Physics Potential of Long-Baseline Experiments

Abstract

The discovery of neutrino mixing and oscillations over the past decade provides firm evidence for new physics beyond the Standard Model. Recently, 𝜃₁₃ has been determined to be moderately large, quite close to its previous upper bound. This represents a significant milestone in establishing the three-flavor oscillation picture of neutrinos. It has opened up exciting prospects for current and future long-baseline neutrino oscillation experiments towards addressing the remaining fundamental questions, in particular the type of the neutrino mass hierarchy and the possible presence of a CP-violating phase. Another recent and crucial development is the indication of non-maximal 2-3 mixing angle, causing the octant ambiguity of 𝜃₂₃. In this paper, I will review the phenomenology of long-baseline neutrino oscillations with a special emphasis on sub-leading three-flavor effects, which will play a crucial role in resolving these unknowns. First, I will give a brief description of neutrino oscillation phenomenon. Then, I will discuss our present global understanding of the neutrino mass-mixing parameters and will identify the major unknowns in this sector. After that, I will present the physics reach of current generation long-baseline experiments. Finally, I will conclude with a discussion on the physics capabilities of accelerator-driven possible future long-baseline precision oscillation facilities.

1. Introduction and Motivation

We are going through an exciting phase when the light of new findings is breaking apart our long-held understanding of the Standard Model of particle physics. This revolution started in part with the widely confirmed claim that neutrinos have mass, and it will continue to be waged by currently running and upcoming neutrino experiments. Over the last fifteen years or so, fantastic data from world-class experiments involving neutrinos from the sun [1–7], the Earth atmosphere [8, 9], nuclear reactors [10–16], and accelerators [17–22] have firmly established the phenomenon of neutrino flavor oscillations [23, 24]. This implies that neutrinos have mass and they mix with each other, providing an exclusive example of experimental evidence for physics beyond the Standard Model.

The most recent development in the field of neutrinos is the discovery of the smallest lepton mixing angle 𝜃₁₃. Finally, it has been measured to be nonzero with utmost confidence by the reactor neutrino experiments Daya Bay [13] and RENO [14]. They have found a moderately large value of 𝜃₁₃: sin²2𝜃₁₃|_{DayaBay(rate-only)}=0.089±0.010(stat)±0.005(syst) [13], sin²2𝜃₁₃|_{RENO(rate-only)}=0.113±0.013(stat)±0.019(syst) [14],which is in perfect agreement with the data provided by the another reactor experiment Double Chooz [15, 16] and the accelerator experiments MINOS [20] and T2K [22]. All the three global fits [25–27] of all the world neutrino oscillation data available indicate a nonzero value of 𝜃₁₃ at more than 10𝜎 and suggest a best-fit value of sin²𝜃₁₃≃0.023 with a relative 1𝜎 precision of 10%. Daya Bay experiment is expected to reduce this uncertainty to a level of 5% by 2016 when they will finish collecting all the data [28]. This recent high-precision measurement of a moderately large value of 𝜃₁₃ signifies an important breakthrough in validating the standard three-flavor oscillation picture of neutrinos [29]. It has created exciting opportunities for current and future neutrino oscillation experiments to address the remaining fundamental unknowns. This fairly large value of 𝜃₁₃ has provided a “golden” avenue to directly probe the neutrino mass hierarchy (two possibilities are there: it can be either normal (NH) if Δ𝑚²₃₁≡𝑚²₃−𝑚²₁>0 or inverted (IH) if Δ𝑚²₃₁<0, as described in Figure 1) using the Earth matter effects and to search for leptonic CP violation (if the Dirac CP phase, 𝛿_CP differs from 0^∘ or 180^∘) in accelerator-based long-baseline neutrino oscillation experiments [30, 31]. Another recent and important development related to neutrino mixing parameters is the hint of nonmaximal 2-3 mixing by the MINOS accelerator experiment [32, 33]. However, the maximal value of 𝜃₂₃ is still favored by the atmospheric neutrino data, dominated by Super-Kamiokande [34]. Combined analyses of all the neutrino oscillation data available [25–27] also prefer the deviation from maximal mixing for 𝜃₂₃; that is, (0.5−sin²𝜃₂₃)≠0. In 𝜈_𝜇 survival probability, the dominant term mainly depends on sin²2𝜃₂₃. Now, if sin²2𝜃₂₃ differs from 1 as indicated by the recent neutrino data, then we get two solutions for 𝜃₂₃: one <45^∘, named as lower octant (LO) and the other >45^∘, named as higher octant (HO). In other words, if the quantity (0.5−sin²𝜃₂₃) is positive (negative), then 𝜃₂₃ belongs to LO (HO). This leads to the problem of octant degeneracy of 𝜃₂₃ [35] which is a part of the overall eightfold degeneracy [36, 37], where the other two degeneracies are (𝜃₁₃,𝛿_CP) intrinsic degeneracy [38] and the (hierarchy, 𝛿_CP) degeneracy [39]. The resolution of the three fundamental issues in the neutrino sector: neutrino mass hierarchy, octant of 𝜃₂₃, and CP violation is possible only by observing the impact of three-flavor effects in neutrino oscillation experiments [40, 41].

Figure 1 

The sign of Δ𝑚231≡𝑚23−𝑚21 is unknown. The hierarchy of the neutrino mass spectrum can be normal or inverted.

The information on neutrino mass hierarchy is very important in order to settle the structure of neutrino mass matrix which in turn can give crucial piece of information towards the underlying theory of neutrino masses and mixing [42]. This is also a vital ingredient for neutrinoless double beta decay searches investigating the Majorana nature of neutrinos. If Δ𝑚²₃₁<0, and yet no neutrinoless double beta decay is observed even in the very far future experiments, that would be a strong hint that neutrinos are not Majorana particles [43]. Another fundamental missing link that needs to be addressed in long-baseline neutrino oscillation experiments is to measure 𝛿_CP and to explore leptonic CP violation. This new source of low-energy CP violation in neutral lepton sector has drawn tremendous interest because of the possibility of leptogenesis leading to baryogenesis and the observed baryon asymmetry in the universe [44]. Leptogenesis demands the existence of CP violation in the leptonic sector; for a recent review, see [45]. The possible link between leptogenesis and neutrino oscillations has been studied in [46–48]. It is likely that the CP violating phase in neutrino oscillations is not directly responsible to generate the CP violation leading to leptogenesis. But there is no doubt that a demonstration of CP violation in neutrino oscillations will provide a crucial guidepost for models of leptonic CP violation and leptogenesis. Precise measurement of 𝜃₂₃ and the determination of its correct octant (if it turns out to be nonmaximal) are also very vital tasks that need to be undertaken by the current- and next-generation neutrino oscillation experiments. These pieces of information will provide crucial inputs to the theories of neutrino masses and mixing [42, 49–51]. A number of excellent ideas, such as 𝜇↔𝜏 symmetry [52–59], 𝐴₄ flavor symmetry [60–64], quark-lepton complementarity [65–68], and neutrino mixing anarchy [69, 70], have been proposed to explain the observed pattern of one small and two large mixing angles in the neutrino sector. Future ultraprecise measurement of 𝜃₂₃ will severely constrain these models leading to a better understanding of the theories of neutrino masses and mixing.

The outline of this review work is as follows. We start in Section 2 by revisiting the phenomenon of neutrino oscillation in the three-neutrino framework. Then, we discuss the importance of matter effect on neutrino oscillation in Section 3. In Section 4, we take a look at our present understanding of neutrino oscillation parameters and we identify the fundamental missing links in the neutrino sector that can be answered in the current- and next-generation long-baseline neutrino oscillation experiments. Section 5 discusses in detail the three-flavor effects in long-baseline neutrino oscillation experiments with the help of 𝜈_𝜇→𝜈_𝑒 oscillation probability, 𝑃_𝜇𝑒. Next, in Section 6, we study the physics reach of current-generation long-baseline beam experiments, T2K and NO𝜈A. In Section 7, we give an overview on the possible options for next-generation accelerator-driven long-baseline oscillation facilities. Finally, in Section 8, we conclude with a summary of the main points.

