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

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

Soebur Razzaque Department of Physics, University of Johannesburg, PO Box 524, Auckland Park 2006, South Africa    A. Yu. Smirnov Max-Planck-Institute for Nuclear Physics, Saupfercheckweg 1, D-69117 Heidelberg, Germany International Centre for Theoretical Physics, Strada Costiera 11, I-34100 Trieste, Italy
###### Abstract

We explore a possibility to measure the CP-violating phase using multi-megaton scale ice or water Cherenkov detectors with low, 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 (track) and (cascade) events in the neutrino energy and zenith angle plane have been computed for different values of . 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 increases with decrease of the energy threshold. The events contribute to the CP-sensitivity as much as the events. While sensitivity of PINGU to is low, we find that future possible upgrade, Super-PINGU, with few megaton effective volume at () GeV and e.g. after 4 years of exposure will be able to disentangle values of from with “distinguishability” ( significance in ’s) correspondingly. Here the intervals of are due to various uncertainties of detection of the low energy events, especially the flavor identification, systematics, etc.. Super-PINGU can be used simultaneously for the proton decay searches.

###### pacs:
14.60.Pq, 14.60.St

## 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 of atmospheric and accelerator neutrinos, high energy cosmic neutrinos, etc. reviews .

The present experimental results have very low sensitivity to giving only weak indications of the preferable interval of its values. Thus, the T2K and reactor data favor the interval with central value t2k . Analysis of the SuperKamiokande atmospheric neutrino data gives preferable range  himmel . The global fit of all oscillation data, e.g. from fogli , agrees with these results: at level and no restriction appears at level. The values around are disfavored. Similar results with the best fit value (NH) have been obtained in valle and with in nufit .

A possibility to measure is generally associated with accelerator long base-line (LBL) neutrino experiments. There is certain potential to improve our knowledge of with further operation of T2K and NOvA ghosh . Proposals of more remote experiments, which will measure with reasonable accuracy, include LBNE lbne , J-PARC - HyperKamiokande hyperk , ESS ess and LBNO lbno . Further developments can be related to the low energy neutrino and muon factories, beta beams, etc., see reviews .

Another possibility to determine 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 hyperk : During 10 years of running with fiducial volume Mton the HK will be able to discriminate the values of phases at about 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 ghosh2 .

Various theoretical and phenomenological aspects of the CP-violation in atmospheric neutrinos have been explored in a number of publications before  Peres:2003wd ; Kimura1 ; GonzalezGarcia:2004wg ; our2 ; our3 ; Mena:2008rh ; ARS ; latimer ; Agarwalla:2012uj ; ohlsson ; ge1 ; ge2 . In particular, pattern of the neutrino oscillograms (lines of equal probabilities in the plane) with CP violation has been studied in details in our3 . It was realized that structure of the oscillograms is determined to a large extent by the grid of the magic lines of three different types Barger:2001yr ; Huber:2003ak ; Smirnov:2006sm ; our3 (solar, atmospheric and interference phase lines). 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 ARS .

Capacities of new generation of the atmospheric neutrino detectors (PINGU, ORCA) have been explored recently pingu ; ARS ; pingu2 ; orca ; Winter:2013ema . It was found ARS ; pingu2 that these detectors with 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 in contrast to the accelerator experiments, but the information on the CP-phase will be rather poor.

The goal of this paper is twofold: (i) detailed study of the CP-violation effects in atmospheric neutrinos, and (ii) tentative estimation of sensitivity to the CP-phase of future large detectors, 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 the energy threshold . This opens up a possibility to measure using multi-megaton scale ice or water Cherenkov detectors with GeV. We study dependence of the energy and zenith angle distributions of events produced by and on the CP phase. We estimate distinguishability of different values of . According to the present proposal pingu2 PINGU will have low sensitivity to and only further upgrades, which we will call Super-PINGU, can measure with potentially competitive accuracy. We discuss requirements for such detectors. We identify problems and challenges of these CP measurements, and propose ways to resolve or mitigate the problems. We formulate conditions, in particular on accuracies of knowledge of external parameters and level of flavor misidentification, to achieve the goal.

