Fate of the sterile neutrino

Fate of the sterile neutrino

21 September 2000 Physics Letters B 489 Ž2000. 345–352 www.elsevier.nlrlocaternpe Fate of the sterile neutrino V. Barger a , B. Kayser b, J. Learned...

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21 September 2000

Physics Letters B 489 Ž2000. 345–352 www.elsevier.nlrlocaternpe

Fate of the sterile neutrino V. Barger a , B. Kayser b, J. Learned c , T. Weiler d , K. Whisnant e a

Department of Physics, UniÕersity of Wisconsin, Madison, WI 53706, USA DiÕision of Physics, National Science Foundation, Arlington, VA 22230, USA c Department of Physics, UniÕersity of Hawaii, Honolulu, HI 96822, USA Department of Physics and Astronomy, Vanderbilt UniÕersity, NashÕille, TN 37235, USA e Department of Physics and Astronomy, Iowa State UniÕersity, Ames, IA 50011, USA b

d

Received 2 August 2000; accepted 4 August 2000 Editor: M. Cveticˇ

Abstract





In light of recent Super-Kamiokande data and global fits that seem to exclude both pure nm ns oscillations of atmospheric neutrinos and pure ne ns oscillations of solar neutrinos Žwhere ns is a sterile neutrino., we reconsider four-neutrino models to explain the LSND, atmospheric, and solar neutrino oscillation indications. We argue that the solar data, with the exception of the 37Cl results, are suggestive of ne ns oscillations that average to a probability of approximately 12 . In this interpretation, with two pairs of nearly degenerate mass eigenstates separated by order 1 eV, the day-night asymmetry, seasonal dependence, and energy dependence for 8 B neutrinos should be small. Alternatively, we find that four-neutrino models with one mass eigenstate widely separated from the others Žand with small sterile mixings to active neutrinos. may now be acceptable in light of recently updated LSND results; the 37Cl data can be accommodated in this model. For each scenario, we present simple four-neutrino mixing matrices that fit the stated criterion and discuss future tests. q 2000 Elsevier Science B.V. All rights reserved.

1. Introduction Accelerator, atmospheric, and solar neutrino data give evidence for neutrino oscillations and thus for neutrino masses and mixing. The LSND accelerator experiment finds a small nm ne appearance probability and a mass-squared difference d m2LSND ) 0.2 eV 2 w1,2x. The atmospheric experiments measure the nm nm survival probability versus both pathlength L and neutrino energy En . The amplitude for nm nx Ž x / e . oscillations is inferred to be maxi-







E-mail address: [email protected] ŽV. Barger..



mal or near-maximal with d m2atm ; 3 = 10y3 eV 2 w3–5x. The combined solar neutrino experiments determine the ne ne survival probability to be ; 0.3 to 0.7, depending on the neutrino energy w6–9x and d m2solar Q 10y3 eV 2 is required for consistency with the null result for the ne ne probability measurement in the CHOOZ experiment w10x. Thus, taken together, the data require three distinct d m2 . With three neutrinos there are only two independent d m2 , so a fourth, sterile, neutrino Ž ns . needs to be invoked in addition to ne , nm , and nt . Having no weak interactions, the sterile neutrino escapes the Nn , 3 constraint from the invisible decay width of the Z-boson w11x.

0370-2693r00r$ - see front matter q 2000 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 0 - 2 6 9 3 Ž 0 0 . 0 0 9 5 0 - 3





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Successful global descriptions of the oscillation data have been made in a four-neutrino framework w12–16x. Compatibility of the LSND result with null results of accelerator w17x and reactor w18x disappearance experiments was found w12,13x to favor a 2 q 2 mass spectrum, with the mass-squared differences d m2solar and d m2atm separated by the larger difference d m2LSND , over a 1 q 3 spectrum, with one mass eigenstate widely separated from the others. In the simplest of the 2 q 2 models, the oscillations are nm nt Žatmospheric. and ne ns Žsolar., or alternatively nm ns Žatmospheric. and ne nt Žsolar.. More generally, sterile neutrinos may be involved in both atmospheric and solar oscillations. In the 2 q 2 mixing schemes, the sterile flavor content must be significant in the solar or atmospheric oscillations or in both. New results from Super-Kamiokande ŽSuperK. w4,7x impact critically on oscillations to sterile neutrinos. The zenith angle dependence of high-energy atmospheric events, along with the neutral current p 0 production rate, exclude pure nm ns oscillations at 99% C.L. The electron energy dependence of solar neutrino events, along with the absence of a significant day-night effect, exclude pure ne ns oscillations at 99% C.L. when all of the solar data, including the 37 Cl results, are taken into account. Thus a dominant involvement of ns in either atmospheric or solar oscillations is brought into question. Also, LSND has reported new results w2x with a somewhat lower average oscillation probability than before and the KARMEN experiment w19x excludes part of the LSND allowed region. Consequently, previous exclusions w12,13x of the 1 q 3 schemes need to be reassessed. In the context of this new data, we reconsider four-neutrino oscillation models and the implications for the existence of a sterile neutrino.

