Long-Baseline Neutrino Experiments

# Long-Baseline Neutrino Experiments

M. V. Diwan V. Galymov X. Qian A. Rubbia Brookhaven National Laboratory, Upton, NY, USA Institut de physique nucléaire de Lyon, Villeurbanne, France Brookhaven National Laboratory, Upton, NY, USA ETH Zurich, Institute for Particle Physics, Zurich, Switzerland
July 12, 2019
###### Abstract

We review long-baseline neutrino experiments in which neutrinos are detected after traversing macroscopic distances. Over such distances neutrinos have been found to oscillate among flavor states. Experiments with solar, atmospheric, reactor, and accelerator neutrinos have resulted in a coherent picture of neutrino masses and mixing of the three known flavor states. We will summarize the current best knowledge of neutrino parameters and phenomenology with our focus on the evolution of the experimental technique. We will proceed from the first evidence produced by astrophysical neutrino sources to the current open questions and the goals of future research.

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## 1 Introduction

Neutrinos are electrically neutral spin-1/2 particles that are emitted in radioactive decays of unstable nuclei and subatomic particles. They were originally proposed by W. Pauli in 1930 [1] as an explanation of the apparent non-conservation of energy in radioactivity and inconsistencies between the exclusion principle and nuclear models of that time. Enrico Fermi was responsible for the modern name of the particle and the theory of beta decay which is the foundation for current understanding of weak interactions [2]. Soon after the detection of free neutrinos from radioactive products in nuclear reactors [3, 4], neutrino properties and their fundamental role in shaping the universe became an important line of inquiry [5, 6].

Important questions regarding neutrinos are: the number and types of neutrinos, the nature and strength of their interactions with matter, their masses, and the origin of these masses. A series of ground breaking experimental observations and theoretical advances since the 1950s have made progress on these issues [7]. While neutrino physics historically had been synonymous with understanding of weak interactions, the observation of neutrino oscillations has opened a new portal to fundamental questions regarding the relationship between quarks and leptons and the origin of fermion masses [8].

After the detection of neutrinos from nuclear reactors, the hypothesis of neutrino-antineutrino mixing was made in analogy to neutral mesons [9, 10] given that only one type of neutrino was known at that time. This hypothesis was naturally extended to mixing of neutrino flavors [11, 12, 13], when the neutrinos produced in pion decays from a proton accelerator were shown to be distinct from neutrinos produced in beta decays [14]. Subsequently, the mixing of fields was realized to be a natural consequence of gauge theories with spontaneous symmetry breaking. Both quarks and neutrinos have now been shown to experience flavor mixing although possibly originating from different mechanisms since neutrinos do not have electric charge. Whereas quarks and charged leptons are Dirac fermions with oppositely charged anti-particles, neutrinos could be either Dirac particles with a conserved leptonic charge or neutral Majorana particles with no conserved leptonic charge.

### 1.1 Neutrino Oscillations

Under the simplest assumption of two-flavor neutrino mixing characterized by a single angle , and are superpositions of two mass eigenstates and with masses and :

 (νeνμ)=(cosθsinθ−sinθcosθ)⋅(ν1ν2). (1)

The time evolution of the state in vacuum in terms of and , the energies of the two mass eigenstates, is given by:

 νμ(t)=−sinθ⋅e−iE1⋅t⋅ν1+cosθ⋅e−iE2⋅t⋅ν2. (2)

The complete quantum-mechanical description of neutrino oscillations requires the wave-packet formalism [15]. Nevertheless, the correct expressions for the oscillation probabilities are obtained from Eq. 2 by assuming that the two states have the same momentum or energy [16]. The appearance probability to detect given an initial state at a distance is:

 P(νμ→νe)=sin22θ⋅sin2(1.27⋅Δm2⋅LE). (3)

Given that the total probability must be conserved, the probability of survival or disappearance is simply: . Here is in units of eV, and in or . In the case of two-neutrino mixing, the same formula applies to oscillations of antineutrinos.

When traveling through dense matter, neutrino oscillation probabilities can be significantly modified by coherent forward scattering off electrons and nuclei[17, 18]. The equation of motion for neutrinos in the simple 2-flavor basis including the effect of matter is written as:

 iddt(νeνμ)=12⎛⎜⎝−(Δm22Ecos2θ−√2GFNe)Δm22Esin2θΔm22Esin2θ(Δm22Ecos2θ−√2GFNe)⎞⎟⎠(νeνμ) , (4)

where is the Fermi constant, is the electron density which could be time (or position) dependent. The product acts as an extra potential due to the difference between and scattering amplitude off electrons. This potential reverses sign for antineutrinos. The eigenstates of this system are not the vacuum mass eigenstates and . Under the assumption of constant density a solution in the form of Eq. 3 can be obtained with the following substitutions:

 sin22θ→sin22θ′=tan22θ(1−NeNrese)2+tan22θ ,Δm2→Δm′2=Δm2√(1−NeNrese)2cos22θ+sin22θ , (5)

where the quantity

 Nrese=Δm2cos2θ2E√2GF≈6.56×106Δm2[eV2]E[MeV]cos2θ⋅NA[cm−3] (6)

is called (for ) the resonance density [18, 19] with as Avogadro’s number.

Neutrino oscillations take place when the phases of the eigenstates ( and ) start to differ from each other as a function of time. Over a long period, even extremely small mass differences can lead to sufficient phase difference to cause flavor transformation. Observation of neutrino oscillations implies that neutrinos mix and at least one of them has mass. Neutrinos in the standard model of electroweak interactions are left-handed partners of the charged leptons and only participate in weak interactions. The right-handed components of the neutrino field are absent by definition making neutrinos massless in the model. Consequently, the individual lepton number is strictly conserved as confirmed in rare decays of muons, mesons, and collider experiments [20, 21]. Inclusion of neutrino mass in the standard model by simply introducing right-handed partners creates the difficult problem of explaining the smallness and the nature of the neutrino mass as either Dirac or Majorana. Observation of oscillations therefore is considered physics beyond the standard model that not only points to violation of separate lepton number conservation but also forces us to consider that the nature of the neutrino mass [8] could be quite different from the other fermions. On the basis of observed oscillations and limits from direct measurements, the electron neutrino mass is found to be at least five orders of magnitude smaller than the electron, the lightest of the charged fermions [22]. But the existing data does not allow determination of neutrinos as Dirac or Majorana fermions.

