WTF? Physics
A layman's guide
A layman's guide
The Atmospheric Neutrino Anomaly In 1968, when the "Solar Neutrino Problem" first hit the physics press, it was actually a nice problem for the neutrino physicists to have. That was because a statistically significant neutrino had not yet been identified using Bruno Pontecorvo’s Cl-37 approach.
But better to have a "Solar Neutrino Problem" than the more fundamental question as to whether neutrinos are detectable at all, because that would have meant that the neutrino physicists have just wasted everyone’s time and money.
Back in 1968, an accurate estimate of solar neutrino flux was critical to prove that solar neutrinos existed. But how well was the solar neutrino flux understood when the "Solar Neutrino Problem" was first introduced? Certainly not well enough to be sure that there was really a problem.
But there always seems to be issues with finding neutrinos, and thus we have another problem: the "Atmospheric Neutrino Anomaly".
In contrast to estimating solar neutrino flux, atmospheric neutrino flux was on much firmer ground, and so was the fact that there was another problem with detecting neutrinos.
And that’s because of the extensive work that went into studying cosmic rays and their collisions with the earth’s atmosphere, which was the cause of the atmospheric neutrino flux. This research allowed for very good estimates of flux and more certainty that there was truly an unexpected problem in detecting neutrinos.
Cosmic rays were once thought to be gamma radiation, and hence the name "rays". They mostly consist of fast-moving protons, alpha particles, and single electrons. They ionize the earth’s atmosphere when they collide with atoms.
The collisions not only ionize atmospheric gases such as oxygen and nitrogen, but they also create secondary particles such as kaons, pions, and muons. The rate of production of these secondary particles varies with observables that are relatively easy to measure, such as altitude, latitude, longitude, and the geomagnetic field.
But the rate of kaon, pion, and muon flux caused by cosmic ray collisions are important to the neutrino physicists because distinct neutrinos are presumably created from the process of their decay. These secondary products and their decay channels are shown below.
Decay products of secondary cosmic ray particles[1]As seen above, neutrinos are either created directly by kaon (K) decay, or indirectly by kaons decaying into pions (p), which subsequently decay into neutrinos. So, every decay listed above will directly or indirectly produce neutrinos.
And since they decay into two types of neutrinos, muon (nm) and electron (ne), the neutrino physicists have been very focused on their ratio of production in the atmosphere.
But why are the neutrino physicists so concerned about ratios instead of the actual numbers?
For one very good reason: ratios minimize the dependence on absolute accuracy, and once again, the funniness that surrounds neutrino measurables raises its ugly head. In other words, it’s safer to use a ratio than the actual raw data.
As can be seen in the above chart, there are many pathways to creating muon and electron neutrinos from cosmic rays, and therefore, there are differences in their relative quantities. For example, the number of muon neutrinos divided by the number of election neutrinos, or (nm / ne), is a function of the fluxes of the decay channels of the primary and secondary cosmic rays.
Early on, to constrain the experimental complexities involved in detecting atmospheric neutrinos, the neutrino physicists focused their efforts on pion decay channels. The energy of neutrinos created from pion production is less than those created in kaon decay chains.
But regardless of the kinds of neutrinos the physicists wanted to try to measure, using the decay channels listed above, along with other relevant observables, they would construct computer programs to estimate the ratio of muon neutrinos to electron neutrinos expected at experimentally relevant locations on Earth.
These computer simulations of neutrino events are usually referred to as "Monte Carlo" simulations, and once again, the strange world of neutrinos has another head scratching concoction.
But Monte Carlo simulations are not uncommon in the study of complex multi-body physical systems, so there’s nothing necessarily sordid going on here.
But as we will discuss later, the neutrino physicists that produce the raw experimental data are at the same time determining the way the computer simulation software is being coded. This might be interpreted by the cynical observer as grading your own homework.
And while this may sound like the usual conspiratorial nonsense being spread by an uninformed outsider, there is an insider who thinks that there is some funny business going on in the collaborations that are searching for the elusive neutrino.
IMB versus KamiokaNDE and the Atmospheric Neutrino Anomaly
The "atmospheric neutrino anomaly" is basically the problem of the experimentally determined atmospheric muon neutrino flux being much lower than its theoretically predicted value. In other words, these neutrinos were once again not showing up like they were predicted to.
But with this "anomaly", we have some very revealing behind-the-scenes commentary involving the resolution of experimental inconsistencies between two geographically separated neutrino detectors, commonly referred to as IMB (USA) and KamiokaNDE (Japan).
