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SOLAR NEUTRINO DATA AND ITS IMPLICATIONS

arXiv:hep-ph/9203213v1 17 Mar 1992YCTP-P8-92Revised March 1992SOLAR NEUTRINO DATA AND ITS IMPLICATIONSEvalyn Gates1, Lawrence Krauss2 and Martin White3Center for Theoretical Physics, Sloane LaboratoryYale University, New Haven, CT 06511, USAAbstractThe complete and concurrent Homestake and Kamiokande solar neutrinodata sets (including backgrounds), when compared to detailed model predic-tions, provide no unambiguous indication of the solution to the solar neutrinoproblem. All neutrino-based solutions, including time-varying models, pro-vide reasonable fits to both the 3 year concurrent data and the full 20 yeardata set.

A simple constant B neutrino flux reduction is ruled out at greaterthan the 4σ level for both data sets. While such a flux reduction providesa marginal fit to the unweighted averages of the concurrent data, it doesnot provide a good fit to the average of the full 20 year sample.

Galliumexperiments may not be able to distinguish between the currently allowedneutrino-based possibilities.1Supported in part by the DOE. Address after October 1: Department of Physics,University of Chicago.2Also Department of Astronomy.

Research supported in part by the DOE, the NSFand the TNRLC. Bitnet: KRAUSS@YALEHEP3Research supported in part by a grant from the TNRLC.

Address after September 1:Center for Particle Astrophysics, University of California, Berkeley.1

Perhaps at no time in the past 20 years has there been more interest inthe solar neutrino problem than at the present moment. The apparent deficitof high energy solar neutrinos observed by the Homestake Solar Neutrino De-tector over two decades [1] has now been confirmed by the Kamiokande largeunderground water Cerenkov detector [2].

Gallium detectors are beginningto come online and the SAGE group has recently published their first results[22] which seem to indicate a neutrino deficit which cannot be explained bysolar physics (for a brief discussion see [21]). Other detectors are approvedor are in the planning stages, and there is hope that a solution to the solarneutrino problem may be at hand.

At the same time a growing number oftheoretical neutrino-based “solutions” have been proposed. Heading the listappears to be the MSW solution [5, 6], which, in a restricted but reasonablerange of neutrino masses and mixing angles, allows significant reduction inthe neutrino signal to be observed.

An alternative solution involves a largeneutrino magnetic moment, either diagonal or transitional, which causes neu-trinos to oscillate into “sterile” partners while traversing the magnetic fieldof the sun [7, 8]. While this seems less theoretically compelling, especiallyin view of the large neutrino magnetic moments required, it has the distinctadvantage of allowing not only the neutrino signal to be time varying overthe solar cycle, but also allows for a different time variation to be observedin different detectors!

[9].It may seem a priori that the simplest solution of the original Cl solarneutrino problem resides in the solar model itself, namely that fewer high en-ergy neutrinos are created in the sun than the standard solar model suggests.It is important to determine if this possibility can be ruled out, although itseems increasingly difficult to accommodate, especially in light of the newSAGE results [22]. Also, astrophysical mechanisms which reduce the high en-ergy neutrino flux are now not supported by any other solar measurements(most importantly the p-mode fine structure [23, 1]).

Meanwhile, severalstudies incorporating recent Cl data into the 20 year observations provide2

very tempting, if not compelling, evidence of time variations in the Cl signal[11, 17, 13] which may be correlated, in some yet to be determined way, withthe solar cycle. On the other hand, the Kamiokande data appears naivelyto show no such time variation.

The presence, for the first time, of two dif-ferent data sets for the solar neutrino signal should allow a number of finertests of solar neutrino models to be made (see e.g. [2]).

Surprisingly, how-ever, rarely have all the data been used. For example, analyses have beenperformed comparing the Homestake 20 year “average” signal with the av-eraged Kamiokande 3 year signal.

It is not clear that such a procedure iscorrect. Until we have a better idea of what is at the root of the solar neu-trino problem, we can make no a priori claims about what the Kamiokandesignal would have been if the detector had also taken data during the 20years Homestake was operational, especially given the apparent variations inthe Cl data during this period.

All of the data points from both experimentsshould be exploited, and the error bars examined. For guidance on how totreat the entire data sets one can first analyse the data during the period inwhich the two detectors were both running concurrently.

It is only duringthis time that we have a direct independent check on the Cl data, and cancheck for consistency between the data sets. One might then be guided onhow to use all the data to test various hypotheses.

This is the spirit of thefollowing work. We have utilized the entire Homestake and Kamiokande datasets, concentrating first on the concurrent data sets and then on all the data,in order to investigate the range of models which may or may not fit thedata.

