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JHEP06(2019)113 f q,r x l c,y u,ac aq,ar f aa f o Springer j f P. Coyle, May 19, 2019 ag,ah June 21, 2019 g g Y. Hello, April 24, 2019 al : am f : u A. Domi, : l P. Fermani, aj R. Mele, December 21, 2018 n q,r e M. Spurio, G. Maggi, : 1 M.C. Bouwhuis, an f, Revised an,g g Accepted L. Caramete, Published B. Belhorma, d C. Nielsen, I. El Bojaddaini, J.-J. Aubert, D. Lef`evre, M. Perrin-Terrin, i T. Gr´egoire, i F. Fassi, C.W. James, q,r d T. Chiarusi, e Received g h h N. Randazzo, n n H. Costantini, A.J. Heijboer, u,ac I. Di Palma, at k af k I. Kreykenbohm, U. Katz, F. Sch¨ussler, S. Basa, S. Bourret, d I. Salvadori, E. Nezri, ab S. Loucatos, T. Eberl, k h n t M. Ardid, h a as A. Capone, d Published for SISSA by J.A. Mart´ınez-Mora, R. Gozzini, https://doi.org/10.1007/JHEP06(2019)113 A. Ettahiri, f C. Racca, R. Le Breton, G. Illuminati, ae f d,f d d C. Pellegrino, ao,ap R. Coniglione, o C. Distefano, M. Kreter, O. Kalekin, H. van Haren, M. Lotze, S. Navas, x g,h d G. Anton, P. Sapienza, ab J. Busto, f D. Drouhin, g,ai aa J. Boumaaza, c f f c,y H. Glotin, L. Quinn, J. Barrios-Mart´ı, A. Salah-Eddine, R. Cherkaoui El Moursli, l,m a g t v ad,g R. Lahmann, A. Marinelli, J. Hofest¨adt, g A. Enzenh¨ofer, d G.E. P˘av˘ala¸s, M. Kadler, u,ac a ab M. Colomer, l S. Hallmann, A. Moussa, M. Lincetto, D. Dornic, J. Brunner, A. Deschamps, P. Gay, d A. Kouchner, B. Baret, aq h,g M. Anghinolfi, T. Pradier, w g,z l,p h g u,ac R. Bormuth, b . J. H¨oßl, M. Sanguineti, o g,ac M. Chabab, 3 k h C. Lachaud, l,m q,r,s K. Graf, D. Els¨asser, A. S´anchez-Losa, M. Jongen, a The Authors. G. Levi, c,ak A. Margiotta, n k R. Bruijn, A. Coleiro, A.F. D´ıaz, c V. Popa, M. Organokov, i L. Fusco,

l,m P. Migliozzi, S. Biagi, M. Andr´e, o v , am g T. Avgitas, k C. Donzaud, k f,i a f g S. Celli, l,p u f Corresponding author. 1 Open Access Article funded by SCOAP A. Nu˜nez, P. Piattelli, G. Riccobene, D.F.E. Samtleben, N.R. Khan-Chowdhury, V. Kulikovskiy, E. Leonora, M. Marcelin, K. Melis, N. El Khayati, G. Ferrara, R. Gracia Ruiz, J.J. Hern´andez-Rey, M. de Jong, V. Bertin, H. Brˆanza¸s, J. Carr, M. Circella, A. Creusot, R. Don`a, The ANTARES collaboration A. Albert, J. Aublin, Measuring the atmospheric neutrinoparameters oscillation and constraining thewith 3+1 ten years model of ANTARES data JHEP06(2019)113 f an,g D. Zaborov, B. Vallage, k aj J. Wilms, A. Trovato, h ´ aq,ar Etoile Site de Chˆateau-Gombert, D. Vivolo, T. Thakore, k n S. Viola, Y. Tayalati, u,ac h c,y F. Versari, M. Taiuti, g,ai an and J. Z´u˜niga h University Mohammed I, Laboratory ofB.P. 717, Physics Oujda of 6000, Matter Morocco andInstitut Radiations, f¨urTheoretische Physik und Astrophysik,Emil-Fischer Universit¨atW¨urzburg, Str. 31, 97074 Germany W¨urzburg, Dipartimento di Fisica e Astronomia dell’Universit`a,Viale Berti Pichat 6/2, 40127 Bologna, Department of Computer Architecture and18071 Technology/CITIC, Granada, University Spain of Granada, UCA,G´eoazur, CNRS, IRD, Observatoire deDipartimento la di Cˆoted’Azur, Sophia Fisica dell’Universit`a,Via Antipolis, Dodecaneso 33,Universit´eParis-Sud, 91405 16146 Orsay Genova, Cedex, Italy France INFN — Sezione di Roma,Dipartimento P.le di Aldo Fisica Moro dell’Universit`aLa Sapienza, 2,Gran P.le 00185 Sasso Aldo Roma, Science Moro Italy Institute, 2, Viale 00185LPHEA, Francesco Faculty Roma, Crispi of Italy 7, Science — 00167 Semlali, L’Aquila,INFN Cadi Italy — Ayyad University, Sezione P.O.B. di 2390, Bologna, Marrakech,INFN Morocco Viale — Berti-Pichat Sezione 6/2, di 40127 Bari, Bologna, Via Italy E. Orabona 4, 70126 Bari, Italy University Mohammed V in4 Rabat, Faculty av. of Ibn Sciences, Battouta, B.P.Institute 1014, of R.P. 10000 Space Science, Rabat, Morocco RO-077125Universiteit Bucharest, Romania van M˘agurele, Amsterdam, InstituutScience voor Park Hoge-Energie Fysica, 105, 1098 XG