Prospects for Neutrino Oscillation Physics

Abstract

Recently the last unknown lepton mixing angle 𝜃₁₃ has been determined to be relatively large, not too far from its previous upper bound. This opens exciting possibilities for upcoming neutrino oscillation experiments towards addressing fundamental questions, among them the type of the neutrino mass hierarchy and the search for CP violation in the lepton sector. In this paper we review the phenomenology of neutrino oscillations, focusing on subleading effects, which will be the key towards these goals. Starting from a discussion of the present determination of three-flavour oscillation parameters, we give an outlook on the potential of near-term oscillation physics as well as on the long-term program towards possible future precision oscillation facilities. We discuss accelerator-driven long-baseline experiments as well as nonaccelerator possibilities from atmospheric and reactor neutrinos.

1. Introduction

Over the last 15 years so huge progress has been made in the study of neutrino oscillations [1–4], and with the recent determination of the last unknown mixing angle 𝜃₁₃ [5–9] a clear first-order picture of the three-flavour lepton mixing matrix has emerged. The results of a global fit [10] to world neutrino oscillation data including data presented at the Neutrino2012 conference are summarized in Figure 1 and Table 1 (for alternative global fits see [11, 12], and combinations of the recent data relevant to the 𝜃₁₃ determination have been presented in [13, 14]). Global data as of June 2012 disfavours 𝜃₁₃=0 with a Δ𝜒²≈100 and its value are determined as 𝜃₁₃=(8.6^+0.44_−0.46)^∘. (An uncertainty about this number at the level of 1𝜎 remains due to a tension between predicted reactor neutrino fluxes and data from reactor experiments with baselines less than 100 m, the so-called reactor anomaly [15].) Establishing such a relatively large value of 𝜃₁₃, comparable to the previous bound [16], opens exciting possibilities for neutrino oscillations. There are fundamental open questions in neutrino physics which can be addressed with neutrino oscillations.(i)Is there CP violation in the leptonic sector, as in the quark sector? The behaviour under the CP transformation is one of the fundamental properties of particles and a violation of the CP symmetry might be linked to the baryon asymmetry of the universe.(ii)What is the hierarchy of the neutrino mass spectrum, normal or inverted? This information is important phenomenologically for the interpretation of other neutrino experiments, for instance, neutrinoless double beta decay. Moreover, together with the absolute mass scale, it is one of the key pieces of information on neutrino masses.(iii)What are the precise values of the neutrino-mixing angles? Do they show an underlying pattern? The answers to these questions are a necessary input in order to solve the flavour problem.With our current knowledge of 𝜃₁₃ answering those questions becomes a realistic possibility.

Table 1 

Three-flavour oscillation parameters from a fit to global data after the Neutrino2012 conference [10]. The normalization of reactor neutrino fluxes is determined by short-baseline reactor data, which are included in the fit.

Figure 1 

Determination of three-flavour oscillation parameters [10]. The red (blue) curves are for NH (IH). Results for different assumptions concerning the analysis of data from reactor experiments are shown: for solid curves the normalization of reactor fluxes is left free, and data from short-baseline (less than 100 m) reactor experiments are included. For dashed curves short-baseline data are not included but reactor fluxes as predicted in [152] are assumed.

The outline of this work is as follows. In Section 2 we review the current status of neutrino oscillations and discuss the phenomenology of long-baseline (LBL), reactor, and atmospheric neutrino experiments. In Section 3 we discuss the potential of currently operating LBL and reactor facilities in the time frame of about 10 years. In Section 4 some nonaccelerator-based methods to determine the neutrino mass hierarchy are discussed. In Section 5 we give an overview on possible long-term experimental strategies towards high-precision oscillation facilities. We do not discuss solar, supernova, or other astrophysical neutrinos which are covered in detail in other chapters of this volume.

We will remain within the three-neutrino mixing framework and will not discuss deviations from it such as sterile neutrinos or nonstandard interactions. We use the standard convention for parametrizing the three-flavour lepton-mixing matrix in terms of the three angles, 𝜃₁₂, 𝜃₂₃, 𝜃₁₃, and one Dirac-type CP phase 𝛿 [17]: 𝑐𝑈=₁₂𝑐₁₃𝑠₁₂𝑐₁₃𝑠₁₃𝑒^−𝑖𝛿−𝑠₁₂𝑐₂₃−𝑐₁₂𝑠₁₃𝑠₂₃𝑒^𝑖𝛿+𝑐₁₂𝑐₂₃−𝑠₁₂𝑠₁₃𝑠₂₃𝑒^𝑖𝛿𝑐₁₃𝑠₂₃+𝑠₁₂𝑠₂₃−𝑐₁₂𝑠₁₃𝑐₂₃𝑒^𝑖𝛿−𝑐₁₂𝑠₂₃−𝑠₁₂𝑠₁₃𝑐₂₃𝑒^𝑖𝛿𝑐₁₃𝑐₂₃, (1) where 𝑐_𝑖𝑗≡cos𝜃_𝑖𝑗 and 𝑠_𝑖𝑗≡sin𝜃_𝑖𝑗. Squared differences of the neutrino masses 𝑚_𝑖 are defined as Δ𝑚²_𝑖𝑗≡𝑚²_𝑖−𝑚²_𝑗. The neutrino mass hierarchy is determined by the sign of Δ𝑚²₃₁, with Δ𝑚²₃₁>0 corresponding to normal hierarchy (NH), and Δ𝑚²₃₁<0 to inverted hierarchy (IH).

2. Present Status as Case Study for Oscillation Phenomenology

In this section we introduce the oscillation probabilities relevant to long-baseline accelerator experiments, reactor experiments, as well as atmospheric neutrinos. We use the present data to illustrate the interplay and complementarity of different types of oscillation experiments.

2.1. The Beam-Reactor Interplay

Since the advent of data on 𝜈_𝜇→𝜈_𝑒 searches from T2K [5] and MINOS [6] on the one side, and 𝜃₁₃ reactor experiments Double Chooz [7], Daya Bay [8], and RENO [9] on the other side, the long anticipated complementarity of beam and reactor experiments [18, 19] is now a reality. In this section we discuss some aspects of that related to deviations of 𝜃₂₃ from maximal mixing, as well as the dependence of the global fit on the CP phase 𝛿.

Recent data seem to indicate a deviation of 𝜃₂₃ from the maximal mixing value of 45^∘, roughly at the level of 1.7𝜎−2𝜎, compare Figure 1 (in [12] a somewhat higher significance is obtained). If confirmed, such a deviation would have profound implications for neutrino mass models based on flavour symmetries. An important contribution to this effect comes from recent MINOS data on 𝜈_𝜇 disappearance. Neglecting effects of Δ𝑚²₂₁ and the matter effect, the relevant survival probability is given by 𝑃_{𝜈𝜇→𝜈𝜇}=1−4𝑈_𝜇3²1−𝑈_𝜇3²sin²Δ𝑚²₃₁𝐿,𝑈4𝐸_𝜇32=sin²𝜃₂₃cos²𝜃₁₃, (2) where 𝐿 is the baseline and 𝐸 is the neutrino energy. Hence, the probability is symmetric under |𝑈_𝜇3|²→(1−|𝑈_𝜇3|²). In the two-flavour limit of 𝜃₁₃=0 this implies that the data is sensitive only to sin²2𝜃₂₃, which for 𝜃₂₃≠45^∘ leads to a degeneracy between the first and second octants of 𝜃₂₃ [20]. Indeed, recent data from MINOS [21] have given a best fit point of sin²2𝜃≈0.94 if analysed in a two-flavour framework.

Since 𝜃₁₃ is large, one can try to explore a synergy between long-baseline appearance experiments and an independent determination of 𝜃₁₃ at reactor experiments in order to resolve the degeneracy [18, 20, 22]. Let us look at the appearance probability relevant to the 𝜈_𝜇→𝜈_𝑒 searches at T2K [5] and MINOS [6]. Expanding to second order in the small parameters sin𝜃₁₃ and 𝛼≡Δ𝑚²₂₁/Δ𝑚²₃₁ and assuming a constant matter density that one finds [23–25]: 𝑃_{𝜈𝜇→𝜈𝑒}≈4sin²𝜃₁₃sin²𝜃₂₃sin²Δ(1−𝐴)(1−𝐴)²+𝛼²sin²2𝜃₁₂cos²𝜃₂₃sin²𝐴Δ𝐴²+2𝛼sin𝜃₁₃sin2𝜃₁₂sin2𝜃₂₃×cosΔ±𝛿_CPsinΔ𝐴𝐴sinΔ(1−𝐴),1−𝐴 (3) with the definitions Δ≡Δ𝑚²₃₁𝐿4𝐸,𝐴≡2𝐸𝑉Δ𝑚²₃₁, (4) where 𝐿 is the baseline, 𝐸 is the neutrino energy, and 𝑉 is the effective matter potential [26]. Note that 𝛼, Δ, and 𝐴 are sensitive to the sign of Δ𝑚²₃₁ (i.e., the type of the neutrino mass ordering). The plus (minus) sign in (3) applies for neutrinos (antineutrinos), and for antineutrinos 𝑉→−𝑉, which implies that 𝐴→−𝐴. It is clear from (3) that in the case of large matter effect, 𝐴≳1, the terms (1−𝐴) depend strongly on the type of the mass hierarchy, and for 𝐴=1 (possible for neutrinos and NH, or antineutrinos and IH) a resonance is encountered [27]. Numerically one finds for a typical matter density of 3 g/cm³|𝐴|𝐸≃0.09GeVΔ𝑚²₃₁2.5×10⁻³eV²⁻¹. (5) Since, for T2K 𝐸∼0.7 GeV, matter effects are of order few percent, whereas in experiments like NOvA [28] with 𝐸∼2 GeV we can have |𝐴|∼0.2. Note that 𝛼²≈10⁻³, which implies that the second term in the first line of (3) gives a very small contribution compared to the other terms.