The construction of the IMB detector, which began in 1979, was the result of a "collaboration" between the University of California at Irvine (UCI), the University of Michigan, and the Brookhaven National Laboratory. Collaborations are the rule with expensive and time-consuming neutrino experiments, but the IMB collaboration originally had little to do with neutrinos. Rather, it was primarily dedicated to discovering "proton decay".
And that brings us back to Frederick Reines, who managed to parlay his fame as a key neutrino physicist to become the first dean of physical sciences at UCI.
After "discovering" the neutrino under suspicious circumstances, Reines was going after the discovery of proton decay. In this endeavor, he pitched the IMB detector as a "nucleon decay detector", although detecting neutrino oscillations was also mentioned in his proposal.
The hypothesis of proton decay was part of several "Grand Unified Theories", and the most commonly proposed proton decay channel was a positron and a neutral pion which would subsequently decay into two photons.
Similar to the Homestake Mine experiment, the IMB detector was constructed in a mine 600 meters underground to minimize the background radiation caused by cosmic rays. But instead of using chlorine, they used an aquarium-like tank filled with 2.5 million gallons of purified water.
And instead of chemically separating the gaseous argon from the chlorine, they used thousands of photomultiplier tubes (PMTs) to capture the signatures of neutrino interactions. And that involved the analysis of photons that were intercepted by the PMTs.
This is analogous to the methodology used in Reines’s Savannah River experiment, which also involved the PMT detection of patterns of photon interaction that were proposed to be the signature of neutrino interactions.
But there is quite a twist to this approach, and that involves the comparison of the detected ratio of muon neutrinos per electron neutrino. But why do they do that? And compare it to what?
Against a computer simulation of the same ratio, of course. In other words, we have left the world of "upper limits" and "squared mass differences", and we are now having to deal with a "ratio of a ratio".
What is the point of comparing a ratio of actual data to a ratio provided by computer simulation? In other words:
(nm / ne)data / (nm / ne)mc
where data = experiment and mc = Monte Carlo computer simulation
Why put so much emphasis on a ratio of actual data compared to a Monte Carlo computer simulation? As we have previously mentioned, this places the overall problems associated with the quality of neutrino detection on the back burner. It also downplays the raw data differences generated by different neutrino laboratories.
So, it's all about the poor quality of the actual data. It’s like they are saying, "listen, we know we might be off on our raw counts, but the ratio between muon and electron neutrinos shouldn’t be impacted by experimental issues."
About the time IMB was completed, an IMB-like detector was being built 1,000 meters underground in the Kamioka zinc mine in Japan. This was the Kamioka Nuclear Decay Experiment, or KamiokaNDE. The original configuration of this detector was smaller than IMB, providing about one-third of the fiducial water volume.
But even with its diminished size, it was a very big player in the small circle of neutrino laboratories. Since KamiokaNDE and IMB were so similar in methodology, they kept a close eye on each other’s work.
As a brief sidenote, in 1989, the KamiokaNDE collaboration announced their preliminary experimental estimates of solar neutrino flux and compared them to Ray Davis’s results. Pointing out the need for an "independent observation of solar neutrino flux"[2] , Hirata et al. announced that their own methodology accounted for 46% of the "best estimate of the standard solar model" with a standard error of 30%.
And that was a big problem. Two expensive experiments of solar neutrino flux reported 28% (radiochemical) and 46% (Cherenkov) of the prevailing theoretical estimate. While these two results weren’t that far off based on their standard errors, both teams thought it cast a shadow upon their respective experiments. So much so that the KamiokaNDE collaborators were picking and choosing a selected handful of the dozens of runs of Davis’s experiments to make their results seem more consistent.
KamiokaNDE, focusing on the recent five runs of Davis’s experiment, which came in at 63% of prediction, said our "result does not seem to be inconsistent with the recent results from the chlorine experiment [Davis]."[3]
But the radiochemical versus Cherenkov methodologies had another issue which involved solar flares. Davis announced that there was a possible relationship between solar flares and neutrino activity, which KamiokaNDE was unable to detect, once again raising questions about the validity of one or both experimental approaches.
But the discrepancies between the radiochemical and Cherenkov methods were constrained to the realm of solar neutrinos. When it came to the topic of atmospheric neutrinos, there were two laboratories employing very similar Cherenkov methodologies. KamiokaNDE and IMB.
