The Super-Kamiokande contained neutrino interaction data consists of events with energy between 50 MeV and 1.5 GeV in deposited energy, and has been analyzed from 300 days of live time. Here we present a preliminary report upon the examination of the identified muon and electron events with energies up to 10 GeV in terms of the hypothesis of neutrinos of atmospheric origin undergoing flavor oscillations. We find that no-oscillations is rejected at the >5 sigma level on the basis of statistics alone. Acceptable fits between the data and simulations are found for muon neutrinos oscillating to either tau neutrinos or another sterile species. No evidence is apparent for electron neutrino oscillations in our sensitive region in mass squared differences (delta m^2 = 0.001 - 0.1 eV^2). We find that the muon neutrino mixing must be nearly maximal, with delta m^2 = 0.005 eV^2 with an error of about a factor of two. This region has been indicated previously as a possible resolution to the ``atmospheric neutrino anomaly''.
Here we present the first evidence that the effect depends upon distance (angle of arrival) and energy, in a manner consistent with neutrino oscillations. We also show that the phenomena seems to be incompatible with all neutrino oscillations scenarios except those with nu_mu oscillations to to nu_taus or nu_x. Note that with our as yet limited statistics we cannot claim certain discovery of neutrino oscillations, but this conclusion is assuredly the most favored interpretation of our present data set. Of course this is predicated upon the lack of discovery of any as yet unrecognized significant systematic bias in the data sample, a matter of intense scrutiny at this time (but no evidence for such an effect has been found).
I. Introduction
For many years now we have known that there is a discrepancy between the predicted and observed ratio of muon neutrinos to electron neutrinos of atmospheric origin in the energy range of around 1 GeV(Svoboda 1985, Clark 1997, Oyama 1989, Oyama 1997, Goodman 1997, Gaisser 1995). This anomaly is seen in not only different channels in the water Cherenkov experiments (identified muon and electron event topologies, and also in the fraction of events containing muon decays), but also (with limited statistics) in the Soudan detector (Goodman 1997), which employs a different target and has completely different systematics. (There were also reports from the two European detectors which had inconclusive statistical conclusions, and some concerns with use of exiting events in any case.) Moreover, with increasing detector size the discrepancy between data and predictions employing standard physics has become more inescapable. Many suggestions of systematic and calculational problems have been put forward, have been investigated, but none have provided an escape(Gaisser 1995, Harrison 1997). Since the first observations of the anomaly we have been aware of the possible interpretation as neutrino oscillations. Yet, because of limited statistics we have not had the ability to find compelling evidence for oscillations in terms of variation with distance and energy. We now have some evidence in hand.
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. Many clever schemes for neutrino mixing have been put forward to interpret the three present neutrino anomalies which are in solar neutrinos, the presently considered atmospheric anomaly and the more recent suggestion of oscillations from the LSND experiment(Athanassopoulous 1997). We cannot review them all here. We focus upon the possibility of simple hierarchical muon neutrino oscillations (Learned 1988, Barger 1988, Hidaka 1988, Acker 1993) for the present, since as we will show, they provide an adequate explanation of what we do observe. In Section II, below, we show that all but these simplest schemes of oscillations seem to be ruled out.
In the past the data has indicated a simple deficit in the muon/electron neutrino ratio (quoted most often in terms of the ratio R of the muon to electron neutrino fluxes observed, divided by the predicted ratio), without our being able to know whether the muons were disappearing, the electrons appearing, or muon neutrinos being transmuted to electron neutrinos. We have not seen previously, due to limited statistics, the expected variation due to oscillations in energy and angle. Herein we present the first evidence in that direction, as discussed in Section III.
In Section IV, we present a measurement of dm2 employing the R as a function of dm2. In Section V, we show the results of varying what seems to be the most sensitive parameter in making the fits to oscillations, the minimum energy. Conclusions are in section VI.
II. Asymmetry Analysis
In Figure 1 we present the variation with energy of an up-to-down asymmetry for muon and electron events (A=(D-U)/(U+D)), separately. The use of this ratio has the virtue that many possible systematics factor out, leaving the physics variation to be exposed with good statistical weight. The data was selected to be between 200 MeV and 5 GeV in visible energy. Below this energy range the angle determination deteriorates, and above the muons are not likely to be contained.
