Super-PINGU for measurement of the leptonic CP-phase with atmospheric neutrinos

We explore a possibility to measure the CP-violating phase $\delta$ using multi-megaton scale ice or water Cherenkov detectors with low, $(0.2 - 1)$ GeV, energy threshold assuming that the neutrino mass hierarchy is identified. We elaborate the relevant theoretical and phenomenological aspects of this possibility. The distributions of the $\nu_\mu$ (track) and $\nu_e$ (cascade) events in the neutrino energy and zenith angle $(E_\nu - \theta_z)$ plane have been computed for different values of $\delta$. We study properties and distinguishability of the distributions before and after smearing over the neutrino energy and zenith angle. The CP-violation effects are not washed out by smearing, and furthermore, the sensitivity to $\delta$ increases with decrease of the energy threshold. The $\nu_e$ events contribute to the CP-sensitivity as much as the $\nu_\mu$ events. While sensitivity of PINGU to $\delta$ is low, we find that future possible upgrade, Super-PINGU, with few megaton effective volume at ($0.5 - 1$) GeV and e.g. after 4 years of exposure will be able to disentangle values of $\delta = \pi/2,~ \pi,~ 3\pi/2$ from $\delta = 0$ with"distinguishability"($\sim$ significance in $\sigma$'s) $S_{\sigma}^{tot} = (3 - 8),~ (6 - 14),~(3 - 8)$ correspondingly. Here the intervals of $S_{\sigma}^{tot}$ are due to various uncertainties of detection of the low energy events, especially the flavor identification, systematics, {\it etc.} Super-PINGU can be used simultaneously for the proton decay searches.


I. INTRODUCTION
Discovery of the leptonic CP violation and measurement of the Dirac CP phase are among the main objectives in neutrino physics and, in general, in particle physics. They may have fundamental implications for theory and important consequences for phenomenology (atmospheric neutrinos, accelerator neutrinos, high energy cosmic neutrinos, etc.) [1].
The present experimental results have very low sensitivity to δ CP giving only weak indications of preferable interval of its values. Thus, the T2K and reactor data favor the interval δ CP = (1 − 2)π with central value δ CP = 1.5π [2]. Analysis of the SuperKamiokande atmospheric neutrino data gives preferable range (1.2 ± 0.5)π [3]. The global fit of all oscillation data e.g. from [4] agrees with these results: δ CP ≈ 1.39 +0. 38 −0.27 π (1σ) and no restriction appears at 3σ level. The values δ CP ∼ π/2 are disfavored. Similar results with the best fit value δ CP = 1.34 (NH) have been obtained in [5].
A possibility to measure δ CP is generally associated with accelerator neutrino beams.
There is certain potential to improve our knowledge of δ CP with further operation of T2K and NOvA experiments [6]. More remote proposals of experiments which will measure δ CP with reasonable accuracy include LBNE [7], J-PARC -HyperKamiokande [8] and LBNO [9].
Further developments can be related to the low energy neutrino and muon factories, beta beams, etc., see [1].
Another possibility to determine δ CP is to use the atmospheric neutrino fluxes and large underground/underwater detectors. Sensitivity of future atmospheric neutrino studies by HyperKamiokande (HK) has been estimated in [8]: During 10 years of running with fiducial volume 0.57 Mton the HK will be able to discriminate the values of phases δ CP = 40 • , 140 • , 220 • , 320 • at about (1 − 1.5)σ CL. ICAL at INO alone will have very low sensitivity, but combined with data from T2K and NOvA, it will reduce degeneracy of parameters, and thus, increase the global sensitivity [10].
Various theoretical and phenomenological aspects of the CP-violation in atmospheric neutrinos have been explored in a number of publications before [11][12][13][14][15][16][17][18][19][20][21]. In particular, pattern of the neutrino oscillograms (lines of equal probabilities in the E ν − cos θ z plane) with CP violation has been studied in details in [14]. It was realized that structure of oscillograms is determined to a large extent by the grid of the magic lines of three different types [14,[22][23][24]. Although at the probability level the effects of the CP-violation can be of order 1, there are a number of factors which substantially reduce the effects at the level of observable events.
Capacities of new generation of the atmospheric neutrino detectors (PINGU, ORCA) have been explored recently [16,[25][26][27]. It was found [16,26] that these detectors with E th ∼ 3 GeV have good sensitivity to the neutrino mass hierarchy and the parameters of the 2-3 sector (the 2-3 mixing and mass splitting). However, the CP-violation effects turn out to be sub-leading. This helps in establishing the hierarchy without serious degeneracy with δ CP in contrast to the accelerator experiments, but the information on the CP-phase will be rather poor.
In this paper we explore sensitivity of future big atmospheric neutrino detectors to δ CP , assuming that the neutrino mass hierarchy is identified. We will show that in spite of averaging of oscillation pattern over the neutrino energy and direction, the CP-violation effects are not washed out and furthermore increase with lowering of the energy threshold E th . This opens up a possibility to measure δ CP using multi-megaton scale ice or water Cherenkov detectors with E th = (0.2 − 0.5) GeV. We study dependence of the energy and zenith angle distributions of events produced by ν e and ν µ on the CP phase. We estimate distinguishability of different values of δ CP . According to the present proposal [26] PINGU will have low sensitivity to δ CP and only further upgrades, which we will call Super-PINGU, can measure δ CP with competitive accuracy.
The paper is organized as follows. In Sec. II we summarize relevant information on the oscillation probabilities and their dependence on CP-phase. We present the probabilities in quasi-constant density approximation. The grid of the magic lines will be described.
In Sec. III we consider a possible upgrade of PINGU -Super-PINGU which will be able to measure δ CP and outline a procedure of computation of numbers of events. In Sec. IV we compute distributions as well as relative differences of distributions of the ν µ events in the E ν − cos θ z plane (the relative CP-differences) for different values of δ CP . We study dependence of these distributions on δ CP before and after smearing over the neutrino energy and direction. In sec. V we perform similar studies of the cascade (mainly ν e ) events. Sec.
VI contains estimations of sensitivity of Super-PINGU to δ CP and discussion of our results.
We conclude in Sec. VII.

