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The recent 71Ga solar neutrino observation is combined with the

arXiv:hep-ph/9208241v1 21 Jul 1992SOLAR NEUTRINOSWITHTHREE FLAVOR MIXINGSDavid Harley1T.K. Kuo2James Pantaleone1∗1Institute for Nuclear TheoryUniversity of WashingtonSeattle, Washington 981952Physics DepartmentPurdue UniversityWest Lafayette, IN 47907AbstractThe recent 71Ga solar neutrino observation is combined with the37Cl and Kamiokande-II observations in an analysis for neutrino massesand mixings.

The allowed parameter region is found for matter en-hanced mixings among all three neutrino flavors. Distortions of thesolar neutrino spectrum unique to three flavors are possible and maybe observed in continuing and next generation experiments.∗Address after August 1, 1992: Physics Department, Indiana University, Bloomington,IN 474050

For two decades the measurement of solar neutrinos in the 37Cl experi-ment [1] has found far fewer neutrinos than predicted by the Standard SolarModel (SSM). This result was supported by the water Cherenkov experimentKamiokande-II [2] and, very recently, by the gallium experiments SAGE [3]and GALLEX [4].

Theoretically, a very appealing explanation is that reso-nant neutrino oscillations (the MSW effect [5, 6], for reviews see [7, 8]) candeplete the νe flux produced in the sun. The most interesting consequence ofthis interpretation is that the observed neutrino flux is tied in with the neu-trino masses and mixing angles.

We have thus available the means to probeneutrino masses as small as 10−6 eV. It should be noted that such small neu-trino masses are expected in the standard model from effective gravitationalcorrections, or they may come from new physics such as Grand Unified The-ories [9].

Thus solar neutrino observations provide an important window intophysics at very high energy scales.However it is premature to draw definitive conclusions as to the presenceor absence of neutrino masses. It is conceivable that the SSM may be in-complete.

Also, the present solar neutrino observations generally suffer fromlow counting rates. New solar neutrino detectors are needed to overcomethese problems.

To aid the design choices for future experiments, we hereexamine what neutrino mass and mixing parameters the present data imply.Indeed, such an analysis has been carried out [4] assuming only two flavorsof neutrinos. The result is that the neutrino masses and mixing angles areconstrained to lie in a very small region, almost pinpointing the neutrino pa-rameters.

This is an extremely important conclusion and one needs to assessits reliability. An obvious question is how good the two-flavor approximationis.

The purpose of this Article is to analyze the problem in the realistic caseof three flavors. We also discuss how the favored parameter regions may beprobed in the future.The preferred neutrino mass and mixing parameter regions are found bycalculating the chi-squared between the predicted and measured results.

The37Cl and GALLEX results given in Table 1 are used. The SAGE result isnot used, pending further analysis of recent data.

The Kamiokande-II mea-surements in 12 different energy bins are included explicitly and their overallsystematic error is correctly accounted for [2]. The SSM fluxes of Bahcalland Pinsonneault [10] are used.

Their estimated uncertainties are neglectedherein since they are small compared to the experimental uncertainties–including them would slightly increase the allowed neutrino parameter re-1

gion.When a neutrino propagates through matter, a resonance can occur be-tween the vacuum neutrino mass and the induced mass from the electronbackground. For three flavors, there are two two-flavor type resonances whichcan occur [11]: a resonance between the νe and the ν3 vacuum mass eigen-state and a resonance between the νe and the ν2 vacuum mass eigenstate.For each resonance there is a range of energies for which the neutrino fluxis rotated into a different flavor that present experiments can not observe.This energy range spans from a sharp, lower (adiabatic) threshold to a moregradual, upper (nonadiabatic) threshold.

To leading order in small mixing,these energy ranges can be qualitatively described as6MeV m2i10−4eV 2!< E < 10MeV m2i × |Uei|210−8eV 2! (1)for the νe −νi resonance.

Here mi is the ith neutrino’s mass in vacuum, and|Uei|2 is the amount of νe in the ith vacuum mass eigenstate. This equationwill aid in understanding the figures, however for the actual calculationsplotted, a more accurate, analytical expression is used to describe the survivalprobability [11] of a solar νe.

