universe

Review Observation of Atmospheric Neutrinos

Yusuke Koshio

Experimental Particle Physics Group, Department of Physics, Okayama University, Okayama 7008530, Japan; [email protected]



Received: 21 April 2020; Accepted: 1 June 2020; Published: 5 June 2020


Abstract: In 1998, the Super-Kamiokande discovered neutrino oscillation using atmospheric neutrino anomalies. It was the first direct evidence of neutrino mass and the first phenomenon to be discovered beyond the standard model of particle physics. Recently, more precise measurements of neutrino oscillation parameters using atmospheric neutrinos have been achieved by several detectors, such as Super-Kamiokande, IceCube, and ANTARES. In addition, precise predictions and measurements of atmospheric neutrino flux have been performed. This paper presents the history, current status, and future prospects of the atmospheric neutrino observation.

Keywords: neutrino oscillation; atmospheric neutrino

1. Introduction

1.1. History Atmospheric neutrinos are generated when primary cosmic rays strike nuclei in the atmosphere and the hadron that results from these collision decays [1]. The principles and expected event rates of neutrino detection were first proposed in the 1960s [2,3]. The first discovery was achieved by two independently working groups in 1965. These groups placed detectors deep underground in the Kolar Gold Mines of South India [4] and a South African gold mine [5], respectively, to avoid cosmic-ray muons. They looked for upward-going muon events, because it must be generated through the interaction of atmospheric neutrino from the other side of the Earth with the surrounding rock, and found the signal clearly. The next generation of neutrino detectors, Kamiokande and IMB (Irvine-Michigan-Brookhaven detector), appeared in the 1980s. These were kiloton-scale water Cherenkov detectors, whose detection principles will be mentioned later. The original purpose of the experiments using these detectors was to search for nucleon decay predicted by the Grand Unified Theory, and atmospheric neutrinos were one of the serious backgrounds of the nucleon decay search. However, nucleon decay was not discovered, thus, neutrinos became the main target of research. The successful of the neutrino observation was, at first, neutrino signals from Supernova 1987A [6]. In addition, Kamiokande detected clear neutrino signal from the Sun [7], called “solar neutrinos”. The first discovery of the solar neutrinos were reported by Homestake experiment in 1968 [8]. The strength of the observed signal was about one third of the predicted amount. It was long standing problem called “solar neutrino puzzle” [9]. At the beginning of the 2000s, this was found to be attributable to neutrino oscillation. The anomalous atmospheric neutrino measurements were also reported by several experiments in this period. The ratio of νµ and νe should be roughly 2:1 since atmospheric neutrinos are generated by the decay of pions (νµ) and muons (νµ and νe); however, the ratio observed was not in agreement with the predicted ratio. The average discrepancy between the data and Monte Carlo simulation +0.08 (MC) was (µ/e)data/(µ/e)MC = 0.57−0.07 ± 0.07 in Kamiokande, though the ratio had incident angle dependence [10,11]. The value reported in IMB was 0.54 ± 0.05 ± 0.12 [12,13] which was consistent with

Universe 2020, 6, 80; doi:10.3390/universe6060080 www.mdpi.com/journal/universe Universe 2020, 6, 80 2 of 24 the results obtained from the Kamiokande detector. However, in Frejus experiment, which deployed a sandwich made of iron plates and plastic flash tubes in a 1780 m underground tunnel between +0.32 France and Italy, it was 1.13−0.25 [14], which was inconsistent with the results of other experiments. The discrepancy between the data and the MC reported by Kamiokande and IMB attributed to neutrino oscillation; however, the statistical evidence was insufficient to support this conclusion. In 1998, Super-Kamiokande reported a clear evidence of the neutrino oscillation using an anomaly of the zenith angle distribution of the atmospheric neutrinos [15]. Several atmospheric neutrino experiments confirmed the result at the same period. One was the MACRO (Monopole, Astrophysics and Cosmic Ray Observatory) experiment, which was a tracking detector composed of liquid scintillator and streamer tubes in Gran Sasso Laboratory in Italy, reported a clear distortion in upward-going muon spectrum caused by neutrino oscillation [16]. The other was Soudan II, which was a fine-grained gas tracking detector in the Soudan Underground Mine State Park, MN, USA. Due to the capability of tracking of low-velocity charged particles generated by neutrino interaction, the direction and energy of incident neutrinos were well reconstructed. Using this information, the Soudan II reported the ratio of the travel distance divided by energy of neutrinos which is sensitive to determine the neutrino oscillation parameters [17]. Neutrino oscillation parameters reported by all these experiments were in good agreement. Recently, several huge volume detectors such as the IceCUBE neutrino detector and ANTARES (Astronomy with a Neutrino Telescope and Abyss environmental RESearch) detector reported the atmospheric neutrino observations. More precise measurements of the three-flavor neutrino oscillation parameters like mass ordering and Charge conjugation Parity symmetry (CP) violation in lepton sector are possible using atmospheric neutrino data due to increasing the statistics and several different observations. In addition, atmospheric neutrino flux has been measured by combining all the experiments and compared with theoretical calculations. Many efforts to reveal unresolved issues in neutrino physics have been made by several groups both theoretically and experimentally using atmospheric neutrino, and will be also made in future.

1.2. Atmospheric Neutrino Flux Prediction Atmospheric neutrinos are generated from the decay of π and K, which are secondary particles resulting from the interaction of cosmic rays with air molecules in the Earth’s atmosphere. They come from all directions into detectors although their flux depends on the zenith and azimuth angles of the direction of their arrival. The flux in the horizontal direction is generally higher than that in the vertical direction, because of the longer path taken by parent particles through the atmosphere. The energy from atmospheric neutrinos exists in a wide range of 100 MeV to PeV scale measurements along a power-law spectrum, as shown in the Figure1. The reason fewer neutrinos are produced at higher energies is mainly due to the falling primary cosmic-ray spectrum. The effect that the π decay lengths are longer than the paths in atmosphere and the parent particles reach the ground before decaying, plays a role. It is also relevant for the energy dependence of νµ and νe ratio and causes the neutrino flux to peak at the horizon as shown in the right Figure1. The energy distribution is suppressed below the GeV energy region due to the rigidity cutoff effect of the primary cosmic rays by Earth’s magnetic field. The trajectories of charged particles in primary cosmic rays are affected by geomagnetic fields. Only particles that interact in the atmosphere before curving back into the space produce atmospheric neutrinos. The trajectory depends on the particles’ momentum and total charge; the ratio of them is called “rigidity”. Geomagnetic fields affect particles with lower energy more strongly; therefore, low-energy atmospheric neutrino flux is suppressed, a process that is called “cutoff”. Since Kamioka is located at a rather low geomagnetic latitude, it has a high local vertical rigidity cutoff. In addition, due to its strong azimuthal asymmetry, the cutoff is higher for particles arriving from the east than from the west. This east-west asymmetry arises from the fact that the primary cosmic rays are positively charged. On the other hand, in high-energy regions, above a few Universe 2020, 6, 80 3 of 24

tens TeV, neutrinos from charm mesons’ decay are considered, instead of π and K decay, due to the much shorter lifetimes of these charm mesons which are on the order of 10−12 s. These are called “prompt” neutrinos [18,19], which are uniformly generated in the atmosphere, with equal fluxes of νµ and νe. Precise predictions of the atmospheric neutrino have been performed by several research groups, including HKKM [20], Bartol [21], and FLUKA group [22]. The differences among these models are choices of hadronic models and measurements of the primary cosmic-ray spectrum. Figure1 shows a comparison of atmospheric neutrino fluxes and neutrino flux ratios calculated by different groups. The flux calculations differ by about 10%. Another feature is that the flavor ratio between νµ + ν¯µ and νe + ν¯e below the GeV scale is approximately two; however, it increases with the energy levels, as shown in Figure1, because the muons produced by π decay reach the ground before decaying. Time variation in the atmospheric neutrino flux is also expected. Long-term variation is due to solar activity with an average period of 11 years. The primary cosmic-ray flux at Earth is anticorrelated with the solar activity because the plasma from the Sun scatters the cosmic rays, and the cosmic-ray flux is reduced during periods of high solar activity. Consequently, the atmospheric neutrino flux is also predicted to be anticorrelated with solar activity. There is also yearly variation due to seasonal temperature variations that affect atmospheric density which increases in summer, when relatively more neutrinos are produced. IMPROVEMENT OF LOW ENERGY ATMOSPHERIC ... PHYSICAL REVIEW D 83, 123001 (2011) 5 2 ν + ν 10 This Work µ µ HKKMS06 1.5xν νe + νe νe µ Bartol Fluka 0.75xνe ( x1.5) 1 νµ νµ − 210− 1 − 1 νµ 2 (m (m sec sr GeV) 2

ν This Work HKKMS06 ν ν x E e e ν Bartol Neutrino Flux Ratios φ Fluka 0 10 1 −1 0 1 −1 0 1 10 10 10 10 10 10 E (GeV) M. HONDA et al. ν PHYSICALE ν (GeV) REVIEW D 83, 123001 (2011) 3 2 4x10 3x10 FIG. 7 (color online). Comparison0.32 GeV of atmospheric neutrino1.0 fluxes GeV calculated for Kamioka3.2 GeV averaged over all directions (left panel), 1 10 and the flux ratios (right panel), with other calculations. The dashed lines are the calculation by the Bartolνµ group [20,21], dotted lines for the FLUKA group [22], and dash dot for our previous calculation (HKKM06). − 1 ) − 1 νµ νµ

νµ ν 2 0

µ − 1 − 1 − 1 − 1 calculation is smoothly− 1 − 1 connected to the 1D calculation10 atmospheric neutrino5x10 fluxes calculatedνe with the modified- − 2 − 2 − 1 − 2 (m s sr sr s (m GeV ) (m s sr GeV 3 (m s sr GeV ) νe ν ν carried out in the previous10 work. As the modifiedνµ ν JAM is JAM show an increase from the previous one below 1 GeV, φ φ ν φ used below 32 GeV, any difference abovee that is due to the as is expected from the increase of atmospheric muon νe difference of the calculation scheme between 3D and 1D. spectraνe below 1 GeV=c. 0 However, the difference between present and previous In the right2x10 panel of Fig. 7, we show the ratios of the νe