### 1.2 Experimental Techniques

Figure 1 shows the natural and man-made sources of neutrinos that are the main contributors to our understanding of neutrino oscillations. To be sensitive to oscillations the neutrino energy and the distance to the detector need satisfy the condition for at least one mass-squared difference. In most cases, moving the detector or the source is impractical to cover a range of L/E, and therefore a broad energy spectrum is beneficial to detect oscillations as a distortion of the expected energy distribution. For solar neutrino experiments (, ), for atmospheric neutrino experiments (, ), for reactor neutrino experiments (, ), and for accelerator neutrino experiments (, ), the sensitivity ranges are and , respectively. In all cases, the event rate per ton of detector is very low because of the small cross section for neutrino interactions, necessitating large detectors. The signal and background characteristics are technique dependent, nevertheless the detectors need to have very good shielding from cosmic ray muons generated in the atmosphere. This can be achieved by placing the detectors deep underground. In the case of accelerator neutrino beams, the pulsed nature of the beam provides additional background suppression.

The Sun is a copious source of with a flux of at the Earth produced by proton-proton and carbon-nitrogen-oxygen fusion reactions [23]. The solar spectrum is calculated using the standard solar model (SSM) with uncertainties ranging from 1% for the component to 14% for the higher energy Boron-8 component (). Solar neutrinos are detected using charged-current neutrino reactions on nuclei with low thresholds or using elastic scattering from electrons. In the presence of oscillations, a depletion of the expected flux is observed.

Nuclear fission reactors produce electron antineutrinos generating per GW of thermal power. As products of decay chains of neutron rich nuclei, these are emitted isotropically from the core with an energy spectrum predominantly below  [24, 25]. They are detected via the inverse beta decay (IBD) reaction, , in detectors that contain free protons. The energy of an incoming antineutrino is determined by measuring the positron energy, while the detection of the recoiling neutron, in coincidence with the positron signal, suppresses backgrounds.

Atmospheric neutrinos, coming from decays of and mesons produced in interactions of cosmic rays in the upper atmosphere, have an approximately isotropic flux of on the Earth. The dominant contributions come from the decay chains, and . The flux has a falling energy spectrum with a peak  GeV, and flux ratio  [26, 27]. Charged-current interactions of atmospheric neutrinos enable the study of oscillations over a broad range of distances ( km from above to  km from below) in large underground detectors. The sensitivity to oscillations is greatly improved by the well-understood flux which must be up-down symmetric (within ) in the absence of oscillations [28].

Accelerator neutrino beams are produced by bombarding high energy protons into a stationary target. The emerging and mesons are collected by a system of magnetic lenses (horns) [29], collimated towards the detector, and allowed to decay within a shielded tunnel into muons and neutrinos. The neutrino energy can be in the range of to . The flux can be sign-selected to be predominantly or with a small contamination () from electron type. The primary proton energy, the geometry of the target, horns, as well as the horn current can be adjusted to tune the beam spectrum with great flexibility.

An important aspect of reactor and accelerator experiments is the ability to place near and far detectors to study the same source. The near detector is placed sufficiently close to the source to measure the neutrino flux before any significant flavor transformation. The far detector can be positioned in the location of the expected oscillation maximum to measure the spectrum and flavor composition after oscillations. The comparison of observations at the two sites allows for accurate determinations of oscillation parameters free of many uncertainties associated with neutrino production and interaction rates.

In Sec. 2, we review the 3-flavor neutrino model and summarize the experimental progress over the past several decades in establishing it. In Sec. 4 and Sec. 5, we examine the current and future planned long-baseline experiments which aim to determine the remaining unknowns and improve the precision of known neutrino parameters. We will describe the motivation for a program of long-baseline neutrino experiments with particular emphasis on determination of the neutrino mass ordering and search for new charge-parity (CP) symmetry violation.

Although the discovery of neutrino oscillations took place using natural sources of neutrinos from the Sun and the atmosphere, future precise measurements are expected to originate from man-made sources such as reactors and accelerators that are well-monitored and controllable. As described in Sec. 5, future precise measurements of CP violation will only be possible in a long-baseline accelerator-based, high-power, and pure neutrino beam directed at a near detector and a very large and capable far detector.

## 2 The Three-flavor neutrino Standard Model

Although a definitive description of massive neutrinos within the standard model does not yet exist, all compelling neutrino oscillation data can be described by mixing among three left-handed neutrino flavors, in analogy with quark mixing via the Cabibbo-Kobayshi-Maskawa (CKM) matrix [7]. This number of active neutrinos is also compatible with the measured invisible decay width of the Z-boson. The data firmly establishes that the 3 neutrino flavors are superpositions of at least 3 light mass states with unequal masses, , all smaller than eV. The oscillations are characterized by two independent mass differences: and with . The unitary mixing matrix, , called the Pontecorvo-Maki-Nakagawa-Sakata (PMNS) matrix, is parameterized by 3 Euler angles and depending on whether the are Dirac or Majorana, 1 or 3 phases potentially leading to CP violation.