An important observation is that the first term in (3) (which dominates for large 𝜃₁₃) depends on sin²𝜃₂₃ and therefore is sensitive to the octant. Reactor experiments with 𝐿∼1km, on the other hand, provide a measurement of 𝜃₁₃ independent of 𝜃₂₃. The relevant survival probability is given by 𝑃_{𝜈𝑒→𝜈𝑒}=1−sin²2𝜃₁₃sin²Δ+𝒪𝛼². (6) Hence, by combining the data from reactor experiments such as Double Chooz [7], Daya Bay [8], and RENO [9] with the appearance data from T2K and MINOS one should be sensitive in principle to the octant of 𝜃₂₃. The situation from present data is illustrated in Figure 2, where we show the determination of 𝜃₁₃ from the beam experiments T2K and MINOS as a function of the CP phase 𝛿 and the octant of 𝜃₂₃, where we have chosen values motivated by the MINOS disappearance result. The resulting regions in sin²2𝜃₁₃ are compared to the reactor measurments from DoubleChooz, DayaBay, and RENO. It is clear from that figure that for present data from beams and reactors it is not possible to distinguish between 1st and 2nd 𝜃₂₃ octants. For both possibilities overlap regions between beams and reactors can be found although they are at different values of 𝛿. Therefore, current data from reactor and long-baseline beam experiments are not able to resolve the 𝜃₂₃ octant degeneracy. The lifting of the degeneracy (at low CL) visible in Figure 1 appears due to atmospheric neutrino data, to be discussed below.

(a)

(b)

Figure 2 

(a) Preferred regions in the sin22𝜃13−𝛿 plane. The contour curves correspond to beams T2K + MINOS, where sin2𝜃23 is fixed to the two degenerate solutions in the 1st (red) and 2nd (blue) octant. The gray region corresponds to the 𝜃13 determination from the reactors Double Chooz, Daya Bay, and RENO. (b) Δ𝜒2(𝛿) for beams (dashed) and beams+reactors (solid) with the same color coding as in the left panels. Upper (lower) panels are for NH (IH).

In principle the reactor-beam combination should also offer some sensitivity to the CP phase 𝛿. This is shown in the right panels of Figure 2. We see that if the octant of 𝜃₂₃ and the neutrino mass hierarchy were known, already present data from the beam and reactor experiments used in that figure would show quite sizeable dependence on the CP phase, depending on which of the 4 degenerate solutions is considered. However, it is also clear from the figure that we marginalize over those four solutions, 𝜒²(𝛿) becomes very flat, and essentially all values of 𝛿 would be consistent within Δ𝜒²≲1. This is a real-life example of how degeneracies can seriously spoil the sensitivity of long-baseline data [29]. The somewhat larger 𝛿 dependence visible in Figure 1 follows again from the global fit including atmospheric neutrinos, as discussed below.

2.2. Subleading Effects in Atmospheric Neutrino Oscillations

Atmospheric neutrinos provide a powerful tool to study neutrino oscillations, which is manifest also by the first evidence for oscillations from Super-Kamiokande in 1998 [1]. In this section we briefly discuss subleading oscillation modes, triggered by Δ𝑚²₂₁ and/or 𝜃₁₃, and comment on using them for addressing some of the open questions in oscillation physics.

An important property of atmospheric neutrinos is the fact that the neutrino source contains 𝜈_𝑒 and 𝜈_𝜇 as well as neutrinos and antineutrinos. Therefore, the contributions to 𝑒-like and 𝜇-like event samples can be written schematically as 𝑁_𝑒∝Φ_𝑒𝑃_{𝜈𝑒→𝜈𝑒}+Φ_𝜇𝑃_{𝜈𝜇→𝜈𝑒}𝜎_𝑒=Φ_𝑒𝑃_{𝜈𝑒→𝜈𝑒}+𝑟𝑃_{𝜈𝜇→𝜈𝑒}𝜎_𝑒𝑁_𝜇∝Φ_𝑒𝑃_{𝜈𝑒→𝜈𝜇}+Φ_𝜇𝑃_{𝜈𝜇→𝜈𝜇}𝜎_𝜇=Φ_𝜇1𝑟𝑃_{𝜈𝑒→𝜈𝜇}+𝑃_{𝜈𝜇→𝜈𝜇}𝜎_𝜇, (7) where Φ_𝛼 and 𝜎_𝛼 are initial flux and detection cross section for neutrino of flavour 𝛼, and we have defined the flux ratio Φ𝑟≡^𝜇Φ_𝑒, (8) with 𝑟≈2 in the sub-GeV range and 𝑟≈ 2.6–4.5 in the multi-GeV range. In (7) we have suppressed energy and zenith angle dependence, as well as detector resolutions and efficiencies. If the detector cannot identify the charge of the lepton, a sum over neutrino and antineutrinos is implicitly assumed; otherwise, analogous relations hold for neutrinos and antineutrinos separately. Hence, the observation of 𝑒-like and 𝜇-like events contains convoluted information on different oscillation channels.

An interesting observable is the excess of 𝑒-like events (relative to the no-oscillation prediction 𝑁⁰_𝑒), since in the two-flavour limit one expects 𝑁_𝑒=𝑁⁰_𝑒, and therefore any deviation of the observed number of events from 𝑁⁰_𝑒 should be due to subleading effects. The excess can be written in the following way; see, for example, [30]: 𝑁_𝑒𝑁⁰_𝑒−1≈𝑟sin²𝜃₂₃−1𝑃_2𝜈Δ𝑚²₃₁,𝜃₁₃+𝑟cos²𝜃₂₃−1𝑃_2𝜈Δ𝑚²₂₁,𝜃₁₂−sin𝜃₁₃sin2𝜃₂₃𝑟Re𝐴^∗_𝑒𝑒𝐴_𝜇𝑒. (9) Here 𝑃_2𝜈(Δ𝑚²,𝜃) is an effective two-flavour oscillation probability governed by a mass-squared difference Δ𝑚² and a mixing angle 𝜃, and 𝐴_𝑒𝑒 and 𝐴_𝜇𝑒 are elements of a transition amplitude matrix. The three terms appearing in (9) have a well-defined physical interpretation. The first term is important in the multi-GeV range and is controlled by the mixing angle 𝜃₁₃ in 𝑃_2𝜈(Δ𝑚²₃₁,𝜃₁₃). This probability can be strongly affected by resonant matter effects [31–36]. Depending on the mass hierarchy the resonance will occur either for neutrinos or antineutrinos. The second term is important for sub-GeV events, and it takes into account the effect of “solar oscillations” due to Δ𝑚²₂₁ and 𝜃₁₂ [37–40]. Via the prefactor containing the flux ratio 𝑟, both the first and second terms in (9) depend on the octant of 𝜃₂₃ though, in oposite directions, the multi-GeV (sub-GeV) excess is suppressed (enhanced) for 𝜃₂₃<45^∘. Finally, the last term in (9) is an interference term between 𝜃₁₃ and Δ𝑚²₂₁ amplitudes, and this term shows also dependence on the CP phase 𝛿 [30, 40].

Three neutrino effects may also show up in 𝜇-like events. This is especially interesting for experiments which can only observe muons, such as: for example, the INO or IceCube experiments. For these types of experiments the multi-GeV region is most interesting, where effects Δ𝑚²₂₁ are very small. Hence, we can approximate Δ𝑚²₂₁≈0, and following [41, 42] one can write the excess in 𝜇-like events as 𝑁_𝜇𝑁⁰_𝜇−1≈sin²𝜃₂₃1𝑟−sin²𝜃₂₃𝑃_2𝜈Δ𝑚²₃₁,𝜃₁₃−12sin²2𝜃₂₃1−Re𝐴₃₃. (10) The first term is controlled by 𝜃₁₃ and is subject to resonant matter effects, similar to the first term in (9), though with a different dependence on 𝜃₂₃ and the flux ratio. In the second term, 𝐴₃₃ is a probability amplitude satisfying 𝑃_2𝜈(Δ𝑚²₃₁,𝜃₁₃)=1−|𝐴₃₃|². In the limit 𝜃₁₃=0 we have Re(𝐴₃₃)=cos(Δ𝑚²₃₁𝐿/2𝐸), such that the second term in (10) just describes two-flavour 𝜈_𝜇→𝜈_𝜇 vacuum oscillations.

