On the robustness of IceCube's bound on sterile neutrinos in the presence of non-standard interactions
PPrepared for submission to JHEP
On the robustness of IceCube’s bound on sterileneutrinos in the presence of non-standardinteractions
Arman Esmaili and Hiroshi Nunokawa
Departamento de Física, Pontifícia Universidade Católica do Rio de Janeiro, C. P. 38097, 22451-900, Rio de Janeiro, Brazil
E-mail: [email protected] , [email protected] Abstract:
The mixing parameters of sterile neutrino(s) preferred by the MiniBooNEand LNSD experiments are in strong tension with the exclusion limit from the IceCubeexperiment. Recently it has been claimed that by considering the non-standard neutrinointeractions (NSI) in addition to the sterile neutrino, the IceCube’s limit can be relaxedand the tension can be reconciled; a baroque scenario as it has been called. We will showthat this claim is just an artifact originating from the energy cuts of the chosen datasets.Contrary to the claim, by turning on the NSI and fixing the NSI parameters to the proposedvalues, not only the IceCube’s limit on sterile neutrino cannot be alleviated, but in factthe tension will be aggravated (or at least keeps its strength). The reconciliation, moreappropriately, can be called surreal . Keywords:
Neutrino Physics, Beyond Standard Model a r X i v : . [ h e p - ph ] O c t ontents (3 + 1)+ NSI: a concert of new physics scenarios 33 Oscillation probabilities in the (3 + 1)+
NSI scenario 54 Conclusions 8
Currently, almost all the neutrino data can be explained consistently in the ν formalism,consisting of three active neutrino flavors and the corresponding mixing parameters (fora global fit to all the available data see [1–3]). However, there are still some anomalies ,coming from the electron neutrino appearance experiments, namely, the LSND [4] andMiniBooNE [5] experiments, which indicate the existence of extra neutrino states, the so-called sterile neutrinos, with the mass ∼ O (1) eV. In particular, the recent update fromthe MiniBooNE experiment [6], which combines the ν e and ¯ ν e appearance data, reportsan excess of . σ in the low energy range that can be increased to . σ if combined withthe LSND data. This excess can be interpreted in the scenario (3 active + 1 sterileneutrino state) with ∆ m ∼ O (1) eV and sin θ eµ (cid:38) − . The allowed region in the (sin θ eµ , ∆ m ) plane can be found in [6].The main obstacle in a happy interpretation of the LSND/MiniBooNE excess in termsof the sterile neutrinos is the strong tension with the disappearance data including MI-NOS/MINOS+ [7] and IceCube [8] experiments (for a global status of the sterile neutrinomixing from various experiments see [9, 10]). Among these the IceCube’s limit has a differ-ent nature: while the MINOS experiment is sensitive to the sterile neutrino mixing throughan averaged effect over the long baseline, the IceCube sensitivity originates from a resonanceeffect, an amplification of the ¯ ν µ → ¯ ν s oscillation probability (or ν µ → ν s for ∆ m < )for atmospheric neutrinos crossing the Earth in the ∼ TeV energy range. The possibilityof exploring the active-sterile neutrino mixing by looking at the energy and zenith angledistributions of the high energy atmospheric neutrinos has been proposed in [11, 12] (seealso [13]). By the realization of the IceCube detector, as the first km -volume neutrinotelescope which is able to detect ∼ TeV atmospheric neutrinos, this possibility has beenstudied in detail: the limit on eV-scale sterile neutrinos has been derived using the datacollected during the construction phase of the IceCube [14] and it has been shown thatfew-years worth of the IceCube data can exclude completely the preferred region by LSNDand MiniBooNE [15]. The effect of the various mixing parameters in the scenariohas been studied in [15] and the search strategy generalized to the cascade event topology– 1 –n [16]. Finally the IceCube collaboration published the result of the sterile neutrino analy-sis [8], considering one-year muon-track data with the reconstructed muon energy proxy inthe range