 νlL=3∑j=1UljνjL (7)
 U=⎛⎜⎝c13c12c13s12s13e−iδCP−c23s12−s13s23c12eiδCPc23c12−s13s23s12eiδCPc13s23s23s12−s13c23c12eiδCP−s23c12−s13c23s12eiδCPc13c23⎞⎟⎠ ×diag(1,eiα212,eiα312) (8)

Here, and , is the Dirac phase, and and are Majorana phases that cannot be observed in oscillation experiments. Including the 3 neutrino masses there are a total of 7 or 9 additional free parameters in the minimally extended standard model with massive neutrinos for the cases of Dirac or Majorana neutrinos, respectively. The full oscillation phenomenology is then described by modifying Eq. (3) for 3- mixing. For neutrinos produced with energy and flavor , the probability of its transformation to flavor after traveling a distance in vacuum is expressed as:

 Pll′≡P(νl→νl′) = =

The best values for the parameters can be obtained from a global fit to the data of neutrino oscillation experiments [7, 30] The measured mass-squared differences are , , where the sign of is unknown and is commonly referred to as the problem of mass ordering or hierarchy (MH). The determined mixing angles are , , and for normal and for inverted mass ordering. Currently is the least known mixing angle with a strong degeneracy around because the measurement is dominated by disappearance of which measures . From global fits [30, 31] to the data the current 1 uncertainties on the parameters , , , , and are about 3%, 4%, 2%, 12%, and 5%, respectively. Finally, the current experimental data does not significantly constrain the Dirac CP phase.

Since the ratio of the two mass differences, , is large, most experimental situations can be analyzed using a 2- model with small corrections. If the experimental situation is such that and then oscillations due to the larger are averaged out due to either the size of the production region or experimental energy resolution. On the other hand, if then the oscillations due to the smaller remain a small correction. Fig. 2 illustrates the distinctive pattern of masses and mixings in the neutrino sector. The normal or NH (inverted or IH) mass ordering is on the left (right). The maximal mixing in is parameterized by . The potentially large CP asymmetry is demonstrated as a variation of flavor content within each mass state.

### 2.1 Measurement of |Δm232| and θ23

The first compelling evidence of neutrino flavor oscillation was obtained by the Super-Kamiokande (Super-K) experiment observing atmospheric neutrinos. The cylindrical Super-K detector contains 50 ktons of pure water 1000 m underground in western Japan. Charged particles with velocities exceeding that of light in water emit Cherenkov radiation which is detected using regularly spaced large photo-multiplier tubes along the walls, sensitive to single photo-electrons [33]. For atmospheric neutrinos, the detection principle relies on observing Cherenkov light from the charged lepton produced in the reaction: . The energy, direction, and flavor of the incoming neutrino is reconstructed from the detected pattern, arrival times, and intensity of Cherenkov photons.

In 1998 Super-K reported a zenith angle dependent deficit of the upward going atmospheric muon neutrinos, inconsistent with the expected nearly isotropic flux [34]. This definitive result followed the earlier hints from previous water Cherenkov detectors, Kamioka Nucleon Decay Experiment (KamiokaNDE) and Irvine-Michigan-Brookhaven (IMB) [35, 36]. The deficit of upward going muon neutrinos was not accompanied by excess of upward going electron neutrinos. This indicated that were oscillating into which could not be directly observed in the detector due to the high energy threshold (3.5 GeV) required for production and the extremely short lifetime.

The discovery of atmospheric neutrino oscillations was possible because of the large available dynamic range in , the necessary event statistics provided by the massive underground detectors, and the favorably large mixing [37]. Super-K obtains the best precision on the parameters through selection of muon type events with good energy and angular resolution to prepare an analysis using as the observable. As demonstrated in [38] the event depletion at corresponds to the first oscillation maximum with . At , the fast oscillations cannot be resolved, and an average factor of depletion of type events is measured corresponding to a precise measurement of or near-maximal mixing. In the 3-neutrino scenario Super-K measures (the atmospheric mass-squared splitting, ) and a product of matrix elements . The latter further converts to a measurement of (the atmospheric mixing angle).

The unusually large neutrino mixing observed in the atmospheric oscillations discovery needed a confirmation by accelerator neutrino experiments in which the neutrino beam is tunable and has high purity. Independent confirmation came from K2K (KEK to Kamioka) [39], MINOS(Main Injector Neutrino Oscillations) [40], and T2K (Tokai to Kamioka) [41] in which disappearance of muon neutrinos was observed with a laboratory produced beam. The most precise measurements of and are currently obtained by MINOS [42] and T2K [43], respectively. Figure 3 shows the muon neutrino spectrum observed in MINOS. MINOS uses an accelerator produced beam from the Fermilab (USA) Main Injector accelerator with 120 GeV protons; the peak of the low energy neutrino beam is adjusted to be about 3 GeV. The MINOS far detector is a magnetized steel scintillator detector with 5 ktons of mass located 735 km away in the Soudan mine; in addition MINOS has a near detector using the same technology with 1 kton of mass placed 1 km away from the beam production target. A unique feature of MINOS is the ability to run the beam with changes to the target position and the horn current. By fitting several different spectra in the near detector, MINOS is able to make predictions with uncertainty  [44] for the far detector event spectrum without oscillations which is important for parameter determination.

T2K uses an accelerator produced beam from the JPARC (Japan Proton Accelerator Research Complex) accelerator with 30 GeV protons; the Super-K water Cherenkov detector is used as the far detector located 295 km away. The direction of the beam is adjusted to be off-axis [45] by 2.5 to obtain a narrow band beam with maximum at 0.6 GeV [46] and width of GeV. The Super-K water Cherenkov detector has excellent energy and particle identification capability at this energy allowing the experiment to see almost complete disappearance of muon neutrino events at . The off-axis technique with excellent energy resolution could provide the best determination of .

