Sterile neutrinos: Are they real? - Inspire HEP

Eberly/Lincoln TPT/sterile neutrinos v1.8

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Sterile neutrinos: Are they real? 1 2

B. Eberly, University of Southern Maine, Portland, ME 041043 4

D. Lincoln, Fermi National Accelerator Laboratory, Box 500, Batavia, IL, 605105 6

Neutrinos are perhaps the least understood of the known denizens of the subatomic 7 world. They have nearly no mass, interact only via the weak nuclear force and gravity 8 and, perhaps most surprising, the three known species of neutrinos can transform from 9 one variant into another. This transformation, called neutrino oscillation, has been 10 demonstrated only relatively recently and has led to speculation that there might be 11 another, even more mysterious, neutrino variant, called the sterile neutrino. While the 12 sterile neutrino remains a hypothetical particle, it is an interesting one and searches for 13 it are a key research focus of the world’s neutrino scientist community. 14

15 Before talking about sterile neutrinos, it is perhaps useful to briefly review the behaviors 16 of ordinary neutrinos and the history that revolves around them. Neutrinos were 17 hypothesized in 1930 by Austrian physicist Wolfgang Pauli. He proposed [1] them to 18 solve some mysteries involving beta decay. At the time, beta decay was thought to be 19 the transformation of a neutron into a proton and an electron. However, careful studies 20 of the process made scientists realize that energy and angular momentum did not seem 21 to be conserved in this process. Pauli’s conjecture was that a third particle, which we 22 now call the neutrino, was created in the decay and carried away the missing energy 23 and angular momentum. 24

25 Neutrinos do not interact via either electromagnetism or the strong nuclear force and 26 therefore are very difficult to detect. It wasn’t until 1956 that neutrinos emitted from 27 nuclear reactors were observed [1,2]. (Technically, antineutrinos were observed in this 28 early experiment.) The following year, violation of parity conservation was observed in 29 weak nuclear interactions [1,3]. Parity is conserved if one can replace (𝑥, 𝑦, 𝑧) →30 (−𝑥, −𝑦, −𝑧) and the equations describing the interaction are unchanged. If one makes 31 this substitution and the change is evident, then parity is not conserved under that 32 interaction. 33

34 Because parity is violated in weak interactions, and because neutrinos interact only via 35 the weak force, this implies all detectable neutrinos are left-handed, while detectable 36 antimatter neutrinos are all right-handed. Right-handed means that the particle’s 37 intrinsic angular momentum (its spin) is parallel to the direction of motion, while left-38 handed means that it is antiparallel. Handedness is illustrated in figure 1, while the 39 implications of parity conservation violation are illustrated in figure 2. 40

FERMILAB-PUB-22-234-PPD

This manuscript has been authored by Fermi Research Alliance, LLC under Contract No. DE-AC02-07CH11359 with the U.S. Department of Energy, Office of Science, Office of High Energy Physics.


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41 Figure 1: Right-handed objects are defined when the angular momentum vector is 42 parallel to the direction of motion, while left-handed objects have an angular momentum 43 vector that is antiparallel to the direction of motion. Alternatively, when viewed so that 44 the object is moving directly toward the observer, right-handed objects rotate in a 45 counterclockwise fashion, while left-handed objects rotate in a clockwise manner. [end 46 caption]47

48 Figure 2: Parity conservation violation implies that certain spin configurations are not 49 observed. For instance, right-handed antineutrinos and left-handed neutrinos are 50 observed. However, if one interchanges right and left, the direction of motion will 51 change, although the spin direction doesn’t change. The result – left-handed 52 antineutrinos and right-handed neutrinos – are not observed in nature. [end caption] 53

54 In 1962, a collaboration led by Leon Lederman, Melvin Schwartz, and Jack Steinberger 55 discovered that neutrinos come in more than one distinct “flavor;” specifically they found 56 that there is an electron-type and a muon-type neutrino [1,4]. Each of these is 57 associated with the named charged lepton. With the discovery of the tau lepton in 1975 58 [1], researchers suspected that there was a third tau-type neutrino, a conjecture that 59 was confirmed in 2000 [5]. In the early 1990s, researchers using data from the LEP 60


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accelerator, located at the CERN laboratory in Europe, were able to prove that there 61 exist precisely three distinct flavors of low-mass neutrinos [6]. 62

