Internal Document, Not for Distribution

Preliminary Draft, 25 August 1997

Evidence for

Neutrino Oscillations

in IMB Upcoming Muon Data

John G. Learned

University of Hawaii

Abstract


Introduction

As the analysis of the contained neutrino interaction data in SK has become more strong it is seen that the deficit in the R (= flux of nu_mu/nu_e, data over prediction) is definitely in the range of 0.5 - 0.7 (as shown earlier in IMB, Kamioka, and Soudan). And, with increasing evidence, it appears that the problem stems from muon disappearance. We have thus looked at the upcoming muon data for support or confrontation.

The contained data indicate large mixing and a mass square difference (dm2 hereafter) of around 0.005 eV^2, but with a fairly large band of acceptability in dm2, ranging from as low as 0.001 to nearly 0.01 eV^2. All tests show a very steep slope in the probability functions on the upper side, pushing us downwards in dm2, but the tests are a little softer on the small side of the dm2 minimum.

The range of dm2 and the implication of muons disappearing, imply necessary and detectable consequences for the flux of upcoming muons from atmospheric neutrinos traversing the earth below the horizon. For present purposes we stick with analysis of muons with a minimum energy of about 2 GeV passing a given location underearth. This muon flux, which has a mean energy of about 20 GeV at the detector, originates in neutrinos with energies ranging from roughly 10 GeV to 1 TeV, with a mean of around 100 GeV, a very broad distribution (the fairly steep neutrino spectrum is sampled with muon range and cross-section rising with energy, flattening the response curve). As it happens, the dm2 indicated by contained data corresponds to an L/E = 250 km/GeV. Thus most atmospheric neutrinos throughout the lower hemisphere of arrival directions are well mixed, reducing the average muon flux. We have no chance to see the actual oscillations in the upcoming muon flux, due to the continuous spectrum and all the convolutions. So we must look for a smooth decrease from the unoscillated flux to the mixed flux versus angle, and, as the calculations indicate, this occurs most strongly just below the horizon.

Unfortunately it is those bins which are most vulnerable to pollution from downgoing muons that contain the most information, an uncomfortable situation. I should note, for the non-expert, that the events in question are assuredly real muons with angles below the horizon. These are not misfit events, they have been multiply scanned and fitted. There have been studies done in the past, looking at upscattered muons, and none indicated (need references here) any problem for muons of more than about 1 GeV. Thus if we are having a problem it is due to an unpredicted large angle scattering of muons.

As reported earlier (jgl note to SK collaboration, 7/97) we have tested the SK upmu data, and find 4.5 sigma equivalent rejection of the null hypothesis (no-oscillations), and acceptable fits for muon neutrino oscillations in a region overlapping that indicated by contained data, though the acceptance basin is shifted a little towards larger dm2 (but not significantly). Hence a test of the IMB data can be crucial, since the rate of muons about 10 degrees above the horizon was about a factor of ten less than in SK, and hence any in-scattered muons must be proportionately less.

The IMB Upcoming Muon Sample

The IMB upcoming muons were analyzed in detail in three dissertations (R. Svoboda, UH, 1985; R. Becker-Szendy, UH 1991; G. McGrath, UH 1993). The most recent full analysis was the Becker-Szendy thesis, drawing on the work of Svoboda. The McGrath thesis was mostly concerned with higher level analysis and with downgoing muons, and mainly brought the Becker-Szendy data set to completion with the final few events from the last IMB runs. Other unpublished work has been done on the sample, particularly for stopping muons, by Svoboda's group at LSU. Much of the focus of all these works was upon attempts to discern any sources of extraterrestrial neutrinos, and so the effort was highly directed towards finding means for accurate muon angular reconstruction.

Neutrino oscillation studies were conducted with the IMB upmu sample, nonetheless. Employing chi-squared tests on the upmu flux in ten angular bins simply did not give enough statistical power to discriminate between the angular distributions with and without oscillations, as long as one normalized the total numbers of events. We did see that the total number of events matched the predicted number with no oscillations, and we employed this to set limits on muon neutrino oscillations, ruling out a large region above dm2 = 10^-2 eV^2. That result was, however, criticized by others as depending upon the choice of flux model and such, and this resulted in a lively debate. This debate may be renewed now, and I find myself on the opposite side than earlier (in times past I was a non-believer in oscillations... the SK contained data has completely won me over).

