Measurement of the atmospheric νμ flux with six detection units of KM3NeT/ORCA

Abstract

A measurement of the atmospheric ${\nu }_{\mu }+{\overline{\nu }}_{\mu }$ flux with energies between 1 and 100 GeV is presented. The measurement has been performed using data taken with the first six detection units of the KM3NeT/ORCA detector, referred to as ORCA6. The data were collected between January 2020 and November 2021 and correspond to 510 days of livetime, with a total exposure of 433 kton$·$years. Using machine learning classification, 3894 neutrino candidate events have been selected with an atmospheric muon contamination of less than 1$\%$. The atmospheric ${\nu }_{\mu }+{\overline{\nu }}_{\mu }$ energy spectrum is derived using an unfolding procedure and the impact of systematic uncertainties is estimated. The atmospheric ${\nu }_{\mu }+{\overline{\nu }}_{\mu }$ flux measured using the ORCA6 configuration is in agreement with the values measured by other experiments.

Introduction

Atmospheric neutrinos originate from extensive air showers (EAS), produced when primary cosmic rays interact with the nuclei of the atmosphere [1]. The energy spectrum of atmospheric neutrinos covers a wide range, from about 100 MeV to PeV energies. They are mainly produced in the decays of charged $\pi$ and K mesons, and they constitute the conventional component of the flux:

1

$$\begin{array}{c}\hfill {\pi }^{+(-)}\to {\mu }^{+(-)}+{\nu }_{\mu }({\overline{\nu }}_{\mu })\text{,}{\text{K}}^{+(-)}\to {\mu }^{+(-)}+{\nu }_{\mu }({\overline{\nu }}_{\mu })\end{array}$$with muons decaying into

2

$$\begin{array}{c}\hfill {\mu }^{-(+)}\to e^{-(+)}+{\overline{\nu }}_e({\nu }_e)+{\nu }_{\mu }({\overline{\nu }}_{\mu }).\end{array}$$At higher energies, neutrinos that are produced in decays of charmed mesons (mainly D mesons) constitute the prompt atmospheric neutrino flux. Due to their longer lifetime (${\tau }_{\pi }\sim {10}^{-8}\text{s}$) compared to that of D mesons (${\tau }_D\sim {10}^{-12}-{10}^{-13}\text{s}$) [2], charged $\pi$ and K mesons travel larger distances before decaying and experience higher energy losses. As a product of cosmic ray collisions in the atmosphere, the flux of atmospheric neutrinos approximately follows a power law, $dN/dE\propto E^{\gamma }$. Conventional neutrinos have a softer energy spectrum (${\gamma }_c\approx -3.7$) than prompt neutrinos (${\gamma }_p\approx -2.7$). Simulations indicate that for vertically incoming neutrinos, the crossover between conventional and prompt fluxes occurs in the energy range of ${10}^5$–${10}^6$ GeV, and that the crossover energy increases with zenith angle [3].

As a result of Eqs. 1 and 2, the ratio of ${\nu }_{\mu }+{\overline{\nu }}_{\mu }$ to ${\nu }_e+{\overline{\nu }}_e$ in the conventional flux is about 2 below 1 GeV. This ratio increases with energy, as higher energy muons can reach the Earth before decaying. The flux of neutrinos in the vertical direction is lower than the flux near the horizon, due to the different path-lengths of the parent particles in the atmosphere before decaying [4]. The effect of the Earth’s magnetic field on the primary cosmic rays can be neglected for neutrinos above a few GeV [5].

The ${\nu }_e+{\overline{\nu }}_e$ flux has been measured by Fréjus [6], IceCube [7, 8], Super-Kamiokande [9] and ANTARES [10], while the atmospheric ${\nu }_{\mu }+{\overline{\nu }}_{\mu }$ flux has been measured by Frejus [6], AMANDA [11], Super-Kamiokande [9], IceCube [12, 13–14], and ANTARES [10, 15]. MACRO studied the ${\nu }_{\mu }+{\overline{\nu }}_{\mu }$ flux to perform neutrino oscillation studies [16]. In ref. [14], IceCube observed a flattening of the flux at high energies, which can be attributed to a diffuse flux of astrophysical neutrinos. Moreover, IceCube has measured the seasonal variation of the atmospheric ${\nu }_{\mu }+{\overline{\nu }}_{\mu }$ flux [17]. While above 100 GeV large neutrino telescopes are sensitive and several different measurements of the the atmospheric ${\nu }_{\mu }+{\overline{\nu }}_{\mu }$ flux have been performed, limited experimental information exists in the region below. The measurement by Frejus does not account for neutrino oscillations, which had not been discovered at the time. The Super-Kamiokande experiment provides precision measurements up to 10 GeV, since the experiment has been designed for neutrino energies between 100 MeV and few GeV. Consequently, the energy region between 10 GeV and 100 GeV would benefit from additional experimental data.

