UvA-DARE (Digital Academic Repository) Sensitivity to light sterile neutrino mixing parameters with KM3NeT/ORCA

: KM3NeT/ORCA is a next-generation neutrino telescope optimised for atmospheric neutrino oscillations studies. In this paper, the sensitivity of ORCA to the presence of a light sterile neutrino in a 3+1 model is presented. After three years of data taking, ORCA will be able to probe the active-sterile mixing angles θ 14 , θ 24 , θ 34 and the eﬀective angle θ µe , over a broad range of mass squared diﬀerence ∆ m 241 ∼ [10 − 5 , 10] eV 2 , allowing to test the eV-mass sterile neutrino hypothesis as the origin of short baseline anomalies, as well as probing the hypothesis of a very light sterile neutrino, not yet constrained by cosmology. ORCA will be able to explore a relevant fraction of the parameter space not yet reached by present measurements.


Introduction
The study of neutrino oscillations has seen remarkable progress in the last three decades.An increasing number of solar, atmospheric and accelerator neutrino experiments have performed precision measurements of the neutrino oscillation parameters [1].The experimental data is consistent with the three weakly-interacting neutrino picture (here referred to as the standard picture).Nevertheless, a number of questions remain unanswered, in particular what is the Neutrino Mass Ordering (NMO) and whether neutrino oscillations violate the CP symmetry.Upcoming experiments such as KM3NeT/ORCA [2], SBN [3], DUNE [4], JUNO [5], Hyper-K [6], IceCube/Gen2 [7] and INO [8] aim to resolve these questions over the next decades.At the same time, several short baseline (SBL) neutrino experiments have reported anomalous experimental results which are inconsistent with the standard picture.A comprehensive review can be found in ref. [9].Such results could be explained by assuming the existence of an additional neutrino (hereafter SBL neutrino).However, the Z-width measurement [10] has demonstrated that only three neutrinos can participate to weak interactions, for which they are referred as active neutrinos.Therefore, the SBL neutrino, JHEP10(2021)180 not being able to participate to weak interactions, is called sterile.The SBL sterile neutrino should be light (∆m 2  41 ∼ 1 eV 2 ) and its presence affects the standard neutrino oscillation probabilities via its mixing with active neutrinos, in the so called 3+1 model.Specifically, oscillations in the presence of a single sterile neutrino can be modelled by extending the standard picture to include four neutrino eigenstates.In this case, six new parameters are introduced in the model: one additional mass square difference ∆m 2  41 , three active-sterile mixing angles θ 14 , θ 24 and θ 34 , and two additional CP-violating phases δ 14 , δ 24 .
The neutrino evolution in matter can be described by the following effective Hamiltonian: where H 0 = diag(0, ∆m 2  21 , ∆m 2  31 , ∆m 2 41 )/2E, and V = √ 2G F diag(N e , 0, 0, N n /2), with G F being the Fermi constant and N e , N n representing the density of electrons and neutrons in the propagation medium.U is an extended 4 × 4 unitary matrix relating flavour and mass eigenstates, which can be parametrised such that: where R jk is a rotation matrix in the j-k plane and, similarly, Rjk is a generalised unitary rotation matrix with an added complex phase.
In the 3+1 model, the active-sterile mixing elements are expressed by U e4 = sin θ 14 e −iδ 14 , ( U µ4 = cos θ 14 sin θ 24 e −iδ 24 , ( U τ 4 = cos θ 14 cos θ 24 sin θ 34 . (1.5) Several experiments have been searching for the SBL sterile neutrino.To date, results are not fully consistent with the 3+1 model: disappearance experiments results are compatible with the standard neutrino scenario while some appearance experiments, such as LSND [11] and MiniBooNE [12], observed significant ν e or νe excesses.The global fit of the experimental data with the 3+1 model results in a poor goodness-of-fit, suggesting the need of additional factors in order to explain all data.Even stronger bounds on the sterile parametric space come from cosmology [13], which indirectly constrains the effective number of relativistic species N eff in our Universe.Theoretically, the three active neutrinos give N eff ∼ 3 [14].If a light sterile neutrino with the mixing parameters determined by SBL oscillations is included in the model, it should have been fully thermalised with the active neutrinos [15].This would require N eff ∼ 4. Cosmological data measure a value of N eff well-compatible with three neutrino species [16], showing a tension with the SBL anomalies.Such a tension is relaxed when cosmological data are combined with astrophysical measurements of cepheids, supernovae and gravitational lensing.In this case, the obtained value of N eff is compatible with four at 68% C.L. [15,16].
More generally, cosmological data alone can be compatible with a sterile neutrino with a mass in the eV range only if its contribution to N eff is very small, or with a somewhat larger N eff only if it comes from a nearly massless sterile particle [17,18].