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 analytical formulas for the probabilities in quasi-constant density approximation. The grid of the magic lines will be described and we will show how the grid determines structure of oscillograms. In Sec.  III we consider a possible upgrade of PINGU, called Super-PINGU, which will be able to measure and outline a procedure of computation of numbers of events. In Sec. IV we compute the distributions as well as relative differences of distributions of the events in the plane (the relative CP-differences) for different values of . We study dependence of these distributions on before and after smearing over the neutrino energy and direction. In Sec. V we perform similar studies of the cascade (mainly ) events. Sec. VI contains estimations of the total sensitivity of Super-PINGU to and discussion of our results. We conclude in Sec. VII.

## Ii Oscillation probabilities, CP-domains

### ii.1 Oscillation amplitudes and probabilities

We will study the CP-violation phase defined in the standard parametrization of the PMNS mixing matrix, , where is the matrix of rotation in the -plane and . We consider evolution of the neutrino states in the propagation basis, determined by the relation . In this basis the CP dependence is dropped out from the evolution and appears via projection of the propagation states back onto the flavor states at the production and detection. Due to this, dependence of the probabilities and numbers of events on is simple and explicit. Therefore the results will be presented in terms of amplitudes in this basis (see our3 for details), where the matrix of amplitudes is defined as

 ||Aαβ||=⎛⎜⎝AeeAe~2Ae~3...A~2~2A~2~3......A~3~3⎞⎟⎠.

Here we have taken into account the equalities and valid for symmetric density profile and in absence of the fundamental CP and T violation in the propagation basis. In the low energy domain, GeV, i.e. below the 1-3 resonance, one can further decrease the number of amplitudes involved down to 3 (see Peres:2003wd and comment 111The basis used in the paper Peres:2003wd differs from the basis considered here by the additional 1-3 rotation on the 1-3 mixing in matter. This basis is useful for description of oscillations at low energies (in the sub-GeV range) since it allows to make certain approximations which simplify description. Namely, neglecting changes of 1-3 mixing in matter with distance one can reduce 3-neutrino evolution problem to 2-neutrino evolution problem. Correspondingly all the probabilities can be expressed in terms of just three real functions , and . The main dependence on 1-3 mixing as well as on is explicit here. The formulas in Peres:2003wd are approximate, and in general they are not valid at high energies (in multi-GeV range). Since the highest sensitivity to CP is at low energies these formulas give accurate description of CP-effects.).

The oscillation probabilities can be written as

 Pαβ≡Pindαβ+Pδαβ, (1)

where and are the -independent and -dependent parts of the probability , respectively. Notice that , since contains terms which are proportional to , generally even on , and these terms do not disappear when . Then the total probability is . The probabilities equal our3

 Pindeμ = c223|Ae~2|2+s223|Ae~3|2, (2) Pindμμ = ∣∣c223A~2~2+s223A~3~3∣∣2.

The amplitude is doubly suppressed by small quantities and our3 . Therefore terms that are quadratic in can be neglected in the first approximation of our analytical study. For the dependent parts we have then our3 ,

 Pδeμ=sin2θ23Re[eiδA∗e~2Ae~3]=sin2θ23|Ae~2Ae~3|cos(ϕ+δ), (3)

where , and

 Pδμμ=−sin2θ23Re[A∗e~2Ae~3]cosδ+D23=−sin2θ23|Ae~2Ae~3|cosϕcosδ+D23. (4)

Here

 D23≡sin2θ23cosδcos2θ23Re[A∗~2~3(A~3~3−A~2~2)].

The term is small if the 2-3 mixing is close to the maximal one, and as we said, in addition the amplitude is small. Let us emphasize that in the phase dependence, , factors out, whereas in it appears in combination with the oscillation phase .