™ ™









2. 2 H 2 models The 2 q 2 models have two nearly degenerate pairs of neutrino mass eigenstates giving d m2atm and d m2solar separated by the LSND scale d m2LSND ; 1 eV 2 . Maximal n 3 , n 2 mixing describes the atmospheric data and maximal n 0 , n 1 mixing describes the solar data. The mass hierarchy may be nor-

mal Ž m 3 4 m1 . or inverted Ž m 3 < m1 .. The d m2solar splitting must be smaller than 10y3 eV 2 to avoid the CHOOZ reactor constraint on ne ne oscillations and larger than the ; 10y1 0 eV 2 of just-so vacuum oscillations w20x so that the oscillations average to give an approximately energy-independent solar ne survival probability.



2.1. Solar neutrinos: maximal ne

™n

s

oscillations

The new SuperK solar neutrino data show a remarkably flat recoil electron energy distribution from 5 MeV to 14 MeV, with average value w7x .016 datarSSMs 0.465q0 y0 .014 ,

Ž 1.

where SSM is the Standard Solar Model prediction w9x. The 8 B flux normalization in the SSM is somewhat uncertain, and the above result is suggestive of rapid oscillations with maximal amplitude that give an average oscillation probability, ² P Ž ne ne .:, of approximately 12 . Regeneration of ne in the Earth, due to the effects of coherent forward scattering of the neutrinos with matter w21x, causes a day-night variation in the measured neutrino flux. The smallness of the day-night effect 2Ž D y N .rŽ D q N . s y0.034 " 0.022 " 0.013 w7x in the SuperK data Ž1.3 s from zero. puts strong constraints on solar solutions with matter effects w7,22–24x. It has been noted that there can be a day-night effect even for maximal mixing w23,24x, but calculations for active neutrinos show that it is small Ž; 1–2 %. for d m2solar R 5 = 10y5 eV 2 or d m2solar Q 10y7 eV 2 w7x. In ne ns oscillations, the difference in coherent scattering amplitudes in matter is '2 GF Ž Ne y Nnr2. , '2 GF Ner2 Žwhere Ne and Nn are the electron and neutron number densities, respectively, and Ne ; Nn in the Earth.. The corresponding amplitude difference for ne nm Žor ne nt . is '2 GF Ne . Therefore, for maximal mixing the day-night effect should be smaller with a sterile neutrino than with an active neutrino, but it would not be zero. The regeneration effect in the earth for neutrinos detected at night raises the value of ² P Ž ne ne .: for maximal mixing with active neutrinos from 0.50 to about 0.54 w23x. For sterile neutrinos, we roughly estimate that ² P Ž ne ne .: , 0.52 in SuperK for maximal mixing with regeneration. Regeneration also