Figure 1: Up-to-down asymmetry for electron-like (left) and muon-like (right) events in Super-Kamiokande as a function of observed energy in GeV. Upper plots are with independent bins, lower show cumulative asymmetry above the indicated energy.
In the presence of no oscillations this plot should show no variation beyond a few percent due to some asymmetry in the earth's magnetic field's effect on incoming cosmic rays(Stanev 1997). One sees no discernible trend in the electron data, while the muon asymmetry is 0.17 +/- 0.05. In Figure 2 we present a theoretical calculation of the asymmetry expected for muon neutrino oscillations with delta m^2 = 0.005 eV^2. The calculation is for charged current events alone and does not contain experimental details, and thus represents what one might expect from an ideal detector.
Figure 2: Up-to-down asymmetry for muon-like and electron-like events in Super-Kamiokande as a function of energy from a theoretical calculation with muon neutrino oscillations inserted with delta m^2 = 0.005 eV^2 and full mixing. The electron asymmetry plotted here is the negative of the asymmetry elsewhere (will fix this), and shows a slight expected excess of upcoming electron events (2.5%) due to the complexity of the earth's magnetic field.
John Flanagan, Sandip Pakvasa and I have calculated the asymmtery predicted by eight different formulations of neutrino oscillations. (This has been submitted to PRL, 9/97, and can be found as preprint hep-ph/9709483). This is illustrated below in Figure 3. Following that is the SK 300 day (offsite) sample, in Figure 4. One sees that this test, with no need for the intercession of a Monte Carlo simulation, already eliminates all models except the simplest nu_mu <-> nu_x.
Figure 3: A plot of the trajectories of the electron asymmetry (vertical axis) versus the muon asymmetry (horizontal axis). The arrow heads show the direction of increasing energy. The parameters are typical of the solutions presented in the references. (See Flanagan, Learned and Pakvasa, hep-ph/9708483, for details.)
Figure 4: The same as in Figure 3, but showing the SuperK 300 day data for muon and electron asymmetries. The error bars are large, but the data overlays only the nu_mu <-> nu_x solution in Figure 3.
I have added an appendix which presents some studies which look for systematics which might have caused the signal we see. These are events entering from the outside (neutrons or gammas, for example), or some differential sensitivity in the upper and lower portions of the detector. The conclusion is that I see nothing to make me suspicious that we are being misled.
This asymmetry data is suggestive, but not compelling statistically, as yet. We note that this is however, the first evidence that the anomaly is surely in the muon channel and not in the electron channel. Note also that the asymmetry in the data does not depend upon any calculated flux or flux ratio: it is an independent test of the two neutrino channels. And this test demonstrates variation in angle and energy, the hallmarks of oscillations. No Monte Carlo simulation is required to demonstrate that the suggestions considered to date for systematic means to explain away the anomaly are strongly rejected. We only require the simulation to interpret the results in terms of some particular oscillation parameters, as we discuss below.
III. L/E Analysis
In Figure 5 we present a plot of the electron and muon data as a function of Log10(L/E). The data spans roughly 5 orders of magnitude because we have ranges of a factor of 25 in energy, and 10^4 in distance from overhead to upcoming. Unfortunately the energy we record from superk is not the true neutrino energy, but is rather widely distributed below the true neutrino energy (typically the visible energy is around 1/2 the neutrino energy). [We use estimators E_nu_e = E_vis + 487 MeV and E_nu_mu = 1.1 times E_vis + 360 MeV]. Moreover we do not reconstruct the neutrino angle very well (at best a few degrees, but often much worse). Also we cannot know the point of origin of the neutrino in the atmosphere. We have employed an approximation to the mean distance (Stanev 1997), and explored the effect of varying the effective production altitude. We conclude that the varying range, which is fractionally large for neutrinos straight downcoming, has little effect on the present analysis because, fortuitously, the neutrino oscillation length we find (~250 km for neutrinos of about 1 GeV) is much longer than the distance from overhead. Our resolution in L/E is shown on a log scale in Figure 6, where one sees that the FWHM of the distribution is about 0.2.
