This memo is an html version, slightly updated, of a memo first distributed to the SK collaboration on 15 July 1997 (jgl, 26 Aug. '97).

Evidence for Neutrino Oscillations Using

Up Coming Muon Data from Super-Kamiokande

John G. Learned

University of Hawaii



Atmospheric neutrinos are produced as a consequence of cosmic ray interactions with the upper atmosphere, and the decay of daughter mesons. The predicted fluxes are hard to calculate from first principles because one must take account of the entire cascade process. The neutrino flux peaks near the horizon at higher energies, where the particles arriving near tangentially to the earth have a better chance to decay. While the muon neutrinos are expected to be twice the numbers of electron neutrinos at energies around 1 GeV, due to decreasing probability for muon decay, the muon neutrinos dominate at higher energies, reaching a factor of 20 (vertical)-30 (horizontal) by 1 TeV. The distance from decay point to detector varies over a range from about 14 km downcoming, to 350 km from near the horizon, to 12,000 km for upcoming neutrinos. For angles below the horizon, the flight distance varies by about 15% due to decay point fluctuations.

For the study of muons produced by neutrino interactions outside the detector, we are restricted to muons below the horizon. This is because there are about 10^5 as many muons coming directly down from the cosmic ray interactions in the atmosphere as there are muons from locally interacting neutrinos. Even though the downgoing muons have higher mean energies (around 300 GeV as compared with muons from neutrinos being around 20 GeV at the detector), we cannot distinguish them in the downward traveling hemisphere. The downgoing cosmic ray muon flux drops below the flux of muons from neutrinos beyond about 85 degrees in our detector.

This variation of the neutrino flux with angle and energy, which is predictable to about 10% in relative fluxes, provides an excellent tool with which to explore for neutrino oscillations. In the standard form of vacuum neutrino oscillations, the probability of neutrino oscillation from one flavor to another varies with sin^2 (delta m^2 L/4E), where the delta m^2 is the mass difference between two mass eigenstates, L is the flight path, and E the neutrino energy. For the present study we do not observe the interaction, but only the muons which have reached the detector, which may have originated distances of many meters away, and even kilometers for muons of very high energies (TeV's), though rarely. The muons which traverse the detector have a mean neutrino source energy of about 100 GeV, and those that stop in the detector of about 20 GeV, though both distributions are very wide (100% FWHM). Consequently, any given direction samples a wide range in neutrino energies, and, if oscillations are present, averages over a wide range in L/E.

The angle measurement for the entering muons is relatively good in contrast, being reliable to about 1 degree in reconstruction. The major source of scattering, more than electromagnetic interactions in traversing the earth, is from the weak interaction itself, and amounts to about 3 degree between muon direction and neutrino direction for the energies in question.

Angular Distribution Analysis

We have analyzed so far 247.4 live days of data, from the period of 29 April 1996 through 20 March 1997, with a net livetime fraction of roughly 80%, including experiment run-in. In this sample we have 343 muons which have detectable entry points in both the outer and inner detectors, and which have more than 7 m of visibly (Cherenkov radiating) path length in the inner detector. The effective flux, averaged over the lower hemisphere is 1.76 +/- 0.10 times 10^-13 muons/cm^2/sec/sr (statistical error only). This may be compared to predicted fluxes of 1.99 (Agrawal 1996) and 1.86 (Honda 1995), in the same units. More about this below.

Note that, as illustrated in Figure 1, while the minimum muon energy in the detector is 1.7 GeV, the effective energy threshold is near to 6 GeV for throughgoing muons, in the inner detector alone. The additional requirement of an entry point in the outer detector adds about 3 m to the range requirement. We have also extracted the entering-stopping muons, which are a substantial fraction of the throughgoing events (about 30%), and these allow us to set a simple muon range (energy) threshold independent of location of the entering muon. Such a data set can then be directly compared with a flux calculation for muons with energy above a given threshold.