And early on, there were significant discrepancies between these two labs. The politics that surrounded their reconciliation provided another dubious chapter in the already dubious history of neutrino research.
IMB’s Mismatch with KamiokaNDE
Whenever one of the neutrino collaborations publishes their results, we are subjected to a smattering of experimental data, odd terminology, and usually a pathway forward for future research. But seldom do we ever get a chance to see what is really going on behind the curtains.
We got this chance with Dr. John LoSecco, a key member of the IMB collaboration. LoSecco wrote a paper in 2005 outlining several experimental discrepancies between IMB and KamiokaNDE, and further, the interesting politics that surrounded them. The paper was published in 2016.[4]
LoSecco highlighted the substantial difference between what IMB and KamiokaNDE respectively reported as muon neutrino flux. And this anomaly was eventually published in an IMB collaboration report in October 1986.[5]
"The simulation predicts that 34%±1% of the events should have an identified muon decay while our data has 26%±3%."[6]
This "shortfall" of the identified muon decay count, which was a proxy for muon neutrino flux, was based on the computer simulation of what IMB expected the muon count to be. In February 1986, before this report was published, LoSecco was invited to give a talk about proton decay, which included a discussion on atmospheric neutrinos. And at that time, he was well aware of this shortfall.
LoSecco alluded to the politics associated with the IMB collaboration, highlighting the fact that they didn’t want to publish their evidence about the problem with the muon neutrino flux:
"It is noteworthy that most collaborations, including IMB can be very conservative. As can be clearly seen in many documents leading up to this period, such as the PhD theses quoted, most people hoped that the effect would just go away since it constituted an uncertainty to the background to proton decay. In fact the muon decay deficiency was not mentioned in early drafts of the 1986 article. It was added, at my insistence, since the topic of the paper, comparing neutrino observations with expectations, seemed appropriate."[7]
LoSecco is taking credit for making public the atmospheric neutrino anomaly, while about the same time, KamiokaNDE reported no such problem. LoSecco references the 1986 Kamiokande account[8] of their experimental advantages:
"The Kamioka detector had much more light collection capability than the IMB detector. This permitted them to utilize the shape of the Cherenkov image to determine if the interaction had produced a muon, M type events, or an electron, S type events. The S stands for showering since the electron would multiple scatter, bremsstrahlung and pair produce; a processes known as an electromagnetic shower. Muon induced events had a much crisper, cleaner image. Kamioka had used this difference in images to distinguish electron from muon type events. While Nakahata et al. had no numbers for data, the data was the same as the Kajita PhD. thesis[46] from earlier in 1986. The figures containing experimental observations are identical. Kajita’s PhD thesis reported the observations from the first phase of Kamioka, known as Kamiokande I. In an exposure of 1.11 kt-yr they reported 141 contained event in 474 days of live time. The ability to distinguish showering from non-showering tracks permitted them to report the event rates in the two categories."[9]
A side note here is the reference to the figures in the Kamioka report as being identical to the data from "Kajita’s PhD thesis".
LoSecco goes about the business of quoting the actual numbers on Kajita’s PhD thesis that were referenced but not explicitly published in the KamiokaNDE 1986 report, then goes on to say that it clearly indicates a problem with the KamiokaNDE muon decay rate.
"But the numbers quoted above clearly indicate a 2.4 standard deviation deficiency of muon decay signals and a 1.6 standard deviation excess of M type events, when compared to expectations. None of these significances is calculated in the thesis. I realized that the low muon decay rate reported, but not noted by KamiokaNDE could provide confirmation of the IMB-1 3.5 standard deviation observation."[10]
So LoSecco suspects that KamiokaNDE made an important omission with their paper. Before the publication of IMB’s report, which included the notification of a muon neutrino deficit, LoSecco actually meets with Kajita and company in Japan and tells them there is something wrong with their M/S (muon and showering) analysis as compared to their muon decay rates. LoSecco discusses the discussion:
"I pointed out the discrepancy between the M/S analysis and the muon decay rates in the Kamiokande analyses. The response was kind, blank stares. I was assured that the M/S analysis was correct. Kajita’s thesis[46] and the Nakahata paper[42] were not unique. All Kamiokande reports stressed how closely the neutrino observations matched expectations."[11]
But it wasn’t until 1988, when the KamiokaNDE collaboration finally agreed with IMB’s analysis, that they corrected their problem. But LoSecco was still suspicious:
"In 1988 the Kamiokande experiment published a paper (Hirata et al.[47]) confirming the reported deficit of muon like neutrinos....The M/S (muon and showering) pattern recognition classification method had been modified to give agreement with the muon decay rate [emphasis added]. The paper concluded that there appeared to be a muon deficiency. Only 59% of the expected number of muon type events were observed....Interpretation of the data in this paper is still difficult since it also shows no directional modulation (figure 21) or energy distortion. A careful reading does indicate that the event rate appeared to be lower than that reported for the Kamiokande I data. A brief review of the reports from Kamiokande from turn on to the 1988 paper shows a significant change in interpretation in the 1988 paper [emphasis added]."[12]
So LoSecco notes that the KamiokaNDE collaboration modified their pattern recognition classification method of the Cherenkov radiation events so that it matched up with the IMB method.
