Seesaw model solutions of the solar neutrino problem

NUCLEAR

P H VS I C S B

Nuclear Physics B 374 (1992) 373-391 North-Holland

________________

Seesaw model solutions of the solar neutrino problem S.A. Bludman

a,b

D.C. Kennedy b,c and P.G. Langacker b

Centerfor Particle Astrophysics, University of California, Berkeley, CA 94720, USA b Department of Physics, University of Pennsylvania, Philadelphia, PA 19104, USA Fermi NationalAcceleratorLaboratory, P.O. Box 500 M5106, Batavia, IL 60510, USA a

Received 30 July 1991 (Revised 23 December 1991) Accepted for publication 30 December 1991

We re-examine solutions to the solar neutrino problem involving neutrino oscillations, decaying neutrinos and a cooler center of the sun. Comparison of the Homestake and Kamiokande II observations implies that low-energy neutrinos are more suppressed than high-energy, disfavoring the large-mass adiabatic MSW resonance and allowing us to exclude non-standard solar 22O 0.004—0.5, ~m2 = 0.08— models with vertical a simplelarge-angle reduction of core= temperature. 10 meV2), (sin2 0.6—0.9, jim2Diagonal 0.08—2,(sin 4—60 meV2), and part of the horizontal (sin22O 0.02—0.6, 4m2 = 80 meV2) MSW solutions are compatible with these data (meV= iO~ eV). The present SAGE results disfavor non-standard solar models and the horizontal MSW solution. Assuming 1’e ~ v~ conversion, we show that seesaw models of neutrino masses imply m~=0.3—9 meV, intermediate-scale symmetry breaking O(109_1012) GeV, and a r-neutririo mass with cosmological ramifications in some cases and laboratory or atmospheric ~ — v~oscillations. While seesaw predictions for neutrino masses are model-dependent and subject to large uncertainties, we prove that leptonic mixing angles are similar to quark mixings for a wide range of models. We also present two specific seesaw models compatible with electroweak neutral current and proton decay constraints, including radiative corrections to the seesaw predictions.

1. Introduction The solar neutrino problem, the persistent discrepancy between theoretical predictions of the sun’s neutrino output and the observed fluxes, has been with us now for two decades [1]. In the last few years, a number of theoretical and experimental developments have moved this problem much closer to solution. In addition to the original 37C1 experiment at Homestake, we now have over three years of observation of solar neutrinos by the Kamiokande II (K-Il) water detector via tie scattering [2]. On the theoretical side now exists a body of “standard solar models” which are consistent with each other and, apart from the neutrino fluxes, agree with solar and main sequence stellar observations quite well [31*~ Following *

Bahcall and Pinsonneault [3] demonstrate the equivalence of existing solar models for identical inputs and present an improved solar model whose results also lead to rejection of the cooler sun solution.

0550-3213/92/$05.O0 © 1992

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Elsevier Science Publishers B.V. All rights reserved

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this literature, we argue that consideration of the available data and of naturalness strongly suggests the Mikheyev—Smirnov—Wolfenstein (MSW) effect of resonant neutrino flavor conversion as the solution of the solar neutrino problem and show that simple properties of the solar neutrino spectrum rule out a non-standard sun as a solution. Our argument does not make use of the recent preliminary data of the SAGE 7tGa collaboration, although their current results are in striking agreement with MSW predictions [4].This conclusion is also reached in a different approach by Bahcall and Bethe [5] We then consider the implications of neutrino masses and mixings in the context of “seesaw” models, both generally and with two specific realizations of the seesaw mechanism. We present in detail two-step grand unified models with intermediate-scale symmetry breaking that lead to neutrino masses overlapping with the MSW range, albeit with large uncertainties. We also prove that leptonic mixing angles are similar to quark mixings for most grand unified seesaw models. An isotropic flux of r’eL from the solar core is a necessary byproduct of the fusion reactions that produce the solar luminosity, with specific fluxes predicted by the standard solar model (SSM). These reactions fall into two chains, pp and CNO, each with a number of processes that produce neutrinos. The full solar neutrino spectrum is a superposition of each of these component sources: pp, pep, hep, 7Be, 8B in the pp chain; and t3N ‘SO, ‘7F in the CNO chain. The SSM depends on standard principles of astrophysics (mechanical and thermal equilibrium, radiative and convective energy transport, fusion production of light elements and luminosity), on constitutive relations (equation of state, radiative opacity), and on the present solar age, composition and luminosity. A non-standard solar model would include new physics and/or modifications of standard physics; the minimal assumption of equilibrium would still obtain, however, as required by the age of the earth and the sun’s place on the main sequence. Thermal equilibrium then implies that, while variations in the solar model may change the total flux of each component (pp, 8B, etc.) of the neutrino spectrum, the shape and position of each spectrum component remain fixed [3]. The rate of detection R of solar neutrinos for a given detector is usually expressed in solar neutrino units (SNU): one neutrino-detector interaction/1036 detector atoms/ second. The predicted rates of Bahcall and Ulrich are given with an “effective 3o- error,” which we re-express as core temperature variation in sect. 2. For chlorine, the predicted rate is R~= 7.9 ±2.6 (3o-), while the observed rate at Homestake is R~S 2.12 ±0.34 (lo-) [1]. The ratio r of observed to predicted rates is then r~ 1= 0.27 ±0.04, where the error is from Robs only. We ignore possible time-variations of this signal, as these have not been confirmed by Kamiokande II [2] and would demand a more complex interpretation of the data. For Kamiokande II, this ratio is rKII = 0.46 ±0.08 [2]. These two ratios do not ~.