The paper is structured as follows. In Sect. 2, the KM3NeT/ORCA detector is introduced. In Sect. 3, the data and Monte Carlo (MC) simulation used in the analysis are presented. In Sect. 4, the procedure to select a high-purity atmospheric neutrino event sample from data collected with a partial configuration of the detector is described. In Sect. 5, the ${\nu }_{\mu }+{\overline{\nu }}_{\mu }$ charged-current (CC) energy spectrum is obtained, exploiting an unfolding procedure. In Sect. 6, the studies to evaluate the impact of systematic uncertainties on the measurement are presented. Finally, the results are shown in Sect. 7 and discussed in Sect. 8.

The KM3NeT/ORCA detector

The KM3NeT Collaboration is currently constructing two detectors with different scientific goals in the Mediterranean Sea [18]. The ARCA (Astroparticle Research with Cosmics in the Abyss) detector is located about 80 km offshore Sicily, Italy, at a depth of 3450 m. The primary goal of ARCA is the detection of high-energy astrophysical neutrinos. The ORCA (Oscillation Research with Cosmics in the Abyss) detector is located 40 km offshore Toulon, France, at a depth of 2450 m. The goal of ORCA is to study atmospheric neutrino oscillations in the energy range of a few GeV, and to determine the neutrino mass ordering [19]. The ARCA and ORCA detectors are built using the same technology.

The ORCA detector is an array of photomultiplier tubes (PMTs) for the detection of Cherenkov radiation emitted along the path of neutrino-induced particles in seawater. The 3-inch Hamamatsu PMTs [20] are packed in groups of 31 into high-resistance glass spheres in order to endure the pressure at the sea depths. The spheres contain also the required electronics for the operation of the PMTs and the data transmission [21], as well as instruments for the orientation [22], position and time calibration [23]. These assemblies are called Digital Optical Modules (DOMs) [24] and provide an almost $4\pi$ solid angle coverage.

The DOMs are evenly distributed across vertical support structures, the detection units (DUs) [25]. Each DU hosts 18 DOMs, is anchored to the seabed and is kept taut and aloft by the buoyancy of the DOMs and of a buoy which is tied to its top. The distance between two consecutive DOMs in ORCA is about 9 m, and the horizontal separation between the DUs is about 20 m. This geometry was optimised to enhance the sensitivity to GeV neutrinos. When completed, the ORCA detector will consist of 115 DUs, instrumenting a mass of about 7 Mton of seawater.

When the voltage amplitude at the PMT anode exceeds a predefined threshold, a hit is recorded and a set of information containing the duration of the pulse over the threshold, the time of the pulse leading edge, and the PMT identifier, are registered and transmitted to shore. On-shore, the data are processed and filtered by a computer farm using dedicated trigger algorithms and the resulting events are stored on disk.

Events recorded in a Cherenkov neutrino telescope have different signatures depending on the physics processes that are involved. Atmospheric muons, residuals of cosmic ray showers, produce long down-going tracks with an abundant flux reaching the detector. Muons mainly from CC interactions of muon neutrinos are selected by applying geometrical cuts – i.e. selecting upgoing track-like events. The electron and tau neutrino CC interactions and the neutral-current (NC) interactions of all neutrino flavours, produce mainly shower-like events, characterised by a more localised area of light emission.

Data and MC simulation

The data used in this analysis have been collected with the 6-DU configuration of the ORCA detector. The data stream is organised in periods of data taking, referred to as runs, with a duration of about 6 h. Runs for which more than half of the PMTs exceed their maximum measurable hit rate as a result of extreme environmental conditions, such as bioluminescence, are not considered in the analysis. Moreover, a fraction of the operational time was devoted to calibration and test periods of data-taking. The corresponding data are removed from the analysis. The livetime of the selected dataset is 510 days, with an equivalent exposure of 433 kton$·$years [26].

