Bounds on non-standard interactions of neutrinos from IceCube DeepCore data

New physics in neutrino sector can reveal itself via non-standard neutrino interactions which can result in modification of the standard picture of neutrino propagation in matter. Experiments with atmospheric neutrinos has been used to probe this scenario. Using publicly available three-year low energy data in IceCube DeepCore we place bounds on the parameters $\epsilon_{\alpha\beta}$ of non-standard neutrino interactions in propagation. We obtain restrictive constraints not only for $\mu\tau$ sector but also for flavor changing interactions involving electron neutrinos.


Introduction
Phenomenon of neutrino oscillations is well established by many experiments [1]. It implies nontrivial mixing between neutrinos and non-zero values of their masses. Explanation of these properties of neutrinos lies beyond the Standard Model and requires new physics. The latter may reveal itself via non-renormalizable interactions involving neutrino fields. In particular, new Fermi-type interactions of neutrinos with SM fermions are of interest and they are known as non-standard neutrino interactions [2,3]. In what follows we consider neutral current (NC) or matter non-standard neutrino interactions (NSI) with the lagrangian Here P L,R are the chirality projectors, f P αβ are the NSI parameters and sum goes over all SM fermions f . Note that the lagrangian (1) contains only operators which do not change flavor of the fermion f . The flavor changing interactions of the type (1) are severely constrained from results on lepton flavor violating and FCNC processes. NC NSI (1) can modify neutrino phenomenology in several ways and these interactions can reveal themselves in scattering processes as well as in neutrino oscillation experiments (see Refs. [4][5][6][7] for reviews). Examples of phenomenologically viable models predicting the matter NSI with sizable values of couplings were discussed e.g. in Refs. [8][9][10][11][12].
One of the consequences of the interactions (1) is modification of neutrino propagation in matter. In the presence of NSI the evolution of neutrino having energy E is described by the Hamiltonian where U is the vacuum Pontecorvo-Maki-Nakagawa-Sakata (PMNS) matrix and ∆m 2 21 , ∆m 2 31 are differences of the neutrino masses squared. The last term in (2) describes matter effects and it depends on the matter density through V e = (−) √ 2G F N e for (anti)neutrinos, where N e is the electron number density. The NSI parameters from the interaction lagrangian (1) enter the Hamiltonian as follows where αβ = f =e,u,d Here N f is the number density of the fermion f in matter. In this study we concentrate on neutrino propagation in the Earth and in this case the expression (4) for NSI parameters transforms into In what follows the notation αβ refers to Eq.(5). In general the parameters αβ are complex-valued numbers. Here we take them real for simplicity (see [13] for recent discussion of the effect of CP violation in NSI).
Atmospheric neutrinos is an important tool to explore neutrino properties and, in particular, to search for new interactions in neutrino sector. The impact of NSI in propagation at experiments with atmospheric neutrinos has been studied extensively (see [14][15][16][17][18][19][20][21][22][23][24][25] for an incomplete list). Probes with atmospheric neutrinos has been used to constrain the matter NSI in the Earth. Results of experiments with atmospheric neutrinos allow to put rather stringent bounds [26][27][28][29][30] on the parameters αβ . Earlier studies of NSI with IceCube data [27,29,30] put constraints on the NSI parameters in µτ sector. Preliminary results with the bounds on more generic NSI models were recently reported in [31][32][33] by IceCube collaboration. In the present study we use the publicly available IceCube DeepCore three-year low energy data sample [34] and perform an independent analysis to constrain the parameters αβ . This data sample is very close to what was used by IceCube to measure the neutrino oscillation parameters in Ref. [35]. We perform an analysis of the NSI effect on atmospheric neutrino propagation in the Earth. Information provided by IceCube with the data release [34] and, in particular, results of Monte-Carlo simulation, expected background from atmospheric muons as well as parametrization of estimated instrumental systematic effects, allows one to made a realistic prediction for number of expected events in models with non-zero matter NSI parameters and compare them against the data. As a result we obtain allowed regions for the parameters αβ under certain model assumptions. We compare them with results from other oscillation experiments and discuss impact of systematic uncertainties.