Figure 5: The distribution in observed events with visible energy between 200 MeV and 5 GeV in Super-Kamiokande in contained events from the first 300 days, plotted as a function of the neutrino flight distance divided by energy, L/E, on a logarithmic scale. The data is shown for electrons (a) and muons (b), with simulation events overlayed by a dashed line. The ratio of observed to predicted flux is presented in (c) and (d). Oscillation expectations for delta m^2 = 0.005 eV^2 are shown by dashes in (d).
Figure 6: The resolution in L/E on a logarithmic scale, as calculated using simulated data for reconstructed and true input values of neutrino flight distance and energy.
In the presence of oscillations one would like to see a ``chirped'' like signal in the L/E plot, Figure 5. In practice we have not achieved sufficient resolution (nor at present is it clear that one could do so even in an ideal detector of atmospheric neutrinos) to see the oscillatory signature, but only a smooth decrease from the expected number at the left (small L/E) as one moves to large neutrino oscillation times (large L/E), to a reduced (oscillated) fraction. In the upper portion of Figure 5 we show the observed data, with simulated data overlayed (dashes), for no oscillations for electrons (a) and muons (b). The Monte Carlo sample is equivalent to about 7.75 years of data, and is normalized to the observed livetime in this figure. We observe that in the ratio of simulated to predicted data, shown below, the electrons (c) are perfectly consistent with predictions whilst the muons (d) are not. In (d) we also show, with a dashed line, the results of injecting oscillations into the simulation with delta m^2 = 0.005 eV^2 and full mixing. One needs no calculation to see that the fit is obviously entirely satisfactory (and it is so numerically).
In Figure 7, we show the results of a calculation which injects neutrino oscillations into the simulated data set and calculates the chi^2 difference between observed and oscillated simulation data. We have used Poisson statistics for calculating equivalent chi^2 values where the bin contents are small. We present here a test of the hypothesis of muon neutrinos oscillating into another unseen neutrino (which could be nu_tau or a new sterile neutrino), with no effect occurring upon electron neutrinos. The dotted line shows the effect of normalizing the event totals to the muons in data and simulation, and testing the electron fit. This is equivalent to a test of R (the ratio of muon to electron neutrinos, observed to predicted) as we have presented previously, and shows a broad minimum in the region around 0.005 eV^2.
Figure 7: The values of equivalent chi^2 between observed and simulated data, with several hypotheses versus the trial value for muon neutrino oscillation with delta m^2 as shown. The dashed curve shows the results for testing the electron distribution in L/E space,while normalizing the real and simulated muon data (R test). The dotted curve shows the results for testing the muon data alone, while normalizing the muon event totals (L/E shape test). Finally, the solid curve shows the result for normalizing the electron total while testing the muons (shape and R test).
The dashed curve show the independent results of normalizing the muon flux whilst testing the chi^2 of muon data to oscillated simulated data. This test then is sensitive to the shape of the oscillation signal in L/E space, and again one sees a broad minimum. The same test with no-oscillations yields a chi^2/DOF = 146/10, a very poor fit.
Finally in the solid curve we show the result of a fit in which we normalize the electron fluxes, and simply test the observed muon flux versus the oscillated muon simulation. The hypothesis thus tested is: given that the simulation describes the electron data adequately, what muon oscillation parameters would be needed to describe the observed data? We find an acceptable fit with maximal mixing and delta m^2 = 0.0048, with the error best expressed as a multiplicative factor of 2.2. This value is gotten by fitting the curvature of the chi^2 near the minimum.
In Figure 8 one sees a plot of the chi^2/DOF of the muon neutrino oscillation hypothesis as we vary delta m^2 and mixing angle, sin^2(2 theta). One sees that the mixing angle must be close to maximal, with one sigma being at about sin^2(2 theta) = 0.8.
Figure 8: A plot showing the contours of chi^2/10DOF of fits to the 300 day Super-Kamiokande contained event data under the hypothesis of muon neutrino vacuum oscillations to an unobserved neutrino.
