Nuclear Physics B (Proc. Suppl.) 66 (1998) 226--230
mmmlmmllmlm PROCEEDINGS SUPPLEMENTS
T h e case for a sterile n e u t r i n o David O. Caldwell a* aPhysics Department, University of California, Santa Barbara, CA 93106-9530, U.S.A. If the solar and atmospheric neutrino deficits and the LSND results are consequences of neutrino mass, and if such mass is needed for a substantial part of dark matter, then there are only two basic patterns of those masses which are possible. The more likely of the two requires a sterile neutrino, which could also play a significant role in the dynamics of, and production of heavy elements by, supernovae. Important to this mass pattern is a correct understanding of the LSND results, now widely misinterpreted.
1, I N T R O D U C T I O N If all the current indications for neutrino mass are correct, the possible patterns are remarkably restricted. While one is free to disbelieve any of these experiments or observations, some papers claim to fit all existing data when they do not, and some experiments are planned which reduce their chances of success by ignoring information. All these indications for neutrino mass lead to just two basic schemes of neutrino mass, the more likely one of which requires introducing a sterile neutrino. Of interest for this case are possible additional indications of a need for that peculiar neutrino, as will be discussed here. Although the evidence for neutrino mass has increased considerably, the possible patterns of masses have not changed much in the past five years. When it seemed likely that a portion of the dark m a t t e r of the universe is in the form of neutrinos , the solar and atmospheric neutrino deficits could be reconciled with this new requirement in only two basic ways : either all three neutrinos had to be nearly degenerate with an appreciable mass, or a fourth (sterile) neutrino was needed. Variations on these schemes, such as inverting the mass hierarchy  or utilizing three-neutrino oscillations  have appeared, but the basic patterns remain the same today despite new information, such as the results of the LSND experiment  or the effect of neutrinos on heavy element nucleosynthesis in supernovae . These *Supported in part by the U.S. Department of Energy. 0920-5632/98/$19.00 © 1998 ElsevierScienceB.V. All fights reserved. PII S0920-5632(98)00042-5
latter two issues will be discussed below, particularly as the nucleosynthesis indicates a need for the sterile neutrino, and the results of the LSND experiment have been widely misinterpreted. 2. S U M M A R Y NEUTRINO
OF EVIDENCE MASS
The case for neutrino mass from observations of solar and atmospheric neutrino deficits is discussed in detail elsewhere in these Proceedings, so here only specific issues relevant to the neutrino mass patterns will be mentioned. The solar ve deficit now provides strong evidence for neutrino mass, since observations of four experiments can be quantitatively understood on the basis of neutrino oscillations, and of the three types of experiments, two have to be incorrect for an astrophysical explanation of the deficit to work . For this paper, it is important to note that a solution to the deficit in terms of neutrino oscillations requires that the mass-squared difference between the ve and whatever it turns into be no more than Am2~ ~ 10 -5 eV 2. Evidence for neutrino mass from the ratio of v~/v~ from atmospheric neutrinos is perhaps less strong but has increased recently with the confirmation by Super-Kamiokande and an improved result from SOUDAN II, giving a check on the water Cherenkov detectors by a calorimetric experiment. The Super-Kamiokande result at present, however, makes less clear the needed range of mass-squared difference, Am21, since the zenith-
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angle (hence source-to-detector distance) dependence is ambiguous. We shall see that this could be a crucial issue in determining the neutrino mass pattern. On the basis of the atmospheric data alone, it is also not clear whether the dominant process is v t, --4 ue or v~ --4 vT. Turning now to the apparent need for some of the missing mass of the universe to be in the form of neutrinos, it is important to distinguish what specific model of such mixed dark matter works. The cold dark matter model (CDM), which was a fair approximation to the structure of the universe, when normalized to the COBE data produced too much structure on small scales, since baryons readily clump around the cold dark matter. The first cold + hot dark matter models (CHDM) had --~ 30% neutrinos and fit structure on all scales very well  because the free streaming of the neutrinos reduced density fluctuations on small scales. Unfortunately, this damping of density perturbations also caused structure to form too late. Reducing the neutrino content to .-~ 20% allowed early enough structure formation . With all the mass in one neutrino species, this otherwise successful model (CvDM) overproduced clusters of galaxies. In other words, the CvDM model worked well at all distance scales except -.. 1 0 h - l M p c , where h is the Hubble constant in units of 100 k m . s - l . M p c -1. If h = 0.5 and ~t = 1 (i.e., a critical density universe), the mass of the neutrino required in the CvDM model is 94h2~F~ = 4.7 eV, for a neutrino fraction of fl, F~ = 0.20. If instead this mass is divided between two nearly degenerate neutrino species, the motivation for which is one of the two neutrino mass patterns mentioned in the Introduction, this Cv2DM model turned out to have a remarkable property . While 4.7 eV in one neutrino species or two makes essentially no difference at very large or very small scales, at .-~ 1 0 h - l M p c the larger free-streaming length of the 2.4 eV neutrinos tends to wash out fluctuations and lowers the abundance of clusters. Thus the model (Cv2DM) with two, 2.4 eV neutrinos fits structure information on all scales. The Cv2DM model works only if ~t = 1 and if h <~ 0.6. While formerly high values of h have been reduced (no longer an age crisis), so that this