*

For combined SSM/MSW predictions see Bahcall and Haxton [5].

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agree with each other, but the neutrino energy thresholds for the two detectors differ: chlorine detects both 8B and 7Be neutrinos (0.814 MeV threshold), while Kamiokande II detects essentially 8B neutrinos only (7.5 MeV threshold). We conclude therefore that comparison of these two rates show that lower-energy neutrinos are more suppressed than higher-energy neutrinos.

2. Non-standard solar and particle physics models 2.1. COOLER SUN

The proposed solutions to the solar neutrino problem involve new nuclear physics, astrophysics, or particle physics. The nuclear physics is well-determined by laboratory measurements, except for the large uncertainty in the 7Be(p, y)8B cross section and its extrapolation to solar energies, which leaves a 9% (hi) uncertainty in the flux of 8B neutrinos. The non-standard solar models that we consider lower the temperature in the solar core, by using a lower heavy-element fraction Z/X, lower elemental radiative cross sections, or by redistributing energy into non-thermal degress of freedom such as a large core-magnetic field, rapid core-rotation, or core accumulation of weakly-interacting massive particles. Indeed, the heavy-element mass fraction is probably the most uncertain input in the SSM neutrino fluxes. The central temperature of the sun T~and the different neutrino fluxes both outputs of the SSM, are approximately correlated [3] by the power laws ~,

~(7Be)-’T~,

4(8B)—T~8,

(1)

so that the ratios r of observed to SSM predicted neutrino rates are (1 ±0.025)[(0,77)(1 ±0.09)Tc~8+ (0.14)(1 ±0.034)T~+ small terms], rKll

=

(1 ±0.09)T~8,

(2)

where T~is the core temperature normalized to the SSM central temperature of 1.56 X iO~’K 1.34 keY [3]. We quote lo- uncertainties in the observed rates and in the nuclear cross section effects on the fluxes, which are properly correlated between r~ 1and rKII, and in the chlorine absorption cross section. The effects of a cooler sun can now be simply parametrized by changing T~,keeping the cross sections within their experimental limits. The cooler sun solution requires the higher-energy neutrinos to be more suppressed, in contradiction with the data. Thus we can fit the K-Il rate alone by a 4 ±1% reduction in T~(T~”= 0.958 ±0.011), but the Homestake rate alone requires an 8 ±1% reduction (T~= 0.915 ±0.011). These two temperature reduc-

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tions differ from one another by nearly 4o-. We can simultaneously fit both rates by an 7 ±1% temperature reduction (T~= 0.927 ±0.009) but only with ~2/d.o.f. = 8.03, which rejects the cooler sun hypothesis at 99.6% confidence. (There is some uncertainty in the temperature exponents in (1). Nevertheless, our conclusion rests only on the fact that the 7Be neutrino flux, which is detected only in the 37Cl experiment, is not more temperature-dependent than the 8B neutrino flux: even if both fluxes had the same temperature dependence, r~ 1= rKJl would be expected. In fact, the observed ratios differ by more than 2o-. Including the cross section 2/d.o.f. = uncertainties, the cooler sun hypothesis is then already rejected at the ~ 4.46 or 96.6% confidence. The enhanced temperature dependence of the 8B neutrino flux merely strengthens this conclusion.)

2.2. NON-STANDARD PARTICLE PHYSICS

The non-standard particle physics solutions proposed for the solar neutrino problem all require the violation of neutrino flavor and/or chirality in some way [6]. Two are based on neutrino mass splittings and flavor mixings: vacuum oscillations [7] and MSW matter-induced resonant flavor conversion [8]. Vacuum oscillations require fine-tuning either mixing fine-tuned to maximum, or the flavor oscillation length fine-tuned to the Earth-Sun distance (“just-so” oscillations). The third solution requires in addition that the neutrino have a magnetic moment: the matter-induced resonant spin flip (LMA [9]). The LMA effect would naturally explain an anti-correlation of the neutrino fluxes with solar magnetic activity, but requires a huge solar magnetic field (i0~—iO~G) and an enormous neutrino magnetic moment (~> 10— 12 p’s) that are difficult to explain except with otherwise unmotivated models of new physics [101. Both matter-induced resonances would occur in the sun itself. The fourth solution is neutrino decay into an undetectable final state [11]. Neutrino decay requires fine-tuning the decay length (neutrino lifetime) to the earth—sun distance to fit the energy dependence and can be reconciled with the SN1987A results only with large flavor-mixing anyway. Of these four, only the MSW effect can be considered natural: it occurs for a wide range of mass splittings and mixings without conflicting with known physics [6]. MSW oscillations can occur in three forms: most clearly presented in the MSW plane by mass-squared splitting .~m2and flavor mixing sin22O, they form a triangle with the horizontal large-mass adiabatic, diagonal non- or semi-adiabatic, and vertical large-angle branches, where the last becomes degenerate with vacuum oscillations as the mixing approaches maximum (fig. 1). We use meV iO~eV as a convenient unit for neutrino masses. Contrary to the data, the horizontal MSW branch and the cooler sun model all suppress high-energy neutrinos. The diagonal MSW branch and neutrino decay suppress low-energy neutrinos, consistent with the data. Maximal-mixing vacuum —