63 In a series of experiments that began in 1964, Ray Davis measured the neutrino flux 64 from the Sun and found that his measurements were below what were expected from 65 calculations [1,7]. This discrepancy was known as the solar neutrino problem. A 66 parallel set of experiments studying neutrinos generated in the Earth’s atmosphere from 67 cosmic rays in the 1980s seemed to indicate a deficit of muon neutrinos [1,8]. This 68 discrepancy is called the atmospheric neutrino anomaly. These observations and 69 others led scientists to speculate that perhaps the different types of neutrinos could 70 transform into each other via a process called neutrino oscillation. The basic process 71 was first proposed by Bruno Pontecorvo in 1957 [1,9], and the phenomenon of neutrino 72 oscillation was definitively confirmed through a series of experiments performed at the 73 end of the 20th century [1,10]. These measurements were performed using the 74 SuperKamiokande detector (atmospheric neutrinos) and the Sudbury Neutrino 75 Observatory (SNO, solar neutrinos). An accessible review of this phenomenon is given 76 in Ref. [11]. Figure 3 shows an image of the Sun taken by the SuperKamiokande 77 experiment, imaged through the Earth and using only neutrinos. 78

79 Figure 3: This figure is an image of the sun taken by the SuperKamiokande detector 80 using only neutrinos that penetrate the Earth and thus illustrating the capabilities of 81 neutrinos to effortlessly penetrate matter. (Figure courtesy the SuperKamiokande 82 collaboration.) [end caption] 83

84 Neutrino oscillations 85

86 Neutrino oscillations solved both the atmospheric and solar neutrino problems, but they 87 are probably the most surprising and counterintuitive of all particle physics phenomena. 88


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Unlike all other fundamental subatomic particles, neutrinos can, over time, change their 89 identity. This process can only proceed if conservation laws that were previously 90 thought to be inviolable are, in fact, violated. 91 92 Lepton number is a quantity that is conserved in essentially the same manner as 93 energy, momentum, and electrical charge. If a photon (which has a lepton number of 94 zero, i.e., L = 0) transforms into a matter/antimatter pair and one half of the pair is an 95 electron (L = +1), the other particle must be a positron (L = -1). Similarly, lepton number 96 is conserved if a positively charged W boson (L = 0) decays into a positron (L = -1) and 97 an electron neutrino (L = +1), i.e., 𝑊+ → 𝑒+ + 𝜐𝑒. (In both these examples, 98 conservation of charge implicitly restricts the possible decay products.) 99 100 For most interactions, lepton number conservation is more restrictive than that. Lepton 101 number is conserved within lepton families, meaning there is an electron-type lepton 102 number, muon-type, and tau-type, and that each of these is conserved in particle 103 decays. For example, the positive W boson decay mentioned above is observed, while 104 the decay 𝑊+ → 𝜇+ + 𝜐𝑒 is not observed. This is because while lepton number is 105 conserved in both example processes, in the second process after the decay there is a 106 muon-type lepton number of -1 and an electron-type lepton number of +1. Because this 107 type of decay is not observed in nature, we say that it is forbidden. 108 109 Neutrino oscillation is the only known phenomenon that does not respect flavor-specific 110 lepton number conservation. If an electron-type neutrino transforms into a muon-type or 111 tau-type, the flavor-specific lepton number changes during the transformation. 112 113 There is another aspect of neutrino oscillation that can be confusing. Unlike most 114 subatomic particles, the observed neutrinos do not have a unique mass. In contrast, 115 any electron will have a mass of 0.511 MeV/c2 and every muon has a mass of 106 116 MeV/c2. The oscillation of neutrinos is essentially a quantum mechanical phenomenon 117 that occurs because the observed neutrinos are a quantum superposition of different 118 mass states. This can be illustrated most easily in a simplified world which has only two 119 neutrino flavors – say the electron- (e) and muon-type (). These form an orthogonal 120 coordinate system in what one might call the “neutrino plane.” In addition, the neutrino 121 plane can be spanned by another orthogonal coordinate system based on the mass 122 states. 123 124 For instance, assume that there are two neutrinos 1 and 2, which have mass m1 and 125 m2, respectively. A coordinate system based on mass can also exist in the neutrino 126 plane, oriented at an angle with respect to the flavor coordinate system. This is 127 essentially identical to how a cartesian plane can be represented by orthogonal (x,y) 128 and (u,v) coordinate systems, which have an angle between them. This idea is 129 illustrated in figure 4. 130 131