requirement on h is likely to be met, there has also been more popularizing of a low-density universe, one that goes on expanding forever. However, as more data come in, the arguments for a low ~ are diminishing, and this particular Cv2DM model is finding added support from structure in the non-linear regime, the frequency of voids in the universe, and the lower than expected amount of interstellar infrared radiation . Since the pattern of neutrino masses (to be discussed in the next section) which motivated the Cv2DM model preceded results from the LSND experiment, it is essential to see if those results are compatible with the model. Recall that the LSND experiment detected 22 events of the type D~p -4 e+n, based on identifying an electron between 36 and 60 MeV using Cherenkov and scintillation light and correlated with a "y from np --9 d')' (2.2 MeV), whereas only 4.6 4- 0.6 were expected from backgrounds . This evidence for ~ -+ De oscillations received support from the LSND observation of the essentially independent process, v~ --+ v~, via about the same number of signal events (but now v~C ~ e - X ) , but about twice the background, making the chance of a fluctuation only about 10 -2, instead of the < 10 -7 of the D~ case. While the v, --+ v~ and D~ -4 P~ results are consistent, only the latter provide restrictions on the value of Am 2. The P~ results interpreted as a two-generation oscillation have been presented  in a plot like Fig. 1, except that comparisons were made to limits from other experiments. Fig. 1 is the correct way to determine favored regions of A m 2 as a function of the mixing angle, 0. The plot utilizes all the information about the events, in particular the neutrino energy, E, and the distance of the event from the source, L. In order to increase the range of L / E , values of E down to 20 MeV were used. Fig. 1 shows contours at 2.3 and 4.5 log-likelihood units from the maximum. If this were a gaussian distribution, which it is not (its integral being infinite), the contours would correspond to 90% and 99% likelihood levels, but in addition they have been smeared to account for some systematic errors. Comparison to the KARMEN experiment , which presents results in a similar way, shows no conflict, but if limits are plotted (as they are
D.O. Caldwell/Nuclear Physics B (Proc. Suppl.) 66 (1998)226-230
,~o 10 10
10 -1 -1 10
1 sin2 20
Figure 1. Mass-squared difference (Am 2) vs. degree of mixing (sin 2 20) for a ~ --~ ~e explanation of the LSND beam-excess data. Shown are regions of Am 2 favored using the energy (from 20 to 60 MeV) and distance from the source of each event. in Ref. 5) on this graph from E776 at BNL  and the Bugey reactor experiment , then one might conclude that the only allowed A m 2 region is 0.2-3 eV 2. If instead an 80% confidence level band is calculated to compare with the 90% confidence level limits of those experiments using, as they do, just numbers of events (i.e., not using the L / E information) and using only the 36-60 MeV region with its much lower background, then there is no conflict with other experiments above 0.2 eV 2, up to the recent limit of about 10 eV 2 from the NOMAD experiment, as shown in Fig. 2. Thus the LSND result is quite consistent with the requirements of the Cu2DM model, which would need Am2~ ,~ 6-8 eV 2 .
3. P O S S I B L E N E U T R I N O TERNS
M A S S PAT-
As mentioned in the Introduction, if all of the evidence discussed above for neutrino mass is correct, then only two basic patterns of neutrino mass are viable. Because of the LSND experiment, the one in which ue, u~,, and ur must be almost degenerate in mass has to be altered slightly
Figure 2. As in Fig. 1, but the LSND p, data here give an 80% C.L. band to be compared with the LSND u, result (solid lines), KARMEN (dashes), E776 (dots), Bugey (dash-dot), and NOMAD. from five years ago to include three-neutrino oscillations . The second pattern remains as it was before: the introduction of a sterile neutrino to accommodate the three A m 2 required. These two schemes will now be addressed in more detail. For the first pattern, the near degeneracy is needed because the mass differences between each pair of neutrinos must be small for the solar and atmospheric oscillations, but the sum of the three masses must be ,~ 5 eV. While fewer detailed simulations have been done for the case of threeneutrino dark matter, this division of the needed mass probably does not work quite as well as the two-neutrino case. A possibly more serious problem is that either the neutrinos must be theoretically unfavored Dirac particles, or it is likely in the Majorana case that there is a conflict with neutrinoless double beta decay limits. The contributions to the effective neutrino mass for this process from two out of the three neutrinos can cancel, but the contribution of the third is likely to be too large. This scheme gets around the problem of too many Am2's by making the one measured by LSND the same as for the atmospheric neutrino
D.O. Caldwell/Nuclear Physics B (Proc. Suppl.) 66 (1998) 226-230