2. Three-Neutrino Mixing and Oscillation Framework

Bruno Pontecorvo was the pathfinder of neutrino oscillation [74, 75]. In 1957, he gave this concept [76, 77] based on a two-level quantum system. Neutrino oscillation is a simple quantum mechanical phenomenon in which neutrino changes flavor as it travels. This phenomenon arises if neutrinos have nondegenerate masses and there is mixing. First, we consider the fact that neutrinos (𝜈_𝑒,𝜈_𝜇,𝜈_𝜏) are produced or detected via weak interactions and therefore they are referred to as weak-eigenstate neutrinos (denoted as 𝜈_𝛼). It means that they are the weak doublet-partners of 𝑒⁻,  𝜇⁻, and 𝜏⁻, respectively. In such a case, if neutrinos are assumed to be massive, then in general, it is not mandatory that the mass-matrix of neutrinos written in this weak (flavor) basis will have to be diagonal. So, it follows that the mass eigenstate neutrinos 𝜈_𝑖,𝑖=1,2,3 (the basis in which the neutrino mass matrix is diagonal) are not identical to the weak or flavor basis (the charged lepton mass-matrix is diagonal in this basis) and for three light active neutrinos, we have 𝜈_𝛼=³_𝑖=1𝑈^∗_𝛼𝑖𝜈_𝑖, (1) where 𝛼 can be 𝑒,𝜇, or 𝜏 and 𝑈 is the 3×3 unitary leptonic mixing matrix known as the Pontecorvo-Maki-Nakagawa-Sakata (PMNS) matrix [23, 78]. This matrix is analogous to the CKM matrix in the quark sector. We use the standard Particle Data Group convention [79] to parametrize the PMNS matrix in terms of the three mixing angles: 𝜃₁₂ (solar mixing angle), 𝜃₂₃ (atmospheric mixing angle), and 𝜃₁₃ (reactor mixing angle) and one Dirac-type CP phase 𝛿_CP (ignoring Majorana phases). The mixing matrix 𝑈 can be parameterized as 𝑈_PMNS=1000𝑐₂₃𝑠₂₃0−𝑠₂₃𝑐_{23Atmosphericmixing}×𝑐₁₃0𝑠₁₃𝑒^{−𝑖𝛿CP}010−𝑠₁₃𝑒^𝑖𝛿CP0𝑐_{13Reactormixing}×𝑐₁₂𝑠₁₂0−𝑠₁₂𝑐₁₂0001_Solarmixing, (2) where 𝑐_𝑖𝑗=cos𝜃_𝑖𝑗 and 𝑠_𝑖𝑗=sin𝜃_𝑖𝑗. The neutrino mixing matrix finally takes the form 𝑈_PMNS=𝑐₁₂𝑐₁₃𝑠₁₂𝑐₁₃𝑠₁₃𝑒^{−𝑖𝛿CP}−𝑠₁₂𝑐₂₃−𝑐₁₂𝑠₂₃𝑠₁₃𝑒^𝑖𝛿CP𝑐₁₂𝑐₂₃−𝑠₁₂𝑠₂₃𝑠₁₃𝑒^𝑖𝛿CP𝑠₂₃𝑐₁₃𝑠₁₂𝑠₂₃−𝑐₁₂𝑐₂₃𝑠₁₃𝑒^𝑖𝛿CP−𝑐₁₂𝑠₂₃−𝑠₁₂𝑐₂₃𝑠₁₃𝑒^𝑖𝛿CP𝑐₂₃𝑐₁₃. (3) It is quite interesting to note that three mixing angles are simply related to the flavor components of the three mass eigenstates as 𝑈_𝑒2²𝑈_𝑒1²=tan²𝜃₁₂,𝑈_𝜇3²𝑈_𝜏3²=tan²𝜃₂₃,|𝑈_𝑒3|²=sin²𝜃₁₃. (4) The transition probability that an initial 𝜈_𝛼 of energy 𝐸 gets converted to a 𝜈_𝛽 after traveling a distance 𝐿 in vacuum is given by 𝑃𝜈_𝛼→𝜈_𝛽=𝑃_𝛼𝛽=_𝑗𝑈_𝛽𝑗𝑒^{−𝑖𝑚2𝑗𝐿/2𝐸}𝑈^∗_𝛼𝑗², (5) where the last factor arises from the decomposition of 𝜈_𝛼 into the mass eigenstates, the phase factor in the middle appears due to the propagation of each mass eigenstate over distance 𝐿, and the first factor emerges from their recomposition into the flavor eigenstate 𝜈_𝛽 at the end. Equation (5) can also be written as 𝑃_𝛼𝛽=𝛿_𝛼𝛽−4^𝑛_𝑖>𝑗Re𝑈^∗_𝛼𝑖𝑈^∗_𝛽𝑗𝑈_𝛽𝑖𝑈_𝛼𝑗sin²𝑋_{𝑖𝑗CPconserving}−2^𝑛_𝑖>𝑗Im𝑈^∗_𝛼𝑖𝑈^∗_𝛽𝑗𝑈_𝛽𝑖𝑈_𝛼𝑗sin²𝑋_{𝑖𝑗CPviolating}, (6) where 𝑋_𝑖𝑗=𝑚²_𝑖−𝑚²_𝑗𝐿4𝐸=1.27Δ𝑚²_𝑖𝑗eV²𝐿/𝐸.𝑚/MeV (7)Δ𝑚²_𝑖𝑗=𝑚²_𝑖−𝑚²_𝑗 is known as the mass splitting and neutrino oscillations are only sensitive to this mass-squared difference but not to the absolute neutrino mass scale. The transition probability (given by (6)) has an oscillatory behaviour with oscillation lengths: 𝐿^osc_𝑖𝑗=4𝜋𝐸Δ𝑚²_𝑖𝑗𝐸()≃2.48mMeVΔ𝑚²_𝑖𝑗eV²𝐸()=2.48kmGeVΔ𝑚²_𝑖𝑗eV², (8) and the amplitudes are proportional to the elements in the mixing matrix. Since neutrino oscillations can occur only if there is a mass difference between at least two neutrinos, an observation of this effect proves that at least one nonzero neutrino mass exists. In a three-flavor framework, there are two independent mass-squared differences between the three neutrino masses: Δ𝑚²₂₁=𝑚²₂−𝑚²₁ (responsible for solar neutrino oscillations) and Δ𝑚²₃₁=𝑚²₃−𝑚²₁ (responsible for atmospheric neutrino oscillations). The angle 𝜃₁₃ connects the solar sector with the atmospheric one and determines the impact of the three-flavor effects. The last term of (6) accommodates the CP violating part, proportional to sin𝛿_CP. This contribution can only be probed in a neutrino oscillation experiment measuring the appearance probability of a new flavor, since for disappearance experiments (𝛼=𝛽) the last term becomes zero identically. Also, the CP violating term changes sign in going from 𝑃(𝜈_𝛼→𝜈_𝛽) to 𝑃(𝜈_𝛼→𝜈_𝛽) (for antineutrino, we have to replace 𝑈 by 𝑈^∗). It also changes sign in going from 𝑃(𝜈_𝛼→𝜈_𝛽) to 𝑃(𝜈_𝛽→𝜈_𝛼), since the CPT invariance ensures that 𝑃(𝜈_𝛼→𝜈_𝛽)=𝑃(𝜈_𝛽→𝜈_𝛼).

3. Neutrino Propagation through Matter

Oscillation probability changes dramatically when neutrino passes through matter [80–82]. During propagation through matter, the weak interaction couples the neutrinos to matter. Besides few hard scattering events, there is also coherent forward elastic scattering of neutrinos with matter particles they encounter along the path. The important fact is that the coherent forward elastic scattering amplitudes are not the same for all neutrino flavors. The ordinary matter consists of electrons, protons, and neutrons but it does not contain any muons or tau-leptons. Neutrinos of all three flavors (𝜈_𝑒, 𝜈_𝜇, and 𝜈_𝜏) interact with the electrons, protons, and neutrons of matter through flavor-independent neutral current interaction mediated by 𝑍⁰ bosons. These contributions are the same for neutrinos of all three flavors, leading to an overall phase which can be subtracted. Interestingly, the electron neutrinos have an additional contribution due to their charged current interactions with the ambient electrons of the medium which are mediated by the 𝑊^± exchange. This extra matter potential appears in the form 𝐴=±2√2𝐺_𝐹𝑁_𝑒𝐸, (9) where 𝐺_𝐹 is the Fermi coupling constant, 𝑁_𝑒 is the electron number density inside the Earth, and 𝐸 is the neutrino energy. The + sign refers to neutrinos while the − to antineutrinos. The connection between the electron density (𝑁_𝑒) and the matter density (𝜌) is given by 𝑉_𝐶𝐶=√2𝐺_𝐹𝑁_𝑒≃7.6𝑌_𝑒𝜌10¹⁴g/cm³eV, (10) where 𝑌_𝑒=𝑁_𝑒/(𝑁_𝑝+𝑁_𝑛) is the relative electron number density. 𝑁_𝑝, 𝑁_𝑛 are the proton and neutron densities in Earth’s matter, respectively. In an electrically neutral, isoscalar medium, we have 𝑁_𝑒=𝑁_𝑝=𝑁_𝑛 and 𝑌_𝑒 comes out to be 0.5. If we compare the strength of 𝑉_𝐶𝐶 for the Earth with Δ𝑚²₃₁/2𝐸, then we can judge the importance of Earth’s matter effect on neutrino oscillations. If we consider a neutrino of 5 GeV passing through the core of the Earth (𝜌∼10g/cm³), then 𝑉_𝐶𝐶 is comparable with Δ𝑚²₃₁/2𝐸 (=2.4×10⁻¹³eV if Δ𝑚²₃₁=2.4×10⁻³ eV²).

In a two-flavor formalism, the time evolution of the flavor eigenstates in matter is given by the following Schrödinger equation: 𝑖𝑑𝜈𝑑𝑡^𝛼𝜈_𝛽=1𝑈𝑚2𝐸²¹00𝑚²₂𝑈^†+𝐴(𝐿)0𝜈00𝜈^𝛼_𝛽. (11) In case of constant matter density, the problem boils down to a stationary one and a trivial diagonalization of the Hamiltonian can provide the solution. In matter, the vacuum oscillation parameters are connected to the new parameters (the new parameters in matter carry a superscript 𝑚) in the following way: Δ𝑚^2𝑚=Δ𝑚²cos2𝜃−𝐴²+Δ𝑚²sin2𝜃²,sin2𝜃^𝑚=sin2𝜃Δ𝑚²Δ𝑚^2𝑚. (12) The famous MSW-resonance [80–83] condition is satisfied at Δ𝑚²cos2𝜃=𝐴. (13) At MSW-resonance, sin2𝜃^𝑚=1 (from (12) and (13) which immediately implies that independent of the value of the vacuum mixing angle 𝜃, the mixing in matter is maximal; that is, 𝜃^𝑚=𝜋/4. This resonance occurs for neutrinos (antineutrinos) if Δ𝑚² is positive (negative). It suggests that the matter potential modifies the oscillation probability differently depending on the sign of Δ𝑚². Following (13), the resonance energy can be expressed as 𝐸_res=10.83GeVΔ𝑚²2.4×10⁻³eV²⋅cos2𝜃⋅0.962.8g/cm³𝜌. (14) When neutrino travels through the upper part of the Earth mantle with 𝜌=2.8g/cm³, the resonance occurs at roughly 10.8 GeV for positive Δ𝑚² of 2.4×10⁻³eV². The MSW potential arises due to matter and not antimatter and this fact is responsible for the observed asymmetry between neutrino and antineutrino oscillation probabilities even in the two-neutrino case. In three-flavor scenario, besides the genuine CP asymmetry caused by the CP phase 𝛿_CP, we also have fake CP asymmetry induced by matter which causes hindrances in extracting the information on 𝛿_CP.