2.3. Interplay of Complementary Data Sets in the Present Global Fit

As mentioned above, while MINOS 𝜈_𝜇 disappearance data prefers a nonmaximal value of 𝜃₂₃, we do not observe any sensitivity to the octant of 𝜃₂₃ from global data without atmospheric neutrinos. In the global analysis of [10] including atmospheric data a weak preference for the 1st octant is obtained in the case of NH; see Figure 1. Similar results are obtained also in [12] with even somewhat higher significance. This can be attributed to a zenith-angle-independent event excess in the sub-GeV 𝑒-like data in SuperKamiokande. Such an excess can be explained by oscillations due to Δ𝑚²₂₁ [37–39]. For sub-GeV events the second term in (9) is relevant. In that energy regime 𝑟≈2 and for sin²𝜃₂₃≈0.5 the prefactor (𝑟cos²𝜃₂₃−1) is suppressed, whereas in the 1st octant with sin²𝜃₂₃<0.5 an excess is induced. Let us mention that in an official SuperKamiokande analysis [43] this effect is not clearly observed although one should take into account that there is no combined analysis with MINOS data performed. It can be seen from (9) and (10) that there can be some features in 𝑒-like or 𝜇-like data samples which exhibit a different dependence on 𝜃₂₃, and which of those subtle effects dominates depends on details of the detector simulation, binning, and treatment of systematic uncertainties. Apparently competing effects become somewhat more important for IH, as in that case the preference for the 1st octant disappears; see Figure 1.

We emphasize the importance of resolving the 𝜃₂₃ octant degeneracy in order to obtain sensitivity to the CP phase 𝛿. This can be seen from Figure 2. By favouring one of the two solution for sin²𝜃₂₃ the beam-reactor combination provides a better sensitivity to 𝛿, visible in the right panels. With current data this effect is still small, given the final sensitivity to 𝛿 shown in Figure 1, which is at level of Δ𝜒²≈3. We emphasize again the crucial interplay of different data sets necessary for this sensitivity to emerge: MINOS 𝜈_𝜇 disappearance prefers sin²2𝜃₂₃<1, atmospheric data slightly disfavours sin²𝜃₂₃>0.5, and the 𝜈_𝜇→𝜈_𝑒 data from beams combined with the 𝜃₁₃ determination from reactors provide sensitivity to 𝛿.

3. The Current Generation of Long-Baseline Beam and Reactor Experiments

The reactor experiments, Double Chooz [7], DayaBay [8], and RENO [9], have obtained spectacular results already after few weeks of data taking. All of them are still statistics dominated, and the precision of the determination of 𝜃₁₃ will improve considerably for higher exposures. According to the results of [44] (based on assumptions on systematics from the proposals of the three experiments) the ultimate precision will be dominated by DayaBay. Also T2K [5] is essentially only in the “startup phase” (which unfortunately has been interrupted by the 2011 earthquake in Japan). In addition the NOvA experiment [28] will come online soon and will provide additional data on 𝜈_𝜇→𝜈_𝑒 appearance. In [45] the expected combined potential of those experiments with their final exposures has been investigated in respect to address the mass hierarchy or a determination of the CP phase 𝛿. Here we review the results obtained there in the light of the by now known value of 𝜃₁₃. The “nominal” exposures are summarized in Table 2. The experimental configurations are based on official documents as of 2009. As a rough rule of thumb those data might be available around 2020.

Table 2 

Summary of the “nominal” luminosities for the current generation of reactor and beam experiments. See [45] for details.

Let us first discuss the prospects for the 𝜃₂₃ measurment, including the determination of the octant in case of a nonmaximal value. Figure 3 shows the ability of T2K + NOvA + DayaBay to reconstruct sin²𝜃₂₃ as a function of its true value. Data on 𝜈_𝜇 disappearance from T2K and NOvA are mainly sensitive to sin²2𝜃₂₃ (see discussion related to (2)), whereas the combination of the 𝜈_𝜇→𝜈_𝑒 appearance data with the 𝜃₁₃ reactor measurment provides sensitivity to the octant [18, 20, 22], as discussed above see also Figure 2. While this mechanism does not work for current data, it can be used to identify the right octant at 3𝜎 with projected exposures if |sin²𝜃₂₃−0.5|>0.1. Note the slight asymmetry of the regions, which is a consequence of the relatively large value of 𝜃₁₃ and can be understood from (2). We also observe that for large deviations from maximality the accuracy on sin²𝜃₂₃ will be quite good, around ±0.02 at 3𝜎, whereas close to maximality the determination will be much worse, with a 3𝜎 range of about 0.45<sin²𝜃₂₃<0.58, due to the flatness of sin²2𝜃 at 𝜃=45^∘.

Figure 3 

Determination of sin2𝜃23 at 1,2,3𝜎 (1 dof) as a function of the true value from T2K, NOvA, and DayaBay with their final exposures according to Table 2. True values of sin22𝜃13=0.1 and 𝛿=0 have been assumed. The vertical dotted line and the shaded regions indicate the present best fit point and 3𝜎 excluded region, respectively. Figure is based on the results of [45].

As we have seen above, already with present data a global fit of all experiments shows a slight dependence on the CP phase 𝛿. On the other hand the mass hierarchy is undistinguishable with a Δ𝜒²≲1 between NH and IH, see Figure 1. Now we address the question whether with near-term data from the experiments listed in Table 2 we may be able to say something on the type of the neutrino mass hierarchy or the CP phase 𝛿.

Possible outcome of a global fit to data from the final exposure of T2K, NOvA, and DayaBay is shown in Figure 4. Left (right) panel correspond to NH (IH), and we adopt the two exemplary values of 𝛿=𝜋/2 (upper panels) and 3𝜋/2 (lower panels) corresponding to maximal CP violation. The colored regions would be obtained under the assumption of known mass hierarchy. In this case some regions of 𝛿 can be excluded at 3𝜎 although CP-conserving values 0 or 𝜋 are always contained in the allowed region, indicating that CP violation cannot be established, even under the assumption of known hierarchy. The detailed study performed in [45] shows that only for less than 30% of all possible values of the phase 𝛿 a hint for CP violation at 90% CL may be obtained. Even for upgraded versions of the beams, with increased beam power, extended running time (up to around 2025), and including antineutrino data from T2K CP violation can be “discovered” at 3𝜎 for about 25% of all possible values of 𝛿 (see also [46] for related results).

(a)

(b)

(c)

(d)

Figure 4 

Fits in the 𝜃13−𝛿 plane for sin22𝜃13=0.1, 𝜃23=𝜋/4, and 𝛿=𝜋/2 (upper row) and 𝛿=3𝜋/2 (lower row) and a true NH (left panels) and IH (right panels), assuming the final exposure from T2K, NOvA, and DayaBay according to Table 2. The contours refer to 1𝜎, 2𝜎, and 3𝜎 (2 dof). The fit contours for the right fit hierarchy are shaded (colored), and the ones for the wrong hierarchy fit are shown as curves. The best-fit values are marked by diamonds and boxes for the right and wrong hierarchy, respectively, where the minimum 𝜒2 for the wrong hierarchy is explicitly shown. Figures are based on the results of [45].

The mass hierarchy determination relies on the matter effect in the 𝜈_𝜇→𝜈_𝑒 transitions. From the experiments considered here only for the NOvA experiment a notable matter effect is present due to the baseline of 812 km. In Figure 4 we illustrate the combined potential to identify the neutrino mass hierarchy. The black contour curves correspond to the allowed regions obtained by fitting the simulated data with the wrong hierarchy. The local 𝜒² minimum is marked with a black box and the 𝜒² value of the local minimum is given in the figure. It turns out that the four examples shown in the figure correspond approximately to the most optimistic and pessimistic cases for the hierarchy determination. The best possible configuration is obtained for a true NH and 𝛿=3𝜋/2 (left-lower pannel) and a true IH and 𝛿=𝜋/2 (right-upper pannel), where the hierarchy can be identified at about 3𝜎. For the opposite combinations (true NH, 𝛿=𝜋/2 and true IH, 𝛿=3𝜋/2) very poor sensitivity is obtained with 𝜒²≈2. This behaviour can be understood from (3) by considering the sign of the interference term in the second line. Good sensitivity is obtained when, for the channel which is enhanced by the matter resonance (neutrinos for NH or antineutrinos for IH), the CP phase has such a value that the sign of the interference term is positive (constructive interference), which leads to a maximum enhancement of the event numbers in the resonant channel, see also [47].

Unfortunately a significant determination of the mass hierarchy is only possible for very special points in the parameter space, close to the ones shown in Figure 4, lower-left or upper-right panels. Even for a 90% CL hint for the true hierarchy can be obtained only for about 50% of all possible values of 𝛿. With the abovementioned upgrades in beam power and extended running times up to 2025 a 3𝜎 mass hierarchy determination can be reached for about 30–40% of all possible values of 𝛿 [45] (see also [48] for a recent analysis).

We conclude that with the upcoming experiments as summarized in Table 2 it will be very hard to address CP violation and the mass hierarchy with reasonable significance, and it seems necessary to consider projects beyond those. Before considering high precision long-baseline facilities designed to address those questions in Section 5 we discuss in the next section alternative ways to determine the neutrino mass hierarchy.

4. Alternative Mass Hierarchy Determinations

The fact that 𝜃₁₃ has been found to be relatively large opens interesting possibilities to identify the mass hierarchy, beyond accelerator-driven long-baseline experiments. In this section we mention some of those possibilities, based on 𝜃₁₃-induced matter effects in atmospheric neutrinos as discussed in Section 2.2 by considering different kinds of atmospheric neutrino detectors: magnetized detectors (Section 4.1), huge nonmagnetized detectors using water or liquid argon (Section 4.2), or the IceCube detector (Section 4.3). In Section 4.4 we briefly mention an interesting method based on vacuum oscillations of reactor neutrinos. We do not discuss the possibility to use supernova neutrinos [49–52] or neutrinoless double beta decay [53, 54] to identify the mass hierarchy.

