PHYSICAL REVIEW D, VOLUME 60, 093010

Seasonal dependence in the solar neutrino flux

P. C. de HolandaInstituto de Fı´sica Corpuscular– C.S.I.C., Departamento de Fı´sica Teo`rica, Universitat of Vale`ncia, 46100 Burjassot, Vale`ncia, Spain

and Instituto de Fı´sica Gleb Wataghin, Universidade Estadual de Campinas, UNICAMP 13083-970– Campinas, Brazil

C. Pena-Garay, M. C. Gonzalez-Garcia, and J. W. F. ValleInstituto de Fı´sica Corpuscular– C.S.I.C., Departamento de Fı´sica Teo`rica, Universitat of Vale`ncia, 46100 Burgassot, Vale`ncia, Spain

~Received 24 March 1999; published 8 October 1999!

Mikheyev-Smirnov-Wolfenstein~MSW! solutions of the solar neutrino problem predict a seasonal depen-dence of the zenith angle distribution of the event rates, due to the nonzero latitude at the Super-Kamiokandesite. We calculate this seasonal dependence and compare it with the expectations in the no-oscillation case aswell as just-so scenario, in the light of the latest Super-Kamiokande 708-day data. The seasonal dependencecan be sizable in the large mixing angle MSW solution and would be correlated with the day-night effect. Thismay be used to discriminate between MSW and just-so scenarios and should be taken into account in refinedfits of the data.@S0556-2821~99!08419-2#

PACS number~s!: 26.65.1t, 14.60.Pq

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The difference in thene fluxes during the day and thnight due to the regeneration of thene in the Earth — theso-called day-night effect — is one of the milestones ofMikheyev-Smirnov-Wolfenstein~MSW! solutions of the so-lar neutrino problem~SNP! @1,2#. This effect is negligible inthe just-so picture@3#. Conversely, the oscillatory behavioof the conversion probability in the just-so scenario leadsseasonal-dependent event rates beyond the simple geomcal factor, due to variation of the Sun-Earth distance in dferent seasons of the year. Though recognized in the edays of the MSW effect@2,4# this seasonal effect has beeneglected in most discussions of the MSW solution toSNP and has even been recently claimed to be absent inMSW picture@5,6#.

Recent Super-Kamiokande data after 708 days@7# exhibitan excess of the number of events during the night@8#.Though not yet statistically significant this provides somhint in favor of the possible existence of a day-night effeOn the other hand there is also some hint for a seasvariation in these data, especially for recoil electron eneabove 11.5 MeV. While the former would be an indicationfavor of the MSW solution, the latter would favor the just-solution.

Here we call the attention to this interesting feature ofMSW solution, namely, that the expected MSW event rado exhibit a seasonal effect due to the different night dution throughout the year at the experimental site, which leto a seasonal-dependentne regeneration effect in the EarthTaking into account the relative position of the SupKamiokande setup in each period of the year, we calcuthe distribution of the events through the year both forlarge mixing angle~LMA ! and the small mixing angle~SMA! solutions to the SNP. We find that the effect canas large as the one expected in the just-so scenario, ecially in the LMA solution, where it amounts to;10% sea-sonal variation@see Eq.~12!# at the best fit point for the solaneutrino event rates given by Ref.@9#. For the SMA solutionwe find that the magnitude of the seasonal MSW effec

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very small at the best fit point increasing as sin22u increaseswithin the 99% C.L. region. We illustrate this behaviorFigs. 1 and 2 and in Table I.

Let us now describe our calculation. For simplicity, letconsider the two-neutrino mixing case

ne5cosu n11sinu n2 ,nm52sinu n11cosu n2 . ~1!

We have determined the solar neutrino survival probabiPee in the usual way, assuming that the neutrino state aring at the Earth is an incoherent mixture of then1 and n2mass eigenstates.

Pee5Pe1SunP1e

Earth1Pe2SunP2e

Earth, ~2!

where Pe1Sun is the probability that a solar neutrino, that

created asne , leaves the Sun as a mass eigenstaten1, andP1e

Earth is the probability that a neutrino which enters the Eaasn1 arrives at the detector asne . Similar definitions applyto Pe2

Sun andP2eEarth.

The quantityPe1Sun is given, after discarding the oscillatio

terms, as

Pe1Sun512Pe2

Sun51

21S 1

22PLZD cos@2um~r 0!# , ~3!

where PLZ denotes the standard Landau-Zener probabi@10# andum(r 0) is the mixing angle in matter at the neutrinproduction point. In our calculations of the expected evrates we have averaged this probability with respect toproduction point assuming the production point distributigiven in Ref.@11#.

In order to obtainPieEarth we integrate the evolution equa

tion in matter assuming a step-function profile of the Eamatter density. In the notation of Ref.@12#, we obtain, forP2e

Earth512P1eEarth

P2eEarth~F!5~Z sinu!21~W1cosu1W3sinu!2, ~4!