So what was KamiokaNDE doing here? Was KamiokaNDE so unsure of their method that they quietly modified it to align with IMB?
That’s exactly what happened, and in doing so, they invalidated their own work. Keep in mind that KamiokaNDE had a problem with the internal consistency of their M/S analysis and their muon decay rates. But they did not retroactively correct their 1986 analysis.
Rather, they did another run, which completed late in 1987, and started fresh with their new "pattern recognition classification method" for the Cherenkov radiation events for their 1988 report. KamiokaNDE adjusted their methodology to match numbers from a competing neutrino laboratory, and did it quietly.
And we’ve seen this post hoc adjustment before. This is similar to how Reines handled the mysterious changing neutrino cross section. This same sleight of hand also occurred in Davis’s Homestake Mine experiment. It resulted in Davis changing the method in categorizing the "event selection procedure" to distinguish AR-37 decay events from background radiation:
"Fortunately it was found that adding a pulse rise time restriction to the event selection procedure could improve signal versus background discrimination sufficiently that 37Ar decay events could be distinguished from background pulses in the solar neutrino samples. Rise time counting was introduced in late 1970 starting with run 18. Early results employing this detection technique were presented at the Irvine conference & Reines (Trimble 1972)."[13]
Note that the much ballyhooed "Solar Neutrino Problem" was based on data prior to 1970, before this addition of "pulse rise time" was added to Davis’s event selection procedure.
In fact, Davis left out the data runs prior to 1970 in his 1998 paper on the Homestake Mine experiment. So, what happened to the data the "Solar Neutrino Problem" was founded upon? In 1998, Davis no longer acknowledged it as relevant.
But we digress, and return to the mysterious behavior of KamiokaNDE. LoSecco continued his critique of their data:
"Kamiokande 1 had reported an event rate of 116±10 events per kiloton-year but the new paper based on combining this data with subsequent data had an event rate of 92.3±5.7 per kiloton-year (figure 22). One can understand that the M/S numbers changed from earlier reports because the M/S fitting method had been revised but why should the event rate drop? [emphasis added] The 2.87 kiloton-year in the paper was the sum of the 1.11 kiloton-year from Kamiokande I and 1.76 kiloton-year from Kamiokande II. By subtraction this means that Kamiokande II had 136 events in 1.76 kiloton-years or an event rate of 77.3±6.8 events/ktonyr, a drop of 38% from Kamiokande 1....It would have been nice to see the muon neutrino fraction independently for these two exposures, Kamiokande I and II. But the data has never been released in a format that would make that possible.[emphasis added]"[14]
Pretty strong stuff from LoSecco, along with his hint that KamiokaNDE was being coy with their raw data regarding the fraction of muon neutrinos. But what’s even more interesting is LoSecco’s presentation at the "History of the Neutrino Conference" in 2018, where he took several shots at the IMB collaboration:
"The IMB collaboration had a secrecy rule to limit rumors of the expected discovery of proton decay. The rules were enforced by a senior management team that had a record of prior mistakes."[15]
LoSecco might be referencing Fred Reines’s problem with the Savannah River experiment, where he announced the discovery of the neutrino based on a cross section that had changed on him before he could publish the details of the experiment. Reines then made some adjustments to his assumptions, and magically matched the revised neutrino cross section again before publication of the experimental details. Reines was also in the senior management team for the IMB collaboration.
LoSecco, in his materials used in his presentation, posted the following tidbits about the IMB collaboration[16]:
"Senior members had very strong control. Careers could be ruined."
"I had been fired from my first post doc job (E310) since I was skeptical of the high Y anomaly."
"Senior members had been involved in recent, very public mistakes."