In addition to confirming the disappearance signal, it was important to explicitly test for as indicated by the atmospheric neutrino result. Such a test requires detection of leptons from a charged current interaction of at high energies. As the lepton decays almost immediately ( m), even a boosted with a momentum of 2 GeV/c would only travel about tenth of a mm before decay. Furthermore, the decays to a variety of final states making it difficult to distinguish from neutral current events. Despite these difficulties Super-K collaboration has statistically identified events at in the atmospheric neutrino data from the backgrounds generated by deep inelastic scattering of and [47]. Detection of individual events requires tracking and detection of decays at very short distances. The direct observation of appearance in a beam was performed by the OPERA (Oscillation Project with Emulsion Tracking) experiment using emulsion films with spatial resolution of m [48]. OPERA has reported observation of 5 events with an expected background of events confirming the dominance of the oscillation mode [49, 50].

### 2.2 Measurement of Δm221 and θ12

Starting in 1967, Ray Davis measured the solar flux with 610 tons of liquid CCl through the reaction ClAr [52]. The solar neutrino rate was determined by periodically counting decays of radioactive Ar extracted from the detector liquid. The measured flux was only about 1/3 [53] of the prediction from the standard solar model (SSM) [54]. This result was further confirmed by the Gallium experiments, SAGE [55] and GALLEX [56], using the GaGe reaction as well as Kamiokande [57] and Super-K [58, 59] experiments using the reaction. Neutrino mixing offered a natural explanation to the solar neutrino puzzle, since some of the generated in the Sun could transform to muon or tau neutrino flavors during flight and thus become undetectable due to the low solar neutrino energies which are below the thresholds for muon and tau production.

The Sudbury Neutrino Observatory (SNO) experiment was designed to measure the flux of all neutrino flavors from the Sun using the neutral current reaction on deuterium, , and separately the flux through the charged-current reaction . SNO was a water Cherenkov detector with an active target mass composed of 1 kton of heavy water (DO) which provides the deuterium targets for solar neutrinos. The final-state neutron from the neutral-current reaction was detected by a variety of techniques with consistent results. Measurement of the electron energy from the final state of the charged current reaction determines the solar neutrino spectrum above MeV [60]. The results [61] are consistent with the predictions of SSM incorporating -oscillations enhanced by the matter effect in the Sun. Borexino in Italy is an extremely high resolution liquid scintillation underground detector and has contributed by measuring the elastic scattering reaction, , for energies below the Be line (0.861 MeV) in the solar neutrino spectrum. The solar survival probability as function of energy is obtained by combining data collected at all energies by the radiochemical experiments(based on chlorine and gallium), Borexino, SNO, and Super-K [62].

The electron density in the core of the Sun is and it drops approximately exponentially to 0 at the surface. An electron neutrino of energy MeV generated in the core of the Sun has a resonance density much larger than and therefore according to Eq. (5) the oscillations are as if in vacuum. The spread due to distance and the source size wash out the oscillations and the survival probability is a constant . At energy MeV, and the eigenstate inside the Sun coincides with the heavier effective mass eigenstate which adiabatically evolves into as the density decreases. In this energy region the produced in the core leave the Sun in the state with the flavor content of . This effect is called the Mikheyev-Smirnov-Wolfenstein (MSW) effect [17, 63, 64, 65]. The energy dependent survival probability additionally provides the constraint and is a confirmation of the solar matter effect. In the 3- picture, the charge current ( only) and the neutral current (sum of all ) solar rate is a constraint on a combination of the PMNS matrix elements or .

Depending on the solar zenith angle solar neutrinos pass through varying Earth matter density before they are detected. The effect of Earth matter on solar neutrinos can produce a day/night asymmetry which was recently measured by Super-K [66] at 2.7 sigma to be (-3.21.2 %) consistent with expectations from the global fit. The measurement provides an indication of the terrestrial matter effect as well as further independent constraint on the solar oscillation parameters.

The Sun is a broadband source of pure , and when combined with the enhancement due to the MSW effect and the very long distance to the Earth, it is not surprising that solar neutrino experiments provided the first hint of neutrino oscillations. Nevertheless, despite consistency with the large mixing angle , solar neutrino experiments included the possibility of multiple solutions at statistical significance of  [60] . The Kamioka liquid-scintillator antineutrino detector (KamLAND) experiment [67] in Japan selected the large mixing angle solution at with detection of reactor over a long distance. There are a large number of nuclear power reactors around the KamLAND site with a flux averaged distance of 180 km. With neutrino energy determined from the measured positron energy in the IBD process, KamLAND was able to observe the oscillation spectrum shown in Fig.  4. Taking advantage of the very different ranges of the atmospheric and reactor neutrinos, KamLAND and the solar neutrino experiments can be understood in the 2- framework and provide (the solar mass-squared splitting, ) and (the solar mixing angle). Within the 3- framework the KamLAND disappearance constrains and a combination of PMNS matrix elements . The latter is mostly sensitive to the solar mixing angle and has a weak dependence on the mixing angle .

### 2.3 Measurement of θ13

The key observation to complete the 3- picture is the determination of oscillations in the same range as indicated by the atmospheric . The main impact of this observation is an explicit demonstration that the state is composed of at least three mass eigenstates or that all elements in the top row of the matrix in Eq. (2) are non-zero. Both the solar and KamLAND measurements have a weak dependence on . A joint analysis of these data provided an initial hint for a non-zero value  [69]. Previous reactor experiments with baseline of km [70, 71, 72] and a single far detector provided upper limits at 90% C.L..

A new campaign of experiments was launched to determine by either disappearance with reactors or () appearance with accelerators. In the case of reactor experiments, the maximum expected depletion was , and so a carefully designed experiment with near and far detectors was needed. In the case of accelerators, the signal was expected to have low statistics with background from both neutral current events and the contamination in the beam, and therefore a well-designed neutrino beam and a large detector with excellent electron identification were needed.