A run-by-run Monte Carlo simulation strategy [27] has been followed, in which each run is simulated in order to account for the variation of the detector data-taking conditions. In-situ calibration constants related to the time offsets and response of PMTs are regularly extracted and applied to data and MC [28].

Atmospheric neutrinos have been generated using the gSeaGen software [29], which is based on the GENIE [30] neutrino generator (both neutrinos and anti-neutrinos are mentioned as neutrinos hereafter, since ORCA can not discriminate between them). The CC neutrino interactions have been simulated for all neutrino flavours, while the NC interactions have been simulated for muon neutrinos and scaled to account also for the other flavours. Event weights have been assigned to account for the atmospheric neutrino flux and neutrino oscillation probabilities. The HKKM14 conventional atmospheric neutrino flux model has been used [31] to weight atmospheric neutrino events. Neutrino oscillation probabilities have been computed with OscProb [32], assuming the mass eigenstates follow a normal ordering, and using the oscillation parameters from NuFIT 5.2 [33] global oscillation fits. Atmospheric muons have been simulated with the MUPAGE software [34, 35].

The GEANT4-based [36] KM3Sim [37], a custom software package, is used for the light propagation of atmospheric neutrino events. In KM3Sim, a full simulation of photons is performed, by tracking in detail the individual secondary particles that are produced at the generation level, as well as the emitted Cherenkov photons. For higher neutrino energies ($E_{\nu }>500$ GeV) and for atmospheric muons, an internal KM3NeT software package is used for the simulation of the light production and propagation, which evaluates the number of photoelectrons recorded by a PMT using tabulated probability distribution functions (PDFs) of the photon arrival time.

Noise hits are simulated on the basis of the measured PMT rates from data. The PMT signals of the simulated hits are digitised. Finally, the stream of simulated events is processed using the same trigger algorithms used for data.

All data and MC simulated events are reconstructed using two KM3NeT-built software packages: one assuming a track-like topology [38] and another assuming a shower-like topology [39].

Event selection

Pre-selection

Pure noise hits (mainly caused by ⁴⁰K decays) can produce triggers that lead to poorly reconstructed events. The contribution of pure noise events is suppressed by requiring a minimum number of hits used in the track reconstruction ($N_{\mathit{hits}}>20$), and a quality cut on the logarithm of the maximum likelihood of the track reconstruction (${log}_{10}L>1.7$). These criteria are hereafter mentioned as anti-noise cuts. The distribution of the cosine of reconstructed zenith angle ($-1$ corresponds to a vertical upward-going direction) for data and MC simulated events is shown in Fig. 1, after the application of the anti-noise cuts.

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Fig. 1

Distribution of the reconstructed cosine zenith angle for data (black line) and MC simulated events (blue line for atmospheric muons, red line for neutrinos) after applying the anti-noise cuts

The main background source in a neutrino Cherenkov detector comes from atmospheric muons. By selecting only upward-going tracks, an atmospheric muon background rejection of $\sim 98\%$ is achieved, while $\sim 54\%$ of the atmospheric neutrinos are preserved, with respect to the sample of events after the anti-noise cuts.

BDT classification

The event selection procedure is eventually reduced to a binary classification problem: the discrimination between atmospheric ${\nu }_{\mu }+{\overline{\nu }}_{\mu }$ CC events and atmospheric muon events misreconstructed as upward-going. As a preliminary detector configuration, ORCA6 provides reduced resolution for distinguishing track-like from shower-like events. Therefore, the final event classification is oriented to suppress atmospheric muons, the vast majority of the background. The contribution of the atmospheric neutrino background in the event selection is subtracted in the unfolding procedure as described in Sect. 5.

For the final event classification, a boosted decision trees (BDT) algorithm with adaptive boosting is used, implemented in TMVA [40]. Dedicated MC simulated event samples are produced in order to train the BDT classifier. The training for the signal is performed with a sample of CC atmospheric muon neutrino events. For the background, the BDT is trained on atmospheric muon events. The pre-selection criteria are applied to both training samples. The MC event samples for signal and background were independently shuffled and then split into equal sizes training and test datasets between the classes. This approach ensured that both the training and test sets contained balanced and representative samples from each class, avoiding any bias due to class imbalance. Variables defined on triggered hits and on the basis of the Cherenkov hypothesis for the origin of the hits, the quality of the event reconstruction, the event topology, as well as the deposited charge in the detector are used as BDT features.