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sin2 2~ Fig. I. Two-parameter plane for the MSW effect, with overlap of separate Homestake and Kamiokande II solar neutrino data fits (each 90% CL). The hole in the large-angle solution is excluded by absence of earth-induced day—night variations in the K-Il signal [2]. Courtesy of K. Lande and A.K. Mann.

oscillations affect all energies equally, while “just-so” oscillations have a complicated sinusoidal energy dependence. The overlap of the separate 90% confidence fits for the Homestake and K-lI data in the MSW plane is shown in fig. 1, where ~.1m2= m~ m~ for two-flavor mixing [1,2,5,6,8]. (Alternatively, one could perform a joint fit of the combined data of both experiments.) As expected from its energy dependence, the horizontal branch is mostly excluded by the overlapping regions. This result is also favored by the K-Il energy spectrum measurement alone [2]. The allowed diagonal solution lies along the line —

z.~m2sin22O

0.04 meV2,

(3)

with ranges sin2 20 0.004—0.5 and 4m2 0.08—10 meV2, not requiring large mixing and consistent also with Cabibbo mixing, sin220~ 0.18. The large-angle vertical solution (sin22O 0.6—0.9) allows two ranges of mass splittings, zlm2 0.08—2 meV2 and 4—60 meV2, the region in-between being excluded by the absence of earth-induced day-night variations in the K-Il data. Finally, a horizontal solution is possible with 4m2 80 meV2: sin22O 0.02—0.6, consistent with Cabibbo mixing. The minimum 7tGa rate consistent with solar luminosity is 80 SNU, while the SSM prediction is 132 ±20 SNU [3]. The current result of the SAGE 17Ga collaboration, R~ < 79 SNU (90% confidence) [4], disfavors any astrophysical solution and rules out the horizontal MSW branch (fig. 2). The lower parts of the

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sin 2 2 9 Fig. 2. MSW iso-SNU plot for 71Ga detectors, with overlap of the Homestake—Kamiokande H allowed regions shaded. Contours mark lines of equal r’e survival probability from 0.1 to 0.9 in steps of 0.1, normalized to the Bahcall—Ulrich predicted rate of 132 SNU. Adapted from fig. I and W.C. Haxton, ref. 18]. diagonal and vertical MSW solutions are still allowed by the combination of all three experiments, and Cabibbo mixing would then be allowed only on the diagonal branch, with the low rate RGa 10 SNU (fig. 2). Future 71Ga observations will distinguish the diagonal from the vertical solutions only if R~ < 20 SNU. Another key test of the MSW solution will be provided by the neutral-current mode of the Sudbury neutrino experiment (SNO).

3. Neutrino mass and mixing: Seesaw models 3.1. GENERAL FEATURES

The MSW solution (3) has important implications for particle physics: it represents new physics beyond the Standard Model (SM). Neutrino masses and mixings can be incorporated into the SM without difficulty, but one needs a definite model for their origin to obtain predictions and explain their smallness [12,13]. Models with ordinary minimal Dirac masses for neutrinos have little predictive power and cannot naturally explain why m~~z m~,where 8 is a charged lepton. A more attractive class of models is based on explicit or spontaneous breaking of lepton number and yields Majorana masses for neutrinos, but these

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too have serious problems. Triplet Majoron models [14] affect the Z width (Z —p Majoron + scalar contributes as two additional neutrino generations) and thus are ruled out. New charged scalars have been proposed to generate Majorana neutrino masses and magnetic moments at loop level, but the predictions depend on unknown Yukawa couplings and scalar masses [15].Only the “seesaw” models appear to have real predictive power, albeit with many uncertainties remaining [12,13,16] A seesaw model requires a right-handed SU(2) singlet Weyl neutrino NR for each ordinary VL. A Dirac mass term *

**~

=

mDVLNR + h.c.,

(4)

is generated by the ordinary Higgs mechanism (Higgs doublet) in analogy with the Dirac masses for the quarks and charged leptons. Because neutrinos are neutral, lepton number violating Majorana masses are also allowed, in contrast to the charged fermions. In the absence of Higgs triplets the ordinary VL cannot acquire such a term at tree level, but the NR can have a Majorana mass, one typically associated with some large scale beyond the SM =