We have carried out extensive numerical model calculations, in whichneutrinos are propagated, with complete phase information, through muchof the sun, in order to estimate the flux in various neutrino species at theearth’s surface. We have also used realistic models of detector sensitivity inorder to turn fluxes into detection rates.We emphasize that using the concurrent Kamiokande data to “check” theCl data is important beyond the strict question of whether or not any solar3

cycle time variation exists. It allows us to understand how best to treat thefull 20 year Cl data to explore solutions to the solar neutrino problem.

Wedisplay in figure 1 (a) and (b) the two full data sets, and the concurrentdata sets. Both data sets are normalized to the Standard Solar model [1]predictions (see section (3)).While quoted averages of the two data sets appear at first sight to differ,when the full data sets are displayed this issue is less clear.

The Cl signalclearly has much more jitter, with several apparently anomalously low points,but aside from this one might not be surprised if told that all data came froma single detector. This may suggest that a simple, energy independent deficitof B neutrinos could be consistent with all of the data.

To properly explorethis possibility, as well as the possibility that the solar neutrino deficit isneutrino related, a more quantitative approach is required.2. Neutrino flux at the earthThe neutrino spectrum predicted by the Standard Solar Model (SSM) isdescribed in detail by Bahcall in [1].

The dominant neutrino flux, that due tothe pp reaction in the sun, with energies less than 0.42 MeV, is unobservablein both the Cl and Kamiokande detectors, due to their thresholds.Thecomponent of the flux which gives the dominant contribution to the Cl signal,and the entire contribution to Kamiokande is the high energy 8B continuousspectrum (8B →7 Be∗+ e+ + νe), with neutrino energies up to 15MeV and atotal predicted flux at the earth of (5.8±2.2)×106cm−2s−1 (“3σ” theoreticalerror). The only other component of the neutrino spectrum contributing tothe Cl signal at greater than the 5% level are the Be neutrinos (7Be+e−→7Li + νe), with fixed energy 0.862 MeV and a predicted flux of 4.7 ± .7(3σ) ×109cm−2s−1.There are two ways one might expect to alter these predicted fluxes.

Firstone might lower the overall flux by a fixed amount by postulating some newsolar physics.For example, if the core temperature is lowered compared4

to the SSM, the B signal can be significantly reduced (such a temperaturereduction is the aim of many non-standard solar models, e.g. see [1]).We have incorporated these possibilities in our analysis by treating thenet B flux as a free parameter in one set of runs, and examining the goodnessof fit with the combined data sets as this parameter is varied compared tothe SSM.

While this is a very simplistic “non-standard solar model” we canuse it to perform straightforward statistical tests of how well the data is fitby models aiming at such a B flux reduction.The other possibility is that the origin of the solar neutrino problem liesin the properties of neutrinos themselves. If neutrinos have non-zero masseigenstates which do not coincide with weak eigenstates, neutrino propa-gation will lead to oscillations between the different weak states, namelybetween electron, muon, and tau neutrinos.

Since the Cl detector is sensitiveonly to electron neutrinos, while the Kamiokande water detector is sensitivepredominantly to electron neutrinos, such oscillations could have the possibil-ity of reducing the observed signal in both detectors. Moreover, the presenceof matter can enhance the oscillations between neutrino species [5] due to thepresence of level crossings which occur as the background electron densityvaries.

If one supplements neutrino masses with large magnetic moments,which in general need not be diagonal in the weak basis, then another pos-sibility arises. Magnetic fields in the sun could cause oscillations betweenleft and right handed neutrino states, with or without induced level cross-ings [7].

In general, left-right mixing can allow neutrino states to oscillateinto antineutrino states, unlike the pure MSW mechanism [8]. In any case,as long as the right handed states have suppressed interaction rates in thedetectors, this can reduce the observed neutrino signal.

Moreover, it allowsfor a possible correlation with the solar cycle, although the required neutrinomagnetic moments, at least for currently envisaged magnetic field strengthsin the sun, are large enough to cause other potential astrophysical problems[14]. Finally, in the most general case, both effects may be operational with5

the different factors dominating in different regimes of mass, mixing angle,and magnetic field space [8]. This allows independent time variations to beobserved in the two detectors [9], and it is this general case which we shallconsider here.We followed explicitly the propagation of neutrinos through the sun bynumerically integrating the Hamiltonian evolution equation for neutrinosthrough matter for a two generation model with Majorana-type transitionmagnetic moment and offdiagonal mass terms [8, 9, 15, 16].i ddtνeνµ¯νe¯νµ= Hνeνµ¯νe¯νµ(1)The Hamiltonian for the system [8] is given byH =ae∆m24Eν sin 2θ0µB∆m24Eν sin 2θ∆m22Eν cos 2θ + aµ−µB00−µB−ae∆m24Eν sin 2θµB0∆m24Eν sin 2θ∆m22Eν cos 2θ −aµ(2)where B is the magnetic field, ae = GF(2Ne−Nn)/√2 and aµ = GF(−Nn)/√2with Ne, Nn the electron and neutron densities as a function of radius in thesolar interior.