In Figures A.8 and A.9, in the appendix, we present the results of the same sort of scan shown in Figure 7, over dm2. Instead of using the distributions in L/E to test the data versus MC with injected oscillations, these tests were made versus zenith angle and versus energy. What you see on the left hand side, which gives the goodness of fit (in chi^2/DOF), is that the fit is not as nice in either angle (L) or energy (E), as in L/E.
IV. Measuring the Oscillation Parameters
[circulated as email note to he group, 28 Sep. '97]
Below you will find a figure illustrating a new way to "measure" dm2. The value I get for nu_mu<->nu_x is (3.09 +2.66 -1.71)*10^-3 eV^2 at the 1 sigma points, under the assumption of maximal mixing. This is explained below.In thinking about how to proceed with the oscillations game it seems to me we might break our process down into steps something like the following:
1) validation of the MC
2) demonstration of the cleanliness of the data sample
3) testing for the inadequacy of the null hypothesis
4) testing for acceptable models
5) measuring the appropriate parameters of acceptable model(s)
We have been working on numbers 1 through 3. Perhaps the up-down asymmetry tests demonstrate 4 (although other methods would work too). Then assuming we are satisfied that we have the right model(s) (for which the only candidate right now seems to be nu_mu <-> nu_x), we need the best tests to squeeze out s2th and dm2 values. (Moving from hypothesis testing to parameter estimation). This may involve the use of different analytical tools, but not necessarily (eg., devotees of maximum likelihood may claim it does everything.
First, it seems that we can use the plateau region in L/E space for a measure of s2th, decoupled from dm2 (as long as we fit in the plateau region alone). We need to work on this, but it is close to 1/2 (see section on threshold cuts below), maximal mixing, which is what we use in the following.
In talking with John Flanagan, we have been thinking that we need some method which puts as many events into as few bins as possible, for maximal statistical power to measure dm2. I will not trouble you with the variations we considered, but in the end we came back to good old R ([nu_mu/nu_e] data/ [nu_mu/nu_e] mc), though with R as a function of the dm2 inserted into the MC. This has the added virtue that is is easy to explain to people, as they are already used to R.
This you will see plotted in the following Figure 9. The leftmost value is actually for dm2 = 0.0 (no oscillations) and you see we get back the R = 0.65 of no oscillations. So, we can calculate the dm2 value by the crossing of R = 1.0, and take the errors to be where the upper and lower statistical errors cross 1.0 as well. Note we only account for statistics here (data and MC). Note also that since the curve crosses 1.0 at a low slope, this illustrates why we get such a wide region of acceptability in dm2. It also points out that the error band will shrink speedily with better statistics, which is good.
It is not obvious to me that this is the optimal combination of measureables to obtain dm2. I welcome suggestions.
V. Concerning the Minimum Energy Cut's Effect on Oscillation Parameters
[this section circulated as email note to he group, 30 Sept. '97.]
In the following we discuss the effects of the minimum energy cut on the resulting oscillations parameters. The summary is that smallest minimum energy cuts seem best (smallest dm2 errors), and that there seems to be something peculiar going on with the data when the cut is around 600 MeV.
Below you will find a picture with four graphs. The horizontal axis for each is the minimum visible energy (in GeV) employed for the oscillation parameter fits. The 300 day off-site data and 7.75 year off-site MC (East) are used.
The top left plot in Figure 10 shows the values of dm2 found from the R(dm2) test (which is to say, where R, as a function of dm2, crosses one). The upper (dots) and lower (dashes) are + and - one sigma statistical errors on dm2. The lower error limit falls away at evis min cuts above 600 MeV.
The top right plot in FIgure 10 shows the R_mu (= mu_data/mu_mc) value of the (presumably) oscillated plateau in the log(L/E) plots, that is above L/E > 10^4 km/GeV. This is a measure of 1 - 0.5sinsq(2theta) (under the assumption of nu_mu<->nu_x mixing). It should not be below 1/2 (which is maximal mixing). Note that it becomes non-physical in the region of 600 MeV in the minimum evis cut.