In matter with symmetric density profile one has for the inverse channels

 Pβα=Pαβ(δ→−δ),

in particular, . For antineutrinos the probabilities have the same form as for neutrinos with substitution:

 δ→−δ,   ϕm32→¯ϕm32,   ϕm21→¯ϕm21,   θmij→¯θmij, (5)

where and are the mixing angles and phases in matter for antineutrinos, and is the matter potential. In particular,

 ¯Pδeμ=sin2θ23|¯Ae~2¯Ae~3|cos(¯ϕ−δ). (6)

### ii.2 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 our3 (see also kimura2 and Blennow:2013vta ). 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 . 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 the three layer model with constant densities in each layer; corrections are computed in our3 .

In the case of constant density our3

 Ae~2 = −ieiϕm21cosθm13sin2θm12sinϕm21, (7) Ae~3 = −ieiϕm21sin2θm13(sinϕm32e−iϕm31+cos2θm12sinϕm21). (8)

The half-phases equal in the high energy range (substantially larger than the 1-2 resonance, GeV):

 ϕm32≈Δm231L4Eν√(1−ϵ)2∓2(1−ϵ)ξcos2θ13+ξ2. (9)

Here with being the radius of the Earth,

 ξ≡2VEνΔm231,    ϵ≡sin2θ12Δm221Δm231,

and the upper (lower) sign corresponds to neutrinos (antineutrinos). For two other phases we obtain

 ϕm21 ≈ Δm231L8Eν[1+ξ−ϵ(2cot2θ12−1)]−12ϕm32, (10) ϕm31 ≈ Δm231L8Eν[1+ξ−ϵ(2cot2θ12−1)]+12ϕm32, (11)

where is given in (9). In practical cases the terms can be neglected. For low energies (close to the 1-2 resonance):

 ϕm21≈Δm221L4Eν ⎷(cos2θ12∓2VEνΔm221)2+sin22θ12. (12)

Notice that in the energy range above the 1-2 resonance and the amplitude (8) is reduced to the two neutrino form, which corresponds to factorization our3 .

Inserting expressions for the amplitudes (7) and (8) into (3) we obtain

 Pδeμ=Jθsinϕm21[sinϕm32cos(δ−ϕm31)+cos2θm12sinϕm21cosδ], (13)

where

 Jθ≡sin2θ23sin2θm12sin2θm13cosθm13 (14)

is the mixing angles factor of the Jarlskog invariant in matter. Using relation we obtain from (13)

 Pδeμ≈Jθsinϕm21[12sin2ϕm32cos(δ−ϕm21)+sin2ϕm32sin(δ−ϕm21)+cos2θm12sinϕm21cosδ]. (15)

Similarly, neglecting we find for

 Pδμμ=−cosδJθsinϕm21[sinϕm32cosϕm31+cos2θm12sinϕm21], (16)

or excluding :

 Pδμμ=−cosδJθsinϕm21[12sin2ϕm32cosϕm21−sin2ϕm32sinϕm21+cos2θm12sinϕm21], (17)

where 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.

### ii.3 Numerical results

We have computed the probabilities by performing numerical integration of the evolution equation for the complete system. We used the PREM density profile of the Earth prem and the values of the neutrino parameters eV, eV, , and , which are close to the current best fit values fogli . We assume the normal neutrino mass hierarchy in the most part of the paper.

In Fig. 1 we show the oscillation probabilities and as functions of the neutrino energy for different values of CP-phase and zenith angles. In the low energy range where sensitivity to is high and consider . 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 GeV. For core crossing trajectories (upper panels) the parametric effects distort the dependence of probability on energy.

The key feature which opens up a possibility to measure is the presence of systematic shift of the oscillatory curves (probabilities) at low energies, GeV, with increase of the phase. The shift occurs in the same way in wide energy interval GeV, and essentially for all trajectories which cross the mantle only. This systematic shift can be understood using analytical expressions for the probabilities. Averaging (15) over fast oscillations driven by the 1-3 splitting we find

 ⟨Pδeμ⟩=Jθ2[cosδcos2θm12sin2ϕm21+12sinδsin2ϕm21]. (18)

The first term does not change the sign with , whereas the second one does. Notice that above the 1-2 resonance , and so

 ⟨Pδeμ⟩≈Jθ2[−cosδsin2ϕm21+12sinδsin2ϕm21]. (19)

The difference of probabilities for a given value and equals:

 ⟨Pδeμ⟩−⟨P0eμ⟩=Jθ2[(1−cosδ)sin2ϕm21+12sinδsin2ϕm21]. (20)

The first term is positive for all and , and it is this term that produces a systematic shift of the probabilities.