4. Global Status of Oscillation Parameters and Missing Links

Oscillation data cannot predict the lowest neutrino mass. However, it can be probed in tritium beta decay [84] or neutrinoless double beta decay [85] processes. We can also make an estimation of the lowest neutrino mass from the contribution of neutrinos to the energy density of the universe [86]. Very recent measurements from the Planck experiment in combination with the WMAP polarization and baryon acoustic oscillation data have set an upper bound over the sum of all the neutrino mass eigenvalues of ∑𝑚_𝑖≤0.23 eV at 95% C.L. [87]. But oscillation experiments are sensitive to the values of two independent mass-squared differences: Δ𝑚²₂₁ and Δ𝑚²₃₁. Recent global fit [27] of all the available neutrino oscillation data in three-flavor framework is given as best-fit Δ𝑚²₂₁=7.5×10⁻⁵eV² and |Δ𝑚²₃₁|=2.4×10⁻³eV² with the relative 1𝜎 precision of 2.4% and 2.8%, respectively. The atmospheric mass splitting is 32 times larger than the solar mass splitting, showing the smallness of the ratio 𝛼=Δ𝑚²₂₁/Δ𝑚²₃₁≃0.03. At present, the 3𝜎 allowed range for Δ𝑚²₂₁ is 7.0×10⁻⁵eV²→8.1×10⁻⁵eV² and the same for |Δ𝑚²₃₁| is 2.2×10⁻³eV²→2.7×10⁻⁵eV². Δ𝑚²₂₁ is required to be positive to explain the observed energy dependence of the electron neutrino survival probability in solar neutrino experiments, but Δ𝑚²₃₁ is allowed to be either positive or negative by the present oscillation data. Hence, two patterns of neutrino masses are possible: 𝑚₃>𝑚₂>𝑚₁, called NH where Δ𝑚²₃₁ is positive and 𝑚₂>𝑚₁>𝑚₃, called IH where Δ𝑚²₃₁ is negative. Determining the sign of Δ𝑚²₃₁ is one of the prime goals of the current- and next-generation long-baseline neutrino oscillation experiments.

As far as the mixing angles are concerned, the solar neutrino mixing angle 𝜃₁₂ is now pretty well determined with a best-fit value of sin²𝜃₁₂=0.3 and the relative 1𝜎 precision on this parameter is 4%. The 3𝜎 allowed range for this parameter is 0.27→0.35. The smallest lepton mixing angle 𝜃₁₃ has been discovered very recently with a moderately large best-fit value of sin²𝜃₁₃=0.023. The relative 1𝜎 precision achieved on this parameter is also quite remarkable which is around 10%. The error in the measurement of sin²𝜃₁₃ lies in the range of 0.016→0.029 at 3𝜎 confidence level. The uncertainty associated with the choice of reactor 𝜈_𝑒 fluxes [88–90] and its impact on the determination of 𝜃₁₃ have been discussed in detail in Section 3 of [27]. Here, we would like to emphasize on that fact that the values of 𝜃₁₃ measured by the recent reactor and accelerator experiments are consistent with each other within errors, providing an important verification of the framework of three-neutrino mixing. These recent onset of data from reactor and accelerator experiments will also enable us to explore the long expected complementarity between these two independent measurements [40, 91–93]. Our understanding of the 2-3 mixing angle 𝜃₂₃ has also been refined a lot in recent years. The best-fit values and ranges of 𝜃₂₃ obtained from the three recent global fits [25–27] are listed in Table 1. A common feature that has emerged from all the three global fits is that we now have hint for nonmaximal 𝜃₂₃, giving two degenerate solutions: either 𝜃₂₃ belongs to the LO (sin²𝜃₂₃≈0.4) or it lies in the HO (sin²𝜃₂₃≈0.6). This octant ambiguity of 𝜃₂₃, in principal, can be resolved with the help of 𝜈_𝜇↔𝜈_𝑒 oscillation data. The preferred value would depend on the choice of the neutrino mass ordering. However, as can be seen from Table 1, the fits of [25] do not agree on which value should be preferred, even when the mass ordering is fixed to be NH. LO is preferred over HO for both NH and IH in [26]. Reference [27] marginalizes over the mass ordering, so the degeneracy remains. The global best-fits in [25, 27] do not see any sensitivity to the octant of 𝜃₂₃ unless they add the information from the atmospheric neutrinos. But, in [26], they do find a preference for LO even without adding the atmospheric neutrino data. At present, the relative 1𝜎 precision on sin²𝜃₂₃ is around 11%. Further improvement in the measurement of 𝜃₂₃ and settling the issue of its octant (if it turns out to be nonmaximal) are also the crucial issues that need to be addressed in current- and next-generation long-baseline experiments. Leptonic CP violation can be established if CP violating phase 𝛿_CP is shown to differ from 0 and 180^∘. We have not seen any signal for CP violation in the data so far. Thus, 𝛿_CP can have any value in the range [−180^∘,180^∘]. Measuring the value of 𝛿_CP and establishing the CP violation in the neutral lepton sector would be the top most priorities for the present and future long-baseline experiments.

Table 1 

1𝜎 bounds on sin²𝜃₂₃ from the global fits performed in [25–27]. The numbers cited from [27] have been obtained by keeping the reactor fluxes free in the fit and also including the short-baseline reactor data with 𝐿≲100 m, with the mass hierarchy marginalized. This table has been taken from [71].

Due to the fact that both 𝛼 and 𝜃₁₃ are small, so far, it was possible to analyze the data from each neutrino oscillation experiment adopting an appropriate, effective two-flavor oscillation approach. This method has been quite successful in measuring the solar and atmospheric neutrino parameters. The next step must involve probing the full three-flavor effects, including the subleading ones which are proportional to 𝛼. These are the key requirements to discover neutrino mass hierarchy, CP violation, and octant of 𝜃₂₃ in long-baseline experiments [94, 95].

5. Three-Flavor Effects in 𝜈_𝜇→𝜈_𝑒 Oscillation Channel

To illustrate the impact of three-flavor effects, the most relevant oscillation channels are 𝜈_𝜇→𝜈_𝑒 and 𝜈_𝜇→𝜈_𝑒. A study of these oscillation channels at long-baseline superbeam experiments is capable of addressing all the three major issues discussed in the previous section. In particular, the use of an appearance channel in which the neutrino changes flavor between production and detection is mandatory to explore CP violation in neutrino oscillations. Earth’s matter effects are also going to play a significant role in probing these fundamental unknowns. The exact expressions of the three-flavor oscillation probabilities including matter effects are very complicated. Therefore, to demonstrate the nature of neutrino oscillations as a function of baseline and/or neutrino energy, it is quite useful to have an approximate analytic expression for 𝑃_𝜇𝑒 (the 𝑇-conjugate of 𝑃_𝑒𝜇) in matter [80, 82, 96], keeping terms only up to second order in the small quantities 𝜃₁₃ and 𝛼 [97–99]:𝑃_𝜇𝑒≃sin²𝜃₂₃sin²2𝜃₁₃sin²1−𝐴Δ1−²_𝐶0𝐴+𝛼²cos²𝜃₂₃sin²2𝜃₁₂sin²(𝐴Δ)²_𝐶𝐴₁∓𝛼sin2𝜃₁₃cos𝜃₁₃sin2𝜃₁₂sin2𝜃₂₃sin(Δ)sin𝐴Δ𝐴sin1−𝐴Δ1−_𝐶−𝐴sin𝛿_CP+𝛼sin2𝜃₁₃cos𝜃₁₃sin2𝜃₁₂sin2𝜃₂₃cos(Δ)sin(𝐴Δ)𝐴sin[(1−𝐴)Δ](1−𝐴)_𝐶+cos𝛿_CP, (15)

where Δ≡Δ𝑚²₃₁𝐿,𝐴4𝐸𝐴≡Δ𝑚²₃₁. (16) Equation (15) has been derived under the constant matter density approximation. The matter effect is expressed by the dimensionless quantity 𝐴. The “−” sign which precedes the term 𝐶₋ refers to neutrinos whereas the “+” refers to to antineutrinos. In (15), 𝛼, Δ, and 𝐴 are sensitive to the sign of Δ𝑚²₃₁, that is, the type of the neutrino mass ordering. Note that the sign of 𝐴 changes with the sign of Δ𝑚²₃₁ as well as in going from neutrino to the corresponding antineutrino mode. The former suggests that the matter effect can be utilized to determine the sign of Δ𝑚²₃₁, while the latter implies that it can mimic a CP violating effect and hence complicate the extraction of 𝛿_CP by comparing neutrino and antineutrino data. For large 𝜃₁₃, the first term of (15) (𝐶₀) dominates and it contains the largest Earth matter effect which can therefore be used to measure the sign of Δ𝑚²₃₁. This term also depends on sin²𝜃₂₃ and therefore is sensitive to the octant of 𝜃₂₃. The subdominant terms 𝐶₋ and 𝐶₊ are suppressed by 𝛼 and provide information on 𝛿_CP. The term 𝐶₋ is the CP-violating part. The term 𝐶₊, although 𝛿_CP-dependent, is CP-conserving. The term 𝐶₁ is independent of both 𝜃₁₃ and 𝛿_CP and depends mainly on the solar parameters, Δ𝑚²₂₁ and 𝜃₁₂.

5.1. Hierarchy-δ_CP Degeneracy

Since the hierarchy and 𝛿_CP are both unknown, the interplay of the terms 𝐶₀, 𝐶₋, and 𝐶₊ in (15) gives rise to hierarchy-𝛿_CP degeneracy [39]. This degeneracy can be broken completely using the large Earth matter effects provided by the baselines which are >1000 km [41, 100, 101]. For these long baselines, we can also observe both the first and second oscillation maxima quite efficiently using the detectors like Liquid Argon Time Projection Chamber (LArTPC) [102]. It helps to evade the problem of (sgn(Δ𝑚²₃₁),𝛿_CP) degeneracy [39] and (𝜃₁₃,𝛿_CP) intrinsic degeneracy [38] which can cause the 𝜋-transit [103] effect even for large values of sin²2𝜃₁₃. Adding data from two different experiments with different baselines can also be very useful to resolve this degeneracy [37, 39, 104–107]. Another elegant way to tackle these degeneracies is to kill the spurious clone solutions at the “magic” baseline [36, 73, 108, 109]. When sin(𝐴Δ)=0, the last three terms in (15) drop out and the 𝛿_CP dependence disappears from the 𝑃_𝜇𝑒 channel, which provides a clean ground for 𝜃₁₃ and sgn(Δ𝑚²₃₁) measurements. Since 𝐴Δ=±(2√2𝐺_𝐹𝑛_𝑒𝐿)/4 by definition, the first nontrivial solution for the condition, sin(𝐴Δ)=0 reduces to 𝜌𝐿=√2𝜋/𝐺_𝐹𝑌_𝑒. This gives (𝜌/[g/𝑐𝑐])(𝐿/[km])≃32725, which for the PREM density profile of the Earth is satisfied for the “magic baseline,” 𝐿_magic≃7690km. The CERN to India-Based Neutrino Observatory (INO) [110] distance corresponds to 𝐿=7360 km, which is tantalizingly close to this “magic” baseline. Performing a long-baseline experiment at “Bimagic” baseline can be also very promising to suppress the effect of these degeneracies [111, 112]. Note that the low-order expansion of the probability 𝑃_𝜇𝑒 given by (15) is valid only for values of 𝐸 and Earth matter density 𝜌 (and hence 𝐿) where flavor oscillations are far from resonance; that is, 𝐴≪1. In the limit 𝐴∼1, one can check that even though the analytic expression for 𝑃_𝜇𝑒 given by (15) remains finite, the resultant probability obtained is incorrect [113, 114]. While we will use this analytical formula to explain our results in some cases, all the simulations presented in this review article are based on the full three-flavor neutrino oscillation probabilities in matter, using the Preliminary Reference Earth Model (PREM) [115].