works is very small in the 2 figure above 1 GeV. Note, 1 we atmospheric neutrino flux averaged over all directions. It is 4x10 3x10 magnify the difference− between1.0 −0.50 0present 0.50and 1.0 previous−1.0 −0.50 0seen 0.50 that 1.0 the differences−1.0 −0.50 in 0 the 0.50 flux 1.0 ratios are very small cos works in the ratio in Fig. 8. The differencecosθ is less than a cosbetweenθ present and previousθ works above 1 GeV as the FIG. 9 (color online). The zenith angle dependence of atmospheric neutrino flux averaged over all azimuthal angles calculated for fewFigure percent 1. aboveKamioka.Comparison 1 Here GeV. is the On of arrival atmospheric the direction other of the neutrino, hand, neutrino with thecos fluxesabsolute1 for vertically at atmospheric Kamioka downward going averaged neutrinoneutrinos, and fluxes.cos over It1 allfor is directions noticeable that vertically upward going neutrinos. ¼the # # = # # ratios are very¼À close to each other (upper left), and the flux ratio of the neutrino flavor (upperð " þ right)"Þð accordinge þ eÞ to HKKM11 (red) [20], including the ones calculated by different authors. We find Bartol (dashed) [21], FLUKA (dotted) [22], and a previous HKKM06 model [23]. Some plots are applied E = 0.32 GeV E = 1.0 GeV there is a visibleE difference= 3.2 GeV in #e=#e between the present by several factors forν easy viewing. Lowerν figures show zenith angleν dependence of atmospheric 1.2 1.2 νµ 1.2 and previous works.1.2 As is seen in the left panel of Fig. 5 in νµ νµ νµ neutrino1 flux averaged1 over all azimuthal angles1 for KamiokaSec. II, the site original calculated1 JAM by interaction HKKM11. modelThese has figures a smaller 0.8 0.8 0.8 are taken0.8 from [20]. %þ=%À ratio, and is responsible for the smaller #e=#e ratio 1.2 νµ 1.2 below 1 GeV.1.2 Note, we do not modify the interaction νµ νµ 1 1 model more than1 the muon flux data require. It is difficult 1.3. Neutrino Interaction0.8 0.8 0.8 1.2 This Work This Work This Work νµ to examine the "þ="À ratio at the momenta correspond- 1 νe Interactions with1.2 nuclei in water (or1.2 ice) anding rocks to theνe surrounding#e=#1.2e below 1 the GeV, detector dueνe to the are small the statistics norm in in 1 1 1 Ratio to HKKM06 Ratio to HKKM06 Ratio to HKKM06 0.8 Bartol Bartol the observation at balloon altitude. 0.8This Work 0.8 0.8 Bartol atmospheric neutrino observations,Fluka and that withFluka electrons is negligible as the cross-section is three The muons at seaFluka level or mountain altitude are not 1.2 νe 1.2 1.2 orders of magnitude1.2 smaller than thatνe with nuclei. Interactionsusefulν toe examine can the be atmospheric classifiedνe into neutrino charged-current of these ener- 1 1 gies, since the1 muons result from higher energy pions± at 0 1 0.8 0.8 0.8 (CC) or neutral-currentRatio to HKKM06 (NC) interactions according to the type of bosons that are exchanged, W or Z , Bartol higher altitude. 0.8 −1.0 −0.50 0 0.50 1.0 −1.0 −0.50 0 0.50 1.0 −1.0 −0.50 0 0.50 1.0 Fluka In Fig. 9, we show the atmospheric neutrino fluxes as a cosθ cosθ cosθ function of the zenith angle averaging over all the azimu- 1.2 FIG. 10 (color online). Ratio of the atmospheric neutrino flux averaged over azimuth angles as the function of zenith angle to the same quantity calculated in HKKM06. Theνe short dash lines stand forthal the calculation angles by at Bartol 3 neutrino group [20,21] energies; and dotted lines 0.32, for the 1.0, and 3.2 GeV 1 FLUKA group [22]. for Kamioka. In Fig. 10, we show the comparison of the 0.8 present and previous works in the ratio as the function of the variation of atmospheric neutrino flux as a function the geomagnetic field [23]. Note, the variation of upward of azimuthal angle. In Fig. 11, we show the variation of goingzenith neutrinos angle. ( There0:2 > cos is a) difference is much more due compli- to the increase of the −1 0 1 À 10atmospheric neutrino10 flux as the10 function of the azimu- catedneutrino than the flux variation itself, of downward but the going ratio neutrinos is almost constant. thal angle averaging them over the five zenith angle ( cos>0:2). This is because the upward-going neutri- E (GeV) Actually, the calculated zenith angle dependences are vir- ranges, 1 > cos>ν 0:6, 0:6 > cos>0:2, 0:2 > cos> nos are produced in a far larger area on the Earth than 0:2, 0:2 > cos> 0:6, and 0:6 > cos> 1, at thetually downward-going the same neutrinos, as for the and calculation are affected by in large Ref. [2]. À À À À À FIG. 8 (color online).1 GeV forAtmospheric Kamioka. It is neutrino seen in that fluxes the variation calculated of the variationIt seems of rigidity that cutoff the and zenith geomagnetic angle field. dependence The of the 3D atmospheric neutrino flux has complex structures at downward-going neutrinos are produced just above the for Kamioka averaged1 GeV dueover to all the directions,rigidity cutoff as and a muon ratio bending to the in detector.calculation is smoothly connected to that of the 1D HKKM06 calculation. The dashed lines are the calculation by calculation just above 3.2 GeV for the average over all Bartol group [20,21] and dotted lines the FLUKA group [22]. azimuth angles. However, this is not true when we study 123001-8

123001-7 Universe 2020, 6, 80 4 of 24 respectively. A CC interaction produces a charged lepton, electron or muon, whose flavor corresponds to that of a neutrino, νe or νµ. Therefore, the original neutrino flavor is identified by distinguishing the flavor of the related charged lepton. However, an NC interaction does not indicate the neutrino flavor since the outgoing lepton is a neutrino. The following neutrino interactions are dominant in the atmospheric neutrino energy region,

• Charged-Current quasi-elastic scattering : ν + N → l + N’ • Charged-Current pion production : ν + N → l + N’ + π • Deep Inelastic Scattering (DIS) : ν + N → l + N’ + hadrons where N and N’ are nucleons (proton or neutron) and l is a charged lepton (CC) or neutrino (NC). Here, pion production is realized via ∆ resonance excitation. To generate neutrino interactions, there are several pieces of simulation software. Figure2 shows the total cross-section of νµ in total and each interactions predicted by NEUT [24] version 5.3.6, which was used in the latest atmospheric neutrino analysis in Super-Kamiokande. In this model, charged-current quasi-elastic interactions are simulated using the Llewellyn-Smith formalism [25] with nucleons distributed according to the 2 Smith-Moniz relativistic Fermi gas [26] assuming an axial mass MA = 1.21 GeV/c and form factors from [27]. Interactions on correlated pairs of nucleons have been included following the model of Nieves [28]. Pion production processes are simulated using the Rein-Sehgal model [29] with Graczyk form factors [30]. The cross-section in this model are consistent with several experimental results.

1.4 0.7 CCQE CC single pion CC Total CCQE CC Total

/GeV) ANL ANL 82 CCFR 90 /GeV) GGM 77 CCFR 90 2 2 1.2 GGM 77 BNL 86 CDHSW 87 0.6 GGM 79 CDHSW 87

cm GGM 79 IHEP-JIRN 96 cm Serpukhov IHEP-JIRN 96

-38 Serpukhov IHEP-ITEP 79 -38 IHEP-ITEP 79 1 CCFRR 84 0.5 CCFRR 84 BNL 82 /E (10 /E (10 σ σ 0.8 0.4

0.6 0.3

0.4 0.2

0.2 0.1

0 0 10-1 1 10 102 10-1 1 10 102 E (GeV) E (GeV)

Figure 2. Total cross-section divided by neutrino energy for νµ (left) and ν¯µ (right) to nucleon charged-current interactions calculated by NEUT version 5.3.6 overlaid with several experiments. Data points are taken from the following experiments: ANL [31], GGM77 [32], GGM79 (left) [33] (right) [34], Serpukhov [35], ANL82 [36], BNL86 [37], CCFR90 [38], CDHSW87 [39], IHEP-JINR96 [40], IHEP-ITEP79 [41], CCFRR84 [42] and BNL82 [43].

1.4. Neutrino Oscillation Neutrino oscillation occurs if flavor eigenstates are mixed with mass eigenstates, and a difference in mass exists. The mixing between flavor eigenstates (να) and mass eigenstates (νi) can be written as

∗ |ναi = ∑ Uαi |νii . (1) Universe 2020, 6, 80 5 of 24

In the three-flavor neutrino framework, U is Pontecorvo-Maki-Nakagawa-Sakata (PMNS) mixing matrix [44–46], and usually parametrized by three mixing angles (θ12, θ23, θ13), and one CP-violating Dirac phase (δCP) in neutrino oscillations as follows:

     −iδ      νe 1 0 0 c13 0 s13e CP c12 s12 0 ν1            νµ  =  0 c23 s23   0 1 0   −s12 c12 0   ν2  (2) −iδ ντ 0 −s23 c23 −s13e CP 0 c13 0 0 1 ν3 where cij and sij represent cos θij and sin θij, respectively. Neutrino oscillation frequencies are 2 2 2 2 2 2 determined by the neutrino mass differences, ∆m21 = m2 − m1 and ∆m32 = m3 − m2, where m1, 2 m2, and m3 are the three mass eigenvalues. Among these oscillation parameters, θ12 and ∆m21 have 2 been measured by solar and reactor neutrino experiments, θ23 and ∆m32 have been measured by atmospheric and accelerator neutrino experiments, and θ13 has been determined by reactor neutrino experiments. The order of absolute mass (m3  m2 > m1 in a normal ordering or m2 > m1  m3 in an inverted ordering) and the CP phase (δCP) are still unknown parameters. The dominant neutrino oscillation in atmospheric neutrinos is a channel between νµ and ντ caused by the parameters of the mass eigenstate between ν2 and ν3. In addition, the appearance of oscillation from νµ to νe is considered to be a sub-leading effect. The dominant neutrino oscillation probabilities in a vacuum for atmospheric neutrinos are expressed as follows: ! ∆m2 L P(ν → ν ) ' 1 − sin2 2θ sin2 31 (3) e e 13 4E ! ∆m2 L P(ν → ν ) ' 1 − 4 cos2 θ sin2 θ (1 − cos2 θ sin2 θ ) sin2 31 (4) µ µ 13 23 13 23 4E ! ∆m2 L P(ν ↔ ν ) ' sin2 θ sin2 2θ sin2 31 (5) µ e 23 13 4E ! ∆m2 L P(ν ↔ ν ) ' sin2 2θ cos4 θ sin2 31 (6) µ τ 23 13 4E ! ∆m2 L P(ν ↔ ν ) ' cos2 θ sin2 2θ sin2 31 (7) e τ 23 13 4E where L is the flight length of neutrinos and E is the neutrino energy. When neutrinos traverse the Earth, their matter potential due to the difference in the forward-scattering amplitudes of νe and νµ,τ, which induces a matter dependent effect on of neutrino oscillations [47], must be taken into account. 2 2 In this scenario, the oscillation value in Equation (7) can be rewritten by replacing ∆m31 and sin 2θ13 for constant matter density, q 2 2 2 2 2 ∆m31,M = ∆m31 sin 2θ13 + (2EVe/∆m31 − cos 2θ13) (8) sin2 2θ sin2 2θ = 13 (9) 13,M 2 2 2 sin 2θ13 + (2EVe/∆m31 − cos 2θ13) √ where Ve = ± 2GF Ne is the effective matter potential, and the sign is positive for neutrinos or negative for anti-neutrinos; Ne is the electron density, which is assumed to be constant; and GF denotes the Fermi constant. The potential is derived from the difference that νe which has both CC with electrons 2 and NC with electrons and nucleons, while νµ and ντ have only NC. In this form, when 2EVe/∆m31 = cos 2θ13, the effective mixing angle is resonantly enhanced. Since cos 2θ13 is positive, the enhancement 2 only occurs for neutrinos if ∆m31 is positive which is as normal mass ordering, while it occurs for anti-neutrinos only in the case of the inverted mass ordering. Figure3 shows the νµ survival and Universe 2020, 6, 80 6 of 24