400 GeV − TeV, demonstrating the exclusion of the preferred parameter spaceregion by the appearance experiments (to be precise, constraining the angle θ in the scenario).Another new physics scenario that can be probed by the high energy atmospheric neu-trinos observed by the IceCube is the non-standard neutrino interactions. This possibilityhas been proposed and used to derive the most stringent bound on the NSI parameters (the ε µτ and ε µµ − ε ττ ) in [17], with consistent results in [18, 19] (see [20] for bounds on theNSI parameters from the global analysis of oscillation data). Recently it has been proposedin [21] that the addition of non-standard neutrino interaction to the picture can relaxthe limit of IceCube on sterile neutrinos and reconcile the appearance and disappearancediscrepant results. The same claim has been repeated in a more recent work [22] whichanalyses the data of MINOS+ and IceCube, including the DeepCore data, and concludesthat a combination of the charged-current and neutral-current NSI can relax the limits ofboth experiments. In this framework, the charged-current NSI is required for the relax-ation of the MINOS+ limit (a nonzero detector NSI, ε Dµµ , has been assumed) while theneutral-current NSI parameters (nonzero ε µµ , ε ττ and ε ss ) will, pretendedly, loosen the Ice-Cube’s limits. The IceCube data analyzed in [22] consists of the publicly available sterilesearch data [8] with the muon energy ∈ [501 GeV ,
10 TeV] and the DeepCore oscillationdata [24] with the muon energy range [6 ,
56] GeV .In this paper we will argue that in fact the apparent relaxation of the IceCube’s limiton the sterile neutrino in the (3 + 1)+
NSI scenario originates from the negligence of the [56 , energy range and has no physical rationale behind. It should be emphasizedthat although the recent IceCube publicization of data on atmospheric neutrinos does notinclude the [56 , energy range, older data are available, such as the IC-79 [25]and IC-40 [26], which cover this energy range and do not show any significant deviationfrom the ν framework expectation. Considering this energy range, contrary to the claimin [21, 22], the limit on sterile neutrino from IceCube will be stronger in the (3 + 1)+ NSIscenario or will keep the strength, depending on the quality of the data in the [56 , energy range. To this aim, actually, a simple oscillation probability calculation is enoughto manifest the argument.The paper is organized as follows: In sec. 2 we summarize the main features of theatmospheric muon (anti-)neutrino oscillation in the (3 + 1)+ NSI scenario and describe ourassumptions. In sec. 3 we discuss in detail, based mainly on oscillation probabilities, why theaddition of NSI to the model cannot help to the reconciliation of the LSND/MiniBooNEand IceCube tension. Finally, in sec. 4, we provide our conclusions. https://icecube.wisc.edu/science/data/IC86-sterile-neutrino The public data is available in the muon energy range [400 GeV ,
20 TeV] . https://icecube.wisc.edu/science/data/2018nuosc – 2 – The (3 + 1)+
NSI: a concert of new physics scenarios
The phenomenology of high energy atmospheric neutrino oscillation in the and NSIframeworks have been studied separately in [15, 17]. The characteristics of oscillation inthe (3 + 1)+
NSI is just a simultaneous consideration of both frameworks. In the followingwe will summarize the evolution equation of neutrinos in the (3 + 1)+
NSI scenario, in orderto fix the notation and remind the main features.The evolution equation of the neutrinos, in the flavor basis, in the presence of mattereffect, in the (3 + 1)+