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132 Figure 4: A two dimensional cartesian plane can be spanned by a number of orthogonal 133 coordinate systems, which are traditionally labelled (x,y) and (u,v), with an angle 134 between the x and u axes. Both are equally good coordinate systems. The u and v 135 axes are each a mixture of x and y components that can be determined by projecting 136 them onto the x and y axes. The situation is similar for neutrinos, for which a two-137 component neutrino plane can be represented by either a flavor coordinate system (e, 138 ) or a mass one (1, 2). [end caption]139

140 The fact that the mass and flavor systems are not the same for neutrinos implies that if 141 one were to identify an electron neutrino and then subsequently attempt to measure its 142 mass, one would obtain either m1 or m2, with the probability depending on the angle 143 between the two coordinate systems. 144

145 Neutrinos oscillate because they are created in flavor states. As neutrinos propagate, 146 the Schrödinger equation predicts that each component mass state evolves in time 147 differently, allowing the neutrino to be detected later in a different flavor state. If m1 ≠ 148 m2, we can use fairly simple quantum mechanics [11] to calculate the probability that 149 two distinct flavor neutrinos can transform into one another. In the simple case where 150 there are only two flavors of neutrinos, if one starts with a beam of (for example) pure 151 electron neutrinos, with an energy, E, expressed in GeV, and a detector located L 152 kilometers from where the neutrino beam is produced, the probability that any electron-153 type neutrino will be detected as a muon-type neutrino is 154

155

𝑃(𝑣𝑒 → 𝑣𝜇) = sin2(2𝜃) sin2 (1.27 𝐿 ∆𝑚12

2

𝐸) (1). 156

157 The mathematics describing the oscillation probabilities between three distinct flavor 158 and mass states is considerably more complicated, and the interested reader can find 159 more information in [12]. Equation (1) is given in an SI form in Ref. [13]. 160

161 One can dissect equation (1). Parameters L and E are determined by the experimental 162 setup. The angle between the flavor and mass states in the neutrino plane is and it 163 determines the maximum probability of oscillation. The mass difference Δ𝑚12

2 = 𝑚12 −164

𝑚22, expressed in eV2, contributes to the frequency at which neutrinos oscillate. The 165

parameters and Δ𝑚122 are constants of nature and can be extracted from the data. 166


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However, it is often true that any specific experiment cannot determine a unique 167 numerical value for both of them. It is far more common that a range of parameters are 168 allowed and, furthermore, the fact that the two parameters are correlated by equation 169 (1) means that the published result is a complicated graph that allows many possible 170 solutions. In the situation where an experiment observes no oscillation signal, then only 171 an exclusion region can be determined. One such graph is shown in figure 5. 172 173 For the three-neutrino flavor situation, there are more oscillation parameters, specifically 174 12, 13, 23, 𝛿𝐶𝑃, Δ𝑚12

2 , Δ𝑚132 , and Δ𝑚23

2 (𝛿𝐶𝑃 is a parameter, forbidden in the simple two-175 neutrino picture, which allows neutrinos to oscillate differently than antineutrinos). In 176 spite of this added complexity, many experimental results fit quite well to the two-177 neutrino oscillation described by equation (1). Measurement of the oscillation 178 parameters is an active research industry, and a review of the current numerical values 179 and their uncertainties can be found in Ref. [12]. 180 181 Disagreement in Data 182 183 Prior to the definitive proof of the phenomenon of neutrino oscillation in 1998 – 2001, 184 scientists were fairly convinced that the phenomenon was the explanation for the myriad 185 unexplained observations in neutrino data. Consequently, accelerator-based 186 experiments were proposed to search for the signature of neutrino oscillation. One 187 such experiment was proposed in 1990 [14]. This experiment was to be performed at 188 the Los Alamos National Laboratory and it was named the Liquid Scintillator Neutrino 189 Detector, or LSND. 190 191 LSND generated both muon neutrinos and antineutrinos and then used their detector to 192 search for the appearance of their electron counterparts. Their method was to use a 193 positively-charged pion beam from the LAMPF (Los Alamos Meson Production Facility) 194 beamline. These pions were then slowed by passing through matter and most came 195 nearly to rest. Positive pions decay into positive muons and muon neutrinos. Positive 196 muons then decay into positrons, electron neutrinos and muon antineutrinos (i.e., 𝜋+ →197 𝜇+𝜈𝜇 → 𝑒+𝜈𝜇�̅�𝜇𝜈𝑒). 198 199 The search strategy was based upon searching for an excess of electron-type 200 antineutrinos. The reason is that no electron antineutrinos (�̅�𝑒) were produced in the 201 decay chain. If they were observed, then one hypothesis is that the muon antineutrino 202 oscillated into an electron antineutrino (�̅�𝜇 → �̅�𝑒). Additionally, electron neutrinos (𝜐𝑒) are 203 only produced by the muon that decays at rest and therefore should have relatively little 204 energy. Detection of higher energy electron neutrinos could be taken as evidence of 205 oscillations from muon neutrinos created by pions that decayed before they could come 206 to rest. 207 208 LSND published a series of papers, beginning in 1995 [15]. The first papers reported 209 searches for electron antineutrino appearance, while subsequent papers included 210 electron neutrino appearance searches. The final LSND paper was published in 2001 211 and it included results gleaned from both searches [16]. Furthermore, this paper was 212