The BDT performance was optimised to obtain the highest classification efficiency, through a grid search over five key configuration parameters: the number of trees, maximum tree depth, adaptive boosting parameter, minimum percentage of training events per node, and the number of grid points for node splitting. Each trained classifier was applied also to a $\sim 10\%$ of data and MC simulated events. The optimal configuration was selected based on overall signal efficiency, background contamination and agreement between data and MC at indicative BDT score cut values. The distribution of the BDT score for the data and MC simulated events fulfilling the pre-selection requirements is presented in Fig. 2.

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Fig. 2

BDT score distribution for data and MC simulated events fulfilling the pre-selection requirements

Table 1. Number of data and MC simulated events at different levels of the selection procedure, corresponding to the 433 kton$·$years exposure

	Anti-noise	+Up-going	+BDT
Data	2.7$\times {10}^8$	4.1$\times {10}^6$	3894
Atm. $\mu$	2.6$\times {10}^8$	4.4$\times {10}^6$	23
Atm. ${\nu }_e+{\overline{\nu }}_e$ CC	2486	1590	552
Atm. ${\nu }_{\mu }+{\overline{\nu }}_{\mu }$ CC	15235	7673	2958
Atm. ${\nu }_{\tau }+{\overline{\nu }}_{\tau }$ CC	327	308	162
Atm. $\nu +\overline{\nu }$ NC	1279	778	222
MC total	2.6$\times {10}^8$	4.4$\times {10}^6$	3917

The signal efficiency and background contamination of the final event selection were scanned over the BDT score cut. A cut value for the BDT score is chosen at 0.56 (Fig. 2), for which the background contamination falls under 1%. The number of selected data and MC simulated events is shown in Table 1. The contribution of atmospheric muon background is less than $0.6\%$, and $75.5\%$ of the MC simulated events that have been selected are ${\nu }_{\mu }+{\overline{\nu }}_{\mu }$ CC events. The distribution of the cosine of reconstructed zenith angle and the distribution of the distance of the reconstructed vertex to the geometrical centre of the instrumented volume is shown in Figs. 3 and 4 respectively, compared to the MC simulation.

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Fig. 3

Distribution of the reconstructed cosine zenith angle for data and MC simulated events after the BDT selection

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Fig. 4

Distribution of the distance of the reconstructed vertex to the geometrical centre of the instrumented volume for data and MC simulated events after the BDT selection as defined in the text

Unfolding of the energy spectrum

Unfolding scheme

The distribution of a measured quantity in n bins can be expressed as:

3

$$\begin{array}{c}\hfill \begin{array}{c}\hfill y_i=\sum _{j=1}^m{A}_{\mathit{ij}}{x}_j+b_i,1\le i\le n\end{array}\end{array}$$where $\overrightarrow{y}$ is the vector of observed values (reconstructed energy), $\overrightarrow{x}$ represents the true neutrino energy distribution in m bins, and $\overrightarrow{b}$ accounts for background contributions. The response matrix $\widehat{A}$ describes the probability of observing a reconstructed event in bin i given a true neutrino energy in bin j. In the case of a neutrino Cherenkov detector in seawater, this matrix represents a complex convolution of effects such as the physics of neutrino interactions, the propagation of secondary particles through seawater, the PMT efficiencies and the reconstruction resolution of the detector. Because the true neutrino energy is not directly measurable, this inverse problem incorporates smearing effects. The purpose of unfolding is to estimate the true spectrum $\overrightarrow{x}$ from the measured distribution $\overrightarrow{y}$, using the detector response described by $\widehat{A}$.

The track and shower reconstruction algorithms have been developed and optimised for the completed configuration of KM3NeT/ORCA. Applying these algorithms to the significantly smaller instrumented volume of ORCA6 has an impact on the reconstruction performance. In the case of the energy reconstruction, the performance of the shower reconstruction algorithm is less affected, and therefore the shower reconstructed energy was chosen as the measured distribution $\overrightarrow{y}$. This distribution is shown in Fig. 5.