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P.C. de HOLANDAet al. PHYSICAL REVIEW D 60 093010

whereu is the mixing angle in vacuum and the Earth mateffect is included in the formulas forZ, W1, andW3, whichcan be found in Ref.@12#. P2e

Earth depends on the amount oEarth matter traveled by the neutrino in its way to the dettor, or, in other words, on its arrival direction which is usally parametrized in terms of the nadir angle,F, of the sun atthe detector site.

It is very important to realize that the daily range of vartion of the nadir angle depends on the period of the year.a result the quantityP2e

Earth is seasonal dependent. This wiin turn, manifest itself as a seasonal dependence of thepected neutrino event rates. The general expression ofexpected signal in the presence of oscillations at a given tt,Sosc(t), is

Sosc~ t !5E dEnl~En!$se~En!Pee~En ,t !1sx~En!

3@12Pee~En ,t !#%, ~5!

where En is the neutrino energy,l is the neutrino energyspectrum@13# with the latest normalization@14#, se(sx) isthe ne(nx ,x5m,t) interaction cross section in the standamodel @15#, and Pee is the ne survival probability, whichvaries in time through the interval of day and night along

FIG. 1. Ratio of predicted event rate to the SSM predictversus time of the year in Super-Kamiokande for various pointthe SMA solution region of the SNP as labeled. We have normized these three curves to the same yearly averaged eventwhich corresponds to8B flux normalization 0.7 for the best fit poinin Ref. @9#. We also show the expectation in the absence of oslations with 8B flux normalization of 0.47~short dashed line! andthe expected effect for vacuum oscillation solution C in Ref.@6#~dash-dotted curve! together with the 708 Super-Kamiokande dapoints.

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year. The expected signal in the absence of oscillatiSno osccan be obtained from Eq.~5! by substitutingPee51.

The cross sectionsse,x are calculated including radiativcorrections and must be corrected for energy thresholdresolution effects. In the calculation of the expected signais understood that thena-e cross sectionssa(E)(a5e,x)have to be properly corrected to take into account the detor energy resolution and the analysis window for eachperiment. In Super-Kamiokande, the finite energy resolutimplies that themeasuredkinetic energyT of the scatteredelectron is distributed around thetrue kinetic energyT8 ac-cording to a resolution function Res (T,T8) of the form@16#

Res~T,T8!51

A2psexpF2

~T2T8!2

2s2 G , ~6!

where

s5s0AT8/MeV, ~7!

and s050.47 MeV for Super-Kamiokande@7,17#. On theother hand, the distribution of the true kinetic energyT8 foran interacting neutrino of energyEn is dictated by the differ-ential cross sectiondsa(En , T8)/dT8, that we take fromRef. @15#. The kinematic limits are

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FIG. 2. Ratio of predicted event rate to the SSM predictiversus time of the year in Super-Kamiokande for various LMsolutions of the SNP labeled in the figure. These three curvesnormalized to the same yearly averaged event rate correspondia 8B flux normalization 1.45 for the best fit point in Ref.@9#. Wealso show the expectation in the absence of oscillations with8Bflux normalization of 0.47~short dashed line! and the expectedeffect for vacuum oscillation solution C in Ref.@6# ~dash-dottedcurve! together with the 708 Super-Kamiokande data points.

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SEASONAL DEPENDENCE IN THE SOLAR NEUTRINO FLUX PHYSICAL REVIEW D60 093010

0<T8<T8~En!, T8~En!5En

11me/2En. ~8!

For assigned values ofs0 , Tmin , andTmax, the correctedcross sectionsa(En) is defined as

sa~En!5ETmin

TmaxdTE

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T8(En)dT8Res~T,T8!

dsa~En ,T8!

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~9!

Finally, in order to compare our results with the recedata from the Super-Kamiokande Collaboration, we malso include the geometrical seasonal neutrino flux variadue to the variation of the Sun-Earth distance (L'1.531013 cm) arising from the Earth’s orbit eccentricity because the neutrino fluxes in Eq.~5! are yearly averages. Inorder to account for this effect we assume a 1/L2 dependenceof the flux. Notice that the Super-Kamiokande data are psented as ratio of observed events over the expected nuin the standard solar model where this expected numbeevents does not include the geometrical variation. Thusmust compare the experimental points with the predictio

Nosc~ t0 ,Dt !

Nno osc(Dt)5

Et02Dt/2

t01Dt/2

dt@Sosc~ t !/L2~ t !#

DtSno osc~10!

where

L~ t !5F12e cos 2pt

TG ~11!

ande50.0167 is the eccentricity of Earth’s orbit around tSun, andT51 year.

We now turn to our results. In order to study the behavof the seasonal variation we have explored the param

TABLE I. Seasonal variation~in percent! of the ratio of pre-dicted event rate in various oscillation scenarios to the SSM pretion.

Point Dm2(eV2) sin2(2u) Var ~%!