4.1. Atmospheric Neutrinos—Magnetized

The determination of the mass hierarchy based on the matter effect relies on the ability to find out whether the resonant enhancement occurs for neutrinos (which would signal NH) or for antineutrinos (IH). Since atmospheric neutrinos contain both neutrinos and antineutrinos, the sensitivity to the hierarchy (for a given total number of events) is much better if neutrino- and antineutrino-induced events can be distinguished, which can be done if the charge of the charged lepton can be identified. In this respect, magnetized iron calorimeters are a promising technology since they offer excellent charge discrimination for muons with few GeV energies. In particular the ICal experiment at the India-based Neutrino Observatory (INO) [55, 56] aims at the measurment of charge-separated atmospheric neutrino-induced muons. (Sensitivities of a hypothetical magnetized liquid argon detector have been estimated in [57, 58].) In such detectors the identification of electrons is difficult and therefore one relies on signals in 𝜇-like events, described by the expression in (10).

Early studies along these lines have been performed in [41, 59]. In Figure 5 we reproduce results obtained recently in [60, 61], where the combined sensitivity of the INO detector with data from T2K, NOvA, and DayaBay has been considered. For other recent studies see, for example, [62, 63]. It has been stressed in [42, 64] that the sensitivity to the mass hierarchy strongly depends on the ability to reconstruct the neutrino energy and direction. The second term in (10) induces characteristic features in the energy and zenith angle distribution of 𝜇-like events. If those features can be resolved by the detector, they provide robust sensitivity to the mass hierarchy. The “low” and “high” resolution scenarios referred to in Figure 5 assume resolutions of 𝜎_𝐸/𝐸_𝜈=0.15, 𝜎_𝜃=15^∘ (low) and 𝜎_𝐸/𝐸_𝜈=0.10, 𝜎_𝜃=10^∘ (high). Furthermore the sensitivity for a 50 kt or 100 kt detector is shown, which is supposed to start data taking in 2011 [55]. We observe that the sensitivity to the mass hierarchy is significantly increased compared to NOvA + T2K + DayaBay only. For all but the low resolution 50 kt detector the hierarchy can be identified for all values of 𝛿 at 2𝜎. However, for a 3𝜎 determination with 100% coverage in 𝛿 the high resolution 100 kt detector seems necessary.

(a)

(b)

Figure 5 

The time evolution of the fraction of values of the CP phase 𝛿 for which the combination of INO, NOvA, T2K, and DayaBay would be sensitive to the mass ordering at 2𝜎 (a) and 3𝜎 (b). Black (red) curves correspond to INO detector mass of 50 kt (100 kt) and dashed (solid) curves correspond to the low (high) resolution scenario; see text. The shaded area is the corresponding result without atmospheric data from INO. The true value of sin22𝜃13 has been assumed to be 0.09. Figures are based on the results of [60, 61].

4.2. Atmospheric Neutrinos—Water/Argon

If charge identification is not possible (as, for instance, in water Cerenkov detectors) the effect of changing the mass hierarchy is strongly diluted by summing neutrino and antineutrino events. However, the total sample is dominated by neutrinos due to higher fluxes and detection cross-sections. Therefore the cancellation is not complete and a net effect remains between having the resonance in neutrinos or antineutrinos. Furthermore, also a statistical separation of neutrino and antineutrino events may be possible. For example, in SuperKamiokande the fraction of single and multiring events is different for neutrinos and antineutrinos, or the probability to observe a decay electron either from a muon or a pion is different [43]. On the other hand, water Cerenkov detectors can be made very big, possibly at the mega ton scale [65, 66] which may allow to explore those subtle signatures. Since for those detectors electron detection is possible, the impact of the matter effect on multi-GeV 𝑒-like events (see (9)) can be explored.

The left panel of Figure 6 shows the sensitivity to the mass hierarchy of atmospheric neutrino data in a 560 kt water Cerenkov detector, the so-called HyperKamiokande project [66]. Depending on the parameter values, a more than 3𝜎 determination of the mass hierarchy seems possible after a few years of exposure. The figure shows that the sensitivity strongly depends on the value of 𝜃₂₃. In general the mass hierarchy sensitivity of atmospheric neutrinos is better for larger values of sin²𝜃₂₃. The same behaviour is also observed for magnetized muon detectors such as INO, see, for example, [42]. This follows from the relations given in Section 2.2, where it can be seen that effects of the oscillation probability 𝑃_2𝜈(Δ𝑚²₃₁,𝜃₁₃)—which encodes the resonant matter effects due to 𝜃₁₃ carrying the information on the hierarchy—are larger for large sin²𝜃₂₃.

(a)

(b)

Figure 6 

(a) Sensitivity to the neutrino mass hierarchy from Hyper Kamiokande atmospheric neutrino data. 𝜃23 and 𝜃13 are assumed to be known as indicated in the figure. Plot from [66]. (b) Sensitivity to the mass hierarchy at 2𝜎 from a combination of beams at relatively short baseline with 4.4 Mt yr atmospheric neutrino data from a water Cerenkov detector. βB and SPL refer to a beta-beam and a 4 MW superbeam from CERN to Frejus (130 km), respectively, whereas T2HK corresponds to a 4 MW beam from JPARC to the HyperKamiokande detector (295 km). Dashed curves are data from beams only, and solid curves are for beams + atmospheric neutrinos. Figure is based on the results of [68].

Many future projects for long-baseline accelerator experiments use large volume detectors which are also able to observe atmospheric neutrinos. Therefore, it is an obvious idea to explore synergies between the data from the beam and atmospheric neutrinos [67]. Figure 6 (right pannel) shows some examples, where information from beam experiments with relatively short baselines is combined with data from atmospheric neutrinos in a 440 kt water Cerenkov detector in order to resolve the mass hierarchy [68]. The beams considered there have baselines which are too short to address the mass hierarchy, and only the combination with atmospheric neutrinos allows to address this question. (Note that the combined data from the beta beam and SPL superbeam provides also some sensitivity to the hierarchy even without atmospheric data (dashed magenta curve). This is based on the combination of data from 𝜈_𝑒→𝜈_𝜇 (beta beam) and 𝜈_𝜇→𝜈_𝑒 (superbeam) oscillations, which allows to break the mass hierarchy degeneracy already at first order in the parameter 𝐴 (see (5)), which works already at the distance of 130 km [69]; see also [70].) Both panels in Figure 6 are based on a water Cerenkov detector, but similar results can be achieved in large (100 kt scale) liquid argon detectors [57]. We mention also that atmospheric data from such big detectors (including also the sub-GeV samples) provide excellent sensitivity to the octant of 𝜃₂₃ (see e.g., [67]) through the effects discussed already in the context of present data in Section 2.3.

4.3. Atmospheric Neutrinos—Ice

The IceCube neutrino telescope in Antarctica is able to collect a huge amount of atmospheric neutrino events. Due to the high energy threshold those data are not very sensitive to oscillations although they provide interesting constraints on nonstandard neutrino properties; see for example, [71]. With the so-called DeepCore extension [72] a threshold of around 10 GeV has been achieved, and the first results on oscillations of atmospheric neutrinos have been presented [73]; see [74] for a study on the neutrino mass hierarchy. With a further proposed extension of the IceCube detector called PINGU [75] the threshold could be even lowered to few GeV, opening the exciting possibility of a multimega ton scale detector exploring the matter resonance region. The most straight forward type of events will be muons without charge identification, and one has to rely on the huge statistic in order to identify the effect of the mass hierarchy. Below we discuss some results obtained recently in [76] focusing on the muon signal. Signatures from 𝜈_𝑒- and 𝜈_𝜏-induced events have also been studied in [76].

In order to identify the difference between normal and inverted mass hierarchy again a crucial issue will be the ability to reconstruct the neutrino energy and direction. In Figure 7 the difference between event numbers for NH and IH (weighted by the statistical error), binned in neutrino energy 𝐸_𝜈 and zenith angle 𝜃_𝑧, is shown for two assumptions on the reconstruction abilities. In the left pannel, with better resolutions, we can observe clearly the effects of the matter resonance. We note also that in different regions in the 𝐸_𝜈−cos𝜃_𝑧 plane the difference between NH and IH changes sign. This means that a worse resolution can easily wash out the effect. This is evident also from the right pannel, where a worse resolution has been assumed, leading to reduced significance per bin. Therefore, aiming for good energy and angular reconstruction will be an important goal in the design of the PINGU project.

(a)

(b)

Figure 7 

Statistical significance per bin of the difference between NH and IH for one year of PINGU data from 𝜈𝜇-induced events, binned in neutrino energy (bin width Δ𝐸𝜈=1 GeV) and cosine of the zenith angle (bin width Δcos𝜃𝑧=0.05). In the left (right) panel neutrino energy and angular reconstruction resolutions of 2 (4) GeV and 11.25∘ (22.5∘) have been assumed. Figures from [76].

Figure 7 shows the quantity 𝑆_𝑖≡(𝑁^IH_𝑖−𝑁^NH_𝑖)/𝑁^NH_𝑖, where 𝑁^NH_𝑖 (𝑁^IH_𝑖) is the number of 𝜇-like events in the case of NH (IH) in a given bin 𝑖. Hence, 𝑆_𝑖 corresponds to the statistical significance (in number of standard deviations) per bin. In the absence of systematical errors the total significance is given by ∑_𝑖𝑆²_𝑖, and the configurations considered in Figure 7 would lead to sensitivities at the level of 16𝜎 (left pannel) or 7𝜎 (right pannel) [76]. Hence, considering only statistical errors, excellent sensitivity to the mass hierarchy is obtained already after one year of PINGU data. Those very promising results are yet to be supported by detailed studies on the achievable energy and angular reconstruction as well as realistic investigations of systematical uncertainties.