In 2012, three new reactor experiments reported results on . Double Chooz (France) reported results with a far detector only [73] disfavoring at 1.6. The Daya Bay experiment (China) reported the discovery of non-zero with significance from an array of three near, and three far identical detectors (each with 20 tons of gadolium-loaded liquid scintillator fiducial mass) placed to optimize sensitivity from three nuclear power stations [74]. RENO (Reactor Experiment for Neutrino Oscillations) also reported results from an array of nuclear power stations in Korea and identical near and far Gd-loaded liquid scintillator detectors confirming Daya Bay’s discovery with a significance [75]. Daya Bay increased their significance to 10 with a larger data set [76] and recently added more target mass to the far and near sites (4 near and 4 far detectors) [77]. The additional statistics and careful energy calibration in Daya Bay has resulted in an independent measurement of which governs this oscillation (Fig. 5). The third neutrino mixing angle is now precisely measured to be 8.4. MINOS [78] and T2K [79] have reported their searches of to oscillation that is also sensitive to . In particular, T2K obtained an early result disfavoring at 2.5. The status of current accelerator experiments addressing will be covered in more detail in the next section.

The picture of masses and mixings illustrated in Fig. 2 was assembled over several decades, mainly using precise observations of or disappearance. By convention , and the strong evidence of matter effect in the Sun indicates leading to  [7]. But the question of neutrino mass hierarchy, whether is heavier or lighter than , remains unresolved  [80]. The sign of and the Dirac or Majorana phases are also unknown. These questions as well as further precision measurements are expected to be addressed by running and future experiments optimized for the known parameters and focused on appearance measurements of conversion.

The precise prediction of 3 neutrino oscillations using the PMNS matrix model in matter is shown in Fig. 6. For the left panel, the flavor evolution of a 3 MeV electron antineutrino – typical of a reactor experiment – is shown over a distance of 100 km. Due to the large difference between and , two different oscillation frequencies are seen with a relative shift dependent on the sign of . Reactor experiments can only observe disappearance as the neutrino energies are too low for muon and tau production in charged-current reactions.

For the right panel in Fig. 6, the flavor evolution of a 1 GeV muon neutrino – typical of an accelerator experiment – is shown over a distance 5000 km. According to the best fit parameters such neutrinos should undergo a complete transformation after about 500 km with a small fraction appearing as . The fraction undergoes a large modulation depending on the unknown CP phase (dashed lines represent ), and is opposite in magnitude for neutrinos and antineutrinos. The CP effect also grows for higher oscillation nodes. A properly designed experimental program using reactor and accelerator sources should allow us to measure these oscillations over several nodes, to uncover if a large CP violation exists in the lepton sector, and to precisely determine the parameters including the mass hierarchy and the possibility of maximum mixing.

A few anomalous experimental results including LSND [81], MiniBooNE [82], and measurements of absolute reactor flux [83] cannot be explained by the 3-neutrino framework. These results suggest that the 33 PMNS matrix might not be unitary, and there might exist sterile neutrino(s) with the mass-squared splitting with small mixing with the three active neutrinos. Details about these data and future direct searches at short baselines ( km/GeV) can be found in Ref. [84]. A joint fit of the current solar, KamLAND, and Daya Bay results provides the first unitarity test of the top row of the PMNS matrix (), insufficient to exclude sterile neutrinos with small mixing.  [85, 86]. The comparison of derived from reactor disappearance measurements and from accelerator appearance measurements also provides a strong indirect test of the unitarity of PMNS matrix [86]. In the rest of this review, we will focus on the standard three-flavor neutrino model and precise measurements of its parameters with current and future long-baseline experiments.

## 3 Phenomenology of CP violation and mass hierarchy

We first examine the appearance probability in the 3-flavor model to understand the experimental optimization:

 Pμe=|Uμ1U∗e1e−iE1t+Uμ2U∗e2e−iE2t+Uμ3U∗e3e−iE3t|2 (10)

where represents the phase advance of the mass eigenstate over the flight time between the source and the detector. The oscillation probability for or can be obtained by simply exchanging the labels . Therefore, if any element of the PMNS matrix is complex, (e.g. ), neutrino oscillations are not invariant under time reversal or charge-parity conjugation. This is a phenomenological consequence of 3-generation mixing with at least one complex phase.

Explicitly, if the PMNS matrix is complex, then the asymmetry is a direct measurement of the CP violation in the neutrino system. In the case of 3-neutrino mixing, . We prefer to use the fractional asymmetry in experimental considerations: . The preferred observable is since it is expected to be large, and the production and detection of and is far easier than . For vacuum oscillations the CP asymmetry to leading order in is:

 AμeCP≈2JPMNSCPsin2(θ23)sin2(θ13)cos2(θ13)×αΔm231L4Eν, (11)

where is the rephasing invariant [7],

 JPMNSCP≡18sin2θ12sin2θ13sin2θ23cosθ13sinδCP≈0.032sinδCP. (12)

This is in sharp contrast to the very small mixing in the quark sector which leads to the very small value for the corresponding invariant: , despite the large value of . Recent studies in leptogenesis [87] have shown that the phases in the PMNS matrix can provide the CP violation large enough for the generation of observed baryon asymmetry of the universe [88, 89]. Therefore observation of CP violation due to could account for a large fraction of the baryon asymmetry [90, 91]. The asymmetry increases as a function of and is shown to decrease with [92, 93] as long as it is not too small, and with current parameters the asymmetry can be as large as for [94]. Due to the linear dependence on , and , the statistical error on is approximately independent of and providing substantial flexibility in experimental considerations [92, 95].

Observation of requires a long-baseline experiment with a pure accelerator generated beam of and crossing the Earth before reaching the detector. The effect of Earth matter must be taken into account using the electron density through the crust or mantle of . Using the well-known values of and , for typical accelerator neutrino energy of large change is expected to the oscillation probability (Eq. (4) and (6)). The effect for neutrinos (anti-neutrinos) leads to an enhancement (suppression) for and a suppression (enhancement) for , and therefore measuring it will determine the order of the neutrino masses. Atmospheric neutrinos are also expected to show significant sensitivity to Earth matter for crossing the core [96, 97]. The effect of Earth matter on neutrino oscillations with accelerator or atmospheric neutrinos has not been demonstrated in a definitive way, therefore such a measurement has interest both for phenomenology and for determining the mass ordering.