NSI scenario can be written as: i ddx ν e ν µ ν τ ν s = ( H + V m ( x )) ν e ν µ ν τ ν s , (2.1)where H corresponds to the Hamiltonian in vacuum, and is given by H = 12 E ν U m m
00 0 0 ∆ m U † . (2.2)Here E ν is the neutrino energy, ∆ m ij ≡ m i − m j ( i, j = 1 , , , ) are the mass-squareddifferences, and U is the mixing matrix for the model, parameterized as U ≡ R ( θ ) R ( θ , δ ) R ( θ , δ ) R ( θ ) R ( θ , δ ) R ( θ ) , (2.3)where R ij ( θ ij ) is the rotation matrix with the angle θ ij in the i - j plane; while the rotation-like matrices R ij ( θ ij , δ ij ) can be obtained from R ij ( θ ij ) by the following replacements: sin θ ij → sin θ ij e − iδ ij and − sin θ ij → − sin θ ij e iδ ij .The V m term in Eq. (2.1) represents the matter potential induced by the standard(coherent) interaction as well as by the NSI, which can be parameterized generally as V m ( x ) = √ G F n e ( x ) ε ee ε eµ ε eτ ε es ε ∗ eµ ε µµ ε µτ ε µs ε ∗ eτ ε ∗ µτ ε ττ ε τs ε ∗ es ε ∗ µs ε ∗ τs κ + ε ss , (2.4)where G F is the Fermi constant, n e is the electron number density of the propagationmedium (in our case is the Earth), the ε αβ are the dimensionless parameters characterizingthe strength of the NSIs , and κ ≡ n n / (2 n e ) ∼ / ( n n being the neutron number density)for the Earth’s matter. For anti-neutrinos, the overall sign of the matter potential inEq. (2.1) as well as that of the CP violating phases in Eq. (2.3) must be flipped. The Hermitian matrix of ε parameters is effectively a sum over the NSI of neutrinos with the mainingredients of the Earth, that is the electron, u and d quarks. See [17] for clarifications. – 3 –he oscillation pattern of the high energy atmospheric neutrinos in scenario is well-known: assuming ∆ m > , where otherwise a strong tension with the cosmological datawill appear, a resonance enhancement of the ¯ ν µ → ¯ ν s oscillation occurs as neutrinos pass theEarth’s core and/or mantle as follows. For core-crossing trajectories, that is cos θ z (cid:46) − . where θ z is the zenith angle, the enhancement is a parametric resonance [27, 28] at the energy (cid:39) (2 . θ (∆ m / eV ) ; while for the mantle-crossing trajectories ( cos θ z (cid:38) − . )the enhancement is an MSW resonance at the energy (cid:39) (4 TeV) cos 2 θ (∆ m / eV ) .The resonance enhancement of ¯ ν µ → ¯ ν s conversion is independent of the θ , and asshown in [15] the enhancement becomes more effective with the increase of θ . In whatfollows, we set sin θ = 0 . and θ = 0 where the latter choice leads to the mostconservative limit on sterile neutrino mixing. For the standard ν -oscillation parameterswe use the best-fit values of the global fit [1] for the normal mass ordering, ∆ m > , whileour discussion will be equally valid also for the inverted mass ordering. It should be noticedthat in the high energy range ( > GeV) no standard oscillation effect is important andby decreasing the energy to ∼ GeV the atmospheric oscillation parameters start to playa role. All the CP-violating phases also will be set to zero, as recommended in [15] for aconservative limit.In the presence of the NSI the resonance energies will be modified. To simplify thediscussion, and also to consider the same setup as in [21, 22], we will assume nonzero ( ε µµ , ε ττ , ε ss ) and setting all the other NSI parameters to zero. By neglecting the ν e -flavor oscillation, see [15], through a simple calculation of the resonance condition in the (3 + 1)+ NSI scenario we can obtain the resonance energy of µ − s conversion. For themantle-crossing trajectories the MSW resonance energy will be modified to E res ν (cid:39) θ (cid:18) ∆ m eV (cid:19) (cid:20) − ε µµ + 2 ε ss (cid:21) TeV . (2.5)For the core-crossing trajectories, where cos θ z (cid:39) − , the numerical factor