The lower left plot in Figure 10 is the difference between upper and lower one sigma dm2 values, on a log_10 scale (data from top left plot). This illustrates that we get the best resolution at the smallest values of minimum visible energy. [Note this may be in conflict with the results of Larry Wai, who studied the MC to see where to make the cut to get best resolution. I do not know why we have a different result].
The lower right plot in Figure 10 show the normalization factor, derived by dividing the number of observed electron events by MC electron events in the sample (assuming 7.75 years for the MC, and 301.4 live days for the data). One sees that the MC predictions are a bit low, by 15-18% or so below 500 MeV in the evis cut, and dropping above that to 6-10%.
One thing that seems clear from this study is that the data is telling us to go for the lowest evis. This maximizes the statistics and it gets one more points at large L/E values.
VI. Conclusions
In summary we are not claiming as yet to have definitively observed neutrino oscillations while simultaneously rejecting all other explanations. However, we now do have evidence which is perfectly compatible with oscillations while incompatible with no-oscillations at a very high statistical level, such that if there is something wrong it is not a statistical fluctuation, but a deep systematic error.
Plus, the data we have in hand behaves correctly in terms of L/E variation as expected from neutrino oscillations (where previous tests were insensitive to this variation). In up/down asymmtery tests which are presented in Section II, we show that the present data is apparently incompatible with all but the simplest of neutrino oscillation scenario of nu_mu <-> nu_x. Schemes such as the maximal mixing scheme of Harrison, Perkins and Scott (Harrison 1997), and schemes which vary with L times E (Pantaleone 1996, Glashow 1997), and all schemes with significant nu_e mixing at this level of dm2 (10^-2 to 10^-3 eV^2) are apparently ruled out. Our present data leads us to delta m^2 =0.005 +0.005 -0.004 eV^2 with near maximal mixing (sin^2(2 theta )>0.8) for muon neutrinos oscillating to another unseen species. The result is potentially compatible with the older models of Pakvasa and others (Learned 1988, Barger 1988, Hidaka 1988), more recent three neutrino mixing schemes such as some of the variations presented in Fogli, et al. (Fogli 1996), and essentially all models with approximately this mass difference and without much electron neutrino mixing at this level.
With the data presented herein representing 300 days of equivalent detector operations, or about one year of calendar time, we can expect that the statistical errors will be halved in a further three years. Further improvements in analysis, gotten through identifying better algorithms for determination of the unseen neutrino's L/E and perhaps better identification of event classifications may help. Meanwhile our main task seems to be to find ways to test the veracity of the simulations and our sensitivity to subtle systematics.
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Appendix: Search for Systematic Effects
Under Construction
We know that the results we are finding are not unique to Super-Kamiokande in terms of the overall deficit of muon events at low energies, since we confirm IMB, Kamioka and Soudan. The new feature is that we have enough volume and data to see higher energies and begin to see the variation with angle, energy and L/E, as discussed above in the main text. We still worry about some sort of detector anisotropy, or some as yet undetected background which could cause what we see. In the following I will employ the down-up asymmetry as the tool with which to search for bias. This parameter, as stated in the main text, has the virtue of being independent of the Monte Carlo simulation, and indicating something of physics significance in the data without need for complex interpretation. The question we want to explore is, does the asymmetry change in any significant way as we make various cuts on the data set?
What we plot for the asymmetry, as indicated in Figure 1 above, is the ratio of the difference between downgoing events and upgoing events divided by the total. The left plots are for electrons and the right are for muons. The upper set, in 600 MeV bins, from 200 MeV to 3.8 GeV, show the asymmetry in individual bins. The bins towards the right have a lot of scatter because of small statistics, and really only the first two have much resolution in asymmetry. The lower plot shows the integral asymmetry above a given energy, in 40 bins from 200 MeV to 4 GeV. The error bars are thus correlated from point to point. The purpose is simply to see the trend with energy. Again, the bins beyond about 2 GeV have large scatter from small statistics.
