Restricting the LSND and MiniBooNE sterile neutrinos with the IceCube atmospheric neutrino data

# Restricting the LSND and MiniBooNE sterile neutrinos with the IceCube atmospheric neutrino data

## Abstract

We study oscillations of the high energy atmospheric neutrinos in the Earth into sterile neutrinos with the eV-scale mass. The MSW resonance and parametric enhancement of the oscillations lead to distortion of the zenith angle distribution of the muon-track events which can be observed by IceCube. Due to matter effect, the IceCube signal depends not only on the mixing element relevant for LSND and MiniBooNE but also on and the CP-violating phase . We show that the case with leads to the weakest IceCube signal and therefore should be used to bound . We compute the zenith angle distributions of the events for different energy intervals in the range (0.1 - 10) TeV and find that inclusion of the energy information (binning in energy) improves the sensitivity to drastically. We estimate that with already collected (during 3 - 4 years) IceCube statistics the bound ( C.L.) can be established and the mixing required by LSND and MiniBooNE can be excluded at confidence level.

## I Introduction

Existence of sterile neutrinos with mass  eV and with mixing required by the LSND (1) and MiniBooNE (2) anomalies is not a small perturbation of the standard scheme. It leads to rich phenomenology and far going consequences for theory. Therefore checks of the existence of these neutrinos become one of the main objectives of the neutrino physics. Recall that the and oscillations interpretation of the LSND and MiniBooNE results implies non-vanishing admixtures of the and in the 4th mass eigenstate, quantified respectively by mixing elements and , and the oscillation depth is given by .

The present experimental situation is rather controversial.

(i) Interpretation of the LSND/MiniBooNE results in terms of oscillations with sterile neutrinos is not very convincing in view of uncertainties in the cross-sections, energy scale calibration as well as backgrounds (3). Moreover, fit of the energy spectra of excess of events in MiniBooNE with one sterile neutrino is rather poor. Good fit can be obtained in the presence of second sterile neutrino with large mass  eV (4). However, the latter is in serious conflict with Cosmology and laboratory observations.

(ii) There is strong tension between the appearance (LSND/MiniBooNE) and disappearance (short baseline) experimental results within both 3+1 and 3+2 schemes (5). Furthermore, the disappearance has not been observed by MiniBooNE itself, thus leading to constraints on  (6). Strong bound follows from the neutral current interaction measurements in near and far MINOS detectors (7). A combination of the negative results of CDHS (8), MiniBooNE and MINOS on the disappearance puts the strongest limit on .

(iii) The appearance signal has not been observed in OPERA (9) and ICARUS (10) experiments. This directly excludes the low part of the LSND/MiniBooNE region.

(iv) The reactor anomaly – disappearance of the flux from reactors (11), is in favor of the eV mass scale sterile neutrinos. Still the claimed deficit of the signal can be due to underestimated uncertainties in the antineutrino flux calculations (12).

In this connection large number of new experiments has been proposed to test existence of sterile neutrinos (see (13) and references therein, see also (14)) and some of them can be realized in the next 5 - 10 years. At the same time, study of the atmospheric neutrinos by IceCube with already collected statistics can contribute in substantial or even decisive way to resolution of the LSND/MiniBooNE anomaly.

Indeed, for  eV implied by LSND, oscillations of neutrinos with energies  TeV in the matter of the Earth will undergo the MSW resonance enhancement (15); (16); (17). Cosmic neutrinos would be affected by these oscillations (15); (16). The resonantly enhanced oscillations lead to partial disappearance of the (or ) flux, and consequently, to distortion of the energy and zenith angle distributions of the muon neutrino charged current events (17). This can be explored in IceCube using the atmospheric neutrino flux (17). Actually, the resonance enhancement of the oscillations occurs for the mantle crossing trajectories with the MSW resonance peak at  (17). For the core crossing trajectories the parametric enhancement of oscillations takes place (18) at about 2 times lower energies. The original consideration for single sterile neutrino was generalized later to the case of two sterile neutrinos (19).