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Therefore, more terrestrial and cosmological observations are necessary to understand the origin of the SBL anomalies.Moreover, new observations able to constrain the notfully-excluded sterile neutrino region from cosmology, at very low sterile mass splittings (∆m 2 41 1 eV 2 ) can further contribute to testing the sterile neutrino hypothesis.In this context, the role of next-generation neutrino detectors, such as KM3NeT, is relevant, given their ability to probe the sterile neutrino hypothesis with atmospheric neutrinos [19,20].KM3NeT is a research infrastructure hosting a network of next generation neutrino telescopes currently under construction in the Mediterranean Sea [2] and built upon the experience from the ANTARES neutrino telescope [21].Once completed, KM3NeT will consist of two detectors: (1) ORCA (Oscillation Research with Cosmics in the Abyss) near Toulon, France, optimised for GeV-scale atmospheric neutrino studies, and (2) ARCA (Astroparticle Research with Cosmics in the Abyss), in Sicily, Italy, optimised for the observation of higher-energy (E ν > 1 TeV) neutrinos from astrophysical sources.
By exploiting the natural source of atmospheric neutrinos, passing through the Earth and interacting within the detector volume, KM3NeT will perform neutrino oscillation studies over a broad range of energies (from few GeV up to PeV) and baselines (up to the Earth diameter).Matter effects, experienced by atmospheric neutrinos during their passage through the Earth, are expected to enhance the effect of the presence of a sterile neutrino.Moreover, the wide L/E range available in KM3NeT increases its potential to investigate the existence of a sterile neutrino in the 3+1 model.
This paper is focused on the ORCA capability to search for a light sterile neutrino.It will be shown that ORCA has a high potential to simultaneously constrain the activesterile mixing angles θ 14 , θ 24 , θ 34 and the effective angle θ µe , with three years of data taking.Particularly, the ORCA sensitivity to such parameters is competitive with other experiments for sterile neutrino mass at the eV scale, indicated by SBL anomalies, and it is able to provide even stronger constraints for extremely low sterile mass splittings (∆m 2 41 down to 10 −5 eV 2 ).
This paper is organised as follows: section 2 describes the KM3NeT/ORCA neutrino telescope.Section 3 discusses the 3+1 flavour model and oscillation probabilities.Section 4 describes the sterile neutrino analysis method, including a brief summary of the ORCA Monte Carlo (MC) simulation flow.Results on the ORCA sensitivity are presented in section 5. Finally, the results are summarised and discussed in section 6.

The KM3NeT/ORCA detector
KM3NeT/ORCA is a deep water neutrino detector under construction in the Mediterranean Sea.Its location is 42 • 48 N 06 • 02 E, about 40 km offshore from Toulon, France, at a depth of about 2450 m.Upon its completion, ORCA will consist of 115 flexible detection units (DUs), 200 m high, each comprising 18 Digital Optical Modules (DOMs).A DOM is a pressure resistant, 17-inch diameter glass sphere containing a total of 31, 3" photomultiplier tubes (PMTs) and their associated electronics.
The primary goal of ORCA is to determine the neutrino mass ordering and to make neutrino oscillation measurements, such as atmospheric parameters (sin 2 θ 23 , ∆m 2 31 ) as -3 -

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well as to search for ν τ appearance [22].Neutrino oscillation studies [2] have demonstrated the presence of a resonance in neutrino oscillation probabilities for few-GeV (2 − 8 GeV) atmospheric neutrinos passing through the Earth.Such a resonance allows the NMO [2] measurement.
The ORCA geometrical configuration is optimised for studies with atmospheric neutrinos in the few GeV range: the horizontal spacing between DUs is ∼ 20 m, whereas the vertical spacing between DOMs in each DU is ∼ 9 m, with the first DOM being about 30 m above the seabed.The total instrumented volume is 6.7 • 10 6 m 3 (about 7 Mt of sea water).
In this energy regime, the events produced by atmospheric neutrinos interacting in water are spatially contained.In particular, two event topologies can be produced: track-like events, characterised by a long muon track, mostly from ν µ charged-current (CC) interactions in water, and shower-like events, characterised by events with no distinguishable tracks, mostly from ν e -CC and all neutral-current (NC) interactions, but with sizeable contributions from ν τ -CC and ν µ -CC events with short tracks.A track-like event in water has a length of ∼ 4 m/GeV, whereas shower-like events have a log(E/GeV) dependence, which corresponds to a size of the order of a few meters.
The ORCA detector is an excellent instrument for the sterile neutrino search due to its dense configuration and to matter effects, whose impact in oscillation probabilities of GeV neutrinos travelling in the Earth is described in the next section.
More details on KM3NeT/ORCA can be found in [2,22].