of Eq. (2.5)should be replaced by . TeV. As can be seen, the nonzero ε µµ and/or ε ss shift the resonanceenergy. The resonance energy is independent of the ε ττ . Thus, neglecting for the momentthe effect of the NSI parameters in the low energy, that is E ν ∼ (10 − GeV, by turningon the neutral-current NSI the resonance will occur at some shifted energy, but does notdisappear . The effect of the NSI parameters on the high energy atmospheric neutrino,including the low energy (10 − GeV range, has been studied in detail in [17] and wewill not repeat it here. The atmospheric oscillation probabilities in low energy range dependdramatically on the ε µµ − ε ττ . Thus, obviously, a large value inserted for ε µµ in Eq. (2.5)should be compensated with a large value for ε ττ such that their difference remains smalland the fit to the DeepCore data does not deteriorate. This is exactly the case chosenin [21, 22] and we will take it also in this paper. Apparently, in Eq. (2.5), it is possible to raise the resonance energy to a very high value by setting ε µµ (cid:39) and ε ss (cid:39) − / , and thus, relax the IceCube bound. However, this setup of NSI parameters leadsto anomalies in the cascade-type events of IceCube and can be constrained. We will leave this setup for afuture study. – 4 – Oscillation probabilities in the (3 + 1)+
NSI scenario
The oscillation probabilities P ( ν µ → ν µ ) and P (¯ ν µ → ¯ ν µ ) can be obtained by the numericalsolution of the neutrino evolution equation in the (3 + 1)+ NSI scenario; i.e. , the Eq. (2.1)and the corresponding one for the anti-neutrinos. In the numerical solution we will use thePREM model [23] for the Earth matter density profile assuming Y e = 0 . , where Y e denotesthe number of electrons per nucleon.By looking at the (3 + 1)+ NSI oscillation probabilities, in this section we will argue andshow that the relaxation of the IceCube’s bound on the sterile neutrinos, as claimed in [21,22], does not happen. To be specific, we will consider two different sets of NSI parameters,motivated by and studied in [22], which come from a scan over the NSI parameter values.These two sets, named case (a) and case (b), are summarized in Table. 1. For the case (a)the ε ss has been set to zero while a scan over | ε ττ | < and | ε ττ − ε µµ | < . has been done .For the case (b) the scan is over | ε ss | < , | ε ττ | < . and | ε ττ − ε µµ | < . . As these twocases have been reported as the best scenarios of an scan over the NSI parameter values,any other choice of the NSI parameters will be less effective in relaxing the IceCube’s boundaccording to [22].The sterile neutrino mixing parameters in the cases (a) and (b) have been fixed to thereported best fitted values in [22]. Although, the P ( ν µ → ν µ ) and P (¯ ν µ → ¯ ν µ ) oscillationprobabilities do not depend on θ , we will just set it to sin θ = 0 . in our numericalcalculation. Table 1 . The two sets of parameters in the (3 + 1)+
NSI scenario considered in this work, whichcome from a scan of the parameter space performed in [22]. All the other parameters of the 3+1model and/or NSI parameters not indicated in the table are assumed to be zero.
Case sin θ sin θ ∆ m (cid:15) µµ (cid:15) ττ (cid:15) ss (a) 0.02 0.063 0.32 eV -4.3 -4.0 0.0(b) 0.02 0.032 0.62 eV -0.7 -0.5 6.0Let us start with the case (a). The upper and lower panels of the Figure 1 show,respectively, the P ( ν µ → ν µ ) and P (¯ ν µ → ¯ ν µ ) oscillation probabilities for mantle-crossingtrajectory cos θ z = − . . The black thick dashed curve shows the oscillation probability inthe ν framework. The red solid curve shows the oscillation probability in the modelwith sterile mixing parameters as in case (a); i.e. , the mixing parameters shown in the firstrow of Table 1 and setting the NSI parameters to zero. As expected, the ∆ m = 0 .