The first results from AMANDA and IceCube-40 experiments motivated further detailed studies of these oscillation effects (21); (20); (22); (23); (24). In (20) the effects of sterile neutrinos have been studied in different mixing schemes. It was observed that in the case of mixing the effects, and consequently, bounds on the mixing angle become weaker. The effect of can also be observed at lower energies, GeV, in the DeepCore experiment (21). The analysis of DeepCore has been extended to and models with two sterile neutrinos in (22). In (23) by scanning the parameter space of the model, which includes , a mixing scheme independent exclusion region has been found in () plane from the IceCube-40 data. The bound on mixing in (23) does not exclude the favored region by MiniBooNE/LSND. In (24) the potential of cascade IceCube events in constraining sterile neutrinos has been studied.

Recently results from IceCube experiment collected during the period from May 2010 to May 2011 have been published (25) (the so-called IceCube-79 data). The zenith angle distribution of events is in very good agreement with the standard oscillations. The statistical errors in each of 10 zenith angle bins are below . The systematic errors are still rather large, however small spread of the experimental points within the statistical errors indicate that systematic errors are correlated. The expected effect from sterile neutrinos is in several vertical bins (21); (22); (23). This indicates that effects of sterile neutrinos, if exist, should be small.

After IceCube-79 the IceCube-86 with larger effective area is taking data. Presently the IceCube-86 exposure is at least 4 times larger than the IceCube-79 one and therefore statistical error is reduced by factor 2. Also systematic error is expected to be smaller. With this one can perform critical test of existence of the LSND/MiniBooNE sterile neutrinos.

To exclude the LSND and MiniBooNE sterile neutrino one needs to explore effects in whole range of relevant parameters including mixing angles and CP-violating phases. In general the mixing is described by 6 mixing angles and 3 CP-violating phases. Therefore complete scanning of the parameter space is very cumbersome. Fortunately, the effect in IceCube depends appreciably only on few parameters. In this paper we identify these relevant parameters and find their values which minimize the sterile neutrino effects in IceCube. We explore ways to improve sensitivity of IceCube to sterile neutrinos using information about energies of events (energy binning).

The paper is organized as follows. In Sec. II we consider in detail dependence of the oscillation probabilities relevant for IceCube on mixing scheme, and in particular on the mixing. We evaluate also effect of CP-violating phases. We show that flavor mixing scheme with provides the weakest IceCube signal and therefore should be used to exclude the sterile neutrino interpretation of the LSND and MiniBooNE results. In Sec. III the zenith angle distributions of events in IceCube for different energy ranges are computed. Also we present the energy distributions of events smeared with the neutrino energy reconstruction function. In Sec. IV we perform simple statistical analysis evaluating sensitivity of IceCube to the sterile neutrino mixing. We explore how the energy information will improve the sensitivity. Conclusions are presented in Sec. V.

## Ii LSND/MiniBooNE and IceCube signals

### ii.1 Generalities

We will consider mixing of 4 neutrinos : , where . The unitary mixing matrix is usually parametrized as

 U4=R34(θ34)R24δ(θ24,δ24)R14δ(θ14,δ14)R23(θ23)R13δ(θ13,δ13)R12(θ12) , (1)

where ( and ) is the rotation matrix in the -plane over the angle . The rotation contains CP-violating phase in such a way that and . In Eq. (1), is the usual CP-violating phase of the scheme; whereas and are the two new phases. In this parametrization:

 Uμ4=cosθ14sinθ24e−iδ24,Uτ4=cosθ14cosθ24sinθ34 .

The Hamiltonian describing propagation of this system in matter is

 H=12EνU4M2U4†+V(r) , (2)

where is the diagonal matrix of mass-squared differences:

 M2≡diag(0,Δm221,Δm231,Δm241),    Δm2ij≡m2i−m2j , (3)

with . The matrix of matter potentials in the flavor basis, , after subtracting the neutral current contribution, takes the following form:

 V(r)=√2GFdiag(Ne(r),0,0,Nn(r)/2) . (4)

Here is the Fermi constant; and are the electron and neutron number densities.