A. Oscillation amplitudes and probabilities
We will study the CP-violation phase δ CP defined in the standard parametrization of the PMNS mixing matrix, U P M N S = U 23 I δ U 13 I * δ U 12 , where U ij is the matrix of rotation in the ij-plane and I δ ≡ diag(1, 1, e iδ CP ). We consider evolution of the neutrino states ν f ≡ (ν e , ν µ , ν τ ) T in the propagation basis, ν prop = (ν e ,ν 2 ,ν 3 ) T determined by the relation ν f = U 23 I δ ν prop . In this basis the CP dependence is dropped out from the evolution and appears via projection of the propagation states onto the flavor states at the production and detection. Dependence of the probabilities and numbers of events on δ CP is simple and explicit. The results will be presented in terms of amplitudes in this basis (see [16,28] for details). The matrix of amplitudes is defined as Here we have taken into account the equalities A eĩ = A˜i e and A23 = A32 valid for symmetric density profile and in absence of the fundamental CP and T violation in the propagation basis. In the low energy domain, E < ∼ (2 − 3) GeV, i.e. below the 1-3 resonance, one can further decrease the number of amplitudes involved down to 3 (see [11] and comment [38]).
The oscillation probabilities P αβ ≡ |A αβ | 2 can be written as where P 0 αβ and P δ αβ are the δ-independent and δ-dependent parts of the probability P αβ , respectively. Notice that P 0 αβ = P δ=0 αβ since P δ αβ contains terms which are proportional to cos δ, generally even on δ, and these terms do not disappear when δ CP = 0. The probabilities The amplitude A23 is doubly suppressed by small quantities ∆m 2 21 /∆m 2 31 and s 13 [14]. Therefore terms that are quadratic in A23 can be neglected. For δ CP −dependent parts we have then [14] P δ ee = 0, where φ ≡ arg(A * e2 A e3 ), and Here The term D 23 is small if the 2-3 mixing is close to the maximal one, and as we said, in addition the amplitude A23 is small. Let us emphasize that in P δ µµ the phase dependence cos δ CP factors out, whereas in P δ eµ it appears in combination with the oscillation phase φ. In matter with symmetric density profile one has for the inverse channels in particular, P δ µe = P −δ eµ . For antineutrinos the probabilities have the same form as for neutrinos with substitution: are the mixing angles and phases in matter for antineutrinos. In particular, B. Quasi-constant density approximation One can further advance in analytical study using explicit expressions for the amplitudes in the constant (or quasi-constant) density approximation [14] (see also [29]). According to this approximation, at high energies for a given trajectory in mantle one can use the mixing angles computed for the average value of the potential V =V (θ z ). For low energies, where adiabaticity condition is fulfilled, the mixing angle is determined by the surface density. The oscillation phases, however, should be computed by integration over the neutrino trajectory.
For core-crossing trajectories one can use three layer model with constant densities in each layer.
In the case of constant density [14] A e2 = −ie iφ m 21 cos θ m 13 sin 2θ m 12 sin φ m 21 , The half-phases equal in the high energy range (substantially larger than the 1-2 resonance, Here L = 2R E cos θ z , , and the upper (lower) sign corresponds to neutrinos (antineutrinos). For two other phases we obtain where φ m 32 is given in (9). In practical cases the −terms can be neglected. For low energies (close to the 1-2 resonance): Notice that in the energy range above the 1-2 resonance cos 2 θ m 12 ≈ 0 and the amplitude A e3 (8) is reduced to the two neutrino form, which corresponds to factorization [14].

into (3) we obtain
where J θ ≡ sin 2θ 23 sin 2θ m 12 sin 2θ m 13 cos θ m 13 (14) is the mixing angles factor of the Jarlskog invariant in matter. Using relation φ m we obtain from (13) Similarly, neglecting D 23 we find for the P δ where δ CP dependence factors out.
For antineutrinos we have the same expressions (15) and (17) with substitution (5) and We will use the analytic expressions (15) and (17) and the corresponding expressions for antineutrinos for interpretation of numerical results.