32 eV leads to an MSW resonance at E ν (cid:39) . TeV in the anti-neutrino channel. The IceCube’ssensitivity to the mixing parameters of case (a) originates mainly from this dip in theanti-neutrino oscillation probability. By turning on the NSI parameters, the oscillationprobability shown by blue dashed curve will be modified. The resonance dip in ¯ ν µ → ¯ ν µ Although this set of values for the NSI parameters are in conflict with the solar neutrino data [20], wewill continue with it as an example of the claimed maximum relaxation of IceCube’s bound reported in [22]. – 5 –scillation, in accordance with Eq. (2.5), shifts to ∼ GeV. In Figure 1 the green andpink shaded regions show, respectively, the energy ranges of the IceCube’s sterile neutrinoanalysis [8] and the DeepCore oscillation analysis [24]. For comparison, the oscillationprobability for ν + NSI scenario, with NSI parameter values of case (a), is also shown bythe brown dot-dashed curve. As can be seen, addition of the sterile neutrino will makes thelow energy part of the ν + NSI oscillation probability compatible with the ν , that otherwiseis completely ruled out. Obviously, if one considers just the DeepCore and IceCube sterileneutrino analyses energy ranges, the limit on sterile neutrino in the (3 + 1)+ NSI scenariocan be relaxed; in the pink and green shaded regions there is a small difference between the ν and (3 + 1)+ NSI oscillation probabilities. However, the huge discrepancy between the ν and (3 + 1)+ NSI scenarios lies in the gap between the two energy ranges.The Figure 2 shows the oscillation probabilities for the case (a) and for the core-crossingtrajectory with cos θ z = − , with the same color code and line type as in Figure 1. For cos θ z = − the muon anti-neutrinos experience the parametric resonance in scenarioat ∼ TeV as expected; while in the (3 + 1)+
NSI scenario the resonance shifts to lowerenergies, again in the gap between the energy ranges of DeepCore and IceCube sterileneutrino analyses. The are more deviations in the low energy part for the core-crossingtrajectories due to the higher matter density in the propagation path of the neutrino whichamplifies the effect of the NSI. In fact this deviation in the low energy part of the Figure 2is the origin of the limit obtained in [22] for case (a); otherwise, by just considering thegreen shaded region, IceCube sterile neutrino analysis is not sensitive to case (a).Figures 3 and 4 show the oscillation probabilities for the case (b), respectively, for themantle and core crossing trajectories. The discussions presented for the case (a) apply alsofor the case (b). Again, the principal effect of the NSI is to shift the resonances into thegap in the energy ranges of the DeepCore and IceCube sterile neutrino analyses.A comment on the energy ranges of the DeepCore and IceCube neutrino sterile analysesis in order: although the respective [6 ,
56] GeV and [501 GeV ,
10 TeV] energy ranges arereferring to the muon energy produced in the ν µ and ¯ ν µ charged-current interactions, weare taking it as a proxy of the neutrino energy and show them in the Figure 1 (and thesubsequent figures) as cuts on the E ν . Needless to say, this is just roughly correct and ina detailed analysis of the data the difference between the muon energy and E ν should betaken into account. However, it is not our goal in this paper to perform an analysis of thedata, basically since there are no publicly available data from IceCube collaboration in theenergy range of [56 , . Instead, we will do a sensitivity analysis, similar to the onedone in [15], just to quantify our argument. The main points of our argument are clearenough that hopefully motivate the IceCube collaboration to release the data in the wholeenergy range and perform an analysis by taking into account all the details.Even without an elaborate calculation it is clear that a dip in the oscillation probabilityat ∼ GeV can be excluded more strongly than or at least at the same level of a dipat ∼ TeV; which is what happening in (3 + 1)+