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Fig. 5

Distribution of the shower reconstructed energy for data and MC simulated selected events of the event selection

For the implementation of the unfolding procedure, the TUnfold software is used [41]. The $\overrightarrow{x}$ vector in Eq. 3 is estimated using a least square method with Tikhonov regularisation [42] and a constraint on the total number of events. TUnfold also allows for a subtraction of background events using the estimation of the MC simulation. The response matrix is constructed using the ${\nu }_{\mu }+{\overline{\nu }}_{\mu }$ CC selected events. The shower-like atmospheric ${\nu }_e+{\overline{\nu }}_e$ CC, ${\nu }_{\tau }+{\overline{\nu }}_{\tau }$ CC, $\nu +\overline{\nu }$ NC, as well as the (negligible, sub-percent) contribution of atmospheric muon events that survive the selection criteria, are treated as background sources within TUnfold. Moreover, considering the limited instrumented volume of ORCA6, the contribution of events with $E_{\nu }>100$ GeV is treated as an overflow to the measured energy spectrum. As a result, the flux is measured in the range between 1 GeV and 100 GeV. Three bins are selected for the true neutrino energy, as ${log}_{10}(E_{\nu }/\text{GeV}):\{0.0,0.7,1.3,2.0\}$. Details on the unfolding scheme can be found in Appendix A.

The unfolded energy spectrum of the ${\nu }_{\mu }+{\overline{\nu }}_{\mu }$ CC events is shown in Fig. 6. The results of the unfolding procedure are summarised in Table 2.

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Fig. 6

Distribution of the unfolded energy spectrum. The distribution of the true energy for the ${\nu }_{\mu }+{\overline{\nu }}_{\mu }$ CC MC simulated events that survive the event selection criteria has been also added as a reference

Table 2. Number of events for the unfolded energy spectrum with statistical errors. The number of MC simulated events for each bin in true (MC) neutrino energy has been also added as a reference

${log}_{10}(E_{\nu }/\text{GeV})$	Unfolded	MC ${\nu }_{\mu }+{\overline{\nu }}_{\mu }$ CC
0.0–0.7	$194\pm 16\%$	202
0.7–1.3	$540\pm 6\%$	680
1.3–2.0	$1055\pm 5\%$	948

The differences between the unfolded distribution and the MC distribution are in agreement with what is shown in Fig. 5. Systematic uncertainties are presented in Sect. 6, where a discussion of the differences between data and MC simulation is presented.

Conversion to flux values

A procedure to convert the unfolded energy spectrum into flux is developed, based on the procedure followed by the Super-Kamiokande experiment [9]. The atmospheric ${\nu }_{\mu }+{\overline{\nu }}_{\mu }$ flux for each energy bin i is therefore calculated as:

4

$$\begin{array}{c}\hfill \begin{array}{c}\hfill {\mathrm{\Phi }}_i={\mathrm{\Phi }}_{\text{MC}}^{{\nu }_{\mu }+{\overline{\nu }}_{\mu }}({\tilde{E}}_i)·\frac{N_i^{\text{unfolded}}}{N_{i,\text{MC}}^{{\nu }_{\mu }+{\overline{\nu }}_{\mu }\text{CC}}}\end{array}\end{array}$$where ${\mathrm{\Phi }}_i$ is the measured value and ${\mathrm{\Phi }}_{\text{MC}}^{{\nu }_{\mu }+{\overline{\nu }}_{\mu }}({\tilde{E}}_i)$ is the flux predicted by the HKKM14 atmospheric ${\nu }_{\mu }+{\overline{\nu }}_{\mu }$ flux model [31], calculated at the energy weighted bin centre ${\tilde{E}}_i$ of the selected events, $N_i^{\text{unfolded}}$ is the number of events after unfolding, and $N_{i,\text{MC}}^{{\nu }_{\mu }+{\overline{\nu }}_{\mu }\text{CC}}$ is the number of MC simulated ${\nu }_{\mu }+{\overline{\nu }}_{\mu }\text{CC}$ events, as in Table 2 and in Fig. 6.