The rest of the paper is organized as follows. In Section 2 we describe methodology used to bound αβ with the IceCube DeepCore three-year data sample. In Section 3 we present our results. Section 4 is reserved for conclusions.

Description of the analysis
For the present study we use publicly available three-year data sample [34] in IceCube DeepCore which is referred to as 'Sample B' in 2 Ref. [37] and which is very close to what was used in the oscillation analysis [35]. This sample contains 40920 events in total which are distributed over 8 × 8 × 2 binned histogram. The latter is composed of equally spaced bins in log 10 E reco ∈ [0.75, 1.75], 8 equally spaced bins in cos θ reco ∈ [−1, 1] as well as 2 bins which correspond to the track-like and cascade-like events. Here E reco and θ reco refer to reconstructed values of neutrino energy and zenith angle. We use the public IceCube Monte-Carlo provided along with the data sample to model the detector response and to relate physical values of the energy, zenith angle and type of neutrino with reconstructed characteristics of the events. The simulated neutrino sample released at [34] allows to calculate for each bin i the effective areas A να,t i (E, cos θ) to be converted with predicted neutrino flux to obtain expected number of events in the bins. The effective areas are obtained as functions of true neutrino energy E in the range from 1 to 1000 GeV, zenith angle θ, neutrino type ν α and type of neutrino interaction t (CC or NC).
For prediction of the neutrino flux at the detector level we start with the atmospheric neutrino fluxes Φ atm να (E, θ) for ν e ,ν e , ν µ andν µ taken from Ref. [38]. To describe propagation of the atmospheric neutrinos in the Earth in presence of NSI one should solve the Schrodinger equation with the Hamiltonian (2) for the case of varying density. In the present analysis we solve it numerically as described in [39,40] using the algorithm presented in [41]. For calculation of the electron number density in the Earth we use PREM [42]. We fix the following values 3 of neutrino oscillation parameters: sin 2 θ 12 = 0.304, sin 2 θ 13 = 0.0217 and ∆m 2 21 = 7.53 · 10 −5 eV 2 and set δ CP = 0. Upon obtaining solution to the Schrodinger equation we calculate the transition probabilities P να→ν β (E, cos θ) to find neutrino of a flavor ν β at the detector level from neutrino of a flavor ν α produced in the atmosphere. It is the probability functions P να→ν β (E, cos θ) which depend on the matter NSI parameters αβ . The resulting neutrino fluxes at the detector level are then obtained as and the expected number of events in i-th bin can be found as follows where T is the lifetime for the data sample under consideration and sum goes over contributions from different neutrino flavors ν α (including antineutrinos) and interaction types t (CC or NC).
To compare the expected event distribution with the IceCube DeepCore data we take into account systematic uncertainties and our procedure includes a set of relevant nuisance parameters η j . They correspond to overall normalization of atmospheric neutrino flux with no prior, the spectral index of the atmospheric neutrino flux with the nominal value γ = −2.66 and a prior σ γ = 0.1, relative normalizations of ν e ,ν e events and NC events with an uncertainty 20%. Also we include additional corrections to the atmospheric neutrino flux at production to take into account uncertainties in the hadron production in atmosphere. This corrections as functions of neutrino energy and zenith angle have been chosen to reproduce the uncertainties estimated in Ref. [47] similar to how it was done in [48]. Also we take into account uncertainty related to the neutrino nucleon cross section with baryon resonance production which is important for low energy part of the neutrino sample. For that we introduce an additional nuisance parameter for the contribution from such type of the events with 40% uncertainty which is close to what was found in [48]. As discussed in [35] uncertainties on the DIS cross section have negligible impact on the results. We include in the analysis the template for the background of atmospheric muons with corresponding uncorrelated error σ uncor ν,µatm provided by IceCube for this data sample [34]. Normalization of the background is taken as a nuisance parameter with no prior. Finally, we account for instrumental systematic uncertainties related to the optical efficiencies of DOMs and relevant properties of the ice. These uncertainties are included in our analysis as described in [34]. In summary, our implementation of the systematic uncertainties is very close to how it was done in the original oscillation analysis [35]. We will discuss the impact of the most important systematic errors in the next Section.