Theoretical background
The general solutions to the Hamiltonian in eq.(1.1) have a rich phenomenology that is difficult to express in analytical form.For the purposes of this analysis, eq.(1.1) is solved numerically in its full form using the software package OscProb [23].Figure 1 shows an example of the impact of the matter potential on the effective values of the squared masses (eigenvalues of eq.(1.1)) as a function of energy, assuming a medium of constant density for illustration purposes.Four resonances can be identified in the sterile neutrino models as regions of minimal distance between consecutive masses: one related to each pair (s 1k ,1 ∆m 2 k1 ) (at ∼0.05 GeV, 4 GeV and 3 TeV), and a second-order resonance connecting s 23 , s 24 , s 34 , and ∆m 2 31 (at ∼100 GeV), and with a strong dependence on δ 24 as explained in section 3.1.2.The eigenvectors of eq.(1.1) define an effective mixing matrix which, in a medium of constant density, can be used to compute oscillation probabilities by direct replacement in the vacuum oscillation formula: where  tioned resonances can be readily identified.While the full numerical solutions exemplified above can already provide some insight, some exploration of common analytical approximations can be enlightening even if not used in the analysis.They are described in the following subsections.
-5 - )) as a function of neutrino energy.The latter is taken as a combination of all three mass-squared difference terms involving the fourth mass state, which are approximately of equal frequency at this scale.Left: magnitudes associated with ν e disappearance probabilities.Right: magnitudes associated with ν µ disappearance probabilities.Top: comparison between 3ν and a sterile neutrino scenario with δ 24 = 0. Bottom: comparison between sterile neutrino scenarios with δ 24 set to either 0 or π.All plots apply to neutrinos in normal ordering.The same parameters as in figure 1 were used.

Large |∆m 2 41 | limit
Anomalous oscillation results, such as LSND and MiniBooNE, are commonly interpreted as oscillations in a higher frequency than the solar and atmospheric scales.Under this scenario, the limit ∆m 2 41 → ∞ can be considered in which all oscillations driven by ∆m 2 41 are averaged out and observable only through scaling factors.Hereafter, this will be referred to as the high frequency (HF) region.
Following ref. [24], the mixing matrix U can be split such that U = U 4ν U 3ν , with U 3ν = R 23 R13 R 12 containing only the active-active mixing elements, and representing the active-sterile mixing.If the Hamiltonian is rotated with U 4ν , it becomes approximately block-diagonal in the limit where ∆m 2  41 → ∞: 2) The evolution matrix can then be expressed as: The remaining problem lies in the diagonalisation of H(3) .For that, further approximations, which are valid in specific energy regimes, are employed.In general, a scale will be used to represent small quantities.The mixing parameters s 34 , s 24 , s 14 , and s 13 will all be considered of O( ).Additionally, ∆m 2 21 /∆m 2 31 ∼ s 2 13 will be treated as O( 2).In this approximation, probabilities can be written to O( 2 ) as: P ee ≈ P (3)  ee cos 2θ 14 , ( P µµ ≈ P (3)  µµ cos 2θ 24 + 2c 2  24 Re[U 4ν µ2 U 4ν µ1 * S (3)  µµ (S (3)   eµ * + S (3)   µe * )], where U 4ν µ1 = −s 14 s 24 e iδ 14 −iδ 24 , U 4ν µ2 = c 24 , and S αβ correspond to elements of the evolution matrix in eq.(3.3).The effect of mixing with sterile neutrinos is given by a scaling of the 3-neutrino submatrix probabilities.Additionally, some interference terms appear if both s 14 and s 24 are non-zero.

The ORCA low energy regime
The ORCA detector is most sensitive to neutrinos in the energy range of 3 − 100 GeV considered in this analysis, crossing the Earth with paths of mean density varying between 3 and 9 g/cm 3 .In the lower part of this energy range (E < 10 GeV), when 2  31 /2E, to leading order in small quantities, H(3) simplifies to:

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This approximately 2-flavour form can be readily solved leading to the well-known MSW resonance of θ 13 : where Rm 13 represents the effective generalised unitary rotation matrix in the 1-3 plane, parametrised by the effective mixing angle θ m 13 and the unchanged phase δ 13 .All effects arising from the presence of sterile neutrino are constrained to vacuum-like mixing through U 4ν as in eq.(3.3).