Amsterdam, The Netherlands u re´rcoitCre38,Rue 13388 Fr´ed´ericJoliot-Curie Marseille Cedex 13,National France Center for EnergyB.P. Sciences 1382, and R.P. 10001 Nuclear Techniques, Rabat, Morocco INFN — Laboratori Nazionali delNikhef, Sud Science (LNS), Park, Via Amsterdam, S. TheHuygens-Kamerlingh Sofia Netherlands Onnes 62, Laboratorium, 95123 Universiteit Catania, Leiden, Italy The Netherlands APC, Universit´eParis Diderot, CNRS/IN2P3, CEA/Irfu, Observatoire de Paris, Sorbonne ParisIFIC Cit´e,France — Instituto de Corpuscularc/ Jos´eBeltr´an2, F´ısica E-46980 (CSIC Catedr´atico Paterna, — Valencia, Universitat Spain deLAM Val`encia), — Laboratoire d’Astrophysique de Marseille, Pˆolede l’ INFN — Sezione Erlangen di Centre Genova,Friedrich-Alexander-Universit¨atErlangen-N¨urnberg, for Via Astroparticle Dodecaneso Physics, 33, 16146Erwin-Rommel-Str. Genova, 1, Italy 91058 Erlangen, Germany Institut d’Investigaci´oper a la Gesti´oIntegradaUniversitat de de Polit`ecnica les Val`encia,c/ Paranimf Zones 1, Costaneres 46730 (IGIC), Aix Gandia, Marseille Spain University, CNRS/IN2P3, CPPM, Marseille, France Universit´ede Strasbourg, CNRS, IPHC UMRTechnical 7178, University F-67000 of Strasbourg, Catalonia, France LaboratoryRambla of Exposici´o,08800 Vilanova Applied Bioacoustics, i la Geltr´u,Barcelona, Spain l i t b c j s e q r o z p g d v y a k x h f u n w m V. Van Elewyck, J.D. Zornoza Th. Stolarczyk, ab ac aa JHEP06(2019)113 has 23 θ and 2 32 m 1812.08650 Neutrino Detectors and Telescopes (experiments), Oscillation The ANTARES neutrino telescope has an energy threshold of a few tens of [email protected] 18071 Granada, Spain GRPHE — Universit´ede Haute AlsaceRue — du Institut Grillenbreit Universitaire 34, de BP Technologie 50568-68008 deE-mail: Colmar, Colmar, France INFN — Sezione di Pisa,Dipartimento di Largo B. Fisica Pontecorvo dell’Universit`a,Largo B. 3,INFN Pontecorvo 56127 3, — Pisa, 56127 Sezione Italy Pisa, di Italy Napoli,Dipartimento Via di Cintia, Fisica dell’Universit`aFederico 80126 II Napoli, diDepartamento Italy y Napoli, de Te´orica del Via F´ısica Cosmos Cintia, & 80126 C.A.F.P.E., Napoli, University Italy of Granada, Leninskie gory, 119991 Moscow, Russia Mediterranean Institute of (MIO),13288, Aix-Marseille Marseille, University, Cedex 9,Universit´edu France; Sud Toulon-Var, CNRS-INSU/IRD UMINFN 110, — 83957, Sezione La Garde di Cedex, Catania,IRFU, France Via CEA, S. Universit´eParis-Saclay, F-91191 Sofia Gif-sur-Yvette, 64, France 95123 Catania, Italy ARC Centre of Excellence forInstitut All-sky Universitaire Astrophysics de (CAASTRO), France, 75005 Australia Dr. Paris, Remeis-Sternwarte France and ECAP,Sternwartstr. Friedrich-Alexander-Universit¨atErlangen-N¨urnberg, 7, 96049 Bamberg, Germany Moscow State University, Skobeltsyn Institute of Nuclear Physics, CNRS/IN2P3, BP 10448, F-63000 Clermont-Ferrand,LIS, France UMR Universit´ede Toulon, AixRoyal Marseille Netherlands Universit´e,CNRS, Institute 83041 for Toulon, France Landsdiep Sea 4, Research (NIOZ) 1797 and SZ Utrecht ’tInternational University, Horntje Centre (Texel), for The Radio Netherlands Bentley, Astronomy Research WA — 6102, Curtin Australia University, Laboratoire de Physique Corpusculaire, Clermont Universit´e,Universit´eBlaise Pascal, al ai at ArXiv ePrint: been performed — which is3+1 consistent neutrino with model world have best-fit been values derived. — and constraints onKeywords: the Abstract: GeV. This allows todue study to the neutrino oscillations. phenomenon of Inforesees a atmospheric similar the way, existence constraints neutrino on of disappearance theANTARES one 3+1 neutrino neutrino sterile telescope model, neutrino, from which can 2007 to be 2016, inferred. a new Using measurement data of collected ∆ by the aj as ae aq ar ao ap ag ad ak ah af an am JHEP06(2019)113 ] ) , 6 3 3 12 –  , ν 1 (1.1) 2 L km of 32 2 4 , ν E from its 