νµ → νe transition probabilities for neutrinos and anti-neutrinos assuming a normal mass ordering. Earth matter effects suppress the disappearance of νµ and enhance the appearance of νe especially in upward-going neutrinos, where the cosine zenith angle is negative, with energies in the range of 2–10 GeV. The appearance of νe in neutrino (b) in Figure3 is enhanced in this energy region, while no clear enhancement appears in the anti-neutrino plot (d). If the mass ordering is inverted, this feature is switched and appears as an anti-neutrino case. Therefore, the difference in this level of enhancement between neutrinos and anti-neutrinos can help to determine the mass ordering in neutrino oscillation. The νµ → νe transition probability is also affected by the δCP parameter, which results in a change of about 2% in the maximum total νe flux observed in the energy region less than 1 GeV. K. ABE et al. PHYS. REV. D 97, 072001 (2018)

FigureFIG. 3. 2.Oscillation Oscillation probabilities probabilities for neutrinos for (upper neutrinos panels) and (upper antineutrinos panels) (lower panels) and anti-neutrinosas a function of energy (lower and zenith panels) angle as assuming a normal mass hierarchy. Matter effects in the Earth produce the distortions in the neutrino figures between two and ten GeV, a functionwhich are of not energy present in and the antineutrino zenith angle figures. Distortions assuming in the aν normalμ survival probability mass ordering. and enhancements The in figures the νe appearance on the left probability occur primarily in angular regions corresponding to neutrino propagation across both the outer core and mantle regions show(cosine the disappearance zenith < −0.9) and propagation of νµ and through those the mantle on the and crust right (−0 show.9 < cosine the zenith appearance< −0.45). For of an invertedνe. Matter hierarchy effects the in 2 −3 2 matter effects appear in the antineutrino figures instead. Here the oscillation parameters are taken to be Δm 32 2.5 ×∼10 eV , the Earth2 produce2 distortions in the neutrino figures with energies in the range¼ of 2 10 GeV in sin θ23 0.5, sin θ13 0.0219, and δCP 0. upward-going¼ neutrinos¼ (where the¼ cosine zenith angle is negative), while there is no such distortion in

the anti-neutrinoquantity of underground figures. water In inverted available to hierarchies, fill the detector theneutrino distortion interactions caused provide by a matter calibration effects point via appears the π0 only in anti-neutrinoand maintain figures. its temperature. The discontinuities These changes impact of the the probabilitymomentum near and stopping the cosine cosmic zenith ray muons angles of ofvarious−0.5 and water transparency and subsequent performance of the momenta are used to measure photoelectron production as −0.8 arisedetector from and therefore neutrino must propagation be corrected through across calibra- the differenta function matter of muon density track length regions (Cherenkov of the angle) crust, for mantle, and core.tions. In Since downward-going neutrino oscillations neutrinos are a function (where of the the cosinemulti-GeV zenith (sub-GeV) angle energies. is positive), Here the muon a 25 track km lengthbaseline of neutrino energy, a thorough understanding of the detector is estimated using the distance between the entering vertex the vacuumenergy scale neutrino is important oscillation for precision is assumed. measurements. Here theand neutrino the position oscillation of the electron parameters produced in its are subsequent taken to be 2 At the same−3 time2 the range of energies of2 interest to decay. The energy spectrum of these Michel electrons ∆m = 2.5 × 10 eV , sin 2θ23 = 0.5, sin θ13 = 0.0219 and δCP = 0. This figure is taken from [48]. 32 atmospheric neutrino analysis spans from tens of MeV to additionally serves as a low energy calibration point. tens of TeV, eliminating the possibility of calibration Figure 3 shows the absolute energy scale measurement 2. Detectorsthrough radioactive isotopes. Accordingly, the energy scale using each of these samples. is calibrated using natural sidebands covering a variety of In the oscillation analysis the absolute energy scale 2.1. Super-Kamiokandeenergies. Neutral pions reconstructed from atmospheric uncertainty is conservatively taken to be the value of the The Super-Kamiokande (SK) detector is a cylindrical tank of 39.3 m diameter and 41.4 m height, 072001-6 filled with 50 kilotons of pure water as shown in Figure4. It is located 1000 m underground (2700 m water equivalent) in the Kamioka mine in Gifu Prefecture, Japan. The detector is divided into two regions called “inner” and “outer”, and lined with 11,129 twenty-inch PMTs in the inner detector and 1885 eight-inch PMTs in the outer detector. The signal detection method of SK is that Cherenkov light Universe 2020, 6, 80 7 of 24 generated in water from the charged particles is observed by PMTs [49]. The fiducial volume used for the data analysis is 22.5 kilotons, which is within 2 m of the inner wall to maintain the detector performance. SK was launched in April 1996, and until now, there have been five experimental phases. The first phase lasted five years until July 2001. There were several successes in the detection of atmospheric neutrinos and solar neutrinos. However, a serious accident occurred in November 2001, in which most of the PMTs were broken. The experiment was resumed in October 2002, with about half the number of PMTs. Fortunately, even with half the number of PMTs, the atmospheric neutrino analysis was not affected greatly. After a three-year operation, full reconstruction was completed in 2006. The third phase, with almost the same number of PMTs as in the first phase, started in July 2006, and ended in August 2008. In the fourth phase, new electronics (QBEE [50]) were installed, which improved the detection efficiency of the decay electron from stopping muon higher. It is also possible to tag a 2.2 MeV gamma-ray, which is produced by the neutron capture by protons in water, even though it is about 20% detection efficiency. Neutron signals are useful for the improvement of atmospheric neutrino analysis. Although it is difficult to separate the neutrino and anti-neutrino events, information about the number of neutrons can be used to statistically differentiate between them because the number of neutrons in anti-neutrinos is larger than that in neutrinos. The fourth phase continued for 10 years until May 2018. SK was refurbished during the rest of 2018 to allow the loading of gadolinium to pure water. If gadolinium is loaded, the efficiency of neutron detection significantly improves by up to 90% depending on the gadolinium concentration. The main purpose of this gadolinium loading is the discovery of Diffuse Supernova Neutrino Background, but it also improves the analysis of atmospheric neutrinos due to the high neutron detection efficiency. After a successful refurbishment, the fifth phase started in January 2019.

Figure 4. SK detector.

In the atmospheric neutrinos observation in SK [51], the data sample is categorized by three distinct topologies: fully contained (FC), partially contained (PC), and upward-going muon (UPMU). FC events produce reconstructed interaction vertices inside the fiducial volume, while PC events also produce interaction vertices within the fiducial volume of the inner detector but are accompanied by considerable light in the outer detector. FC can be further sub-divided into single- and multi-ring and electron- and muon-like events. UPMU events are caused by muon-neutrino interactions in the surrounding rock, which produce penetrating muons. These muons either stop in the inner detector volume (stopping events) or continue through the inner detector (through-going events). The leptons, muons or electrons, generated by neutrino interactions preserve information from the original neutrinos, including the neutrinos’ flavor, energy, and direction. To identify muons or electrons, the Cherenkov ring pattern is used. An electron produces diffused ring patterns because of the electromagnetic shower and multiple scattering, while a muon produces a ring with a sharp VOLUME 81, NUMBER 8 PHYSICALREVIEWLETTERS 24AUGUST 1998

cally, final state leptons with p , 100 MeVyc carry 65% of the incoming neutrino energy increasing to ,85% at p ≠ 1 GeVyc. The neutrino flight distance L is esti- mated following Ref. [18] using the estimated neutrino energy and the reconstructed lepton direction and flavor. Figure 4 shows the ratio of FC data to Monte Carlo for e-like and m-like events with p . 400 MeV as a func- tion of LyEn, compared to the expectation for nm nt oscillations with our best-fit parameters. The e-like$ data show no significant variation in LyEn, while the m-like events show a significant deficit at large LyEn. At large LyEn, the nm have presumably undergone numerous os- cillations and have averaged out to roughly half the initial rate. The asymmetry A of the e-like events in the present data is consistent with expectations without neutrino oscilla- Universe 2020, 6, 80 tions and two-flavor ne nm oscillations are not favored.8 of 24 This is in agreement with$ recent results from the CHOOZ FIG. 2. The 68%, 90%, and 99% confidence intervals are experiment [22]. The LSND experiment has reported the shownedge. for The sin SK2 2u atmosphericand Dm2 for n neutrinon two-neutrino analysis uses oscil- eventappearance categories, of n reconstructede in a beam of energynm produced and direction, by stopped m $ t lationsand particle based on types33.0 kton (muon-like yr of Super-Kamiokande or electron-like). data. The pions [23]. The LSND results do not contradict the 90% confidence interval obtained by the Kamiokande experi- present results if they are observing small mixing angles. ment is also shown. With the best-fit parameters for n n oscillations, we Discovery of the Neutrino Oscillation m $ t expect a total of only 15–20 events from nt charged- case overlapped at 1 3 1023 ,Dm2 , 4 3 1023 eV2 current interactions in the data sample. Using the current 2Figure5 shows several plots of zenith angle dependence on the first results obtained from SK in for sin 2u ≠ 1. sample, oscillations between nm and nt are indistinguish- 1998 [15]. The electron-like events were consistent with the predicted results, while the number of As a cross-check of the above analyses, we have re- able from oscillations between nm and a noninteracting constructedupward-going the best muon-like estimate of events the ratio wereLyE significantlyn for each smallersterile neutrino. than that of the downward-going events. event.This was The evidence neutrino ofenergy the neutrino is estimated oscillation by applying that amuon-neutrinosFigure 2 shows converted the Super-Kamiokande to other flavor resultsof neutrinos overlaid correction to the final state lepton momentum. Typi- with the allowed region obtained by the Kamiokande through the flight inside the Earth.

FIG. 3.Figure Zenith 5. angleZenith distributions angle distributions of m-like and ofe- muon-like events and for electron-like sub-GeV and events multi-GeV at sub-GeV data sets. and Upward-going multi-GeV particles have cos Q,0 and downward-going particles have cos Q.0. Sub-GeV data are shown separately for p , 400 MeVyc and p . 400scalesMeVy inc. the Multi-GeV first 535e-like days distributions of SK data are produced shown for p [15,].2.5 Upward-goingand p . 2.5 GeV eventsyc and theare multi-GeV at less thanm-like zero are shown separatelyand for downward-going FC and PC events. events The hatched are at region more shows than the zero Monte on theCarlo horizontal expectation axis. for no The oscillations vertical normalized axis shows to the data live time with statistical errors. The bold line is the best-fit expectation for nm nt oscillations with the overall flux normalization fitted asthe a free number parameter. of events in each bin. The hatched regions show$ the expected results without neutrino oscillation. The lines show the best-fit expectations with neutrino oscillation.

