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First Draft, 12 December 1997

The Case for Nu_mu Oscillations in SuperK

John G. Learned

University of Hawaii


There are three steps in the process of claiming the discovery of muon neutrino oscillations, as now envisioned:

We will discuss these individually now.

Eliminating Alternative Hypotheses

All alternative hypotheses so far presented as explanations of the atmospheric neutrino anomaly are listed below in Table 1 below, in summary notation which we will explain in the following. The anomaly is over a decade old, having been first found by the IMB group as a deficit in the number of muon decays seen in contained neutrino interactions, as compared with calculations.

The dominant contained-neutrino interaction in underground detectors originates from the quasi-elastic interactions of neutrinos made by cosmic rays impinging upon the high atmosphere. The anomaly was strongly reinforced by evidence from Kamioka and IMB when techniques were developed to distinguish between these single track events, resolving them into showering (electron-neutrino-like) and non-showering (muon-neutrino-like). The anomaly as it existed up until Super-Kamiokande, was in the double ratio R of muon-like events to electron-like events, data divided by expectations. The effect is not small, being about 0.6, a deficit of nearly a half.

There have been three other detectors which have had data bearing upon this anomaly, the Frejus, NUSEX and Soudan instruments. The exposures for the first two were small, and there are considerable questions about entering events in those instruments (now highlighted by the Soudan detector, which has a surrounding veto shield). New results from Soudan that better account for entering tracks yield a value for the ratio of ratios of R=0.61(+/-0.15 +/-0.05) (see talk of T. Kafka at TAUP97).

In the pre-Super-Kamiokande data, labeled ``old'' in the Table below, we saw no angular or energy dependence of the effect, and could not distinguish between an excess of electron events or a deficit of muon events or some combination of the two. It was, thus, impossible to distinguish between the possibilities of there being more electron-like events from some peculiar form of proton decay, or an extraterrestrial source of electron neutrinos, and the many proposed variant forms of neutrino oscillations.

In Table 1, the xx refers to items which have been doubly confirmed in Super-Kamiokande, and generally made much stronger by the new data. It must be emphasized that the size of the anomaly being as large as it is, does not permit a subtle problem to be the cause of the anomaly. As we will discuss later, it is true that subtle effects can have significant influence on the parameters measured, but not enough to dismiss the phenomena.

In the Table 1, we show the ``problem'' hypotheses summarized as atmospheric neutrino problems, cross section difficulties, systematic misidentification of particles, entering (generally downgoing) backgrounds, and inherent detector asymmetric response. We discuss each of these.

Table 1: Summary of alternative hypotheses to explain the atmospheric neutrino anomaly, with new evidence from Super-Kamiokande.

 | Evidence        |         Old        |            New            |
 |                 |--------------------|---------------------------|
 |                 |   R  | mudk | Vol  |  R   |  A_e | A_mu |R(L/E)|
 | Hypothesis      |E .lt.| Frac | Frac |E .gt.|   ~0 |.gt. 0| ~0.5 |
 |                 | 1 GeV|      |      | 1 GeV|      |      |      |
 |                 |      |      |      |      |      |      |      |
 | Atm. Flux Calc. |  xx  |      |      |   x  |      |   x  |   x  |
 |                 |      |      |      |      |      |      |      | 
 | Cross Sections  |  xx  |      |      |   x  |      |   x  |      |
 |                 |      |      |      |      |      |      |      | 
 | Particle Ident. |      |  xx  |  xx  |      |      |      |      |
 |                 |      |      |      |      |      |      |      | 
 | Entering Bkgrd. |      |      |  xx  |      |      |   x  |      | 
 |                 |      |      |      |      |      |      |      | 
 | Detector Asym.  |      |      |  xx  |      |      |      |      |
 |                 |      |      |      |      |      |      |      |
 |                 |      |      |      |      |      |      |      |
 | X-Ter. nu_e     |      |      |      |      |      |   x  |   x  |
 |                 |      |      |      |      |      |      |      |
 | Proton Decay    |      |      |      |   x  |      |   x  |      |
 |                 |      |      |      |      |      |      |      |
 | nu_mu Decay     |      |      |      |      |      |      |   x  |
 |                 |      |      |      |      |      |      |      |
 | nu_mu Abs.      |      |      |      |      |      |      |   x  |
 |                 |      |      |      |      |      |      |      |
 | nu_mu -  nu_e   |      |      |      |      |   x  |      |      |
 |                 |      |      |      |      |      |      |      |
 | nu_mu -  nu_s   |      |      |      |      |      |      |      |
 |                 |      |      |      |      |      |      |      |
 | nu_mu -  nu_tau |      |      |      |      |      |      |      |
 |                 |      |      |      |      |      |      |      |