4.4. Mass Hierarchy from Reactors

All the possibilities to identify the neutrino mass hierarchy discussed above are based on the matter effect in oscillations due to 𝜃₁₃. In [77] an alternative has been pointed out, based on oscillations of reactor neutrinos, where matter effects are negligible. The three-flavour survival probability of 𝜈_𝑒 in vacuum is easily obtained as 𝑃_{𝜈𝑒→𝜈𝑒}=1−cos⁴𝜃₁₃sin²2𝜃₁₂sin²Δ𝑚²₂₁𝐿4𝐸−sin²2𝜃₁₃×cos²𝜃₁₂sin²Δ𝑚²₃₁𝐿4𝐸+sin²𝜃₁₂sin²Δ𝑚²₃₂𝐿.4𝐸 (11) The spectrum of reactor experiments ranges from neutrino energies of about 1.3 MeV to 12 MeV with a peak around 4 MeV. Consider now a baseline 𝐿≃60km. Then we obtain for the arguments of the oscillating terms: Δ𝑚²₂₁𝐿≈𝜋4𝐸2𝐸4MeV⁻¹,Δ𝑚²₃₁𝐿≈4𝐸Δ𝑚²₃₂𝐿𝐸4𝐸≈504MeV⁻¹. (12) Hence, considering the spectrum obtained in a reactor experiment at about 60 km, the first term in (11) gives a “slow” oscillation in 1/𝐸, with a large amplitude of cos⁴𝜃₁₃sin²2𝜃₁₂≈0.8. These are the oscillations due to the “solar” frequency as observed by the KamLAND experiment. For an experiment at 60 km the first minimum of the survival probability occurs close to 𝐸∼4MeV, at the peak of the expected number of events.

The terms in the second line of (11) lead to fast oscillations in 1/𝐸 (see (12)) on top of the slow “solar” oscillation, with a small amplitude proportional to sin²2𝜃₁₃≈0.1. As evident from (11) there are actually two fast frequencies, one due to Δ𝑚²₃₁ and one due to Δ𝑚²₃₂, which differ by Δ𝑚²₂₁ (about 3%). The sensitivity to the mass hierarchy appears as follows. First, note that depending on the hierarchy we have |Δ𝑚²₃₁|>|Δ𝑚²₃₂| for NH or |Δ𝑚²₃₁|<|Δ𝑚²₃₂| for IH. Second, the amplitudes of the two fast frequencies are different because of the nonmaximal value of 𝜃₁₂: the amplitude of the Δ𝑚²₃₁ frequency is sin²2𝜃₁₃cos²𝜃₁₂≈0.07 while the one of the Δ𝑚²₃₂ frequency is sin²2𝜃₁₃sin²𝜃₁₂≈0.03. Hence, if an experiment can measure the fast frequencies and find out which one of the two fast frequencies has the larger amplitude (the larger or the smaller frequency), the mass hierarchy is determined. The effect can be illustrated by performing a Fourier transform of the event spectrum, where the two frequencies appear as a high and low peaks in the transformed spectrum [78].

The experimental requirements are obvious from the above discussion: (i) a good energy resolution is required in order not to wash out the fast oscillations and (ii) because of the small amplitude of the fast oscillations one needs enough statistics to be able to establish their presence. Numerical studies have been performed in [78–80]. The results of [80] indicate that exposures of order few 100 kt GW yr and energy resolutions of order 3% are required, which makes this measurement challenging. (We recall that the KamLAND experiment has about 1 kt, the LENA proposal [65] is for 50 kt, and typical modern reactor neutrino experiments have an energy resolution of 5-6%.) The DayaBay collaboration has identified a suitable detector location at a distance of 60 km to several reactor cores with a total of 17.4 GW power (and another 17.4 GW in the planning stage) and is pursuing the possibility of a mass hierarchy measurment as the DayaBay-II project [81]. Let us also mention that such a big reactor experiment at 60 km would provide ultimate precision on the determination of 𝜃₁₂ and Δ𝑚²₂₁; see, for example, [82].

5. High-Precision Long-Baseline Facilities

As discussed above, the next generation of neutrino experiments will have some sensitivity to matter effects, and it may be possible to have the mass hierarchy determined by 2025. The search for CP violation is more challenging, and it is unlikely that its discovery can be achieved in the same time frame. Upgraded long baseline experiments with larger statistics and better control of systematics will be needed.

In order to understand how these experiments will be sensitive to CP violation and will achieve precise measurements of the oscillation parameters, let us consider the approximate formula for the oscillation probability 𝑃_{𝜈𝜇→𝜈𝑒} given in (3). The first term in the probability is the “atmospheric term” which is dominant as 𝜃₁₃ is large, sin²2𝜃₁₃≈0.09. This is the term which is most sensitive to matter effects and drives the ability of coming and next-generation experiments to establish the mass hierarchy. The second line is the “CP term” which contains the dependence on the CP violating phase 𝛿. As we see, this term becomes more important at lower energies, and for this reason access to the low energy part of the spectrum is critical to achieve good sensitivity to CP violation. It should also be noted that for large 𝜃₁₃ the “CP term" is a small correction with respect to the dominant “atmospheric term,” and in fact the CP asymmetry, defined as (𝑃_{𝜈𝜇→𝜈𝑒}−𝑃_{𝜈𝜇→𝜈𝑒})/(𝑃_{𝜈𝜇→𝜈𝑒}+𝑃_{𝜈𝜇→𝜈𝑒}), scales as sin⁻¹𝜃₁₃ and is suppressed for large 𝜃₁₃. Therefore, despite the fact that large 𝜃₁₃ implies large number of events at future LBL facilities, the discovery of CP violation remains very challenging and requires precise measurements of the probabilities, with small statistical and systematic errors, (It has been shown that CP violation can also be searched for in short baseline experiments, such as (a Decay-At-rest Experiment for CP studies At the Laboratory for Underground Science) [83] DAEδALUS. This uses high-power proton accelerators to produce a 𝜈_𝜇 beam with energies in the few tens of MeVs. The appearance oscillation 𝑃_{𝜈𝜇→𝜈𝑒} will be detected via inverse beta decay by a large WC detector doped with gadolinium, to reduce the backgrounds, or scintillator detectors. Further details are provided elsewhere in this volume).

Moreover, as it can be seen from the probability equation, CP-violating and matter effects are entangled, and the extraction of the parameters of interest, namely, the sgn(Δ𝑚²₃₁), the phase 𝛿, and 𝜃₂₃, is affected by the widely studied problem of degeneracies: different sets of parameters give the same probabilities in the neutrino and in the antineutrino channels at fixed 𝐿/𝐸 [20, 29, 47, 84, 85]. Therefore, even a very precise reconstruction of the probabilities does not allow to determine the true parameters, and the physics reach is severely affected. In vacuum three degeneracies can be identified. (i) The intrinsic degeneracy: 𝜃₁₃, 𝛿 have fake solutions which strongly depend on energy. For large 𝜃₁₃ and in vacuum, the "fake" solutions are given by [84] 𝜃^′₁₃≃𝜃₁₃+cos𝛿sin2𝜃₁₂Δ𝑚²₂₁𝐿4𝐸cot𝜃₂₃cotΔ𝑚²₂₁𝐿.4𝐸 (13)

(ii) The sign degeneracy: in absence of matter effects, it is possible to change the sign of Δ𝑚²₃₁ and 𝛿 to 𝜋−𝛿 without affecting the probabilities. In matter this degeneracy is broken. (iii) The octant of 𝜃₂₃: if the angle is not maximal as currently suggested by the data, see Section 1.

The problem of degeneracies has significant impact on the precision of the oscillation parameter measurements, and in particular on the ability to establish CP violation. A lot of effort has gone into devising strategies to weaken the impact of the degeneracies; see, for example, [47, 70, 86–102]. Long enough baselines (>800–1000 km) have strong matter effects and can be used to solve the sign degeneracy; information at several energies, for example, by using wideband beams, is important, with the one coming from the low-energy part of spectrum being critical for CP violation and the octant degeneracy. Other techniques have also been explored, for example, combining different channels which have different dependence on the parameters, or different baselines and/or typical energies. Several of these studies have been performed focusing on small values of 𝜃₁₃. Thanks to the large value of 𝜃₁₃, some of the degeneracies become less important or more easily solved. For instance, determining the mass hierarchy will be easier than previously expected, and it is even possible that it will be achieved prior to the start of the next generation of long baseline experiments, as discussed in Section 4.

In this section we briefly review future LBL experiments which will provide improved sensitivity to the mass hierarchy and CP violation and the possibility to measure with various degree of precision the oscillation parameters. Several type of setup are under consideration, some of them being at the design study or proposal level and others more advanced: superbeams (LBNE, T2HK, LAGUNA-LBNO), betabeams, and neutrino factory (NF). In the following, we briefly review these three different types of facilities. We focus on their physics reach and we defer the readers to the relevant chapters in this volume for a detailed description of the experimental facilities.