15662.2. IceCube The IceCube is a cubic-kilometer neutrino detector buried in the Antarctic ice, as shown in Figure6. It comprises 5160 digital optical modules (DOMs) along 86 vertical strings, with 60 DOMs per string. Each DOM houses a downward-facing 10-in PMT with electronics in a glass pressure sphere. Of the 86 strings, 78 are deployed with an inter-string distance of ∼125 m and a space of ∼17 m between each DOM at depths between 1450 and 2450 m below the surface. This part of the detector is optimized for the neutrino energy range of 100 GeV to 100 PeV. The remaining eight strings, located at the bottom center of the detector, are set more densely with DOMs. This detector, called DeepCore, comprises 647 DOMs with high-quantum-efficiency PMTs placed 2100 m under the clearest ice. DeepCore has a volume of ∼107m3, and is optimized for the detection of lower-energy neutrinos down to 5.6 GeV. IceCube observes Cherenkov light generated in ice by charged particles resulting from neutrino interactions. As the target neutrino energy for IceCube is high, deep inelastic scattering is a dominant feature in neutrino interaction. The observed events are categorized into two types of patterns: long, straight tracks produced by muons (track-like) and spherical cascades produced by electromagnetic and/or hadronic showers (cascade-like). IceCube observation was fully commissioned in 2011. One important result was the discovery of two ultrahigh-energy (∼PeV) neutrinos in 2013 [52]. Compared to the expected number of atmospheric +0.041 neutrinos, 0.082 ± 0.004−0.057, these two events were possibly of astrophysical origin. SEARCH FOR A DIFFUSE FLUX OF ASTROPHYSICAL ... PHYSICAL REVIEW D 84, 082001 (2011) give rise to a detectable extraterrestrial flux. In this angular distribution of the conventional atmospheric neu- paper, we present results from a search for a diffuse flux trino flux are summarized in Refs. [9,10]. The conventional of astrophysical muon neutrinos performed with the atmospheric neutrino spectrum approximately follows an 3:7 IceCube detector using data collected in its half completed EÀ spectrum in the TeV energy range. The prompt configuration between April 2008 and May 2009. We first component of the atmospheric neutrino flux is yet to be summarize astrophysical and atmospheric neutrino models measured, but full calculations of the prompt flux are given in Sec. II and describe the IceCube detector in Sec. III.We in Refs. [11–13]. The prompt component of the atmos- outline in Sec. IV how our final neutrino sample was pheric neutrino flux is predicted to follow the primary obtained. The analysis methodology is discussed in detail cosmic-ray energy spectrum which is approximately 2:7 in Sec. V and we present our final results in Sec. VI. EÀ . Since a hypothetical diffuse astrophysical neutrino flux would have a harder energy spectrum than atmos- II. ASTROPHYSICAL AND ATMOSPHERIC pheric neutrino backgrounds, evidence for a diffuse flux NEUTRINO FLUXES would appear as a hardening of an energy-related observ- able distribution. The benchmark diffuse astrophysical #" search pre- sented in this paper assumes an astrophysical flux, È, III. THE ICECUBE DETECTOR 2 with a spectrum È EÀ resulting from shock accelera- / 2 IceCube consists of three detectors operating together. tion. In addition to the EÀ spectral shape, astrophysical models of varying normalization and spectral shapes were The main in-ice array is composed of 4800 digital optical tested as well. The Waxman-Bahcall upper bound [1] was modules (DOMs) arranged in 80 strings which are de- 2 ployed vertically with 60 DOMs per string. The detector derived for optically thin sources assuming a È EÀ primary cosmic-ray spectrum. Becker, Biermann,/ and is deployed deep in the Antarctic ice between a depth of Rhode [2] calculated the diffuse astrophysical neutrino 1450 and 2450 m. The vertical spacing between each DOM flux from active galactic nuclei using observations from is 17 m and the horizontal spacing between each string of DOMs is 125 m giving a total instrumented volume of Fanaroff and Riley Class II (FR-II) radio galaxies. These 3 sources were used to normalize the flux of neutrinos by 1 km . The design is optimized for the energy range of assuming a relationship between the disk luminosity, the 100 GeV to 100 PeV [14]. The DeepCore extension is luminosity in the observed radio band, and the calculated deployed within the main in-ice array and consists of six neutrino flux. Mannheim [3] and Stecker [4] derived mod- specialized strings which lower the energy reach to els for optically thick active galactic nuclei sources assum- 10 GeV. IceCube was deployed in stages with the first ing the sub-TeV diffuse gamma-ray flux observed by the string deployed during the 2005–2006 Austral summer. Compton Gamma Ray Observatory [5] is produced by the This analysis is based on 1 yr of data taken with the decay of neutral pions. BL Lacertae objects that emit TeV 40-string configuration (Fig. 1) which was deployed during gamma rays can be interpreted to be opticallyUniverse thin2020, to6, 80 9 of 24 photon-neutron interactions. The model calculated by Mu¨cke et al. [6] assumes that charged cosmic rays are produced in these sources through the decay of escaping neutrons. An average spectrum of neutrinos from the pre- cursor and prompt phases of gamma-ray bursts is calcu- lated in Ref. [7] by correlating the gamma-ray emission to the observed flux of ultrahigh-energy cosmic rays. The primary backgrounds in the search for diffuse as- trophysical #" are the atmospheric muons and neutrinos arising from cosmic-ray-induced extensive air showers. The substantial downward-going atmospheric muon back- ground persists over a wide energy range from primary cosmic-ray energies of around a GeV to the highest mea- sured extensive air showers of 100 EeV [8]. These events were removed by using the Earth as a filter in order to select upward-going neutrinos traversing through the Earth. Two classes of atmospheric neutrinos were consid- ered: neutrinos arising from the decay of pions and kaons (the conventional atmospheric neutrino flux) and neutrinos FIG. 1 (color online).Figure Three-dimensional 6. IceCube detector [ view53]. of the IceCube arising from the decay of charm-containing mesons (the 2.3. ANTARES detector layout. This work was based on the 40-string configu- prompt atmospheric neutrino flux). Detailed three- ration which was half of the completed detector. The 40-string dimensional calculations of the energy spectrumThe and ANTARESconfiguration neutrino telescope was operational is in the from Mediterranean April 2008 to SeaMay at 2009. a depth of 2475 m [54]. A schematic view of the detector is shown in Figure7. It comprises 12 detection lines: 11 equipped with 25 storeys of three optical modules and one line with 20 storeys of optical modules, giving a total of 885 optical modules. Each optical module has a 10-in PMT, whose axis points 45◦ downward. The detector082001-3was completed in 2008, and a total of 2830 days of data had been analyzed for atmospheric neutrino data by 2016.

Figure 7. Schematic of the ANTARES detector [54].

2.4. Detector Summary Table1 shows the summary of three detectors for atmospheric neutrino research. Universe 2020, 6, 80 10 of 24

Table 1. Summary of the future atmospheric neutrino detectors.

Detector Type Mass (MegaTon) Future Possibility Hyper-K Water Cherenkov (Underground) 0.187 (fiducial) Second tank IceCube Upgrade Water Cherenkov (Ice) 2 (instrumented) PINGU KM3NeT/ORCA Water Cherenkov (Deep-Sea) 8 (instrumented) DUNE Liquid Argon TPC 0.01 (fiducial for 1st module) 40 kTon (in final)

3. Oscillation Analysis of Atmospheric Neutrinos

3.1. Oscillation Parameter Determination

3.1.1. Super-Kamiokande The main contribution to the understanding of neutrino oscillation from atmospheric neutrinos 2 is the determination of θ23 and ∆m32. Recently, all sub-leading effects of neutrino oscillation can be investigated due to precise observation and large statistics of data. Here, the earth matter effect plays an important role in neutrino oscillation schemes, which resolves mass ordering, two possible θ23 regions, and δCP. It has different behavior of appearance between νe and ν¯e for the mass ordering. However, SK is insensitive to the charge sign of particles; therefore, CC neutrino and anti-neutrino interactions cannot be distinguished on an event-by-event basis. Instead, they are statistically separated based on the number of decay electrons, number of Cherenkov rings, and transverse momentum. In the most sensitive energy region between 2 and 10 GeV, not only CC quasi-elastic interactions but also single pion production via ∆ resonance excitation and the deep inelastic scattering process should be considered. In single pion production, π− generated in an anti-neutrino reaction, such as + − 16 ν¯e + n → e nπ , will be captured on an O nucleus, leaving the positron as the only detected particle and no delayed electron signal. In neutrino reactions, on the other hand, π+ is generated via − + νe + n → e nπ . It is not captured in this manner and produces a delayed electron signal through its decay chain. Thus, an anti-neutrino tends to produce a single-ring event without any delayed electron, while a neutrino event has the opposite effect. Due to the V-A structure of the weak interaction, the angular distribution of the leading lepton from ν¯ is more forward than those from ν reaction, which means that the transverse momentum in ν¯ is expected to be smaller than that in ν. The statistical separation of νe and ν¯e is performed by a likelihood method using the above variables. For a constraint on the neutrino oscillation parameter, the data obtained in SK are fitted to expectations by MC simulation using a binned χ2 method. There are 520 analysis bins in total (energy and zenith angles in each event category) for each SK phase and 155 systematic uncertainties. Figure8 2 2 2 shows the chi-square differences from the minimum as a function of ∆m32 (or ∆m13), sin θ23 and δCP, for the normal and inverted mass ordering cases using 5326 days of data from SK measurements [48]. From the results, the normal mass ordering showed better agreement than the inverted mass ordering 2 2 2 with ∆χ ≡ χNH,min − χIH,min = −3.48. The best-fit neutrino oscillation parameters in the normal 2 +0.13 −3 2 +0.036 mass ordering were ∆m32 = 2.50−0.31 × 10 , sin θ23 = 0.587−0.069, δCP = 4.18 radians. In addition, a neutrino oscillation analysis with constraints from other experiments was provided. One concerned reactor short-baseline neutrino experiments, Daya Bay, RENO, and Double Chooz, for θ13, and 2 the central value of these experiments was sin θ13 = 0.0219 ± 0.0012 [55]. The other concerned Tokai-to-Kamioka (T2K) long-baseline neutrino experiment [56]. Since SK is the far detector of T2K, many experimental aspects, for example, the detector simulation, the neutrino interaction generator (NEUT), and the event reconstruction tools, are common between them. Thus, it is possible add published binned T2K data to the SK atmospheric neutrino fit for determination of the neutrino oscillation parameters. When the constraints on θ13 from the reactor experiments were applied to atmospheric neutrinos, the preference for the normal mass ordering was ∆χ2 = −4.33. When the T2K constraints were added, it became stronger at ∆χ2 = −5.27. K. ABE et al. PHYS. REV. D 97, 072001 (2018)

constrained to 0.0219 0.0012 (discussed below) and globally preferred value of 0.0219 the constraints are weak introducing an additionalÆ scaling parameter on the electron and include this value at the 1σ level. In the normal 2 density in Eq. (5), α. This parameter is allowed to range in hierarchy fit the point at sin θ13 0.0 is disfavored at 20 steps from 0.0 to 1.9, with α 1.0 corresponding to the approximately 2σ indicating the data¼ have a weak prefer- standard electron density in the¼ Earth. ence for nonzero values. A summary of the best fit infor- mation and parameter constraints is presented in Table V. 2 A. Results and discussion The data’s preference for both nonzero sin θ13 and the normal mass hierarchy suggest the presence of upward- Figure 9 shows one-dimensional allowed regions for 2 2 going electron neutrino appearance at multi-GeV energies Δm 32;31 , sin θ23, θ13 and δCP. In each plot the curve is driven by matter effects in the Earth (cf. Fig. 2). Figure 10 j j 2 drawn such that the χ for each point on the horizontal axis shows the up-down asymmetry of the multi-GeV single- is the smallest value among all parameter sets including that and multiring electronlike analysis samples. Here the point. When the atmospheric neutrino data are fit by asymmetry is defined as NU − ND=NU ND, where themselves with no constraint on θ , the normal hierarchy þ 13 NU ND are the number of events whose zenith angle hypothesis yields better data-MC agreement than the ð Þ satisfy cos θz < −0.4 (cos θz > 0.4). Small excesses seen K. ABE et al. PHYS.2 REV.2 D 97, 072001 (2018) inverted hierarchy hypothesis with χNH;min − χIH;min between a few and ten GeV in the multi-GeV e-like νe and 2 ¼ −3.48. The preferred value of sin θ13 is 0.018(0.008) the multiring e-like νe and ν¯e samples drive these constrained to 0.0219 0.0012 (discussed below)assuming and the formerglobally (latter). preferred Though value both of 0.0219 differ thefrom constraints the preferences. are weak introducing an additionalÆ scaling parameter on the electron and include this value at the 1σ level. In the normal 2 density in Eq. (5), α. This parameter is allowed to range in hierarchy20 fit the point at sin θ13 0.0 is disfavored20 at 20 steps from 0.0 to 1.9, with α 1.0 corresponding to the approximately 2σ indicating the data¼ have a weak prefer- standard electron density in the¼ Earth. ence for nonzero values. A summary of the best fit infor- mation and parameter constraints is presented in Table V. 15 2 15 A. Results and discussion The data’s preference for both nonzero sin θ13 and the normal mass hierarchy suggest the presence of upward- Figure 9 shows one-dimensional allowed regions for