The transition probability 𝑃_𝜇𝑒 as a function of the neutrino energy is shown in Figure 2. We allow 𝛿_CP to vary within the range −180^∘ to 180^∘ and the resultant probability is shown as a band, with the thickness of the band reflecting the effect of 𝛿_CP on 𝑃_𝜇𝑒. Inside each band, the probability for 𝛿_CP=90^∘ (𝛿_CP=−90^∘) case is shown explicitly by the solid (dashed) line. In each panel, the blue (red) band is for NH (IH). Left panel (right panel) depicts the probability for neutrino (antineutrino). In upper panels, we take the baseline of 295 km which matches with the distance of Tokai-to-Kamioka (T2K) experiment [116, 117] in Japan. In lower panels, we consider the baseline of 810 km which is the distance between Fermilab and Ash River, chosen for the NuMI Off-Axis Neutrino Appearance (NO𝜈A) experiment [118–120] in the United States. Matter effect increases 𝑃(𝜈_𝜇→𝜈_𝑒) for NH and decreases it for IH and vice versa for 𝑃(𝜈_𝜇→𝜈_𝑒). For 𝛿_CP in the lower half-plane ((LHP), −180^∘≤𝛿_CP≤0), 𝑃(𝜈_𝜇→𝜈_𝑒) is larger and for 𝛿_CP in the upper half-plane ((UHP), 0≤𝛿_CP≤180^∘), 𝑃(𝜈_𝜇→𝜈_𝑒) is smaller. Hence, for the combination (NH, LHP), the values of 𝑃(𝜈_𝜇→𝜈_𝑒) are much higher than those for IH (and 𝑃(𝜈_𝜇→𝜈_𝑒) values are much lower). Similarly, for the combination (IH, UHP), the values of 𝑃(𝜈_𝜇→𝜈_𝑒) are much lower than those of NH (and 𝑃(𝜈_𝜇→𝜈_𝑒) values are much higher). Thus, LHP is the favorable half-plane for NH and UHP is favorable for IH [72, 121]. For T2K baseline, the matter effect is very small and therefore the 𝛿_CP bands drawn for NH and IH overlap for almost entire range of 𝛿_CP (except for the most favorable combinations like: NH, 𝛿_CP=−90^∘ and IH, 𝛿_CP=90^∘) for almost all the choices of 𝐸. NO𝜈A has better chances to discriminate between NH and IH compared to T2K because of its larger baseline causing larger matter effect. But for the unfavorable combinations like NH, 𝛿_CP=90^∘ and IH, 𝛿_CP=−90^∘, the NH and IH bands still overlap with each other. In the upper panels of Figure 3, we study the same for the Long-Baseline Neutrino experiment (LBNE) [122–125] baseline of 1300 km which is the distance between the Fermilab and the Homestake mine in South Dakota in the United States. The lower panels of Figure 3 depict the hierarchy-𝛿_CP degeneracy pattern for the Long-Baseline Neutrino Oscillation experiment (LBNO) [126–130] baseline of 2290 km which is the distance between the CERN and the Pyhäsalmi mine in Finland. For both the LBNE and LBNO baselines, the matter effects are substantial and they break the hierarchy-𝛿_CP degeneracy completely.

(a)

(b)

Figure 2 

The transition probability 𝑃𝜇𝑒 as a function of neutrino energy. The band reflects the effect of unknown 𝛿CP. Inside each band, the probability for 𝛿CP=90∘ (𝛿CP=−90∘) case is shown by the solid (dashed) line. The blue (red) band is for NH (IH). The left panel (right panel) is for 𝜈 (𝜈). The upper panels are drawn for the T2K baseline of 295 km. The lower panels are for the NO𝜈A baseline of 810 km. Here, we take sin22𝜃13=0.089 and sin2𝜃23=0.5.

(a)

(b)

Figure 3 

𝑃𝜇𝑒 as a function of neutrino energy. Here, the bands correspond to different values of 𝛿CP from −180∘ to 180∘. Inside each band, the probability for 𝛿CP=90∘ (𝛿CP=−90∘) case is shown by the solid (dashed) line. The blue (red) band is for NH (IH). The left panel (right panel) is for 𝜈 (𝜈). The upper panels are drawn for the LBNE baseline of 1300 km. The lower panels are for the LBNO baseline of 2290 km. Here, we take sin22𝜃13=0.089 and sin2𝜃23=0.5.

5.2. Octant-δ_CP Degeneracy

There is a similar octant-𝛿_CP degeneracy also in the 𝑃_𝜇𝑒 channel, which limits our ability to determine the correct octant of 𝜃₂₃ [35]. The upper left (right) panel of Figure 4 shows 𝑃_𝜇𝑒 versus 𝐸_𝜈 (𝑃_𝜇𝑒 versus 𝐸_𝜈) for all possible values of 𝛿_CP and for the two different values of sin²𝜃₂₃, assuming NH to be the true hierarchy. These plots are drawn for the T2K experiment. The lower panels show the same for the NO𝜈A baseline. As can be seen from the upper and lower left panels of Figure 4, for neutrino data, the two octant bands overlap for some values of 𝛿_CP and are distinct for other values. The combinations of octant and 𝛿_CP which lie farthest from overlap will be favorable combinations for octant determination. For example, LO and 𝛿_CP of 90^∘ and HO and 𝛿_CP of −90^∘ form the favorable combinations. For the combinations with overlap, HO and 𝛿_CP of 90^∘ and LO and 𝛿_CP of −90^∘, it is impossible to determine octant using neutrino data alone. However, as we see from the upper and lower right panels, these unfavorable combinations for neutrino case are the favorable ones for the antineutrino case. Thus, a combination of neutrino and antineutrino data will have a better capability to determine octant compared to neutrino data alone. This is in contrast to the hierarchy-𝛿_CP degeneracy, where for a given hierarchy, the favorable 𝛿_CP region is the same for both neutrino and antineutrino. Thus, we draw the conclusion that a balanced neutrino and antineutrino data is imperative for resolving the octant ambiguity of 𝜃₂₃ for all values of 𝛿_CP [71]. The octant-𝛿_CP degeneracy pattern for the LBNE (LBNO) experiment can be seen from the upper (lower) panels of Figure 5.

(a)

(b)

Figure 4 

𝑃𝜇𝑒 as a function of neutrino energy. Here, the bands correspond to different values of 𝛿CP ranging from −180∘ to 180∘. Inside each band, the probability for 𝛿CP=90∘ (𝛿CP=−90∘) case is shown by the solid (dashed) line. The red (blue) band is for HO with sin2𝜃23=0.59 (LO with sin2𝜃23=0.41). The left panel (right panel) is for 𝜈 (𝜈). The upper panels are drawn for the T2K baseline of 295 km. The lower panels are for the NO𝜈A baseline of 810 km. Here, we consider sin22𝜃13=0.089 and NH.

(a)

(b)

Figure 5 

𝑃𝜇𝑒 as a function of neutrino energy. Here, the bands correspond to different values of 𝛿CP ranging from −180∘ to 180∘. Inside each band, the probability for 𝛿CP=90∘ (𝛿CP=−90∘) case is shown by the solid (dashed) line. The red (blue) band is for HO with sin2𝜃23=0.59 (LO with sin2𝜃23=0.41). The left panel (right panel) is for 𝜈 (𝜈). The upper panels are drawn for the LBNE baseline of 1300 km. The lower panels are for the LBNO baseline of 2290 km. Here, we consider sin22𝜃13=0.089 and NH.

6. Present-Generation Beam Experiments: T2K and NO𝝂A

With the aim to unravel the 𝜃₁₃-driven 𝜈_𝜇→𝜈_𝑒 appearance oscillation, the T2K experiment [116, 117] started its journey in 2010 and the NO𝜈A experiment [118–120] in the United States is now under construction and will start taking data near the end of this year. The detection of electron neutrino appearance in a 𝜈_𝜇 beam is the prime goal of these experiments and their experimental setups are optimized to achieve this target. Both the T2K and NO𝜈A experiments use the classic off-axis beam technique [131] that delivers a narrow peak in the energy spectrum, tuned to be at the expected oscillation maximum. Furthermore, this off-axis technology helps to reduce the background coming from the intrinsic 𝜈_𝑒 contamination in the beam and a smaller fraction of high-energy tails reduces the background coming from neutral current events. As a result, it improves the signal-to-background ratio a lot. With the recent discovery of a moderately large value of 𝜃₁₃, these current-generation experiments are now poised to probe the impact of full three-flavor effects to discover neutrino mass hierarchy, CP violation, and octant of 𝜃₂₃. But to achieve these goals, they need to have very high proton beam powers of order 1 MW and detectors with huge fiducial masses (of order 10 kilotons) and therefore, these experiments are known as “superbeam” experiments. Next, we briefly describe the main features of the T2K and NO𝜈A experiments and then we present the physics reach of these experiments in light of the recently discovered moderately large value of 𝜃₁₃.

6.1. T2K

T2K uses the 50 kilotons Super-Kamiokande water Cherenkov detector (fiducial volume 22.5 kilotons) as the far detector for the neutrino beam from J-PARC. The detector is at a distance of 295 km from the source at an off-axis angle of 2.5^∘ [116]. The neutrino flux is peaked sharply at the first oscillation maximum of 0.6GeV. The experiment is scheduled to run for 5 years in the neutrino mode with a power of 0.75 MW. Because of the low energy of the peak flux, the neutral current backgrounds are small and they can be rejected based on energy cut. The signal efficiency is 87%. To estimate the physics sensitivity, the background information and other details are taken from [132, 133].