Since the size of the effect on due to the CP phase and the Earth matter is similar, both effects have to be considered for actual experiments. For accelerator experiments, assuming a constant density of matter, a sufficiently accurate expression to leading order in has been derived [98](also see Fig. 6):

 P(νμ→νe) = + − +

where

 Δij=Δm2ijL/4Eν, A=√2GFNe2Eν/Δm231.

For anti-neutrinos the signs of and are reversed. The last two terms in the expression are proportional to . The dependence on the mass hierarchy in and the CP phase can be disentangled with precise measurement of and appearance as a function of energy [95, 99].

The precise measurement of by reactor neutrino experiments [100, 101] has also provided a unique opportunity to determine the neutrino mass hierarchy in a medium-baseline high resolution reactor neutrino oscillation experiment [102, 103, 104, 105, 106, 107, 108, 109, 110, 111, 112, 113]. The energy and distance for reactor experiments is sufficiently low that the vacuum formula for the survival probability of an electron antineutrino can be used (see Fig. 6):

 P(νe→νe) = − − −

This probability depends on the mass hierarchy as for NH and for IH. The measured reactor neutrino energy spectrum for (or ) is expected to exhibit oscillations with a slow () and a fast component. The fast component arising from the and terms has amplitude proportional to . The two choices for mass hierarchy produce a small energy dependent shift (of the order of ) in the oscillation pattern which can be measured with sufficient energy resolution. From above formula, well-optimized reactor experiments can access five neutrino mixing parameters: , , , , and the mass hierarchy (or the sign of ). Such high precision measurements will enable tests of neutrino mass and mixing models. For example, models of quark-lepton complementarity and others [114, 115], have predicted and .

Based on Eq. (3) and Eq. (3), an optimized program of measurements for appearance for using an accelerator beam, and disappearance for using reactor flux cover the remaining issues within neutrino oscillation physics. They also provide redundancy for measurements of mass hierarchy, , and . These measurements depend on man-made pure beam sources with optimized range. Neutrino physics phenomenology has proven to be quite rich and any future program should not only be guided by measurements motivated by the current best model, but should also be sufficiently redundant and with enough dynamic range to allow identification of new physics effects if they exist. In the next sections we will examine the current and future planned programs for these measurements.

## 4 Current-generation of Experiments

The accelerator disappearance experiment MINOS+ [116], and the reactor experiments Daya Bay, Double Chooz, and RENO are expected to continue operations to improve precision on , and . The error on the value of the is expected to reach 3% [117]. The sign and magnitude of is of significance for the theoretical work aimed at explaining underlying symmetries responsible for neutrino mixing [115]. The measurement of is correlated to since, to leading order (Eq. (3)), the appearance probability depends on . Precision measurement of is therefore an important aspect of the current program. The accumulation of atmospheric and solar neutrino data is also expected to continue with Super-K, MINOS+, IceCube [118], and Borexino.

The main focus of the current program, however, is the transition with the T2K and NOA experiments. These experiments were both optimized before the discovery of to obtain evidence for this transition with the best signal to background ratio. The two principal backgrounds to an electron neutrino event in the few GeV range are: the intrinsic contamination present in the accelerator-produced beam at level, and weak neutral current interactions that produce photons and particles. The background in the beam comes from and -meson decays in the decay tunnel. The background is indistinguishable from the signal except that it has a broader energy spectrum. The neutral current processes produce photons from decays that could be misidentified as single electrons in case of asymmetric or overlapping electro-magnetic showers. To limit the impact of both of these backgrounds with broad energy spectra, both T2K and NOA have adapted the strategy of using a narrow neutrino energy spectrum. The narrow energy spectrum cuts down the contributions from neutrinos outside of the energies of interest as set by the oscillation probability. The narrow-band beam is achieved with the off-axis neutrino beam technique [45] in which the detector is placed at a small angle to the beam to exploit the kinematic momentum peak in the 2-body decay. The T2K and NOA experiments utilize neutrino beams with a peak energy of 0.6 GeV and 2 GeV, at off-axis angles of 44 and 14 mrad, respectively.

The T2K experiment in Japan, running since 2010, has analyzed data from integrated exposure of protons with the beam polarity in the neutrino mode. The top plot in Fig. 7 shows the energy spectrum of 28 events identified as in T2K [119]. The background expectation for these data from beam contamination and neutral current were 3.2 and 1.0 events, respectively. This observation conclusively establishes the presence of oscillations driven by at confidence level. The bottom panel of the figure shows the energy spectrum of the T2K candidate events [120] compared to no oscillation expectation. The deficit of in T2K is consistent with maximal mixing angle . A combined fit using both appearance and disappearance data results in with a small dependence on the assumed mass hierarchy.

T2K has an excess of events compared to the prediction for and the best fit value for from reactor data. While T2K data alone cannot provide statistically significant constraints on , the addition of the reactor measurement gives a preference for the negative values of with the best-fit point near (Fig. 10). Future addition of antineutrino data from T2K as well as the NOA data should further constrain the parameter range.

The NOA (NuMI Off-axis Appearance) experiment uses the Fermilab Main Injector 120 GeV proton beam. NOA also follows the two-detector approach – both off-axis at 14 mrad – with the near detector (at 1 km) characterizing the neutrino beam close to the production point and the far (at 810 km) measuring the flavor composition after oscillations. The detectors use segmented liquid scintillator and are structurally and functionally identical with the near and far active masses of 0.3 and 14 kton, respectively. The NOA collaboration reported their first measurement of appearance and disappearance signals in the summer of 2015 [121] after 400 kW beam operation for about a year. Depending on the event reconstruction method, the experiment has found 6 to 11 events with background expectation of event. The event rate appears to be consistent with with normal hierarchy and the disappearance result is consistent with the current picture of maximum mixing.