For the values of -phase shown in Fig. 1 we obtain from (18)

 ⟨P0eμ⟩ = −⟨Pπeμ⟩=Jθ2cos2θm12sin2ϕm21, (21) ⟨Pπ/2eμ⟩ = −⟨P3π/2eμ⟩=Jθ4sin22ϕm21.

These equations show that is the smallest one. The probability increases with and reaches maximum at . For the trajectory with , the oscillation phase equals . That leads to , and consequently, to equal total probabilities. For the phase equals which gives different values of probability: , and furthermore . These results are in agreement with plots shown in Fig. 1.

Although there is certain phase shift with change of , the sizes of energy intervals where the difference 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 curve is around GeV. For the core-crossing trajectories () due to the parametric effects the transition probability first increases with increase of , it reaches maximum at and then decreases.

The probability, (17), averaged over the 1-3 oscillations equals

 ⟨Pδμμ⟩=−Jθ2cosδsin2ϕm21cos2θm12, (22)

where the term is neglected. Notice immediately that the CP-effect in the channel has an opposite sign with respect to that in the channel (18). Therefore the presence of both and original fluxes weakens the total CP-effect, and consequently, the sensitivity to which is unavoidable. We will call this the flavor suppression.

According to (22) dependence of the probability on factors out and therefore turns out to be very simple. The maximal effect is for ,

 ⟨P0μμ⟩=−⟨Pπμμ⟩≈Jθ2sin2ϕm21,

and , so that the total probabilities are equal for and which in perfect agreement with result of Fig. 1.

The difference of probabilities for a given value of and zero phase equals

 ⟨Pδμμ⟩−⟨P0μμ⟩=Jθ2(1−cosδ)sin2ϕm21cos2θm12≈−Jθ2(1−cosδ)sin2ϕm21. (23)

Only CP-even contribution is present.

The probabilities in antineutrino channels are shown in Fig. 2. Their dependencies on and can be immediately understood from our analytical treatment. According to (5) the averaged probabilities equal

 ⟨¯Pδeμ⟩ = ¯Jθ2[cosδcos2¯θm12sin2¯ϕm21−12sinδsin2¯ϕm21], (24) ⟨¯Pδμμ⟩ = −¯Jθ2cosδsin2¯ϕm21cos2¯θm12. (25)

For energies far above the 1-2 resonance, the expressions are further simplified since (recall, for neutrinos ):

 ⟨¯Pδeμ⟩=¯Jθ2[cosδsin2¯ϕm21−12sinδsin2¯ϕm21],     ⟨¯Pδμμ⟩=−¯Jθ2cosδsin2¯ϕm21. (26)

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 amplitude is for and the smallest one is for which is opposite to the 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 and signals are separated at least partially (see Sec. IV C).

As follows from Fig. 2 for the mantle crossing trajectories, only the largest CP effect on is in the range GeV where maximal values equal and for and correspondingly. These numbers are about 2 times smaller than for neutrinos. The reason is that, in the case of NH for neutrinos both and are enhanced in matter whereas for antineutrinos both and are suppressed. The antineutrino probabilities decrease with increase of energy above 0.8 GeV. This, as well as smaller cross-sections suppresses number of events and therefore reduces cancellation of the CP-effect.

Similar consideration can be performed for the channel for which . Notice that in vacuum , i.e. the probability is even function of . In the matter dominated region we have due to change of sign of the potential. The differences of the antineutrino probabilities for a given and equals at

 ⟨¯Pδeμ⟩−⟨¯P0eμ⟩ = −¯Jθ2[(1−cosδ)sin2¯ϕm21+12sinδsin2¯ϕm21], (27) ⟨¯Pδμμ⟩−⟨¯P0μμ⟩ = ¯Jθ2(1−cosδ)sin2¯ϕm21. (28)

They 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.