6.2. NO𝜈A

NO𝜈A is a 14 kilotons totally active scintillator detector (TASD) placed at a distance of 810 km from Fermilab, at a location which is 0.8^∘ off-axis from the NuMI beam. Because of the off-axis location, the flux of the neutrinos is reduced but is sharply peaked around 2 GeV, again close to the first oscillation maximum energy of 1.7 GeV in 𝑃(𝜈_𝜇→𝜈_𝑒). The most problematic background in NO𝜈A experiment is neutral current interactions which mostly consist in the single 𝜋⁰ production. However, the measured energy of this background is shifted to values of energy below the region where the flux is significant. Hence, this background can be rejected using a simple kinematic cut. The experiment is scheduled to have three-year run in neutrino mode first and then later, three-year run in antineutrino mode as well with a NuMI beam power of 0.7 MW, corresponding to 6×10²⁰ protons on target per year. The details of the experiment are given in [120]. In light of the recent measurement of large 𝜃₁₃, NO𝜈A has reoptimized its event selection criteria. Relaxing the cuts, they now allow more events in both signal and background. Additional neutral current backgrounds are reconstructed at lower energies and can be managed by a kinematical cut. In our calculations, we use these reoptimized values of signal and background, the details of which are given in [72, 134].

6.3. Mass Ordering and CP Violation Discovery

In this section, we describe the capabilities of the T2K and NO𝜈A experiments for the determination of mass hierarchy and CP violation. We use GLoBES [135, 136] software to simulate the data for these experiments. For the atmospheric/accelerator neutrino parameters, we take the following central (true) values: Δ𝑚²_eff=2.4⋅10⁻³eV²,sin²2𝜃₂₃=1.0, (17) where Δ𝑚²_eff is the effective mass-squared difference measured by the accelerator experiments in 𝜈_𝜇→𝜈_𝜇 disappearance channel [32, 33]. It is related to the Δ𝑚²₃₁ (larger) and Δ𝑚²₂₁ (smaller) mass-squared differences through the expression [137, 138] Δ𝑚²_eff=Δ𝑚²₃₁−Δ𝑚²₂₁×cos²𝜃₁₂−cos𝛿_CPsin𝜃₁₃sin2𝜃₁₂tan𝜃₂₃. (18) The value of Δ𝑚²₃₁ is calculated separately for NH and for IH using this equation where Δ𝑚²_eff is taken to be +ve for NH and −ve for IH. For 𝜃₂₃, we take the maximal mixing as still favored by the Super-Kamiokande atmospheric data [34, 139, 140]. For 𝜃₁₃, we take the best-fit value of sin²𝜃₁₃=0.026. The uncertainties in the above parameters are taken to be 𝜎(sin²𝜃₁₃)=13% [25], 𝜎(|Δ𝑚²_eff|)=4%, and 𝜎(sin²2𝜃₂₃)=2% [116]. In the calculation, these pieces of information are included in the form of priors. In our 𝜒² fit, we marginalize over all oscillation parameters within their ±3𝜎 ranges, as well as the mass hierarchy, by allowing these parameters to vary in the fit and picking the smallest value of the 𝜒² function. We take the solar parameters to be Δ𝑚²₂₁=7.62⋅10⁻⁵eV²,sin²𝜃₁₂=0.32. (19) We keep the solar parameters to be fixed throughout the calculation because varying them will have negligible effect. We also take the Earth matter density to be a constant 2.8 g/cm³ because the variations and the uncertainties in density can be neglected for the T2K and NO𝜈A baselines. While calculating the sensitivity for T2K, we include a 2% systematic error on appearance signal events and a (uncorrelated) 5% systematic error on backgrounds. For NO𝜈A, we have assumed 5% systematic error on appearance signal events and a (uncorrelated) 10% systematic error on background events. These pieces of information on the systematic errors are included in the 𝜒² function using the pull method as described in, for example, [103, 141]. In our definition of the 𝜒² function, we have assumed that the neutrino and antineutrino channels are completely uncorrelated, all the energy bins for a given channel are fully correlated, and the systematic errors on signal and background are fully uncorrelated. We perform the usual 𝜒² analysis using a Poissonian likelihood function adding the information coming from 𝜈_𝑒 appearance and 𝜈_𝜇 disappearance channels.

For long-baseline experiments, the measurement of the mass hierarchy is easier than a measurement of 𝛿_CP because matter effects enhance the separation between the oscillation spectra, and therefore the event rates, of a NH and an IH. Additionally, this measurement is one that is “discrete” as we only need to differentiate between two possibilities. A “discovery” of the mass hierarchy is defined as the ability to exclude any degenerate solution for the wrong (fit) hierarchy at a given confidence level. A “discovery” of CP violation, if it exists, means being able to exclude the CP-conserving values of 0^∘,180^∘ at a given confidence level. Clearly, this measurement becomes very difficult for the 𝛿_CPvalues which are closer to 0^∘, 180^∘. Therefore, whilst it is possible to discover the mass hierarchy for all possible values of 𝛿_CP, the same is not true for CP violation.

In Figure 6, we plot the hierarchy discrimination sensitivity of the old NO𝜈A, the new NO𝜈A (with reoptimized event selection criteria for large 𝜃₁₃), and the combined sensitivity of new NO𝜈A and T2K, as a function of the true value of 𝛿_CP. In the left (right) panel, we have assumed NH (IH) to be the true hierarchy. We see that the wrong hierarchy can be ruled out very effectively for 𝛿_CP in the favorable half-plane, which is LHP (UHP) for NH (IH). The new event selection criteria of NO𝜈A make the experiment even more effective in ruling out the wrong hierarchy for 𝛿_CP in the favorable half-plane. In the unfavorable half-plane, both the old and the new criteria are equally ineffective. However, the addition of T2K data improves the situation significantly and Δ𝜒² increases from 0 to ≥2 for all the true values of 𝛿_CP, thus making it possible to get a 90% CL hint of hierarchy with some additional data. We have checked that a further increment in the exposure of T2K or addition of antineutrino data from T2K does not improve the hierarchy sensitivity much.

(a)

(b)

Figure 6 

Left panel (right panel) shows the Δ𝜒2 for the mass hierarchy discovery as a function of true value of 𝛿CP assuming NH (IH) as true hierarchy. This figure has been taken from [72].

The prospects of determining the neutrino mass hierarchy with the combined data from T2K, NO𝜈A, Double Chooz, RENO, Daya Bay, and the atmospheric neutrino experiment ICAL@INO [110] have been studied in detail in [143]. With 10 years of atmospheric ICAL@INO data collected by 50 kilotons magnetized iron calorimeter detector combined with T2K, NOvA, and reactor data, a 2.3𝜎–5.7𝜎 discovery of the neutrino mass hierarchy could be achieved depending on the true values of sin²𝜃₂₃[0.4--0.6], sin²2𝜃₁₃[0.08--0.12], and 𝛿_CP[0--2𝜋] [143].

Reoptimization of the event selection criteria of NO𝜈A has the most dramatic effect on the CP violation discovery potential of the experiment. In Figure 7, we plot the sensitivity to rule out the CP conserving scenarios, as a function of true 𝛿_CP in the left (right) panel for NH (IH) being the true hierarchy. We notice that, while in the case of old NO𝜈A there is no CP violation sensitivity at all at 90% CL, there is such a sensitivity in new NO𝜈A, for about one-third fraction of the favorable half-plane. Addition of T2K data leads to CP violation sensitivity for about half the region in both favorable half planes at 90% confidence level. It can be shown that, T2K by itself, has no CP violation sensitivity. But the synergistic combination of NO𝜈A and T2K leads to much better CP violation sensitivity compared to the individual capabilities. Here, we would like to mention that a large value of 𝜃₁₃ always does not help for CP violation discovery. As 𝜃₁₃ becomes large, the number of electron appearance event increases, reducing the statistical error. However, the large atmospheric term acts as a background in the measurement of CP phase. In fact, the CP asymmetry term is proportional to 1/sin2𝜃₁₃ [144, 145]. These two contradictory issues make the measurement of CP phase quite complicated. The systematic uncertainties are also going to play a crucial role for CP violation discovery in light of large 𝜃₁₃ [146].

(a)

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Figure 7 

Left panel (right panel) shows the Δ𝜒2 for the CP violation discovery as a function of true value of 𝛿CP assuming NH (IH) as true hierarchy. This figure has been taken from [72].

A summary of our results is given in Table 2 in terms of the fraction of 𝛿_CP values for which mass hierarchy can be determined/CP violation can be detected. Please note that in deriving the results given in Table 2, we have considered the best-fit value of sin²𝜃₁₃=0.026 and maximal mixing for 𝜃₂₃. In Table 3, we present the same for the rather conservative choices of neutrino mixing angles: sin²𝜃₁₃=0.023 (the best-fit value suggested by the Daya Bay experiment) and sin²𝜃₂₃=0.413 (the LO value of sin²𝜃₂₃ as indicated by the recent MINOS accelerator data).

Table 2 

Fractions of true values of 𝛿_CP for which a discovery is possible for mass hierarchy and CP violation. The numbers without (with) parentheses correspond to 90% (95%) CL. Here we take the central values: sin²𝜃₁₃=0.026 and sin²𝜃₂₃=0.5. The results are shown for both NH and IH as true hierarchy. This table has been taken from [72].

Table 3 

Fractions of true values of 𝛿_CP for which a discovery is possible for mass hierarchy and CP violation. The numbers without (with) parentheses correspond to 90% (95%) CL. Here we take the central value for sin²𝜃₁₃ to be 0.023 as predicted by Daya Bay. For sin²𝜃₂₃, the best-fit value that we consider is 0.413. The results are shown for both NH and IH as true hierarchy. This table has been taken from [72].

In [72], we further explore the improvement in the hierarchy and CP violation sensitivities for T2K and NO𝜈A due to the addition of a 10 kilotons LArTPC placed close to NO𝜈A site and exposed to the NuMI beam during NO𝜈A running. It is expected, of course, that such a detector will come on line much later than NO𝜈A. The capabilities of such a detector are equivalent to those of NO𝜈A in all aspects. We find that combined data from 10  kilotons LArTPC (3 years of 𝜈 + 3 years of 𝜈 run), NO𝜈A (6 years of 𝜈 + 6 years of 𝜈 run), and T2K (5 years of 𝜈 run) can give a close to 2𝜎 hint of hierarchy discovery for all values of 𝛿_CP. With this combined data, we can achieve CP violation discovery at 95% CL for roughly 60% values of 𝛿_CP. A similar proposal considering 6 kilotons LArTPC has been studied recently in detail in [147].