MNNJ~NR +

h.c.,

(5)

where N~ CN~ is the left-handed CPT-conjugate of NR and C the charge conjugation matrix, with MN >> mD. The neutrinos then have a 2 X 2 mass matrix 0

mD

where

4

P~

MN)(NR)’

~~tiLNL)(T

(6)

Cv~ is the ordinary right-handed antineutrino and m~= mD. If there are two Majorana mass eigenstates i.’~ and ~~‘2 with masses

=

MN >> mD,

m~, m

1

—

MN

~

m2MN,

mD

(7)

thus explaining why ordinary neutrinos are so much lighter than the charged fermions (m~~ mD). The light (m1) and heavy (m2) neutrinos coincide with VL and NR, up to small O(mD/MN) mixings, which we henceforth ignore.tiL’ N~,4, be extended to three generations, if we take matrix and andThe NRseesaw each model to be can a three-component vector, mD a 3 X 3 Dirac *

**

For a review of the seesaw mechanism see ref. [131. Previous applications of the seesaw for the MSW results include ref. [16] and ref. [211 of ref. [13].

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MN = M~a 3 x 3 symmetric Majorana matrix. If the three eigenvalues of MN are all much larger than the components of mD, then there are three light Majorana neutrinos with mass matrix m~mDMNmD,

(8)

and three heavy Majorana neutrinos with mass matrix MN. The limit with MN proportional to the identity is the quadratic seesaw, because the m~then vary as m~.Alternatively, the eigenvalues of MN may follow the same hierarchy as mD (that is, MN approximately proportional to mD), thus leading to the linear seesaw: the m~vary as mD in this case. The matrix MN can also be generated at ioop level; in some of these cases, MN is proportional to mD, leading to a linear seesaw [17]. Another complication is that the simple seesaw models assume a vanishing tiL Majorana mass, but important terms of this type can be generated at ioop level in extended models [12]. 3.2. GRAND-UNIFIED SEESAW MODELS

Seesaw models have been proposed for many scales MN of new physics, from MN TeV up to MN 1016 GeY [12,13]. For MN > 0 (iO~)GeV and mD m~or m~,where m, and m 0 are the mass matrices for the charged leptons and up-type quarks, the seesaw model can generate a hierarchy of neutrino masses m~~ m~ ~ m~ in the range relevant to the solar neutrino problem and cosmology and that is consistent with all laboratory and astrophysical limits. We will take the four cases of linear and quadratic seesaws with mD = m~or m, as spanning the likely range of theoretical possibilities. To discuss the seesaw model predictions in any greater detail requires a definite model. The most predictive are the seesaws of grand unified theories [18], which naturally give m0 = m~at tree level, and generally predict CKM mixing; i.e. that the quark and leptonic mixings are the same. In other models, one often expects mD mj. For m~,choosing m1 or m~clearly makes a substantial difference. The Majorana matrix MN is another source of uncertainty. The tree-level relations (7) are modified by renormalization group corrections if we run them down from some unification scale to low energy. The mixings will also require renormalization group corrections that are generally minor [19]. For these reasons, seesaw calculations should be viewed as a semi-quantitative guide to possible neutrino masses and mixings rather than as rigorous predictions. Two models consistent with the measured weak mixing angle, the minimal supersymmetric Standard Model (MSSM) and an SO(10) grand unified theory breaking in two steps, are presented in appendix A. In the CKM quark mixing matrix, the mixings of the first, second and third families follow the definite hierarchy [20]: sin 6~3 0.001—0.02
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10

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I

~

I 1111111

I

I 1111111

I

I I III

1o~ io~ 1o2 10_i sin2 29 I

11111111

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11111111

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Fig. 3. MSW fits as in fig. 1, with predictions of seesaw models from appendix A superimposed and assuming V 1~~1 VCKM; ~ f—a v~for SUSY GUT; v~a-a v~ for SO(10). The superheavy SO(10) scale MN shown is from appendix A, but can be adjusted.

0.06
zontal MSW branch (fig. 1, upper left corner) would imply m~ 3—10 meY; with the up-quark seesaw, the third-generation MN 3 m~/m~ 0(1015) GeV, the grand unification scale. The MSSM seesaw of appendix A is illustrative of this possibility (fig. 3), but the small-mixing solution is already excluded by comparison of the Homestake and K-Il data and also by the SAGE measurement. 3.3. SECOND FAMILY: p-NEUTRINO MASS

Cabibbo mixing, on the other hand, is allowed by the MSW fit, as shown above, with high-mass horizontal 2and low-mass diagonal solutions. The Depending seesaw mass = m~ and m~ 0.5 meV or 9 meV. on hierarchy then implies z.1m the model (linear or quadratic, lepton or up-quark), the second generation MN 2 0(109_1012) GeY, the so-called “intermediate” scale, distinguished from grand unified scales > 1014 GeV (table 1). This coincides with part of the allowed range (108_1012 GeV) for the breaking of the Peccei—Quinn U(1) symmetry in invisible axion models [211. (This upper part is favored by the SN1987A constraints on axions.) This might be coincidental, or the two scales, intermediate and Peccei— Quinn, could be related: models are possible in which the same Higgs field plays both roles. The intermediate scale also occurs in the spontaneous breakdown of

