29 November 2001

Physics Letters B 521 (2001) 287–290www.elsevier.com/locate/npe

Electron–neutrino survival probability from solar-neutrino data

V. Berezinskya,b, M. Lissiac

a INFN, Laboratori Nazionali del Gran Sasso, I-67010 Assergi (AQ), Italyb Institute for Nuclear Research, Moscow, Russia

c Istituto Nazionale di Fisica Nucleare, Sezione di Cagliari and Dipartimento di Fisica dell’Università di Cagliari,I-09042 Monserrato (CA), Italy

Received 30 August 2001; accepted 8 October 2001Editor: G.F. Guidice

Abstract

With SNO data [SNO Collaboration, nucl-ex/0106015] on electron–neutrino flux from the sun, it is possible to derive theνe

survival probabilityPee(E) from existing experimental data of Super-Kamiokande, gallium experiments and Homestake. Thecombined data of SNO and Super-Kamiokande provide boronνe flux and the total flux of all active boron neutrinos, givingthusPee(E) for boron neutrinos. The Homestake detector, after subtraction of the signal from boron neutrinos, gives the flux ofBe+CNO neutrinos, andPee for the corresponding energy interval, if the produced flux is taken from the Standard Solar Model(SSM). Gallium detectors, GALLEX, SAGE and GNO, detect additionallypp-neutrinos. Thepp flux can be calculated sub-tracting from the gallium signal the rate due to boron, beryllium and CNO neutrinos. The ratio of the measuredpp-neutrino fluxto that predicted by the SSM gives the survival probability forpp-neutrinos. Comparison with theoretical survival probabilitiesshows that the best (among known models) fit is given by LMA and LOW solutions. 2001 Published by Elsevier Science B.V.

The recent measurement of boronνe flux by SNO[1], combined with Super-Kamiokande data [2], givesstrong evidence for neutrino oscillations [1]. It is basedon the fact that theνe flux measured by SNO throughthe charged current (CC) interactionνe + d → e +p +p induces in the Super-Kamiokande detector lesselectrons than observed. The excess of the electronscan be produced only by active neutrinos of otherflavors:νµ andντ . This flux of active neutrinos is thusdetermined and found to be 3.3σ above the zero value.Such analysis was suggested earlier in Ref. [3], andrecently it was further developed in Ref. [4] (for otherrecent analysis of SNO data, see [5,6]).

E-mail addresses: venya.berezinsky@lngs.infn.it(V. Berezinsky), marcello.lissia@ca.infn.it (M. Lissia).

In this Letter we shall demonstrate that the new re-sults obtained by SNO allow us to derive the survivalprobability for boron, beryllium andpp electron neu-trinos (for general analysis, see [7] and for calculationof survival probability for boron neutrinos [4]).

The probability of electron neutrinos to survive onthe way from the production point inside the Sun tothe detection site on the Earth is referred to as survivalprobability, Pee . In case of oscillations, 1− Pee isthe probability of electron–neutrino conversion intoneutrinos of other flavors.

The flux of 8B electron neutrinos determined bySNO via CC-events isΦνe = 1.75 ± 0.148 ×106 cm−2 s−1 [1], where we used the upper valuefor systematic error and summed errors quadrati-cally. SK measurements provide the fluxΦν = Φνe +0.154(Φνµ + Φντ ), where 0.154 is a ratio of cross-

0370-2693/01/$ – see front matter 2001 Published by Elsevier Science B.V.PII: S0370-2693(01)01212-6

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sections σνµe/σνee. The comparison of these twofluxes allows us to findΦνµ + Φντ and Φtot =Φνe + Φνµ + Φντ , which is equal to 5.44± 0.99×106 cm−2 s−1 [1]. Thus the survival probability for8Belectron neutrinos can be found asPee = Φνe/Φtot =0.32± 0.065. Note that this derivation ofPee does notdepend on the SSM flux. The partial oscillation to ster-ile neutrinos,νe → νs , diminishes furtherPee . Thispossibility is somewhat disfavored by the observationthatΦtot is close to prediction of the SSM [1].

The error in the value ofPee indicated above is cal-culated by adding all errors in quadrature and it needsfurther discussion. The uncertainties ofΦνe andΦtotare correlated and therefore the usual interpretationof the indicated error is allowed only in the limitΦνe � Φtot. In fact, the 2σ - and 3σ -equivalent inter-vals are asymmetric and greater than 1σ error 0.065multiplied by factor of 2 and 3, respectively. In partic-ular it can be shown that 1−Pee interval has real 3.3σdeflection from zero value.