2 2 2 going electron neutrino appearance at multi-GeV2 energies Δm 32;31 , sin θ23, θ13 and δCP. In each plot the curve is drivenχ 10 by matter effects in the Earth (cf. Fig. 2). Figureχ 10 10 j j 2 ∆ ∆ drawn such that the χ for each point on the horizontal axis shows the up-down asymmetry of the multi-GeV single- is the smallest value among all parameter sets including that and multiring99% electronlike analysis samples. Here99% the point. When the atmospheric neutrino data are fit by asymmetry5 is defined as NU − ND=NU ND,5 where themselves with no constraint on θ , the normal hierarchy 95% þ 95% 13 NU N90%D are the number of events whose zenith angle90% hypothesis yields better data-MC agreement than the satisfyð cosÞ θ < 0.4 (cos θ > 0.4). Small excesses seen 68% z − z 68% Universeinverted2020 hierarchy, 6, 80 hypothesis with χ2 − χ2 between a few and ten GeV in the multi-GeV e-like ν and 11 of 24 NH;min IH;min ¼ 0 0 e 3.48. The preferred value of sin2 θ is 0.018(0.008) the0.001 multiring 0.002 e-like 0.003ν and 0.004ν¯ samples 0.005 drive0.2 these 0.4 0.6 0.8 − 13 2 e 2 2e 2 assuming the former (latter). Though both differ from the preferences. | ∆ m32 | , | ∆ m31 | eV sin θ23

20 2020 20

Inverted Hierarchy Normal Hierarchy 15 1515 15 2 2 2 2 χ χ χ 10 χ 1010 10 ∆ ∆ ∆ ∆

99% 99% 99% 99%

5 55 5 95% 95% 95% 95% 90% 90% 90% 90%

68% 68% 68% 68% 0 00 0 0.001 0.002 0.003 0.004 0.005 00.2 0.02 0.4 0.04 0.6 0.06 0.8 0246 2 2 2 sin22 δ | ∆ m32 | , | ∆ m31 | eV sin θθ1323 CP

20 FIG. 9. Constraints20 on neutrino oscillation parameters from the Super-K atmospheric neutrino data fit with no external constraints. Figure 8. Constraints on neutrinoOrange lines oscillation denote the inverted parameters hierarchy result, which from has been atmospheric offset from the normal neutrino hierarchy result, data shown in in SK blue, [ by48 the]. 2 2 difference in their minimum Invertedχ values. Hierarchy 2 2 The difference in the χ distributions as a functionNormal Hierarchy of ∆m32 (left), sin θ23 (center), and δCP (right) 15 15 assuming normal (blue) and inverted (orange) hierarchies. The minimum χ2 value in the normal mass 072001-12 2 2

ordering isχ 10 set to zero. χ 10 ∆ ∆ 3.1.2. IceCube 99% 99% 5 5 95% 95% 90% 90% 2 Measurements of the68% atmospheric neutrino68% oscillation parameters of θ23 and ∆m32 in IceCube 0 0 0 0.02 0.04 0.06 0246 were reported in 2018 [572 ]. The unique characteristics of IceCube detector for the atmospheric sin θ13 δCP

neutrinoFIG. 9. Constraints search on neutrino are, oscillation at first, parameters the from the high Super-K atmosphericstatistics neutrino due data fit with to no the external huge constraints. volume. It is also possible Orange lines denote the inverted hierarchy result, which has been offset from the normal hierarchy result, shown in blue, by the to detectdifferencedownward-going in their minimum χ2 values. muon events induced by atmospheric neutrino interaction using the surrounding IceCube detector to distinguish with cosmic-ray muons. To provide a constraint for the neutrino oscillation parameters,072001-12 the reconstructed neutrino energy (8 bins) and zenith angle (8 bins), both track-like and cascade-like, were fit to the expectations using a binned χ2 method. Here, the reconstruction was performed by calculating the likelihood of the observation of photoelectrons by DOMs as a function of the neutrino interaction position, direction and energy. The best-fit neutrino 2 +0.11 −3 2 +0.07 oscillation parameters were ∆m32 = 2.31−0.13 × 10 , sin θ23 = 0.51−0.09 assuming a normal mass ordering. Figure9 shows the L/E distribution along with the corresponding predicted counts given the best-fit neutrino oscillation parameters broken down by event type for both track-like and cascade-like events. The two peaks corresponded to down-going and up-going neutrino trajectories. As the track-like sample was enriched in νµ CC events, up-going events were strongly suppressed in track-like events due to neutrino oscillation. While the cascade-like sample was evenly divided between νµ CC PHYSICAL REVIEWevents and LETTERS interactions120, without071801 a muon (2018) in the final state, some suppression could also be observed. photons near the DOMs, modulating the relative optical efficiency as a function of the incident photon angle. The effect of the refrozen ice column is modeled by two effective parameters controlling the shape of the DOM angular acceptance curve. The first parameter controls the lateral angular accep- tance (i.e., relative sensitivity to photons traveling roughly 20° above versus below the horizontal) and is fairly well constrained by LED calibration data. Five MC data sets were generated covering the −1σ to 1σ uncertainty from the LED calibration, and were parametrizedþ in the same way as the overall optical efficiency described above. A Gaussian prior based on the LED data is used. The second parameter controls sensitivity to photons traveling vertically upward and striking the DOMs head Figure 9. L/E distribution of data (black dots) and expectations with the best fit to the neutrino on, which is not well constrained by string-to-string LED FIG. 2. Data projected onto L=E for illustration. The black dots oscillation parameters (hatched histograms) in IceCube [57]. The red dotted line shows the expectations calibration. That effect is modeled using a dimensionlesswithout neutrinoindicate oscillations. the data The along bottom with plots their show corresponding the ratio of the datastatistical to the fitted errors. expectations. parameter ranging from −5 (corresponding to a bubble The dotted line shows the expectation in the absence of neutrino column completely obscuring the DOM face for vertically oscillations. The stacked hatched histograms are the predicted incident photons) to 2.5 (no obscuration). Zero corresponds to counts given the best-fit values of all parameters in the fit for each constant sensitivity for angles of incidence from 0° to 30° from component. The bottom plots show the ratio of the data to the vertical. Six MC sets covering the range from 5 to 2 were fitted prediction. The bars indicate statistical uncertainties, and − the shaded region corresponds to the σuncor uncertainty in the ν μatm used to parametrize this effect. No prior is applied to this expectation, as defined in Eq. (2), whichþ is dominated by the parameter due to lack of information from calibration data. uncertainty in μatm. The last nuisance parameter controls the level of atmos- pheric muon contamination in the final sample. As described above, the shape of this background in the To better visualize the fit, Fig. 2 shows the results of the analysis histogram, including binwise uncertainties, is fit projected onto a single L=E axis, for both the track-like derived from data. Since the absolute efficiency for tagging and cascade-like events. The two peaks in each distribution background events with this method is unknown, the correspond to down-going and up-going neutrino trajecto- normalization of the muon contribution is left free in the fit. ries. Up-going νμ ν¯μ are strongly suppressed in the An illustration of how these nuisance parameters are track-like channel dueþ to oscillations. Some suppression constrained in the fit is provided as Supplemental Material of up-going cascade-like data is also visible, due to the [47]to this Letter. In addition to the systematic uncertainties disappearance of lower-energy νμ which are not tagged as discussed above, we have considered the impact of seed track-like by our reconstruction. dependence in our event reconstruction, different optical 2 2 Figure 3 shows the region of sin θ23 and Δm32 allowed models for both the undisturbed ice and the refrozen ice by our analysis at 90% C.L., along with our best fit and columns, and an improved detector calibration currently several other leading measurements of these parameters being prepared. In all these cases the impact on the final [12–14,16].Thecontoursarecalculatedusingthe result was found to be minor, and they were thus omitted approach of Feldman and Cousins [48] to ensure proper from the fit and the error estimate. coverage. Results and conclusion.—The analysis procedure Our results are consistent with those from other experi- 2 0.11 described above gives a best fit of Δm32 2.31−þ0.13 × ments [12–16], but using significantly higher-energy neu- −3 2 2 0.07 ¼ 10 eV and sin θ 0.51þ , assuming normal neu- trinos and subject to a different set of systematic 23 ¼ −0.09 trino mass ordering (NO). For the inverted mass ordering uncertainties. Our data prefer maximal mixing, similar to (IO), the best fit shifts to Δm2 −2.32 × 10−3 eV2 and the result from T2K [13]. The best-fit values from the 32 ¼ sin2 θ 0.51. The pulls on the nuisance parameters are NOνA experiment [14] are disfavored by Δχ2 8.9 (first 23 ¼ ¼ shown in Table I. Though IceCube’s current sensitivity to octant) or Δχ2 8.8 (second octant), corresponding to a the mass ordering is low, dedicated analyses are underway significance of ¼2.6σ using the method of Feldman and to measure this. Cousins, although there is considerable overlap in the The data agree well with the best-fit MC data set, with 90% confidence regions of the two measurements. χ2 117.4 for both neutrino mass orderings. This corre- Further improvements to our analysis are underway, sponds¼ to a p value of 0.52 given the 119 effective degrees including the incorporation of additional years of data, of freedom estimated via toy MCs, following the procedure extensions of our event selections, and improved calibra- described in Ref. [27]. tion of the detector response.

071801-6 Universe 2020, 6, 80 12 of 24

3.1.3. ANTARES

2 The measurement of the atmospheric neutrino oscillation parameters of θ23 and ∆m32 in ANTARES was reported in 2019 [58]. The energy for the atmospheric neutrino analysis in ANTARES ranged from a few tens of GeV up to 100 GeV. Track-like events originating from the penetrating muons produced via CC interactions of νµ were used for the analysis. On the other hand, shower-produced events, for example, electromagnetic showers from νe CC interactions or hadronic showers from NC interactions, were regarded as the background events. Here, muon-track reconstruction was essential, and two different algorithms were used: one in which PMT hits were selected to find the best muon track and another involving fitting to a chain at each step to improve the track estimation. Once the muon track was reconstructed, the muon energy was estimated from its track length, given a constant energy loss of muons in the sea at 0.24 GeV/m in the energy range of 10–100 GeV. To obtain a constraint for the neutrino oscillation parameters, a logarithmic base-10 scale of reconstructed neutrino energy in GeV was divided into eight bins, seven from 1.2 to 2.0 plus an additional underflow bin for log10(E/GeV) < 1.2. The cosine zenith angle was divided into 17 bins from 0.15 to 1.0. The fit was performed using a log-likelihood approach. The best-fit neutrino oscillation 2 +0.4 −3 +12 parameters were ∆m32 = 2.0−0.3 × 10 , θ23 = 45−11 degree. Figure 10 shows the reconstructed energy divided by the cosine of the reconstructed zenith angle. The lowest bin of this plot is expected to be affected by neutrino oscillation. The data showed good agreement with neutrino oscillation assumptions.