The first, simplest test is to make a larger cut on the fiducial volume, as we have done in figure A.1 below, where we cut 5 m from the walls instead of the usual 2 m. There is no big change, no hint that the effect is lessening. This single test indicates the lack of any trouble from entering particles, such as gammas or neutrons as suggested by others. Note that the 5m from the wall is on top of the roughly 3 m of surrounding volume, for a total of at least 8 strong interaction length from the nearest surface, and about 24 radiation lengths! I think it is safe to dismiss any large anisotropy such as we see from muons as being caused by downgoing backgrounds.
Another possibility we have thought about is that since the water flows into the bottom of the detector from the filter system, and is taken off the top, perhaps we could have some bias due to a gradient in the water clarity (eg. more scattering makes muons into electrons near the top). This sounds rather unlikely, and indeed the measurements of water clarity do not see any such effect, at least since the water clarity became stable in the Spring of 1996. Nonetheless, as a test of any such class of detector asymmetry, we have plotted the data in four ways: with the entire sample in the detector upper half (Figure A.2), lower half (Figure A.3), in the upper half looking down compared to the lower half looking up (Figure A.4), and the remaining combination, top looking up, bottom looking down (Figure A.5). The first two of these are perhaps foolish to plot since the detector is not symmetric (one direction looking into the wall, and the other way a long way from the wall). Still one seen no gross change in the muon asymmetry in any of these cases.
There does seem to be a peculiarity in the lowest energy electrons with asymmetry at the couple of percent level, which we are endeavering to understand at present. The electron asymmetry is predicted to be slightly negative (-2.5%) at low energies (few hundred MeV) by the calculations, but we see a little more, perhaps -5%. We do not yet know if this is some small low energy detector bias, or if the effect is due to the magnetic field of the earth not properly being accounted for in the flux calculations. Perhaps this is telling us not to take the asymmetries more seriously than to about 5%. The muon asymmetry is huge in comparison to this. In fact the muon asymmetry is, if anything becoming worrisomely large with energy (simple mixing will not allow it to be more than 1/3).
In the final two plots, Figures A.6 and A.7, we plot the same asymmetries but for the first and second half of the data set (before and after run number 3160). One sees no gross change to suggest detector instability. One can make endless plots with various cuts on the data sample. However, I think the foregoing are adequate to demonstrate the robustness of the asymmetry in the muon data, and the lack of anything consistent taking place with the electron asymmetry. This effect will not go away easily.
Figure A.1: Asymmetry plots for a cut requiring the vertex to be more than 5 m from any wall.
Figure A.2: Asymmetry plots for events in the detector upper half.
Figure A.3: Asymmetry plots for events in the detector lower half.
Figure A.4: Asymmetry plots for events with vertices in the upper half, going downwards versus events in the lower half traveling upwards.
Figure A.5: Asymmetry plots for events with vertices in the upper half, going upwards, versus data in the lower half, going downwards (opposite of Figure A.6).
Figure A.6: Asymmetry plots for the data in the first half of the run sample period.
Figure A.7: Asymmetry plots for the data in the second half of the run sample period.
Figure A.8: The values of equivalent chi^2 between observed and simulated data, with several hypotheses versus the trial value for muon neutrino oscillation with delta m^2 as shown. The dashed curve shows the results for testing the electron distribution in cos(zenith angle) space, while normalizing the real and simulated muon data (R test). The dotted curve shows the results for testing the muon data alone, while normalizing the muon event totals (cos(zenith angle) shape test). Finally, the solid curve shows the result for normalizing the electron total while testing the muons (shape and R test). Compare this with figure 5, for the sweep while comparing distributions in L/E, and the following figure A.9 for the same sweep while examining the energy distributions.
Figure A.9: The values of equivalent chi^2 between observed and simulated data, with several hypotheses versus the trial value for muon neutrino oscillation with delta m^2 as shown. The dashed curve shows the results for testing the electron distribution in energy, while normalizing the real and simulated muon data (R test). The dotted curve shows the results for testing the muon data alone, while normalizing the muon event totals (energy distribution shape test). Finally, the solid curve shows the result for normalizing the electron total while testing the muons (shape and R test). Compare this with figure 5, for the sweep while comparing distributions in L/E, and the previous figure A.8 for the same sweep while examining the angle variation.