6.4. Resolving the Octant Ambiguity of 𝜃₂₃

In this section, we study whether the expected appearance data from the ongoing T2K experiment and the upcoming NO𝜈A experiment can resolve the octant ambiguity of 𝜃₂₃ or not.

In Figure 8, we show neutrino events versus antineutrino events for various octant-hierarchy combinations. In each case, with varying values of 𝛿_CP, the plot becomes an ellipse. The left panel depicts these ellipses for T2K whereas the right panel shows the same for NO𝜈A. Here, we assume that T2K will have equal 𝜈 and 𝜈 runs of 2.5 years each. In the right panel, we see that the ellipses for the two mass orderings overlap whereas the ellipses of LO are well separated from those of HO. Hence, we can expect that NO𝜈A will have better octant resolution capability than hierarchy discrimination. This situation is even more dramatic in the left panel where there is large overlap between the two hierarchies but clear separation between the octants. Thus, it is very likely that antineutrino data from T2K may play an important role in the determination of octant.

(a)

(b)

Figure 8 

Neutrino and antineutrino appearance events for all possible combinations of hierarchy, octant, and 𝛿CP. The left (right) panel is for T2K (NO𝜈A). Here, sin22𝜃13=0.089. For LO (HO), sin2𝜃23=0.41(0.59). Note that for T2K, equal 𝜈 and 𝜈 runs of 2.5 years each have been assumed. This figure has been taken from [71].

In Figures 9 and 10, we study the behavior of Δ𝜒² between the true and the wrong octants as a function of true 𝛿_CP. Here, the Δ𝜒² is estimated in the following way. First, we fix the true value of 𝛿_CP. We take sin²𝜃₂₃ to be its best-fit value in the true octant: 0.41 for LO and 0.59 for HO. If the LO (HO) is the true octant, the test values of sin²𝜃₂₃ in the HO (LO) are varied within the range [0.5,0.63] ([0.36,0.5]), where 0.63 (0.36) is the 2𝜎 upper (lower) limit of the allowed range of sin²𝜃₂₃. The Δ𝜒² is computed between the spectra with the best-fit sin²𝜃₂₃ of the true octant and that with various test values in the wrong octant and is marginalized over other neutrino parameters, especially the hierarchy, sin²2𝜃₁₃, and 𝛿_CP. Figures 9 and 10 portray the minimum of this Δ𝜒² versus the true value of 𝛿_CP.

(a)

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Figure 9 

Octant resolving capability as a function of true 𝛿CP for various setups. In these plots, LO is assumed to be the true octant. The left (right) panel corresponds to NH (IH) being the true hierarchy. This figure has been taken from [71].

(a)

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Figure 10 

Octant resolving capability as a function of true 𝛿CP for various setups. In these plots, HO is assumed to be the true octant. The left (right) panel corresponds to NH (IH) being the true hierarchy. This figure has been taken from [71].

From Figure 9, we observe that the NO𝜈A data by itself can almost rule out the wrong octant at 2𝜎, if LO is the true octant. If HO is the true octant, then NO𝜈A data is not sufficient to rule out the wrong octant as can be seen from Figure 10. In fact, the wrong octant can be ruled out only for about half of the true 𝛿_CP values. As illustrated in Figures 9 and 10, addition of T2K data improves the octant determination ability significantly. From Figure 9, we see that the combined data from NO𝜈A and T2K (5-year 𝜈 run) give a 2𝜎 octant resolution for all values of true 𝛿_CP if LO is the true octant. From Figure 10, we see that this combined data can rule out the wrong octant at 2𝜎 for HO-IH, but not for HO-NH. The problem of HO-NH can be solved if the T2K has equal 𝜈 and 𝜈 runs of 2.5 years each. This change improves the octant determination for the unfavorable values of true 𝛿_CP (where Δ𝜒² is minimum) for all four combinations of hierarchy and octant. In particular, for the case of HO-NH, it leads to a complete ruling out of the wrong octant at 2𝜎 for all values of true 𝛿_CP. Thus, balanced runs of T2K in 𝜈−𝜈 mode are preferred over a pure 𝜈 run because of better octant determination capability. We observe that this feature of LO being more favorable compared to HO is a consequence of marginalization over the oscillation parameters (mainly 𝛿_CP) and the systematic uncertainties. We checked that in the absence of any kind of marginalization Δ𝜒²_HO is consistently larger than Δ𝜒²_LO.

In the discussion so far, we have assumed the true values of sin²𝜃₂₃ to be 0.41 for LO and 0.59 for HO. These are, of course, the best-fit points from the global analyses. But, we must consider the octant resolution capability for values of true sin²𝜃₂₃ in the full allowed range (0.34 to 0.67). In Figure 11, we plot the 2𝜎 and 3𝜎 octant resolution contours in true sin²𝜃₂₃-true 𝛿_CP plane. Octant resolution is possible only for points lying outside the contours. These figures clearly show that octant resolution is possible at 2𝜎 for global best-fit points and at 3𝜎 for MINOS best-fit points. The results for the two hierarchies are quite similar. From Figure 11, we can see that if T2K experiment would have equal neutrino and antineutrino runs of 2.5 years each, a 2𝜎 resolution of the octant becomes possible provided sin²𝜃₂₃≤0.43or≥0.58 for any value of 𝛿_CP. In [148], the possibility of determining the octant of 𝜃₂₃ in the long-baseline experiments T2K and NO𝜈A in conjunction with future atmospheric neutrino detectors has been studied. The combined data from T2K and NO𝜈A can provide a ∼ 2% precision on sin²𝜃₂₃ at 2𝜎 using the information coming from the disappearance channel [71].

(a)

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Figure 11 

Octant resolving capability in the true sin2𝜃23-true 𝛿CP plane for the combined 3𝜈+3𝜈 runs of NO𝜈A and 2.5𝜈+2.5𝜈 runs of T2K. Both 2𝜎 and 3𝜎 CL contours are plotted. The vertical lines correspond to the best-fit values of global data and those of MINOS accelerator data. The left (right) panel corresponds to NH (IH) being the true hierarchy. This figure has been taken from [71].

In this section, we have discussed in detail the physics reach of current generation long-baseline beam experiments: T2K and NO𝜈A to unravel the neutrino mass hierarchy, CP violation, and octant of 𝜃₂₃. Given their relatively short baselines, narrowband beams, and limited statistics, these experiments suffer a lot from the hierarchy-𝛿_CP and octant-𝛿_CP degeneracies. They can provide a hint for these unknown issues only for favorable ranges of parameters at limited confidence level. Hence, new long-baseline experiments with intense neutrino beam sources and advanced detector technologies are mandatory [100, 149–151] to fathom the hitherto uncharted parameter space of the neutrino mixing matrix well beyond the capabilities of T2K and NO𝜈A.

7. Next Generation Long-Baseline Beam Experiments

In this section, we briefly review the possible options for future high-precision long-baseline beam experiments with a special emphasis on high-power superbeam facilities using liquid argon and water Cherenkov detectors. As discussed in the literature (see, e.g., [149, 152]), the ability of future long-baseline neutrino experiments to discover mass hierarchy, octant of 𝜃₂₃, and CP violation depends on the achievable event statistics and hence strongly on the value of 𝜃₁₃. The fact that 𝜃₁₃ is large will have a significant impact on the realization of future long-baseline neutrino oscillation experiments, for which planning till 2011 was focused on a staged approach to achieve sensitivity to increasingly smaller values of 𝜃₁₃. This approach was exemplified in the optimization of the neutrino factory [142, 153] and beta-beam [154, 155] experiments for which it was possible to discover a value of sin²2𝜃₁₃ as small as 10⁻⁴. However, following the recent discovery of a moderately large value of 𝜃₁₃, the focus of future optimizations will be on the possibility to explore mass hierarchy, octant of 𝜃₂₃, and CP violation for a given value of 𝜃₁₃. A relatively large value of 𝜃₁₃ also allows us to pursue an incremental program, staged in terms of the size of the experiment [100], producing significant new results at each stage. Out of the three major unknowns, the discovery of CP violation is the most toughest goal to achieve. Hence, the determination of mass hierarchy and octant should be considered as the first step towards the discovery of leptonic CP violation. Future facilities must be developed with the requirements that they should have the capability to determine hierarchy and octant at 3𝜎 confidence level or better for any possible value of 𝛿_CP during the first stage and discover CP violation and measure 𝛿_CP during the second stage. An incremental approach is also justified in view of the challenges (some of them are unknown) involved in operating very-high-power superbeams and in building giant underground neutrino detectors, which makes such an approach effectively safer and possibly more cost-effective. Both the proposed LBNE and LBNO facilities have adopted this staged approach in light of large 𝜃₁₃. First, we present a comparative study of the physics reach that can be achieved during the first phase of these two proposed experiments. Then, we talk about the proposals of J-PARC to Hyper-Kamiokande (T2HK) long-baseline superbeam experiment [156] and CERN to MEMPHYS (at Fréjus) SPL superbeam experiment [157–160]. We also discuss the possibility to explore leptonic CP violation based on the European Spallation Source (ESS) proton linac which can deliver very intense, cost-effective, and high-performance neutrino beam in parallel with the production of spallation neutron [161, 162]. Megaton-size water Cherenkov detector is one of the key components of these three experimental setups. Next, we discuss the physics prospects of Low-Energy Neutrino Factory (LENF) [163–166] in conjunction with magnetized iron detector (MIND) [142, 167, 168] which seems to be a very promising setup to explore leptonic CP violation for large 𝜃₁₃. Finally, we mention few proposals based on monoflavor beta-beam concept [154, 155, 169–171].