To avoid conflict with cosmological constraints we assume , and therefore the resonance enhancement of oscillations takes place in the anti-neutrino channel, which leads to a dip in at energies .

The appearance signals of LSND/MiniBooNE depend on and and do not depend on . The disappearance depends on only. This is because the effects in these experiments are due to short baseline oscillations in vacuum. In contrast, the IceCube signal depends besides on the admixture  (20); (23). Therefore to get limit on the LSND/MiniBooNE sterile neutrino one needs to select the value of which leads to the smallest effect in IceCube for fixed . The current upper limit on is rather weak: at 90% C.L. (7). The global analysis of the oscillation data, which includes also the atmospheric neutrinos gives moderately stronger upper limit: at  (5). IceCube itself can constrain down to by using cascade events induced by the atmospheric neutrinos (24). Indeed, leads to the oscillation transitions , ’s produce leptons and then decays of ’s generate cascades. Using the cascade events, it is also possible to disentangle the effects of and . Indeed, results in a deficit of cascades, whereas via the oscillations leads to an excess of cascades. In what follows we will explore the range .

Let us consider the differences between the LSND/MiniBooNE and IceCube signals in more details. We will concentrate on the charged current events. There are two simplifying circumstances at high energies:

• Strong suppression of the mixing everywhere apart from the resonance in the TeV range. Even in the resonance the oscillations relevant for the events will be suppressed by small .

• Smallness of the original atmospheric flux at high energies.

Under these circumstances one can exclude the flavor from consideration and neglect the 1-2 mass splitting. As a result, the system is reduced to the system of three flavors mixed in three mass states : . Here is the submatrix of after removing the first row and column and setting () to zero. In the next subsection we will make further simplification developing a single approximation. We will use these simplifications in our qualitative analysis which will allow to understand various results of numerical computations. The latter have been done for the complete system.

### ii.2 Single Δm2 approximation

At very high energies  TeV, when also can be neglected, the system is described by and the vector of mixing parameters . We can perform rotation in the plane by the angle , determined by

 tanθ′=Uμ4Uτ4 , (5)

such that in new basis, , the first component of the vector vanishes:

 →U′T=(0,√U2τ4+U2μ4,Us4) . (6)

In this basis the Hamiltonian becomes

 H=Δm2422Eν→U′→U′T+V . (7)

The state decouples and the problem is reduced to two neutrinos problem with the mixing parameter in vacuum

 sin22θx=4U2s4(U2τ4+U2μ4)=4U2s4(1−U2s4) , (8)

and the potential .

Let us introduce the matrix in the basis :

 S′=⎛⎜⎝1000Aτ′τ′Asτ′0Aτ′sAss⎞⎟⎠ . (9)

In terms of the matrix in the original flavor basis equals

 S=Rμτ(θ′)†S′Rμτ(θ′) . (10)

From Eqs. (9), (10) and (5) we find the survival probability

 Pμμ≡|Sμμ|2=∣∣ ∣∣1−U2μ4U2τ4+U2μ4(1−Aτ′τ′)∣∣ ∣∣2 . (11)

It can be rewritten as

 Pμμ=|1−κ(1−Aτ′τ′)|2 , (12)

where the prefactor of the amplitude term, , equals

 κ≡U2μ4U2τ4+U2μ4=sin2θ′ . (13)

Let us consider properties of the probability . Independently of the , the maximal value is achieved if . The minimal possible value (maximal oscillation effect) do depend on :

• For , that is , we have

 Pminμμ=0 ,   if  Aτ′τ′=−U2τ4U2μ4 . (14)
• For or ,

 Pminμμ=|1−2κ|2=∣∣ ∣∣U2τ4−U2μ4U2τ4+U2μ4∣∣ ∣∣2,   if    Aτ′τ′=−1 . (15)

Notice that the amplitude depends on the neutrino energy, zenith angle, etc., and conditions on the amplitude in the Eqs. (14) and (15) may not be satisfied. So, extrema may not be realized and .