The size of the Earth matter effect depends on the energy of the oscillating neutrino (Eq. (6)). The T2K and NOA experiments have the oscillation maximum at and GeV, respectively. The resulting large difference between matter effects, since , provides complementarity between the two experiments. The increase (or suppression) of due to the matter effect can be either masked or enhanced by the CP asymmetry, creating an ambiguity if the asymmetry due to matter is smaller than CP. The ambiguity can be reduced by a combination of measurements from the two experiments which will also provide information on the neutrino mass hierarchy and [122].

## 5 Next-generation Experiments

The next generation experiments fall in two categories: detectors for high statistics atmospheric neutrino studies, and well-optimized experiments using man-made accelerator and reactor neutrinos. In both cases the detectors need to be very large with excellent energy resolution and particle identification, and well-shielded from cosmic ray muons.

### 5.1 Atmospheric Neutrinos

The next-generation atmospheric neutrino experiments will focus on resolving the neutrino mass hierarchy (MH)  [123, 124, 125, 126, 127]. In the normal (inverted) hierarchy there is a resonant effect for () at energy GeV and zenith angle (corresponding to traversal through the outer core of the Earth). The effect can be measured through a large distortion in and . Very large statistics (and correspondingly large detectors) are needed because of the limited solid angle and the atmospheric flux power law spectrum, , in this energy range. The measurement requires excellent zenith angle resolution and an energy threshold , relatively low for very large detectors optimized for very high energies () such as IceCube. The capability to differentiate positive and negative muons and muon tracks from showers would allow enhanced sensitivity. Muon charge can be determined either by magnetizing the detector or by utilizing the difference in absorption and decay of and in dense materials. The latter requires very low energy threshold to be able to measure muon decay.

The proposed next-generation experiments include PINGU [128], ORCA  [129], INO  [130], and Hyper-Kamiokande [131]. PINGU is a proposed multi-megaton high density array as an upgrade for IceCube at the South Pole Station. ORCA is proposed as a deep-sea neutrino telescope in the Mediterranean as part of KM3NeT (multi-km neutrino telescope). INO is a planned to be an underground magnetized iron calorimeter with kton mass and 1.5 Tesla magnetic field in southern India. Hyper-Kamiokande [131] is a proposed giant water Cherenkov detector near the site of the current Super-Kamiokande detector in Japan. These experiments are expected to be limited by systematics because of the energy and angular resolution needed. Their capability to distinguish between the two mass hierarchies, which is correlated to , can reach above 3 independent of the CP phase given enough exposure. As remarked in Sec. 4, there is a large ambiguity between the CP phase and the mass hierarchy in current accelerator based experiments. This can be further reduced by combining with atmospheric neutrino data [132].

The oscillations patterns for atmospheric neutrinos also exhibit dependence on , predominantly at low energies (below a few GeV), raising a possibility of obtaining a constraint of this parameter complimentary to CP-violation searches with accelerator beams [133] if the mass hierarchy is well-known. A multi-Mton scale detector with low energy detection threshold (0.5 GeV) and good angular and energy resolution would, however, be required to realize a measurement of this phase with high degree of significance [134] using atmospheric neutrinos.

### 5.2 Reactor Neutrino Oscillations

The next generation reactor experiment will be optimized to resolve the mass hierarchy by detecting the shift (Sec. 3) in energy dependent oscillations in the range. As usual the choice of the baseline distance is important. The fast oscillations due to the last two terms in Eq. (3) ( and ) will exhibit the shift at any distance, however at short distances (defined by ), the shift is ambiguous with the measurement of itself, and at long distances () the oscillations are too fast to be observable with achievable energy resolution. Using Eq. 3 we can write the asymmetry in obvious notation:

 PIH(¯νe→¯νe)−PNH(¯νe→¯νe)=sin22θ13cos2θ12sin2Δ32sin2Δ21. (15)

At or the maximum of solar oscillations, the asymmetry between IH and NH vanishes. For , the nodes of oscillations are shifted to lower (higher) energies for normal (inverted) hierarchy, but for , the converse is true. This phenomenology allows us to choose so that the node is around MeV or the maximum of the reactor spectrum to avoid ambiguities [110]. This node can be shown to be at . The oscillations on either side of the node at can be compared in a single detector to resolve for IH versus NH, reducing dependence on the absolute energy calibration (Fig. 8). Nevertheless, the detector must have sufficient statistics ( events after oscillations), energy resolution( at 1 MeV), and linearity () across the reactor energy range. Otherwise, low-energy oscillation pattern will be smeared out. The detector also must be equidistant from all reactor cores so that the oscillation pattern is not obscured. The large event sample from such an arrangement would also enable precision () measurements of , , and .

There are two proposed next generation reactor experiments: the Jiangmen Underground Neutrino Observatory (JUNO) [135, 136] in China and RENO-50 [137] in South Korea. JUNO (RENO-50) design calls for a 20 kton (18 kton) fiducial mass liquid scintillator detector placed 700 m (900 m) underground, 52.5 km (47 km) away from a set of reactors with total thermal power of (). At about 50 km, the IBD rate is with (without) oscillations indicating a need for 3000  ktonGWyear of exposure.

The needed energy resolution and linearity lead to requirements on the scintillator brightness, attenuation length, photo-detector coverage, and quantum efficiency so that the photo-electron yield is per MeV. With the required energy resolution JUNO will achieve separation of the two mass hierarchies with 6 years of operation[136].

### 5.3 Accelerator Neutrino Oscillations

Precision studies of neutrino oscillations with accelerators will focus on the search for CP violation with an optimized combination of neutrino beam energy, distance to the detector, and detector technology. The preferred mode for future accelerator studies is and the charge conjugate (See Sec.3); this requires production of an intense beam. Using the known parameter values the expected size of the signal at first maximum is , well above the the contamination background of in a horn focused beam. The large expected signal makes background reduction as described in Sec. 4 a secondary concern. And therefore a broadband beam can be utilized to maximize statistics and the dynamic range. Furthermore alternative extremely low background methods using very pure intense beams of and using decay such as the Neutrino Factory [138, 139] or decays of radioactive beta beams [140] are no longer favored.

