Measurement of the atmospheric ν_e and ν_μ energy spectra with the ANTARES neutrino telescope
A. Albert, S. Alves, M. André, M. Anghinolfi, G. Anton, M. Ardid, J.-J. Aubert, J. Aublin, B. Baret, S. Basa, B. Belhorma, M. Bendahman, V. Bertin, S. Biagi, M. Bissinger, J. Boumaaza, M. Bouta, M.C. Bouwhuis, H. Brânza?, R. Bruijn, J. Brunner, J. Busto, A. Capone, L. Caramete, J. Carr, V. Carretero, S. Celli, M. Chabab, T. N. Chau, R. Cherkaoui El Moursli, T. Chiarusi, M. Circella, A. Coleiro, M. Colomer-Molla, R. Coniglione, P. Coyle, A. Creusot, A. F. Díaz, G. de Wasseige, A. Deschamps, C. Distefano, I. Di Palma, A. Domi, C. Donzaud, D. Dornic, D. Drouhin, T. Eberl, N. El Khayati, A. Enzenhöfer, P. Fermani, G. Ferrara, F. Filippini, L. Fusco, R. García, Y. Gatelet, P. Gay, H. Glotin, R. Gozzini, K. Graf, C. Guidi, S. Hallmann, H. van Haren, A.J. Heijboer, Y. Hello, J.J. Hernández-Rey, J. Hö?l, J. Hofestädt, F. Huang, G. Illuminati, C.W. James, B. Jisse-Jung, M. de Jong, P. de Jong, M. Jongen, M. Kadler, O. Kalekin, U. Katz, N.R. Khan-Chowdhury, A. Kouchner, I. Kreykenbohm, V. Kulikovskiy, R. Lahmann, R. Le Breton, D. Lefèvre, E. Leonora, G. Levi, M. Lincetto, D. Lopez-Coto, S. Loucatos, L. Maderer, J. Manczak, M. Marcelin, A. Margiotta, A. Marinelli, J.A. Martínez-Mora, K. Melis, P. Migliozzi, M. Moser, A. Moussa, R. Muller, et al. (42 additional authors not shown)
MMeasurement of the atmospheric ν e and ν µ energy spectra with theANTARES neutrino telescope A. Albert a,b , S. Alves s , M. Andr´e c , M. Anghinolfi d , G. Anton e , M. Ardid f , J.-J. Aubert g ,J. Aublin h , B. Baret h , S. Basa i , B. Belhorma j , M. Bendahman k,h , V. Bertin g , S. Biagi l ,M. Bissinger e , J. Boumaaza k , M. Bouta m , M.C. Bouwhuis n , H. Brˆanza¸s o , R. Bruijn n,p ,J. Brunner g , J. Busto g , A. Capone q,r , L. Caramete o , J. Carr g , V. Carretero s , S. Celli q,r ,M. Chabab t , T. N. Chau h , R. Cherkaoui El Moursli k , T. Chiarusi u , M. Circella v , A. Coleiro h ,M. Colomer-Molla h,s , R. Coniglione l , P. Coyle g , A. Creusot h , A. F. D´ıaz w , G. de Wasseige h ,A. Deschamps x , C. Distefano l , I. Di Palma q,r , A. Domi d,y , C. Donzaud h,z , D. Dornic g ,D. Drouhin a,b , T. Eberl e , N. El Khayati k , A. Enzenh¨ofer g , P. Fermani q,r , G. Ferrara l ,F. Filippini u,aa , L. Fusco h,g , R. Garc´ıa n , Y. Gatelet h , P. Gay ab,h , H. Glotin ac , R. Gozzini s,e ,K. Graf e , C. Guidi d,y , S. Hallmann e , H. van Haren ad , A.J. Heijboer n , Y. Hello x ,J.J. Hern´andez-Rey s , J. H¨oßl e , J. Hofest¨adt e , F. Huang a , G. Illuminati s,h , C.W. James ae ,B. Jisse-Jung n , M. de Jong n,af , P. de Jong n , M. Jongen n , M. Kadler ag , O. Kalekin e , U. Katz e ,N.R. Khan-Chowdhury s , A. Kouchner h , I. Kreykenbohm ai , V. Kulikovskiy d,aj , R. Lahmann e ,R. Le Breton h , D. Lef`evre ak , E. Leonora al , G. Levi u,aa , M. Lincetto g , D. Lopez-Coto am ,S. Loucatos an,h , L. Maderer h , J. Manczak s , M. Marcelin i , A. Margiotta u,aa , A. Marinelli ao ,J.A. Mart´ınez-Mora f , K. Melis n,p , P. Migliozzi ao , M. Moser e , A. Moussa m , R. Muller n , L. Nauta n ,S. Navas am , E. Nezri i , A. Nu˜nez-Casti˜neyra g,i , B. O’Fearraigh n , M. Organokov a , G.E. P˘av˘ala¸s o ,C. Pellegrino u,ap,aq , M. Perrin-Terrin g , P. Piattelli l , C. Pieterse s , C. Poir`e f , V. Popa o , T. Pradier a ,N. Randazzo al , S. Reck e , G. Riccobene l , F. Salesa Greus s , D.F.E. Samtleben n,af ,A. S´anchez-Losa v , M. Sanguineti d,y , P. Sapienza l , J. Schnabel e , J. Schumann e , F. Sch¨ussler an ,M. Spurio u,aa, ∗ , Th. Stolarczyk an , M. Taiuti d,y , Y. Tayalati k , T. Thakore s , S.J. Tingay ae ,B. Vallage an,h , V. Van Elewyck h,ah , F. Versari u,aa,h, ∗ , S. Viola l , D. Vivolo ao,ar , J. Wilms ai ,A. Zegarelli q,r , J.D. Zornoza s , J. Z´u˜niga s , (The ANTARES Collaboration) a Universit´e de Strasbourg, CNRS, IPHC UMR 7178, F-67000 Strasbourg, France b Universit´e de Haute