That expression is valid for both small and largemixing, and has the proper level crossing behaviour in the convective zone.The possibility of enhanced mixing from propagation through the Earth isignored–this distorts the allowed parameter region slightly.In certain limits, one neutrino decouples and then the approximationof assuming only two flavors is reasonable. This is the case at the top ofFigs.

(1) and (2). There the plotted allowed regions are in agreement withthose found in ref.

[4]. Generally, the region at small mixing corresponds toresonant conversion of some of the 7Be and predominantly the lower energy8B solar fluxes.

For the region at large mixing, resonant conversion occursfor all of the 8B and varying amounts of the 7Be solar fluxes–but becauseof the large vacuum mixing the observability of these fluxes is increased andthat of the pp flux is decreased. These two two-flavor solutions can be easilydistinguished by future experiments.

In SNO [13] or Super-Kamiokande [14],the small mixing solution produces a ”nonadiabatic” distortion of the 8Bflux while the large mixing solution produces day-night differences [15]. InBOREXINO [16], we find that the small and large mixing solutions predict0.2-0.8 and 0.45-0.75 of the 7Be SSM flux, respectively.

However these two-flavor conclusions can be modified by three-flavor effects.2

One way that three-flavors can be relevant is if parts of both of the twotwo-flavor conversion energy ranges lie in the experimentally observable re-gion. This is illustrated in Fig.

(1). At the top of the figure, the 3 neutrinodecouples because the |Ue3|2 mixing is small and also m23 is large enough sothat the conversion energy range of the νe −ν3 resonance, Eq.

(1) with i=3,lies above that experimentally accessible with solar neutrinos. But as m23 ap-proaches 1 × 10−4 eV2 the νe −ν3 resonance rotates away more of the higherenergy neutrinos.

For the region at small |Ue2|2, this can be compensated forby decreasing |Ue2|2 which decreases the energy range where νe −ν2 resonantconversion occurs. Since a single, two-flavor, adiabatic threshold in the 8Bflux can not explain the K-II data, |Ue2|2 can not become too small.

Howeverthe two-flavor range of |Ue2|2, 9 × 10−4 to 3 × 10−3, now extends down to1 × 10−4, producing the ”foot” visible on the region at small |Ue2|2 [17].As m23 decreases further, too much 8B flux is rotated away and no solutionsare possible. But at m23 ≈5 × 10−6 eV2 the νe −ν3 resonance explains thedata with the νe −ν2 resonance decoupled.

Hence the lower allowed regionextends to arbitrarily small |Ue2|2 and m22 values. This region terminates atm22 × |Ue2|2 ≈2 × 10−10 eV2 where the nonadiabatic threshold of the νe −ν2resonance converts too much of the low energy solar pp flux.

m23 can increasenear this border because rotating away more of the lower energy pp flux withthe νe −ν2 resonance can be compensated for by removing less of the 7Beflux with the νe −ν3 resonance. BOREXINO would then observe the fullSSM 7Be flux.Fig.

(1) illustrates a general theme for three flavor effects. Parts fromboth of the two two-flavor energy ranges (also vacuum oscillation distor-tions [18]) can lie in the experimentally observable region at the same time.The resulting spectral distortions are different from those of the two-flavorcase and, depending on the values of the parameter, can be very compli-cated.

New experiments will be able to search for these spectral distortions.Small distortions of the high energy 8B flux will be observable in SNO andSuper-Kamiokande. Discerning distortions of the low energy pp neutrinosmay be possible with high statistics in the 71Ga experiments in combinationwith accurate measurements of the other fluxes.

Additionally, expanding theenergy range of observations would expand the constraints on the ”third”neutrino. This may be done by observing the hep neutrinos (SNO and Super-Kamiokande) and by using new techniques to observe the large flux of lowenergy pp neutrinos (e.g.

[19]).3

Another way that all three flavors can be relevant to the solar neutrinoflux is through the mixing. Precision measurements at an energy thresholdof a single resonance can show the presence of a ”third” neutrino if themixing with it is nonzero (see Sect II.D.4 of ref.

[7]). Large mixings mustbe considered a likely possibility since neutrino masses are obviously verydifferent from the known Dirac masses.The effects of large mixings areillustrated in Fig.

(2).In Fig. (2), m23 is large enough so that the energy range where νe −ν3 res-onance conversion occurs is above that relevant to current solar neutrino ex-periments.