It is a measure of how far we have come in data processing technologya decade or so, that a major limiting factors in the upmu analysis in IMB were the availability of adequate computing power and affordable storage media. The on-line computer in the initial IMB runs was partially programmed in machine language, and was filled to the extent that not a single statement could be changed without major difficulty. We could not afford to store the bulk of the data which was from the downgoing muons, so we had to sort though the data on-line to identify upcoming muons. Much effort by a number of collaborators went into fast muon fitters and a fairly good efficiency was achieved, but it was far from 100% efficient, particularly near the horizon. On the positive side, we saved muon events with zenith angles beyond 82 degrees, and were able to produce a neutrino sample which extended to 5 degrees above the horizon.

In the end we cut the sample exactly at the horizon. Having made the initial cuts higher, on automatically fitted data, this means that the inevitable migration of events from slightly below the horizon to slightly above is compensated by events moving in the opposite direction upon final fitting. (In SK there is a slight, perhaps 1%, net migration of events OUT of the upmu sample). The rectangular geometry of the IMB detector also made for strong discrimination between events near the horizon, to about one degree, whereas the overall reconstruction accuracy was no better than 2.5 degrees (compared to about 1 degree in SK for all muons). In sum, the IMB sample is clean, though deficient in events near the horizon, from data lost by the initial upmu tagging algorithms (mostly FASTMU, see Becker-Szendy thesis for details).

The analysis strategy for the IMB data was to generate Monte Carlo simulated muons, run them through the same on-line selection routines and compare with the real data. (This method has been used in Kamioka and SK as well). While the technique, common to accelerator based HEP analysis, nicely accounts for the details of the detector, data selection and fitting (to the extent that the MC is correct), it suffers from the fact that the analysis and interpretation of data requires the MC. One generally does not thus produce some simple result which anyone may then interpret in their own way. [This has caused erroneous interpretation by several groups attempting to use IMB and other data for comparison of results among detectors, as in the Gaisser, Halzen and Stanev review (Physics Reports 258, 174, 1995) and neutrino oscillation studies, as in the papers by Lisi, Fogli and company (eg., hep-ph/9708213, 1997). I think we experimenters are largely at fault for not being very clear about what we were presenting.] In any case, this kind of analysis, where one puts the putative physics into the MC, grinds it through and then compares with the data sample, certainly restricts the kinds of tests one can make upon the data sample.

In the SK data we have the opportunity to make nice clean cuts on the upmu sample, by requiring only some minimum track length in the detector (say 10 m), and keeping all events which pass that cut, whether they subsequently stop in the detector or exit. The result is that is is as though the instrument had a constant thickness, independently of zenith angle (though with varying capture area). We can thus make (and have made) clean calculations with fixed muon threshold energy of flux versus zenith angle. [Please note that while we have plotted a flux versus zenith angle in the past for all these detectors, it has typically been with the weighting of the particular instrument for throughgoing muons. This has meant, for example, that the mean muon momentum sampled by the instrument, varied from maxima near the horizon and nadir, through a broad flat minimum at slanting angles. ]

With the above in mind, I went back to Becker-Szendy's thesis. There he has presented the effective area versus zenith angle for the IMB detector (his Figure 4.17). This curve is a weighted and smoothed average between that for IMB-1,2 and that for IMB-3 (roughly 2/3 of the data). This is shown in the following :

The IMB data sample is shown below in terms of the zenith angle distribution for upcoming muons with greater than 2 GeV of energy at the detector. The upper figure is the raw number of events as a function of zenith angle in degrees (90 = horizontal, 180 = straight up coming). The lower figure shows the calculated flux, employing the foregoing effective area, the livetime of 1315 days and the data sample. The units are 10^-14/cm^2/sec/sr. The data is presented (in opposite order to the first plot) in 10 bins in the cosine of the zenith angle (0.0 = horizontal, -1.0 = straight up coming). The error bars represent Poisson fluctuations.

The K-S Test of the Angular Distribution

I have written a routine for examining the SK upmu angular distribution (7/97). This program calculates the expected atmospheric neutrino induced muon flux versus angle for muons above some prescribed energy energy threshold. It does this calculation employing the Bartol neutrino flux model, with an interpolation subroutine supplied by Todor Stanev. For each energy the routine calculates the neutrino flux and then applies oscillations (or no-oscillations, as appropriate). The flight distance is taken to be that of the geometrical distance to an altitude of 20 km at production. The flight distances do vary a few percent, but the approximation is good for events below the horizon (particularly given the convolutions that follow). Standard nu_mu <-> nu_tau (or nu_sterile) oscillations (with maximal mixing angle) are generated.