Alsace, F-68200 Mulhouse, France c Technical University of Catalonia, Laboratory of Applied Bioacoustics, Rambla Exposici´o, 08800 Vilanova i la Geltr´u,Barcelona, Spain d INFN - Sezione di Genova, Via Dodecaneso 33, 16146 Genova, Italy e Friedrich-Alexander-Universit¨at Erlangen-N¨urnberg, Erlangen Centre for Astroparticle Physics, Erwin-Rommel-Str. 1,91058 Erlangen, Germany f Institut d’Investigaci´o per a la Gesti´o Integrada de les Zones Costaneres (IGIC) - Universitat Polit`ecnica de Val`encia. C/Paranimf 1, 46730 Gandia, Spain g Aix Marseille Univ, CNRS/IN2P3, CPPM, Marseille, France h Universit´e de Paris, CNRS, Astroparticule et Cosmologie, F-75006 Paris, France i Aix Marseille Univ, CNRS, CNES, LAM, Marseille, France j National Center for Energy Sciences and Nuclear Techniques, B.P.1382, R. P.10001 Rabat, Morocco k University Mohammed V in Rabat, Faculty of Sciences, 4 av. Ibn Battouta, B.P. 1014, R.P. 10000 Rabat, Morocco l INFN - Laboratori Nazionali del Sud (LNS), Via S. Sofia 62, 95123 Catania, Italy m University Mohammed I, Laboratory of Physics of Matter and Radiations, B.P.717, Oujda 6000, Morocco n Nikhef, Science Park, Amsterdam, The Netherlands o Institute of Space Science, RO-077125 Bucharest, M˘agurele, Romania p Universiteit van Amsterdam, Instituut voor Hoge-Energie Fysica, Science Park 105, 1098 XG Amsterdam, The Netherlands q INFN - Sezione di Roma, P.le Aldo Moro 2, 00185 Roma, Italy r Dipartimento di Fisica dell’Universit`a La Sapienza, P.le Aldo Moro 2, 00185 Roma, Italy s IFIC - Instituto de F´ısica Corpuscular (CSIC - Universitat de Val`encia) c/ Catedr´atico Jos´e Beltr´an, 2 E-46980 Paterna,Valencia, Spain t LPHEA, Faculty of Science - Semlali, Cadi Ayyad University, P.O.B. 2390, Marrakech, Morocco. u INFN - Sezione di Bologna, Viale Berti-Pichat 6/2, 40127 Bologna, Italy v INFN - Sezione di Bari, Via E. Orabona 4, 70126 Bari, Italy a r X i v : . [ h e p - e x ] J a n Department of Computer Architecture and Technology/CITIC, University of Granada, 18071 Granada, Spain x G´eoazur, UCA, CNRS, IRD, Observatoire de la Cˆote d’Azur, Sophia Antipolis, France y Dipartimento di Fisica dell’Universit`a, Via Dodecaneso 33, 16146 Genova, Italy z Universit´e Paris-Sud, 91405 Orsay Cedex, France aa Dipartimento di Fisica e Astronomia dell’Universit`a, Viale Berti Pichat 6/2, 40127 Bologna, Italy ab Laboratoire de Physique Corpusculaire, Clermont Universit´e, Universit´e Blaise Pascal, CNRS/IN2P3, BP 10448, F-63000Clermont-Ferrand, France ac LIS, UMR Universit´e de Toulon, Aix Marseille Universit´e, CNRS, 83041 Toulon, France ad Royal Netherlands Institute for Sea Research (NIOZ), Landsdiep 4, 1797 SZ ’t Horntje (Texel), the Netherlands ae International Centre for Radio Astronomy Research - Curtin University, Bentley, WA 6102, Australia af Huygens-Kamerlingh Onnes Laboratorium, Universiteit Leiden, The Netherlands ag Institut f¨ur Theoretische Physik und Astrophysik, Universit¨at W¨urzburg, Emil-Fischer Str. 31, 97074 W¨urzburg, Germany ah Institut Universitaire de France, 75005 Paris, France ai Dr. Remeis-Sternwarte and ECAP, Friedrich-Alexander-Universit¨at Erlangen-N¨urnberg, Sternwartstr. 7, 96049 Bamberg,Germany aj Moscow State University, Skobeltsyn Institute of Nuclear Physics, Leninskie gory, 119991 Moscow, Russia ak Mediterranean Institute of Oceanography (MIO), Aix-Marseille University, 13288, Marseille, Cedex 9, France; Universit´edu Sud Toulon-Var, CNRS-INSU/IRD UM 110, 83957, La Garde Cedex, France al INFN - Sezione di Catania, Via S. Sofia 64, 95123 Catania, Italy am Dpto. de F´ısica Te´orica y del Cosmos & C.A.F.P.E., University of Granada, 18071 Granada, Spain an IRFU, CEA, Universit´e Paris-Saclay, F-91191 Gif-sur-Yvette, France ao INFN - Sezione di Napoli, Via Cintia 80126 Napoli, Italy ap Museo Storico della Fisica e Centro Studi e Ricerche Enrico Fermi, Piazza del Viminale 1, 00184, Roma aq INFN - CNAF, Viale C. Berti Pichat 6/2, 40127, Bologna ar Dipartimento di Fisica dell’Universit`a Federico II di Napoli, Via Cintia 80126, Napoli, Italy