Atmospheric Flux:

Atmospheric flux calculation problems cannot be the source of the anomaly, as was concluded by many who studied the situation over the last 15 years. Simply, the ratio of atmospheric muon neutrinos to electron neutrinos is largely determined at low energies (below several GeV) by the decay scheme and kinematics to be 2:1. Different models give significantly different spectra, but the ratio is robust. There is some complication at the few hundred MeV energy range due to the lopsided nature of the earth's magnetic field. This causes a predicted up/down asymmetry in the electron neutrino flux of about 5%, which is observed.

Cross Section:

A mistaken double ratio due to a difference in electron neutrino and muon neutrino cross sections was an idea pursued for a while as an explanation for the anomaly. The well established fact of lepton universality prevents any significant difference, except at energies near the muon mass, where calculable kinematic differences exist. The fact that the anomaly now is seen to persist to energies above a GeV, dismisses this possibility.

Particle Identification:

Biased particle identification was another concern. We have the independent cross check of a sample of events where the muon is observed to stop and decay, which gives completely consistent results in IMB, Kamioka and now Super-Kamiokande. Nonetheless, a beam test was carried out by both the Kamioka and the IMB groups at KEK, which validated the particle identification methods already in use. We are concerned about particle identification as a function of energy, particularly as we move to higher energies now available. This is a matter about which the Collaboration is cautious, since while it will not make the anomaly go away, it can significantly pull the values determined for delta m^2.

Entering Backgrounds:

Cosmic ray induced backgrounds that enter the detector have long been suggested as a problem. In smaller detectors, in particular the more compact and dense instruments such as the Frejus, NUSEX and Soudan projects, leakage is a serious matter. However, the water Cherenkov detectors are so large that these problems are not significant. In particular, in Super-Kamiokande we have more than 2.5 meters of material outside the inner volume and, in addition, we make a software fiducial cut requiring vertices to be 2 meters inside the inner detector walls. There are typically at least 5 strong interaction lengths of shielding and roughly 14 radiation lengths of active material before something can get into the inner volume. Beyond that we make successive data cuts restricting the analysis to more deeply shielded inner regions, and find no trend to suggest any problem with entering background. This was true in Kamioka and IMB, and is made much stronger by the new and vastly larger detector.

Detector Bias:

Inherent detector asymmetries have been a worry in the past, but more so now that we have enough of a data sample to see up/down asymmetries, as we observe. Additional tests not reported here, indicate nothing within an order of magnitude of being able to explain the effects seen. A summary argument is that we do not see a strong up/down asymmetry in the electron data, but do so in the muon data. We have been unable to invent systematic mechanisms which would only effect the muon data. (This is also an argument against entering backgrounds, which would be mostly downgoing and would generally pose as showering events).

Now we discuss some of the physics explanations for the anomaly.

Extraterrestrial Electron Neutrinos:

The possibility of an extraterrestrial electron neutrino flux causing the R to be low is dismissed by the observation of an up/down asymmetry in the muon data, which indicates that the muons are in deficit and the cause of the anomaly. (We discuss the up/down asymmetry later.)

Nucleon Decay:

The possibility that nucleon decay could have been the source of the anomaly is multiply eliminated, because of the low R value extending to beyond 1 GeV and angular variation in the muons.

We have added two seemingly unlikely hypotheses to the list for completeness:

Muon Neutrino Decay:

We can reject a new and unknown mode of muon neutrino decay by the observation that the plot (see below) of R(L/E) descends from 1.0 to a plateau at around 0.5, and does not continue exponentially to zero.

Muon Neutrino Absorption:

Another seemingly unlikely hypothesis might be that the muon neutrinos are somehow being absorbed by a new and unknown process or transformed by some sort of spin flip to a sterile state. Such processes should track the column density of the earth. The core of the earth occupies only about 0.1 of the solid angle at the detector, and the anomaly is seen in much of the solid angle. There is nothing in the angular variation of R that suggests a correlation with the column density of the earth.