5.1. Future Superbeams

Superbeams are based on currently used technology and require an upgrade in neutrino flux and detector size. The beam is constituted mainly by muon neutrinos which are produced by pion and kaon decays. The experiments search for the 𝜈_𝜇→𝜈_𝜇 probability, sensitive to Δ𝑚²₃₁ and 𝜃₂₃, and, importantly, for the subdominant oscillation 𝜈_𝜇→𝜈_𝑒. Detectors with excellent 𝜈_𝑒 reconstruction are needed, the technologies of choice being Water-Cherenkov, Liquid Argon (LAr), or scintillator (LSc) ones. A wide range of energies is currently under consideration, going from 200 MeV of SPL to several GeVs for CN2PY (CERN to Pyhäsalmi), with corresponding distances from 100 km to 2300 km. The detector can be located on-axis or off-axis: in the first case it sees a wide spectrum, while in the latter the beam is peaked at low energies and its high energy tail is suppressed. They typically have an excellent reach for the mass hierarchy, if 𝐿>1000 km or so, and very good sensitivity to CP violation. The main limiting factor is the intrinsic 𝜈_𝑒 contamination of the beam, at a level of 0.5%–1%. Another very important experimental issue is the background due to the misidentified 𝜋⁰ produced in neutral current (NC) interactions, as one of the 𝛾s from the pion decay 𝜋⁰→𝛾𝛾 is missed. This background is particularly important at low energies for CP violation searches and impacts differently beams at different energies and with different detectors; for instance, LAr ones have an excellent NC rejection. For the antineutrino channel, a significant contribution to the signal and background can also come from the 𝜈_𝜇 and 𝜈_𝑒 components of the beam, in absence of detector magnetisation. Systematic errors are an important factor: for large 𝜃₁₃, as the “atmospheric term" dominates the appearance oscillation probability, those on the signal are more relevant than those on the backgrounds and need to be controlled at the few % level. Various superbeam options are under study or being proposed for the future.

LBNE [103]: in the US the Long-Baseline Neutrino Experiment (LBNE) is the most advanced proposal for a next generation long baseline option. According to the 2010 LBNE Interim Report [103], the beam is sourced at the Main Injector at Fermilab using a new neutrino beamline with 700 kW of power. Its main requirements include a broad beam which covers both first and second oscillation maxima located at 2.4 and 0.8 GeV, respectively, an increased flux at low energy in order to compensate for the lower detection cross the sections, a suppression at energies above 5 GeV in order to reduce the NC backgrounds which pile up at low energy, and the lowest level of 𝜈_𝑒 contamination possible. The detector is located at the DUSEL site, at a distance of 1300 km from Fermilab. Various options were contemplated in the 2010 LBNE Interim Report: two or three 100-kton fiducial mass Water Cherenkov detectors with 15% or 30% PMT coverage and with or without gadolinium loading, or multiple 17-kton fiducial mass LAr detectors or a combination of them. The location could be at 4850, 800, or 300 feet depth, depending on the emphasis put on nonaccelerator physics, such as proton decay, supernova, and other astrophysical neutrinos. The experiment was assumed to run for 5 years in neutrinos and 5 in antineutrinos. In this configuration, it could achieve the determination of the mass hierarchy in less then a year at 3𝜎 as shown in Figure 8. LBNE could also have very good sensitivity to CP violation with a 60% coverage at 3𝜎 in the allowed range of values of sin²2𝜃₁₃, for a 200 kton Water Cherenkov or 34 kton LAr detectors, see Figure 9.

(a)

(b)

Figure 8 

LBNE sensitivity to the mass hierarchy at 3𝜎 as a function of the exposure for WC (a) and LAr (b) detectors. The value of 𝛿 is fixed at 0 but a similar behaviour is expected for other values of the phase. Figures taken from [103].

(a)

(b)

Figure 9 

LBNE sensitivity to CPV at 3𝜎 (red curves) and 5𝜎 (blue curves) for 5 years of running in the neutrino channel plus 5 in the antineutrino one. A 200 kton WC (a) or a 34 kton LAr (b) are used in a 700 kW beam. Figure is taken from [103].

In 2011, the technology choice was made, favouring a LAr detector thanks to its excellent performance in energy resolution, efficiency, and background reduction for the range of energies of interest. In 2012, due to funding restrictions, the LBNE configuration was reconsidered, and a severe reduction of the detector size was necessary at a first stage, keeping open the possibility of an upgrade to a large/multiple detector at a later time. Three options were considered for the first step: (i) a beam from the the existing NuMI beamline in the low-energy configuration with a 30-kton LAr detector located at the surface 14 mrad off-axis at Ash River in Minnesota, 𝐿=810km, (ii) using the beam above but with a 15-kton detector at the Soudan mine in Minnesota, 𝐿=735km as MINOS, (iii) a new low-energy LBNE beamline aimed at a 10-kton LAr detector at Homestake (on-axis) in South Dakota, 𝐿=1300km. A report by the Steering Committee [104] was prepared and the reach of each option is analysed in detail, see Figure 10. The report favoured option (iii). This configuration offers the best opportunities for a long-term programme with a 20–25 kton underground detector at Homestake and a Project X sourced beam. The recommendation was very well received and on 29 June 2012 DOE confirmed that CD1 will be reviewed towards the end of October 2012.

(a)

(b)

(c)

(d)

Figure 10 

LBNE sensitivity to the mass hierarchy (upper plots) and CPV (lower plots) at 3𝜎 as a function of 𝛿 for the three reconfiguration options, as described in the text. The sensitivities are reported for the experiment alone (left) and when combined with NOvA for 3𝜈+3𝜈 years and T2K. The Ash River and Soudan options use the NuMI beam line, and therefore additional 5𝜈+5𝜈 years for NOvA are included, assuming that this detector will be kept in operation in parallel to the LAr detector. Figure taken from [104].

LAGUNA-LBNO (CN2PY) [105–107]: in Europe a next-generation superbeam experiment with a beam sourced at CERN is being considered in the LAGUNA and LAGUNA-LBNO FP7 Design Studies, funded by the European Commission. The LAGUNA project, which is finished in 2011, considered seven possible locations for a European large underground laboratory which could host a megaton-scale detector for neutrino, astroparticle physics, and proton decay searches. The Design Study focussed on site investigations and on the development of the design of a facility for the neutrino underground observatory. Three detector technologies were considered: 100 kton liquid argon [108], 50 kton liquid scintillator [109, 110], and 440 kton Water Čerenkov [111] detectors. The study concluded that all locations would in principle allow to host the facility. Importantly, the chosen detector could also be the target for a superbeam from CERN. Depending on the site, the available distances, see Table 3, range from 130 km for Fréjus to 2300 km for Pyhäsalmi, the longest baseline considered at present for superbeams. The Design Study LAGUNA-LBNO, which started in October 2011, is further developing the study of the beam and the physics reach of the long baseline setup, with focus on the CERN to Pyhäsalmi option for the first phase and various options for a second stage.

Table 3 

The seven potential locations for an underground neutrino observatory under consideration in the LAGUNA Design Study: the distance from CERN and the energy of the first oscillation maximum, in the absence of matter effects, are given. From [106].

A detailed study of the sensitivity to the mass hierarchy and CP violation has been performed in [107]. In the lowest energy configuration of the beam (the 𝐿=130km baseline), the simulation assumed 5.6×10²² protons on target (PoT) per year, with an energy of 4.5 GeV, for 2 (8) years of running for neutrino (antineutrinos). In the multi-GeV regime, used for baselines with 𝐿>130km, the CERN high-power PS2 configuration was considered with 3×10²¹ PoT per year with 50 GeV, corresponding to 2.4 MW with 10⁷ useful second per year (or 1.6 MW assuming 1.5×10⁷ seconds per year). Given the large value of 𝜃₁₃, most of the configurations can determine the ordering of neutrino masses at high confidence level, as shown in Figure 11, with increased sensitivity for longer baselines. In view of this, the possibility of staging the detector, starting from a “pilot” detector of 10 to 20 kton mass, to be later upgraded to reach the baseline configuration of 100 kton, has been considered [112]. The study shows that the mass hierarchy can be reached in few years of data taking; see Figure 12.

(a)   LAr

(b) WC

(c) LSc

Figure 11 

LAGUNA hierarchy discovery at 3𝜎 as a function of sin22𝜃13 and 𝛿. The different lines correspond to different baselines, as indicated in the legend in km. The details of the beams used are reported in the text. The detectors are 100 kton LAr TPC, 50 kton LSc, and 440 WC detector. The shaded region reports the 3𝜎 allowed values from Daya Bay. Figure is taken from [107].

(a)

(b)

(c)

(d)

Figure 12 

CP fraction for which a mass hierarchy (upper plots) and CP violation (lower plots) discovery at 3𝜎 (left) and 5𝜎 (right) is possible as a function of exposure for a staged LAGUNA setup. The different lines correspond to true normal (inverted) hierarchy for solid (dashed) lines and for a baseline of 2290 km (1540 km) for red (blue) lines. Figure is taken from [112] where more details about the simulations can be found.

These setups can also provide excellent sensitivity to CP violation both for the short baselines with a Water Cherenkov detector and the longer ones with a LAr detector. The LSc option could provide similar reach but only if the NC background could be controlled at a similar level. Current studies seem to indicate that NC could not be rejected at more than the 10%–20% level, severely affecting the sensitivity to CP violation for this type of detector. Typically, for the relevant range of values of 𝜃₁₃, CP violation can be established at 3𝜎 for ~70% of the values of 𝛿 phase and good reach is obtained even at the 5𝜎 level; see Figures 12 and 13. It should be noted that sufficiently long baselines, such as the 2300 km one, and a broad spectrum with good energy resolution allow to have an excellent separation of the asymmetry due to matter effects (i.e., the mass hierarchy measurement) and the CP asymmetry and thus to break the parameter degeneracies discussed above. Therefore, the existence of matter and CP violation-induced effects will be tested explicitly, without overrelying on theoretical modelling and assumptions.