Let us note, that in general interactions of the type (1) result in changes of the NC neutrinonucleon cross section and thus can modify expected number of events. However, this effect is very model dependent. Not only it depends on other combinations of the parameters f P αβ than those in Eq. (5), but it is also affected by microscopic model behind the effective lagrangian (1) and the resulting cross section will be different fore models with light and heavy mediators In the present analysis we conservatively do not take into account the impact of NSI on the modification on the NC neutrino-nucleon cross sections.
To obtain bounds on the parameters of non-standard neutrino interactions we define (c.f. Eq.(2) in [35]) where n ν i (n µatm i ) is the expected number of events from atmospheric neutrinos (muons), n data i is the number of data events in the i-th bin, σ data i = √ n ν i , and the first sum goes over all bins in the data sample. The second term accounts for contribution from the nuisance parameters η j , given their default valuesη j and uncertainties σ η j . We fix sin 2 θ 12 , sin 2 θ 13 and ∆m 2 21 as described above. We checked that the uncertainties in their values produce negligible effect on the final results. Also we assume δ CP = 0. Other neutrino oscillation parameters, sin 2 θ 23 and ∆m 2 31 , are not fixed to any a priori value but were determined from the analysis itself in most part of the study. As a consistency check we reproduce confidence regions for sin 2 θ 23 and ∆m 2 31 from the IceCube DeepCore data sample assuming αβ = 0. In particular, we find for the case of normal mass ordering sin 2 θ 23 = 0.52 +0.07 −0.08 , ∆m 2 32 = 2.29 +0.16 −0.15 eV 2 which is very close to those intervals obtained in Ref. [35]. In our analysis with non-vanishing matter NSI parameters we consider sin 2 θ 23 and ∆m 2 31 as nuisance parameters with no prior unless stated otherwise. In Section 3 we discuss how the bounds on αβ are robust if sin 2 θ 23 and ∆m 2 31 are fixed in the analysis. The IceCube DeepCore low energy data sample of the atmospheric neutrinos [34] contains (contrary to the earlier IceCube DeepCore oscillation analyses [49]) not only track-like but also cascadelike events which results mainly from CC interactions of ν µ and ν e (see Fig. 1 in Ref. [35]). This makes the data sample sensitive not only to changes in the muon neutrino flux but also to modification in the flux of electron neutrinos. Most of the theoretical studies discuss impact of the NSI in propagation mainly on the muon neutrino flux Φ det νµ . This is mostly sufficient for the models with non-zero values of the corresponding parameters in µτ sector where the NSI effect on the electron neutrino flux is very mild given current bounds on the NSI parameters. However, this is not the case for models with non-zero eτ or eµ . For illustration, in Figs. 1 and 2 we show oscillograms for ∆P µµ and ∆P µe which are the differences between the transition probabilities with and without NSI, i.e. ∆P αβ = P N SI να→ν β − P noN SI να→ν β for neutrinos. Fig. 1 and 2 correspond to eτ = 0.3 and eµ = 0.3, respectively. Right panels on both Figures correspond to ∆P µµ while left are reserved for ∆P µe . We observe that the off-diagonal matter NSI involving electron neutrinos may considerably modify not only the fluxes of muon neutrinos but also that of electron neutrinos. Note that the effect of non-zero  eτ decreases with increase of neutrino energy but this is not the case for eµ . Thus we expect that IceCube data for neutrinos at high energies will be also sensitive to the parameter eµ .