The muon flux is then generated with the aid of a function also supplied by Todor Stanev, which function gives the probability of a given neutrino resulting in a muon of more than the stated threshold energy at some point underground (see the GHS review paper, op. cit., for details). This function includes the details of cross-sections and muon propagation. Neutrinos and anti-neutrinos are treated separately. Finally the muon flux is integrated over the neutrino spectrum, for each zenith angle. In order to minimize effects due to beating of the binning frequency with the oscillations, many steps are employed (1000 in energy, and 10 for each zenith angle bin, of which 100 are used).

The resulting muon flux is integrated over angle to provide a smooth integral distribution of the number of muons expected below a given zenith angle. An interpolation routine is employed for making values available for any zenith angle. With this function normalized to the total expected flux in the lower hemisphere, this provides the test function for the Kolmogorov-Smirnov test.

The data set is tested in the usual way, checking the difference between the cumulative data fraction at each zenith angle and the test function, and employing the maximal difference in the statistical probability for the two samples being drawn from the same distribution. The K-S test is good down to as few as 10 events, so there is no problem with small numbers. The K-S test, like all statistical tests, is more sensitive to some differences between the data and test function than others. For example, the K-S test works best for shifts in the mean between statistical distributions, but is not very sensitive to variations in width. It is also most sensitive to variations in the middle range of the distributions, and single "bulges". (By the way, it is a rank test, which is invariant under any conformal transformation of the scale, as long as it preserves the data order). The test is less sensitive to variations near the ends of the distributions, which is also a good feature here, it seems. This test thus seems to be a very nice match to our needs. Note that the K-S test totally ignores the absolute normalization.

Results

We employ the final IMB muon data sample of 591 events with zenith angles below the horizon, and with more than 2 GeV of energy seen in the IMB detector. The sweep in dm2 is illustrated in the following figure, and discussed below:

The K-S functions are illustrated in the following figure, where the data is the wiggly line, the best fit for oscillations is the solid line, and the no-oscillations case is the dotted line.

The next figure shows the difference between the data and test functions for the same two cases. The vertical axis is the negative log of the K-S probability, point-by-point. Only the maximum value is used (the number of trials is already accounted for in the K-S probability). The plot illustrates however where the maximum difference occurs, and we see that the peak is at a cosine zenith angle of around -0.36.

The dm2 scan above shows that there is a broad maximum in the probability of the data and test function representing the same data set, in the range on dm2 = 0.005 eV^2. The vertical axis is -log10(P_KS). Thus one represents a 10% probability, two a 1% probability, etc. The null hypothesis is represented by the left most point, plotted at dm2 = 6E-4 eV^2, which is actually for no-oscillations, and which has P_KS = 6E-5, or about 5 sigma equivalent. We may thus conclude that the no-oscillation hypothesis is rejected with a high statistical probability, and that oscillations with the stated dm2 are perfectly acceptable, under this K-S test. Note that we do not, of course, rule out other hypotheses. We can only say that the one tested is acceptable, and the null hypothesis is unlikely. Note also two caveats. First, this is a statistical test only and makes no statement about systematic errors. Second, we have ignored the absolute normalization.

The difficulty with absolute predicted number of events is illustrated below, which shows the number of events predicted as a function of dm2. We see 591, and we expect about 500 with no-oscillations. Note that the observed numbers are already nearly 20% above the predicted flux, even without oscillations. At the minimum in the dm2 sweep, we expect about 350 events. Thus if the interpretation of oscillations is correct (and there are no significant errors in my calculations), the neutrino flux calculation must be increased by a whopping 70%, or putting it more gently, the presently predicted flux is only 60% of the true muon neutrino flux.

This situation is similar to what I found for SK. Indeed every underground detector since the early 1980's has found a flux that agrees within errors with the predicted upcoming muon fluxes. Since there are a variety of underground detectors (Baksan, IMB, Kamioka, Frejus, NUSEX, Soudan, MACRO and LVD), and a variety of flux calculations (which vary amongst themselves by around 20%) and several neutrino interaction cross section calculations (which makes little difference for these energies), and all yield approximately the same results, the conflict not to be dismissed lightly. We discuss this further below.

I really do not know how to handle the systematic error question easily for this IMB analysis. Ideally I would employ a number of different flux models, cross-sections, and muon propagation routines. Also I would carry out the test with extremal models of the effective area versus zenith angle for IMB. Unfortunately at this time I have access to none of these (not having an available IMB Monte Carlo muon data set), so I must wave the hands and discuss potential sensitivity. First, I think it is reasonable that the results are insensitive to the flux model and neutrino to muon conversion algorithm. This is because the effect of oscillations is quite broad in energy, and a slow function of angle. Having the slope of the flux slightly incorrect, or the crossection (y distribution) off a bit, will only make slight shifts in the point of the minimum in dm2 space, and should generally be present only in second order, remembering that the absolute normalization is not relevant here.