C. Numerical results
We have computed the probabilities P αβ = P αβ (E ν , θ z ) by performing numerical integration of the evolution equation for the complete 3ν−system. We used the PREM density profile of the Earth [30] and the values of the neutrino parameters ∆m 2 32 = 2.35 · 10 −3 eV 2 , ∆m 2 21 = 7.6 · 10 −5 eV 2 , sin 2 θ 23 = 0.42, sin 2 θ 12 = 0.312 and sin 2 θ 13 = 0.025, which are close to the current best fit values [4]. We assume the normal neutrino mass hierarchy in the most part of the paper.
In Fig. 1 we show the oscillation probabilities ν e → ν µ and ν µ → ν µ as functions of the neutrino energy for different values of CP-phase and zenith angles. Let us focus first on the low energy range where sensitivity to δ CP is high and consider P eµ . In Fig. 1 the resonantly enhanced probability due to the 1-2 mixing and mass splitting is modulated by fast oscillations driven by the 1-3 mass and mixing. The 1-2 resonance energy in the mantle is at E ν = E R 12 ≈ 0.12 GeV. For core crossing trajectories the parametric effects distort the dependence of probability on energy.
The key feature which opens up a possibility to measure δ CP is the presence of a systematic shift of the oscillatory curves (probabilities) at low energies, < ∼ 2 GeV, with increase of the phase. The shift occurs in the same way in wide energy range E ν = (0.2−2) GeV and zenith angle ranges -essentially for all trajectories which cross the mantle only. This systematic shift can be understood using analytical expressions for the probabilities. Averaging P δ eµ (15) over fast oscillations driven by the 1 − 3 splitting we find The first term does not change the sign with φ m 21 , whereas the second one does. Notice that above the 1-2 resonance cos 2θ m 12 ≈ −1, and so The difference of probabilities for a given δ CP and δ CO = 0 equals: The first term is positive for all φ m 21 and δ CP , and it is this term that produces a systematic shift of the probabilities.
Although there is certain phase shift with change of δ CP , the sizes of energy intervals where the difference P (δ 1 ) − P (δ 2 ) has positive and negative signs are strongly different.
One sign dominates, and therefore there is no averaging over energy. Maximal relative upward shift of the probability curves compared to the δ CP = 0 curves is around (0.4 − 1) GeV. For the core-crossing trajectories (cos θ z < −0.83) due to the parametric effects the transition probability first increases with increase of δ CP , it reaches maximum at δ CP ∼ π/2 and then decreases.
The ν µ − ν µ probability P δ µµ (17) averaged over the 1-3 oscillation mode equals where the D 23 term is neglected. Notice immediately that CP-effect in the ν µ − ν µ channel has an opposite sign with respect to ν e − ν µ channel. Therefore the presence of both ν e and ν µ original fluxes weakens the total CP-effect, and consequently, the sensitivity to δ CP which is unavoidable. We will call this the flavor suppression.
According to (22) dependence of the ν µ − ν µ probability on δ CP factors out and turns out to be very simple. The maximal effect is for δ CP = 0, and P π/2 µµ = P 3π/2 µµ = 0, so that the total probabilities are equal for π/2 and 3π/2 in perfect agreement with result of Fig. 1.
The difference of probabilities for a given value of δ CP and zero phase equals Only CP-even contribution is present.
The probabilities in antineutrino channels are shown in Fig. 2. Dependencies of the probabilities on E ν and cos θ z can be immediately understood from our analytical treatment.
According to (5) the averaged probabilities equal For energies far above the 1-2 resonance, the expressions are further simplified since cos 2θ m 12 ≈ 1 (recall, for neutrinos cos 2θ m 12 → −1): Comparing this with (20) and (22) we find that for antineutrinos the probabilities have opposite sign with respect to the probabilities for neutrinos. Indeed, according to Fig. 2 for mantle trajectories the biggest amplitudeP eµ is for δ CP = 0 and the smallest one is for δ CP = π which is opposite to the P eµ case. This means that summation of signals from neutrinos and antineutrinos reduces the effect of CP-phase, and consequently, the sensitivity to this phase. This C-suppression can be reduced if ν andν signals are separated at least partially (see sec. IV C). The reason is that in the case of NH for neutrinos both θ m 12 and θ m 13 are enhanced in matter whereas for antineutrinos bothθ m 12 andθ m 13 are suppressed. The antineutrino probabilities decrease with increase of energy above 0.8 GeV.
Similar consideration can be performed for the ν µ − ν µ channel for which P δ µµ = −J θ 2 cos δ CP sin 2φm 21 . Notice that in vacuum P δ µµ = P δ µµ , i.e. the probability is even function of δ CP . In the matter dominated region P δ µµ ≈ − P δ µµ due to change of sign of the potential. For difference of the antineutrino probabilities we obtain at cos 2θ m 12 ≈ 1 which also have an opposite sign with respect to the differences for neutrinos (20) and (23), and equal up to change of mixing angles and phases in matter.