deficit; i.e., ~ 0.3 eV 2. This value is larger than usually assumed to explain the latter observations, but is possible if the zenith angle dependence of the Super-Kamiokande data indeed goes away. The second difficulty with a three-neutrino scheme, too few neutrino flavors (i.e., r~ ~ re is needed for both the solar and LSND cases), is avoided by invoking three-neutrino, or indirect, neutrino oscillations. Many papers have been written on three-neutrino oscillation schemes, but the one described here  is the only one which can be compatible with all the manifestations of neutrino mass described in the previous section. Some papers claim such compatibility, but these claims are incorrect , often requiring energy independence of the solar neutrinos, for example. While this three-neutrino scheme  is not really in conflict with data yet, it can easily be eliminated in the near future. On the other hand, the other possible scheme has no experimental or observational problems. The solar deficit is due to re -+ rs, a sterile (i.e., SU(2) singlet) neutrino, the atmospheric r~,/re ratio is explained by r~ --+ r r , LSND is observing r~ --+ re with A m ~ .-~ 6-8 eV 2, and the dark matter is shared between the nearly degenerate r~ and rr, with re and r, being much lighter. The re ~ r8 solution to the solar neutrino problem works for either small-angle MSW or for vacuum oscillations, and both avoid bringing the sterile neutrino into equilibrium in the early universe, eliminating problems with nucleosynthesis. Whether ve -4 r , is correct can be tested by a combination of SNO and Super-Kamiokande results. 4. S U P E R N O V A
While the three-neutrino scheme is compatible, the four-neutrino pattern apparently conflicts with the production of heavy elements in the outer neutrino-heated ejecta of Type II supernovae. This r process, the rapid capture of many neutrons, can provide a limitation on the mixing of r , and re. The limit comes about because energetic r~ ((E) ~ 25 MeV) could convert via an MSW transition to re inside the region of the r process, producing re of much higher energy than the thermal re ((E) ~ 11 MeV). The latter,
having charge current interactions, emerge from farther out in the supernova where it is cooler. Since the cross section for yen ~ e - p rises as the square of the energy, these converted energetic ve would deplete the neutrons, stopping the r process. Calculations  of this effect limit sin 2 20 for v~ -~ ve to <~ 10 -4 for Am2~ >~ 2 eV 2, conflicting with the LSND A m 2 needed for the four-neutrino scheme but being compatible with that for the three-neutrino one. The sterile neutrino, however, can not only rescue the four-neutrino scheme, but also the r process itself, since it has been found recently that the r-process region is not sufficiently neutron rich in conventional simulations. The sterile neutrino would produce two effects. First, there is a zone, outside the neutrinosphere (where neutrinos can readily escape) but inside the v~ --+ ve MSW ("LSND") region, where the v, interaction potential goes to zero, so a v~ -4 r8 transition can occur nearby, depleting the dangerous highenergy v, population. Second, because of this v~ reduction, the dominant process in the MSW region reverses, becoming ve -4 v~, which reduces the ve flux into the r-process region, making it more neutron rich because of fewer yen -+ e - p reactions. Detailed calculations  even show that a high A m 2 like 6-8 eV 2 is needed in order that the MSW region be at a sufficiently small radius to reduce the re flux enough. The sterile neutrino may have two further uses in supernovae. While the r process goes on at rather late times (~ 10 s post bounce), another nucleosynthesis process goes on at early times (~ 1 s) and requires a proton-rich region. This p process could be aided by two regions where the ve interaction potential goes to zero, one inside the core and the other outside the neutrinosphere. The first converts re --+ rs, which then escape the dense core and reconvert at the second, producing high energy res which produce protons via t e n e - p . These regions are effective only at these very early times and may also provide the needed extra energy deposit to blow off the supernova mantle, since in present calculations the shock is stalled at an early time (0.15 s post bounce) close to the radius at which the rs -9 re reconversion would occur at that time.
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Thus there may be evidence for the sterile neutrino which had to be invoked in a rather ad hoc fashion to provide the small mass difference scale needed to explain the solar ve deficit, if the other active neutrinos are needed for the atmospheric v~/ve ratio and both the hot dark matter component and the LSND result.
5. C O N C L U S I O N S Evidence is increasing for neutrino mass from solar, atmospheric, and LSND observations, as well as detailed simulations of cold + hot dark matter. If all these indications for mass are true, only two mass patterns remain: either the three active neutrinos are nearly degenerate in mass, with three-flavor oscillations being important, or the u~ and ur are nearly degenerate in mass (for the atmospheric u~ deficit) and constitute the hot dark matter, while a light ve and a sterile neutrino, us, explain the solar ue deficit. The former scheme may have problems with neutrinoless double beta decay limits and requiring the Am 2 needed for LSND and the atmospheric observation to be the same, ~ 0.3 eV 2. In contrast, heavy element nucleosynthesis in supernovae, and possibly even the explosion itself, may provide evidence for the sterile neutrino. The supernova neutrino laboratory actually favors a larger value of Am 2 in the LSND case, compatible with that needed in the four-neutrino scheme. It is unfortunate that no experiment in the near future will provide a direct check for that value of Am 2.
8. 9. 10. 11.
Appreciation is due for extensive contributions to different parts of this work provided by G.M. Fuller, R.N. Mohapatra, J.R. Primack, Y.Z. Qian, and S.J. Yellin.
1. While cold + hot dark matter has been advocated since 1984, it was not studied in detail until much later, the first impressive quantitative results I saw was at a 1992 conference, since published as J.R. Primack and J. Holtzman, Gamma-Ray Neutrino
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