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published late enough that it included information from the SuperKamiokande and SNO 213 measurements which established that neutrino oscillations were real [10]. Figure 5 is 214 taken from this reference. 215 216 And that’s where the situation becomes interesting. When several experiments attempt 217 to measure the same quantity, one can overlay the various experiments’ results and get 218 a more accurate estimate of the parameters being measured by combining 219 measurements. 220

221 Figure 5: Final measurement of the LSND experiment [16] analyzed using the two-222 neutrino oscillation approximation. LSND estimates that the true values for sin2 2 and 223 m2 is within the blue region with 90% probability and within the blue and yellow region 224 with 99% probability. Also included are measurements from four other neutrino 225 oscillation experiments at the time. For each experiment, the region to the right of the 226 red curve is excluded at a 90% confidence level, while the region to the left is allowed. 227 References for the other experiments can be found in [16]. We see that the Karmen 228 experiment excludes nearly all of the parameter space favored by LSND. [end caption] 229 230 However, it is also occasionally true that measurements might disagree. Figure 5 231 shows such a situation. This plot shows the values of for sin2 2 and m2 that are 232 allowed according to the LSND data (yellow + blue at 99% confidence), as well as 233 exclusion curves for four other experiments. In all cases, those experiments exclude 234 the region to the right of the respective red curve with a 90% confidence and allow the 235 parameter space to the left. All experiments exclude at least a portion of the LSND 236 preferred parameter space, with the Karmen and Bugey experiments excluding nearly 237 all of it. The Karmen experiment was qualitatively similar to LSND, and the differences 238 are detailed in [16]. 239 240 It is possible in the current day to perform a global fit to all oscillation data, excluding 241 LSND data. That analysis finds that two of the mass states (m1 and m2) are nearly the 242 same (Δ𝑚12

2 = (7.53 ± 0.18) × 10−5 eV2), while the third mass state (m3) is quite 243 different (Δ𝑚23

2 = (2.453 ± 0.033) × 10−3 eV2) [12]. The LSND result for m2 is several 244


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orders of magnitude larger than the values determined from other, more recent 245 measurements. If we assume that the measurements are accurate, and that the LSND 246 result is caused by neutrino oscillations, then clearly the standard three-neutrino picture 247 is not complete. 248

249 Sterile neutrinos 250

251 In order to resolve the discrepancies between LSND and the other neutrino oscillation 252 experiments, scientists have considered the possibility that there might be four 253 neutrinos. A four-neutrino configuration in which the neutrino can freely oscillate 254 between all four flavors can be made to fit much of the existing data. 255

256 However, there are issues. First, experiments at LEP have demonstrated that there 257 exist exactly three low-mass neutrinos that interact via the weak force [6]. Furthermore, 258 in order for the four-neutrino theoretical paradigm to explain any discrepancies caused 259 by LSND data, the fourth neutrino must have a mass of order 1 eV or larger. This runs 260 afoul of cosmological estimates of the sum of all active neutrino species, which 261 conclude that ∑ 𝑚𝜈 < 0.13 eV [17].262

263 This has led scientists to propose [18] the existence of what are called “sterile 264 neutrinos.” Sterile neutrinos are currently hypothetical and are defined to be a class of 265 low mass particles that participate in neutrino oscillation, but do not participate in the 266 weak force. 267