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TABLE 1 Seesaw predictions for the second family superheavy Majorana neutrino mass MN

2 = m~/m, or m~/m,, and for m~,assuming m~=0.5 meV and 9 meV. Quadratic up-quark predictions for m,. include radiative enhancement factor of two to four mD—mt m~=0.5meV (quadratic) I’ (linear)

MN2(GeV) m~(eV) m,, (eV)

2 5x10’ 3—21 0.03 —0.05

2X10’° 0.1 0.01

1 50—380 2x10’ 0.5—0.9

1x109 2 0.2

m~=9meV MN rn,2(GeV) (eV) rn, (eV)

(quadratic) (linear)

hidden-sector supergravity, with

“

~/m3/2MPIanck

lO~GeV, where m

3/2 TeV is the gravitino mass [22]. The intermediate-scale SO(10) GUT model of appendix A predicts a m,, range that overlaps with the high-mass solution (fig. 3) and, with realistic modifications, might be made consistent with the SAGE-favored low-mass solution as well. 3.4. EXTRAPOLATION TO THIRD FAMILY: r-NEUTRINO MASS

With rn, the i--neutrino mass can be predicted directly from the seesaw. The ranges of n~ spanned by the linear and quadratic up-quark and leptonic models, given m,, 0.5 or 9 meV, are shown in table 1 (see also fig. 4). These ranges are derived l~’yrescaling m~ by the first or second power of m1/m~or m~/m,1and including a radiative enhancement factor of two to four for the quadratic up-quark case (see appendix A), which then overlaps with the range of cosmological and laboratory interest. ,

102

I

I

10 >

a)

i

UP-QUARK

2 10 0

LEPTON

100

) 200

mt (GeV) Fig. 4. Ranges of the tau neutrino mass rn~, predicted by linear and quadratic seesaw models, for m~=0.5 meV: up-quark models (open), leptonic models (hatched). Following table 1.

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The dynamical age of the universe and its present rate of expansion together constrain the present normalized mass density of the Friedmann universe (without cosmological constant) to 120h2 = 0.03—0.25, with h = 0.5—0.7 [231. Since rn,, = 92[2,,h2 eV, m~<23 eV. The mass density in relic neutrinos will equal that in baryonic matter (Q~h2 0.01—0.02) if m~= 1—2 eY; the universe can be closed with neutrinos comprising 90% of the mass density (fl 0 1, fl, = 0.9) if m,, 21 eV. In that case, massive neutrinos provide a natural source of hot dark matter [231. Together with non-gaussian primordial fluctuations, massive neutrinos could then be responsible for large-scale structure, especially if the canonical cold dark matter scenario proves unable to account for the very large cosmological structures now observed. The observed small-scale structure would probably still require cold dark matter [24]. Majorana neutrino masses will also drive lepton number violating processes in the early universe, but the range of masses and mixings predicted by the MSW effect and the seesaw are probably too small to produce any difficulties with cosmological B L generation [25]. The high-mass solution (rn, 9 meV), already disfavored by the SAGE data, predicts rn~ 50—380 eV in tF’e quadratic up-quark case (table 1), which is cosmologically excluded. On the other hand, the low-mass quadratic up-quark prediction (rn,, 3—21 eV) coincides with the cosmologically important range almost exactly (table 1 and fig. 4). Although tie ti~ oscillations are too see CKM-type either in the laboratory or be in 2>small 1 eV2toand mixings should the sun, v~ p~ oscillations with z.1m observable experimentally. For example, the existing 90% limit for oscilla—

~

+-*

~

tions is sin220,~~ <0.004 for ~im2> 20 eV2 [26] which already excludes the central prediction sin220~ 0.009. Alternatively, CKM mixing of v~ and v~ requires m~<1 eY, too small to be of cosmological significance. Atmospheric neutrino experiments are sensitive to much smaller /~rn2,but only for large mixing, sin226~>0.4 [271. *~

3.5. COMMENTS ON POSSIBLE 17 keV NEUTRINO

A mass 17 keV neutrino with 02 1% mixing in /3-decay has been reported in some crystalline detector experiments [28], but not in magnetic spectrometer experiments [29]. Such a neutrino, if it exists, must impart only a small (<1 eV) Majorana mass to v~ in order to evade double /3-decay constraints, and needs to decay invisibly and fast to satisfy cosmological and astrophysical limits [30]. If the double /3-decay and laboratory neutrino oscillation constraints are to be satisfied, a 17 keV neutrino must be a Majorana ti~ mixture accompanied by a Majorana partner either degenerate in mass or ten to sixteen times more massive [31,32]. The possibility of accelerated cooling of the supernova SN1987A remnant —

*

~

Future experimental prospects are surveyed by Harari [26).