D. Magic lines and CP-domains
In what follows we will study difference of probabilities and distributions of events in the E ν − cos θ z plane for different values of δ CP . The patterns of distributions are determined to a large extent by the grid of the magic lines [14,[22][23][24]. The lines fix the borders of the CP-domains -the regions in the E ν − cos θ z plane of the same sign of the CP-difference.
Recall that the magic lines are defined as the lines in the E ν − θ z plane along which the oscillation probabilities do not depend on phase δ in the so called factorization (quasi 2ν) approximation [14]. Correspondingly, the CP-differences vanish along these lines.
1. The solar magic lines are determined by the condition where in neutrino channels φ S is given by the expression (12) for φ m 21 (valid in 3ν framework below 1-3 resonance but) extended to all the energies. For antineutrinos in the NH casē φ S =φ m 21 everywhere. Along these lines |A e2 | = 0 below the 1-3 resonance. That would be the line of zero solar amplitude in the 2ν approximation. The minimum of the total probability at 0.15 GeV at cos θ z = −0.4 ( Fig. 1) corresponds to the first magic line with The minimum at 0.17 GeV at cos θ z = −0.8 ( Fig. 1) corresponds to the second magic line with φ m 21 = 2π. Notice that the energy of minimal level splitting (maximal oscillation length) is given by GeV which is much bigger than E R 12 = 0.12 GeV. So, below 0.7 GeV the splitting increases and correspondingly the oscillation length decreases. Therefore the same phase can be obtained for smaller | cos θ z | and therefore the solar magic lines bend toward smaller | cos θ z |. At energies much above the 1-2 resonance these lines do not depend on energy and are situated at 2. The atmospheric magic lines are determined by the equality Along these lines |A e3 | ≈ 0. It would vanish exactly in the 2ν approximation, when cos 2 θ m 12 ≈ 0. Zeros of probability at E ν ≥ 2 GeV (see Fig. 1) which do not depend on δ CP are situated on the atmospheric magic lines. E.g. for cos θ z = −0.4 these points are at E ν = 2 GeV and E ν = 3.2 GeV. For cos θ z = −0.8, E ν = 2.3, 2.9, 4.1 GeV. For P µµ the solar and atmospheric magic lines coincide with those for P eµ in the limit D 23 = 0.
The magic lines determined by (29) and (31) do not not coincide with lines where |A e2 | = 0 and |A e3 | = 0 in the 3ν framework. But they play the role of asymptotics of the true lines of where dependence of probabilities on δ CP disappears.
3. The interference phase lines are important for distinguishing different values of the CP-phase: a given value δ CP and a different value δ 0 . Along these lines P δ αβ − P δ 0 αβ = 0. According to (3) for P eµ the condition reads where φ ≈ −φ 31 and the latter is the vacuum phase. This condition corresponds to intersection of probability curves for different values of phases δ and δ 0 in Fig. 1. For δ 0 = 0 the condition can be written as φ 31 + δ CP = −φ 31 or For the inverse channel, ν µ → ν e , the sign of δ CP should be changed. According to (4) dependencies of the ν µ → ν µ probability on φ and δ CP factor out in the approximation, and the corresponding interference phase line is determined by the condition cos φ = 0, or The condition can be written as where R E is the Earth radius and in general φ(δ CP ) should lead to the vanishing CPdifference.
The exact value of interference phase φ does not coincide with −φ 31 . In the constant density approximation φ equals the phase of the expression in brackets of A e3 (8): Notice that φ would be equal −φ m 31 , if cos φ m 31 = 0. The latter is satisfied for high energies E E R 12 , where cos 2 θ m 12 ≈ 0. However, if φ m 31 ≈ π/2 we can not neglect the second term in the denominator of (34). Notice that in the limit cos 2 θ m 12 = 0 we obtain from (16) where one can see immediately all three "magic" conditions.
Notice that magic lines could be introduced immediately in the 3ν framework as the lines along which P δ αβ − P δ 0 αβ = 0. In this case they would, indeed, determine the borders of domains with different sign of the CP-difference of the probabilities. We use the original definitions of magic lines to match with previous discussions. Still as we said before, the solar, atmospheric and interference lines nearly coincide with the exact lines of zero CP-differences in certain energy intervals or in asymptotics. The corresponding phases are related as

III. PINGU, SUPER-PINGU AND CP
The key conclusion of the previous section is that integration over the neutrino energy and direction does not suppress the CP effect significantly. Furthermore, for all trajectories which cross the mantle of the Earth only, the CP violation effect is similar: the same sign and the same change with δ CP . Effect is different in the core, for cos θ z < −0.83. So, it could be partial cancellation due to smearing over the zenith angle. Furthermore, the relative CP effect at the probability level increases with decrease of energy. In this connection we will explore a possibility to measure δ CP using multi-megaton scale neutrino detectors with low energy threshold. As it was already realized in [16], sensitivity of PINGU to δ CP is low.
So, we will consider future possible upgrades of PINGU. We will also quantify capacity of PINGU to obtain information about δ CP .

A. PINGU and Super PINGU
We calculate event rates for the proposed PINGU detector and for possible future PINGU upgrade which we will call Super-PINGU. The PINGU detector [26] will have 40 strings additional to the DeepCore strings and 60 digital optical modules (DOM's) at 5 m spacing in each string. A compact array like PINGU could detect neutrinos with energies as low as (1 -3) GeV. Strict criteria allow over 90% efficiency of event reconstruction for all 3 flavors [26]. We parametrize the PINGU effective volumes as and respectively for ν µ and ν e . These parametrizations well represent simulated volumes [26] from > ∼ 1 GeV up to 25 GeV. We will use an accuracy of the energy and angle reconstruction similar to those in [26].
Along with the PINGU proposal the idea has been discussed to construct "ultimate" i.e. one order of magnitude below the threshold in the present PINGU proposal. For this, a denser array of DOM's is required which will lead to an increase of the effective volume of a detector at low energies. For definiteness we will take the effective volume which corresponds to the PINGU detector simulations with a total of 126 strings and 60 DOM's per string each [31]. The effective mass can be parameterized as for both ν µ and ν e events. We will call this version Super-PINGU. According to (36) and (38)  To evaluate sensitivity of Super-PINGU to δ CP we will compute the (E − cos θ z ) distributions of events of different type and explore their dependence on δ CP , evaluating distinguishability. The numbers of events N α , produced by neutrinos ν α (α = e, µ) with energies and zenith angles in small bins ∆(E ν ) and ∆(cos θ z ) marked by subscript ji equal Here T is the exposure time, N A is the Avogadro's number, ρ is the density of ice and V eff is the effective volume of the detector. The density of events of the type α, d α , that is, the number of events per unit time per target nucleon, is given by where Φ α andΦ α are the fluxes of neutrinos and antineutrinos at the detector which produce events of the type α, and σ α andσ α are the corresponding cross-sections. In turn, the fluxes at the detector equal are the original muon and electron neutrino fluxes at the production. We use the deep inelastic cross-sections for E > ∼ 1 GeV: and we take the cross-sections for electron and muon neutrinos of the same energy to be the same at E > ∼ 1 GeV. With decrease of energy also inelastic one -few pions production as well as quasi-elastic processes will contribute. For simplicity, in our estimations we will use the total neutrino-nucleon cross-sections down to (0.2 -0.3) GeV as they are parametrized in [26], assuming that different contributing processes would produce visible effect at the detector with the same efficiency. For antineutrinos there is no data below 1 GeV and we use similar extrapolation as for neutrinos. Clearly in future these computations should be refined.