V. Barger et al.r Physics Letters B 489 (2000) 345–352

causes a weak dependence on energy of the solar neutrino suppression. For maximal mixing of ne with an active neutrino, ² P Ž ne ne .: is slightly higher at higher neutrino energies w23x, consistent with the weak trend of the SuperK data; less energy dependence should be present with ne oscillations to ns . Assuming ² P Ž ne ne .: s 0.52 for maximal ne ns oscillations, the SuperK result in Eq. Ž1. would be reproduced by a 8 B flux normalization of n s 0.89. The SSM predictions Žin SNU. for the Gallium experiments are Ž70, pp ., Ž34, 7 Be., Ž3, pep ., Ž10, CNO., and Ž12, 8 B., giving a total of 129 SNU; the .8 Ž q7 .5 observed values are 75.4q7 SAGE., 77.5y7.8 y7 .4 q1 0.8 ŽGALLEX. and 65.8y1 0.2 ŽGNO.. With the above 8 B normalization and ² P Ž ne ne .: s 12 , the predicted rate in the Gallium experiments is 63.8 SNU. Thus all of the Gallium measurements are consistent within 2 s of the value that would be found for maximal amplitude ne ns oscillations that average to 12 . The new data from GNO are especially suggestive of this interpretation. A least-squares fit to the SuperK and Gallium rates with the 8 B flux normalization n as a free parameter gives a best fit value of n s 0.90 with x 2 s 5.5 for 3 degrees of freedom, which corresponds to a 14% goodness of fit. In this calculation we have not taken into account any regeneration effect for the Gallium data, which may improve the fit. The discrepant measurement in the above interpretation is the 37 Cl value of datarSSMs 0.33 " 0.03 from the Homestake mine experiment, which would require a significant energy dependence of the solar ne flux suppression. Most global fits to neutrino oscillation data include the 37 Cl data Žand often disregard the LSND data. and then ne ns oscillations are excluded. We instead suggest the possibility that the solar datarSSM flux ratio is relatively flat over the entire 0.233 MeV to 14 MeV energy range and is described by ne ns oscillations with maximal mixing. An important test of approximate constant suppression of the solar neutrino spectrum will be the BOREXINO w25x experiment, which can measure the 7 Be component of the solar neutrino flux. Nearly complete suppression of the 7 Be component is needed for the 37 Cl data to be consistent with the suppression of the 8 B component measured by SuperK.















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However, a suppression of the 7 Be flux similar to the datarSSM measured by SuperK would favor maximal ne ns mixing of solar neutrinos. The SNO w26x and ICARUS w27x experiments will also provide critical tests of the 8 B flux suppression, and SNO will test maximal ne ns oscillations through the NCrCC ratio Žwhich should be the same as the value without oscillations..





2.2. Mixing matrix for the 2 q 2 model We assume a 2 q 2 scenario in which one pair of nearly degenerate mass eigenstates has maximal ne ns mixing for solar neutrinos and the other pair has maximal Žor nearly maximal. nm nt oscillations for atmospheric neutrinos. Small off-diagonal mixings between ne , ns and nm , nt can accommodate the LSND results. Neglecting CP-violating phases for the present, an approximate mixing with these properties is





°'

1

1

2

'2

ns 1 y '2 ne , e nm nt 0

0

1

0

0

e

e

'2 ye

¢

0

1

1

'2

'2

y

1

1

'2

'2



ß

n0 n1 , Ž 2. n2 n3

0

2 < where < d m10 < < d m 232 < < < d m221 <. In the notation of Ref. w28x, we have chosen the mixing angles u 01 s u 23 s pr4, u 02 s u 03 s 0, and u 12 s u 13 , with e s sin u 13 . The oscillation probabilities are given in Table 1. The parameter e is determined from the LSND data to be approximately

e,

ž

0.016 eV 2 < d m2LSND <

0.91

/

,

Ž 3.

where d m2LSND is restricted to the range 0.2 to 1.7 eV 2 by the BUGEY and KARMEN experiments; for d m2LSND , 6 eV 2 , e , 0.022 is marginally possible. Other forms of the mixing matrix Žsee, e.g., Refs. w12–16,28x. are also acceptable, provided that the

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Table 1 Oscillation probabilities in the 2 q 2 model defined by Eq. Ž2., to leading order for each oscillation scale. The oscillation arguments are defined by D j ' 1.27d m2j LrE, with d m2j in eV 2 , L in km, and E in GeV s

2

1 y sin Dsolar

s e m t

m

e 2

sin Dsolar 1 y 8 e 2 sin2D LS ND y sin2Dsolar

ne – ns mixing is nearly maximal. Future short and long-baseline experiments will be useful in determining the mixing matrix. For example, there are no ne nt oscillations at short baselines for the mixing in Eq. Ž2., but there could be if the Ue2 and Ue3 mixing matrix elements are not equal in magnitude or have different phases Žsee, e.g., Refs. w14,28–30x., in which case unitarity requires Ut 0 / 0 andror Ut 1 / 0. Also, both ne nm and ne nt oscillations are possible at long baselines, and there can be CP violation in these channels if the four-neutrino mixing matrix is not real Žsee, e.g., Refs. w28,30–32x..