And what seems to be a minor issue, but one that LoSecco had to point out, was an irregularity with the publication of the an IMB paper in 1986:
"The published observed value of the muon decay fraction was 26±3%. Drafts of the paper distributed to the collaboration for approval had the correct 26±2%. The ±2 is based on binomial statistics. An event had a muon decay or it did not. I do not know how the ±3 got in the paper."[17]
So, in between the time the IMB collaborators were given a draft of the paper and its final publication, the standard error got bumped up from 2% to 3%.
As we have noticed before, the neutrino experimentalists don’t seem to feel very sure of their data. Quietly bumping up the standard error is indeed suspicious behavior, and a window into the unseen world behind the curtains of the neutrino collaborations.
But LoSecco’s snipping at IMB senior management is rather mild compared to his take on Kajita and the KamiokaNDE collaboration:
"As mentioned above, in June 1986 I asked Kamiokande to confirm our evidence of a muon deficit. At the time they were reporting a 1.6σ muon excess. At the 2018 conference, I asked Takaaki Kajita to confirm the time line. In his response he indicated that Kamiokande had relied on scanning to classify events as muon or electron. It wasn’t until Fall of 1986 that an automated method was developed. This confirms what is indicated in Takita’s 1989 PhD thesis. But the thesis doesn’t give specific dates. Kajita indicated there had been no formal particle identification in Kamiokande before I mentioned the muon deficit. The Kamiokande work was not independent. It was a confirmation of the IMB result."[18]
LoSecco’s accusation of sloppy science is certainly not limited to the KamiokaNDE collaboration. He takes a shot at Lee and Koh, and blames them for some problems with IMB’s data:
"All IMB-3 contained atmospheric neutrino comparisons are inaccurate since they were modeled with the Lee and Koh neutrino spectrum which is flawed by a programming bug which underestimates the muon neutrino flux."[19]
So, it was a "programming bug". And this brings us back to the "Monte Carlo" computer simulations that the neutrino physicists felt compelled to use in justifying neutrino oscillations.
The KamiokaNDE collaboration had deep hooks in the specs for the computer simulation, dating back to the beginning. This, on the face of it, might be a conflict of interest. After all, the computer simulation is used to validate the actual experimental data.
And that leads to something that the KamiokaNDE collaboration calls NEUT, the software wing of the collaboration. Here is their brief description (bolded added):
"NEUT has a long rich history, originally developed in the 1980s as a tool to study atmospheric neutrinos and nucleon decay in the Kamiokande experiment [2], and some of the original FORTRAN77 code is still in use. NEUT continues to be predominantly developed and maintained by members of the Kamiokande series of experiments (Super-Kamiokande, T2K, Hyper-Kamiokande) and many source files contain comments messages from the numerous physicists who have contributed to the simulation over the past 35 years-including those working on the Nobel prize-winning Super-Kamiokande (SK) analysis [3]. Recent development has targeted the improvements most-needed for precise and robust analyses of SK and T2K neutrino oscillation data and neutrino cross-section measurements. At T2K energies, charged-current quasi-elastic interactions dominate. It has become clear over the last decade that such interactions can only be precisely predicted by incorporating detailed models of relevant nuclear dynamics. For the multi-GeV samples used in SK analyses, shallow and deep inelastic scattering channels are critical for modelling the expected rate of multi-ring events seen in the detector. The transition region between resonance excitation and deep inelastic scattering has proven particularly difficult to model well. Because of the in-house nature of NEUT development and analysis usage, it is not yet open source. We do not yet have the resources to migrate to an open source model, but, access to the code and usage instructions are available upon request."[20]
So, the source code for the simulation was developed by the same physicists that produced the data. And in 1986, KamiokaNDE’s reported that their experiment and their computer simulation worked like a charm:
"our Monte Carlo program reproduces the KAMIOKANDE’s data quite well."[21]
So, what could be better than having your experimental data matching your very own computer simulation of that data? Since the same physicists had their hands in both results, how much can we trust that they’re not being professionally motivated to make things work out? After all, we’ve already seen the KamiokaNDE collaboration do it once.
And there was a lot of funniness going on during LoSecco’s interaction with the KamiokaNDE team, which LoSecco noted:
"Kajita indicated there had been no formal particle identification in Kamiokande before I mentioned the muon deficit."[22]
And in the cleanup of this problem, LoSecco was equally suspicious.