We used the following fit to the electron and neutron densitiesin the standard model sun [1]Ne =(2.45 × 1026 exp (−10.54x)0.2

The free parameters in the calculation are the neutrino energy E, mass-squared difference ∆m2, vacuum mixing angle sin2(2θ) and Zeeman energyµB: the product of the transition magnetic moment and solar magnetic field.For reasons of simplicity we assumed this to be uniform over the radiationand convection zones in the sun, falling sharply to zero at the exterior.The evolution in the interior was performed using a Runge-Kutta algo-rithm with adaptive step size control [15] in double precision arithmetic. Onorder of 105 steps were taken for the higher mass gaps, the neutrinos beingevolved from just before the resonance [16]√2GFNeres = ∆m22E cos(2θ)(5)to the edge of the Sun.In the exterior of the sun, where the magnetic field is assumed to be zero,the neutrino and anti-neutrino sectors decouple and the vacuum oscillationscan be computed using standard analytic formulae (see [16]).Since eachdetector signal averages over a period of at least 2 months (though not nec-essarily weighting times evenly) we modelled the motion of the Earth in asimple way by averaging the vacuum oscillations over an Earth-Sun distanceof d(1 −e/2) to d(1 + e/2), where the semi-major axis is d = 1.496 × 108kmand the eccentricity of the Earth’s orbit is e = 0.0167.

This corresponds tothe variation in the Earth-Sun distance over 3 months.The general form of the propagation matrix for neutrinos allows for theconversion of electron neutrinos into muon neutrinos and also into muonand electron antineutrinos. The latter conversion can occur in two steps,either by a magnetic moment induced oscillation followed by an MSW typeoscillation, or the reverse.

Assuming initially electron neutrinos are emitted,the probability Pi of finding each of the 4 species at the Earth was computedfor a grid of the 4 parameters. For the continuum spectra we calculated theprobabilities for 30 energies ranging from 0.5MeV to 15MeV in 0.5MeVsteps, and we also calculated the probabilities at 0.862MeV and 1.442MeV7

corresponding to the 7Be and pep neutrino lines respectively. The mass gap,∆m2, ranged from 10−5eV 2 to 10−8eV 2; for higher mass gaps the Zeemanenergy plays no role and pure MSW/vacuum mixing, results.

This case hasbeen well studied and the higher mass gaps in the so called “adiabatic regime”may already be ruled out by experiment [2].The vacuum mixing angle,sin2(2θ), ranged from 0.01 to 1.00 and the Zeeman energies, µB, from 0 to 5×10−10µBkG. The best limit on neutrino transition moments is astrophysical,coming from the luminosity of red giant stars before and after the He flash[14],µ < 3 × 10−12µB (3σ)(6)and the best lab limits (from ¯νe −e scattering) are [20]|κe| < 4 × 10−10, |κµ| < 10−9;µi = κiµB(7)so the larger Zeeman energies require enormous fields in the solar interior.The expected event rates in the detectors were calculated by convolvingknown neutrino cross sections with published detector efficiencies [1, 17, 18].3.

The DataThe 90 Homestake data points between the years 1970 and 1991 were ob-tained with about 2 months of integration time per point. The time shownfor each Homestake data point in figure 1 is the mean time of productionof the radioactive Argon atoms (see [1] for a description).

For each pointthe experiment reported an upper limit on the production rate, a lower limiton the rate, and the mean value of the rate, all determined by a maximumlikelihood fit to the data [4]. The errors about the mean were generally sym-metric, except in the case where the lower limit would have become negative,in which case the reported error bars were quoted as half the difference be-tween the upper limit and zero, and were thus sometimes artificially small.This suppression of the errors would artificially increase the weighting ofthese points in any fit to the data.

In order to remove this effect, we utilized8

fully symmetric error bars on all points. The size of 1σ error bars was fixedto be the difference between the reported upper limit and the mean valuefor each point.

It has been calculated that .08 ± .03 argon atoms/day areproduced [1] by the (muon induced) background. In determining the aver-age Homestake signal it is appropriate to subtract this background after theaverage Ar rate has been computed from the total data set, and add errorsin quadrature.

When performing a point by point fit of theory to the data,however, it is appropriate to subtract this background from each data pointand add its uncertainty to the rate uncertainty for each point in quadrature.4Figure 1 displays the values divided by the standard solar model (SSM) pre-dicted rate. Because of the unusually small errors on many of the pointswith small rates, the treatment of errors in the Homestake experiment hasbeen an issue of some debate.

In particular the “error” determined by themaximum likelihood fit is not a Gaussian 1σ error for points with small num-bers of counts (N ≤5) and the use of a χ2 analysis will not weight thesepoints correctly (see e.g. [3]).