Abstract
This letter presents a combined measurement of the energy spectra of atmospheric ν e and ν µ inthe energy range between ∼
100 GeV and ∼
50 TeV with the ANTARES neutrino telescope. Theanalysis uses 3012 days of detector livetime in the period 2007–2017, and selects 1016 neutrinosinteracting in (or close to) the instrumented volume of the detector, yielding shower-like events(mainly from ν e + ν e charged current plus all neutrino neutral current interactions) and startingtrack events (mainly from ν µ + ν µ charged current interactions). The contamination by atmosphericmuons in the final sample is suppressed at the level of a few per mill by different steps in the selectionanalysis, including a Boosted Decision Tree classifier. The distribution of reconstructed events isunfolded in terms of electron and muon neutrino fluxes. The derived energy spectra are comparedwith previous measurements that, above 100 GeV, are limited to experiments in polar ice and, for ν µ , to Super-Kamiokande. Keywords:
Neutrino telescope, Atmospheric neutrinos, ANTARES ∗ Corresponding authors
January 29, 2021 . Introduction
Atmospheric neutrinos are secondary particles produced by cosmic ray (CR) primaries interact-ing in the Earth’s atmosphere. Due to the need of very large detectors, only a few measurementsof the differential flux exist, namely from the AMANDA [1, 2], IceCube [3, 4, 5, 6, 7] and Super-Kamiokande [8] Collaborations, and a historical measurement from the Frejus Collaboration [9].The ANTARES Collaboration has reported a measurement of the atmospheric ν µ energy spectrumin [10].Different theoretical frameworks are available to estimate atmospheric neutrino fluxes [11, 12,13, 14]. At energies from 100 GeV to 1 PeV, the main source of ν µ are semi-leptonic and three-bodydecays of charged kaons, while the contributions from pion and muon decays dominate below 100GeV. This conventional neutrino flux tends towards a power law Φ cν ∝ E − γ CR − ν , where γ CR is thespectral index of the primary CRs.Above 100 GeV and up to some tens of TeV, atmospheric ν e ’s come mostly from decays ofneutral and charged kaons, and have the same spectral index of conventional ν µ . Below 100 GeV, ν e are predominantly produced by muon decays. The ν µ /ν e flux ratio is ∼ ∼
20 at 1 TeV.At high energies, equal fluxes of ν µ and ν e are produced by the decays of charged and neutralD-mesons. Because of the very short lifetime of these mesons, the resulting flux is called prompt neutrino flux [15, 16] and its energy spectrum, Φ pν ∝ E − γ CR ν , follows the primary spectrum up tovery high energies. The transition from the region in the spectrum dominated by conventionalneutrinos to prompt neutrinos is expected to occur at E ν ∼ ν µ and around E ν ∼
30 TeVfor ν e . As a rule of thumb, the primary CR energy is about 20 times higher than the energy ofthe secondary neutrino. Uncertainties on the conventional flux models at neutrino energies above1 TeV are mainly due to a poor knowledge of primary CR energy spectrum and composition, andof hadronic interactions, in particular of strange quark production mechanisms [11]. For a recent,detailed description of the hadronic interactions leading to inclusive lepton fluxes, refer to [17].Finally, ν τ production in the atmosphere is rare: this is dominated by the decay D + s → τ + ν τ ,followed by τ decay. As oscillation effects for atmospheric ν ’s are negligible above ∼
100 GeV, thecontribution from tau neutrinos is not considered in this analysis.This letter describes a strategy to select shower-like and starting track events ( §
2) over thebackground of atmospheric muons ( § § ν µ and ν e , taking into account the detectoracceptance ( § § ν e + ν e and of ν µ + ν µ , averaged over the zenith region 90 ◦ –180 ◦ .