For the mixing scheme, when , we obtain from Eq. (12)

 Pμμ=|Aτ′τ′|2 . (16)

If (the “ - mass” mixing scheme),

 Pμμ=14|1+Aτ′τ′|2 . (17)

In the LSND and MiniBooNE experiments the oscillations occur in vacuum. For the Hamiltonian in Eq. (7) has the eigenvalues and . Using Eq. (7) it is straightforward to find that

 Avacτ′τ′=1−(U2τ4+U2μ4)(1−e−iϕ4) , (18)

where the vacuum phase equals

 ϕ4≡Δm242L2Eν . (19)

Inserting Eq. (18) into Eq. (12) we obtain

 Pμμ=∣∣1−U2μ4(1−e−iϕ4)∣∣2 . (20)

Here the dependence on disappears and Eq. (20) coincides with the standard oscillation probability which depends on only.

In the case of IceCube, neutrinos oscillate in the matter of Earth and one should use the amplitude of oscillation in matter . Cancellation of the factors which depend on does not occur anymore and the probability depends on . Let us elucidate this dependence considering oscillations in matter with constant density. The amplitude is function of , and . Both the oscillation depth and length depend on in non-trivial ways:

 Aτ′τ′=cos2θmxe−iϕm3+sin2θmxe−iϕm4=e−iϕm3[cos2θmx+sin2θmxe−iΔϕm] , (21)

or

 1−Aτ′τ′=1−e−iϕm3+sin2θmx(e−iϕm3−e−iϕm4) . (22)

Here

 sin2θmx=12⎡⎢ ⎢⎣1−1+~V−2s2x√(1+~V)2−4s2x~V⎤⎥ ⎥⎦ , (23)

and

 s2x≡sin2θx=(U2τ4+U2μ4) , (24)
 ~V≡2EνVΔm242 . (25)

The phases equal

 ϕm3,4=ϕ42[1+~V∓√(1+~V)2−4s2x~V] . (26)

(At the same time the prefactor in Eq. (12) is function of mixing parameters in vacuum.)

Let us analyze these expressions in two cases.

1. Tails of resonances (energy range far from resonance) and . In the lowest order, neglecting under squared root in Eq. (23), we have

 sin2θmx≈s2x1+~V , (27)
 ϕm4=ϕ4(1+~V),    ϕm3=0 . (28)

Then the probability equals

 Pμμ=∣∣ ∣∣1−U2μ41+~V(1−e−iϕm4)∣∣ ∣∣2 , (29)

which does not depend on .

Taking the first approximation in in the denominator of Eq. (23) we obtain

 sin2θmx≈s2x(1+~V)2−2s2x~V , (30)
 ϕm3=ϕ4~V1+~Vs2x,    ϕm4=ϕ4(1+~V)−ϕm3 . (31)

If , we have

 1s2x(1−Aτ′τ′)=iϕ4~V1+~V+1−e−iϕ4(1+~V)−iϕ4~V1+~Vs2x[1+e−iϕ4(1+~V)](1+~V)2−2~Vs2x . (32)

Dependence on appears only in corrections via . In the neutrino (non-resonance) channel we have the same expression with .

Notice that is the standard probability which depends on the phase difference . In contrast, is generically the probability of system even in the limit . As a result it depends on both phases and separately. Nonzero value of leads, in particular, to the fact that in Eq. (21) cannot be and to deviation of maximal value of the probability from 1 (see Fig. 1). From these formulae one can see that decreases with the increase of , and therefore, .

2. The resonance region. For we have

 sin2θmx≈12(1+sx) , (33)
 ϕm3,4=∓ϕ4sx . (34)

(In resonance .) Inserting these expressions into Eq. (22) and Eq. (12) we obtain

 Pμμ=∣∣∣1−U2μ41s2x[1−cos(ϕ4sx)+isxsin(ϕ4sx)]∣∣∣2 . (35)

With the increase of , and consequently , the probability in Eq. (35) in resonance increases; i.e., sterile neutrino effect decreases.

For small phases, , Eq. (35) leads to

 Pμμ=∣∣1−U2μ4(ϕ24+iϕ4)∣∣2 , (36)

where again dependence on disappears. Notice that in the dip of the phase equals , so for small () the dependence of probability on the mixing scheme is very weak. It appears when becomes large.