Results
In this Section we present results on the allowed regions for several matter NSI parameters αβ from the analysis of the low energy three-year IceCube DeepCore data sample [34]. In this study we limit ourselves to a constrained analysis in the matter NSI parameter space assuming only some of αβ to be non-zero. As neutrino oscillation probabilities depend on differences between the diagonal elements ee − µµ and τ τ − µµ we fix µµ = 0 in what follows. Let us start with the µτ sector where the NSI parameters, µτ and ≡ τ τ − µµ , are known to be severely constrained from experiments with atmospheric neutrinos [27,29,30]. Firstly, we take a single non-zero matter NSI parameter, µτ or τ τ , at a time and perform minimization of χ 2 given by Eq. (8) with respect to all other variables (including ∆m 2 31 , sin 2 θ 23 and other nuisance parameters discussed in previous Section). We consider the cases of normal (NO) and inverted (IO) neutrino mass ordering independently. The results for ∆χ 2 ≡ χ 2 − χ 2 min , where χ 2 min is an absolute minimum of χ 2 for each case, are shown in Fig. 3 for non-zero µτ (left panel) and τ τ (right panel). We find the following single parameter allowed ranges at 90% C.L.
Note that the obtained regions for µτ are consistent 4 with those −0.020 < µτ < 0.024 (NO) obtained by IceCube collaboration [30] in a similar single parameter analysis using three years of their data with upward going track events selection. The regions (9), (10) are also close to the preliminary IceCube bounds | µτ | < ∼ 0.17 and | τ τ | < ∼ 0.04 (NO) from an analysis of DeepCore data reported 5 in [33].
Next, we consider the NSI models in which both parameters in the µτ sector, i.e. µτ and τ τ . are not equal to zero. Corresponding allowed regions on ( µτ , τ τ ) plane are presented in Fig. 4 for NO (left panel) and IO (right panel). After marginalization with respect to each of these parameters we obtain the following allowed ranges at 90% C.L.
which are only slightly wider than those found in the single parameter analysis and shown in (9), (10). The marginalized bounds on µτ are weaker than the allowed range −0.018 < µτ < 0.016 (NO), 90% C.I. obtained in Ref. [29] from the analysis of a one year high energy IceCube data sample [50] and than the earlier bound −0.018 < µτ < 0.017 (NO) at 90% C.L. obtained in Ref. [27] using the data from 79-string IceCube configuration and DeepCore. At the same time the marginalized bounds (12) on the parameter ≡ τ τ are somewhat better than the allowed range −0.11 < < 0.09 at 90% C.L. from Ref. [27] and the constraint | | < 0.15 obtained by Super-Kamiokande experiment [26]. Comparable bound on was obtained in Ref. [29] from a combination of high energy IceCube data and Super-Kamiokande results. Let us note that the effect of on neutrino propagation decreases with neutrino energy (see e.g. [27]) and thus to probe this parameter one should rely on low energy part of the atmospheric neutrino spectrum. Now let us turn to the NSI models with non-zero parameters involving electron neutrinos. In Fig. 5 we show ∆χ 2 for NSI models with non-zero flavor changing parameters eτ (left panel) and eµ (right panel) taking a single non-zero NSI parameter at a time. Corresponding allowed regions at 90% C.L. read The allowed single parameter bounds for eµ and eτ are consistent with the preliminary results | eµ | < ∼ 0.16 and | eτ | < ∼ 0.2 (NO) reported by IceCube [33]. Note that based on the behaviour of ∆P µµ and ∆P µe shown in Fig. 2 one can expect an improvement in the allowed ranges on eµ with data on atmospheric neutrino at high energies. It is well known that the effect of eτ on neutrino transition probabilities strongly depends on the values of other matter NSI parameters. Following the analysis [26] and earlier studies [15,17] let us consider the NSI models with non-zero parameters in eτ sector, i.e. ee , eτ and τ τ . In Fig. 6 we show allowed regions for eτ and τ τ within these models assuming ee = 0. The parabolic form of the allowed region comes from the approximate relation τ τ ∼ | eτ | 2 1+ ee , which should be satisfied for consistency with high energy part of the atmospheric neutrino spectrum, see [15,17]. Similar allowed regions from preliminary analysis of tree-year DeepCore data were presented in [31]. The sensitivity of the considered dataset to ee is relatively weak, but still its non-zero value may affect the bounds on eτ and τ τ in this class of NSI models through the above relation. As an illustration, in Figs. 7 and 8 are shown the allowed regions in ( τ τ , eτ ) plane assuming ee = 0.2 and −0.2, respectively. We see the dependence of the bounds for eτ on the assumption about the value of ee . The same is valid for allowed regions for τ τ which can be considerably modified as compared to those given by Eq. (12) obtained for the NSI models with other αβ = 0. The found allowed regions for eτ and τ τ are compatible to the latest constraints on these parameters obtained with the Super-Kamiokande data in Ref. [28] using somewhat different anzatz for the parameter space.