2. The ANTARES detector and neutrino reconstruction algorithms
The ANTARES telescope [18] is a deep-sea Cherenkov neutrino detector, located 40 km offshoreToulon, France, in the Mediterranean Sea. The detector comprises a three dimensional array of885 optical modules [19], each one housing a 10-inch photomultiplier tube (PMT). The opticalmodules are distributed along 12 vertical strings anchored to the sea floor at distances of about70 m from each other and at a depth of about 2500 m. The detection of light from upwardgoing charged particles is optimised with the PMTs facing 45 ◦ downward. Particles above theCherenkov threshold induce a coherent radiation emitted in a cone with a characteristic angle3 C (cid:39) ◦ in water. For high-energy muons ( E µ > E µ , and the resulting electromagnetic showersproduce additional light. Completed in 2008, the telescope aims primarily at the detection ofneutrino-induced through-going muons.The signals induced in the PMT by detected photons are referred to as hits [20]. The position,time, and collected charge of the hits are used to reconstruct the direction and energy of eventsinduced by neutrino interactions and atmospheric muons. Trigger conditions based on combinationsof local coincidences are applied to identify signals due to physics events over the environmentallight background due to K decays and bioluminescence [21]. For astronomy studies, atmosphericmuons and atmospheric neutrinos constitute the main source of background.This analysis focuses on events induced by neutrinos whose interaction vertices are containedinside (or close to) the instrumented detector volume. These events include: • ν e charged current (CC) interactions producing electromagnetic and hadronic cascades, andneutral current (NC) interactions of neutrinos of all flavors inducing hadronic cascades. Due to theradiation and nuclear interaction lengths in water, the cascades extend up to a maximum distanceof ∼
10 m from the interaction vertex, much shorter than the distance between detector strings.These events are thus almost point-like at the scale of the detector and are referred to as shower-like events in the following. • ν µ CC interactions, with a hadronic cascade near the vertex and a starting muon. Most ofthese muons are minimum ionising particles, and can travel in water about 4 m per GeV of energy,inducing Cherenkov light over large distances with respect to the interaction vertex position. Theseevents with a cascade and a track are referred to as starting track events in the following.The track reconstruction algorithm used in off-line ANTARES analyses is called AAFit [22]and it is based on a likelihood fit that exploits a detailed parametrisation of the probability densityfunction for the time of the hits. The algorithm provides the track direction with its estimatedangular uncertainty; the number of hits used for the reconstruction; and a quality parameter,referred to as Λ. The track direction and the Λ quality parameter are used to remove the largestfraction of atmospheric muons, which are downward going [23].An auxiliary algorithm denoted as GridFit [24] searches for tracks in 500 different directionscovering the full solid angle. The number of hits compatible with a muon track coming fromeach direction is evaluated and a likelihood fit is performed. The GridFit method defines qualityparameters for reconstructed tracks that are usually used in ANTARES studies to improve therejection of downward going atmospheric muon events.Shower-like events occurring in the proximity of the instrumented volume are reconstructedwith a dedicated algorithm, TANTRA [25]; this likelihood-based method allows to reconstruct thevertex coordinates, the neutrino direction, and the neutrino energy. A parameter associated to thequality of the reconstructed event, M est , is also provided.All ANTARES analyses follow a blinding policy to avoid possible biases. The cuts and theselection criteria are studied and optimised on a sample of Monte Carlo (MC) simulated eventsand only at the end of the full selection chain, these cuts are applied to data. A small samplecontaining 10% of the real data uniformly distributed over livetime is used to verify the agreementwith MC events along the selection.The simulation chain [26] starts with the generation of the event and comprises the generationof Cherenkov light, the inclusion of the environmental optical background extracted from real data,and the digitisation of the PMT signals following a run-by-run strategy. This strategy accounts4or seasonal variations related to biological activities and for inefficiencies due to the ageing of thePMTs and to biofouling [27].At the end of the full simulation chain, a set of MC files is available for each run of real data,stored in the same format. Simulated files are processed with the same reconstruction algorithmsand analysis procedures used for the corresponding data. Monte Carlo neutrino events have beengenerated in the energy range 10 ≤ E ν ≤ GeV, separately for ν e , ν µ and their antineutrinos,and for CC and NC processes. Details on the simulation chain, hadronic model for cross sections,interaction kinematics and parton distribution functions are given in [26]. The same MC sample canbe differently weighted to reproduce the conventional atmospheric neutrinos, the prompt neutrinosand theoretical astrophysical signals. In the present letter, the atmospheric ν e and ν µ fluxes arerepresented with the same models used in [28], namely, the conventional component follows thespectrum described in [12], extrapolated at higher energy as in [5], and the prompt contributionas calculated in [15]. The MC statistics for the atmospheric neutrino sample corresponds to morethan two orders of magnitude than for real data.Finally, for each data run, a file with simulated atmospheric muons (CR µ ) is produced withthe MUPAGE package [29, 30]; in this case, the equivalent MC livetime corresponds to 1/3 of thereal run livetime.