1 4 m ν 10 L ∆ ∼  2 ) sin 2 9 | 3 ) and mass ( µ 11 τ U , ν µ − | , ν e (1 2 ν ) is studied. The vacuum sur- | µ 3 ν µ → U ]. | µ 4 ν 4 interacting at a distance P 4 12 − E 1 13 ∼  16 10 km of vertically down-going to 4 L – 1 – ji 2 ∼ E 4 m ∆  2 sin 2 | µi 5 U | 2 | µj U | 1 16 j>i X 4 − = 1 µ ν In this paper the muon disappearance channel ( Neutrino oscillations have been detected by a variety of experiments, studyingAtmospheric solar are produced through the interaction of cosmic rays with nuclei → 5.2 Treatment of systematics for the sterile oscillation analysis 6.1 Results for6.2 the standard oscillation Results analysis for the sterile oscillation analysis 5.1 Treatment of systematics for the standard oscillation analysis µ ν P vival probability for acreation point muon is neutrino given of by: energy in the ’s atmosphere. Theirto flux spans hundreds many of orders of TeV. magnitudeof in baselines Being energy, from on isotropic GeV the tovertically Earth’s first up-going surface, order, neutrinos. from it allows to investigate a large range bility. as well as atmosphericparticle neutrinos, accelerators. but For also a neutrinos comprehensive review produced see from [ nuclear reactors and 1 Introduction Neutrino oscillations arise from theeigenstates. mixing between The flavour mixing ( parameters of(PMNS) the Pontecorvo-Maki-Nakagawa-Sakata matrix and [ the differences between the mass eigenvalues regulate the oscillation proba- 7 Conclusions A Sterile neutrinos and matter effects 6 Results 3 ANTARES data and Monte Carlo4 samples Event reconstruction 5 Analysis Contents 1 Introduction 2 The ANTARES neutrino telescope JHEP06(2019)113 ] ) 2 8 U m 1.1 . For a , a new ]. In the 2 32 34 10 θ [ m ] and Mini- are the mass 23 11 2 i θ and m and ∆ 24 − θ and 2 j 13 , θ ] (see “supplementary 2 32 . Six new real mixing m 14 = 0 is assumed, which θ ij m 14 δ is the rotation matrix for = cos 14 θ ij 2 ji 23 θ R ], m 16 . In line with other analyses of – = sin 24 14 δ where 3 µ , and ∆ 12 U U R and 13 25 GeV. The formalism given in eq. ( 14 R δ 23 ∼ R ] has been designed and optimised for the ex- – 2 – ] as the neutrinos propagate through the Earth. 9 ], the additional neutrinos have to be sterile, i.e., 14 7 – R 13 . 5 24 14 R δ , the first minimum of the survival probability described 34 µ ν R = also contains a CP-violating phase, ij 3+1 R U , and two new phases, 1, 2 41 m i > ) is reached at energies of are elements of the PMNS matrix − 1.1 j survival probability depends only on µj µ ] collaborations, seem to indicate a deviation from the standard 3-flavour picture. ,U ν . If 12 µi ij U θ Even though a does not interact as the active flavours, its presence The 3+1 neutrino model foresees the existence of one sterile neutrino in addition to Despite the fact that is a well established phenomenon, some ob- A previous analysis of ANTARES data, covering the data acquisition period from The ANTARES neutrino telescope [ sterile neutrinos in thealso muon eliminates any disappearance dependency channel on [ would still modify the oscillation patternstandard of neutrino the flavours could standard oscillate neutrinos, into due these to additional the sterile fact species. that In the particular, from three to fourmaterials”) families. is In adopted: this analysisangle the convention from [ parameters have to bemass accounted splitting, for: ∆ three new mixing angles, neutrino state. However, since theis number limited of to weakly three interacting by familiesthey the of do LEP light not results neutrinos undergo [ weak interactions. the three standard ones. A choice has to be made, how to extend the mixing matrix that also includes a more comprehensive treatment of various systematic effects. served experimental anomalies, suchBooNE [ as the onesThese discrepancies reported could be by partially the explained LSND by introducing [ in the model an additional contributions from both neutrinos and antineutrinos. 