The ORCA high energy regime
At higher energies (E 10 GeV), the matter potential starts to dominate.However, a new resonance can still be found when ∆ 31 /V n is of O( 2).In this regime, H(3) is expressed in leading order as: where U 4ν s2 = −c 34 s 24 e iδ 24 and U 4ν s3 = −s 34 .Once again, the Hamiltonian is approximately block diagonal and can be easily solved to give: This new resonance corresponds to a second order effect that couples the 2−3 sector indirectly via s 24 and s 34 .It provides a very rich structure having two main features: a resonance when sin 2θ m 23 → 1 and an antiresonance when sin 2θ m 23 → 0 at finite V n .The resonance conditions are: ) JHEP10(2021)180 A pole exists when both conditions are satisfied, as ∆ m 32 → 0 and no mixing is possible.The structure of these resonances is shown in figure 3. Since θ 23 is close to maximal, the antiresonance of eq.(3.17) is the most noticeable effect in this regime.The antiresonance occurs for neutrinos when cos δ 24 ∆m 2 31 < 0 or for antineutrinos when cos δ 24 ∆m 2 31 > 0, and is only exact for δ 24 = 0 or π.Hence, there is a degeneracy between mass ordering and sign(cos δ 24 ), enhanced by the maximal value of sin 2θ 23 , which suppresses NMO contributions from the resonance term in eq.(3.16).

Finite |∆m 2 41 | regime
At values of ∆m 2 41 for which the associated oscillations cannot be averaged out, no simplifying approximations are known to us at the time of writing.In ORCA, this corresponds to values of ∆m 2 41 0.1 eV 2 , this regime will be referred to as the low frequency (LF) region.In this case, many interference terms are present and the probability formulas can become exceedingly complex.Nevertheless, a full numerical solution is possible on all regimes considered in the analysis, and it is used to extend the results through six orders of magnitude in ∆m 2  41 .For simplicity, ∆m 2 41 will be restricted to positive values.