3. Event selection: signal and background
Data collected from 2007 until the end of 2017 have been used. Only runs without high biolumi-nescence level have been selected. The total livetime corresponds to 3012 days. The background isalmost entirely due to CR µ ’s: after trigger and reconstruction, the expected signal-to-backgroundrate is ∼ − . The background suppression is organised in three different steps.An initial preselection of shower-like and starting track events combines information from boththe track (AAFit) and shower (TANTRA) reconstruction algorithms, according to the followingrequirements: i ) the direction of the event as reconstructed by AAFit must be upward going (i.e.,zenith angle > ◦ ), to reject the largest fraction of CR µ ’s; ii ) the TANTRA’s reconstructedneutrino interaction vertex must be contained in a cylindrical volume of axial radius of 300 mand height of 500 m, centred at the centre-of-gravity of the detector modules; iii ) the TANTRAestimated angular uncertainty on the event direction must be < ◦ and the quality parameter M est < > − . ν µ spectrum [10]. After this preselection,the MC signal is reduced by a factor of two with respect to the trigger and reconstruction level,with ∼
350 survived CR µ ’s for each atmospheric neutrino candidate.The second step, following [10, 23, 31], uses the AAFit quality parameter, Λ. The best com-promise to suppress the largest percentage of background while keeping a large enough fractionof signal events is obtained by removing events with Λ ≤ − .
7. After this cut, 25% of the signalsurvives, with about 30 remaining background events for each atmospheric neutrino. Table 1 sum-marises the number of events passing the preselection and the Λ cut for each MC sample. The lastrow shows the events in the data sample, after unblinding.The final classification of events as signal or background is performed with a Boosted DecisionTree (BDT), defined on a multidimensional parameter space. A BDT is an algorithm that belongsto the family of supervised machine learning techniques. To build the classification function,5 reselection +BDT > .
33+ Λ > − . µ ∼ ν e CC 242 96Atmospheric ν e NC 22 9Atmospheric ν µ CC 3780 620Atmospheric ν µ NC 400 180Cosmic ν Table 1:
Number of events in different Monte Carlo samples surviving the preselection and the cut on trackquality parameter Λ (second column) and the final BDT cut (third column). The last row shows the numberof data events in 3012 days of livetime. The expectation for cosmic neutrinos is computed assuming thediffuse flux presented in [28]. training samples are necessary. CR µ events generated with MUPAGE constitute the backgroundsample; CC+NC interactions of atmospheric ν e are used as signal. This choice is motivated by thefact that the ν e flavor produces the cleanest case of shower-like events and it is the most difficultchannel to measure in neutrino telescopes.For each CR µ or ν e event, the classifier is trained using the following 15 quantities. From theTANTRA shower algorithm, the reconstructed 1) zenith angle and 2) azimuth angle in the localreference frame; 3) interaction vertex coordinates; 4) quality parameter estimator, M est ; 5) numberof detector lines with at least one hit; 6) total number of hits used to reconstruct the event; 7)angular resolution associated to a shower-like event. From the AAFit track-like algorithm, thereconstructed 8) zenith angle and 9) azimuth angle in the local reference frame; 10) track lengthinside the detector volume; 11) quality parameter estimator, Λ; 12) angular resolution associated toa track-like event. From the GridFit track-like algorithm, 13) the quality parameter; 14) the CR µ veto parameter, a likelihood variable based on time sequence and charge of the hits in differentstoreys of the detector, causally-connected under the assumption of a downward going, minimumionizing particle; 15) the number of on-time hits, which assumes that the photons are produced atthe Cherenkov angle and arrive at the PMT unscattered.A ranking of the BDT input variables is derived by counting how often each variable is usedto split decision tree nodes, and by weighting each split occurrence by its squared separation gainand by the number of events in the node [32]. None of the variables is found to be significantlydominant; the variable with the highest ranking is the TANTRA zenith angle (1) with score 0.12,followed with the GridFit quality (13) with score 0.10; in the last two positions, the estimators ofthe angular resolution from TANTRA (8), with score 0.04, and that from AAFit (12), with score0.023.As shown in Fig. 1, the BDT output is an excellent discriminator between events in theatmospheric ν e and background CR µ samples. The BDT distribution obtained from events inducedby atmospheric ν µ CC+NC interactions is also included in the plot. As expected, this distributionresembles that of the ν e signal. The BDT condition > µ ’s present in run-by-run MC events is chosen as selection criteria. An extrapolation of the BDT distribution tail,assuming a Gaussian shape, yields a conservative extrapolation of (at most) ∼ igure 1: BDT output for events passing the preselection + Λ cut. The histograms correspond to different MCsamples: training CR µ (green), training atmospheric ν e (red), atmospheric ν µ (blue). The ν µ events are not usedfor BDT training. The green line corresponds to a Gaussian extrapolation of the CR µ histogram. Both ν e and ν µ fluxes include conventional [12] and prompt neutrinos [15]. The magenta histogram is the expected contribution fromdiffuse cosmic ν ’s, as parameterised in [28]. The orange histogram is the sum of all MC contributions and the blackcrosses are real data (3012 days livetime), after unblinding. The BDT cut value is denoted with a black arrow. expected, the neutrino sample is still dominated by atmospheric ν µ producing starting tracks: only ∼
10% of the selected events originate from ν e . At 1 TeV, the expected flux ratio is Φ ν µ / Φ ν e ∼