MC no osc evt

N ANTARES MC world best fit 1000 MC best fit Data

800

600

400

200

0 50 100 150 200 θ Ereco/cos reco [GeV] Figure 10. Ratio of the reconstructed energy and cosine of the reconstructed zenith angle distribution [58]. The black line shows the data, the red line shows expectations without neutrino oscillations, and the blue and green line show the expectations with neutrino oscillations assuming the world’s best-fit value and the ANTARES best-fit value, respectively.

2 3.1.4. Constraints on θ23 and ∆m32 2 2 Figure 11 shows the allowed region of neutrino oscillation parameters (sin θ23 and ∆m32) at 90% C.L. overlaying several experiments both for atmospheric and accelerator neutrinos. Even though there are different sources of neutrinos, and the sensitive energy range is different in atmospheric neutrino experiments, all results showed good agreement with each other. Here, the MINOS far detector, which is 5.4 kTon mass of iron-scintillator tracking calorimeter, is used for the study of atmospheric neutrinos as well as neutrinos originating from the Fermilab NuMI accelerator beam. The far detector is located at underground (2070 m water equivalent) in Soudan mine, MN, USA. The detector is magnetized and it enables the separation of νµ and ν¯µ on an event-by-event basis using the curvature of the produced charged muon. The contour of MINOS in the figure was combined results of accelerator neutrinos and 60.75 kt-year data of atmospheric neutrinos. Universe 2020, 6, 80 13 of 24

2 2 Figure 11. Contours of the 2D allowed region of ∆m32 and sin θ23 at 90%C.L. from several experiments. Atmospheric neutrinos in SK (green) [48], IceCube/DeepCore (red) [57] and ANTARES (black) [58] are described in this paper. The accelerator neutrino results from T2K (blue) [59], NOvA (purple) [60] and MINOS (light blue) [61] are also shown. This figure is taken from [58].

3.2. Tau-Neutrino Appearance

The dominant channel of the atmospheric neutrino oscillation is transition between νµ and ντ in the standard scenario. Actually, the long-baseline neutrino beam experiment from CERN to GranSasso, OPERA, which is sensitive to the same neutrino oscillation parameters as atmospheric neutrinos, found the appearance from νµ to ντ neutrino oscillation [62]. Finally, OPERA observed 10 ντ events with a background expectation of 2.0 ± 0.4, which was equivalent to 6.1σ level of discovery significance. The ντ appearance should also be seen in the atmospheric neutrino data.

3.2.1. Super-Kamiokande

In the atmospheric neutrino observation by the SK detector, the deficit of upward-going νµ events due to the neutrino oscillation passing through the Earth was observed. The original νµ is considered to change to ντ. The appearance of ντ was also searched by SK [63], but the detection was challenging. 3 Since the atmospheric neutrino flux falls as 1/E and ντ charged-current interactions only occur above the τ lepton production threshold, 3.5 GeV, the expected rate at SK is only one event per kiloton per year. Furthermore, τ events are difficult to identify individually as they tend to produce multiple visible particles in the SK detector, as shown in Figure 12. An analysis was performed that employed a neural network technique to discriminate between “tau-like” and “non-tau-like” events from the hadronic decays of τ detected by SK from atmospheric νe and νµ background events. The following seven variables were used as inputs to the neutral network: (1) total visible energy: τ signal events are expected to have higher visible energy compared to the background events; (2) shower-like events: hadronic decay of τ events tend to make a shower in the ring pattern; (3) number of decay electron candidates: pions produced by the hadronic decay of τ produce more decay electrons than the background events; (4) distance between the primary interaction point and decay electron vertex: pions are expected to have smaller momentum compared to the background; (5) sphericity, which is the evaluation whether isotropic or not: hadronic decay of τ is more isotropic than the background; (6) number of Cherenkov ring fragments: τ events are expected to have more ring candidate; (7) ratio of the observed photons and the most-energetic ring in an event: τ events are expected to be small because energy is carried by multiple particles in the hadronic decay of τ. When “tau-like” events are selected from neural network output in this analysis, 76% of signal events and only 28% of background events remain which is estimated by the simulation. MEASUREMENT OF THE TAU NEUTRINO CROSS SECTION … PHYS. REV. D 98, 052006 (2018)

visible energy than background. The Evis spectrum of the CC ντ signal peaks around 4 GeV, as shown in Fig. 7. By contrast, the Evis spectrum of the back- ground falls with increasing Evis. (2) The particle identification likelihood parameter of the ring with maximum energy. Tau leptons decay quickly to daughter particles after production through leptonic and hadronic decays. Except for the leptonic decay to a muon, most decay channels have at least one showering particle. A showering particle has a negative value in the definition of particle identification likelihood, compared with a positive value for a nonshowering particle. The particle identification of the most energetic ring for the signal has a distribution mostly in the negative region, while the background has a broad distribution in both negative and positive regions. (3) The number of decay electron candidates in the event. Naively, we expect more decay electrons for signal from pion decays which are produced in hadronic tau decays. This variable does not depend on ring reconstruction, so it is relatively independent of most other variables. (4) The maximum distance between the primary inter- action point and any decay electron from a pion or muon decay. Energetic muons can travel a long distance in water. Therefore, CC νμ background involving a high energy neutrino is expected to have

Universe 2020, 6, 80 14 of 24 a large distance between the primary interaction point and the decay electron from the muon. In comparison, the pions from hadronic tau decay are expected to have smaller momentum, resulting in a smaller value of the variable. (5) The clustered sphericity of the event in the center of mass system. Sphericity is a measure of how spherical an event is. A perfectly isotropic event has sphericity 1, while a perfectly one-directional event has sphericity 0. We follow the definition from [36], defining the spherical tensor as

pαpβ Sαβ i i i ; 2 ¼ p2 ðÞ P i i where α,β 1, 2, 3 are threeP Cartesian momentum vectors pointing¼ to binned photoelectric charge in

Figure 12. Typical event pattern of a tau signal with 3.3 GeV visible energy in a simulation [63]. the event. Sphericity is then constructed by finding FIG. 6. Simulation of a single-ring CC background event the eigenvalues, λ1 > λ2 > λ3, of the tensor: To evaluate(top) whether with 2.8 a τ signal GeV was visible being energy observed, in the the maximum ID, a likelihood multiring method NC was used. The outputbackground of a neural network event and (middle) reconstructed with zenith 2.2 angles GeV were visible used to construct energy the in probability density function (PDF) for both signals and the background using simulation. As ν is generated via 3 the ID, and a CC ν event (bottom) with 3.3 GeV visibleτ S λ λ : 3 neutrino oscillation through the Earth,τ signals tend to be an upward-going event, which is why the ¼ 2 ð2 þ 3Þ ðÞ zenith angleenergy was used in for the the ID. PDF. The In this tau analysis, signal the event 5326 produces days of atmospheric multiple neutrino rings, data in SK was fitted tomaking the following it different function: from the single-ring background event. The background event with multirings has a similar pattern The hadronic decay of the heavy tau lepton is more PDF + α × PDF + e × PDF , (10) to the signal event,BG and requirestau more∑ i efforti to statistically isotropic than a typical νμ or νe background. The spectrum of sphericity is centered near 0.8 for signal, where α isdistinguish. a normalization factor of tau signal, ei is a i-th systematic error. According to the normalization factor (α), the failure of tau to appear is zero and the appearance of a tau signal is one. It was found to be 1.47 ± 0.32, assuming the normal mass ordering of neutrino mass splitting, which is equivalent to a 4.6σ level of ντ appearance. Assuming an inverted mass ordering, it was 1.57 ± 0.31, and the significance was 5.0σ. Figure 13 shows the zenith angle distribution for both tau- and non-tau-like events, along with the expectation fitted to tau and the background. These052006-7 plots show good agreement between the data and MC simulations. Based on the significant discovery of the τ event, the charged-current tau-neutrino cross-section was also calculated. The measured cross-section was expressed as (1.47 ± 0.32)× < σtheory >, where < σtheory > is the flux-averaged theoretical cross-section. It was calculated by the integral of the differential charged-current ντ cross-section weighted with the energy spectrum of atmospheric ντ from neutrino oscillations, and was found to be 0.64 × 10−38 cm2. Finally, the measured cross-section integrated from 3.5 to 70 GeV was calculated as (0.94 ± 0.20) × 10−38 cm2.

3.2.2. IceCube

A search for ντ appearance by neutrino oscillation using atmospheric neutrino samples by IceCube/DeepCore was performed [64]. The energy region ranges from 5.6 to 56 GeV, which was the same as that in the neutrino oscillation analysis. The identification of individual ντ events by DeepCore is difficult because of the tau lepton produced via CC interaction decays with a small track length of ∼1 mm, compared to the position resolution of the detector. To search for the appearance of ντ by Universe 2020, 6, 80 15 of 24 neutrino oscillation, the distortion of neutrino energy and direction in cascade-like events compared to non-neutrino oscillation assumption were investigated. In the data analysis, two independent methods were applied: one targeted a high acceptance of all neutrino events, whose background estimation was simulation-driven, and the other was optimized for a higher rejection of non-neutrino events, with data-driven background estimation. Both methods used a boosted decision tree for event selection and background rejection. The neutrino energy and direction were reconstructed by the maximum likelihoodMEASUREMENT method using the charge OF THE and TAU time NEUTRINOobserved by DOMs. CROSS SECTION … PHYS. REV. D 98, 052006 (2018)

300 BG after fit Tau after fit 1000 Data 800 200 600

Events 400 100 200 Tau-like Upward 0 0 -1 -0.5 0 0.5 1 0 0.2 0.4 0.6 0.8 1 1200 600 1000 800 400 600 Events M. G. AARTSEN et al. PHYS. REV. D 99, 032007 (2019) 200 400 200 Non tau-like Downward 0 0 -1 -0.5 0 0.5 1 0 0.2 0.4 0.6 0.8 1 CosΘ NN output