Summarizing, the difference between LSND/MiniBooNE and IceCube signals is due to matter effect. The dependence of probability on in IceCube signal can be understood in the following way. For fixed with the increase of the mixing angle , which determines the strength of resonance (its width) and also the oscillation length, increases. However, for small dependence of the oscillation probability on is weak. With increase of the mixing parameter increases and dependence of probability on becomes strong: It suppresses the peak in the resonance region, since reaches maximal value and stops to increase, whereas the prefactor continues to decrease. In contrast, beyond the resonance, in the tails, continue to increase due to widening of the resonance. As a result, here the disappearance probability increases with and the minimal effect is for (flavor mixing scheme). This approximate analytical consideration helps to understand various features of exact numerical results.

### ii.3 Survival probabilities

We find the () survival probabilities, as functions of neutrino energy and zenith angle, by solving the evolution equation with the Hamiltonian in Eq. (2) numerically. We take the best-fit values of active-active mixing angles as well as and according to (26). We use as a benchmark value or which is at the border of IceCube sensitivity region (see Sec. IV). This value is substantially lower than the one required by LSND: .

In Fig. 1 we show dependence of the survival probability (resonance channel) on the neutrino energy for different values of and for two different zenith angles. In the case of (mantle crossing trajectory, top panel) the resonance dip appears at  TeV (for  eV) as a result of resonance enhancement of oscillations. For  TeV the dependence is well described by the single approximation. Below  TeV effect of oscillations driven by becomes important. It increases with the decrease of energy as a result of interference of the and modes of oscillations. The dashed line in Fig. 1 shows the standard probability. Maximal effect of sterile neutrinos is in the resonance. Notice that the line describing probability in the presence of sterile neutrino for touches the probability (the latter is the upper bound for the probability with sterile neutrino). With the increase of the sterile neutrino effect decreases in the resonance dip. It increases in the tail above  TeV (this can be seen from the analytical formulas obtained in Sec. II.2), and it decreases again below  TeV – in the region were standard oscillations become important. Also, with the increase of the oscillatory curve shifts slightly to higher energies.

For (core crossing trajectory, bottom panel) the parametric enhancement of oscillations takes place. The parametric dip at  TeV is larger than the resonance dip. The dependence of probability on is similar to that for the mantle crossing trajectories.

In Fig. 2 we show the survival probabilities (non-resonance channel). In this channel oscillations are matter suppressed. Again below  TeV the oscillation effect is driven by . Mixing with sterile neutrinos enhances this effect, especially for large . With increase of the probability decreases for all energies, so that the weakest effect is for .

Summarizing, the effect of sterile neutrino increases with the increase of at all energies apart from the region of resonance and parametric dips in the antineutrino channel, and as we will see, the former dominates in the integral effect. So, to exclude sterile neutrino mixing with IceCube one should consider the case .

The highest acceptance of the IceCube detector for events is in the range (0.3 - 1) TeV. Thus, for  eV the maximum sensitivity is in the region of low energy tail of the survival probability, where the sterile neutrino effect increases with . For smaller the resonance dip shifts to the region of the maximal IceCube acceptance and dependence of effect on can be opposite at least in the restricted energy range. However this opposite dependence disappears after smearing over the neutrino energies.

### ii.4 CP-violation effects

Let us consider dependence of the sterile neutrino effects on the CP-phases. In approximation of the parametrization of Eq. (1) the mixing depends on one CP-phase :

 Uf=R34R24δ(δ24)R23. (37)

Clearly, in the cases where mixing (in our approximation) can be parametrized by two rotations, there is no CP-violation, and no dependence on the effects of CP-phase. (Beyond this approximation the CP-violation will show up, however, its effects will be suppressed.)

In the case of the mixing matrix in Eq. (37) becomes . The CP-phase (as well as any other phase introduced here) can be eliminated by redefining the neutrino mass (), or/and flavor fields. Therefore the CP-violation (in approximation) should be proportional to or .