4. Unfolding procedure and detector acceptance
In order to derive from data the ν e and ν µ energy spectra, an unfolding method is used. Thetwo true distributions are deconvolved from the experimentally measured one, based on the bestknowledge of the detector and on assumptions made on the interaction rates of the different neutrinoflavors.In counting experiments, events are grouped into certain regions of phase-space, called bins .The main observable quantities in neutrino telescopes are the neutrino direction and energy, whichare measured only with finite precision due to inevitable detector effects. Consequently, an eventmay be assigned to a wrong bin. In addition, due to the presence of background, only a fractionof the events observed in a given bin originates from the searched signal.The outcome of the unfolding procedure, folded with the detector acceptance and livetime,results in a spectrum that allows a direct comparison with other experiments. Two major classesof unfolding methods exist: algorithms based on matrix inversion or singular value decomposition,such as the TUnfold [33] algorithm used in this analysis; algorithms based on iterative methodsor on the use of Bayes’ theorem [34]. A Bayesian approach has been used, e.g., by the Super-Kamiokande experiment [8] and in our previous measurement of the ν µ energy spectrum usingthrough-going muons [10]. For an overview of the commonly used unfolding algorithms, see also[35, 36].The TUnfold algorithm [33] is a widely tested and validated algorithm in the context of high-energy physics and it can handle one or more background sources. The algorithm allows to estimatethe number of events in m bins of a true distribution x j , given an observed distribution of y i in n y i = m (cid:88) j =1 A ij · x j + b i , ≤ i ≤ n , (1)where each bin has a background contribution b i . A ij is a matrix of probabilities describing themigrations from bin j to any of the n bins. The method, interfaced to the ROOT analysis package[37], uses a least square method with Tikhonov regularisation [38] and an optional constraint to fixthe total number of events. The least square minimisation requires a number of degrees of freedomsuch that n − m >
0, meaning that the data y i have to be measured in finer bins than extractedby the unfolding procedure.The energy estimated by the TANTRA reconstruction algorithm, E reco , is used to construct thedistribution of y i . Events in Fig. 1 with BDT > .
33 are atmospheric ν µ or ν e , with a contaminationof less than a few per mill from CR µ and a ∼
1% fraction of cosmic neutrinos; both samples areconsidered as background. The unfolding method requires a ( n × m ) matrix for the ν µ and ν e energies, with n bins of E reco and m bins of true energy E ν . Monte Carlo samples allow theconstruction of: • A eij , a (6 ×
3) matrix obtained with the simulated samples of ν e CC+NC interactions; • A µij , a (15 ×
5) matrix obtained with the simulated samples of ν µ CC+NC interactions.The chosen number of bins in ( E reco , E ν ) for the two samples provides the highest stability interms of unfolding results applied on MC samples with the same number of events as real data. Inthe unfolding procedure, the use of E reco is limited to energies between ∼
100 GeV and ∼
50 TeV.The lower bound is determined by the fact that our reconstruction algorithm cannot reliablyreconstruct neutrino energies below 100 GeV. Above 50 TeV, the event statistics are significantlyreduced by the requirement of the containment of the interaction vertex within, or near to, theinstrumented volume. In addition, cosmic neutrinos, whose flux suffers large uncertainties, startto be the dominant “background”.Figure 2 shows the distribution of E reco for the ν µ sample (blue) with the bin size used forthe construction of the A µij matrix. For completeness, the distribution of E reco for the ν e sample(red) is superimposed, although the distribution used for the construction of the A eij matrix hasa different binning. The expected contribution in the sample of cosmic ν ’s of all flavors is alsoshown.Concerning the background terms, it includes an extrapolated contribution of 3 track-like CR µ events. Based on the behaviour of atmospheric muons before the BDT cut, the b CRµi terms areassumed to affect only the ν µ sample, and uniformly in the E reco range. The background fromthe cosmic neutrino flux (terms b ci ), assuming equipartition ( ν e : ν µ : ν τ ) = (1 : 1 : 1), contributesabout equivalently to the ν µ and ν e samples, following the E reco distribution shown in Fig. 2.The unfolding procedure is divided into two parts, whose structure is the same: when focusedon the ν µ distribution, the background corresponds to b CRµi , b ci / ν e fraction b ei . Whenfocused on the ν e distribution, the background corresponds to b ci / b µi . Thealgorithm assumes that (cid:80) i ( b µi + b ei + b CRµi + b ci ) is equal to the total number of events. The events b µi and b ei are assumed to be produced with the fluxes given in [12], as in the default Monte Carlosimulation, with free normalization. Possible variations in their spectral indexes are accounted forin the treatment of systematic effects (see § ν µ ( ν e ) sample.The first two columns contain the energy range of the corresponding bin and the weighted centralvalue of the neutrino energy bin, calculated taking into account the steep decrease of the energy8 igure 2: Distribution of E reco with the binning used for the construction of the response matrix for the ν µ sample (blue histogram). The red histogram, with the same binning, refers to the ν e sample. The magentahistogram is the expected contribution from a cosmic neutrino flux, as estimated in [28], while the orangehistogram includes the sum of all MC contributions. The black crosses correspond to real data. Events inthe shaded region are used for unfolding. spectrum and the detector response. The third column shows the unfolded number of data eventsas obtained by the algorithm.