The reference flux ${\mathrm{\Phi }}_{\text{MC}}^{{\nu }_{\mu }+{\overline{\nu }}_{\mu }}(E_{\nu })$ in Eq. 4 is calculated as the zenith and azimuth average of the following expression:

5

where $E_{\nu }$ is the neutrino energy, $\theta$ and $\varphi$ are the zenith angle and the azimuthal angle of the neutrino direction respectively, and $O^{{\nu }_i\to {\nu }_j}$ is the oscillation probability of a flavour i to a flavour j, assuming azimuthal symmetry. The fluxes ${\mathrm{\Phi }}_{\text{MC}}^{{\nu }_i},{\nu }_i={\nu }_e,{\nu }_{\mu },{\overline{\nu }}_e,{\overline{\nu }}_{\mu }$ are the predicted fluxes from the HKKM14 conventional atmospheric neutrino flux model [31], and the neutrino oscillation parameters are the ones from NuFIT 5.2 [33]. The flux value for each bin i is calculated with an interpolation of the ${\mathrm{\Phi }}_{\text{MC}}^{{\nu }_{\mu }+{\overline{\nu }}_{\mu }}$ obtained using Eq. 5, at the weighted energy centre ${\tilde{E}}_i$, and subsequently using the Eq. 4.

Systematic uncertainties

The systematic uncertainties are evaluated by repeating the unfolding procedure described in Sect. 5, varying input parameters of the MC simulations and taking into account modifications in the response matrix and in the background. This produces modifications to the unfolded flux, which are taken as estimates of the systematic uncertainties following the approach of the ANTARES measurement in [10]. The following uncertainties have been considered:

1. The PMT efficiencies and the light absorption length in seawater are modified by $\pm 10\%$ independently, and referred to as detector response in the following.

2. The effect of uncertainties on the hadronic interaction models used in the simulation of EAS is considered as representative of the uncertainties $\delta$ on the ratio of the neutrino flavours, $({\nu }_{\mu }+{\overline{\nu }}_{\mu })/({\nu }_e+{\overline{\nu }}_e)$, as well as on the ratios of neutrinos and antineutrinos, ${\nu }_e/{\overline{\nu }}_e$ and ${\nu }_{\mu }/{\overline{\nu }}_{\mu }$ [43]. In order to estimate this effect, the corresponding ${\mathrm{\Phi }}_i$, ${\mathrm{\Phi }}_j$ in each ratio are changed as ${\mathrm{\Phi }}_i\to {\mathrm{\Phi }}_i(1+\delta )$ and ${\mathrm{\Phi }}_j\to {\mathrm{\Phi }}_j(1-\frac{{\mathrm{\Phi }}_i}{{\mathrm{\Phi }}_j}\delta )$, keeping the total neutrino flux constant.

3. Uncertainties on the neutrino cross section are considered by rescaling the number of ${\nu }_{\mu }+{\overline{\nu }}_{\mu }$ CC simulated events with energy dependent factors taken from [44], and ${\nu }_{\tau }+{\overline{\nu }}_{\tau }$ CC simulated events by $\pm 20\%$.

4. To evaluate the uncertainties introduced by the unfolding procedure, 2000 pseudo-data sets are generated from the reconstructed energy distribution of Fig. 5. Pseudo-unfoldings are performed after replacing the measured reconstructed energy distribution with the random generated ones. The associated systematic uncertainties are considered as the $68\%$ quantiles of the residuals between the toy-unfolding results and the nominal result. Additionally, the unfolding is repeated with the model flux modified as ${\mathrm{\Phi }}_{\text{sys}}={\mathrm{\Phi }}_{\text{nom}}{\left(\frac{E}{10\text{GeV}}\right)}^{\mathrm{\Delta }\gamma }$, where $\mathrm{\Delta }\gamma =\pm 0.05$.

5. A scaling is applied to the MC simulated CC neutrino events with $E_{\text{true}}>500$ GeV, and NC neutrino events with $E_{\mathit{true}}>100$ GeV, to account for the use of different software packages at the light propagation level of MC simulations.

The resulting systematic uncertainties are summarised in Table 3. Contributions are considered independent and summed in quadrature.