Sterile neutrino analysis
The analysis presented here is based on detailed Monte Carlo (MC) simulations as described in ref. [22].Neutrino interactions are generated with gSeaGen [25], which is based on GENIE [26].Secondary particles and their emitted Cherenkov light are propagated with KM3Sim [27], a software package based on GEANT4 [28].The atmospheric neutrino flux is computed from the Honda model [29] for the Gran Sasso site without mountain over -9 -JHEP10(2021)180 the detector, assuming minimum solar activity.Atmospheric muons are generated with MUPAGE [30,31], and propagated with KM3 [32].
Event reconstruction is performed via a maximum likelihood fit to shower and track hypotheses.Background events arising from noise and atmospheric muons are rejected with two independent Random Decision Forests (RDF) trained on MC simulations.A third RDF was used to separate neutrino candidates into three topology classes defined by the output score of the RDF, trained to identify track-like events.Events with a track score larger than 0.7 are labelled as track-like, track scores less than 0.3 are labelled as shower-like, and other values are labelled as an intermediate topology.Moreover, as in ref. [22], only upgoing events are considered in order to get rid of the atmospheric muon contamination.
Instead of using parametrised response functions as in ref. [22], the analysis reported here is based on the aforementioned MC simulations to directly model the detector response.The two approaches have been compared and found consistent.
The MC-based modelling of the detector response is implemented in the KM3NeT framework Swim [33].The detector response is represented by a 4-dimensional matrix, as a function of true and reconstructed neutrino energy E, E , and zenith angle θ, θ , for each interaction channel ν x , R [νx→i] (E, θ, E , θ ).Each entry of this matrix summarises in a single dimensionless coefficient the efficiency of detection, classification and probability of reconstruction for a given true bin (E, θ).Therefore, R incorporates all the effects related both to the detector and to the event selection.More details on this approach, can be found in ref. [33].The binning scheme, for the detector response matrix, used in this analysis is shown in table 1.Since the atmospheric neutrino flux follows a power law in energy, equalwidth bins in log 10 E are chosen.The same choice is adopted for reconstructed events histograms, as the relative energy resolution in ORCA is, to first order, constant above ∼ 10 GeV, δE/E δ(log 10 E) 15% [34].
A binning of constant width in cos θ Z is used.This is motivated by the fact that the solid angle covered by an interval of zenith angle and, considering to first order the atmospheric neutrino flux as isotropic, this choice yields equally populated bins along the zenith angle axis.
For reconstructed event histograms, the choice of binning granularity is dominated by the detector resolutions.The bin width should be comparable with the typical error on the reconstructed variable.Moreover, it should account for a sufficiently smooth sampling of the detector response, to minimise the finite MC statistics issues, which can result in overestimations of sensitivity [33].Statistical fluctuations due to the sparse MC effect are taken into account by following the "Beeston and Barlow method" [35].The values of the standard neutrino parameters used in this analysis is taken from the NuFit v4.1 global fit result with Super-Kamiokande (SK) data [36] and summarised in table 2, for both normal (NO) and inverted ordering (IO).Current fits have large errors on δ CP .The impact of such variable in the analysis has been tested and found to be negligible.For this reason, its value is fixed to the ones reported in table 2.Moreover, ∆m 2  41 > 0 is always assumed.Oscillation probabilities are evaluated with the software package OscProb [23], and to account for Earth's matter effects the PREM model [37] with 44 layers is used.
The above information can be used to define the distinguishability S σ , as a quick estimator of sensitivity of measurements, with the goal of illustrating the impact of a sterile neutrino in the event distributions, as where N Sterile and N Standard are the number of events, as a function of reconstructed energy and zenith angle, in the sterile and standard hypothesis respectively.Figure 4 shows the distinguishability distribution for a sterile neutrino in the HF region, for non-zero θ 24 and θ 34 , assuming three years of ORCA data taking.The presence of the sterile neutrino mainly impacts the track-like events in the form of a deficit of upgoing events at higher energies (E ≥ 40 GeV).Therefore, this region of the sterile parameter space can be well constrained also by neutrino telescopes whose energy threshold is higher than that of ORCA, such as ANTARES [38] and IceCube/DeepCore [39].
-11 -JHEP10(2021)180  To understand the sensitivity to θ 14 in particular, the distinguishability for a sterile neutrino in the HF region and sin 2 θ 14 = 0.05 is shown in figure 5.In this case, the showerlike events are the most affected and mainly for energies < 20 GeV.It follows that ORCA is well suited to test θ 14 .
Finally, figure 6 shows the impact of a sterile neutrino with ∆m 2 41 = 10 −4 eV 2 .In this case, the energy region E < 10 GeV is the most significant, and all the three event topologies are highly impacted.This applies also for ∆m 2 41 = 10 −2 , 10 −3 , 10 −5 eV 2 .
The sensitivity evaluation is based on the minimisation of a negative log-likelihood function describing the agreement between a model prediction and observed data.This is done with the Asimov approach [40]  squared distribution.Specifically, the negative log-likelihood function is defined as: where N model ijt and N data ijt represent the number of expected and measured events in bin (i, j) respectively and the sum over t runs over the three event topologies: tracks, intermediates and showers.η represents the model parameters, which comprise both the oscillation parameters listed in table 2, and nuisance parameters η , which are related to systematic uncertainties.The second sum runs over the nuisance parameters and η k is the assumed prior of the parameter k and σ η k its uncertainty.The set of free parameters considered in this analysis, together with the assumed gaussian priors with mean µ and standard deviation σ, is summarised in table 3.Where the uncertainties on the neutrino flux are taken from ref. [41] and the uncertainty on the detector energy scale follows the investigations reported in ref. [2] (section 3.4.6).Specifically: 1. the ratio between the total number of ν e and νe is allowed to vary with a standard deviation of 7% of the parameter's nominal value, JHEP10(2021)180 2. the ratio between the total number of ν µ and νµ is allowed to vary with a standard deviation of 5% of the parameter's nominal value, 3. the ratio between the total number of ν e and ν µ is allowed to vary with a standard deviation of 2% of the parameter's nominal value, 4. the number of NC events is scaled by the NC scale factor, to which no constraint is applied, 5. the absolute energy scale of the detector, which depends on the knowledge of the PMT efficiencies and the water optical properties, as discussed in ref. [22], is allowed to vary with a standard deviation of 5% around its nominal value, 6. the energy slope of the neutrino flux energy distribution is allowed to vary without constraint, 7. the ratio of upgoing to horizontally-going neutrinos, the zenith angle slope, is allowed to vary with a standard deviation of 2% of the parameter's nominal value, 8. the number of events in the three classes is allowed to vary without constraints, 9. ∆m 2 31 and θ 23 are allowed to vary without constraints, 10. θ 13 is allowed to vary within a 1σ window of the parameter's nominal value, which corresponds to 0.13 • for both NO and IO.
In the following section, the ORCA sensitivity to the active-sterile parameters is presented.