In Figure 2 we present the angular distribution of upcoming events with path length more than 7 m (including stopping muons) in the detector (corresponding to a muon energy threshold of 2.50 GeV), and in dashed lines with more than 20 m in the detector (corresponding to a muon energy threshold of 5.36 GeV). The upper figure gives numbers of events, and the lower gives the derived fluxes. Note that these fluxes are different than presented in the past since they require only one threshold muon range for all zenith angles.

We have examined the effect of simple neutrino oscillations on the upcoming muon angular distribution. We have made a muon flux calculation employing the Bartol flux (Agrawal 1995), injection of putative neutrino oscillations, and Stanev's (Stanev 1997) conversion probability between a neutrino and a muon above a given muon energy threshold, yielding a muon flux. We employ a simple geometric approximation to the Super-Kamiokande detector area for events with more than the specified muon path length in the detector (7 m) to then get predicted numbers of events. We create a table of the integrated predicted flux (starting from the nadir) versus 100 cosine zenith angle bins for each dm2 value tested, and then use an interpolation routine to yield the running value for each real event's cos(theta) value. The energy and angle integral for each zenith bin (of which we use 100) is in 10 costheta steps and 1000 energy steps, up to 1 TeV. We have explored the stability of the calculation to the number of steps: running the calculation with more steps gives essentially the same results.

The implicit assumption here is that the muon angle gives us the neutrino angle. This is not precisely the case, but our studies indicate it not to affect the results. There are so many convolutions (in the spectrum, energy transfer, and range), as to make the calculations insensitive to angle imprecision. Source decay altitude fluctuations do not count for much in these events either, as all flight paths are long compared to the decay distance range, and for the foregoing reasons as well. All this said, we need the MC muon sample to do the job right, but this is still in preparation as it requires huge amounts of computer time.

The data and predicted angular distribution of events are subjected to the Kolmogorov-Smirnov test for each delta m^2 value tested. This test operates on shape only, and is essentially self-normalizing for each delta m^2 value. Figure 3 shows -Log_10(K-S Probability) versus delta m^2, from 10^-3 - 10^-1 eV^2. The first point, below 10^-3 eV^2, shows the value for delta m^2 = 0.0. The null hypothesis is rejected at the level of 8.5 times 10^-4, which corresponds to 4.5 sigma. There is a broad minimum in delta m^2 space, and the result is perfectly consistent with that seen in the Super-Kamiokande contained events.

It is not yet clear whether we can tighten the precision of delta m^2 measurement with this data set. We will use the MC to examine the benefits of more data and make quantitative studies of methods to improve he delta m^2 resolution.

Apparently the reason the test works so powerfully is that the data has a rather flat region at angles nearer the nadir, and then rises strongly near the horizon. Qualitatively this is the same trend as observed in the IMB (Svoboda 1985) and Kamioka (Oyama 1989) data samples. Simple chi^2 does not do so well for such a test as presented above. Since the chi^2 test operates bin-by-bin independently, it has no sensitivity to the overall shape.

One important caveat is that the calculation predicts too few events with oscillations, giving close to the observed number with no oscillations. It has long been remarked that the absolute flux calculations are extremely difficult, and cannot be trusted for absolute rates. The authors claim absolute magnitude of the fluxes to be reliable to around 20%. Of course all neutrino flux calculators roughly knew the answer from the earliest experiments (dating from 1965) before presenting their results. Still the absolute normalization discrepancy is a concern. Eventually we hope to reconcile the predictions between throughgoing and contained events, not an easy business. For now we think it justifiable to ignore the absolute magnitude of the flux predictions for upgoing muons.

Note inserted 10/97: Improved calculations, particularly with crossections in the lower energy regime have to a large extent alleviated this concern. Quantitative statements herein await checking of the numbers.

We have not yet discussed potential systematic errors. First, there can be some selection bias on the muons. Triggering bias is not an issue as these events are all far above threshold, so one should look to selection inefficiency. We have studied the comparison of data samples between on-site and off-site analysis groups, and we find no detectable bias in angular sensitivity, with an upper limit of 1%. We may mis-measure the track length systematically at the level of <1 m. This corresponds to an uncertainty in the incoming muon energy threshold of about <10%. This bias, if it exists, is however independent of angle. In any case the dependence of the muon flux on energy cut is slow, since the average atmospheric neutrino induced muon energy at the detector is about 20 GeV. In sum we find nothing of concern for systematic error in the sample preparation.