C. CP-asymmetry and distinguishability
As in [16], we will employ the distinguishability S σ as a quick estimator of sensitivity of measurements. For a given type of events (we omit the index α) and each ij-bin we define the relative CP-difference as where N δ ij and N 0 ij are the numbers of events computed for a given value of δ CP and for δ CP = 0 correspondingly, and is the total "error" in the ij-bin. If N 0 ij is interpreted as a result of measurement, the first term would correspond to the statistical error and the second one to the uncorrelated systematic errors. As in [16] we assume that the latter is proportional to the number of events: f N 0 ij , and for the coefficient f we will use f = 5%, 10% and 20%. In general f is a function of neutrino energy and zenith angle. Notice that since here the contribution from the systematic error is proportional to (N 0 ij ) 2 , for the same f the role of this error decreases with decreasing size of the bin.
If N δ ij is considered as the fit value, the moduli |S ij | would give the standard deviation and so the statistical significance. However, in contrast to real situation the "measured" value N 0 ij does not fluctuate. Therefore we will not interpret it as number of sigmas, but just use |S| as independent characterization -the distinguishability.
Considering the effect in each bin as an independent measurement (which is possible after smearing), we can define the total distinguishability as where the sum is over all the bins.
The correlated systematic errors, e.g., those of the overall flux normalization and of the tilt of the spectrum, obviously can not reproduce the pattern of the distribution of events similar to the difference of distributions for two different values of δ CP . Their effect to a large extent can be accounted by a reduction of the exposure time and statistics. Moreover, these correlated systematic errors can be parameterized and reduced with better determination of the atmospheric neutrino fluxes.
We will avoid precise statistical interpretation of distinguishability and just consider that it gives some idea about significance and sensitivity. Still, in various cases |S| turn out to be close to the significance as follows from comparison of our previous estimations with results of complete MC simulations [16,26]. It gives reasonable absolute values of significance and reproduces rather precisely relative changes of sensitivities with variations of characteristics of detector and neutrino parameters.
Apart from total distinguishability the sensitivity can be characterized also by maximal positive and negative CP-differences for individual bins in a given range of energies and zenith angles.
For these events the energy of the muon E µ and the direction of its trajectory characterized by the angles θ µ and φ µ as well as the total energy of the hadron cascade (for deep-inelastic scattering) E h can be measured. Using this information one can reconstruct the neutrino energy as where m N is the nucleon mass. Also the direction of cascade can be determined to some extend.
At low energies processes with one -few pion production and then quasi-elastic scattering become dominant. For these events procedure of reconstruction of the neutrino energy and direction becomes different. So, the detection of the low energy events should be considered separately, and such a study is beyond the scope of this paper. There are also some contributions from ν τ which produce τ leptons with subsequent decay into muons.
The δ-dependent part of the number density of the ν µ events in a single bin equals where The ratios r,r depend both on the neutrino energy and zenith angle, e.g., in the range E ν = (2 − 25) GeV and for cos θ z = −0.8 the ratio can be roughly parameterized as r = 1.2 · (E ν /1 GeV) 0.65 . Below 1 GeV one has r ≈ 2.
From eqs. (3) and (4) we find for neutrino contribution This shows that in the case D 23 = 0 the difference d δ µ −d 0 µ should vanish whenever A e2 = 0 or A e3 = 0, i.e. along the solar and atmospheric magic lines considered above. The antineutrino contribution can be written similarly. It is suppressed in comparison with the neutrino contribution by factor ∼ 0.2 due to smaller probabilities (factor of 2 at small energies, see Fig. (2)) and smaller cross-section.
Notice that the main sensitivity to the mass hierarchy searches comes via P e3 which is screened at low energies where r = 2. In contrast, no screening of the CP-dependent terms appears. The phase δ CP affects relative contribution of the ν e − ν µ and ν µ − ν µ channels. Contributions from ν andν are summed up.
Let us consider dependence of the distributions of events on δ CP given in (44) which is explicit and exact. The first term in brackets of (44) as function of δ CP is symmetric with respect to δ CP = π, whereas the second one is antisymmetric. The relative contributions of the two terms are determined by the phase φ. As we will see the first term dominates and the second one produces shift of maximum of S σ to δ CP > π.
Patterns of distributions are determined by the domain structure formed by the magic lines. The borders of domains are inscribed in the grid of magic lines with certain interconnections in the resonance regions. Also non-zero value of D 23 produces further difference.
Let us consider the magic lines for difference of densities of events for a given δ CP and δ CP = 0. Since A e3 and A e2 appear as common factor (in the approximation of D 23 = 0) the solar and atmospheric magic lines are the same as for the probabilities (if neutrinos and antineutrinos are considered separately). The true interference phase condition for number of events corresponds to zero value of the terms in the brackets of (44) which gives At high energies the ν µ flux dominates, r 1, and the pattern of the d µ distribution follows dependence of the probability P µµ on E ν and cos θ z , in particular, the P µµ domain structure.
For At low energies, r ≈ 2 and the difference of the densities equals The effect of averaging can be obtained using constant density approximation. From (20) we obtain for the neutrino part (the first term in (43)) At low energies, when r ≈ 2, this expression reduces to where additional factor 1/2 is due to the flavor suppression. Comparing (48) and (46) we find that averaging is reduced to substitution in the CP-factor cos φ → sin 2 φ m 21 and sin φ → 0.5 sin 2φ m 21 . Expression (48) can be considered as a combination of two functions sin 2 φ m 21 and sin 2φ m 21 with weights determined by the phase δ CP . The first function is even and the second is odd in δ and both functions are zero along the magic lines. For several specific values of δ CP we have (in units 1 4 Similarly for antineutrinos, (27), (28) the difference equals If mixings and phases in neutrino and antineutrino channels would be approximately equal, the effect of inclusion of antineutrinos could be accounted by the the overall suppression factor (1 − P / P κ µ ) ≈ 0.8 to that of neutrino only without change of the shape of the distribution. Differences of the phases and mixing angles in the neutrino and antineutrino channels lead to distortion of the neutrino distribution in the regions around the magic lines.
The ν τ − flux appears at the detector due to the ν µ − ν τ oscillations. In turn, the ν τ interactions ν τ + N → τ + h → µ + ν + ν + h will contribute to the sample of ν µ −events with a muon and a hadron cascade in the final state. However, the number of these events is relatively small due to its branching ratio and small cross-section near the energy threshold.
Also these events have certain features which can be used to discriminate them from the true ν µ −events [16]. Furthermore, as we will see the highest sensitivity to δ CP is in the sub-GeV region where τ leptons are not produced. Effect of the τ decays can be accounted by adding a systematic error.