2007 to 2010, representedand the measured first the study atmospheric of neutrinopresent oscillation this work, parameters, data kind ∆ collected performed during by 10 a years neutrino have telescope, been studied with a new analysis chain ploration of the high-energyenergy Universe threshold by of using about neutrinosfirst 20 as atmospheric GeV oscillation cosmic is minimum, probes. sufficient, makingsible. even also However, if As the its at study neutrinos the of andneutrino edge, neutrino antineutrinos telescopes, oscillations to are in pos- be indistinguishable the sensitive on following to an muon the event-by-event (electron) basis neutrinos in are refered to the sum of is further modified byThroughout matter the effects paper, oscillation [ probabilities arewhich treats calculated matter with effects the for OscProb an packageapproximations. arbitrary [ number of neutrino families numerically without splittings between two massdominance” eigenstates. approximation, The relevant rightmost inHere term the the describes energy the domainvertically “single up-going considered ∆ atmospheric for thisin equation analysis. ( where JHEP06(2019)113 = 2 31 (1.2) (1.3) with m which survival 4 1 τ , ∆ µ U 2 ν 5 eV and . 4 µ = 0 on the U 2 41 24 δ ° m ° , while the event recon- ,270 ° ° ° ° of the 3+1 neutrino model 3 = 90 = 180 = 0 = 0 = 0 24 24 24 24 24 δ δ δ δ δ [GeV] 34 and different combinations of ν with ∆ θ = 0.05, = 0.05, = 0.00, = 0.10, = 0.05, E 34 34 34 34 34 23 ) and modifies the event rate up θ θ θ θ θ 2 2 2 2 2 24 θ δ 24 and δ . , = 0.05, sin = 0.05, sin = 0.10, sin = 0.00, sin = 0.05, sin 24 24 24 2 24 24 24 24 the ANTARES neutrino telescope is and θ θ θ θ θ θ 24 is non-zero, the survival probability of 2 2 2 2 2 θ as a function of neutrino energy (calculated θ 10 sin No Sterile sin sin sin sin 2 µ 34 34 ν θ cos θ sin , is dedicated to the event selection and the 34 24 24 θ θ iδ 5 – 3 – − e being non-zero leads instead to a significant shift of = = sin . If only 4 4 34 24 τ µ θ δ U U . Section 4 and and 24 34 θ disappearance, the same analysis chain and data sample can be , θ = 1. µ 24 ν θ 23 θ 2 2 10 will be smeared out by detector resolution effects, therefore no sensitivity in the energy range of 20–100 GeV, non-zero values of 1.0 0.8 0.6 0.4 0.2 0.0 2

µ

survival probability for maximal mixing of µ µ

ν ν -> -> and sin ν P µ 2 5 eV ν . eV . Survival probability of vertically up-going 3 ]) for different values of mixing angles = 0 − 8 The paper is organised as follows: in section Since the effect of an additional sterile neutrino would be visible in the same energy 10 2 41 · with respect to the non-sterile hypothesis is only modified close to the first oscillation m 5 µ . are also reported. briefly described and itswell detection as principle the is illustrated; Montestruction the Carlo is ANTARES (MC) discussed data in chain sample section are as presented in section to this parameterprobability, is neglected expected. in all similar The analyses surprisingly so far, strong is effectand further zenith of detailed range in as the the exploited appendix. to constrain the 3+1investigation neutrino aiming model to parameters. constrain the In mixing this angles paper, the results of an ν minimum. The case of both the first oscillation minimum into energy energies of (depending few on hundred∆ GeV, easily accessible with ANTARES. The fast wiggles due to can lead to distortionsshows in the their survivalthe probability. mixing This parameters is illustrated in figure 2 for up-going Figure 1 with [ JHEP06(2019)113 K, naturally 40 60 m, while the vertical ∼ , while conclusions are given 6 ]. Finally, the acoustics based 22 downwards. All signals from the ] performs a hit selection based on ◦ 20 – 4 – K coincidence rates [ 40 ]. A typical run lasts few hours. Several time dependent 21 ], whose axis points 45 17 ]. The on-shore trigger system [ 19 , 18 . 