To consider the effect of this, for the analysisof the non-standard solar models and the MSW neutrino model we also usedthe method of [3] to analyze the Homestake data, while still using χ2 for theKamiokande data.The Kamiokande data is more straightforward. Over the period 1987-1990, five data points have been obtained, based on real time measurementsof the directional solar neutrino signal, averaged over a period of severalmonths.

These data points, along with errors, were presented as a fractionof the rate predicted by the SSM [2], and, as shown in figure 1, were used4The average rate (which converts to .26 ± .04 SSM) quoted by the Homestake groupcomes from a maximum likelihood fit of N=61 runs to a constant background plus 1decaying species.The division of the counts into (counter) background and signal isdifferent if the runs are analysed separately or collectively, the (counter) background in arun by run analysis being quite variable. We calculate our average rate as the average ofthe values quoted per run, for N=90 runs.

Note that .26 is bracketed by our weighted andunweighted values. The larger error, .04, is consistent with the smaller number of runsanalysed by the Homestake group.9

directly in this analysis.Finally, we decided not to additionally weight the Kamiokande and Cldata points in terms of the length of the measuring interval associated witheach point. In the first place, longer runs in the Cl experiment do not meanmore data.

Because the produced Ar atoms decay with a 35 day half lifethey will eventually reach an equilibrium abundance after several monthsexposure.Secondly, the small error bars on the Kamiokande data pointspresumably reflect the longer exposure times for each point in this exper-iment, and thus measuring time will in this case naturally be taken intoaccount in any weighting by errors of the data.4. AnalysisIn an effort to determine how the current solar neutrino data constrainsthe various possible models discussed in (2) we compared the predicted sig-nals in both detectors to the data by means of a χ2 goodness-of-fit procedure.For each model we computed the predicted signal over a range of model pa-rameters, and for each combination calculated the value of χ2 for the signalcompared to the data.

We then examined the parameter space for χ2 valuescorresponding to confidence levels of 68% and 95%.The different models we considered are:(a) Non-standard solar model: B flux reduction(b) Non-standard solar model: (B+Be) flux reduction(c) Neutrino Mass model (no magnetic moments-constant flux)(d) Neutrino Mass model (with magnetic moments-variable flux)In case (c), each combination of the neutrino mass-squared difference andvacuum mixing angle produced a constant fit to each of the detector signals.In case (d) in addition to these parameters, the quantity µB was assumed to10

have the form µB = A+Cf(t). In this case f(t) was set to either cos(φ+kt),where φ and k were determined from sunspot data, or to a sawtooth functionof unit amplitude with a net period equal to the solar cycle and the positionof the cusp given by time τ.This latter model was chosen based on anearlier suggestion by Bahcall and Press [11] that the neutrino time variationcould be well described by such a function.

We considered τ = 8.05 years,based on their fit to the Ar data, and τ = 6.65 years based on their fit tosunspot data. Thus in case (d) there are two additional parameters, A andC, involved.

The translation of χ2 values into confidence levels depends uponnumber of degrees of freedom. In determining the goodness of fit of models(a,b,c) with various sets of parameters, the number of degrees of freedom wasset equal to the number of data points, since the model predictions are fixedonce the parameters are fixed, and no parameter in this test is minimized tofit the data.

In model (d) the number of degrees of freedom was reduced by2 since A and C were fit to the data before goodness of fit was evaluated.Our results are displayed in Tables 2,3 and figures 2 - 7.The tableslist the “best fit” (i.e. smallest χ2) model parameters along with degrees offreedom (df)5 for fits to:(i) the concurrent 3 year data, and the averaged 3 year data,(ii) the complete 20 year data set, and the averaged 20 year dataWe do not place much significance on the actual value of the best fit param-eters, rather we would emphasize the regions in ∆m2 −sin2(2θ) space forwhich the model fits the data at a given confidence level.Let us review the fits to each of the data sets in turn.

(i) Concurrent data set: In spite the apparent similarity of the two sig-nals during this period, the simplest apparent resolution of the solar neutrino5Note the number of degrees of freedom to be used for a goodness-of-fit and the num-ber quoted for a “best fit” are not the same, the latter being smaller by the number ofparameters varied in the fit.11

problem, that obtained by reducing the B neutrino flux alone, is ruled outat greater than the 4σ level based on a comparison with the weighted datapoints (including a Be reduction by the same amount allows a fit at the 3σlevel). This discrepancy is because the small error bars on the low Homestakepoints heavily skew any fit.