Sensitivity results
The ORCA sensitivity to the active-sterile mixing angles is here presented.The Asimov dataset is obtained using the parameters in table 2, assuming no sterile neutrino in NO and IO.No assumption is made on NMO: the fit is marginalised over NMO.This allows to conservatively take into account degeneracies between NMO and the sterile parameters.At the SBL neutrino mass scale, ∆m 2 41 ∼ 1 eV 2 , correlated constraints in the θ 24 − θ 34 parameter space are obtained.And, for a more general analysis, sensitivities to the mixing elements θ 14 , θ 24 , θ µe and θ 34 over the range ∆m 2  41 ∈ [10 −5 , 10] eV 2 are presented.
5.1 Sensitivity to θ 24 − θ 34 in the large ∆m 2 41 limit As shown in figure 4, in this sterile mass region, the track channel appears to be the most effective in constraining θ 24 and θ 34 .
As stated in section 3, δ 24 highly impacts the analysis due to matter effects.Therefore, δ 24 is kept free in the fit.Whereas, we investigated the impact of θ 14 and found it to be negligible, therefore θ 14 and δ 14 are fixed to zero in this part of the analysis.
Figure 7 shows the 90% and 99% C.L. ORCA sensitivity on sin 2 θ 24 and sin 2 θ 34 cos 2 θ 24 for three years of data taking.The ORCA sensitivity is compared to upper limits from The obtained sensitivity is compared with current upper limits from ANTARES [38], IceCube/DeepCore (IC) [39] and SK [42].
If not explicitly stated, δ 24 is free in the fit: this applies to the results from ORCA and ANTARES.The excluded region is the one on the top right of the lines.
other neutrino experiments, namely ANTARES [38], IceCube/DeepCore [39] and SK [42].In order to highlight the impact of δ 24 in the final constraints, ANTARES has presented upper limits [38] with δ 24 fixed to 0 and free.Allowing δ 24 to be free worsens the constraints on θ 24 and θ 34 and it needs to be considered as a free parameter by all the analyses in which Earth matter effects are not negligible.Here, only the analysis with δ 24 free is presented.The impact of this quantity in the ORCA sensitivity can be found in ref. [43]: it is maximal when sin 2 θ 24 = sin 2 θ 34 cos 2 θ 24 , for which case it worsens the sensitivity by about a factor of two for sin 2 θ 24 and a factor three for sin 2 θ 34 cos 2 θ 24 .
Due to the degeneracy driven by NMO and δ 24 , discussed in section 3, the ORCA Asimov dataset in NO and δ 24 free (blue line) can be directly compared with Ice-Cube/DeepCore IO and δ 24 = 0 (red line).For SK, upper limits with IO are not available, therefore the ones with NO and δ 24 = 0 are here reported.
From figure 7 it can be concluded that ORCA is competitive in constraining the mixing elements θ 24 and θ 34 , and it is expected to improve the sensitivity to sin 2 θ 34 cos 2 θ 24 by over a factor of two with respect to current limits.

Sensitivity to θ 24 for different ∆m 2 41 values
Figure 8 shows the 90% and 99% C.L. ORCA sensitivity to sin 2 θ 24 assuming three years of data taking.For this analysis, θ 14 , θ 34 , δ 14 and δ 24 are set free in the fit, since their effects on the results of the analysis are expected to be not negligible.
The ORCA sensitivity is compared with upper limits from cosmology [13] for which only 95% C.L. are available, and upper limits from MINOS/MINOS+ [44], IceCube [45] and SK [42].The obtained sensitivity is compared with current upper limits from cosmology [13], MINOS/MINOS+ [44], IceCube (IC) [45] and SK [42].The excluded region is the one on the right of the lines, for IceCube at 90% C.L. it is the external region to the closed contour line.
Both plots show that ORCA is less competitive than MINOS/MINOS+ and IceCube for HF.KM3NeT/ARCA would be better suited to test sin 2 θ 24 in this region.In the LF region, ORCA is able to improve current limits on sin 2 θ 24 by more than one order of magnitude. 2 41 values Figure 9 shows the 95% C.L. ORCA sensitivity to sin 2 θ 14 after three years of data taking.The choice to show the sensitivity at such a level of confidence is motivated by the goal to have a fair comparison with the other experiments, for which the majority of the available upper limits and sensitivity is reported at 95% C.L.For this analysis, θ 24 , θ 34 , δ 14 and δ 24 are free in the fit, since their effects on the results of the analysis are expected to be not negligible.Figure 5 shows that, in the HF region, shower-like events are the most affected by θ 14 and in the optimal energy region for ORCA (E < 10 GeV).However, they are concentrated in the nearly-horizontal region (−0.1 < cos θ Z < −0.6).Nevertheless, ORCA has a competitive sensitivity to Daya Bay+Bugey-3 [44] and STEREO [46] in the HF region.Moreover, ORCA will also be able to test part of the Neutrino-4 allowed region [47].On the contrary, the global fit regions can not be reached with three years of data taking., for three years of data taking.Sensitivity results are compared with current upper limits from cosmology [13], STEREO [46], and Daya Bay+Bugey-3 [44].Current anomaly regions are also reported, from Neutrino-4 [47], global fits [9] and reactors global fits [48].The excluded region is the one on the right of the lines. of 10 −6 .Figure 10 shows the 90% and 99% C.L. ORCA sensitivity to |U µe | 2 , compared with current upper limits from Daya Bay+Bugey-3+MINOS/MINOS+ [44], KARMEN [49], and NOMAD [50].