7 The aim of the MC production is to reproduce in the most realistic way the events The main sources of optical background registered by the ANTARES PMTs are rep- approximately weekly basis from position calibration, performedassure every an few authentic description minutes, of the isuncertainties detector are applied. response small for and each can All individual beThey run. these handled are as Remaining detailed included global in inputs parameters which the are analysis discussed as below. systematic uncertainties. run MC approach isconditions applied are [ taken from realor data permanently and non-operational applied OMs to arelight the masked conditions, run-by-run in MC. the which simulation. First, mightfor temporarily Secondly, vary background each due individual to OM.light bioluminescence, simulation. are These Thirdly, measured samples individual every are OM 104 efficiencies directly ms are used considered, as as input calculated for on the an background been evaluated. expected at the detector,events. as well In as orderthe the to different response account operational of for status the changes of apparatus of the when the detector recording environmental and these conditions, its as components well over time, as a for run-by- 3 ANTARES data and MonteANTARES data Carlo collected samples fromexcluding 2007 to data 2016 acquired have under been adverse considered conditions, in a the analysis. total of After 2830 days of live time has resented by Cherenkov light frompresent decay in products of sea-water, the byby radioactive light isotope energetic emitted atmospheric through , which bioluminescencedetector can from by penetrate above. living deeply organisms, under and the sea and reach the PMTs that pass a thresholdshore of station 0.3 single [ photoelectrons (hits)causality are relations digitised and and builds sent tofrom events the Cherenkov under radiation the induced hypothesis byneutrino that relativistic interactions charged the particles close selected as to hits they the originate are ANTARES produced instrumented in volume. coast of Toulon, France,pleted at in 2008. a mooring ANTARESstoreys is depth of composed of 3 of about optical 12total 2475 detection of modules m. lines, 885 (OMs), each OMs. The exceptspacing one detector line equipped between The was with 12 the horizontal 25 com- spacing with storeys(PMT) among only is from the 20 Hamamatsu 14.5 m. lines storeys [ is of Each OMs, OM for hosts a a 10-inch photomultiplier tube in section 2 The ANTARES neutrino telescope The ANTARES neutrino telescope is located in the , 40 km off the minimisation procedure. The results are presented in section JHEP06(2019)113 A CC [20– µ ν ∈ ] pack- consists ν 23 B E ]; the energy 25 , is computed. This is done, µ L CC interactions an electromag- e ν CC events can be produced as the result of τ ν – 5 – ] for the Fr´ejussite is used in this work. 24 CC interactions, as well as neutral-current interactions (NC) e ν a hit selection, based on time and spatial coincidences of hits, is ) fit. A L ], which takes into account all relevant physics processes and com- 23 -fit is performed in order to find the best track. Events can have a single- 2 , respectively. Both are optimised for events induced by GeV-scale χ B ]. In the following discussion these algorithms will be referred to as method ] package [ oscillations with and without muons in the final state. All these events constitute 26 τ 28 ν , Once the muon track has been reconstructed, its length, Events have been reconstructed using two different algorithms, described in detail Particle propagation and Cherenkov light production are simulated using a GEANT- Even though the sub-marine location of ANTARES provides a good shielding against Neutrino interactions of