5. The unfolded energy spectrum
To transform the unfolded number of events, N evt , given in Table 2 into a differential energyflux in the proper units (GeV − cm − s − sr − ), the following steps are required: i ) divide eachbin by the livetime of 3012 days, obtaining the event rate integrated in the log of the neutrinoenergy over the bin; ii ) divide by the width of the bin (0.54 for ν µ and 0.9 for ν e ); then, transformthe dN evt d log E ν distribution into the dN evt dE ν one; iii ) divide by the integrated value of the observationsolid angle, i.e., 2 π sr; iv ) divide by the detector effective area, A eff ( E ν ), averaged over the zenithangle of the atmospheric neutrino distribution.The effective area is the figure of merit for a neutrino telescope, representing the size of a 100%efficient hypothetical target that the detector offers to a certain simulated neutrino flux. It iscalculated as A eff ( E ν ) = N sel ( E ν ) N gen ( E ν ) · V gen · ρN A · σ ( E ν ) · P Earth ( E ν ) , (2)where N sel ( E ν ) and N gen ( E ν ) are, respectively, the number of selected and generated events of agiven neutrino energy E ν in the generation volume V gen ; ρ and N A are the matter density andthe Avogadro’s number; σ ( E ν ) is the neutrino cross section; P Earth ( E ν ) is the probability of theneutrino to traverse the Earth without being absorbed. Above 100 GeV, there are no correctionsneeded for oscillation effects. Figure 3 shows the effective area obtained from the selection of eventsdescribed in this work.The fourth column of Table 2 presents the differential flux obtained with the overall procedure.The reported statistical error is determined by the TUnfold method.9 log E ν log E ν N evt E ν Φ ν stat. syst.Atmospheric muon neutrinos2.00–2.54 2.32 232 2.4 × − ± ± × − ± ± × − ± ± × − ± ± × − ± ± × − ± ± × − ± ± × − − ± Table 2: The table shows, from the first to last column: ∆ log E ν = log E minν GeV –log E maxν GeV ; the weighted centre ofthe bin, log E ν = log (cid:104) E ν (cid:105) GeV ; the number of unfolded events assigned to the bin, N evt ; the differential flux (times E ν ) computed in the centre of the bin, E ν Φ ν , in units of GeV cm − s − sr − ; the statistical error; and the totalsystematic uncertainty. The result of the unfolding process depends on the MC simulation via the construction of theresponse matrix. In turn, the simulation depends on a number of parameters with associateduncertainties. The effects inducing systematic uncertainties on the measurement of the ν µ fluxusing through-going events have been extensively described in [10]; these also affect the presentanalysis. The impact is estimated by varying each time only one of the following parameters, andproducing dedicated simulation datasets: • Overall sensitivity of the optical modules, changed by +10% and − • The uncertainties on water properties, by scaling up and down by 10% the absorption length oflight in water with respect to the nominal value. • The uncertainties related to the neutrino fluxes used in the default response matrix of theunfolding procedures, including a slope change of ± . ν e and ν µ .Each modified MC sample was then used as pseudo-data to construct a new response matrix,used for unfolding. The deviation in the content of each E ν bin from the spectrum obtained withthe default response matrix, A eij or A µij , corresponds to the systematic uncertainty associated withthe parameter variation. For each energy bin, the total uncertainty is computed as the quadraticsum of each contribution, and the resulting value is reported in the last column of Table 2.