B. Smearing
We will describe the uncertainties of reconstruction of the neutrino energy and direction by smearing functions where E ν and θ z (E r ν and θ r z ) are the true (reconstructed) energy and zenith angle of the neutrinos.
We use for G E and G θ the Gaussian functions G y (y − y r , σ y ) = N y √ 2πσ y e − (y−y r ) 2 2σ 2 y , y = E ν , θ z with the normalization constant N y and widths The numerical coefficients in these expressions were selected to reproduce closely the energy and angular reconstruction functions of PINGU [26]. We do not use exact PINGU functions since characteristics of Super-PINGU will be different and anyway better. The distributions are normalized in such a way that and slightly diminishes total distinguishability.
We obtained the unbinned distribution of events after smearing in the (E r ν − cos θ r z ) plane as α = e, µ, and then the binned them as with ∆(E r ν ) = 0.5 GeV and ∆(cos θ r z ) = 0.025. In Fig. 4 we show the relative CP-difference distributions, S ij (f = 0) (42), computed with smeared distributions of the ν µ events. Smearing leads to disappearance of fine structures and merger of regions of the same sign CP-differences. There is dominant connected region of the negative CP-differences and two separate regions with S ij > 0. The large one is at high energies and small cos θ z , and another one is an island at cos θ z < −0.6 and energies E = (6 − 13) GeV. With increase of δ CP the region with S ij > 0 expand and asymmetry between maximal positive and negative CP-differences increases.
Smearing also leads to a substantial decrease of the sensitivity to δ CP . This reduction is a consequence of the integration over regions with different values and signs of S ij . The bins with the highest CP-difference in the E − cos θ z plane are at the lowest energies and core crossing directions. The decrease of sensitivity is characterized by factor 1.4 -3 and strongly depends on values of δ CP (see Sec. VI). E.g., for δ CP = π, the total distinguishability for super-PINGU is reduced after smearing by a factor 1.5 (see Fig. 9).
To evaluate contributions to the total distinguishability S σ from different energy regions we have computed S σ (E th ) for 1 year of exposure and fixed V ef f (E ν ) using different minimal energies of integration, E th . No systematic errors have been included (f = 0). We find that with decrease of E th from 1 GeV down to 0.5 GeV increases S σ by factor 1.3 -1.6 depending on value of δ CP . Decrease of E th from 1 to 0.2 GeV leads to increase of S σ by factor 1.8 -2.4, again depending on value of δ CP . In particular, for δ CP = π/2 we obtain S σ = 1.3, 2.1, 3.1 for E th = 1, 0.5, 0.2 GeV. The corresponding numbers for δ CP = 3π/2 equal S σ = 2.7, 3.8, 5.3. So, the lowest energy bin of the size 0.8 GeV can contribute as much as whole range from 1 to 20 GeV. Significance increases with decrease of the thresholds and this increase is similar for smeared and unsmeared distributions.

C. Neutrinos and antineutrinos
Measurements of the inelasticity (y distribution) allow to make partial separation of the neutrino and antineutrino signals on statistical ground [34]. Since the CP-differences have opposite signs for neutrinos and antineutrinos at least at low energies one expects improvement of sensitivity to the CP-phase if the ν andν signals are separated. To assess the possible improvement we consider first the ideal situation of complete separation. In Fig.   5 we show the unsmeared CP-difference plots for neutrinos and antineutrinos separately. We take δ CP = π/2 and δ CP = π.
For neutrinos (resonance channel) the distribution is rather similar to that for the sum of signals. Maximal negative single-bin CP difference becomes 1.5 times larger, whereas maximal positive single-bin CP difference is the same, so that asymmetry between positive and negative contributions increases with exclusion ofν. As a result, the significance for neutrinos alone increases by factor 1.5 -1.6 in comparison with significance for the sum of signals.
In contrast, distribution for antineutrinos has different pattern: it has the same sign of the CP difference S ij (positive or negative in the same regions) as for neutrinos in high energy region, E > 8 GeV. Whereas at low energies the antineutrino and neutrino distributions have opposite signs, and so cancellation is strong in the absence of separation. For antineutrinos (especially at low energies), the positive asymmetry dominates. This can be immediately understood on the basis of our analytic considerations in Sec. IIC.
The distinguishability without separation is larger than the difference of ν andν distinguishabilities. For E th = 0.5 GeV and δ CP = π we have S σ,ν = 12.3, S σ,ν = 7.0 and S σ,ν − S σ,ν = 5.3, which is smaller than total S σ = 7.5. This means that cancellation is not complete and reflects the fact that in high energy regions the sign of CP-difference is the same for ν andν. There is an asymmetry between the neutrino and antineutrino contributions related to difference of the cross-sections and fluxes. In the case of ideal separation we would have S σ = 14.2, instead of 7.5, i.e. almost 2 times larger than without separation.
Smearing and partial separation will reduce this enhancement factor down to 1.2 -1.5. Sep-aration of neutrinos and antineutrinos, i.e. reconstruction of y-distributions, at low energies may be problematic.