t 2

2

2 e sin Dsolar 8 e 2 sin2D LSND y 2 e 2 sin2Datm y 2 e 2 sin2Dsolar 1 y 8 e 2 sin2D LS ND y sin2Datm

0 2 e 2 sin2Datm sin2Datm 1 y sin2Datm

a higher datarSSM in SuperK than in the 37 Cl experiment. By way of example, consider the approximate mixing

° ne nm

0 nt ns

1

,

1

¢

2

1

1

d

2

'2 1

y

1

y2

2

d

e

1

'2

y2

e

'2

d



0

1

'2

y2 y

0

'2 e

'2

y

d

'2

ß

1

n1 n2 , n3 n0

0 Ž 4.

3. 1 H 3 models Another way to circumvent the SuperK exclusion of sterile neutrinos is to assume that the solar and atmospheric oscillations are approximately described by oscillations of three active neutrinos; then the LSND result can be explained by a coupling of ne and nm through small mixings with a sterile neutrino that has a mass eigenvalue widely separated from the others. This 1 q 3 model was previously disfavored by incompatibility of the LSND result with null results of the CDHS and BUGEY experiments w12,13x. However, in newly updated results from the LSND experiment w2x, the LSND allowed region is slightly shifted, and this opens a small window for the 1 q 3 models. The solar neutrino data can be explained by oscillations of three active neutrinos in the usual way; the results of the 37 Cl experiment can be reconciled with those of SuperK in part because oscillations of ne to active neutrinos in SuperK also show up as neutral current events Žwith a rate of about 1r6 of the charge-current rate., thereby giving

where e and d are small and both the flavor and mass eigenstates are reordered to reflect the fact that n 0 Žwhich is predominantly ns . is the heaviest state Ž < d m102 < 4 < d m232 < 4 < d m221 < . . In the notation of Ref. w28x, we have chosen the mixing angles u 01 s u 12 s pr4, u 02 s u 23 s 0, sin u 03 s e , and sin u 13 s d . Here the 3 = 3 submatrix that describes the mixing of the three active neutrinos has the bimaximal form w33x. The oscillation probabilities are given in Table 2. For this mixing matrix the leading oscillation amplitudes at the d m2LSND scale in the 1 q 3 scheme are Žwhen e , d < 1. A BUGEY , 4e 2

Ž 5.

for ne disappearance in the BUGEY experiment, A CD HS , 4d 2

Ž 6.

for nm disappearance in the CDHS experiment, and A LSND s 4e 2d 2

Ž 7.

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Table 2 Oscillation probabilities in the 1 q 3 model defined by Eq. Ž4., to leading order for each oscillation scale. The oscillation arguments are defined by D j ' 1.27d m2j LrE, with d m2j in eV 2 , L in km, and E in GeV

m

e e m t s

2

2

2

1 y 4e sin D LSND y sin Dsolar

t 2 2

2

1 2

2

4e d sin D LSND q sin Dsolar 1 y 4d 2 sin2D LS ND y sin2Datm y 14 sin2Dsolar



for nm ne appearance in the LSND experiment. Including the subleading oscillation at the d m 2atm scale, and assuming that the leading oscillation averages, P Ž nm

™ n . , Ž 1 y 2 d . Ž1 y sin 1.27d m 2

m



2

2 atm

LrE .

Ž 8.

for nm nm disappearance in atmospheric neutrino experiments, where d m2atm is in eV 2 , L in km, and E in GeV. The relative normalization of the atmospheric nm and ne fluxes is well known w34x; the predicted effective change in this normalization at the d m2atm oscillation scale is Ž1 y 2 d 2 .rŽ1 y 2 e 2 .. In the region where the CDHS constraint on d is weak or nonexistent Ž d m2LSND Q 0.4 eV 2 ., BUGEY constrains e to be less than about 0.1; conservatively assuming that the atmospheric data constrains the relative normalization to within 10%, we find d 0.24. The upper limits on e Žfrom A BU GEY . and d Žfrom A atm and A CDHS . and the allowed values of A LSND vary with d m2LSND . With the previously reported LSND data w1x, there was no value of d m2LSND for which the constraints on e and d were consistent with the LSND 99% C.L. allowed range of A LSND ; the maximum allowed values of e and d always implied an upper limit on A LSND that was below the LSND measured value. However, with the recent shift of A LSND to lower values, there are now three small d m2LSND islands where the LSND 99% C.L. region is compatible with the BUGEY, CDHS, and KARMEN constraints: d m2LSND , 0.9 eV 2 and