268 While it is often said that neutrinos are only left-handed particles, it is more correct to 269 say that the weak nuclear force only couples to left-handed particles and right-handed 270 antiparticles. Since neutrinos interact only via the weak force, it is therefore possible 271 that right-handed neutrinos exist, but these particles do not participate in the known 272 interactions of the Standard Model. It is not, strictly speaking, necessary that sterile 273 neutrinos are right-handed neutrinos, but most sterile neutrino models propose that they 274 are. 275

276 The existence of right-handed neutrinos is not so implausible, given that all other known 277 fundamental subatomic particles come in both a right- and left-handed spin state. 278 However, the existence of right-handed neutrinos would require an extension of the 279 Standard Model. 280

281 Alternative searches for sterile neutrinos 282

283 Given that the existence of sterile neutrinos is outside the existing Standard Model 284 paradigm, it is worth revisiting the data to consider the possibility that the data is in 285 error. There are other experiments that see discrepancies consistent with 1-eV2-scale 286 neutrino oscillations, including nuclear reactor experiments and radioactive source 287 experiments that observe fewer neutrinos than expected. However, the dominant 288 evidence for sterile neutrinos is the LSND measurement, thus it is imperative that the 289 experiment be confirmed. 290


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291 In 1997, an experiment called MiniBooNE was proposed at Fermilab to check the LSND 292 result [19]. Compared to LSND, the MiniBooNE detector observed neutrinos that were 293 about ten times more energetic, while using dramatically different detection technology. 294 Many of the LSND collaborators were also on MiniBooNE, although the experiment 295 mandated at its outset that no more than 50% of the MiniBooNE author list could be 296 scientists who were also on LSND. This was to protect against the possibility that the 297 LSND measurement could have arisen from some sort of algorithmic error that might be 298 transported from LSND and appear in MiniBooNE analyses. 299

300 MiniBooNE’s first results were announced in 2007 and the experiment concluded that 301 they did not see evidence for the kind of neutrino oscillation reported by LSND [20]. 302 However, a subsequent follow-on analysis reported in 2018 [21] that they saw a 303 significant excess of electron-like events in the data and that the derived parameters 304 were consistent with LSND and furthermore that the data was consistent with oscillation 305 behavior requiring four neutrino flavors; i.e., sterile neutrinos were needed. 306

307 This could be strong evidence for sterile neutrinos, and another experiment was 308 performed at Fermilab to study the MiniBooNE results in greater detail [22]. This 309 experiment is called MicroBooNE, and it uses a different technology than LSND and 310 MiniBooNE. MicroBooNE uses liquid argon as the detector medium, allowing much 311 better reconstruction of interactions in the detector. Initial results from the MicroBooNE 312 experiment [23] show no indication of electron neutrino appearance, which is 313 particularly puzzling because MicroBooNE and MiniBooNE share the same neutrino 314 beam. The experiment continues to analyze data as part of Fermilab’s focus on 315 neutrino physics and its final analysis is not yet available. Figure 6 shows a 316 photomultiplier tube used by MicroBooNE to trigger when a neutrino interacts with the 317 argon in the detector. This interaction creates electrically charged particles which then 318 move and cause scintillation light in the liquid argon. Figure 7 shows the MicroBooNE 319 detector prior to final installation, when it was being tested. Because the detector uses 320 liquid argon as a detection medium, it is wrapped in insulative material to optimize 321 detector performance and minimize cooling costs. 322

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324 Figure 6: A photomultiplier tube used by the MicroBOONE experiment to trigger on 325 scintillator light that is generated when a neutrino interacts in the detector. (Figure 326 courtesy Fermilab/R. Hahn.) [end caption] 327

328 Figure 7: The MicroBooNE detector in the test facility. Because MicroBooNE uses 329 liquid argon as a detection medium, the cryostat is shrouded in insulative material. 330 (Figure courtesy Fermilab/R. Hahn.) [end caption] 331 332 There are other experiments that have searched for signatures that can confirm or 333 falsify the LSND result. These are neutrino disappearance experiments which 334 complement the LSND appearance observations – after all, if one species of neutrino 335 appears during oscillation, then another must disappear. In one experiment at a nuclear 336 reactor complex in France, scientists detect a 6% deficit in electron neutrino flux [24]. 337 This is consistent with a fourth neutrino species with mass of about 1.2 eV/c2. In 338


