygivenbyj1i=cosjei sinji;j2i=sinjei+cosji;(1)whereisthevacuummixingangle.Themass-squareddierenceisdenedasm2m22 m21.Wearein-terestedintherangeofparametersthatencompassesallphysicallydierentsituations.First,observethatEq.(1)isinvariantunder!+,jei! jei,ji! ji,andhencetheranges[ 2;2]and[2;32]arephysicallyequivalent.Next,notethatitisalsoinvariantunder! ,ji! ji,j2i! j2i,henceitissucienttoonlyconsider2[0;2].Finally,itcanalsobemadeinvariantunder!2 ,ji! jibyrelabelingthemasseigenstatesj1i$j2i,i.e.m2! m2.Thus,wecantake(m2>0)withoutlossofgeneral-ity.Allphysicallydierentsituationsareobtainedbyallowing02.Forthecaseofoscillationsinthevacuum,thesurvivalprobabilityisgivenbyP(e!e)=1 sin22sin21:27m2EL:(2)Here,m2isgivenineV2=c4,EinGeV,andLinkm.Inthiscasetheoscillationphenomenoncanbeparam-eterizedbym2andsin22,sinceand2 yieldidenticalphysics.Thereforewecanrestrictourselvesto04,andusetheparameterspace(m2;sin22)withoutanyambiguity.Thisisindeedanadequatepa-1rameterization for reactor antineutrino oscillation exper-iments, short-baseline accelerator neutrino oscillation ex-periments, and $ atmospheric neutrino oscillationexperiments.In certain cases, however, neutrino-matter interactionscan dramatically change the oscillation probability [4].These matter eects are particularly important in ex-plaining the solar eux decit in terms of neutrino os-cillations. In the presence of matter eects, Eq. (2) ismodied toP (e! e) = P1cos2 + (1  P1) sin2+2pP1(1  P1) sin  cos  cos2:54m2EL + ; (3)where P1is the probability of the neutrino exiting theSun in the 1mass eigenstate and  is a phase inducedby the matter eects, which is not important for ourpurposes. See Ref. [5,6] for more details. Because P1depends on m2cos 2, the half of the parameter spacetraditionally considered 0   4(the light side) isphysically inequivalent to the other half4<  2(thedark side). However, all data analysis have been reportedon the (m2; sin22) plane with positive m2only forsolar neutrino experiments and hence only half of theparameter space has been analyzed. Even though thedark side has been studied in the context of three-avor[7] and four-avor [8] neutrino oscillations, the impor-tance of studying both halves for the simplest case oftwo-avor oscillations has been largely ignored in the lit-erature. There also appeared to be a misconception inthe literature that physics was discontinuous at maximalmixing  =4. For instance, matter eects in the Earthwere once thought to disappear as the mixing approachedmaximal. However, the authors of Ref. [9] emphasizedthat the matter eects remain important even for themaximal mixing, and the present authors further showedthat physics is completely continuous beyond  =4[6].One can still retain the dark side with only 0   4if a separate parameter space with m2< 0 is added.This is indeed what Super-Kamiokande did in the case of$ soscillations of atmospheric neutrinos [10]. How-ever, as argued in [6], it is more natural to use 0   2with the xed sign of m2to exhibit the continuity of thephysics between the two halves of the parameter space.Part of the reason why the dark side has been neglectedin the literature is that it is impossible to obtain esur-vival probabilities less than one half when the two masseigenstates are incoherent, i.e., when the last term inEq. (3) is absent. (This occurs in the so-called \MSWregion" 10 8<m2<10 3eV2[11].) Indeed, the datafrom the Homestake experiment [3] used to be about aquarter of the SSM prediction, and this could have beenan argument for dropping the dark side entirely in theMSW region. However, the change from BP95 [12] toBP98 [13] calculations increased the Homestake result toabout a third of the SSM with a relatively large theoret-ical uncertainty. Therefore it is quite possible that the\MSW solutions" extend to the dark side as well. More-over, some people question the SSM and/or the Homes-take experiment, and perform ts by ignoring either (orboth) of them [14]. We show below that some of theMSW solutions to the solar neutrino puzzle indeed ex-tend to the dark side and hence it is necessary to explorethe dark side experimentally. If we further relax the the-oretical prediction on the8B solar neutrino ux and/orignore one of the solar neutrino experiments in the globalt, the preferred regions extend even deeper into the darkside.Another possible reason for disregarding the dark sideis that the so-called \vacuum oscillation region" (m2<10 9eV2) was believed to be the same in the lightand dark sides. This is because P1approaches cos2for m2 10 9eV2(E=MeV) and Eq. (3) reducesto Eq. (2). It is remarkable, however, that low-energy(especially pp) neutrinos do not reach this limit form2>10 10eV2and hence the preferred regions aredierent in the light and