1 2

s 2

sin Dsolar sin2Datm y 14 sin2Dsolar 1 y sin2Datm y 14 sin2Dsolar

4e 2 sin2D LSND q 12 Ž d 2 y 2 e 2 .sin2Dsolar 4d 2 sin2D LSND y d 2 sin2Datm q 14 Ž2 e 2 y d 2 .sin2Dsolar d 2 sin2Datm q 14 Ž2 e 2 y d 2 .sin2Dsolar 1 y 4Ž e 2 q d 2 .sin2D LS ND yd 2 Ž2 e 2 q d 2 .sin2Datm y 14 Ž d 2 y 2 e 2 . 2 sin2Dsolar

d m2LSND , 1.7 eV 2 , for which the BUGEY constraint is somewhat less restrictive, and d m2LSND , 6 eV 2 , for which both the BUGEY and KARMEN constraints are somewhat less restrictive. The E776 experiment at BNL w35x gives a somewhat tighter constraint than KARMEN at d m2LSND s 6 eV 2 , but a small region is still allowed here. Except for a small region near d m2LSND , 0.2 eV 2 Žsee below., the combined data are still inconsistent with oscillations having d m2LSND - 0.9 eV 2 , and for all d m 2LSND when the LSND 90% C.L. allowed region is used. The above analysis is summarized in Table 3; acceptable values of d m 2LSND are those for which Ž4e 2d 2 . max lies within the range of A LSND allowed by LSND and KARMEN. There is no CDHS constraint for d m2LSND - 0.25 eV 2 , which suggests that d m2LSND , 0.2 eV 2 may also be allowed. However, at these d m 2LSND values d must be more than 0.5 to reconcile the BUGEY limit with the LSND measurement, which is not consistent with the relative normalization of the nm and ne fluxes Žsee above.. Table 3 Summary of constraints on four-neutrino oscillation parameters in the 1q3 scheme. The upper limit on A LSND at d m2LSND ,6 eV 2 is from the E776 experiment at BNL w35x. All experimental limits are at 90% C.L., except for LSND, which is at 99% C.L. Ž4e 2d 2 . max Ž A LSND . min Ž A LSND . max d m2LS ND emax dmax ŽeV 2 . ŽBUGEY. ŽCDHS. ŽLSND. ŽKARMEN. 6.0 1.7 0.9 0.3

0.19 0.16 0.12 0.10

0.14 0.12 0.16 0.45

2.8=10y3 1.5=10y3 1.5=10y3 8=10y3

1.5=10y3 0.8=10y3 1.4=10y3 10=10y3

2.0=10y3 1.0=10y3 3.0=10y3 30=10y3

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The model of Eq. Ž4. has bimaximal mixing in the 3 = 3 active sector. Similar results for the 1 q 3 model can be obtained for any 3 = 3 submatrix that can describe the atmospheric and solar data for active neutrinos, as long as e and d are small. This is easily seen by realizing that the general expressions for the short-baseline amplitudes in the 1 q 3 model in Eqs. Ž5. – Ž7. depend only on the magnitudes of the mixing matrix elements Ue0 and Um 0 w e and d , respectively, in Eq. Ž4.x. If Ut 0 / 0, then there can be nm nt and ne nt oscillations at the d m2LSND scale. If Ue3 / 0, then there can be ne nm and ne nt oscillations at the d m2atm scale. Because the allowed windows are already severely constrained by accelerator and reactor data, future experiments at short baselines could easily test or rule out the 1 q 3 scenarios. MiniBooNE w36x will search for nm ne oscillations and test A LSND . Reactor experiments such as at Palo Verde w37x, KamLAND w38x, and the proposed ORLaND w39x, will test A BU GEY , but only ORLaND is expected to significantly improve the bound on e . A future precision measurement of nm disappearance at short baselines could test A CD HS .