There is however one dangerous area, in fact crucial, which is the question of muons near the horizon being scattered into the near horizon sample. This issue was mentioned earlier, in that we find no event clustering near the thin side of the mountain. This issue has been studied previously by the Kamioka group (Oyama 1989) as well. Estimates from colleagues at Tohoku University (Hatakayama 1997) suggest an upper limit of 2 events scattered into this sample from above, which is negligible. Moreover, we have made the K-S test, as illustrated in Figure 3, with a 20 m pathlength minimum (5.4 GeV minimum muon energy) instead of the 7 m minimum discussed above. Because of the steep energy dependence of any hypothetical muon scattering, this 20 m sample should have less of any pollution. The oscillation effect survives despite significantly decreased statistics, and strongly rejects the no-oscillation hypothesis. Still, although the near horizon bins are just where the physics effect should be due to these oscillations, the concern for being fooled gives us caution.

Conclusions on Oscillation Effects in Upcoming Muons

While the Super-Kamiokande is vastly more massive than predecessor solar neutrino experiments, such that it overwhelmed previous experiments statistically within one month of operation, Super-Kamiokande is only about a factor of three larger in muon area than the previous largest detector. It will be a few years before Super-Kamiokande dominates the upgoing muon event total. However, because of the greater dimensions and complete outer detector, we have the ability to set a higher and more uniform muon energy threshold (by muon range). This has proven crucial, as detailed in the foregoing analysis to being able to analyze the angular distribution in terms of consistency with predictions, in terms of distribution shape alone.

Rather remarkably the present analysis rejects the hypothesis of consistency with the expected muon angular distribution with no-oscillations at a level equivalent to 4.5 sigma. At the same time we find that the data does fit well to the calculated angular distribution of the neutrino flux when one includes the hypothesis of simple neutrino oscillations to an unseen neutrino (possibly nu_tau). The region of acceptability is lamentably large at this time, spanning more than an order of magnitude, centered about 0.01 eV^2. This region of acceptability overlaps the region found in the Super-Kamiokande contained event sample. Given the > 5 sigma statistical rejection of no-oscillations in that sample, and the present results, there would seem to be no sensible probability for a statistical fluctuation. Further, given that the data set and systematics of both results are quite independent, we get a check against undetected systematic errors.

That said, one should be cautious about the scientific interpretation of the atmospheric neutrino conundrum. What we have found is that straight-forward expectations for upward moving neutrino interaction in the vicinity of the detector are in deficit by nearly a factor of 50% over an energy range from around <1 to about 20 GeV. It appears that the deficit has some of the hallmarks of muon neutrino oscillations to an unseen neutrino. However, we have assuredly not rejected all other scenarios. Further studies are underway to define the class of acceptable models and to attempt to refine our determination of the mass square difference.


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Figure 1: The effective area for muons versus zenith angle in Super-Kamiokande. The filled region is for muons with more 7 m of range upon entering the inner detector, including those that stop within the detector. The upper curve shows the mean pathlength in the detector for throughgoing muons.

Figure 2: The zenith angle distribution of upcoming muons in the 247 day Super-Kam sample. Numbers of events per cos(theta) bin are shown in the upper plot, and the equivalent flux values in the lower. The upper points are for muons with more than 7 m pathlength in the detector, and the lower for muons with more than 20 m.

Figure 3: The plot shows -Log_10(KS Probability) versus delta m^2, from 10^-3 - 10^-1 eV^2. The first point, below 10^-3 eV^2, shows the value for delta m^2 = 0. The null hypothesis is rejected at the level of 8.5 times 10^-4, which corresponds to 4.5 sigma. The solid curve is for a 2.5 GeV muon energy threshold, and the dashed curve for 5.4 GeV.

latest ed jgl 8/25