"In 1988 the Kamiokande experiment published a paper (Hirata et al.[47]) confirming the reported deficit of muon like neutrinos....The M/S (muon and showering) pattern recognition classification method had been modified to give agreement with the muon decay rate."[23]
KamiokaNDE obviously back peddled after being confronted by LoSecco’s analysis of their data. And further, they changed their "pattern recognition classification", which confirmed the IMB data and addressed LoSecco’s complaint.
KamiokaNDE was a giant in the field of neutrino research. But given this incident, can we really trust all the players in the KamiokaNDE collaboration?
LoSecco doesn’t seem to think so. Not to mention the fact that he didn’t seem to trust all the players in the IMB collaboration, especially senior management.
Where does this leave us?
To date, we have examined three milestone neutrino experiments, and each was tainted in its own way. And each engaged in post hoc adjustments to their neutrino event categorization methodologies.
We have seen an entire "solar neutrino problem" constructed before our very eyes, when there had yet to be any authoritative evidence that solar neutrinos even existed.
We have seen neutrino physicists modify their interpretations of background radiation to better fit their data to changing theoretical derivations of the neutrino cross section.
We have seen a theoretical physicist quietly revising his theories using pre-published data, and then pretending that those theoretical adjustments had nothing to do with his advanced look at that unpublished data.
We have seen the community of neutrino physicists embrace the contorted concept of "oscillation". Without oscillation, much of the experimental evidence can be interpreted as if neutrinos aren’t detectable at all. Or they don’t exist.
In other words, with neutrinos, we have seen politics on a surprising scale.
And as a layman, this is difficult to witness. Scientists are not supposed to be politicians. They are supposed to advance the cause of human understanding, whatever its personal cost. Even if that means admitting they were wrong and undermining their careers in the process. But unfortunately, they are human, and very few are willing to take that bullet.
So, we can expect the ever-complexifying world of the neutrino to get even more bizarre, if that’s possible. As we have seen, the neutrino theorists will always add layers of convolutedness with each unexpected experimental result.
They will still consider every option about the neutrino, other than it being fundamentally undetectable. Or even worse, that it doesn’t exist.
Please send comments to Charles Brack at brack@wtfphysics.com
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[1] Honda, M. et al. Calculation of the flux of atmospheric neutrinos. Phys. Rev. D52, 4985 1995. https://arxiv.org/pdf/hep-ph/9503439.pdf
[2] Hirata, K.S. et al. Observation of solar neutrinos and search for 'solar-flare neutrino' events in KAMIOKANDE, Part of Tests of fundamental laws in physics. Proceedings, 9th Moriond Workshop, 24th Rencontres de Moriond, Les Arcs, France, January 21-28, 1989, 391-400. https://inspirehep.net/files/3f4183c2216c7865ecb217d1427b7b67
[3] Ibid.
[4] LoSecco, J.M. The History of ’Anomalous’ Atmospheric Neutrino Events: A First Person Account, Physics in Perspective 18, 209 (2016)
[5] Haines, T.J. et al. Calculation of Atmospheric Neutrino-Induced Backgrounds in a Nucleon-Decay Search Physical Review Letters, Volume 57, Number 16. October 1986.
[6] Ibid.
[7] LoSecco, J.M. (2016)
[8] Nakahata, M. et al., "Atmospheric neutrino background and pion nuclear effect for Kamioka nucleon decay experiment", J. Phys. Soc. Japan 55 (1986) 3786.
[9] LoSecco, J.M. (2016)
[10] Ibid. p. 39.
[11] Ibid. p. 40.
[12] Ibid.
[13] Davis, R. et al. Measurement of the Solar Electron Neutrino Flux with the Homestake Chlorine Detector, The Astrophysical Journal, 496, 1998 March 20.
[14] LoSecco, J.M. (2016)
[15] LoSecco, J.M. (2019) https://arxiv.org/pdf/1902.01757.pdf
[16] Ibid.
[17] Ibid.
[18] Ibid.
[19] Ibid.
[20] Hayato, Y. and Pickering, L. "The NEUT neutrino interaction simulation program library" The European Physical Journal of Special Topics (2021) https://arxiv.org/abs/2106.15809
[21] Nakahata, M. et al., "Atmospheric neutrino background and pion nuclear effect for Kamioka nucleon decay experiment", J. Phys. Soc. Japan 55 (1986) 3786.
[22] LoSecco, J.M. (2019) https://arxiv.org/pdf/1902.01757.pdf
[23] LoSecco, J.M. (2016)
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