4. Discussion and conclusions The recent Super-Kamiokande data and global fits present new constraints on the mixing of light sterile neutrinos. It is important to assess the viability of a light sterile neutrino, because its existence is required if the solar, atmospheric, reactor, and accelerator data are all to be understood in terms of neutrino oscillations. We have examined at a qualitative level two schemes for four-neutrino mass and mixing which nearly accommodate all present data. In the 2 q 2 scheme, two neutrino pairs are separated by the LSND mass scale. One pair maximally mixes nm and nt to explain the atmospheric data, while the other pair maximally mixes ne and ns to explain the solar deficit with energy-independent oscillations. All data except the low ne capture rate on 37 Cl are explained in this model. A very small day-night effect is expected, at the level of a per cent or less. Suppression of the 7 Be component of the solar neutrino flux by approximately 50% is expected for the BOREXINO experiment.

In the 1 q 3 scheme, the three active neutrinos are separated by the LSND scale from a sterile neutrino state. The active neutrino 3 = 3 submatrix explains the atmospheric and solar data, including the 37 Cl rate. In our example we assigned the bimaximal model w33x to this submatrix, but any 3 = 3 active submatrix that describes the atmospheric and solar data may be used. A small mixing of ne and nm via the sterile state explains the LSND data. The smaller oscillation amplitude recently reported by the LSND collaboration allows marginal accommodation with the null ne and nm disappearance results previously obtained in reactor and accelerator experiments. The above two classes of models allow a viable sterile neutrino having either large mixing with ne or small mixing with all active neutrinos. In both classes we made the simplifying assumption that nt-ns mixing is negligible. Present data may allow considerable nt-ns mixing, in which case both solar and atmospheric neutrino oscillations could have a sizable sterile component. Quantitative fits to atmospheric and solar data are needed to determine how much nt-ns mixing is allowed Žsee, e..g, Ref. w40x.. Neutrinos also provide a hot dark matter component, which is relevant to large-scale structure formation w41x. From the upper limit of mb s 2.5 to 3 eV on the effective ne mass obtained from tritium beta decay endpoint measurements w42x, the mass m max of the heaviest neutrino is bounded by w43x

(d m

2 LSND

(

Q m max Q mb2 q d m2LSND .

Ž 9.

The contribution of the neutrinos to the mass density of the universe is given by Vn s ÝmnrŽ h 2 93 eV., where h is the present Hubble expansion parameter in units of 100 kmrsrMpc w44x; with h s 0.65, both the 2 q 2 and 1 q 3 models give Vn G 0.02. For an inverted neutrino mass spectrum Žwhere the ne is associated predominantly with the heavier neutrinos., the bound is d m2LSND Q m max Q mb , which yields Vn G 0.02 and Vn G 0.07 for the 2 q 2 and 1 q 3 models, respectively. The MAP w45x and PLANCK w46x satellite measurements of the cosmic microwave background radiation may be sensitive to these neutrino densities. The existence of the LSND mass gap will be tested by the MiniBooNE experiment w36x. A defini-

(

V. Barger et al.r Physics Letters B 489 (2000) 345–352

tive test of the presence of the sterile state in the 2 q 2 model will be the measurement of the NCrCC ratio for solar neutrinos by the SNO experiment. With the solar solution given by maximal ne ns mixing in this model, the NC should show the same suppression as the CC. The existence of the sterile neutrino in the 1 q 3 model will not be tested by SNO NCrCC data since the ratio is approximately that of ne oscillations to active neutrinos. The sterile state in the 1 q 3 model can be tested by searches for small amplitude oscillations at short baselines. Observation of the flavor ratio of extragalactic neutrinos may serve as a test of the models. Oscillations of neutrinos from distant sources will have averaged, leaving definite predictions for flavor ratios. If cosmic neutrinos are mainly produced in pionrmuon decay, their initial flavor ratio is nt :nm :ne :ns f 0:2:1:0. A simple calculation w47x then gives the asymptotic ratios 1:1:0.5:0.5 for the 2 q 2 model with two maximally-mixed pairs, and 1:1:1:0 for the 1 q 3 model with bimaximal mixing of the active neutrinos. A virtue of active-sterile neutrino oscillations is that they may aid r-process nucleosynthesis of heavy elements in neutrino-driven supernovae ejecta. The basic requirement is that the ne flux be diminished in the region where inverse b-decay would otherwise transform neutrons into protons. The 2 q 2 and 1 q 3 mass spectra and mixing matrices presented here are of the forms previously discussed to enhance r-process nucleosynthesis. In the 2 q 2 model, this is accomplished with a two-step process: first, a matter-enhanced nmrnt ns transition beyond the neutrino sphere removes the energetic nmrnt before they can convert to energetic ne , and then a large ne nm rnt transition reduces the ne abundance w48x; the required mass-squared parameter is d m 2LSND R 1 eV 2 . In the 1 q 3 scheme, the relevant oscillation is a matter-enhanced ne ns , obtained with d m2LSND R 2 eV 2 and sin2 2 ue s R 10y4 w49x. It appears that the r-process enhancements discussed in Refs. w48,49x may be accomplished by tuning the small parameters in our mixing schemes.