NSI scenario (see, for example, the lowerpanel of Figure 1). The atmospheric muon neutrino flux drops roughly as ∝ E − . ν with theincrease of energy. By taking into account the energy dependence of the cross section andthe increase of effective volume by the increase in energy, the statistics is higher roughly– 6 –y a factor of few at E ν ∼ GeV with respect to E ν ∼ TeV. For example, in theIC-79 dataset the peak of the statistics is at ∼ GeV, and the dataset contains as muchevents at ∼ GeV as in ∼ TeV (see figure 1 of [25]). Other elements, such as thebetter energy resolution in the low energy due to the shorter muon range, also help towarda better constraining power. Thus, for sure, one can conclude that the (3 + 1)+
NSI forboth cases (a) and (b) is more strongly excluded than the model with the same sterilemixing parameters (or pessimistically it is excluded at the same level). This can be seenfrom the figures in this paper: by adding the NSI to the model not only the resonancedip in the muon neutrino survival oscillation probabilities slides to ∼ GeV but alsothe oscillation probabilities will be modified at the DeepCore energy range. The sum ofthese two can lead to a strong bound from IceCube data. A detailed quantification of thisstatement is not possible for us since there is no recent public data from IceCube in thegap energy range. Thus, let us quantify a bit this statement by performing a sensitivityanalysis using the information from the older IceCube datasets.For our sensitivity analysis we use the last public effective area of the IceCube in thewhole range of energy, which goes back to the construction period of the detector: the IC-40 [26] and IC-79 [25] configurations. Of course, at least 10 times more data are availablenow and so we just increase the data-taking period correspondingly. For fixed values of the (sin θ , ∆ m ) we calculate the χ value (which is the same as ∆ χ with respect to the ν framework) by marginalizing the following χ function over the parameters α and β : χ (∆ m , sin θ ; α, β ) = (cid:88) i,j (cid:110) N i,j − α [1 + β (0 . θ z ) i )] N i,j (cid:111) σ i,j, stat + σ i,j, sys + (1 − α ) σ α + β σ β , (3.1)where α and β are the pull parameters taking into account the correlated uncertainties ofthe atmospheric neutrino flux normalization and its zenith dependence (tilt), respectively.Using the Honda flux of atmospheric neutrinos [29], these uncertainties are roughly σ α =0 . and σ β = 0 . . The N i,j is the number of events in the ν scenario in the i th bin ofenergy and j th bin of cos θ z , which can be calculated by convoluting the effective area withthe neutrino flux and oscillation probability. The N i,j is the number of events in the (3 +1)+ NSI scenario with the sterile mixing parameters (sin θ , ∆ m ) . The σ i,j, stat = (cid:112) N i,j is the statistical error and we have added an uncorrelated systematic error σ i,j, sys = f N i,j where f quantifies it.We have verified that by using the [400 GeV ,
20 TeV] energy range, 4 energy bins(uniformly distributed in log), 10 linearly distributed bins of cos θ z in [ − , , and anuncorrelated systematic error f = 10% we can reproduce the exclusion plot of the Ice-Cube analysis in [8] almost exactly. Focusing on the case (b), the χ value for the (∆ m , sin θ ) = (0 .
63 eV , . in the model using the [500 GeV ,
10 TeV] en-ergy range is ∼ . By turning on the NSI parameters and going to (3 + 1)+ NSI scenario,using the same energy range, the χ value drops to ∼ . This is in agreement with theclaim in [21, 22] that the IceCube’s limit on sterile neutrino relaxes in the presence of NSI.However, by extending the energy range to [10 GeV ,
10 TeV] the χ value increases to ∼ ,– 7 –s we expected. The same pattern, that is a significant increase of the χ value, occurs alsofor the case (a) if we extend the energy range as done for the case (b). The IceCube’s bound on the active-sterile neutrino mixing(s) strongly excludes the param-eter space preferred by the appearance experiments LSND and MiniBooNE, such that aglobal fit of the data in the scenario shows strong tensions. Recently it has beenclaimed that by adding non-standard neutrino interaction to the scenario it is possibleto relax the IceCube’s bound and weaken the tension [21, 22].We have revisited the (3 + 1)+