The mean value of the Homestake data duringthis period rises from .25 to .36 of the SSM prediction if each point is equallyweighted and the fit to a non-standard solar model improves dramatically.In this case, if the SSM B flux is reduced by a constant factor, the fit to theunweighted averages is acceptable over a small range at the 99% confidencelevel. Whether or not to ignore the heavy weighting of the apparent anoma-lously low Cl data points therefore becomes an important issue if one is toclaim non-standard solar models are ruled out by the combination of Cl andKamiokande data, at least during the period in which the data was takenconcurrently.

If the procedure of [3] is used the non-standard solar modeljust fits the concurrent data at the 99% confidence level6, with the favoredboron flux reduction at 37% of the SSM.If no model fit the complete fully weighted concurrent data sets, thiswould provide strong evidence in favor of the assumption that the jitter inthe Cl signal precludes its use directly in constraining models, and mightprovide motivation for ignoring the quoted error bars on the data. As can beseen, however, all the models with neutrino masses, including those with atime variability, provide reasonable fits to the data (at 95% confidence level).The range of fit of the MSW model to this concurrent sample is shown infigure 2 (a), along with the claimed fit to the 20 year averaged data byBahcall and Bethe [6] (solid line).

We see that the Bahcall and Bethe linepasses through the arm of the 95% confidence level region. If the unweightedaverages of the Homestake data sets and the Kamiokande average rate are6The procedure of [3] makes use of the likelihood ratio test in which the test statisticis χ2 distributed in the limit of a large number of data points.

In applying this test to theconcurrent Homestake data we should bear in mind that there are only 20 data points.12

compared to the MSW prediction, the allowed regions are shown in figure 2(b). Notice that the fit to the unweighted average is good at the 68% levelover a range of parameters and the 68% region coincides with the Bahcalland Bethe best fit line.

An almost identical region is obtained for the fit tothe weighted averages of the data, suggesting the poorer fit in the case ofthe individual points is due to “jitter” in the data. If the analysis is doneusing the method of [3] the MSW model still fits, though the goodness-of-fitis slightly worse than for the case of the straightforward χ2 fit.We now switch to the time dependent fits, involving a non-zero transitionmagnetic moment.

The “best fit” magnetic field peak Zeeman energy hasa value of 4.6 × 10−10µBkG for the cosine and 4 −5 × 10−10µBkG for thesawtooth fits, which are essentially as good as the MSW fits. Because the20 year data provides more compelling evidence of time variability, we alsoinvestigated the goodness of fit of the 20 year “best fit” parameters to the 3year concurrent set in the time varying models.

The “best fit” values differsomewhat from the best fit to the 3 year data, but they are still comparablygood.This indicates that there is no evidence from the concurrent dataagainst the same time variation inferred from the 20 year Cl sample. (ii) 20 year data set: A non-standard solar model doesn’t fit the full datamuch worse or much better than the 3 year data.

The disagreement withthe complete weighted data sample, allowing only the B flux to be reduced(in this case to 0.1 SSM! ), is still at ≈4.5σ.

Now however that now thedisagreement with the unweighted average rate (requiring a flux reduction to0.15 SSM) is comparably bad. Allowing the Be flux to change as well reducesthe disagreement, but the fit to the unweighted average in this case is at bestonly marginal (99% confidence level).The procedure of [3] decreases thegoodness-of-fit dramatically, with the best fit (at 20% of the SSM boronflux) ruled out at > 5σ.The MSW model fit to the 20 year data is shown in figures 3 and 4.13

Notice the line of best fit is shifted slightly from the Bahcall and Bethe linedue to the inclusion of the latest Homestake data but the fit is still good atthe 95% confidence level. The fits to the weighted and unweighted averagesare good (better than 68%) as one might expect.

If the method of [3] isused to compute the χ2, thus taking account of the Poisson statistics of thelow points, the best fit is only acceptable at the ≈5σ level! The fact thatboth the non-standard solar model and MSW fits, in which the predictionis a constant, are worse using the method of [3] than using a normal χ2procedure suggests that this latter method is much more sensitive to “jitter”in the data.Since it is perhaps the simplest and most elegant of the proposed neutrinobased “solutions” to the solar neutrino problem we feel the MSW modeldeserves a closer inspection.

In this regard we have developed a new wayof presenting the comparison between theory and observation. For the 680(∆m2, sin2 2θ) parameter pairs we calculated in our study, we display in figure5 a plot of the MSW predictions for Homestake vs Kamiokande.

While apriori one might expect such a plot to “fill” much of the plane, one can seethat the allowed region is in fact a narrow band passing from bottom leftto top right. This behaviour is due to the fact that high energy 8B electronneutrinos make up most of the signal for both detectors, leading to a strongcorrelation in the signals for an energy dependent νe flux reduction.

(We thusexpect that adding the neglected contributions from 15O and hep neutrinos tothe Homestake signal will broaden this band slightly.) Still the narrowness ofthe band is a surprising indication of the strong constraints on the predictionsof the MSW solution.