However this ”third” neutrino influences the observations throughthe vacuum mixing. At the top of Fig.

(2), the mixing with the 3 neutrino issmall enough that the νe −ν3 resonance decouples. However with increasing|Ue3|2 the vacuum mixing rotates away more of the solar neutrinos.

Thenthree qualitatively new allowed regions appear. On the right, the allowedregion gradually grows an extension along m22 ≈10−4 eV2 where only thehigher energy 8B neutrinos are resonantly rotated away.

On the left, thereabruptly appears a thin wall of an allowed region along m22×|Ue2|2 ≈6×10−9eV2. This corresponds to resonant conversion of only the lower energy 8Bneutrinos.

For both of these new, allowed regions the 7Be and pp neutrinofluxes are reduced by the large, three-flavor, vacuum mixing. In addition,Fig.

(2) shows a large allowed region at |Ue3|2 ≈0.5. Generally, this cor-responds to partial resonant conversion of all of 8B, varying amounts of the7Be, and 0.5 vacuum mixing of the pp solar fluxes.Fig.

(2) is partially motivated by recent measurements of the atmosphericneutrino flux [20]. One explanation of those observations is with m23 = 1.0 ×10−2 eV2 and large |Ue3|2.That parameter range can be further probedthrough long baseline experiments using accelerator neutrinos [21].

Howevernote that results similar to Fig. (2) are obtained in a plot of m23, |Ue3|2,and |Ue2|2, while keeping m22 ”small”, 10−11 eV2 < m22 < 10−8 eV2.

Thisparameter region can be probed by looking for seasonal variations in thesolar 7Be line flux [22, 18].BOREXINO will observe this flux at a highcounting rate, 10-50 events/day, so that even very small seasonal variationsmay be discernable.There are generally two types of genuine three-flavor effects. If 10−9 eV2

If one |Uei|2 ≥0.05, then vacuum mixing effects becomeimportant. We have illustrated these two possibilities by fixing one of the4

four parameters so that we may make 3-D plots of the allowed values of theneutrino parameters. Other choices yield differences in detail, and sometimesqualitatively new solutions.

We find that assuming only two neutrino flavorsis a simplifying assumption that may easily be inadequate, especially whenaccurate data become available. When three flavors are included, predictionsfor future experiments become much broader.For BOREXINO, the 7Beflux can take any value between 0.2 to 1 of the SSM, and large seasonalvariations are possible.

For SNO or Super-Kamiokande, the mean 8B fluxis already determined by the KII measurements, but spectral distortions farmore intricate than in the two-flavor analysis are possible.We would like to thank Sandip Pakvasa and Archie Hendry for usefulcommunications. JP is grateful to the theory group at Brookhaven Labora-tory for their hospitality and partial support during the completion of thiswork.5

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Table 1. Results of solar neutrino experiments.

The flux is given as afraction of the SSM [10] prediction.ExperimentProcessEthresholdExpt./SSMDavis et al.νe+37Cl→e+37Ar0.81 MeV0.27 ± 0.04Kamiokande-IIν + e →ν + e7.5 MeV0.46 ± 0.05 ± 0.06SAGEνe+71Ga→e+71Ge0.24 MeV0.15 ± 0.14 ± 0.24GALLEXνe+71Ga→e+71Ge0.24 MeV0.63 ± 0.14 ± 0.068

Figure CaptionsThe red surface [23] surrounds the region allowed at 90% confidence level(for three variables) by the solar neutrino observations. The lines thereoncorrespond to the axes tic marks.

The ”bottom” coordinate system is analo-gous to the usual two-flavor type plot for the νe −νi resonance with mi2 (outof page) versus the mixing element |Uei|2 (horizontal axis).1. UPPER FIGURE.

The vertical axis is m23. The constraints |Ue3|2 =2 × 10−3, 10−3eV 2 > m23 > m22 > 10−8eV 2, and 10−5 < |Ue2|2 < 0.5 areassumed.2.

LOWER FIGURE. The vertical axis is |Ue3|2.

The constraints m23 =1 × 10−2 eV2, |Ue2|2 < 0.5, and 10−2 < |Ue3|2 < 0.5 are assumed.9

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