Table 3. Uncertainties for each systematic source category

	${log}_{10}(E_{\nu }/\text{GeV})$
Systematic	0–0.7	0.7–1.3	1.3–2.0
Detector response	$_{-22\%}^{+4\%}$	$_{-21\%}^{+19\%}$	$_{-10\%}^{+15\%}$
Hadronisation in EAS	$_{-13\%}^{+14\%}$	$\pm 10\%$	$\pm 9\%$
Neutrino cross section	$\pm 5\%$	$\pm 3\%$	$_{-4\%}^{+3\%}$
Unfolding	$_{-18\%}^{+21\%}$	$_{-7\%}^{+8\%}$	$_{-22\%}^{+18\%}$
Light simulation	$+3\%$	$-3\%$	$-13\%$

Results

The measured flux values and the statistical and systematic uncertainties are presented in Table 4. The overall systematic uncertainty for each bin is extracted considering all systematic uncertainty sources presented in Sect. 6 as uncorrelated.

Table 4. Final results on the atmospheric neutrino flux measurement with six DUs of KM3NeT/ORCA. The content of the rows is the following: energy range, energy weighted bin centre, measured flux multiplied by the squared energy weighted bin center, measured in $\text{GeV}\times {\text{s}}^{-1}\times {\text{sr}}^{-1}\times {\text{cm}}^{-2}$, statistical uncertainty, systematic uncertainty

${log}_{10}(E_{\nu }/\text{GeV})$	0.0–0.7	0.7–1.3	1.3–2.0
${log}_{10}({\tilde{E}}_i/\text{GeV})$	0.41	0.87	1.50
${\tilde{E}}_i^2{\mathrm{\Phi }}_i$	$1.29·{10}^{-2}$	$4.49·{10}^{-3}$	$1.83·{10}^{-3}$
Stat. error	$\pm 16\%$	$\pm 6\%$	$\pm 5\%$
Syst. error	$_{-32\%}^{+26\%}$	$_{-25\%}^{+23\%}$	$_{-29\%}^{+25\%}$

The measured atmospheric ${\nu }_{\mu }+{\overline{\nu }}_{\mu }$ flux is presented in Fig. 7, superimposed on the atmospheric ${\nu }_{\mu }+{\overline{\nu }}_{\mu }$ flux predicted by the HKKM14 model, according to Eq. 5. The error bars represent the quadrature sum of the statistical uncertainty and the overall systematic uncertainty for each bin.

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Fig. 7

Atmospheric ${\nu }_{\mu }+{\overline{\nu }}_{\mu }$ flux measured with ORCA6, multiplied by ${\tilde{E}}^2$ and superimposed on the atmospheric ${\nu }_{\mu }+{\overline{\nu }}_{\mu }$ flux predicted by the HKKM14 model

The atmospheric ${\nu }_{\mu }+{\overline{\nu }}_{\mu }$ flux measured with ORCA6 is presented in Fig. 8, compared to measurements from other experiments, namely from Super-Kamiokande [9], ANTARES [10], and Frejus [6]. The measurement by Frejus was performed in 1995, before the discovery of neutrino oscillations. For $E_{\nu }>100$ GeV, several measurements have been performed by neutrino telescopes such as AMANDA [11], IceCube [12, 13–14], and ANTARES [15], not depicted in the plot.

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Fig. 8

The atmospheric muon neutrino flux measurement with ORCA6 compared to measurements from other experiments [6, 9, 10]

Conclusions

The ${\nu }_{\mu }+{\overline{\nu }}_{\mu }$ flux measured with the ORCA6 configuration is in agreement with the theoretical model, as well as with the experimental results of Super-Kamiokande. For $0.7<{log}_{10}(E_{\nu }/\text{GeV})<1.3$, the measured value is about $20\%$ lower with respect to the HKKM14 flux, although within errors. A similar trend is observed in $0.6<{log}_{10}(E_{\nu }/\text{GeV})<1.0$ by Super-Kamiokande. This will be further investigated with the ORCA detector, as additional DUs have been deployed and are taking data.