Also shown in figure 5 are the averages of the actualrates seen in the detectors. In this way one can obtain a clear and immediategraphical picture of how well the MSW solution as a whole can reproduce theobserved averages.

As can be seen, the fair overlap between (the constrained)theoretical phase space and the observations is suggestive.The low points in the pre-1987 sample can be well accommodated, as14

has been previously noticed, by a time varying neutrino signal.In addi-tion, as stressed earlier, resonant spin-flavor transitions also allow “arbi-trary” Kamiokande time variation for a given variability in the Cl data. Asexpected, therefore, we find that the complete data sample can be well fitover a wide range of parameter space by a time varying magnetic field cou-pled with a large neutrino transition magnetic moment.

Shown in figure 6(a) and (b) are the regions of mass-mixing angle space allowed at the 68 and95% confidence levels when the magnetic field time variation is fixed at thevalue which provides the minimal χ2 fit to the data for a (a) cosine or (b)sawtooth time dependence. (The actual region of parameter space allowedin this case is a 4 dimensional space in mass, mixing angle, and magneticfield time variation – difficult to draw, but whose boundary in the extremelimit of zero magnetic field splitting would reduce to the MSW plot alreadypresented.) The cosine fit to the data at this optimum magnetic field valueis obviously better than the zero field MSW fit, while the sawtooth fit is evenbroader, and slightly better than the cosine fit at the optimum magnetic fieldvalue.The apparent jitter and/or the occurrence of anomalously low data pointsin the Cl data sample, which dominates over the Kamiokande sample in the20 year fits (by about 4 to 1 in the χ2 determinations), cannot be dismissedbased purely on statistical grounds alone.

We have investigated whether onemight be forced to ignore or rescale the error bars in order to reduce thiseffect by examining the variance of both the weighted and unweighted Cl 20year samples. The mean value of the of the Cl signal for the complete 20year weighted sample is 1.70 ± .22 SNU.

This is significantly smaller thanthe unweighted average of 2.21 ± .24 SNU. Nevertheless, the χ2 per degreeof freedom for this weighted average is 1.07.

This indicates that there is nonecessity to rescale errors to account for the variance of the sample from themean. Alternatively, the unweighted sample has a mean variance per pointof 1.7 SNU.

This is comparable to the error per point in the weighted sample,15

indicating again that there is no evidence that the errors are skewed in anyway.Finally we stress a somewhat non-intuitive result. In the 20 year sample,the Cl data clearly dominates in any fit.

One may feel that comparing modelpredictions to average values may alleviate this problem by treating the twodata sets with equal weight.However, the relative errors determined forthe Homestake mean values are small enough so that the Homestake resultdominates the fit to average values (weighted or unweighted) more than itdoes a fit to the complete sample. Thus, if the Cl data is suspect, for anyreason, using average values rather than the full data set will only exacerbatethis problem.One way in which we might hope to proceed further in distinguishing be-tween models is to examine the predictions for the Ga solar neutrino experi-ments (SAGE and GALLEX collaborations) which are currently beginning torun.

Estimates of gallium rates predicted by the models we have consideredare summarized in the last two columns of Table 2. For a given model, wehave computed the range of neutrino rates that would be seen in a Ga-baseddetector for the region of parameter space not already excluded at the 68%and 95% confidence levels by the present Homestake and Kamiokande data.The time dependence of the predicted Gallium rates for the time-dependentmodels varied widely (including no significant time variation) for equally al-lowed parameter sets.Thus measuring the time dependence of the ratesin Gallium detectors might help further constrain these models, althoughif uncertainties in the data are on the same order as the Cl data, a clearmeasurement of time dependence is unlikely in the short term.

Moreover, anobservation of no time variation in the Gallium detectors would once againnot provide definitive evidence against time variation in the Cl signal. Inthe context of neutrino based models then the SAGE result, (20 ± 38) SNU,is perhaps the least enlightening result one could obtain from a theoreticalpoint of view.16

Kamiokande itself now provides another constraint on resonant spin-flavorconversion models.Electron neutrinos can be converted to electron anti-neutrinos in the sun, and these contribute to the isotropic background signalin the Kamiokande detector. Thus, the flat background of isotropic eventsseen by the Kamiokande detector can place a limit on the flux of electronanti-neutrinos [9].

Although a careful analysis of the data in this regard hasnot yet been performed, estimates of the flux of electron anti-neutrinos forneutrino energies greater than or equal to 10.6 MeV for the time period June1988 through April 1989 are less than approximately 10% of the expectedelectron neutrino flux predicted by the SSM [19]. For the models discussed inthis paper, the predicted electron anti-neutrino fluxes ranged from 0 to 30%of the SSM νe flux.