Acknowledgements We appreciate the hospitality of the Aspen Center for Physics, where this work was initiated. We thank

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Matt Lautenschlager and Ben Wood for useful interactions. This research was supported in part by the US Department of Energy under Grants No. DEFG02-95ER40896, No. DE-FG05-85ER40226, and No. DE-FG02-94ER40817, and in part by the University of Wisconsin Research Committee with funds granted by the Wisconsin Alumni Research Foundation. References w1x LSND Collaboration, C. Athanassopoulos et al., Phys. Rev. Lett. 77 Ž1996. 3082; Phys. Rev. Lett. 81 Ž1998. 1774. w2x G. Mills, talk at Neutrino-2000, XIXth International Conference on Neutrino Physics and Astrophysics, Sudbury, Canada, June 2000. w3x Super-Kamiokande Collaboration, Y. Fukuda et al., Phys. Lett. B 433 Ž1998. 9; B 436 Ž1998. 33; Phys. Rev. Lett. 81 Ž1998. 1562; Phys. Rev. Lett. 82 Ž1999. 2644; Phys. Lett. B 467 Ž1999. 185. w4x Super-Kamiokande Collaboration, H. Sobel, talk at Neutrino2000 w2x. w5x Kamiokande Collaboration, K.S. Hirata et al., Phys. Lett. B 280 Ž1992. 146; Y. Fukuda et al., Phys. Lett. B 335 Ž1994. 237; IMB Collaboration, R. Becker-Szendy et al., Nucl. Phys. ŽProc. Suppl.. B 38 Ž1995. 331; Soudan-2 Collaboration, W.W.M. Allison et al., Phys. Lett. B 391 Ž1997. 491; MACRO Collaboration, M. Ambrosio et al., Phys. Lett. B 434 Ž1998. 451. w6x Super-Kamiokande Collaboration, Y. Fukuda et al., Phys. Rev. Lett. 81 Ž1998. 1158; Phys. Rev. Lett. 82 Ž1999. 1810, 2430. w7x SuperKamiokande Collaboration, Y. Suzuki, talk at Neutrino-2000 w2x. w8x B.T. Cleveland et al., Nucl. Phys. B ŽProc. Suppl.. 38 Ž1995. 47; Kamiokande Collaboration, Y. Fukuda et al., Phys. Rev. Lett. 77 Ž1996. 1683; GALLEX Collaboration, W. Hampel et al., Phys. Lett. B 388 Ž1996. 384; SAGE Collaboration, J.N. Abdurashitov et al., Phys. Rev. Lett. 77 Ž1996. 4708; GNO Collaboration, M. Altmann et al., hep-exr0006034. w9x J.N. Bahcall, M.H. Pinsonneault, Rev. Mod. Phys. 67 Ž1995. 781; J.N. Bahcall, S. Basu, M.H. Pinsonneault, Phys. Lett. B 433 Ž1998. 1. w10x CHOOZ Collaboration, M. Apollonia et al., Phys. Lett. B 420 Ž1998. 397. w11x Review of Particle Physics, Particle Data Group, D.E. Groom et al., Eur. Phys. J. C 15 Ž2000. 1. w12x S.M. Bilenky, C. Giunti, W. Grimus, Eur. Phys. J. C 1 Ž1998. 247; S.M. Bilenky et al., Phys. Rev. D 60 Ž1999. 073007. w13x V. Barger, S. Pakvasa, T.J. Weiler, K. Whisnant, Phys. Rev. D 58 Ž1998. 093016; V. Barger, T.J. Weiler, K. Whisnant, Phys. Lett. B 427 Ž1998. 97. w14x S.C. Gibbons, R.N. Mohapatra, S. Nandi, A. Raychoudhuri, Phys. Lett. B 430 Ž1998. 296.

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