all flavours have been simulated with the GENHEN [ 27 → µ selection, a first prefit,distributed based directions, is on performed. athe directional The final best likelihood scan (log 9 with directions a are large used number asfor of starting ML isotropically points events, by for projecting back to the track the first and last selected hit. For SL events, and method interactions. In method applied and a line topology (SL), if alla multi-line the topology selected (ML), hits when haveof hits been a belong recorded chain to in of OMs the fits, of same aimed different detector lines. to line, improve Method or at each step the track estimation. Starting from a hit netic shower is produced asν well. Moreover, an additional source of background for this study. in [ Charged-current (CC) interactions of muon neutrinosthe produce a detector muon and propagating through inducingevent Cherenkov reconstruction light. and selection They used areevents. in On identified the the as other analysis track-like hand, haveof events. been all optimised The flavours to produce select hadronic such showers. In the case of run of the detector,as and described the above. time evolution of the detector configuration is4 accounted for Event reconstruction based [ putes the probability that photonsa emitted hit. by Finally a the particlehits. detector reach response the At is OM this simulated, surface, stage including producing a the realistic digitisation optical and noise filtering is of added on each OM for each data acquisition atmospheric muons, still aerator large used amount of in them ANTARESand will to angular reach simulate distributions, the atmospheric as detector.are muons well parameterised. The is as The event MUPAGE the contribution gen- [ fromitself. multiplicity this of background muons is propagating also evaluated in from sea the water data age, developed inside theteractions ANTARES in Collaboration. the GeV Itto to allows reproduce multi-PeV to different energy reproduce physical100] range. GeV, neutrino expectations. a in- MC MC For neutrino sample atmosphericable. almost events neutrinos The three can with model hundreds be times by weighted larger Honda than et the al. data [ sample is avail- JHEP06(2019)113 e ν CC CC 720 and (SL) (4.1) 40 µ µ A ∼ ν ν A ∼ 20 events, CC) (about ∼ µ ν ( for method /σ 2 χ , for selected 60%) which contains T CC) ∼ τ ν ( σ (ML), while methods and passing the corresponding , A A CC events has been optimised by µ (SL) sample ( ν A 24 GeV/m . 0 × – 6 – µ 100 GeV. About 7590 well-reconstructed L = . T reco CC events reduce this oscillation signal by ]. This quantity is used in the following as estimator for E τ 29 ν : . As can be seen, atmospheric neutrino oscillations affect the µ 2 L . Events reconstructed by method eV B 3 − 10 ]. the distribution of the MC true neutrino energy, E × ; they are kept in the analysed sample if the corresponding selection criteria 2 10 B 46 . for method L both contribute with approximately 25% to this event sample. Further, = 2 B In figure The main parameter on which the selection is based is the reduced The muon energy estimation is based on the fact that muons in the few-GeV energy 2 32 m CC events areevents. selected. This reduction Oscillations isthe reduce dominantly lowest seen the energetic and in number most the contribute of vertical each events, expected about while the events 20%. taking other by into two reconstruction account methods the energy-dependent cross section ratio while the dashed one∆ refers to aexpected 2-flavour event oscillation distribution scenario withevents for are maximal E expected mixing in and one a half live of time of theseand 2830 events days are when reconstructed oscillations with are method neglected. Roughly are passed. Only eventsminimum which number are of reconstructed as fivebackground up-going storeys induced are with by used