7.1. Discovery Reach of LBNE and LBNO

LBNE. The Long-Baseline Neutrino Experiment (LBNE) [125] is one of the major components of Fermilab’s intensity frontier program. In its first phase (LBNE10), it will have a new, high-intensity, on-axis neutrino beam directed towards a 10  kilotons LArTPC located at Homestake with a baseline of 1300 km. This facility is designed for initial operation at a proton beam power of 708kW, with proton energy of 120GeV that will deliver 6×10²⁰ protons on target in 230 days per calendar year. In our simulation, we have used the latest fluxes being considered by the collaboration, which have been estimated assuming the smaller decay pipe and the lower horn current compared to the previous studies [172]. We have assumed five years of neutrino run and five years of antineutrino run. The detector characteristics have been taken from Table 1 of [100]. To have the LArTPC cross-sections, we have scaled the inclusive charged current cross-sections of water by 1.06 (0.94) for the 𝜈 (𝜈) case [173, 174].

LBNO. The Long-Baseline Neutrino Oscillation experiment (LBNO) [130] plans to use an experimental setup where neutrinos produced in a conventional wide-band beam facility at CERN would be observed in a proposed 20  kilotons (in its first phase) LArTPC housed at the Pyhäsalmi mine in Finland, at a distance of 2290 km. The fluxes have been computed [175] assuming an exposure of 1.5×10²⁰ protons on target in 200 days per calendar year from the SPS accelerator at 400GeV with a beam power of 750kW. For LBNO also, we consider five years of neutrino run and five years of antineutrino run. We assume the same detector properties as those of LBNE.

Event Spectrum at LBNE10 and LBNO. Figure 12 portrays the expected signal and background event spectra in the 𝜈_𝑒 appearance channel as a function of reconstructed neutrino energy including the efficiency and background rejection capabilities for LBNE10 (left panel) and LBNO (right panel) setups. In both panels of Figure 12, one can clearly see a systematic downward bias in the reconstructed energy for neutral current background events due to the final state neutrino included using the migration matrices. The blue dot-dashed and the orange dotted vertical lines display the locations of the first and second oscillation maxima. The green double-dotted-dashed histogram shows the signal event rate. Although, we have some statistics around the second oscillation maximum for both the baselines, its impact is limited due to the fact that the event samples are highly contaminated with neutral current and other backgrounds at lower energies.

(a)

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Figure 12 

Expected signal and background event rates in the 𝜈𝑒 appearance channel as a function of the reconstructed neutrino energy including the efficiency and background rejection capabilities. Here, we consider sin22𝜃13=0.0975 and 𝛿CP=0∘. Left panel (right panel) is for LBNE10 (LBNO). A normal hierarchy has been assumed. In both panels, the blue dot-dashed and the orange dotted vertical lines display the locations of the first and second oscillation maxima.

Physics with Bievents Plots. In Figure 13, we have plotted 𝜈_𝑒 versus 𝜈_𝑒 appearance events, for LBNE10 and 0.5*LBNO for the four possible combinations of hierarchy and octant. Since 𝛿_CP is unknown, events are generated for [−180^∘,180^∘], leading to the ellipses. Here, we take sin²2𝜃₁₃=0.089. For the lower octant (LO) of 𝜃₂₃, the value sin²𝜃₂₃=0.41 is chosen and for higher octant (HO), it is taken to be 0.59. Note that in Figure 13 we have plotted the total number of events, whereas the actual analysis will be done based on the spectral information. Nevertheless, the contours in Figure 13 contain very important information regarding the physics capabilities of the experiments. An experiment can determine both the hierarchy and the octant, if every point on a given ellipse is well separated from every point on each of the other three ellipses. The larger the separation, the better the confidence with which the above parameters can be determined.

Figure 13 

Bievents (𝜈𝑒 and 𝜈𝑒 appearance) plots for the four octant-hierarchy combinations and all possible 𝛿CP values. The experiments considered are LBNE10 and 0.5*LBNO. Here, sin22𝜃13=0.089. For LO (HO), sin2𝜃23=0.41(0.59). This figure has been taken from [41].

For 0.5*LBNO, the two (LO/HO)-IH ellipses are well separated from the two (LO/HO)-NH ellipses, in their number of neutrino events. Hence, 0.5*LBNO has excellent hierarchy determination capability with just neutrino data. However, only neutrino data alone will not be sufficient to determine the octant in case of IH because various points on (LO/HO)-IH ellipses have the same number of neutrino events. Likewise, only antineutrino data cannot determine the octant in case of NH. Therefore, balanced neutrino and antineutrino data is mandatory to make an effective distinction between (LO/HO)-IH ellipses and also between (LO/HO)-NH ellipses.

For LBNE10, 𝜈 data alone cannot determine hierarchy because various points on LO-NH and HO-IH ellipses have the same number of 𝜈_𝑒 events. Thus, 𝜈 data is also needed. Even with 𝜈 data, hierarchy determination can be difficult to achieve, if nature chooses LO and one of the two worst-case combinations of hierarchy and 𝛿_CP which are (NH, 90^∘) or (IH, −90^∘). In such a situation, the 𝜈_𝑒 and 𝜈_𝑒 events are rather close to each other and it will be very difficult for LBNE10 to reject the wrong combination. Regarding octant determination, the capability of LBNE10 is very similar to that of 0.5*LBNO because the separations between the ellipses, belonging to LO and HO, are very similar for the two experiments.

Hierarchy and Octant Discovery with LBNE10 and LBNO. Measurement of hierarchy and octant should be considered as a prerequisite for the discovery of leptonic CP violation. To present the results for mass hierarchy and octant discovery, we consider three experimental setups: LBNE10, LBNO, and a possible LBNO configuration with a detector half the mass (10 kilotons), which we denote as 0.5*LBNO. Scaling down the detector size of LBNO by half makes the exposures of these two experiments very similar. Therefore, considering 0.5*LBNO enables us to make a direct comparison between the inherent properties of the two baselines involved. Both LBNE and LBNO will operate at multi-GeV energies with very long baselines. This will lead to a large enough matter effect to break the hierarchy-𝛿_CP degeneracy completely. They are also scheduled to have equal neutrino and antineutrino runs, tackling the octant-𝛿_CP degeneracy. These experiments are planning to use LArTPCs [102, 176] which have excellent kinematic reconstruction capability for all the observed particles. This feature helps in rejecting quite a large fraction of neutral current background.

We study the hierarchy discovery potential for two true values of sin²𝜃₂₃: 0.41 (LO) and 0.5 (MM). If HO is true, the results will be better than those for the case of MM. This gives us four true combinations of 𝜃₂₃-hierarchy: LO-NH, LO-IH, MM-NH, and MM-IH. Δ𝜒² is calculated for each of these four combinations, assuming the opposite hierarchy to be the test hierarchy. In the fit, we marginalize over test sin²𝜃₂₃ in its 3𝜎 range and Δ𝑚²₃₁ and sin²2𝜃₁₃ in their 2𝜎 ranges. We considered 5% uncertainty in the matter density, 𝜌. Priors were added for 𝜌 (𝜎=5%), Δ𝑚²₃₁ (𝜎=4%), and sin²2𝜃₁₃ (𝜎=5%, as expected by the end of Daya Bay’s run). Δ𝜒² is also marginalized over the uncorrelated systematic uncertainties (5% on signal and 5% on background) in the setups, so as to obtain a Δ𝜒²_min for every 𝛿_CP(true).

Figure 14 shows the discovery reach for hierarchy as a function of 𝛿_CP(true). We see that even 0.5*LBNO has ≳10𝜎 hierarchy discovery for all values of 𝛿_CP(true) and for all four 𝜃₂₃-hierarchy combinations. The potential of LBNO is even better. The LBNO baseline is close to Bimagic which gives it a particular advantage [111, 112]. For LBNE10, a 5𝜎 discovery of hierarchy is possible for only ∼50% of the 𝛿_CP(true), irrespective of the true 𝜃₂₃-hierarchy combination. For the unfavorable hierarchy-𝛿_CP combinations, that is, NH with 𝛿_CP in the upper half-plane or IH with 𝛿_CP in the lower half-plane, the performance of LBNE10 suffers. In particular, for LO and the worst-case combinations ((NH, 90^∘) and (IH, −90^∘)), LBNE10 will not be able to provide even a 3𝜎 hierarchy discrimination. Therefore, LBNE10 must increase their statistics, if NO𝜈A data indicate that the unfavorable combinations are true. The discovery potential for all three setups will be better if 𝜃₂₃ happens to lie in HO. But, we checked that, even then, a 5𝜎 discovery is not possible with LBNE10 for ∼30% of the upper half-plane of 𝛿_CP for HO-NH true and ∼70% of the lower half-plane of 𝛿_CP for HO-IH true.

Figure 14 

Hierarchy discovery reach for LBNO, 0.5*LBNO and LBNE10. Results are shown for the four possible true 𝜃23-hierarchy combinations.

We next consider the discovery reach of the same setups for excluding the wrong octant. We consider the true values of sin²𝜃₂₃=0.41(LO) and sin²𝜃₂₃=0.59(HO), so that we have the following four true combinations of octant and hierarchy: LO-NH, LO-IH, HO-NH, and HO-IH. Δ𝜒² is calculated for each of these four combinations, assuming test sin²𝜃₂₃ values from the other octant. For LO (HO) true, we consider the test sin²𝜃₂₃ range from 0.5 to 0.67 (0.34 to 0.5). The rest of the marginalization procedure (over oscillation parameters as well as systematic uncertainties) is the same as that in the case of hierarchy exclusion except with another difference: the final Δ𝜒² is marginalized over both the hierarchies as the test hierarchy to obtain Δ𝜒²_min.

Figure 15 shows the discovery reach for octant as a function of 𝛿_CP(true). It can be seen that for (LO/HO)-IH true, the sensitivities of LBNE10 and 0.5*LBNO are quite similar whereas they are somewhat better for 0.5*LBNO if (LO/HO)-NH are the true combinations. For LO-(NH/IH), both LBNE10 and 0.5*LBNO have more than 3𝜎 discovery of octant while for HO-(NH/IH), the Δ𝜒²_min varies from ∼6 to ∼11. However, with full LBNO, we have more than 3.5𝜎 discovery of octant for all octant-hierarchy combinations. A 5𝜎 discovery of octant is possible only for LO-NH true for 𝛿_CP(true)∈(∼20^∘,∼150^∘).

Figure 15 

Octant resolving capability for LBNO, 0.5*LBNO, and LBNE10. Results are shown for the four possible true octant-hierarchy combinations.