Oscillations Involving Electron Neutrinos:

Getting to the bottom of the list, we are left with only neutrino oscillations. In our data, the L/E variation of the electron neutrino events is consistent with expectations, and we have no sign of electron neutrino oscillations (in this energy range). If the muon neutrinos were oscillating to electron neutrinos we would see dramatically different up/down asymmetries that we do observe. This can be made more quantitative, but in light of the restrictive new limits on electron neutrino oscillations from the CHOOZ experiment, covering just the range where electron neutrinos would be seen to oscillate if they partook of the atmospheric anomaly, we can safely dismiss that hypothesis. However, three neutrino oscillations schemes can have some odd effects, and this statement requires quantitative examination (again, see below).

Muon Neutrino Oscillations to Sterile or Tau Neutrinos:

Finally we are left with only the hypotheses of muon neutrino oscillations to either tau neutrinos or a new sterile species. The energies are sufficiently low that we cannot observe tau neutrino interactions (at least not yet), so the sterile and tau neutrino hypotheses are (so far) indistinguishable. There is one way to test the sterile neutrino hypothesis, however, which we are working on, and that is via the up/down ratio in the neutral currents. Unfortunately the best telltale signals so far tagged, with pi^o's, have not yielded sufficient statistics to make a distinction. Another possibility has to do with matter induced differences in the ratio of stopping to through going muons (discussed below, but again difficult).

Consistency with Muon Neutrino Oscillations

All tests so far indicate consistency of the data with muon neutrino oscillations with nearly maximal mixing and a mass squared difference in the range between 0.001 eV^2 and 0.01 eV^2.

Further tests upon the angular distribution of up-coming muons yield consistency with the indications from contained neutrino interactions. At this stage the up-coming muon data would not stand on its own merits to claim oscillations. This is discussed below in the up-going muon analysis section.

There are no further implications in any other experiment or situation which speak for or against the conclusions drawn from the contained event data. There appear to be no contradictions.

Measuring the Oscillation Parameters

We discuss in more detail later (in the oscillation analysis section) the shift from hypothesis testing to parameter estimation. These are different activities, and one may employ significantly different tools. In the latter case, we take as a working hypothesis the reality of oscillations of muon neutrinos, and attempt to extract the best values for the parameters (delta m^2 and sin^2(2theta) for two neutrino oscillations). The Collaboration is still struggling to find the best, least biased means for making reliable parameter determinations. We can be certain that the mixing is very nearly maximal. We can also be confident that the delta m^2 value is in the range between 0.001 eV^2 and 0.01 eV^2, but we have not yet settled on the best method and, hence, the best value.

Where Do We Go From Here?

Obviously the next major step for Super-Kamiokande is to work very hard on trying to find or dismiss an up/down asymmetry in the neutral current interactions. (If muons are oscillating to taus, then the taus will have the same NC interactions, and the NC events will be up/down symmetric. If the muon neutrinos are oscillating to a new sterile species, then this sterile species will not have NC interactions, and the NC events will have an up/down asymmetry which is the same as the muons diluted by the electron NC events.) The first look at statistics from this search are not very encouraging, but it is an area in active development and we may be able to make a distinction in a few years.

As mentioned above, we may have a chance to make a distinction between sterile and tau neutrinos via matter effects upon the up-coming muons. This will involve effort to eliminate systematic uncertainties, and also some luck as to where the delta m^2 value settles (no sensitivity for some values).

The direct solution would be to observe tau neutrino interactions in the detector. This is difficult, and so far we are not encouraged by the possibility, as tau neutrino interactions look very much like ordinary NC or electron neutrino interactions. One might hope that the appearance of cascading contained events at tens of GeV would be a signal, but unfortunately even as the electron neutrinos are becoming more scarce with energy, so also is the oscillation length for the muon neutrinos becoming long.

In summary, we have outlined in the preceding, in a somewhat qualitative discussion, the case we are formulating for muon neutrino oscillations in Super-Kamiokande. We believe that we are now in a position to make a convincing case that only the hypothesis of muon neutrino oscillations fits the data we have obtained during the last 1.5 years in Super-Kamiokande.

written 19 December 1997, last edited for html by jgl, 12 Jan 98