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neutron star appears only if the 17 keV neutrino or its partner is a new sterile neutrino [331. A 17 keV neutrino appearing with 1% mixing in /3-decay cannot be responsible for MSW solar neutrino oscillations and cannot arise naturally in three-family seesaw models. Non-seesaw models can be contrived, but small neutrino masses do not then occur naturally and require a symmetry-breaking scale much smaller than the intermediate or GUT scales described above. The models of ref. [34] predict a 17 keV Dirac neutrino that satisfies the double /3-decay limits and decays fast enough to not overdose the universe, but not fast enough to allow time for large-scale cosmological structure to form [30]. *

4. Conclusions We conclude that, unless the 7Be(p, y)8B cross section or the solar radiative opacities have been grossly overestimated, there is a deficit of neutrinos coming from the sun. In any case, the energy dependence of these neutrinos appears to rule out a cooler sun. A minimal extension of the Standard Model leads to neutrino masses and mixings and to MSW neutrino flavor conversion in the sun. Comparison of the Homestake and Kamiokande II data fit to the oscillation parameters allows a mass-squared splitting ~im2 0.08—80 meV2 and is consistent with Cabibbo mixing. The current SAGE results favor the lower part of this Ltm2 range. We have seen how such neutrino masses can be easily accomodated by the seesaw mechanism in realistic grand unified models, which generally require CKIM mixing of the neutrinos and most likely imply a superheavy right-handed Majorana mass MN O(109_1012) GeV arising from intermediate-scale symmetry breaking. For quadratic up-quark seesaw models, the i--neutrino mass is cosmologically significant and will be accessible to future laboratory neutrino oscillation experiments.

We would like to thank Ray Davis, Jr., Ken Lande, Ming Xing Luo, Alfred Mann and Alycia Weinberger of the University of Pennsylvania and John Bahcall of the Institute for Advanced Study for their assistance and suggestions; and the Aspen Center for Physics and the Center for Particle Astrophysics for their support. This work was supported in part by the US Department of Energy under contract DOE-ACO2-76-ERO-3071.

*

MSW oscillations into a light sterile (SU(2)-singlet) neutrino possible.

~

with m~=0.3—9 meV would still be

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Appendix A. Two seesaw models The general prediction of seesaw models is eq. (8), relating the very light Majorana masses for the ordinary neutrinos ti to the superheavy Majorana masses of the associated right-handed neutrinos N mVmDMNmD,

(A.1)

where rnD and MN are 3 X 3 matrices in flavor space. We use the generic result of many grand unified seesaw models that rnD is simply rn~,the Dirac mass matrix for the up-type quarks (u, c, t), and that the charged lepton (e, ~)mass matrix rn~is equal to that of the down-type quarks (rnd) (d, s, b). First, we discuss the uncertainties and radiative corrections in the neutrino flavor mixings. In particular, we prove the significant result that the leptonic mixings are similar to the quark CKM mixings in a wide class of up-quark seesaw models. Second, we consider the tree-level uncertainties in eq. (A.1); and last, the general method for including radiative corrections in the seesaw mass predictions, exemplified by two specific GUT seesaw models consistent with the weak neutral-current data. One of these, an intermediate-scale S0(10) GUT, is encouraging and should be viewed as the first approximation of a more detailed realistic model. ~,

Al. SEESAW MIXINGS

The neutrino mixing angles are more stable against tree-level variations and radiative corrections in the grand unified seesaw than are the masses. We prove this assertion by considering the general case of (A.1). Let m~,,m~,m~,m°~ and M~ be the undiagonalized tree-level mass matrices. Then: m°D rn°~, rn°1= rnd, and rn°~ = (rn~)T = rn~(M~)_l(rn~)T in a grand unified seesaw. The light ti and the charged u, d, 8 fermion masses are the eigenvalues of rn~,m°~, m~and rn~, respectively. Diagonalize each of these four fermion matrices by seven independent, simultaneous L, R transformations: mf = (V~)tm~V~, where the m~are real and diagonal In the GUT case, V~R= V~R.Then: ~ = (V1~)tV~ and Viepi = (V~)tV(. With additional rotations, we can choose left- and right-handed fields such that rn°~ =m~ is diagonal and real, and V~R=1”~ept 11; and then quark ~ = V~ and VCKM mix~ept = (V~)tVcKM.The relation between ings is then controlled by V~, which is inthe turnleptonic determined by the nature of M~. Clearly, for M~diagonal, V~= J1; however, V~is close to the identity for a wide range of possible ~ because of the large hierarchy of up-type quark masses: rn~: rn~: rn~ 25000:300: 1. The sufficient requirements are only that (a) M~./M~
0~ is symmetric, so that V~= V~. *

There are seven independent transformation matrices because rn

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quark masses that is, the hierarchy of successive eigenvalues of M~varies no more than linearly with the same hierarchy of rn°D eigenvalues; and (b) I V~ I O(M~/M~,.), for i
—