CP Violation Discovery with LBNE10 and LBNO. In their first phases, both LBNE10 and LBNO will have very minimal CP violation reach. Figure 16 depicts the CP violation discovery reach for LBNE10 and LBNO. In the left panel (right panel), we have considered NH (IH) as true hierarchy. In Table 4, we mention the fraction of 𝛿_CP values for which CP violation can be detected for these two experimental setups. Both LBNE10 and LBNO have CP violation reach for around 50% values of true 𝛿_CP at 2𝜎 confidence level. At 3𝜎, their CP violation reach is quite minimal: only for 10–20% of the entire range.

Table 4 

Fractions of 𝛿_CP(true) for which a discovery is possible for CP violation. The numbers without (with) parentheses correspond to NH (IH) as true hierarchy. The results are presented at 2𝜎 and 3𝜎 confidence level.

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Figure 16 

Left panel (right panel) shows the Δ𝜒2 for the CP violation discovery as a function of true value of 𝛿CP assuming NH (IH) as true hierarchy.

7.2. T2HK, CERN-MEMPHYS, and ESS LINAC Proposals

The T2HK proposal [156] plans to use a 1.66 MW superbeam from the upgraded J-PARC proton synchrotron facility directed to a 1 megaton water Cherenkov detector (with fiducial mass of 560 kilotons) located at a distance of 295 km from the source at an off-axis angle of 2.5^∘. A proposed location for this detector is about 8 km south of Super-Kamiokande at an underground depth of 1,750 meters water equivalent. The main purpose of this experiment is to achieve an unprecedented discovery reach for CP violation. The water Cherenkov detector provides an excellent energy resolution for sub-GeV low-multiplicity final-state events. Both the high-power narrowband beam and the megaton-size detector play an important role to have large numbers of 𝜈_𝑒 and 𝜈_𝑒 appearance events at the first oscillation maximum. This experiment plans to have 1.5 years of neutrino run and 3.5 years of antineutrino run with one year given by 10⁷ seconds. The baseline of this experiment is too short to have any matter effect. Therefore, its mass hierarchy discovery reach is very limited. The expected accuracy in the determination of CP phase is better than 20^∘ at 1𝜎 confidence level assuming that mass hierarchy is known. For sin²2𝜃₁₃=0.1, the CP violation can be established at 3𝜎 CL for 74% of the 𝛿_CP values provided that we fix the hierarchy in the fit. This CP coverage reduces to 55% if we marginalize over both the choices of hierarchy in the fit [156].

Like T2HK proposal, there is a superbeam configuration which is under consideration in Europe [157, 159, 160] using the CERN to Fréjus baseline of 130 km. The aim of this proposal is to send a 4 MW high-power superbeam from CERN towards a 440 kilotons MEMPHYS water Cherenkov detector [158] located 130 km away at Fréjus. Using this setup, CP violation can be established for roughly 60% values of 𝛿_CP at 3𝜎 confidence level [158] assuming two years of neutrino run and eight years of antineutrino run, with uncorrelated systematic uncertainties of 5% on signal and 10% on background.

Another interesting possibility to explore leptonic CP violation using a very intense, cost-effective, and high-performance neutrino beam from the proton linac of the ESS facility currently under construction in Lund, Sweden, has been studied recently in detail in [161, 162]. A high-power superbeam from this proton linac in conjunction with a megaton-size water Cherenkov detector located in the existing mines at a distance of 300 to 600 km from the ESS facility can discover leptonic CP violation at 5𝜎 confidence level for 50% of the 𝛿_CP values [162].

7.3. Neutrino Factory

The term “Neutrino Factory” [149, 150, 177, 178] has been associated to describe neutrino beams created by the decays of high-energy muons (obtained via pion decay) which are circulated in a storage ring with long straight sections. The decay of muons in these straight sections produces an intense, well-known, and pure beam of 𝜈_𝜇 and 𝜈_𝑒. If 𝜇⁺ are stored, 𝜇⁺→𝑒⁺𝜈_𝑒𝜈_𝜇 decays generate a beam consisting of equal numbers of 𝜈_𝑒 and 𝜈_𝜇. The most promising avenue to explore CP violation, neutrino mass hierarchy, and octant of 𝜃₂₃ at a neutrino factory is the subdominant 𝜈_𝑒→𝜈_𝜇 oscillation channel which produces muons of the opposite charge (wrong-sign muons) to those stored in the storage ring and these can be detected with the help of the charge identification capability of a magnetized iron neutrino detector (MIND) [97, 167]. In the most recent analysis [167] of MIND, low-energy neutrino signal events down to 1GeV were selected with an efficiency plateau of ∼60% for 𝜈_𝜇 and ∼80% for 𝜈_𝜇 events starting at ~5 GeV, while maintaining the background level at or below 10⁻⁴. For the neutral current background, the impact of migration is nonnegligible and it is peaked at lower energies. This feed-down is the strongest effect of migration and thus has potential impact on the energy optimization, since it penalizes neutrino flux at high energies, where there is little oscillation but a large increase in fed-down background.

In light of recently discovered moderately large value of 𝜃₁₃, shorter baselines and lower energies are preferred to achieve high performance in exploring CP violation. Even 𝐸_𝜇 as low as 5 to 8GeV at the Fermilab-Homestake baseline of about 1300 km is quite close to the optimal choice (see upper left panel of Figure 17), which means that the MIND detector approaches the magnetized totally active scintillator detector performance of the Low-Energy Neutrino Factory (LENF) [163]. With the present baseline choice of the Neutrino Factory (𝐸_𝜇=10GeV and 𝐿=2000km) and a 50 kilotons MIND detector, CP violation can be established for around 80% values of 𝛿_CP at 3𝜎 CL considering 5×10²¹ useful muon decays in total.

Figure 17 

Fraction of 𝛿CP(true) for which CP violation will be discovered at 3𝜎 CL as a function of 𝐿 and 𝐸𝜇 for the single-baseline Neutrino Factory. The different panels correspond to different true values of sin22𝜃13, as given there. Here, we consider 5×1021 useful muon decays in total with a 50 kilotons MIND detector. The optimal performance is marked by a dot: (2200, 10.00), (2288, 13.62), (3390, 20.00), and (4345, 22.08) with regard to their best reaches of the fraction of 𝛿CP(true) at 0.77, 0.84, 0.67, and 0.42. This figure has been taken from [142].

7.4. Beta-Beams

Zucchelli [169] put forward the novel idea of a beta-beam [73, 170, 171, 179], which is based on the concept of creating a pure, well-understood, intense, collimated beam of 𝜈_𝑒 or 𝜈_𝑒 through the beta-decay of completely ionized radioactive ions. Firstly, radioactive nuclides are created by impinging a target by accelerated protons. These unstable nuclides are collected, fully ionized, bunched, accelerated, and then stored in a decay ring. The decay of these highly boosted ions in the straight sections of the decay ring produces the so-called beta-beam. It has been proposed to produce 𝜈_𝑒 beams through the decay of highly accelerated ¹⁸Ne ions (¹⁸₁₀Ne→₉¹⁸F+𝑒⁺+𝜈_𝑒) and 𝜈_𝑒 from ⁶He(⁶₂He→⁶₃Li+𝑒⁻+𝜈_𝑒) [169]. More recently, ⁸B(⁸₅B→⁸₄Be+𝑒⁺+𝜈_𝑒) and ⁸Li(⁸₃Li→⁸₄Be+𝑒⁻+𝜈_𝑒) [180–182] with much larger end-point energy have been suggested as alternate sources since these ions can yield higher energy 𝜈_𝑒 and 𝜈_𝑒, respectively, with lower values of the Lorentz boost 𝛾 [154, 183–186]. Details of the four beta-beam candidate ions can be found in Table 5. It may be possible to store radioactive ions producing beams with both polarities in the same ring. This will enable running the experiment in the 𝜈_𝑒 and 𝜈_𝑒 modes simultaneously. In the low 𝛾 design of beta-beams, the standard luminosity taken for the ¹⁸Neand⁶He is 1.1×10¹⁸ (𝜈_𝑒) and 2.9×10¹⁸ (𝜈_𝑒) useful decays per year, respectively.

Table 5 

Beta decay parameters: lifetime 𝜏, electron total end-point energy 𝐸₀, 𝑓-value, and decay fraction for various ions. This table has been taken from [73].

Within the EURISOL Design Study [171], the 𝛾=100 option with ⁶He and ¹⁸Ne ions has been studied quite extensively. The energy spectrum of the emitted neutrinos from these radioactive ions with 𝛾=100 suits well the CERN to Fréjus baseline of 130 km. Compared to superbeam, the main advantage of using beta-beam is that it is an extremely pure beam with no beam contamination occurs at the source. Combining beta-beam with superbeam, we can study the T-conjugated oscillation channels [187, 188] and this combined setup can provide an excellent reach for CP violation discovery.

8. Summary and Conclusions

The discovery of neutrino mixing and oscillations provides strong evidence that neutrinos are massive and leptons flavors are mixed with each other which leads to physics beyond the Standard Model of particle physics. With the recent determination of 𝜃₁₃, for the first time, a clear and comprehensive picture of the three-flavor leptonic mixing matrix has been established. This impressive discovery has crucial consequences for future theoretical and experimental efforts. It has opened up exciting prospects for current and future long-baseline neutrino oscillation experiments towards addressing the remaining fundamental questions, in particular the type of the neutrino mass hierarchy, the possible presence of a CP-violating phase in the neutrino sector, and the correct octant of 𝜃₂₃ (if it turns out to be nonmaximal establishing the recent claims). In this paper, we have made an attempt to review the phenomenology of long-baseline neutrino oscillations with a special emphasis on subleading three-flavor effects, which will play a crucial role in resolving these unknowns in light of recent measurement of a moderately large value of 𝜃₁₃. We have discussed in detail the physics reach of current-generation long-baseline experiments: T2K and NO𝜈A which have very limited reach in addressing these unknowns for only favorable ranges of parameters. Hence, future facilities are indispensable to cover the entire parameter space at unprecedented confidence level. A number of high-precision long-baseline neutrino oscillation experiments have been planned/proposed to sharpen our understanding about these tiny particles. In this review, we have discussed in detail the physics capabilities of few of such proposals based on superbeams, neutrino factory, and beta-beam.

Conflict of Interests

The author declares that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

Sanjib Kumar Agarwalla would like to thank all his collaborators with whom he has worked on long-baseline neutrino oscillation physics. Sanjib Kumar Agarwalla acknowledges the support from DST/INSPIRE Research Grant (IFA-PH-12), Department of Science and Technology, India.