6. Results and conclusions
Figure 4 shows the ( ν e + ν e ) and ( ν µ + ν µ ) fluxes measured in this work, together with the resultsfrom previous experiments. Our unfolded atmospheric neutrino spectra, whose statistical errorsare largely dominant over the systematic ones, are about 20% below the most recent computationsusing the SIBYLL-2.3c hadronic interaction model [17, 39].The measurement of the electron neutrino flux at high energy is challenging, because verylarge detectors are needed to collect sufficient statistics, and due to large systematic uncertainties.10 igure 3: Effective area of the ANTARES neutrino telescope for the events with a vertex inside the instru-mented volume and selected by the analysis cuts described in this work: ν e CC+NC (red line), ν µ CC+NC(blue line). The black solid line is the sum of all the interaction channels and neutrino flavors.
Each measurement of IceCube-DeepCore [4] and IceCube [5] rely on about 200 interacting ν e in the polar ice medium. The present measurement is performed in seawater, under completelydifferent environmental conditions and systematic uncertainties, yielding consistent results withthe ones obtained in polar ice. During a livetime of 3012 days, about 130 ν e interactions have beenreconstructed within the instrumented ANTARES volume. The statistics of the ν e sample is notsufficient to test models above a few tens of TeV, where a significant cosmic flux is present and thetransition from the conventional to the prompt flux is expected. Below 100 GeV, the PMT densityof the ANTARES detector is insufficient to reconstruct a significant number of events.Concerning the unfolded ν µ flux, our previous measurement [10] with a sample of through-goingevents generated by neutrino interactions external to the instrumented volume almost superimposedthe SIBYLL-2.3c model. The present measurement is based on a totally independent data sample,provided by neutrinos whose reconstructed interaction vertex is inside (or nearby) the instrumentedvolume of the detector. The present unfolded flux is more close to that of IceCube with 40 lines [6]and is about 25% below both the flux reported in our previous measurement and the one reportedby IceCube using 59 strings [7], although consistent within errors. Acknowledgements
The authors acknowledge the financial support of the funding agencies: Centre Na-tional de la Recherche Scientifique (CNRS), Commissariat `a l’´energie atomique et aux ´energies alternatives(CEA), Commission Europ´eenne (FEDER fund and Marie Curie Program), Institut Universitaire de France(IUF), LabEx UnivEarthS (ANR-10-LABX-0023 and ANR-18-IDEX-0001), R´egion ˆIle-de-France (DIM-ACAV), R´egion Alsace (contrat CPER), R´egion Provence-Alpes-Cˆote d’Azur, D´epartement du Var andVille de La Seyne-sur-Mer, France; Bundesministerium f¨ur Bildung und Forschung (BMBF), Germany; Isti-tuto Nazionale di Fisica Nucleare (INFN), Italy; Nederlandse organisatie voor Wetenschappelijk Onderzoek(NWO), the Netherlands; Council of the President of the Russian Federation for young scientists and leadingscientific schools supporting grants, Russia; Executive Unit for Financing Higher Education, Research, De-velopment and Innovation (UEFISCDI), Romania; Ministerio de Ciencia e Innovaci´on (MCI) and Agencia igure 4: Measured energy spectra of the atmospheric ν e and ν µ using shower-like and starting track events inthe ANTARES neutrino telescope (black). The measurements by other experiments (Frejus [9], AMANDA-II [2], IceCube [6, 7, 4, 5], and Super-Kamiokande [8]), as well as the previous ν µ flux measurement using adifferent ANTARES data sample [10], are also reported. The vertical error bars include all statistical andsystematic uncertainties.Estatal de Investigaci´on: Programa Estatal de Generaci´on de Conocimiento (refs. PGC2018-096663-B-C41,-A-C42, -B-C43, -B-C44) (MCI/FEDER), Severo Ochoa Centre of Excellence and MultiDark Consolider,Junta de Andaluc´ıa (ref. SOMM17/6104/UGR and A-FQM-053-UGR18), Generalitat Valenciana: Grisol´ıa(ref. GRISOLIA/2018/119) and GenT (ref. CIDEGENT/2018/034) programs, Spain; Ministry of HigherEducation, Scientific Research and Professional Training, Morocco. We also acknowledge the technicalsupport of Ifremer, AIM and Foselev Marine for the sea operation and the CC-IN2P3 for the computingfacilities. References [1] A. Achterberg et al., Phys. Rev. D76, 042008 (2007).[2] R. Abbasi et al., Astropart. Phys. 34, 48 (2010).[3] M. G. Aartsen et al., Phys. Rev. D89, 102001 (2014).[4] M. G. Aartsen et al., Phys. Rev. Lett. 110, 151105 (2013).[5] M.G. Aartsen et al., Phys.Rev. D91, 122004, 2015.[6] R. Abbasi et al., Phys. Rev. D83, 012001 (2011).[7] M. G. Aartsen et al.,Eur. Phys. J. C75, 116 (2015).[8] E. Richard et al., Phys.Rev. D94, 052001 (2016).[9] K. Daum et al., Z. Phys. C66, 417 (1995).
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