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addition, an experiment using the Daya Bay nuclear reactor in China initially noted a 339 deficit in electron neutrino flux, but a recent paper suggests that this deficit can be 340 traced to incorrect modeling [25]. Furthermore, Daya Bay performed a joint search for 341 sterile neutrinos with Bugey-3 reactor experiment in France and the MINOS and 342 MINOS+ experiments at Fermilab [26]. Figure 8 shows their result, in which they saw no 343 evidence of neutrino oscillations. This creates yet another puzzle to resolve, since these 344 experiments completely exclude the region of oscillation parameters most favored by 345 the LSND and MiniBooNE data. 346

347

348 349

Figure 8: The results of a joint search for sterile neutrinos conducted by the MINOS, 350 MINOS+, Daya Bay, and Bugey-3 experiments [26]. They see no evidence of sterile 351 neutrinos and exclude at a 90% confidence level all oscillation parameters to the right of 352 the red line. Also shown are exclusion curves from the NOMAD and KARMEN 353 experiments. Taken together, the LSND and MiniBooNE region seems to be entirely 354 excluded. Figure adapted from [26], which also contains the relevant references. [end 355 caption] 356

357 There are other experiments searching for sterile neutrinos, including one [27] from the 358 IceCube detector, which uses a cubic kilometer of ice in Antarctica as the detection 359 medium. IceCube reports no evidence for the existence of sterile neutrinos. 360

361 362


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Conclusion 363 364

If sterile neutrinos were to exist, it would require a substantial rewrite of our 365 understanding of the subatomic world. There are a few experiments that have reported 366 evidence for their existence; however, there have been many others (with higher 367 sensitivity) that have ruled out the parameter space favored by LSND. At this point, the 368 case for the existence of sterile neutrinos is by no means a strong one. The conflicting 369 results might be because they exist, but are hard to find; or the handful of positive 370 results may simply be anomalies indicative of the difficulty associated with studying 371 neutrinos – ghosts of the subatomic world. 372

373 Fermilab, America’s flagship particle physics laboratory, continues to expand their 374 already world-class neutrino research program and relevant experiments are underway 375 to perform precise tests of the LSND and MiniBooNE results [28]. We can’t say yet 376 whether these efforts will definitively answer the question of the existence of the sterile 377 neutrino, or introduce even more puzzles to resolve, but it is safe to say that results will 378 add another fascinating chapter to the saga of the enigmatic sterile neutrino. 379

380 References 381

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Scientific, Singapore (2012, rev.); R. Crease and C. Mann, The Second Creation:384 Makers of the Revolution in Twentieth-Century Physics, Rutgers University Press, 385 Rutgers, (1996); F. Close, Neutrino, Oxford University Press, Oxford, (2012); L. 386 Lederman and D. Teresi, The God Particle: If the Universe Is the Answer, What Is 387 the Question?, Mariner Books, New York, (2016). 388

389 [2] C. L. Cowan, et al., Detection of the Free Neutrino: A Confirmation, Science 124,390

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[8] K. S. Hirata, et al. [The Kamiokande Collaboration], Experimental study of the 409 atmospheric neutrino flux, Phys. Lett. B205 (1988) 416; D. Casper, et al. 410 Measurement of the atmospheric neutrino composition with the IMB-3 detector, 411 Phys. Rev. Lett. 66 (1991) 2561. 412

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Prog. Theor. Exp. Phys. 2020, 083C01 (2020) (section 14). 429 430 [13] While the form of equation (1) is what is used by particle physicists, it can also be 431

written as 𝑃(𝑣𝑒 → 𝑣𝜇) = sin2(2𝜃) sin2 [( ∆𝑚12

2

4𝐸) (

𝑐3

ℏ) 𝐿], where L is in meters, 432

m2 is in kg, and E is in Joules. 433 434 [14] X. Q. Lu, et al., A Proposal to Search for Neutrino Oscillations with High Sensitivity 435

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445 [16] A. Aguilar et al., Evidence for neutrino oscillations from the observation of �̅�𝜇 446

appearance in a �̅�𝑒 beam, Phys. Rev. D 64, 112007 (2001). 447 448 [17] See reference 12, (Neutrinos in cosmology section). 449 450 [18] D. O. Caldwell and R. N. Mohapatra, Neutrino mass explanations of solar and 451

atmospheric neutrino deficits and hot dark matter, Phys. Rev. D 48, 3259 (1993); J. 452 Peltoniemi and J. W. F. Valle, Reconciling dark matter, solar and atmospheric 453

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