~‘,

~

~ept IL + + £9 VCKM, since I I I for each i ~#j.We are therefore justified in assuming that Viept VCKM for a wide variety of seesaw models and in using the CKM matrix for the tree-level neutrino mixings. The neutrino mixing angle predictions from the seesaw will also receive radiative corrections as they are run down from the GUT scale M~.Because of the GIM mechanism, the gauge boson corrections do not contain large logarithms such as ln(Mx/Mw); rather only mass threshold corrections due to intergenerational splittings ln(m~/rn1),and these are small. The Higgs interactions, on the other hand, do not respect the GIM mechanism and can induce significant additional corrections for large fermion Yukawa couplings, as in the case of the fermion masses. For large m~,the tie v~and u~ v~mixings will then contain non-negligible corrections. Even for rn~ 180 GeV, however, these VCKM matrix elements (V~band VCb) are changed by factors of two or less; the Cabibbo angle remains virtually unchanged [19]. ~

~

~

A.2. SEESAW MASSES

The seesaw predictions for neutrino masses, unlike the mixings, depend directly on the absolute magnitude of the superheavy matrix MN, which can take on a wide range of forms. In any specific model, MN will be equal to an unknown Yukawa coupling matrix times a superheavy vacuum expectation value (v.e.v.), with these Yukawa couplings introducing one uncertainty into the masses. Another important tree-level uncertainty occurs in rn1, from the unknown top quark mass rn~.From precision fits of electroweak data, m1 can be bounded with some accuracy; we choose a range of m, = 124 ±34-0eV [35,36]. We continue to work in the convenient basis for which mD = rn~is diagonal and consider only the quadratic seesaw case, where MN is proportional to the identity, so that Viept VCKM. The simultaneous running of neutrino and quark masses and of the gauge couplings induces ioop corrections in the relationship (A.1). The one-loop renormalization group equations (RGEs) for the masses and couplings are: d In rn(p.) d ln ~ dg,~(1L)= —2b~g,~(/L),

(A.2a) (A.2b)

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where the first line describes the running of fermion masses due to gauge bosons (fermion self-energies) and the second the running of gauge couplings themselves (gauge boson self-energies). n is the gauge group index and is implicitly summed in (A.2a). p. is the renormalization scale, defined in a particular renormalization scheme; we use the modified minimal subtraction (Ms) scheme. For fermions with large masses, Yukawa couplings to Higgses will be important. For ordinary fermions, only the top quark fits in this category; its self-energy receives Higgs contributions proportional to m~,so that (A.2a) is modified dlnrn(p.) dlnp.

rn2(p.) b,~gn(p.)+b~ M~

(A.2c)

Ignoring the Higgs contributions temporarily, the solution of the coupled eqs. (A.2) is m(p.)

—

=

~

b(”)/b

a.

(A.3)

rn(p. 0)

,,

g,~(p.0)

p.0 is the renormalization point for the physical fermion mass. For the top quark, we separate the evolution of the mass due to gauge forces from that due to the Higgs: rn~(p.)= m1(p.)°f(p.),where rn~(p.)°satisfies eqs. (A.2a, b) and f(p.) incorporates the Higgs contribution, satisfying 2(p.) dlnf(p.) rn dlnp. =b~ M~ (A.4) with solution: i I

1f2(p.)

where we have chosen

2i \0_Ju/i ,.Lrn~/.L) =2b,~f ~ ~

(A.5)

f(p.

0) = 1 for convenience. A Landau pole is possible in (A.5) and in some models in fact occurs; that is, the right-hand side reaches unity for some finite p. (rn1
(A.6)

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The ROE solutions (A.3), (A.5) then relate rn~(Mx),m0(Mx) to their physical values at low energies; substituting these relationships into (A.6) yields the radiatively-corrected predictions of the seesaw model. The full solution for the running masses will be the product of factors (A.3), one for each gauge group; and the factor f(p.) for rn~.For large m1, the Higgs-induced correction to rn~,and thusthe to 2/rn~from m~,significantly alters the tree-level seesaw scaling m~/m,, = m, second to third families and is a general feature of grand unified seesaw models. We now present two specific grand unified seesaw models, both of which incorporate the general predictions rnD = rn~and ‘/~epI VCKM, and derive the radiatively corrected seesaw neutrino masses. A.3. TWO MODELS