Finally we study impact of different nuisance parameters discussed in Section 2 on our results. Firstly, in Fig. 9 we present ∆χ 2 obtained in the single-parameter analyses (for NSI models with nonzero µτ , τ τ , eτ or eµ and assuming normal neutrino mass ordering) for the case with all systematic uncertainties included (which is our default analysis) in comparison with those obtained with (a) fixed nuisance parameters related directly to the uncertainties in atmospheric neutrino flux, i.e. spectral index and uncertainties in the hadron production in atmosphere, (b) fixed nuisance parameters related to the experimental systematic uncertainties and (c) fixed nuisance parameters which determine normalizations of different contributions to expected number of events including those related to the neutrino nucleon cross section and atmospheric muon background. For illustration we present results for the case of normal mass ordering. One observes that the most important systematic uncertainties are the experimental ones and those related to atmospheric neutrino flux, specifically for the NSI models with non-zero eτ and µτ . We also study impact of neutrino oscillation parameters, i.e. sin 2 θ 23 and ∆m 2 31 . As an example, we fix them to their optimal values without NSI for the case of normal mass ordering (see Section 2). Corresponding allowed ranges for µτ , τ τ , eτ , eµ (a single non-zero NSI parameter at a time) becomes By comparison with (9), (10) and (13), (14) we see that the oscillation parameters affect very little these bounds.

Conclusions
Let us summarize results of our study. Here we used the three-year IceCube DeepCore data sample [34] of low energy atmospheric neutrinos to constrain the parameters of non-standard neutrino interactions in propagation. This data sample contains track-like as well as cascade-like events which makes it sensitive not mainly to muon but also to electron neutrino flux at the detector level. Using this dataset we found the bounds on several αβ under different model assumptions. In particular, we presented allowed regions for the matter NSI parameters, assuming 1) single non-zero parameter at a time, 2) non-zero parameters in µτ sector and 3) non-zero eτ and τ τ for several fixed values of ee . The bounds on µτ and τ τ were found to be close to those [27][28][29][30] extracted from different data samples of the IceCube/DeepCore as well as from Super-Kamiokande results. Obtained singleparameter bounds on µτ , τ τ , eτ and eµ are consistent with the preliminary IceCube results [31][32][33] from a similar analysis of three-year DeepCore data. We studied the impact of different sources of systematic uncertainties and found that the obtained bounds are mainly stable with respect to the nuisance parameters of the analysis. The main errors in the bounds results from uncertainties in the atmospheric neutrino flux and from experimental uncertainties. They are specifically important for single parameter bounds on eτ and eµ . Note that from energy dependence of neutrino transition probabilities we can expect an improvement in the bound on eµ using studies with high energy part of atmospheric neutrino spectrum.
In the main text we made a comparison mainly with the results obtained from oscillation experiments with atmospheric neutrinos. The obtained bounds on µτ , τ τ , eτ and eµ are also consistent with the results of global analysis of neutrino oscillation experiments [51] (see also [22,52,53] for earlier studies) obtained under assumptions that the matter NSI are given by neutrino interactions with quarks only and that their flavour structure is independent of the quark flavour. Further insight on possible strength of NSI can be obtained by combining with the results of scattering experiments [46,54] (see also [55,56]). Here we were interested in the impact of the low energy IceCube DeepCore data [34] solely and leaved a combined analysis for future study.
I am grateful to Philipp Eller for helpful correspondence. The work was supported by the RSF grant 17-12-01547.