Figure 7 outlines regions of parameter space excluded forvarious flux limits, for Zeeman energies of 2.0 × 10−10 and 5.0 × 10−10µBkGrespectively. (Indicative of average and peak Zeeman energy values whichappear in the best fit solutions.

)Note that some regions favored by thetime varying models are eliminated by the 10% -of-background cut, but noneof the time varying models are completely eliminated on the basis of thisconstraint alone. As the energy threshold for the Kamiokande backgroundsubtraction is reduced, more of the parameter space for magnetic momentinduced oscillations can be probed.

However, it is worth noting that ourresults suggest that none of the present “allowed regions” for the time varyingmodels would be eliminated even if a background cut at the 5% level weremade. It is possible that the SNO heavy water detector may eventually beable to distinguish the antineutrino signal more clearly from the neutrinosignal, and thus could further improve these bounds.5.

ResultsFor convenience we summarize the above analysis and restate the mainresults:17

1. Non-standard solar models which result in a reduced boron flux areruled out, for the concurrent weighted data sample, at the 4σ confidencelevel.

This limit is basically unchanged when the rest of the Cl data istaken into account, though the required flux reduction is more extreme.If the unweighted Cl average signal is utilized instead, this simplest non-standard solar model fits at the 98% confidence level for the concurrentdata sample. In this case, however, the fit to the unweighted averageof the full 20 year sample is incompatible at the ≈4 −5σ level, dueto the low long-term Homestake average.

The SAGE results now alsoappear to argue against this possibility.2. The MSW neutrino mass solution of the solar neutrino model overmuch of the range claimed by Bahcall and Bethe fits the concurrentand 20yr weighted data at only the 95% confidence level.

We have nostatistical evidence that the error bars in the Cl data are anomalous,but if the unweighted mean is utilized instead, the MSW fits improvesignificantly. This suggests the jitter in the Homestake data may be thecause of the higher χ2/dof.

On a Homestake vs. Kamiokande plot theMSW prediction appears as a thin band which overlaps the averageddata. In this way, the agreement between theory and averaged data ismore easily pictured.3.

Models with resonant spin-flavor conversion due to a varying magneticfield in the sun fit the data with a confidence level which is at leastcomparable to the MSW fits – even for the 3 year concurrent samplein which no time variation in the Kamiokande signal is obvious. As ex-pected, the time-varying models provide acceptable fits to the completeweighted data set much more broadly than the MSW models do, andin the case of a sawtooth time-dependence the best fit is also greatlyimproved.The maximum Zeeman splitting needed in these cases israther large, of order 2 −5 × 10−10µBkG.18

4. Most neutrino based solutions to the solar neutrino problem not ex-cluded at the 95% confidence level predict roughly comparable rates inGa, between 5-65 SNU.

Non-standard solar models which are not ex-cluded predict rates greater than 90 SNU. Hence, Ga can decisively ruleout non-standard solar models, but cannot distinguish well betweenneutrino based solutions.Acceptable time-varying models predict awide range of possible time variation in Gallium, including almost noobservable variation.5.

Kamiokande can restrict the allowed parameter range for spin-flavorconversion models, and already rules out ∆m2 in the range 10−8 −10−7eV 2, for mixing angles greater than sin2(2θ) ∼0.3.This limitcomes from the isotropic background in the experiment and will im-prove with time. The SNO detector might improve these further.6.

ConclusionsThe Kamiokande experiment can provide a useful check on the Homes-take experiment, and the combined data from both experiments during theirconcurrent running is consistent with a wide variety of models. Unfortu-nately, however, the specifics of which model and what parameters appearto be favored depend upon how one treats the data, so that no categoricalconclusions can yet be made.Future experiments at Kamiokande and with Ga may not allow muchfiner distinctions between neutrino-based models to be made, but they coulddefinitively rule out non-standard solar model based solutions of the solarneutrino problem.

At this point 20 years of experiments have at least firmlyestablished the existence of the solar neutrino problem and pointed to newmicrophysics as the likely solution. To gain the information necessary tocompletely resolve this issue it will be necessary to measure the solar neu-trino spectrum itself.If neutrino mixing is indeed the cause of the solar19

neutrino problem then a knowledge of which energies are most suppressedwould give us a better handle on the underlying mechanism and parameters(for example in simple MSW mixing, in the regions considered here, lower-ing ∆m2 for a given mixing angle lowers the threshold energy below whichνe →νx conversion takes place).Experiments with this goal in mind (i.e. [24, 25]) are important to pursue.In this way a new window on physics at scales beyond those accessible atpresent accelerators may be fully explored.We thank Ken Lande for providing us with the complete sets of Chlorineneutrino data and for useful discussions on both the Cl and Ga experiments,and M. Smith for informing us of the work of Filippone.We also thankC.