Sensitivity to θ 14 for different ∆m
Figure 10 shows that, after three years of data taking, ORCA will be able to test the majority of the LSND [11] and MiniBoone [12] anomaly region.Moreover, current limits on sin 2 2θ µe will be improved by 1-2 orders of magnitude in the LF region. 2  41 values Figure 11 shows the ORCA sensitivity at 99% C.L. to sin 2 θ 34 after three years of data taking.Here, θ 14 , θ 24 , δ 14 and δ 24 are set free in the fit.Upper limits from cosmology [13], IceCube/DeepCore [39] and SK [42] are also reported.In the LF region there are no upper limits on θ 34 coming from other experiments.

Sensitivity to θ 34 for different ∆m
ORCA is able to constrain θ 34 over a broad range of ∆m 2 41 .In the HF region, consistently with figure 7, ORCA can improve current upper limits on sin 2 θ 34 by about a factor two.

Summary and conclusions
KM3NeT/ORCA, a neutrino detector under construction in the Mediterranean Sea, is optimised for oscillation studies with atmospheric neutrinos in the GeV energy range.In this paper, it has been shown that the ORCA detector has a great potential to search for the presence of a light sterile neutrino in the range ∆m 2  41 ∈ [10 −5 , 10] eV 2 , by fitting the expected number of observed events classified in three topologies, namely track, intermediate  Sensitivity results are compared with current upper limits from Daya Bay+Bugey-3+MINOS/MINOS+ [44], KARMEN [49] and NOMAD [50].Current anomaly regions from LSND [11] and MiniBooNE [12] are also reported.The excluded region is the one on the right of the lines., for three years of data taking.Sensitivity results are compared with current upper limits from cosmology [13], IceCube/DeepCore [39] and SK [42].The excluded region is the one on the right of the lines.

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and shower events.With this methodology, ORCA can probe regions in the active-sterile mixing elements θ 14 , θ 24 , θ 34 and the effective parameter θ µe , not yet constrained by current experiments.Particularly, after three years of data taking, ORCA can improve current limits on sin 2 θ 34 cos 2 θ 24 by about a factor of two, in case of null result, for an eV-mass sterile neutrino.For lower sterile neutrino masses, down to ∆m 2 41 → 10 −5 eV 2 , ORCA will be able to test the unexplored region of the sin 2 θ 24 parameter, and sin 2 2θ µe effective parameter space down to about two orders of magnitude with respect to current limits.The ORCA sensitivity to sin 2 θ 14 is comparable to current upper limits.Finally, in case of null result, ORCA will able to improve current limits on sin 2 θ 34 by about a factor two for an eV-mass sterile neutrino, and it is the first experiment, to date, able to constrain θ 34 in the very low sterile mass region.

Figure 2
shows examples of effective values of the magnitude of some terms from eq. (3.1) as a function of neutrino energy.The impact of the aforemen-JHEP10(2021)180

Figure 1 .
Figure 1.Effective mass-squared values, representing the eigenvalues of eq.(1.1) for neutrinos in both normal (left) and inverted (right) orderings, as a function of neutrino energy.Three models are shown: in the upper panels, the standard picture with three active neutrinos (3ν), is compared to a model with one light sterile neutrino where the CP violating phase δ 24 is set to 0. In the lower panels, two sterile neutrino models are compared with δ 24 set to either 0 or π.The absolute mass scale has been chosen so that the lightest neutrino is massless in vacuum.The oscillation parameters were set to ∆m 2 21 = 7.5×10 −5 eV 2 , |∆m 2 31 | = 2.5×10 −3 eV 2 , |∆m 2 41 | = 1 eV 2 , s 2 12 = 0.3, s 2 13 = 0.02, s 2 23 = 0.57, s 2 14 = 0.01, s 2 24 = s 2 34 = 0.04, and δ 13 = δ 14 = 0.The matter density is set to 8.5 g/cm 3 with a ratio N n /N e = 1.08.

Figure 2 .
Figure 2. Effective oscillation magnitudes associated with ∆ 21 , ∆ 31 , ∆ 32 , and ∆ 4k (see equation (3.1)) as a function of neutrino energy.The latter is taken as a combination of all three mass-squared difference terms involving the fourth mass state, which are approximately of equal frequency at this scale.Left: magnitudes associated with ν e disappearance probabilities.Right: magnitudes associated with ν µ disappearance probabilities.Top: comparison between 3ν and a sterile neutrino scenario with δ 24 = 0. Bottom: comparison between sterile neutrino scenarios with δ 24 set to either 0 or π.All plots apply to neutrinos in normal ordering.The same parameters as in figure1were used.