### ii.4 Magic lines and CP-domains

In what follows we will study differences of probabilities as well as distributions of events in the plane for different values of . The patterns of distributions are determined to a large extent by the grid of the magic lines Barger:2001yr ; Huber:2003ak ; Smirnov:2006sm ; our3 . The lines fix the borders of the CP-domains – the regions in the plane of the same sign of the CP-difference.

Let us summarize relevant information about properties of the magic lines. Recall that the magic lines are defined as the lines in the plane along which the oscillation probabilities do not depend on phase in the so called factorization (quasi ) approximation our3 . Correspondingly, the CP-differences vanish along these lines.

1. The solar magic lines are determined by the condition

 ϕS=ϕm21=nπ,    n=1,2,3,..., (29)

where in neutrino channels is given by the expression (12) for valid in framework below 1-3 resonance but extended to all the energies. For antineutrinos in the NH case everywhere. Along these lines below the 1-3 resonance. That would be the line of zero solar amplitude in the approximation. The minimum of probability at GeV for in Fig. 1 corresponds to the first magic line with . The minimum at GeV for (Fig. 1) is on the second magic line with .

Notice that the energy of minimal level splitting (maximal oscillation length) is given by GeV which is much bigger than GeV due to large 1-2 mixing. So, below 0.7 GeV the splitting increases and correspondingly the oscillation length decreases. Therefore the same phase can be obtained for smaller , and consequently, the solar magic lines bend toward smaller . At energies much above the 1-2 resonance these lines do not depend on energy and are situated at

 cosθz=−0.60, −0.86, −0.97, (30)

for , and correspondingly.

2. The atmospheric magic lines are determined by the equality

 ϕA=ϕm23=nπ,    n=1,2,3,... (31)

Along these lines . It would vanish exactly in the approximation, when that is far above the 1-2 resonance. Zeros of the probability at GeV (see Fig. 1) which do not depend on are situated on the atmospheric magic lines. E.g., for these points are at GeV and GeV. For , zeros are at GeV. For the solar and atmospheric magic lines coincide with those for in the limit .

The magic lines determined by (29) and (31) do not coincide with lines where and in the framework. But they play the role of asymptotics of the true lines where dependence of probabilities on disappears. The latter interpolate between different magic lines.

3. The interference phase lines are important for distinguishing different values of the CP-phase: a given value and a different value . Along these lines . According to (3) for the condition reads

 cos(ϕ+δ)=cos(ϕ+δ0),

where and the latter is the vacuum oscillation phase. This condition corresponds to intersection of probability curves for different values of phases and in Fig. 1. For the condition can be written as or

 ϕ31=Δm231L4Eν=−δ2+nπ. (32)

For the inverse channel, , the sign of should be changed. According to (4) dependencies of the probability on and factor out in the approximation , and the corresponding interference phase line is determined by the condition , or

 ϕ≈ϕ31=π2+nπ.

The condition can be written as

 Eν=−A(ϕ)cosθz=−REΔm2312ϕ(δ)cosθz, (33)

where is the Earth radius and in general should lead to the vanishing CP-difference of probabilities.

The exact value of interference phase does not coincide with . In the constant density approximation equals the phase of the expression in brackets of (8):

 tanϕ=−sinϕm32sinϕm31cosϕm31sinϕm32+cos2θm12sinϕm21. (34)

Notice that would be equal , if . The latter is satisfied for high energies , where . However, if we can not neglect the second term in the denominator of (34). Notice that in the limit we obtain from (16)

 Pδμμ=−cosδJθsinϕm21sinϕm32cosϕm31, (35)

where one can see immediately all three “magic” conditions.

Notice that magic lines could be introduced immediately in the framework as the lines along which . 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

 ϕm21≈{ϕmS,   E≪ER31ϕ0A,   E≥ER31,    ϕm31≈{ϕ0A,   E≪ER31ϕS,   E≫ER31,    ϕm32≈ϕmA,

where is the phase in vacuum. So, the true lines of zero difference of probabilities interpolate between the magic lines (see our3 for details).