The value of Pee is plotted in Figs. 1–3. Thecalculated value refers to the whole energy intervalof boron neutrinos measured in Super-Kamiokande: inFigs. 1–3 the horizontal error bars show the width ofthis interval. The value ofPee is plotted in the middleof this interval and looks asymmetric in logarithmicscale. As already mentioned above, one should notinterpret the many standard deviations which separatesPee from unity in the usual way: the probability thatPee = 1 corresponds to about 3.3σ and not to≈ 10σas it might appear from the figure.

The Homestake detector is sensitive to the boron,beryllium and CNOνe-neutrinos. Subtracting the con-tribution of the measured flux of boronνe-neutrinoswith the standard spectrum to the chlorine detector, wedetermine the contribution of Be+ CNO neutrinos tothe signal. For the production fluxes we use that of theSSM [8]: note that the prediction for the Be neutrinoflux is reliable and the contribution of CNO neutrinosis small. In this case the extracted survival probabil-ity Pee is not affected by possibility ofνe → νs oscil-lation. The survival probability is plotted in Figs. 1–3with errors calculated taking into account uncertaintiesin fluxes together with statistical and systematic er-rors of detection. The horizontal error bar refers to theenergy interval of Be+ CNO neutrinos. The error isstrongly anticorrelated with the error in the boron flux:

Fig. 1. Survival probability of electron neutrinos as a function ofenergy. Data points are extracted from the gallium, chlorine andboron–neutrino signals. The horizontal error bars give the energywindows of each datum. For the interpretation of vertical errorbars see text. The three solid curves show the theoretical survivalprobabilities for the Gribov–Pontecorvo (GP) solution, the LOWMSW solution (LOW) and for the large mixing angle MSW (LMA)solution. The solid LMA curve corresponds to neutrino productionpoint at the peak of the boron/beryllium production zone; the dashedcurve—at the peak of thepp production zone.

a higher (lower) boron flux implies a lower (higher)beryllium flux.

The detectedνe-neutrino flux ofpp-cycle is foundfrom gallium experiments (GALLEX, SAGE andGNO), subtracting the contributions from boron,beryllium and CNO neutrinos found from the Kamio-kande, SNO and gallium experiments. The productionflux is taken from SSM calculations [8]. The calcu-latedpp flux is robust and agrees with the flux foundindependently from solar-luminosity sum rule. Thesurvival probability is plotted in Figs. 1–3. The hori-zontal error bar refers to thepp-neutrino energy inter-val. The survival probability is not affected by oscilla-tion to sterile neutrinos. We shall summarize now theassumptions involved in the calculations of survivalprobabilities at different energies. For boron neutrinoswe used only experimental data, neglecting possibleoscillation to sterile neutrinos. For the other two en-ergy intervals (pp- and Be-neutrinos) we used for to-tal fluxes the production fluxes calculated in the SSM,

V. Berezinsky, M. Lissia / Physics Letters B 521 (2001) 287–290 289

but the calculations of these fluxes are reliable. In ad-dition, we assumed that the measured suppression ofhigh energy boron neutrinos (roughly forE > 6 MeV)is valid also for the lower part of the boron–neutrinospectrum. Since in both cases only electron neutrinosproduce the signal, the extracted survival probabilityis not affected by oscillation to sterile neutrinos.

In Fig. 1 the observed survival probabilities arecompared with the predictions of the LMA MSW,LOW MSW and Gribov–Pontecorvo (GP) [9] solu-tions. For LMA we use one of the best fit solutionsfrom Ref. [10]:�m2 = 3.7× 10−5 eV2 and sin2 2θ =0.79; for LOW:�m2 = 1.0×10−7 eV2, and sin2 2θ =0.97 [10], and for the Gribov–Pontecorvo solution [9]sin2 2θ = 1. The LMA and LOW survival probabili-ties shown are not averaged over the production pointsin the sun. The LOW curve is practically indepen-dent of the production point. The LMA solid curve isshown for a production point in the boron/berylliumproduction zone and the dashed curve shows the sur-vival probability for a neutrino produced at the peakof the pp region. Averaging over the production re-gion gives a curve close to the solid one for boron andberyllium neutrinos, and a curve between the solid anddashed one forpp neutrinos (in fact there is little dif-ference between these curves in the energy region ofpp neutrinos).

Only for illustration, we presentχ2/d.o.f. valuesfor the fits of the data by different oscillation solutions:they are 4/3, 5/3 and 10/3 for the LMA, LOW and GPsolutions, respectively. In fact, thisχ2 analysis couldbe misleading because when the survival probabilityis used as a variable, the probability distribution is notGaussian.