Acknowledgements

The authors acknowledge the financial support of: KM3NeT-INFRADEV2 project, funded by the European Union Horizon Europe Research and Innovation Programme under grant agreement No 101079679; Funds for Scientific Research (FRS-FNRS), Francqui foundation, BAEF foundation. Czech Science Foundation (GAČR 24-12702S); Agence Nationale de la Recherche (contract ANR-15-CE31-0020), Centre National de la Recherche Scientifique (CNRS), Commission Européenne (FEDER fund and Marie Curie Program), LabEx UnivEarthS (ANR-10-LABX-0023 and ANR-18-IDEX-0001), Paris Île-de-France Region, Normandy Region (Alpha, Blue-waves and Neptune), France, The Provence-Alpes-Côte d’Azur Delegation for Research and Innovation (DRARI), the Provence-Alpes-Côte d’Azur region, the Bouches-du-Rhône Departmental Council, the Metropolis of Aix-Marseille Provence and the City of Marseille through the CPER 2021-2027 NEUMED project, The CNRS Institut National de Physique Nucléaire et de Physique des Particules (IN2P3); Shota Rustaveli National Science Foundation of Georgia (SRNSFG, FR-22-13708), Georgia; This work is part of the MuSES project which has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 Research and Innovation Programme (grant agreement No 101142396). This work was supported by the European Research Council, ERC Starting grant MessMapp, under contract no. 949555. The General Secretariat of Research and Innovation (GSRI), Greece; Istituto Nazionale di Fisica Nucleare (INFN) and Ministero dell’Unoversità e della Ricerca (MUR), through PRIN 2022 program (Grant PANTHEON 2022E2J4RK, Next Generation EU) and PON R&I program (Avviso n. 424 del 28 febbraio 2018, Progetto PACK-PIR01 00021), Italy; IDMAR project Po-Fesr Sicilian Region az. 1.5.1; A. De Benedittis, W. Idrissi Ibnsalih, M. Bendahman, A. Nayerhoda, G. Papalashvili, I. C. Rea, A. Simonelli have been supported by the Italian Ministero dell’Università e della Ricerca (MUR), Progetto CIR01 00021 (Avviso n. 2595 del 24 dicembre 2019); KM3NeT4RR MUR Project National Recovery and Resilience Plan (NRRP), Mission 4 Component 2 Investment 3.1, Funded by the European Union – NextGenerationEU,CUP I57G21000040001, Concession Decree MUR No. n. Prot. 123 del 21/06/2022; Ministry of Higher Education, Scientific Research and Innovation, Morocco, and the Arab Fund for Economic and Social Development, Kuwait; Nederlandse organisatie voor Wetenschappelijk Onderzoek (NWO), the Netherlands; The grant ”AstroCeNT: Particle Astrophysics Science and Technology Centre”, carried out within the International Research Agendas programme of the Foundation for Polish Science financed by the European Union under the European Regional Development fund; The program: ’Excellence initiative-research university’ for the AGH University in Krakow; The ARTIQ project: UMO-2021/01/2/ST6/00004 and ARTIQ/ 0004/2021; Ministry of Research, Innovation and Digitalisation, Romania; Slovak Research and Development Agency under Contract No. APVV-22-0413; Ministry of Education, Research, Development and Youth of the Slovak Republic; MCIN for PID2021-124591NB-C41, -C42, -C43 and PDC2023-145913-I00 funded by MCIN/AEI/ 10.13039/501100011033 and by “ERDF A way of making Europe”, for ASFAE/2022/014 and ASFAE/2022 /023 with funding from the EU NextGenerationEU (PRTR-C17.I01) and Generalitat Valenciana, for Grant AST22_6.2 with funding from Consejería de Universidad, Investigación e Innovación and Gobierno de España and European Union - NextGenerationEU, for CSIC-INFRA23013 and for CNS2023-144099, Generalitat Valenciana for CIDEGENT/2018/034, /2019/043, /2020/049, /2021/23, for CIDEIG/2023/20, for CIPROM/2023/51 and for GRISOLIAP/2021/192 and EU for MSC/101025085, Spain; Khalifa University internal grants (ESIG-2023-008, RIG-2023-070 and RIG-2024-047), United Arab Emirates; The European Union’s Horizon 2020 Research and Innovation Programme (ChETEC-INFRA - Project no. 101008324).

Data Availability Statement

Data will be made available on reasonable request. [Authors’ comment: The datasets generated during and/or analysed during the current study are available from the corresponding author on reasonable request.].

Code Availability Statement

Code/software will be made available on reasonable request. [Authors’ comment: The code/software generated during and/or analysed during the current study is available from the corresponding author on reasonable request.].

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