No-oscillation 6

MSW SMA

Best Fit Point 531026 3.531023 6831026 831023 10831026 1.231022 20

MSW LMA

Best Fit Point 1.631025 0.57 101.31025 0.6 223.231025 0.6 9

Vacuum Solutions

C 4.4310210 0.93 15D 6.4310210 1 12A 6.5310211 0.7 9

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space around the small and large mixing angle solutioSMA and LMA, respectively. We find that depending on tvalues of the mass and mixing angle, one may get a sizeenhancement of the geometrical effect.

In Fig. 1 we present the expected event numbers inrecoil electron energy rangeTmin511.5 MeV up to Tmax520 MeV, plotted versus the period of the year for differepoints in the SMA solution region of the SNP divided by thBahcall-Basu-Pinsonneault 1998~BBP98! standard solarmodel ~SSM! predictions in the absence of neutrino convesions @14#. We plot the expected behavior for three pointhe best fit point obtained by@9# with an arbitrary8B flux,Dm255.31026 eV2 and sin22u53.531023, a point insidethe 99% confidence level allowed region withDm25831026 eV2 and sin22u5831023 and a near point withDm25831026 eV2 and sin22u51.231022. We have nor-malized these three curves to the same yearly averaged erate. This corresponds to a8B flux normalization 0.7 for thebest fit point as obtained from the global fit with free8B fluxin Ref. @9#. For the sake of comparison we also plot texpected behavior in the absence of oscillations with8B fluxnormalization of 0.47 as well as the best fit point for tvacuum solution C of Ref.@6#. As seen in the figure theseasonal effect is comparable to the expectation in thesence of oscillation at the best fit point of the SMA solutiand it increases as the mixing angle increases. In Tableshow the seasonal variation~in percent! defined as

Var[2Rmax2Rmin

Rmax1Rmin~12!

for the different MSW and vacuum solutions of the SNwhereR(t)5Nosc(t)/NSSM. We find that for the SMA solu-tion the effect increases as one increases sin22u. For examplefor sin22u50.008, still within the 99 % C.L. allowed regionit reaches 10% and for sin22u50.012 it gets to be as large a20%. Of course, since the seasonal effect is induced byvariation of the regeneration in the Earth along the year,effect is large only in the parameter region where the dnight effect is not negligible, which corresponds to largmixing angle values@18#. Note that in the SMA region thepoints we have chosen in order to illustrate the possible ssonal variation in the MSW picture are consistent with tmeasured yearly average day-night asymmetry.

Now we turn to the LMA solution of the SNP where theffects are potentially larger. Our results for this casedisplayed in Fig. 2. Again, we plot the expected behaviorthree characteristic points: the best fit point obtained by@9#with an arbitrary 8B flux (sin22u50.57,Dm251.631025 eV2), a point inside the 99% confidence level alowed region withDm25131025 eV2 and sin22u50.6 anda point inside the allowed region where the expected averday-night asymmetry is smaller,Dm253.231025 eV2 andsin22u50.6. We have normalized these three curves tosame yearly averaged event rate. This corresponds to a8Bflux normalization 1.45 for the best fit point as obtained frothe global fit with free8B flux in Ref. @9#. We also plot theexpected behavior in the absence of oscillations with8B fluxnormalization of 0.47 and the best fit vacuum solution

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from Ref. @6#. In Table I we show the variation~in percent!corresponding to these points. As seen in the table the eat the best fit point of the LMA solution~10%! is comparablewith the corresponding effect in some of the favored vacuoscillation solutions. In the LMA solution region the sesonal variation is very mildly dependent on the mixing anwhile presents an oscillatory variation withDm2. We mustbear in mind, however, that in the lowerDm2 part of theLMA solution region, the expected yearly average day-niasymmetry is in conflict with the existing data@18#. Finallylet us comment on the effect of an enhanced hep neutflux as suggested in Ref.@19# in order to account forthe recent Super-Kamiokande measurements of the enspectrum. We find that even with large hep enhancemfactors of 20 or more, the expected modifications of oursults near the best fit points both for the SMA and LMA asmall.

To summarize, we have shown that MSW solutions ofsolar neutrino problem can lead to sizeable seasonal de

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dence of the event rates at the Super-Kamiokande detectthe large mixing angle region and this should be taken iaccount in refined fits of the data where the day-night anasis is also performed. The MSW seasonal effect is correlawith the day-night asymmetry@18# and may potentially beuseful in order to pinpoint the underlying mechanismvolved in the explanation of the solar neutrino anomaly, dcriminating between different solutions. For example, tnon-observation of the day-night effect and the confirmatof seasonal-dependent rates would provide an indicationthe just-so picture. Conversely, a possible confirmation oseasonal dependence accompanied by the day-night ewould point towards a LMA MSW-type solution.

We are grateful to E. Akhmedov and C. Yanagisawauseful comments. This work was supported by SpanDGICYT under Grant No. PB95-1077, by the EuropeUnion TMR network ERBFMRXCT960090. P. C. dHolanda was supported by FAPESP and PRONEX~Brazil!.

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