Figure 3 .
Figure 3. Effective parameters ∆ m 32 /∆ 31 (left) and sin 2 2θ m 23 (right) as a function of real values of U µ4 and U τ 4 , for a neutrino energy of 20 GeV.Here, ∆m 2 31 = 2.5 × 10 −3 eV 2 , s 2 23 = 0.57, and a matter density of 8.5 g/cm 3 with a ratio N n /N e = 1.08 were assumed.The resonance and antiresonance described in equations (3.16) and (3.17) are visible on the right as regions of maximum and minimum sin 2 2θ m 23 .At the point where they seem to meet, a pole exists where ∆ m 32 → 0 and sin 2 2θ m 23 becomes undefined.

Figure 4 .
Figure 4. S σ distribution of the three topologies considered in the analysis (tracks, intermediates and showers) assuming three years of data taking.The colour scale denotes the S σ value for each bin, whereas the total S σ is reported on top of the plots: the high value obtained is due to the normalisation.The sterile neutrino parameters are sin 2 θ 14 = 0, sin 2 θ 24 = 0.03, sin 2 θ 34 = 0.05, ∆m 2 41 = 1 eV 2 .

Figure 5 .
Figure5. S σ distribution of the three topologies considered in the analysis (tracks, intermediates and showers) assuming three years of data taking.The colour scale denotes the S σ value for each bin, whereas the total S σ is reported on top of the plots: the high value obtained is due to the normalisation.The sterile neutrino parameters are sin 2 θ 14 = 0.05, sin 2 θ 24 = sin 2 θ 34 = 0, ∆m 2 41 = 1 eV 2 .

Figure 6 .
Figure 6.S σ distribution of the three topologies considered in the analysis (tracks, intermediates and showers) assuming three years of data taking.The colour scale denotes the S σ value for each bin, whereas the total S σ is reported on top of the plots: the high value obtained is due to the normalisation.The sterile neutrino parameters are sin 2 θ 14 = sin 2 θ 24 = sin 2 θ 34 = 0.01, ∆m 2 41 = 10 −4 eV 2 .

Figure 7 .
Figure 7.The 90% (left) and 99% C.L. (right) KM3NeT/ORCA to the mixing parameters θ 24 −θ 34 , with ∆m 2 41 = 1 eV 2, for three years of assumed data taking.The obtained sensitivity is compared with current upper limits from ANTARES[38], IceCube/DeepCore (IC)[39] and SK[42].If not explicitly stated, δ 24 is free in the fit: this applies to the results from ORCA and ANTARES.The excluded region is the one on the top right of the lines.

Figure 8 .
Figure8.The 90% (left) and 99% C.L. (right) KM3NeT/ORCA sensitivity to the mixing parameter θ 24 , assuming three years of data taking.The obtained sensitivity is compared with current upper limits from cosmology[13], MINOS/MINOS+[44], IceCube (IC)[45] and SK[42].The excluded region is the one on the right of the lines, for IceCube at 90% C.L. it is the external region to the closed contour line.

5. 4
Sensitivity to |U µe | 2 for different ∆m 2 41 values Since ORCA can observe both ν e and ν µ disappearance, the effective mixing element|U µe | 2 = sin 2 2θ µe = 4|U e4 | 2 |U µ4 | 2 canbe constrained directly.In this case, θ 14 and θ 24 are left free in the fit, however, their combination is constrained to match the appropriate θ µe value by introducing a penalty term in the likelihood with a very small prior uncertainty

Figure 9 .
Figure 9.The 95% C.L. KM3NeT/ORCA sensitivity to the mixing parameter θ 14 , for different values of ∆m 2 41, for three years of data taking.Sensitivity results are compared with current upper limits from cosmology[13], STEREO[46], and Daya Bay+Bugey-3[44].Current anomaly regions are also reported, from Neutrino-4[47], global fits[9] and reactors global fits[48].The excluded region is the one on the right of the lines.

Figure 11 .
Figure 11.The 99% C.L. KM3NeT/ORCA sensitivity to the mixing parameter θ 34 , for different values of ∆m 2 41, for three years of data taking.Sensitivity results are compared with current upper limits from cosmology[13], IceCube/DeepCore[39] and SK[42].The excluded region is the one on the right of the lines.

Table 1 .
Bin choice for the MC-based response matrix, R, used in this analysis.Energy bins are in log 10 space.

Table 3 .
assuming the negative log-likelihood follows a chi-List of fitted values and relative gaussian priors considered in this analysis.θ 13 refers to the values listed in table 2.