The simplest grand unified theories, those which break in one step to the Standard Model (SM) or its minimal supersymmetric extension (MSSM), predict that the running SU(3), SU(2) and U(1) gauge couplings should meet at the unification scale (Mg). Equivalently, they predict the value of the weak mixing angle sin20w from the experimental value of a/as. The electroweak neutral current data are now sufficiently precise to test this prediction [36],with the result that ordinary (non-supersymmetric) single-step GUTs are excluded, while GUTs based on the MSSM are in spectacular agreement with the observed sin20~.The ordinary single-step GUTs are also excluded by the absence of proton decay. The proton decay rate in supersymmetric GUTs (SUSY GUTs) is suppressed by the larger unification scale 0(10 16) 0eV. Another possibility involves GUTs which break to the SM in two or more stages. Such models do not predict the value of sin2O~because they require extra parameters. For example, in SO(10) models breaking to the SM in two stages, the GUT scale M~ 0(1016_10t7) GeV and the intermediate scale O(10’°)GeV are determined from the experimental values of a/as and sin20~,.Despite the fact that the intermediate-scale GUTs cannot predict sin20~,they can be consistent with sin20~ and are necessary if the neutrino masses needed to solve the solar neutrino problem are to be explained naturally. The first model we examine is the SUSY GUT based on the MSSM. The three running gauge couplings meet at a unification point M~ with common value g 5(M~), yielding a successful prediction unify for the weak SUSY neutral current 2e~,[36]. The three gauge interactions to form SU(5), but mixing SUSY sin SU(5) does not contain the necessary right-handed neutrino NR. The full GUT must therefore be a larger group (such as SUSY S0(10)) that contains SUSY SU(5) as a subgroup, yielding the same predictions for M~as SUSY SU(5). Now M~= g 5(M~)v~~iand MN = hNvGUT, where hN is the unknown right-handed neutrino Yukawa coupling. Thus: MN = (hN/g5(M~))M~. To quantify the uncertainty in hN, we take the ratio h~/g5(M~) in the range 0.01 to 1. The unification

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scale M~has been computed using the two-loop generalization of eq. (A.2b) [36]. It is uncertain mainly because of the SU(3) coupling ~~(M~) = 0.12 ±0.012 and the SUSY scale M~,which we take over the range from M~ to a TeV. The combined uncertainties give M~= 1.6 x 1016 ±0.4 GeV, large enough to suppress proton decay from dimension-6 operators below the current observational limits. The final range for the right-handed neutrino mass is MN = 1.6 x iO’~±14 0eV. The radiatively corrected SUSY GUT seesaw predictions are 2 rn~= (0.05) rn MN

rn~ rn~= (0.09)—, MN

—~-,

rn,,

=

rn~ (0.38)—. MN

(A.7)

Using: rn~(p.= 1 GeV) = 5 MeV, rn~(p.= 2m~)= 1.55 GeV and m~= 124 ±34 GeV, we obtain: (rn~,rn~ rn~)= (7.8 X i0’3, 1.4 X iO~,(0.27—3.5) X 10~)X 10 ±14 eV = (< 2 x 10- “,~6x 10~—4x 10~, 1 x 10-~—0.9)eV. The additional uncertainty in the v~. mass is due to the uncertainty in rn 1, both tree- and loop-level. The Landau pole occurs at m~ 180 GeV, so that larger values of rn1 have a disproportionate effect on rn~.This SUSY GUT model gives too small a ti~2 massmixing for theatMSW solution. Ifcorner we useofthe ~ ti~ mixing instead, we now obtain a z~rn and the upper-left thev~MSW triangle, a region excluded ,

(fig. 3). The second model [13] is the non-SUSY S0(10) GUT, breaking down to the SM in two steps: SO(10) SU(3)~>< SU(2)R>< U(l)BL at M~ (left—right symmetric model); then breaking to the SM at some intermediate scale MR. The SU(2)R breaking at MR is caused by a Higgs triplet that also produces the Majorana mass MN Because of the extra scale MR, sin20~cannot be predicted; rather, sin20~and a/as together determine M~and MR. The major uncertainty is again due to a~(Mz).One obtains [38]: M~= 9.4 x 1016±0.450eV and MR = 1.5 x 1010 ±0.3 GeV. The SU(2)R breaking gives MN = (h~/g~(M~))M~. (Alternatively, we could have fit MN to the MSW data, then tried to construct a model that unifies at some Mg.) Assuming the same range for hN and rn~and the same rn~ and m~as in the first model, the radiatively corrected SO(10) seesaw predictions are —~

~.

rn~= (0.05)

rn2 ~

rn~= (0.07)

rn~ ~-~--,

rn~= (0.18)

rn~ ~—.

(A.8)

The predicted m~are: (mu, m~,m~)= (8.3 x iO~, 1.1 >< 10_I, (1.2—5.0) X 10~) x 10±1.3 eV = (<2 x i0~, 5.5 x 10~—2,60—10~)eV. Again, the second uncer*

This SU(2)R Higgs triplet is a singlet under SU(2)L. Such models also contain SU(2)L Higgs triplets, but we assume with zero v.e.v., thus causing no phenomeriological difficulties and no tree-level contribution to the upper left-hand corner in (6). There is no Majoron in this model, as B — L is gauged.

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tainty in the v~mass is due to the top quark mass at tree and loop level. The Landau pole for rn~occurs at rn~ 380 0eV. The lowest value of rn~ 60 eV somewhat exceeds the cosmological bound of 23 eV, while m~ overlaps with the high-mass solution on the horizontal MSW branch (fig. 3). This two-step S0(10) model is particularly promising in that it includes the seesaw in a natural way and predicts masses partly within the desired range. To overlap with the SAGE-favored low-mass solution on the diagonal MSW branch, rn, need only be lowered by a factor of ten, which might be arranged by varying the intermediate-scale Higgs content to raise MR. Thus increasing MR would also reduce the bottom of the m~ range to a cosmologically acceptable value of 6 eV.

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