NSI scenario and studied the impact of the NSI on theIceCube’s bound on the sterile neutrino. We have shown that in the presence of NSI theresonance enhancement of the ¯ ν µ → ¯ ν s occurs at a shifted energy, but does not disappear.The reason behind the relaxation of the IceCube’s bound claimed in [21, 22] is that for thechosen NSI parameters the resonance enhancement occurs in the gap [56 , GeV betweenthe energy ranges of the two considered datasets (the IceCube sterile neutrino analysis [8]and the DeepCore oscillation analysis [24]). However, although the recent public data of theIceCube do not cover this gap, the older data such as IC-79 [25] and IC-40 [26] include thisenergy range and do not show any significant deviation from the ν oscillation framework.By performing a sensitivity analysis, covering the energy range [10 GeV ,
10 TeV] , we haveshown that in fact the IceCube’s bound on the sterile neutrino mixing becomes stronger inthe presence of the NSI.We therefore conclude that the NSI with the parameter values reported in [22] doesnot relax the IceCube’s bound on the sterile neutrino. In the presence of NSI, the tensionbetween the IceCube’s limit and the LSND/MiniBooNE preferred region persists or evencan get stronger.A detailed analysis by the IceCube collaboration is required to provide the correct limiton the (3 + 1)+
NSI scenario. We hope that this work motivates the IceCube collabora-tion to publicize the atmospheric neutrino data in the full range of energy to avoid suchmisinterpretations. – 8 – ceCube Sterile AnalysisDeepCore n m Æ n m Sin q = D m = e mm = - e tt = - e ss = q z = - + + + NSI3 n NSI
10 50 100 500 1000 5000 1 ¥ E n @ GeV D P H n m Æ n m L IceCube Sterile AnalysisDeepCore n m Æ n m Sin q = D m = e mm = - e tt = - e ss = q z = - + + + NSI3 n NSI
10 50 100 500 1000 5000 1 ¥ E n @ GeV D P H n m Æ n m L Figure 1 . The ν µ → ν µ (upper panel) and ¯ ν µ → ¯ ν µ (lower panel) oscillation probabilities as afunction of the neutrino energy for cos θ z = − . . The black thick dashed curve corresponds tothe ν oscillation, while the red solid curve corresponds to the model with sin θ = 0 . , sin θ = 0 . and ∆ m = 0.32 eV (all the other parameters of the model are set to zero).The blue dashed curve indicates the case where the NSI is added on top of the model, the (3 + 1)+ NSI scenario, with the parameters fixed to the case (a) shown in Table 1. For completeness,the case where only the NSI effect is added to the standard ν oscillation is also shown by the browndot-dashed curve. The energy ranges used by the IceCube’s sterile neutrino analysis [8] and theDeepCore oscillation analysis [24] are indicated by the green and pink shaded regions, respectively. – 9 – ceCube Sterile AnalysisDeepCore n m Æ n m Sin q = D m = e mm = - e tt = - e ss = q z = - + + + NSI3 n NSI
10 50 100 500 1000 5000 1 ¥ E n @ GeV D P H n m Æ n m L IceCube Sterile AnalysisDeepCore n m Æ n m Sin q = D m = e mm = - e tt = - e ss = q z = - + + + NSI3 n NSI
10 50 100 500 1000 5000 1 ¥ E n @ GeV D P H n m Æ n m L Figure 2 . The same as Figure 1 but for cos θ z = − . – 10 – ceCube Sterile AnalysisDeepCore n m Æ n m Sin q = D m = e mm = - e tt = - e ss = q z = - + + + NSI3 n NSI
10 50 100 500 1000 5000 1 ¥ E n @ GeV D P H n m Æ n m L IceCube Sterile AnalysisDeepCore n m Æ n m Sin q = D m = e mm = - e tt = - e ss = q z = - + + + NSI3 n NSI
10 50 100 500 1000 5000 1 ¥ E n @ GeV D P H n m Æ n m L Figure 3 . The same as Figure 1 but for the case (b) shown in Table 1. – 11 – ceCube Sterile AnalysisDeepCore n m Æ n m Sin q = D m = e mm = - e tt = - e ss = q z = - + + + NSI3 n NSI
10 50 100 500 1000 5000 1 ¥ E n @ GeV D P H n m Æ n m L IceCube Sterile AnalysisDeepCore n m Æ n m Sin q = D m = e mm = - e tt = - e ss = q z = - + + + NSI3 n NSI
10 50 100 500 1000 5000 1 ¥ E n @ GeV D P H n m Æ n m L Figure 4 . The same as in Fig. 2 but for the case (b) shown in Table 1. – 12 – cknowledgments
A. E. thanks the computing resource provided by CCJDR, of IFGW-UNICAMP withresources from FAPESP Multi-user Project 09/54213-0, and the partial support by theCNPq fellowship No. 310052/2016-5. H. N. was supported by the CNPq research grants,No. 432848/2016-9 and No. 312424/2017-5.
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