Furthermore, if , the 2-3 rotation can be removed, so that the mixing matrix in Eq. (37) is reduced to two rotations: , and again the CP-phase becomes unphysical. The CP-violation effect should be proportional to the phase induced by or more precisely to and to . The CP-violation in the probability appears as an interference of the main term induced by , which is close to 1 beyond the resonance, and the term induced by . Therefore the CP-violating part of the survival probability should be approximately equal to

 Pδμμ ∼ 2cosδ24sin2θ34sin2θ24sinϕ31≈2cosδ24sin2θ34sin2θ24ϕ31 (38) = cosδ24(Δm241LEν)sin2θ24sin2θ34 .

(It also is proportional to which is omitted above.) Here we have taken into account also that the survival probability is a CP-even function of . According to Eq. (38) the CP-violation effect increases linearly with and also increases with decrease of energy. For high energies, where the 1-3 oscillation phase is very small, the effects are suppressed. These dependences can be seen in Fig. 3. In the top (bottom) panel of Fig. 3 we show the dependence of () survival oscillation probability on neutrino energy for various values of CP-phase . The CP-violation effect appears below  TeV where the driven oscillations become important. Notice that for the and survival probabilities at low energies are nearly the same. For the and probabilities switch with each other in comparison with case:

 P(νμ→νμ;δ=0)≈P(¯νμ→¯νμ;δ=π) .

Varying from to we obtain from Eq. (38)

 ΔPδμμ=2(Δm241LEν)sin2θ24sin2θ34 .

For TeV it gives in agreement with Fig. 3.

Nonzero have opposite effects in neutrino and antineutrino channels: it enhances the oscillations due to in anti-neutrino channel and suppresses oscillations in neutrino channel. This leads to partial cancellation of the CP-phase effect at IceCube where signals from neutrino and antineutrino sum up. As a result (see Sec. III), CP-violation effect is subleading with respect to the effect of 3-4 mixing and our conclusion that the weakest limit on is realized for still holds.

## Iii Zenith angle and energy distributions of the νμ events

The number of -track events in IceCube originating from the () CC interactions with the reconstructed neutrino energy and direction in the ranges and is given by

 Ni,j=TΔΩ∫Δicosθzdcosθz∫ΔjErνdErν∫dEν (39) G(Erν,Eν)Aνμeff(Eν,cosθz)[∑α=e,μΦνα(Eν,cosθz)Pαμ]+(ν→¯ν),

where is the muon (anti)neutrino effective area and is the flux of atmospheric  (27); (28). As we mentioned above, in the energy range we are considering the contribution of can be neglected since it is double suppressed: first, by small original flux, and second, by small oscillation probability which is proportional to . In Eq. (39) is the exposure period and is the azimuthal acceptance of the IceCube detector. For estimations we use the IceCube-79 effective area in the energy range of  TeV obtained by rescaling of the IceCube-40 effective area (23); (29). In our estimation we assume the same ratio of as for the IceCube-40 (23). We take 3 times larger statistics than IceCube-79; that is in Eq. (39), where  days.

In Eq. (39), and are the reconstructed and true neutrino energies, and is the resolution (reconstruction) function. The observable quantities in the IceCube detector are the energy and direction of muons produced in the CC interaction of and with nuclei. Therefore, there are two contributions to the width of the reconstruction functions of energy and direction of neutrinos: 1) the finite resolutions in measurement of the muon energy and direction; 2) the kinematic uncertainty related to the difference between muon energy and direction and the neutrino ones. Let us consider these contributions in order. In the TeV range the IceCube detector measures the muon direction with precision of less than  (29). The average angle between the neutrino and muon momenta in the CC interactions is which decreases from at  GeV down to at  TeV. Thus, we can identify the measured zenith angle of muon with the zenith angle of neutrino and consider 20 bins in without any smearing of the distributions before binning.

The reconstruction of neutrino energy is by far less precise. The observable quantity in the IceCube detector is the energy loss of muons, , which is related to the muon energy by

 dEμdx=−α−βEμ .