the dark sides [15]. This ob-servation also implies that the separation of the MSWregion and the vacuum oscillation region as traditionallydone in the global ts is articial and misleading. It isimportant to study the entire range of m2continuously.If sin22 is not a good parameter, what is the alterna-tive? Two suggestions have been made in the literature.One is sin2, which is natural since the matter eect de-pends directly on sin2 [6]. If plotted on the linear scale,pure vacuum oscillations would yield physics reection-symmetric around sin2 = 0:5. If plotted on the logscale, the reection symmetry is lost, but it is still a usefulparameterization as physics is completely continuous andsmooth from the light to the dark side. Another possibleparameterization is tan2, which retains the reectionsymmetry for pure vacuum oscillation around tan2 = 1if plotted on the log scale [7,6]. We employ tan2 for theanalysis below because we would like to use the log scaleto present the MSW solutions as well as the importanceof the matter eect on the \vacuum oscillation" regionat the same time. Note that the Jacobian from sin2or tan2 to sin22 is singular at  =4and plots withsin22 will display unphysical singular behavior there [6].We next present the results of global ts to the cur-rent solar neutrino data from water Cherenkov detectors(Kamiokande and Super-Kamiokande) [16], a chlorinetarget (Homestake) [3] and gallium targets (GALLEXand SAGE) [17] on the full parameter space. We do notinclude the spectral data from Super-Kamiokande [18] asit appears to be still evolving with time. The t is tothe event rates measured at these experiments only. Incomputing the rates we include not only the pp,7Be, and8B neutrinos, but also the13N,15O, and pep neutrinos.We use Eq. (3) with P1computed in the exponential2approximation for the electron number density prolein the Sun, and properly account for neutrino interac-tions in the Earth during the night with a realistic Earthelectron number density prole by numerically solvingSchrodinger equation as described in [6]. Since the mix-ing angle at the production point in the Sun's core de-pends on the electron number density, we integrate overthe production region numerically. We treat the corre-lations between the theoretical uncertainties at dierentexperiments following Ref. [7]. To insure a smooth tran-sition between the MSW and the vacuum oscillation re-gion, we integrate over the energy spectrum (includingthe thermal broadening of the7Be neutrino \line") form2 10 8eV2and average the neutrino uxes overthe seasons. For m2> 10 8eV2we treat the two masseigenstates as incoherent. Results are completely smoothat m2= 10 8eV2, as expected. This allows us to tthe data from m2= 10 11{10 3eV2all at once, un-like previous analyses which separate out the \vacuumoscillation region" from the rest.As was mentioned earlier, we take the global t to thecurrently available data only as indicative of the ulti-mate result because we expect much better data to becollected in the near future to eventually supersede thecurrent data set. We would like to keep our minds open tosurprises such as the possibility that one of the earlier ex-periments was not entirely correct or that the theoreticaluncertainty in the ux prediction was underestimated.In this spirit, we employ more conservative attitudes inthe global t than most of the analyses in the literaturein the following three possible ways. (1) We allow highercondence levels, such as 3 . (2) We relax the theoret-ical prediction on the neutrino ux. (3) We ignore someof the experimental data in the t.The global t results are presented in Fig. 1 at the 2 (95% CL) and 3  (99.7% CL) levels dened by 2 2minfor two degrees of freedom. It is noteworthy that boththe LMA and LOW solutions (we use the nomenclatureintroduced in [19]) extend to the dark side at the 3 level. At 99% CL, however, the LMA solution is connedto the light side. This result is consistent with the two-avor limit of the three-avor analysis in [7] and the four-avor analysis in [8], where the spectral data is includedand the LOW solution extends into the dark side at 99%CL. Another interesting fact is that the LOW solutionis smoothly connected to the VAC solution, where thepreferred region is clearly asymmetric between the lightand the dark sides. Note that, at m2 10 9eV2,the allowed region is bigger in the dark side. The region10 9< m2< 10 8eV2was, to the best of the authors'knowledge, never studied fully in the literature and thisresult demonstrates the need to study the entire m2region continuously without the articial separation ofthe \MSW region" and \vacuum oscillation region," astraditionally done in the literature.We next present a t where the theoretical prediction0.001 0.01 0.1 1 10tan2 θ10-1010-910-810-7101010---654∆m  (eV  )22LMAVACLOWSMAFIG. 