The asymmetry has contributions from CP violation of , and from the matter effect which grows as a function of the oscillation energy (or the length of baseline) as , with the sign dependent on MH. At the first oscillation maximum, , the asymmetry due to matter effect exceeds the maximum from CP violation at or  [94, 141]. Therefore, two different strategies for experimental optimization exist: distances of would allow measurement of CP violation with low dependence on MH, while distances of would allow resolution of MH and also be sensitive to CP violation. A minimum distance is determined from the requirement that , a condition that ensures high enough probability for and interactions and good particle reconstruction and identification. An additional consideration is to obtain events at the second oscillation maximum, where CP effects are a factor of 3 larger, but event rate is one-ninth as large due to kinematics. A broadband beam designed for baselines of will also have sensitivity at second maximum [95, 141], an important consideration in resolving ambiguities.

The expected event rate for a broadband beam can be calculated to lowest order by integrating the muon neutrino flux, with L in km and with the appearance probability , and the charged current cross section . For vacuum oscillations the event rate is found to be independent of the baseline distance because of the increase of oscillation probability and the cross section so that , with the rate for approximately of . For off-axis beams and distances , the event yield and the ratio of events is smaller. For most modern high energy accelerators, the available beam power is limited to [142], therefore detectors with efficient mass (mass times efficiency) of are needed independent of distance to obtain a few hundred appearance events. At such large scales, water Cherenkov and liquid argon time projection chambers are considered cost effective technologies [143, 144]. At low energies ()– distances of – the charged current cross section is dominated by quasi-elastic interactions with low multiplicity final states, well-reconstructed by water Cherenkov detectors. However, at higher energies () – distances of – charged current events with multiple final state particles must be reconstructed to retain high efficiency; a high granularity detector such as a liquid argon time projection chamber is therefore preferred.

The Hyper-Kamiokande (Hyper-K) experiment [133, 131] in Japan with a baseline of 295 km and the Deep Underground Neutrino Experiment (DUNE) at the Long-Baseline Neutrino Facility (LBNF) in the U.S. with a baseline of  [145, 144, 146] have chosen the two complementary approaches as outlined above.

Hyper-K is planned to be a water Cherenkov detector with 560 kton (1 Mton) fiducial (total) mass near the current Super-K site in western Japan at a depth of . The Hyper-K detector will be placed in the same mrad off-axis neutrino beam matched to the first oscillation node (Fig. 9) as T2K but with a much larger detector and beam power of . Hyper-K will also have a complex of near detectors to monitor and measure the beam to predict background and signal rates at the far detector.

DUNE consists of a horn-produced broad band beam with 60-120 GeV protons with beam-power of 1.2 MW from Fermilab, 40 kton fiducial volume liquid argon time projection far detector 1450 m underground at Sanford Underground Research Laboratory in South Dakota, and high-resolution near detector [146]. The baseline of DUNE [145, 94] is 1300 km which is optimized to measure the matter effect and CP violation simultaneously. The DUNE broad-band flux (Fig. 9) is designed to cover the first () and second () oscillation nodes sufficiently so that the CP phase can be measured using the distortion of the energy spectrum as well as the asymmetry.

The liquid argon time projection technology [147, 148, 149, 150], chosen for DUNE, could be implemented by two approaches: the single phase in which the drifting charge is detected by electrodes in the liquid argon [145] or the double phase in which the charge is drifted to the surface of the liquid argon and amplified in the gas phase before detection [141].

The Hyper-K and DUNE experiments require precise prediction of signal and background spectra in the far detector. The prediction is calculated by extrapolating the measured event spectrum from the near detector, using the known beam geometry and constraints on the cross sections, and near and far detector efficiencies. The systematic errors associated with neutrino nucleus cross sections require further measurements and modeling  [151]. Since the maximum expected asymmetry from CP violation is , the requirement on the allowed systematic error is less than a few percent so that a effect can be measured with events. A joint fit to the four spectra – , , , – is expected to cancel some uncertainties, and the remaining largest contribution is expected to come from the relative energy scale for and events [152]. For DUNE, if the matter and CP asymmetries have the same sign, the combined effect is very large and the mass hierarchy will be determined to within a few months of running; in case of the opposite, exposure of a few years is needed [145]. Figure 10 shows the current global fit for versus while marginalizing over all other oscillation parameters [30, 31] including the reactor constraint on . The figure also shows the expectation at  and  for Hyper-K (DUNE) assuming 10 yrs of exposure at 1.3 MW (1.2 MW) [153, 146]. Note that the Hyper-K and DUNE expectations do not include the constraint. The independent measurements of from accelerators and reactors can be compared to test for new physics [86].

## 6 Conclusion

We have reviewed the current status of the physics of long-baseline neutrino oscillations with emphasis on the experimental technique. The picture of 3-flavor oscillations has now been established by observation of neutrino oscillations with solar, atmospheric, reactor, and accelerator neutrinos. Remarkably, a consistent picture of mass differences and mixing has emerged so that extremely precise experiments, using well-characterized and controlled terrestrial sources of neutrinos using reactors and accelerators, are now feasible. The main objectives of such an experimental program are the determination of potentially large CP violation in the neutrino sector, resolution of the mass ordering, and measurement of the mixing parameters with greater precision. The precision and redundancy of measurements will allow constraints on new physics beyond the 3-flavor model as well as neutrino mass and mixing models.

The writing of this review was supported by the US Department of Energy for M.V.D and X.Q., the Swiss National Science Foundation (SNF) for A.R., and the French National Centre for Scientific Research (CNRS) for V.G..

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