D. Inverted mass hierarchy
For the inverted neutrino mass hierarchy (IH) the pattern of distributions is inverted in comparison with that for normal mass hierarchy at high energies and it is the same for low energies, see Fig. 6. The difference is related to the 1-3 resonance whose effect is different for normal and inverted hierarchies. At low energies sensitivity to the mass hierarchy disappears. We find that ν −ν separation is more important for inverted mass hierarchy since in this case the difference of signals from neutrinos and antineutrinos is smaller.

A. Density of events
The cascade events are produced by the CC ν e interactions ν e + N → e + X,ν e + N → e + + X and several other processes (see discussion below). The density of the CC ν e events is given by where κ e ≡ (σ e /σ e )(Φ 0 µ /Φ 0 µ ), and its δ-dependent part, is determined by the ν µ → ν e oscillation probability only since P ee andP ee are δ-independent.
Consequently, there is no flavor suppression of the CP-violation even for low energies.
The difference of the densities of the ν e -events equals plus smaller antineutrino contribution. As we discussed before, the antineutrino contribution is suppressed by factor 1/4. Comparing this with (44) and (46) we find that for r ≈ 2 and So, the CP difference of the ν e events has an opposite sign with respect to the CP difference of the ν µ events and its size is two times larger. As a result, the cascade events can give even bigger contribution to distinguishability of different values of δ CP than the ν µ events.
The reason is the flavor suppression of CP-differences for the ν µ events, which is absent for the ν e events. With increase of energy the ratio r increases, the flavor suppression becomes weaker. Consequently, (d δ µ − d 0 µ ) increases and the factor in equation (54) becomes smaller, approaching 1. This also depends on value of δ CP .
Explicit expression for the density of events averaged over φ m 32 (which is valid at low energies) can be obtained using constant density approximation. Since P µe = P eµ (δ CP → −δ CP ) we have from (18) Correspondingly, the difference of probabilities for a given δ CP and δ CP = 0 equals when cos 2θ m 12 ≈ −1, For antineutrinos such a difference has similar expression but with overall minus sign and mixing angles and phases taken for antineutrinos.
For difference of the density of events we obtain: In the range where the phases of neutrinos and antineutrinos are approximately equal we with the factor in brackets describing the C-suppression.
In Fig. 7 we show the unsmeared distribution of the CP differences S ij (f = 0) of ν e events for different values of δ CP . As we marked, the transition probability ν µ → ν e gives unique contribution to the CP difference, ν e → ν e contribution and D 23 are absent. As a result, the distributions follow closely the domain structure of ν µ → ν e probability determined by the magic lines. Now the interference phase lines depend on δ CP . From (53) we obtain the condition for the phase (zero value of the expression in the brackets): This condition does not depend on r in contrast to (45), since only one transition probability enters. For δ CP = π/2 we obtain tan φ = 1 or φ = π/4+nπ. Now the pattern of distributions changes with δ CP , since the interference phase and CP phase dependencies do not factor out.
Essentially the domain structure for the CP-difference repeats the corresponding structure of the ν µ → ν e probability. One can see in Fig For the ν e − events the positive CP-difference dominates. The total distinguishability from ν e − events is higher than from ν µ , e.g., for E th = 0.5 GeV we have S σ = 4.8, 9.3, 12.5 for δ CP = π/4, π/2, π. Decrease of the threshold from 1 GeV down to 0.2 enhances distinguishability by a factor of 2.
The problem here is that the cascade events are not only due to ν e interactions but also due to other processes which should be taken into account: (ii) The ν τ CC interactions which produce τ leptons. This generates cascades in all the decays of τ leptons but µ. At low energies contribution of these events is suppressed due to high threshold of the τ lepton production.
(iii) Contribution of CC ν µ events with faint muons (close to threshold of Cherenkov radiation). Fraction of these events is higher at low energies. A part of the CC ν e events can be confused with the ν µ events when one of the pions will be misidentified with muon.
This problem may be cured at least partly by introduction of additional cuts.

B. Smearing of the cascade events
In Fig. 8 we show results of smearing of cascades with the energy and angle reconstruction functions. Recall that information about the angular resolution can be extracted from the PINGU proposal ( Fig. 7(b) and Fig. 8(a) in [26]). It shows that median value of angle is very similar for both cases, though the distribution deviates from the Gaussian one. So, for estimations we use the same characteristics of the smearing functions as for ν µ events.
According to Fig. 8 effect of smearing for the ν e events is weaker than for the ν µ events.
More structures survive and pattern is sharper especially at high energies. One can clearly see the domain structure with the same magic lines. The reason is that the pattern before smearing has bigger variations of values of S ij . At low energies due to loss of angular resolution, structures become horizontal. Asymmetry between the positive and negative CP-differences increases with δ CP and S > 0 becomes dominant for δ CP > ∼ π. Depending on δ CP the distinguishability of ν e events is 1.3 -3 times larger than the distinguishability of ν µ events. This enhancement will be however diluted by other contributions to cascade events, in particular from NC interactions. Decrease of the threshold energy from 1 to 0.2 GeV leads to increase of distinguishability by factor 1.5 -2.