In Fig. 2 the observed survival probabilities arecompared with those calculated in SMA MSW. Oneof the best fit solutions, with spectral and tempo-ral information included, is given by�m2 = 4.6 ×10−6 eV2 and sin2 2θ = 1.36× 10−3. The solid curvecorresponds again to the neutrino produced in theboron/beryllium production region, while the dashedcurve shows the low-energy part of the survival prob-ability for the neutrino born in the peak ofpp pro-duction region. The confidence level is characterisedillustratively by χ2/d.o.f. ≈ 12/3.

In Fig. 3 the observed survival probabilities arecompared with predictions of the Just-So (�m2 =4.6 × 10−10 eV2 and sin2 2θ = 0.83) and Just-So2

Fig. 2. Survival probability of electron neutrinos as function ofenergy. The solid curve shows the theoretical survival probabilityfor the small mixing angle (SMA) MSW solution for neutrino bornat the peak of the boron/beryllium production region; the dashedcurve—at the peak of thepp production region.

Fig. 3. Survival probability of electron neutrinos as function ofenergy. The solid curve shows the theoretical survival probabilityfor the vacuum oscillation solution; for energy below 0.53 MeV,only the average probability is shown. The dashed curve is for theJust-So2 solution.

290 V. Berezinsky, M. Lissia / Physics Letters B 521 (2001) 287–290

(�m2 = 5.5 × 10−12 eV2 and sin2 2θ = 0.96) solu-tions. The qualitatively worse agreement can be char-acterised byχ2/d.o.f. ≈ 11/3 andχ2/d.o.f. ≈ 13/3for Just-So and Just-So2, respectively.

We repeat again that the quantitiesχ2 calculatedabove have only an illustrative character: the obtainedvalue of χ2 for each solution is not connected withprobability in a usual way. This analysis indicateshowever the preferred solutions.

The preferred solutions are LMA and LOW whichare characterised by the lowest values ofχ2.

In principle the survival probabilities for boronneutrinos can be given for several energy bins. Sincethe observed spectrum is well described by the SSMspectrum, the energy bin analysis will further decreasethe probability of SMA MSW, which produces asignificant spectrum distortion in the boron energywindow, and will favour the solutions with a flatsuppression of boron neutrino spectrum, such as theGP, LMA and LOW solutions. Day–night dependencewill not change the analysis in a significant way, sincethese solutions have little day–night dependence in theboron high-energy window.

In conclusion, using the data of all solar-neutrinoexperiments we have derived the electron–neutrinosurvival probabilities. For boron neutrinos only experi-mental data are used, and the SSM flux is not involved.Survival probability decreases in presence ofνe → νs

oscillation. For Be+ CNO andpp neutrinos the SSMcalculations are used for production (i.e., total) fluxes.This is a plausible assumption, because beryllium and

pp neutrino fluxes are reliably calculated and CNOfluxes are small. Oscillation to sterile neutrinos doesnot affect the extracted survival probabilities for Be+CNO andpp neutrinos.

LMA and LOW solutions give better fits, in com-parison with other solutions. It is also possible, inprinciple, to calculate for boron neutrinos the survivalprobabilities for several energy bins: the likelihood ofsolutions with an energy independent suppression forE > 6 MeV, i.e., LOW, LMA and GP, will increase ascompared with the ones that predict an energy depen-dent suppression.

References

[1] Q.R. Ahmad et al., SNO Collaboration, nucl-ex/0106015.[2] S. Fukuda, Super-Kamiokande Collaboration, Phys. Rev.

Lett. 86 (2001) 5651.[3] F.L. Villante, G. Fiorentini, E. Lisi, Phys. Rev. D 59 (1999)

013006;G.L. Fogli, E. Lisi, A. Palazzo, F.L. Villante, Phys. Rev. D 63(2001) 113016.

[4] G.L. Fogli, E. Lisi, D. Montanino, A. Palazzo, hep-ph/0106247.

[5] C. Giunti, hep-ph/0107310.[6] J.N. Bahcall, hep-ph/0108147.[7] V. Barger, D. Marfatia, K. Whisnant, hep-ph/0106207.[8] J.N. Bahcall, M. Pinsonneault, astro-ph/0010346.[9] V.N. Gribov, B. Pontecorvo, Phys. Lett. B 28 (1969) 493;

V. Berezinsky, M.C. Gonzalez-Garcia, C. Pena-Garay, hep-ph/0105294.

[10] J.N. Bahcall, M.C. Gonzalez-Garcia, C. Pena-Garay, hep-ph/0106258.