## 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: it has the same sign and the same change with . Effect is different for the core crossing trajectories, . So, there could be partial cancellation due to smearing over the zenith angle. Another important feature is that the relative CP effect at the probability level increases with decrease of energy. In this connection we will explore a possibility to measure using multi-megaton scale neutrino detectors with low energy threshold. As it was already realized in ARS , sensitivity of PINGU to is low. So, we will consider future possible upgrades of PINGU. We will also quantify capacity of PINGU to obtain information about . For definiteness we will speak about PINGU for which more information is available. Similar upgrades can be considered for ORCA detector orca .

### iii.1 PINGU and Super-PINGU

We calculate event rates for the proposed PINGU detector and for possible future PINGU upgrade which we call Super-PINGU. The PINGU detector pingu2 will have 40 strings additional to the DeepCore strings with 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 GeV. Strict criteria allow over efficiency of event reconstruction for all 3 flavors pingu2 . We parametrize the PINGU effective mass as

 ρVeff,μ(Eν)=3.0[log(Eν/GeV)]0.61Mt (36)

and

 ρVeff,e(Eν)=3.1[log(Eν/GeV)]0.60Mt, (37)

respectively for and . Here is the effective volume and is the density of the ice. These parametrizations well represent simulated volumes pingu2 from GeV up to 25 GeV. We will use an accuracy of the energy and angle reconstruction for PINGU from pingu2 .

Along with the PINGU proposal the idea has been discussed to construct “ultimate” multi-megaton-scale detector MICA with a threshold about 10 MeV allowing to detect the solar and supernova neutrinos mica . Clearly reducing the threshold by more that 2 orders of magnitude is very challenging. In this connection we would like to consider a kind of intermediate step - the detector with an effective energy threshold about (0.1 - 0.2) GeV, 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 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 cowen . The effective mass can be parameterized as

 ρVeff(Eν)=2.6[log(Eν/GeV)+1]1.32Mt, (38)

for both and events. We call this version Super-PINGU. According to (36) and (38) the effective mass, , in the range (1 - 2) GeV equals Mton for PINGU and Mton for Super-PINGU, i.e. 4 times larger. For the bin below 1 GeV the corresponding numbers are 0.3 and 2.2 Mton (7 times larger). This can be compared with MICA, which may have 220 strings and 140 DOMs per string. We will extrapolate to lower energies some PINGU characteristics from the proposal pingu2 .

Going to further upgrade has double effect:

• increase of the effective volume, especially in the low energy bins, and

• improvements of reconstruction of the neutrino energy and direction as well as the flavor identification of events for all energies.

Super-PINGU will have three times denser DOM array than PINGU. Therefore it will collect about 3 times more photons from the same event (with the same neutrino energy). Recall that, in PINGU the average distance between DOM’s is smaller than the photon scattering length (50 m).

We describe uncertainties of reconstruction of the neutrino energy and direction by smearing functions

 GE(Erν,Eν),   Gθ(θrz,θz),

where and ( and ) are the true (reconstructed) energy and zenith angle of the neutrinos. For PINGU we use and from pingu2 determined down to energies GeV. The distributions are normalized in such a way that

 ∫dEνdθzGE(Erν,Eν)Gθ(θrz,θz)=1.

Notice that PINGU distributions have longer tails than the Gaussian functions.