VI. SENSITIVITY TO THE CP-PHASE
In Figs. 9 and 10 we show dependencies of the total distinguishability of a given phase δ CP from δ CP = 0 on values of δ CP for the ν µ and ν e events after 1 year exposure. We use Super PINGU thresholds E th = 0.5 and 1 GeV, and PINGU thresholds E th = 1 and 3 GeV.
Shown are the dependencies before smearing over E ν , θ z , and after smearing with 0, 5%, and 10% correlated systematic error.
The dependencies of distinguishability on δ CP without smearing is nearly symmetric with respect to π. Maximum is slightly shifted to δ CP > π for the ν µ events and to δ CP < π for the ν e events. For Super-PINGU with decrease of threshold from 1 GeV down to 0.5 GeV the total distinguishability increases by factor 1.25 -1.50 depending on the value of δ CP .
Without smearing, ν e events would contribute to the distinguishability about 2 times bigger than ν µ events at low energies due to the absence of flavor suppression.
Smearing diminishes S σ which depends on δ CP and is different for the ν µ and ν e events.
For δ CP = π the suppression is about a factor 2 and it becomes relatively week 1.4 for Systematic uncorrelated errors do not affect strongly the shape of S σ (ν µ ) dependence on δ CP . Furthermore the relative effects of errors decreases with increase of threshold for ν µ events. For Super-PINGU with E th = 0.5 GeV at δ CP = π the 5% level systematics reduces S σ (ν µ ) by 20% in comparison with f = 0 case, etc..
The CP-distinguishability from the ν e events is affected by systematics similarly. The effect also depends on values of δ CP . For Super-PINGU with E th = 0.5 GeV at δ CP = π the 5% systematics reduces S σ (ν e ) by 20% and 10% systematics reduces S σ (ν e ) by 40%. For PINGU with 1 GeV threshold the reduction factors are 10% and 20% for the corresponding level of systematics.
• Decrease of the energy threshold. The increase of S σ can be about 30% for E th = 0.2 GeV. In comparison to PINGU with E th = 3 GeV, the Super-PINGU improves the distinguishability by factor 3.
• Increase of the exposure time or/and increase of the effective volume. E.g. after 6 -8 years the distinguishability for δ CP = π/4 can reach S σ ≈ 2 − 4.
• Partial (statistical) separation of the neutrino and antineutrino signals can enhance distinguishability up to 30%.
We have explored also distinguishability of different values of δ CP from δ 0 = 3π/2 (favored now). We find, e.g., that distinguishability of δ CP = π from 3π/2 is as high as 3π/2 from 0 discussed above. The highest S δ is for δ CP = π/2 and 3π/2 which is comparable with distinguishability of π and 0.
These results show that Super-PINGU can be competitive with other proposals. Indeed, -T2K plus NOvA after 5 years operation in the ν mode and 5 years inν mode will be able to distinguish δ CP = π/2 from 0 at about 2σ CL, and δ CP = 3π/2 from 0 at 3σ (NH) [6].
-LBNE long-baseline experiment with 35 kTon of Liquid Argon detector after 5 years of operation in ν mode and 5 years inν will be able to exclude δ CP = π/2 at 5σ if true phase is zero [7].
-LBNO with CERN-SPS (0.75 MW beam power) and 20 kt LAr TPC at distance 2600 km after 10 years will be able to distinguish δ CP = π/2 from zero with more than 3σ C.L.
About 25% of possible values of δ CP can be identified at 3σ level [9].
Notice that accelerator experiment have good sensitivity at small values of δ CP but degeneracy of δ CP = 0 and π. In contrast, Super-PINGU has relatively low sensitivity at δ CP < π/2, but distinguishability of δ CP = 0 and π is nearly maximal.
Interestingly the strongest difference is for δ CP = 0 and ∼ 3π/2 (both for beams and Super PINGU), so that if the present indications to the δ CP are true it will be easy to establish the CP-violation in lepton sector.
We have assumed that by the time of operation of Super-PINGU the mass hierarchy will be established (probably by PINGU itself). Let us underline that degeneracy with hierarchy is not strong anyway, the hierarchy asymmetry and the CP-difference have different patterns in the E ν − cos θ z plane. The most sensitive region for hierarchy is E ν > ∼ 6 GeV, whereas for CP-phases it is below (3 -4) GeV.

VII. CONCLUSIONS
Assuming that the neutrino mass hierarchy is identified, we have explored a possibility to measure the CP-phase with future multi-megaton scale atmospheric neutrino detectors having low energy threshold. The method consists of comparison of the E ν − cos θ z distributions of events produced by ν e and ν µ for different values of δ CP . We use the relative CP-difference of the distributions to quantify distinguishability.
We presented simple analytic formalism which allows to describe properties of the distri- Most of computations have been made for assuming that by the time of these experiments hierarchy will be established and for definiteness we took the normal mass hierarchy. We estimated that significance of measuring δ CP in the case of inverted mass hierarchy is about 30% lower.
Super-PINGU with large volume at (0.1 -0.2) GeV can be used also for proton decay searches.
The presented study of sensitivity to CP phase should be considered as a very preliminary since various experimental features are not known yet. There is a number of issues related detection of low energy events and determination of their characteristics (flavor, energy, direction etc.). At the same time one can expect that new experimental developments will further improve the sensitivity. In any case, the results obtained here look very promising and certainly show that "Super-PINGU for CP violation and proton decay" deserves further detailed study.          Fig. 3, but separately for ν µ andν µ events and for values of the CP phase δ CP = π/2 (upper panels) and δ CP = π (lower panels).   Fig. 3, but for the ν e +ν e events. Normal mass hierarchy is assumed. FIG. 9: Integrated Super-PINGU distinguishabilities between a given value of δ CP and δ CP = 0 as functions of δ CP for the ν µ +ν µ events (upper panels) and for ν e +ν e events (lower panels).
The dependencies have been computed for the energy thresholds E th = 0.5 GeV (left panels) and E th = 1 GeV (right panels). Different lines show distinguishabilities without smearing, with smearing and different levels of the uncorrelated systematic errors: f = 0, 5% and 10%. Normal mass hierarchy is assumed.