I think the most dangerous systematic is likely to be in the model of effective area versus zenith angle. You will note the dip in effective are for IMB near the horizon, which is about 30% below the geometric area relative to middle angles. The results are indeed sensitive to this, because the corrected flux versus angle, shown above, peaks up near the horizon, and as in SK, this is just where the oscillation effects are most prominent. Given that there are about 600 events, and the effect is strongly dependent upon the 10% bin nearest the horizon, this means that we are only sensitive to changes of the area test function which are large compared to the statistical fluctuations in something like 60 events, or around 15%. Again, this seems, but is surely not proven by such handwaving argument, to be safe as one recalls that we are only concerned with the shape of the area versus angle. Of course, all the above assumes no egregious errors in my code, that of Stanev and friends, and of Becker-Szendy in the MC analysis. I have argued that potential inaccuracies in the flux, crossection and and propagation are not very critical in this test. The Becker-Szendy MC (of which he was only one of a number of contributing authors) was widely worked upon in the IMB collaboration and cross checked against other codes. My calculations should certainly be repeated independently, but at least they are consistent in comparison to other calculations: I find approximately the correct flux distribution, mean muon momentum and the stopping fraction, in comparison with other published no-oscillation results.

Conclusion

The results presented above certainly strengthen the case for muon neutrino oscillations, in my view. I fear the result presented is not publishable, however, without significant further effort in resurrecting the IMB simulation code and checking the approximations employed herein, most notably the effective area versus zenith angle. Indeed, with a good set of MC data, we could do the test in another way, inserting weights for the individual events (probability of a muon neutrino remaining a muon neutrino) and generating test distributions for various dm2's. We can then employ the K-S test for two samples being drawn from the same distribution. Without doing this I do not now see how we can present any sensible quantitative estimate of systematic uncertainty in the results.

While the foregoing may give us confidence in the reality of muon neutrino oscillations, a cloud on the horizon is the confrontation with phenomenological calculations of the absolute magnitude of atmospheric neutrino flux. The test flux employed herein (Bartol model) predicts too few neutrinos by about 40% in the presence of oscillations. Yet, the claimed maximum in systematic error of the calculations is 20%. Given the history of flux predictions following the experimental results in the early 1980's, we suggest herein to simply await the reassessment of the fluxes.

Perhaps there is some particle interaction physics to be learned from this conflict? I have discussed this question with Sandip Pakvasa and others, and we see no simple loophole. One may imagine a new particle (with mass in the few GeV range so as not to perturb the contained event rate too much) which decays to neutrinos, thus increasing the atmospheric flux. Not only does this seem artificial, we do not see how to avoid major changes to the neutral current cross section, and avoid other tests from accelerator experiments. We have also considered the seemingly unlikely coincidence of an extraterrestrial flux of muon neutrinos almost equal to the atmospheric flux. This however, would work the wrong way: we know from plotting the events in equatorial coordinates that there is no major anisotropy in the data. Thus the addition of cosmic neutrinos in significant numbers to the production of underground muons would serve to make the muon zenith angle distribution more flat, the opposite to the effect we observe.

So we return to the confrontation: either muon neutrino oscillations are taking place and the flux calculations are too low, by about 40%, or there are no muon neutrino oscillations with the dm2 indicated by the contained event data! As an experimentalist, and in light of the IMB, Kamioka, Soudan and SK data, I put my money on there being something awry with the neutrino flux calculations.

Note inderted 10/97: Improvements in calculations including the use of crossections from Paolo Lipari with a better treatment of the QE, one pion as well as DIS, have made this magnitude inconcistency largely disappear. Numbers will be presented when we are more sure of them. It seems that the last barrier, as far as I can see, to muon neutrino oscillations as a completely consistent and indeed demanded conclusion to be drawn from the data stands as now inescapable.

Acknowledgments:

Many people have contributed to the works described: all the collaborators from IMB and Super-Kamiokande, who built and operate(d) the experiments, and reduced the data. Important help has come from Sandip Pakvasa and Todor Stanev. Consultations with Ralph Becker-Szendy were important. The updated data sample came to me compliments of Bob Svoboda. Of course the UH team have all contributed, particularly Shige Matsuno (in charge of preparing the SK upmu sample, off-site version) and John Flanagan (and all the off-site HE group, in preparing the SK contained event sample). If the foregoing is all nonsense, I am to blame; if there is merit to it then everyone deserves credit.


written 24 August 1997, last edited by jgl, 26 August 1997