(a) LAr  

(b) WC

Figure 13 

LAGUNA CP fraction for which a CP violation discovery at 3𝜎 (left) and 5𝜎 (right) is possible as a function of sin22𝜃13 for the LAr (upper plots) and WC (lower plots) detector. Figure is taken from [107].

J-PARC to Hyper-Kamiokande long baseline experiment (T2HK) [66]: recently a letter of intent (LoI) has been published for a long baseline experiment which uses a 1.66 MW beam from the J-PARC accelerator to a 1 Mton Water Cherenkov detector located 2.5^∘ off-axis at 295 km distance in the Kamiokande site. Its main goal is the discovery and/or measurement of CP violation in the leptonic sector. One of the advantages of this configuration is the excellent energy resolution provided by the WC detector at these energies, the large number of events, and, thanks to the off-axis location and the beam configuration, the low level of intrinsic background, <1%. A running of 1.5 (3.5) years for neutrinos (antineutrinos) is assumed, with one year given by 10⁷ seconds. Systematic errors play an important role and, based on foreseen improvements with respect to T2K, a level of 5% is assumed for the neutrino flux uncertainty, the neutrino interaction cross-section, the near detector efficiency, and the far detector systematics. The baseline is too short to provide a good reach for the mass hierarchy, with some sensitivity only for favourable values of 𝛿. Additional information can be obtained from atmospheric neutrino events [67], as discussed in Section 4. The knowledge of the mass hierarchy plays an important role as it cannot be determined by the experiment itself but can induce significant degeneracies for large 𝜃₁₃. If the mass hierarchy is known, CP violation can be established at 3𝜎 for ~70% of the values of 𝛿 for sin²2𝜃₁₃>0.03. In the opposite case, there is a loss of ~20% of the coverage in 𝛿 for sin²2𝜃₁₃=0.1. A summary of the reach is reported in Figure 14.

(a)

(b)

Figure 14 

T2HK sensitivity to CP violation at 1, 2, and 3𝜎 as a function of sin22𝜃13. The mass hierarchy is assumed known (a) or not (b). Figure is taken from [66].

SPL [68, 113]: another superbeam configuration is under consideration in Europe within the EUROnu Design study. This setup exploits a 4 MW beam to produce a very low energy superbeam aimed at a 440 kton MEMPHYS Water Cherenkov detector located 130 km away at Fréjus. The very high intensity of the beam and very large detector compensate for the low detection cross-section and excellent sensitivity to CP violation can be reached; see Figure 15. CP violation can be found at 3𝜎 for ~67% of the values of 𝛿 for sin²𝜃₁₃=0.1 [68] (see also [113]). Due to the short distance, no matter effects arise and no sensitivity to the mass hierarchy can be achieved from long baseline neutrino oscillations. However, given the recently discovered large value of 𝜃₁₃, taking into account atmospheric neutrino events will allow to find the hierarchy for sufficient exposure; see Figure 6 [68].

Figure 15 

CP violation discovery at 3𝜎 as a function of 𝛿 for a betabeam, SPL, and T2HK. The combination of SPL and a betabeam is also shown. The betabeam lines correspond to 5.8(2.2)×1018 decays per year of 6He (18Ne). Details of the simulations can be found in [68]. The width of the bands reflects the change of the systematic errors from 2% to 5%. The dashed curves are obtained for a betabeam flux reduced by a factor of two.

5.2. Betabeams

Betabeams [114–116] have been proposed as an alternative type of setup which uses a very pure beam of electron neutrinos produced by beta-decays of highly accelerated ions. In this case, the main oscillation channel is the 𝜈_𝑒→𝜈_𝜇, one which provides sensitivity to the mass hierarchy and CP violation. The neutrino spectrum is very well known and depends on the 𝑄 value of the beta-decay and on the 𝛾 factor of the ions. In a given accelerator, for example, the Main Injector at Fermilab or the SPS at CERN, fully stripped ions can be accelerated to a maximum 𝑍/𝐴 times the proton energy, with 𝑍 and 𝐴 the number of protons and of nucleons, respectively. After the initial idea, subsequent studies were performed in the context of EURISOL and EUROnu Design Studies. The ions, which have suitable lifetimes and can be copiously produced, are the combinations: ⁶He, ¹⁸Ne, and ⁸Li, ⁸B, for 𝜈_𝑒 and 𝜈_𝑒 beams. The former has 𝑄 values of 3.5 MeV and 3.3 MeV, respectively, while ⁸Li, ⁸B of 13.0 MeV and 13.9 MeV, respectively. The latter ions will yield higher neutrino energies for a given 𝛾, but the flux will be lower for the same energy as it scales as 𝛾⁻². All of these isotopes need to be produced artificially, and the production rate turns out to be a limiting factor for the physics reach of the facilities. ⁶He, ¹⁸Ne pose less-significant challenges from the production point of view but do not allow to reach very high energies, while the other ions could provide higher energies without the need for high 𝛾-factors, but, due to the challenges of production, it is still not clear what fluxes could be achievable.

The 𝛾=100 option for a betabeam which uses ⁶He, ¹⁸Ne has been studied in detail within the EURISOL Design Study. Given the very low energies, the most suitable baseline is the CERN to Fréjus one of 130 km. In principle, higher 𝛾 factors could be achieved if a significant upgrade of the present accelerators is envisaged [117]; see also [70, 118–125]. In this case, higher energies and consequently longer distances could be used which provide sensitivity not only to CP violation but also to matter effects. The ideal detector, given the low energies of the beam, is MEMPHYS, a one megaton Water-Cerenkov detector, which has excellent energy resolution and efficiency. Compared to superbeams, betabeams have an extremely pure beam, with no contamination from other flavours at the source. On the other hand, the absence of a 𝜈_𝜇 component implies that a betabeam cannot provide a precision measurement of 𝜃₂₃. Due to the short distance, no sensitivity to the mass hierarchy is achievable, as in the case of the SPL, unless atmospheric neutrinos are included [68]. Excellent reach for CP violation could be obtained, especially if the betabeam is combined with a superbeam from CERN to Fréjus. The two setups are sensitive to the T-conjugated channels, providing a clean measurement of the CP-violating phase 𝛿; see Figure 15. Moreover the betabeam-superbeam combination offers also improved sensitivity to the mass hierarchy, even in the case of short baselines [69]; see Figure 6.

5.3. Neutrino Factory

In a Neutrino Factory [126–128] neutrinos are produced by highly accelerated muons which decay producing a highly collimated beam of muon and electron neutrinos. The spectrum is very well known and high energies can be achieved: the wide beam and high energies allow to reconstruct with precision the oscillatory pattern and typically achieve a superior performance with respect to the other options. Let us consider the decay of 𝜇⁻ (𝜇⁺): it will generate an initial beam with two neutrino components, 𝜈_𝜇 and 𝜈_𝑒 (𝜈_𝜇 and 𝜈_𝑒). These will oscillate inducing also 𝜈_𝑒 and 𝜈_𝜇 (𝜈_𝑒 and 𝜈_𝜇). At the detector, for muon-like events, two different signals will be present: the right-sign muon events which derive from the observation of 𝜈_𝜇 coming from the disappearance channel, 𝜈_𝜇→𝜈_𝜇, and the wrong-sign muon events which are due to 𝜈_𝑒→𝜈_𝜇 oscillations. As the appearance oscillation is sensitive to matter effects and CPV, it is necessary to distinguish the two signals. This is achieved by means of magnetized detectors which can distinguish 𝜇⁺ from 𝜇⁻ events. The mis-Id rate is typically very low at a level of 10⁻⁴−10⁻³, depending on the detector technology. The detector of choice [129] is an iron-magnetized detector (MIND) which provides excellent background rejection and very good energy resolution but low detection efficiency for neutrinos with energies in the few GeV range. This detector performs very well for high energies and is the default choice for muon energies above 8 GeV. For lower energies, detectors with lower-𝑍 would be preferred, such as a magnetized Totally-Active Scintillator Detector (TASD) or LAr. The latter detectors provide excellent efficiency for neutrinos with low energies, excellent energy resolution, and low backgrounds, but their magnetization is extremely challenging and not proven yet for the mass scales of interest. (In [130] the possibility to use nonmagnetized detectors for a neutrino factory has been put forward, which may become an option for large 𝜃₁₃ and offers an interesting synergy with large-scale detectors for nonaccelerator physics. More detailed studies along these lines would be required, and this option is currently not considered within the context of neutrino factory study groups.)

The initial baseline configuration of the NF [129] used muons with an energy of 25 GeV and two different baselines, at approximately 4000 and 7500 km, with two MIND detectors, a 100 kton one at the shorter baseline and a 50 kton one at the “magic” baseline [131]. This second baseline was designed to provide a very powerful determination of the mass hierarchy and a clean determination of 𝜃₁₃, thanks to the strong suppression of the “CP term" due to sinΔ𝐴∼0, and to complement the shorter baseline in the search of CP violation, helping to resolve the degeneracies. Additional studies can be found, for example, in [84, 132–141]. This setup was optimised assuming small values of 𝜃₁₃, and several studies showed that it would outperform all other options for small 𝜃₁₃, thanks to its high number events, very low backgrounds, and small systematic errors [129, 140].