Characteristics of the Super-PINGU reconstruction are expected to be better. We estimate parameters of and for Super-PINGU using the DeepCore resolutions and the simulated PINGU resolutions dcres , gross in the following way. For a given event the number of photons collected is proportional to the density of DOMs, that is for fixed total volume of the detector. Therefore the relative statistical error in determination of characteristics is proportional to , so we can assume that

 σθ∝1√NDOM,    σE∝1√NDOM. (39)

Estimations of resolutions of the DeepCore and PINGU confirm (39). Indeed, DeepCore has about DOM’s, while PINGU (40 strings with 60 DOM’s per string) will have DOM’s (also with higher quantum efficiency), that is, . Since the density of DOM’s in PINGU is about 4.5 times larger, amount of light detected from the same event will be about 4.5 times larger. According to dcres , gross and pingu2 for the events the ratio of resolutions (median errors)

 σPINGUθσDCθ≈0.5. (40)

The ratio equals 0.66 at GeV, however estimation of DC parameters become not very reliable at low energies. For the events the improvement is even better: The ratio of median errors (40) is () in the interval GeV and it becomes 0.6 at 5 GeV.

For neutrino energy reconstruction (median energy resolution) of the events we have

 σPINGUEσDCE≈0.58−0.61 (41)

in the interval GeV. It decreases down to 0.52 at GeV. Similar improvement is expected for the events.

The Super-PINGU will have 3 times larger number (and therefore density) of DOM’s, than PINGU. Therefore according to (39) the resolutions will be further improved by factor . So for Super-PINGU we use the resolution functions from Fig. 7 and 8 of pingu2 , scaling their widths as

 σSuperPINGUθ=1√3σPINGUθ,   σSuperPINGUE=1√3σPINGUE. (42)

We extrapolate these functions down to GeV and for simplicity neglect possible dependences of the factors in Eq. (42) on energy. (Notice that according to pingu2 the median value of angle is very similar for cascades and tracks.)

This estimation of improvement can be considered as conservative. Indeed, the DeepCore characteristics have been obtained after stringent kinematical cuts which allows one to select a sample of high quality events. That reduces efficiency of reconstruction (fraction of reconstructed events) down to , whereas PINGU characteristics have been obtained with efficiency. With stronger cuts in PINGU the reconstruction characteristics could be even better. Also, developments of electronics may lead to further improvements. Clearly, configuration of Super-PINGU should be optimized taking into account also the cost of construction. For large density of strings the issue of the ice stability may become important. One can reduce number of strings by increasing number of DOMs per string (decreasing vertical spacing). Since typical size of an event is about 100 m, for distances between strings (17 m) the total number of DOMs in the unit volume matters and geometry plays only secondary role. Another option is to consider underwater detector, i.e., an upgrade of ORCA.

### iii.2 Distributions of events in the neutrino energy and zenith angle plane

To evaluate sensitivity of Super-PINGU to we will compute the distributions of events of different types and explore their dependence on . The numbers of events , produced by neutrinos () with energies and zenith angles in small bins and marked by subscript equal

 Nij,α=2πNAρT∫Δicosθzdcosθz∫ΔjEνdEν Veff,α(Eν)dα(Eν,θz). (43)

Here is the exposure time, is the Avogadro’s number. The density of events of type , , (the number of events per unit time per target nucleon) is given by

 dα(Eν,θz)=dνα+d¯να=[σαΦα+¯σα¯Φα], (44)

where and are the fluxes of neutrinos and antineutrinos at the detector which produce events of the type , and and are the corresponding cross-sections. In turn, the fluxes at the detector equal

 Φα=Φ0μPμα+Φ0ePeα,

and are the original muon and electron neutrino fluxes at the production.

With decrease of energy, resonance processes (pion production) and quasi-elastic processes will contribute, and the latter dominates below 1 GeV. In our estimations we use the total neutrino-nucleon cross-sections down to (0.2 - 0.3) GeV as they are parametrized in pingu2 , We assume 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 extrapolation given in pingu2 . Clearly in future these computations should be refined.

We use the atmospheric neutrino fluxes, and (and corresponding fluxes of antineutrinos) from Refs. Honda:1995hz ; Athar:2012it . At low energies the geomagnetic effects become important which break azimuthal symmetry.

After smearing in the () plane, we obtained the unbinned distribution of events as

 Nα(Er,cosθr)=2πNATρ∫dcosθz∫dEν GE(Erν,Eν) Gθ(θrz,θz) Veff(Eν) dα(Eν,cosθz), (45)

, and then binned them according to

 Nij,α=