FigureFIG. 13. 14.Zenith Fit angle results, distribution assuming the along normal with hierarchy, the expectation, showing assuming binned projections normal mass in the ordering, NN output and zenith angle distribution for tau- for tau-likelike (NN (upper)> 0. and5), upward-going non-tau-likeFIG. 15. Distributions (lower) [cos Θ < events of− the0. data2 [],63 withnon-tau-like]. The best-fit fitted neutrino (NNtau and signal< 0.5 is) and shown downward-going in gray. [cos Θ > 0.2] events for both the two- dimensional PDFs andmuon data. backgrounds The PDFs subtracted and and signal data simulation. sets have Statistical been combined from SK-I through SK-IV. The fitted tau signal is errors are shown, which include contributions from the full data Figureshown 14 shows in gray. the L/E setdistribution and all background along simulation with added in the quadrature. corresponding The best- predicted counts, given fit neutrino spectrum shows good agreement with the background the best-fit neutrino and cosmic-ray muon broken down for the first method of analysis.2 The plot subtracted data in the reconstructed energy axis (left), the cosine FIG. 17. Observed Δχ from the best fit CC NC (CC) ντ þ showed good agreement betweenof the the reconstructed data and zenith the angle model. (middle)The and PID normalization categories normalization factor of 0.75of τ (0.62)appearance as a function of inthe ντ normalization (right)+ for analysis A. (black lines). Shaded bands show the 68% ranges of the expected 0.30 2 −38 2 DeepCore wasTo found calculate to be the0.73 flux-averaged−0.24 which theoreticalis equivalent cross to a section, 3.2σ level ofdistributionντ appearance. of Δχ values0 obtained. The94 result from0.20 pseudo-experiments× 10 cm 9 assuming nominal valuesð for oscillationÆ parametersÞ and a ντ ðÞ was consistentthe differential with those CC obtainedντ cross for section SK measurements. as a function of neutrino normalization of 1.0. energy is weighted with the energy spectrum of atmos- pheric tau neutrinos from neutrino oscillations. Because CC The corresponding measured values cross for the section nuisance parameters is shown can together with the be found in Table II. theoretical cross sections and2 the MC simulations in ντ interactions are not distinguishable from CC ν¯τ inter- Figure 17 shows the expected and observed Δχ values actions in Super-K, the theoretical cross section is a flux forFig. a ντ15normalization. The measured ranging from and 0 to 2.0.theoretical The band of cross section values expected values assumes standard oscillations with a ντ average of ντ and ν¯τ cross sections. The flux-averaged normalizationare consistent of 1.0. at Our the main1 result.5σ level. for the CC NC þ theoretical cross section, σtheory , is calculated as measurement has a best fit value of 0.73 with the 68% con- h i fidence interval (C.I.) covering the range (0.49,1.07) and the 90% C.I. covering (0.34,1.30). For the CC-only dΦ Eν normalization, we observe the best fit at 0.57 with the ν ;ν¯ dEð Þσ Eν dEν 68% C.I. (0.30,0.98) and the 90% C.I. (0.11,1.25). τ τ ν ð Þ 10 10 σtheory ; 8 These) measured values are compatible with correspond- dΦ Eν 2 h i¼ ð ÞdE ðÞ ing values obtained from analysis B within less than 1σ P ντR;ν¯τ dE ν ν standardcm deviation. These confirmatory results of analysis

-38 0.31 0.36 B are 0.591 þ (0.43þ ) for the CC NC (CC-only) 1 Figure 14. L/E distribution along withP theR expectation of the best-fit neutrino and cosmic-ray−0.25 muon−0.31 [64]. þ dΦ Eν measurement, also see Fig. 18. where ð Þis the differential flux of tau neutrinos as a The bottom portiondEν shows the ratio of the data and the expectation. All values are also compatible within the 90% confidence function of neutrino energy as shown in Fig. 2,andσ E interval10 with-1 expectations assuming the three-flavor neu- 10-1 ν trino oscillation paradigm (i.e., ν normalization 1.0) and 3.3. Sterile Neutrino Analysis ð Þ τ ¼

is the differential charged-current tau neutrino cross the assumed ντ CC cross sections. The significance at Bin per Events which we can reject the null hypothesis of no ντ appearance The existencesections ofused neutrino in NEUT oscillation code as has seen been in Fig. established3.Therange by a wide range10-2 of experiments using 10-2

is 3.(10 Section Cross 2σ and 2.1σ for the CC NC and the CC-only case for þ different sourcesof the of integral neutrinos: is determined the atmosphere, to be the between Sun, nuclear 3.5 reactors,and analysis and accelerators.A, respectively. The However, confirmatory analysis B yields slightly weaker limits of 2.5σ (1.4σ). 70 GeV from the tau neutrino energies in the simulation. -3 The10 confidence-3 intervals for the measurements presented 10 As shown in Fig. 15, the neutrinos have energies more here, shown0 in Fig. 18, are calculated 20 using the approach 40 of 60 than 3.5 GeV in the CC ν interactions because of the Feldman and Cousins [69] toNeutrino ensure proper Energy coverage. (GeV) τ The presented results are of a comparable precision to energy threshold, and the expectation of CC ντ inter- those of SK and OPERA (see Fig. 18), and complementary FIG. 16. Distribution of the data as a function of reconstructed toFIG. those 15. measurements Measured in terms flux-averaged of energy scale, L=E charged-currentrange, tau neutrino actions with more thanL=E, overlaid 70 GeV with the is best less fit neutrino than and one cosmic-ray in the muon histograms for analysis A (top) and B (bottom). The bottom systematiccross section uncertainties, (black), and together statistics. Specifically, with theoretical the differential cross entire run period. portion of each shows the ratio of the data to the predicted SK measurement is based on lower-energy events where sections (ντ in red and ν¯τ in blue), flux-averaged theoretical cross The flux-averageddistribution theoretical at the best charged-current fit point, with black points tau representing neu- roughly 50% interact via CC quasielastic or resonant data and the height of the shaded band the uncertainty of the best scattering,section(dashed while the IceCube gray) and data are tau dominated events by after selection in MC −38 2 trino cross section isfit (statistical calculated errors only). to be 0.64 × 10 cm higher-energysimulations events (gray that histogram). interact primarily The viahorizontal the deep bar of the meas- between 3.5 and 70 GeV, and thus the measured flux- urement point shows the 90% range of tau neutrino energies in

averaged charged current tau neutrino cross section: 032007-18the simulation.

052006-13 Universe 2020, 6, 80 16 of 24 not all neutrino experiments show the three standard flavors of neutrinos. For example, an excess of electron neutrinos in a muon-neutrino beam with ∆m2 ∼ 1eV2 was found in the LSND [65] and MineBooNE [66] experiments. Additional anomalies appeared in the ν¯e and νe rates in several reactor experiments [67] and gallium-based experiments [68]. These results made a hint of neutrino oscillations driven by a ∆m2 > 1eV2. On the other hand, the number of neutrinos was measured as 2.980 ± 0.0082 light neutrino flavors by large electron-positron(LEP) collider using the width of the Z0 mass peak [69]. This result required that its mass of an additional neutrino is either heavier than half the Z0 boson; however, it is difficult to be a player of neutrino oscillations, or not interact via weak interactions, which is called sterile neutrinos. To introduce N additional sterile neutrinos, U in Equation (1) should be (3 + N) × (3 + N) matrix. The matter effect in neutrino oscillation for sterile neutrinos should also be considered. Since sterile neutrinos√ have no CC nor NC interactions, the effective matter potential is expressed by Vn = ±GF/ 2Nn. It is derived from the difference from νµ and ντ which have only NC. Here, NC depend only on neutron density (Nn) because the interactions with electrons and protons are equal and opposite, and their densities are identical in neutral matter. Atmospheric neutrino samples can also make a constraint on sterile neutrinos due to a wide range in both energy and travel distance. The results of single additional sterile neutrino scenario, called the “3+1” model, have been reported by SK [70], IceCube [71], and ANTARES [58]. Actually, it hard for this model to explain all the anomalies consistently; however, it can be extended to models with more than one sterile neutrino although it needs more parameters. The “3+1” model, denoted as U, is that a neutrino flavor eigenstate νs with mass eigenstate ν4 should be added in the mixing matrix defined in Equation (2). In this model, six new parameters are added: three mixing angles 2 θ14, θ24, θ34, two CP-violating phases, and one mass differences ∆m41 To search for sterile neutrinos using atmospheric neutrinos, several assumptions were made. First, since the assumed mass difference 2 2 between the mass eigenstate ν4 and the other is large, sin (∆m41L/4E) is averaged as 1/2. Second, 2 2 2 2 2 the other parameters, except for |Uµ4| = sin θ24, |Uτ4| = cos θ24 sin θ34,(δ24 only for ANTARES), are ignored since they have negligible impact on atmospheric neutrino oscillation. The effect of these neutrino oscillation parameters results in a different νµ disappearance pattern compared to the standard neutrino oscillation scheme. The analysis method is similar to the standard three-neutrino hypothesis, i.e., fitting the 1010 reconstructed energy and cosine zenith angle from the experimental atmospheric neutrino data to the 8 88 prediction but using66 only the νµ disappearance channel. No clear evidence of sterile neutrinos was

LogL 2 found in these experiments.! 44 Figure 15 shows the exclusion region at 90% and 99% C.L. in the |Uµ4| -2

2 2 and |Uτ4| parameters.2 00

24 90% C.L. 99% C.L. θ

2 This work This work

This work NH, δ24 = 0° This work NH, δ24 = 0° cos IceCube (2017) NH, δ24 = 0° IceCube (2017) NH, δ24 = 0° 34 IceCube (2017) IH, δ = 0° IceCube (2017) IH, δ = 0° θ 24 24 2

SK (2015) NH, δ24 = 0° SK (2015) NH, δ24 = 0° sin 10−1

10−2 0 2 4 6 8 −3 2 1 −3 2 1 10 10 10− 10− 2 10 10− 10− 2 0 2 4 6 8 10 sin θ24 sin θ24 -2!LogL 2 2 2 2 2 Figure 15. Contours in 2D exclusion regions of |Uµ4| = sin θ24 and |Uτ4| = cos θ24 sin θ34 at 90% (left) and 99% (right) C.L. obtained by the atmospheric neutrino data in SK (blue) [70], IceCube/DeepCore (red) [71] and ANTARES (black) [58]. The dashed lines show the normal mass ordering, while the solid lines show the inverted mass ordering (IceCube/DeepCore) and unconstraint

δ24 (ANTARES). The markers show the best-fit values for each experiment. This figure is taken from [58]. Universe 2020, 6, 80 17 of 24