In the case of large 𝜃₁₃, a more conservative setup, named the Low-Energy Neutrino Factory (LENF), was proposed as a less-challenging option [142, 143] which used a single baseline of 1300km, corresponding to the Fermilab to DUSEL distance, and, consequently, a lower muon energy, at ~4.5 GeV [144]; see also [145, 146]. Given the low energy, a detector with good-energy resolution and low-energy threshold was needed in order to exploit the rich oscillatory pattern. The detector of choice was a Totally-Active Scintillator Detector (TASD) magnetized by means of a large magnetic cavern or a magnetized LAr TPC, which would be ideal due to the large size and the excellent detector performance, especially at low energy. This initial study showed that excellent reach could be achieved for the mass hierarchy and CP violation; see Figure 16. A subsequent study of the LENF using a Magnetized Iron Neutrino Detector has also shown a promising performance [147], and its reach is reported in Figure 17. A similar study using a TASD and LAr detector [148] found a rather flat performance as a function of 𝐿 and muon energy, for large 𝜃₁₃, as seen in Figure 18. Based on these analyses and in view of the discovery of large 𝜃₁₃, the International Design Study on a Neutrino Factory (IDS-NF) reviewed the baseline configuration in April 2012 and chose a LENF with MIND detector with muon energy of 10 GeV and baseline of 2000 km.

(a)

(b)

Figure 16 

(a) 3𝜎 sensitivity to the mass hierarchy, in terms of the fraction of 𝛿 values for which 𝛿=0, 𝜋 can be excluded, as a function of sin22𝜃13 with a LENF with 20 kton TASD, LENF with 100 kton LAr detector (the band corresponds to the varying detector performance), high energy neutrino factory, a wide band beam, 3 betabeam configurations and for T2HK. Figure is taken from [144] where further details of the simulations are reported. (b) Same as the left but for CP violation.

Figure 17 

Fraction of the values of 𝛿 for which there is sensitivity to CP violation at 3𝜎 as a function of 𝐿 and muon energy. A single baseline Neutrino Factory is used with a 50 kton MIND detector. The optimal performance is marked by a dot. Figure is taken from [147].

(a)

(b)

Figure 18 

CP violation discovery at 3𝜎 fraction as a function of baseline, 𝐿, and stored-muon energy using the TASD detector (a) and LAr one with optimistic performance (b). The lines indicate the fraction of values of 𝛿 for which sensitivity can be achieved. Further details can be found in [148] from which the figures have been taken.

A summary of the results for the LENF and a comparison with other facilities are given in Figure 19 [149]. A table which summarises the setups described and their reach for CP violation is given in Table 4. A word of caution is necessary as the precise reach of each setup is affected by the assumption made on the beam, detector, and systematic errors. Nevertheless, thanks to the intense flux, pure beam, excellent background rejection, and long baselines, an NF has been shown to achieve the best physics reach in search for CP violation and the mass hierarchy.

Table 4 

Summary of the setups described in the text. From left to right, the columns list the names of the setups, the beam power for superbeams or 𝛾 factor for beta beams or muon energy for neutrino factories, the baseline, the detector choice, the running time in years for each polarity (the neutrino and antineutrinos runs are simultaneous for the neutrino factory), the systematic error on the signal and background, the fraction of the values of 𝛿 for which CP-violation could be determined at 3𝜎 at sin²2𝜃₁₃=0.1, and the reference from which the information has been collected. For LBNO with 10 kton and 30 kton LAr detectors, the values of the fraction of 𝛿 are given separately for a true normal (NH) or inverted (IH) hierarchy. The number of seconds per year is 1.7×10⁷ for LBNO and 1×10⁷ for T2HK. BB + SPL considers the combination of the betabeam with 1.1×10¹⁸ (2.8×10¹⁸) ¹⁸Ne (⁶He) useful decays per year together with the SPL described above (with a 500 kton WC detector instead of a 440 kton one). The LENF uses 1.4×10²¹ useful muon decays per year per polarity, and NF10 7×10²⁰. The results quoted depend significantly on the assumptions made in the analysis and should be treated carefully.

Figure 19 

δ-fraction for 3𝜎 CP violation sensitivity for an upgrade of NOvA with a 30 kton LAr detector (NOvA+), T2HK, a WBB with a baseline of 2300 km and a 100 kton LAr detector (WBB), a low energy betabeam combined with the SPL superbeam and aimed at WC detector at 130 km, a higher energy betabeam (BB350), a neutrino factory with 10 GeV (NF10) and 5 GeV (NF5) muons and a 100 kton MIND detector, as well as the expected reach by 2020 by combining NOvA, T2K, and reactor data. Figure is taken from [149].

Once CP violation is discovered, it will be important to measure the values of the phase and of 𝜃₁₃ with high precision. In fact, in many models of leptonic flavour, these values are correlated with the deviations from maximality of 𝜃₂₃ and/or with parameters in the quark sector. Examples of the precision achievable have been typically included in the analysis of the setups, but a comprehensive and detailed study is still at its beginnings. A first comparison between different experiments has been performed in [46]. The main results are reported in Figure 21 and indicate that for 𝜃₁₃ reactor experiments, and in particular Daya Bay, will achieve the best precision, marginally improved by a LENF. The precise measurement of the 𝛿 phase depends significantly on the true value itself, with a significant loss of precision around 𝜋/2 for experiments such as T2HK and beta beams. If instead matter effects are relevant, as it is the case for CN2PY and LENF, the error on 𝛿 tends to become more uniform in 𝛿, and the best performance is given by the LENF which can typically achieve an error of around 5^∘.

5.4. Precision Measurements

With the discovery of large 𝜃₁₃, the focus of future long baseline experiments has shifted not only to the discovery of the mass hierarchy and CP violation as discussed above but also to the precise measurements of the oscillation parameters. Among these, determining if 𝜃₂₃ is maximal or not is of great theoretical importance together with establishing its octant, if nonmaximal. The experiments discussed in the previous subsections, except beta beams, will have sensitivity to these parameters mainly via the disappearance channels 𝜈_𝜇→𝜈_𝜇 and 𝜈_𝜇→𝜈_𝜇. Typically superbeams have a very good reach and a LENF can perform better especially if the low energy part of the spectrum can be reconstructed. It has been shown that the contamination from 𝜈_𝜏 events, coming from 𝜈_𝜇→𝜈_𝜏 oscillations, can have a significant impact on the high energy neutrino factory [150, 151]. It is expected that a lower muon energy will reduce the number of 𝜏 events but their impact in the 10 GeV LENF needs to be fully explored. In Figure 20 we report two examples of the capability of LBNE in the 2010 configuration and of the 4.5 GeV LENF for studying 𝜃₂₃.

(a)

(b)

Figure 20 

(a) LBNE sensitivity to resolve the 𝜃23 octant degeneracy. The standard configuration in 2010 has been used. The blue (green) band shows the results for 200 kt WC (34 kt LAr) and its width is obtained by varying 𝛿, ranging from 10% to 90% CP fraction. In the region above the bands, the 𝜃23 octant can be determined at 90% CL (lower bands) and 3𝜎 (upper bands). Taken from [103]. (b) Nonmaximal 𝜃23 discovery potential, with 𝛿𝜃23≡𝜃23−45∘, as a function of sin22𝜃13, for the LBNO superbeam with 100 kton LAr, 440 kton WC and 50 kton LSc detectors. In the region enclosed by each couple of lines, 𝛿𝜃23=0 cannot be ruled out at a statistical significance of 3𝜎. Results are shown for the detector placed at Pyhäsalmi and for a true normal hierarchy. Figure taken from [107].

(a)

(b)

Figure 21 

1𝜎 precision on 𝜃13 (a) and 𝛿 (b) for the CN2PY and T2HK superbeams, the 𝛾=350 beta-beam, and the LENF. On the left plot, the empty triangle represents the current precision for Daya Bay, and the star the ultimate attainable precision, assuming as true value the present Daya Bay best fit value. The width of the bands represent the dependence of the error on 𝜃13 on 𝛿 and vice versa. For further details, see [46] from which the figure is reproduced.

6. Conclusions

Since the discovery of neutrino oscillations huge progress has been made and this phenomenon is now well established. Yet new important questions are open for the future, including what is the ordering of neutrino masses? Is there CP violation in the leptonic sector? What are the precise values of the neutrino mixing parameters? Are there new phenomena beyond the three-neutrino framework? With the recently discovered relatively large value of the mixing angle 𝜃₁₃, addressing those questions by upcoming oscillation experiments becomes a realistic possibility. In this paper, we have reviewed the phenomenology of oscillation experiments by discussing some aspects of the determination of neutrino oscillation parameters by present global data, and we have tried to give an outlook for possible future developments. In the near term (the next 10 to 15 years), the interplay of complementary data sets will be important, such as long-baseline accelerator experiments, reactor experiments, and atmospheric neutrino experiments. We have discussed the potential to address questions like the nonmaximality and the octant of 𝜃₂₃ and the determination of the neutrino mass hierarchy. In order to address CP violation, it seems from the current perspective that a more long-term program will be necessary. Future high-precision long-baseline neutrino experiments can provide crucial answers to the above questions by studying the subdominant 𝜈_𝜇→𝜈_𝑒 transitions. A wide experimental program for the future is underway or at the discussion stage and includes superbeams, betabeams and neutrino factory. A table which summarises various setups and their reach for CP violation is given in Table 4.

Acknowledgment

The authors acknowledge partial support from the European Union FP7 ITN INVISIBLES (Marie Curie Actions, PITN-GA-2011-289442).