1. A global t to the solar neutrino event rates atchlorine, gallium and water Cherenkov experiments. The re-gions are shown at 2  (light shade) and 3  (dark shade)levels. The region tan2 > 1 is the dark side  >4.of the8B ux is relaxed. Even though the helioseismol-ogy data constraints the sound speed down to about 5%of the solar radius [13], the core region where8B neutri-nos are produced is still not constrained directly. Giventhe sensitive dependence of the8B ux calculation on thecore temperature 8B T22, we may consider it as a freeparameter in the t. This can be done within the formal-ism of Ref. [7] by formally sending the error in CBetoinnity. The result is presented in Fig. 2. The preferredregion extends more into the dark side than the previ-ous t. Even though the LMA and LOW solutions areconnected in this plot, the lack of a large day-night asym-metry at Super-Kamiokande would eliminate the range3  10 7<m2<10 5eV2for 0:2<tan2 < 1 [16].It is important for Super-Kamiokande to report their ex-clusion region on the dark side.Finally, we present a t where the event rate measuredat the Homestake experiment is not used in Fig. 3. Thismay be a sensible exercise given that the neutrino captureeciency was never calibrated in this experiment. Thepreferred region extends into the dark side even at the95% CL. Note also the asymmetry between the dark andthe light sides even for m2< 10 9eV2.We expect the data of the current and next gen-eration of solar neutrino experiments, such as Super-Kamiokande, SNO, GNO, Borexino, KamLAND, toeventually supersede the current data set. Therefore weregard the above global ts only as estimates of the ulti-mate results. The most important point is that all exper-30.0010.010.1110tan2θ10-1010-910-810-710 10 10- - -6 5 4∆m  (eV  )2 2FIG.2.Aglobalttothesolarneutrinoeventratesatchlorine,galliumandwaterCherenkovexperiments,wherethe8Buxistreatedasafreeparameter.Contoursareshownat2(lightshade)and3(darkshade).0.0010.010.1110tan2θ10-1010-910-810-710 10 10- - -6 5 4∆m  (eV  )2 2FIG.3.AglobalttothesolarneutrinoeventratesatthegalliumandwaterCherenkovexperimentsbutnotatthechlorineexperiment.Contoursareshownat2(lightshade)and3(darkshade).imentalcollaborationsshouldreporttheirresults,bothexclusionandmeasurements,onbothsidesoftheparam-eterspace,withoutunnecessarytheoreticalbiastowardsthelightside.Westronglyurgetheexperimentalcollab-orationstoconsiderthispoint.ThisworkwassupportedinpartbytheU.S.DepartmentofEnergyunderContractDE-AC03-76SF00098andinpartbytheNationalScienceFoundationundergrantPHY-95-14797.[1]Super-KamiokandeCollaboration(Y.Fukudaetal.),Phys.Rev.Lett.81,1562(1998),hep-ex/9807003.[2]SolarNeutrinos:TheFirstThirtyYears,eds.,J.N.Bah-call,R.Davis,P.Parker,A.Smirnov,R.Ulrich,Reading,USA,Addison-Wesley,1995,FrontiersinPhysics.Vol92.[3]B.T.Clevelandetal.,Astrophys.J.496,505(1998).[4]L.Wolfenstein,Phys.Rev.D17,2369(1978);S.P.MikheyevandA.Yu.Smirnov,Yad.Fiz.(Sov.J.ofNucl.Phys.)42,1441(1985).[5]A.deGouve^a,A.Friedland,andH.Murayama,Phys.Rev.D60,093011(1999),hep-ph/9904399.[6]A.deGouve^a,A.Friedland,andH.Murayama,hep-ph/9910286.[7]G.L.Foglietal.,hep-ph/9912231.[8]C.Giunti,M.C.Gonzalez-Garcia,andC.Pe~na-Garay,hep-ph/0001101.Theyclaimthatthereisnosolutioninthedarksidebasedontherates-onlytat99%CL.Thissmalldiscrepancymaybeduetotheirtwo-stepapproxi-mationtotheEarthelectronnumberdensityprole.[9]A.H.Guth,L.RandallandM.Serna,JHEP9908,018(1999),hep-ph/9903464.[10]T.Kajita,talkpresentedatPASCOS99,Granlibakken,LakeTahoe,California,December10-16,1999.[11]Forinstance,integrationoverneutrinoenergieswashesouttheoscillatoryfactor.Inthecaseof7Beneutrinos,thisoccursform2>10 8eV2duetothethermalbroadeningeect.Forneutrinoswithcontinuousspec-tra,thewashoutoccursatevenlowerm2.See[5,15]formoredetails.[12]J.N.BahcallandM.H.Pinsonneault,Rev.Mod.Phys.67,781(1995),hep-ph/9505425.[13]J.N.Bahcall,S.Basu,andM.H.Pinsonneault,Phys.Lett.B433,1(1998),astro-ph/9805135.[14]Forarecentstudyalongthisline,seeR.Barbierietal.,JHEP9812,017(1998),hep-ph/9807235.[15]A.Friedland,UCB-PTH-00/04,LBNL-45037.[16]Super-KamiokandeCollaboration(Y.Fukudaetal.),Phys.Rev.Lett.82,1810(1999),hep-ex/9812009.[17]GALLEXCollaboration(W.Hampeletal.),Phys.Lett.B447,127(1999);SAGECollaboration(J.N.Ab-durashitovetal.),Phys.Rev.C60,055801(1999).[18]Super-KamiokandeCollaboration(Y.Fukudaetal.),Phys.Rev.Lett.82,2430(1999),hep-ex/9812011.[19]J.N.Bahcall,P.I.Krastev,andA.Yu.Smirnov,Phys.Rev.D58,096016(1998),hep-ph/9807216.4

The Dark Side of the Solar Neutrino Parameter Space

The Dark Side of the Solar Neutrino Parameter Space

Abstract

Similar works

Full text

Available Versions

Crossref

CERN Document Server