4. Atmospheric Neutrino Flux Measurements Predictions and observations of the atmospheric neutrino flux have been reported by several groups. Its energy distribution is derived from charged particles, mainly muons or electrons, induced by neutrino interaction. Figure 16 shows several predictions and measurements of the energy distribution in atmospheric neutrinos. In this figure, SK applies the so-called “unfolding” method to all experimental phases [72]. The event rate of the observed charged particles induced by atmospheric neutrinos is expressed as the convolution of neutrino flux, neutrino oscillation probability, neutrino cross-section, and detector efficiency. The obtained atmospheric neutrino spectrum is derived from the deconvolution of the above observed values, which is the unfolding method. The observed flux of atmospheric νµ and νe is indicated by the red square and blue circle in Figure16, and are consistent with the predictions that include an assumption of neutrino oscillation. The χ2 values (p-value), including νe and νµ together, are 22.2 (0.51) for HKKM, 30.7 (0.13) for Bartol, and 25.6 (0.32) for FLUKA, with a degree of freedom of 23. IceCube reported the results of atmospheric νµ and νe flux. In the νµ analysis, the observable value is the deposit energy per track length of muons in the detector (dE/dX). It is a convolution of the atmospheric neutrino flux and the response matrix, which accounts for the effect of propagation through the Earth, neutrino interaction, detector response, and event selection. Unfolding of the atmospheric muon-neutrino flux from 100 GeV to 400 TeV was performed [73], and the results are indicated by the pink triangle in Figure 16. In addition, the so-called forward-folding analysis was applied, in which the dE/dX distribution was tested against the hypotheses of muons arising from atmospheric νµ by π or K decay, prompt νµ, and astrophysical νµ [53]. At first, no evidence was found for an astrophysical or prompt νµ. In a comparison with HKKM, the normalization factor of the absolute atmospheric neutrino flux was found to be 0.96 ± 0.16 and the spectra index was found to be steeper by E−0.032±0.014. The allowed regions of these parameters from 332 GeV to 84 TeV are indicated by the pink band in Figure 16. As for the atmospheric νe measurements by IceCube, two results were reported: one was a low-energy region (80 GeV to 6 TeV) using DeepCore [74] and the other was a high-energy region (0.1 to 100 TeV) using the full IceCube detector [75]. After several steps of background reduction, related to νµ in the main sample and cosmic-ray muons from the cascade sample, the data were fitted to the prediction to obtain the atmospheric νe flux. The observed flux was consistent with the prediction, as shown in Figure 16. In addition, no prompt neutrino signal was found for νe. ANTARES reported the atmospheric νµ energy spectrum in the energy range between 0.1 and 200 TeV [76]. It was derived from the measured muon energy distribution through a response matrix, determined from simulations, and an unfolding method. The results are shown in the figure, although it is admittedly busy. The overall normalization factor is 25% higher than the prediction in [21], and the flux is compatible with the IceCube results. The prompt neutrino was not observed in the ANTARES data. An east-west flux asymmetry of atmospheric neutrinos due to the rigidity cutoff is predicted. SK reported the measurement of this effect [72] using an FC sample, selecting both electron- and muon-like events with a single reconstructed Cherenkov ring. Figure 17 shows the azimuthal distribution of the subsample event selected to optimize the significance of the east-west dipole asymmetry. The effect is observed at a significance level of 6.0 (8.0) sigma for the muon-like (electron-like) samples. The dependence of the asymmetry on the zenith angle was also investigated and was observed at the 2.2 sigma level. These effects were consistent with the prediction within the given uncertainties. SK also reported long-term and seasonal atmospheric neutrino flux variations [72]. An anti- correlation with solar activity was predicted for the long-term variation and a weak preference for a correlation was observed at the 1.1 sigma level using 20 years’ observation. The seasonal variation is considered to be occurred by the change in the atmospheric density profile over the year; however, no such correlation was observed in SK. E. RICHARD et al. PHYSICAL REVIEW D 94, 052001 (2016)

The νe and νμ fluxes are also separated into upwardgoing number of iterations. We therefore perform a further post and downwardgoing data sets, using their reconstructed hoc check using the same validation method as before, but direction, and measured in Fig. 12. This is in order to check using our actual unfolded spectra from the data as the any possible bias in the flux calculation due to the pseudodata truth input. Figure 13 shows the unfolded differences of the data sets; for example, neutrino oscil- spectrum as a function of number of iterations, from lation has a stronger effect in the upwardgoing data above 1 up to 10, on top of the previously estimated regularization GeV energies, and the UPMU data is an upwardgoing error. It is seen that at around five iterations, which we had sample only. As seen in the figure, no obvious difference adopted based on the pseudodata test, the unfolded spectra exists between the fluxes measured using these two are stable. Furthermore, the spectra are reproduced approx- data sets. imately within the estimated regularization uncertainties. Although we thoroughly validated the accuracy of the Therefore, we conclude that the shapes of our unfolded unfolding procedure using the HKKM-like pseudodata and spectra are not due to an unexpected additional bias from included the estimated regularization bias as a systematic the unfolding procedure. uncertainty, there has still been some concern about the In Fig. 14 this flux measurement is shown with the ability of the unfolding procedure to accurately reproduce results from other experiments. Our measured data provide more complicated spectral shapes, such as the wavy shape significantly improved precision below 100 GeV. The Universe 2020, 6that, 80 was eventually obtained, and, in particular, whether or minimum of the observed energy range is extended 18 of 24 not such a shape would be more strongly affected by the below 1 GeV, and at higher energies overlaps with νμ

10-1

10-2

10-3 ] -1 sr -1 10-4 sec -2 10-5 Super-Kamiokande I-IV ν µ Frejus ν µ IceCube ν µ unfolding

[GeV cm [GeV -6 IceCube ν µ forward folding

Φ 10 AMANDA-II ν µ unfolding 2 AMANDA-II ν µ forward folding E ANTARES ν µ HKKM11 ν +ν (w/ osc.) 10-7 µ µ Super-Kamiokande I-IV ν e Frejus ν e -8 IceCube/DeepCore 2013 ν e 10 IceCube 2014 ν e HKKM11 ν e+ν e (w/ osc.)

10-9 -1 0 1 2 3 4 5 Log (E /GeV) 10 ν

FIG. 14. The measured energy spectra of the atmospheric νe and νμ fluxes by SK, shown with measurements by other experiments, Figure 16.i.e., FrejusEnergy[39], spectraAMANDA-II of[40,41] the, atmospheric IceCube [42–45], andνe ANTARESand νµ fluxes[75]. The according phrase “forward to folding several” used byexperiments IceCube and with an overlayAMANDA-II of HKKM is synonymous predictions with forward for fitting. the The Kamioka HKKM11 flux site model in solid predictions (with for the neutrino Kamioka site oscillation) are also shown in and solid dashed (with oscillation) and dashed (without oscillation) lines. The error bars on the SK measurement include all statistical and systematic (withoutuncertainties. oscillation) lines. SK used the unfolding method [72]. Both forward-folding and unfolding methods were reported by IceCube in the energy range of 100 GeV to 400 TeV for νµ and 100 GeV to 100 TeV for νe [73–75]. ANTARES used the unfolding052001-14 method in the energy range of 100 GeV to 200 TeV [76]. This figure is taken from [72]. MEASUREMENTS OF THE ATMOSPHERIC NEUTRINO FLUX … PHYSICAL REVIEW D 94, 052001 (2016)

80 700 e e

650 60

600 40

550 20

500 [deg]

B 0 450 -20 400 -40 350 parameter 0 60 120 180 240 300 360 -60

850 µ -80 800 1 0.6 0.2 -0.2 -0.6 -1 750 80 µ 700 60 650 40 600

550 20 [deg] B 500 0 0 60 120 180 240 300 360 -20 Reconstructed Azimuth φ rec [deg] parameter parameter Figure 17. AzimuthalFIG. distribution 22. Azimuthal of a subsample distributions of electron-like of a subselection (top) and of muon-likee-like (bottom) events-40 from the data with statistical(left) and errorμ-like [72 (right)]. The events, boxes from are prediction the SK-I–SK-IV with systematic data (points error. A subsample-60 is selected as it is optimizedwithstatistical to obtain error) the highest and MC significance simulations by (boxes using with only systematic events with 0.4 < E < 3.0 GeV error). The subselection is optimized to obtain the highest -80 and | cos θzenith| < 0.6. significance of the final A parameters, by using only events with 1 0.6 0.2 -0.2 -0.6 -1 0.4

052001-21 Universe 2020, 6, 80 19 of 24

5. Future Prospects In this section, the search for atmospheric neutrinos by next-generation neutrino detectors is briefly described. Table2 shows the summary of the detectors. Succession to the current water Cherenkov detector experiments are now proposed. Here, an enlargement of the volume in underground detector and denser photo-sensor in string type detector are crucial to determine unknown neutrino oscillation parameters such as mass ordering and CP-violating phase. Hyper-Kamiokande (HK) [77] is underground (1750 m water equivalent) water Cherenkov detector, located 8 km from the SK site. It is based on the well-established technology with a fiducial volume of 187 kilotons, which is 8.3 times larger than that obtained for SK, and will start operation in 2027. It consists of 40,000 high-quantum-efficiency PMTs for inner detector. The second HK detectors in future is also proposed [78]. The IceCube plans detector upgrade as “IceCube-Gen2” [79]. The first step of the upgrade is scheduled for deployment in the 2022/2023 polar season [80]. It is proposed that 7 additional strings with 125 optical modules spaced 2.4 m apart, which is denser than IceCube/DeepCore, will set in a small part of IceCube observatory (2 MegaTon of ice). A denser detector, PINGU [81], that consists of 26 strings with 192 optical modules spaced 1.5 m apart, is proposed as a goal of IceCube-Gen2. The denser photo-sensor enables the neutrino oscillation parameter determination with high precision [82]. The successor of the ANTARES is KM3NeT [83]; it is a deep-sea neutrino detector in the Mediterranean Sea. There are two installation sites for different purpose; one is “ARCA” at 3500 m depth in Italy which aims an observation of high-energy neutrino sources in the Universe, the other is “ORCA” at 2500 m depth in France which aims a neutrino oscillation parameter determination using atmospheric neutrinos. The multi-PMT is set in the DOM as an optical sensor, and 18 DOMs (spaced 9 m apart for ORCA) are hosted in one string that is called “Detection Units (DU)” The detector construction was started, and its performance has been checked [84]. The first DU for ORCA had already been installed, and 5 additional DUs will be installed soon, and finally 115 DUs will be installed. Apart from water Cherenkov technique, several different types of detector available for atmospheric neutrinos are proposed. DUNE [85] is a long-baseline neutrino experiment using a massive liquid argon time-projection chamber (LAr TPC). The far detector is located at a depth of 4300 m water equivalent at the Sanford Underground Research Facility, in Lead, SD, USA. One of the advantages of LAr TPC is excellent particle identification and better angular and energy resolutions than water Cherenkov detectors. The well reconstructed momentum of each final state particles enables the determination of the energy and direction of the incident neutrinos. The fiducial volume of one LAr TPC module is 10 kilotons. The first module construction has started, and 4 modules are the goal. The DUNE operation together with the neutrino beam will start in 2026; however, the atmospheric neutrino observation will be possible to start as soon as the first module completes when it is expected before 2026. Main topics of the atmospheric neutrino observation in neutrino oscillations is determination of mass ordering. Any future detectors described here will reach more than 3σ level to reject the incorrect mass ordering, assuming that either the normal or inverted mass ordering is true, after 5–10 years of operation, although this depends on other parameters of the neutrino oscillation. The precise tau appearance observation and studies of any exotic model are also possible.

Table 2. Summary of the detector characteristics.

SK IceCUBE/DeepCore ANTARES Target Pure water Ice Deep-sea water Volume 22.5 kTon fiducial 15 MTon instrumented 10 MTon instrumented PMT 11,129 of 20” PMTs for inner 647 of 10” PMTs 885 of 10” PMTs 1885 of 8” PMTs for outer 5160 of 10” PMTs as veto Attenuation length ∼90 m ∼45 m ∼50 m Universe 2020, 6, 80 20 of 24

6. Summary The research into atmospheric neutrinos has had great success in the last two decades, notably the discovery of neutrino oscillation by the SK detector. This paper reviews the recent achievements made in atmospheric neutrino observations by SK, IceCube, and ANTARES. First, the standard three-neutrino oscillation scheme, expressed by the PMNS matrix, was established and the parameters were determined. The appearance of tau neutrinos in atmospheric neutrinos, predicted by this theory, was confirmed. An exotic scenario, such as the appearance of sterile neutrinos, was tested; however, no clear signal was observed. The observed atmospheric neutrino flux was consistent with the predictions. Among the unknown neutrino oscillation parameters, the normal mass ordering is preferred in the current measurements; however, it is not significantly determined. Atmospheric neutrino observations in the next generation of neutrino detectors are expected to provide final determination of the mass ordering. Funding: This research received no external funding. Acknowledgments: The author gratefully acknowledge the members of the Super-Kamiokande, IceCube and ANTARES Collaborations for their great works and lots of useful discussions. Conflicts of Interest: The authors declare no conflict of interest.

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