Here and are nearly energy-independent coefficients describing the energy losses due to ionization and radiation respectively. For energies  TeV the radiation (-term) dominates and the energy loss of muons is proportional to energy. Therefore the muon energy can be determined by measuring the energy loss even over a part of the muon track provided that the point (vertex) of muon production is known. However, in the high energy range for most of events the vertex of neutrino-nucleon interaction is outside of the geometrical volume of the detector which severely restricts the energy reconstruction and only lower bound can be established on the energy. For the low energy range, where the ionization (-term) dominates, the energy loss of muons is independent of their energy and the muon energy can be inferred from the energy loss measurements only when the whole track of muon is inside the geometrical volume of detector. At low energies most of the muons are produced inside the detector and therefore the muon energy reconstruction improves. The energy of neutrino is related to the muon energy through the inelasticity (fraction of the neutrino energy transferred to hadrons): . The average inelasticity , is nearly constant in the range  TeV, however, has wide distribution. When the vertex of CC interaction is inside the detector, measurement of the hadronic cascade energy would improve the neutrino energy reconstruction. Putting all these factors together, the IceCube collaboration claimed the resolution of the neutrino energy reconstruction in units of in the  TeV range (see (29)).

We compute the zenith angle distributions of events with and without sterile neutrinos using Eq. (39). We take as the reconstruction function the normalized Gaussian distribution with width . The estimations show that variations of within do not produce significant changes of the IceCube sensitivity. The reason is that although energy smearing decreases the depth of resonance dip in , it also widen the dip which can partially compensate the former. Since the acceptance of IceCube detector changes with energy, the smearing of distributions leads to a moderate weakening of bounds. After the smearing of events, we integrated the number of events over the energy bins , as described below.

The zenith angle dependences of events are determined by the probabilities discussed in the previous section and the product . The function has maximum at  TeV. It decreases by one order of magnitude at  TeV and by another factor of 5 down to  TeV. On the other hand, features in the oscillation probabilities have scale. So, to take into account their contributions to the integral effect, the correct factor would be , which has maximum in the range  TeV. In the resonance this product is only 2 times smaller than the maximum of .

In Fig. 4 we show distortion of the zenith angle distribution of the -track events, that is, the ratio of the distributions with and without sterile neutrinos. We take the benchmark value and . The events are integrated over whole energy range  TeV. Fig. 5 shows distortion of the distributions in two energy bins:  TeV and  TeV. Fig. 6 is for three energy bins:  TeV,  TeV and  TeV. The numbers of events in these regions are 9532, 14277 and 9139 correspondingly for the IceCube-79 exposure. This gives an idea about the relative contributions of these three regions.

With the decrease of the resonance and parametric dips shift to lower energies towards the maximum of function . This leads to stronger distortion of the distributions. Notice that for  eV the center of the parametric dip is at  TeV. Therefore in the case of three bins (see Fig. 6) the center is in the second bin, whereas for  eV it is in the third bin. As a result, the effect for core crossing trajectories is stronger in the second bin for  eV and in the third bin for  eV.

The suppression of the number of muon-track events increases with the decrease of ; i.e., with approaching to vertical trajectories. A jump in the distributions at is related to turning on the oscillation effect at low energies. The position of the first oscillation minimum in the survival probability is determined by the condition , where is the radius of the Earth and is the vacuum oscillation length (at low energies the matter effect is small). With further decrease of (increase of the length of neutrino trajectory) the oscillations become quickly averaged. For TeV and the condition gives for the position of minimum, in agreement with result of Fig. 5a (red histogram). For  TeV the minimum is at (see blue histogram). For and low energies the position of minimum is at , so that strong effect develops already in the first zenith angle bin. In the high energy range ( TeV) the minimum is at in agreement with blue histogram of Fig. 5b.

The break of the dependence at corresponds to trajectories which start to cross the core. For stronger suppression is due to the parametric enhancement of oscillations. The strongest relative effect is in the high energy bin which covers the resonance dip. For low energy bins the distortion is rather weak (distribution is almost flat), the effect is reduced to nearly uniform suppression of number of the events due to averaged oscillations. This can be absorbed by the uncertainty in the normalization of the atmospheric neutrino flux. Strong distortion comes from high energy bins where resonances are situated.