Baltay for useful discussions, P. Langacker for helpful advice on errorhandling, and D. Gelernter and D. Kaminsky of the Linda group of theDepartment of Computer Science at Yale for running our evolution code ontheir complex.20

References[1] J. N. Bahcall, “Neutrino Astrophysics”, Cambridge University Press(1989); R. Davis, Jr., in “Proceedings of the Seventh Workshop onGrand Unification”, Toyama, 1986, p.237, ed J. Arafune, World Sci-entific, (1986)[2] K. Hirata et al., Phys. Rev.

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ExperimentAveraging MethodAverageKamiokande:0.4600 ± 0.0781Homestake:20 year weighted:0.2153 ± 0.028420 year unweighted:0.2799 ± 0.0309concurrent weighted:0.2475 ± 0.0436concurrent unweighted:0.3602 ± 0.0528Table 1: Average values for solar neutrino data23

Modelχ2 (d.f. )Parameters∗Ga(68%)Ga (95%)Concurrent Data:MSW32.6(23)1.58,0.25,—,—5-56Cosine30.9(21)1.26,0.10,2.3,2.35-66Sawtooth (6.65)31.7(21)0.25,0.20,2.0,2.05-66Sawtooth (8.05)31.0(21)0.16,0.45,2.4,2.45-66Cos (20yr)31.7(23)1.58,0.20Saw (20yr-6.65)32.1(23)1.58,0.15Saw (20yr-8.05)31.4(23)1.26,0.10Concurrent Data (averages):MSW (weighted)0.76(0)1.26,0.35,—,—6-565-56MSW (unweighted).002(0)0.79,0.90,—,—6-576-57All Data:MSW101 (93)2.51,0.20,—,—4-58Cosine99.7(91)3.16,0.15,1.1,1.18-124-58Sawtooth (6.65)97.8(91)1.26,0.10,1.8,1.87-205-66Sawtooth (8.05)97.4(91)1.26,0.05,2.0,2.05-275-66All Data (averages):MSW (weighted)1.64(0)5.01,0.10,—,—5-205-55MSW (unweighted)0.15(0)2.51,0.04,—,—6-566-56∗Parameters: ∆m2/10−7eV 2, sin2(2θ), A, B(/10−10µBkG), for Zeeman en-ergy = A + B[cos(t) or saw(t)].Table 2: Neutrino Data χ2 Fits and Ga Predictions24

Modelχ2Flux reductionGaConcurrent Data:B67.50.25 of SSM122B+Be49.40.30 of SSM98Concurrent Data (averages):B (weighted)20.30.18 of SSM121B+Be (weighted)9.740.25 of SSM96B (unweighted)7.730.30 of SSM122B+Be (unweighted)2.780.36 of SSM101All Data:B1660.09 of SSM119B+Be1310.20 of SSM93All Data (averages):B (weighted)30.40.07 of SSM119B+Be (weighted)14.60.18 of SSM93B (unweighted)20.00.15 of SSM120B+Be (unweighted)8.630.25 of SSM96Table 3: Non standard solar model χ2 fits and Ga predictions25

Figures1. Shown in (a) is the complete Homestake, and Kamiokande data setused in this analysis, with neutrino signal shown as a fraction of thatpredicted in the Standard Solar Model.

Error bars for the Cl data arediscussed in the text. In (b) the subset of the sample containing thedata obtained concurrently by the two detectors is shown.2.

Those regions in the MSW parameter space (mass-squared differenceand mixing angle) which are allowed by the 3 year concurrent datasample at the 95% confidence levels based on a comparison to (a) allthe weighted concurrent data, and (b) the unweighted averages of thetwo concurrent data sets, are shown. The line shows the solar neutrinoproblem “solution” described by Bahcall and Bethe.3.

Those regions in the MSW parameter space (mass-squared differenceand mixing angle) which are allowed by the full 20 year weighted data.4. Same as the last figure, except based on (a) the weighted average sig-nals, (b) the unweighted average signals.5.

MSW predictions for Homestake and Kamiokande experiments and ex-perimental rates.6. Those regions in ∆m2 −sin2 2θ space which are allowed at the 68 and95% confidence levels for non-zero transition magnetic moments basedon the 20 year weighted data sample, when the Zeeman energy is fixedto its “best fit” value, with time dependence:(a) (1.1 × 10−10 + 1.1 × 10−10 cos(f + kt)) µBkG,(b) (2 × 10−10 + 2 × 10−10saw(t, τ = 8.05)) µBkG.7.

The predicted electron anti-neutrino signal in Kamiokande as a fractionof the observed background for incident anti-neutrinos of energy >26

10.6MeV , for resonant spin conversion models, if the Zeeman energyin the sun has value: (a) 2 ×10−10µBkG, (b) 5 ×10−10µBkG, is shownas a function of ∆m2 and sin2(2θ).27

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