atmospheric selected in muons. hits the is following. required, A inevents is order shown. to For minimise the histogram the with the solid line no neutrino oscillations are assumed, analysis chain to data. the log event selection are kept. Theby events method discarded by this procedure are further reconstructed 5 Analysis To achieve the best sensitivityquality to criteria the has measurement beenperforming of applied. the a oscillation preliminary The parameters, Monte selection Carlo a of (MC) set sensitivity of study, before applying the whole spacing of the detector elementsdetector and only is a found lower tolength limit be inside for around their the 5 energy GeV. instrumentedbe can volume. For found muons be in More leaving derived, [ the details corresponding on to the their muon visible energy resolution can where the factorenergy 0.24 GeV/m range represents of 10–100 the GeVthe neutrino [ energy energy. loss The of energy muons resolution of in fully sea contained muons water is in dominated the by the length is estimated fromhave the recorded z-coordinates the of selected the hits and uppermost taking and into lowermost account storeyrange the which can reconstructed zenith be angle. treatedfrom their as track minimum length ionising particles, and their energy can be estimated since a vertex estimation is not possible due to the lack of azimuth information, the track JHEP06(2019)113 . 2 2 SL eV χ 3 (ML) − A 10 × 46 . 4 = 2 10 (SL and ML) and (SL) events where 2 32 m A A [GeV] T CC - no osc CC - with osc µ µ E ν ν + + µ µ ν ν decay and the resulting soft MC MC τ has been parameterised in the 3 for method (SL), 3682 from method CC events: assuming no oscillations 3 10 µ A 2 SL ν χ 8) with four different exponential fits . 0 and ¯ > µ ν 2 SL χ – 7 – GeV), and the cosine of the reconstructed zenith 2 / 10 the event distribution as a function of the logarithm reco 4 E ( , for selected T 10 . In figure B 10 ANTARES 0

80 60 40 20

shows the distribution of the reduced evt 160 140 120 100 N 3 , and combining the corresponding errors in quadrature, a total background of . MC neutrino energy, E B 120 atmospheric muons has been determined. This value is subsequently used as a After applying the event selection criteria described above on the data sample, a total Figure  directly from MC. of 7710 events have beenand selected, 2078 from 1950 method fromof method the reconstructed energy, log computed from theof errors this on method the formethod fitted events function that have parameters.740 been reconstructed Summing by up method Gaussian the prior mean results value and uncertaintydirection in the distribution minimisation of procedure. the The atmospheric energy and muon background has been, instead, estimated determined from data itself.region dominated The by distribution atmosphericby in muons varying figure ( the fitcorresponding range. integral has Each beento fit estimate computed. has the been The number extrapolated mean of into atmospheric of the muon these signal background, integrals region, and has and its been its uncertainty used has been data are compared to simulated atmospheric neutrinosWhile and the background MC atmospheric reproduces muons. quite wellsignal, the data a in the disagreement signal regionBoth between dominated data by the the and neutrino MC MC followthis expectation an reason, and exponential the law data number in is this of region, background visible but atmospheric for with muons different larger in slopes. the For signal region has been (dashed line). 0.5 at 25 GeV),spectrum the of 17% the produced branching muons. ratio of the muonic Figure 2 (solid line) and a 2-flavour oscillation scenario with maximal mixing and ∆ JHEP06(2019)113 . , is A 4 reco (5.1) θ data i,j N , that are , p , reco 2 θ ) i 2 SL k χ η k 2 η σ − h k η ( k GeV) and cos X / + 2; there are 17 bins in cos .  reco 1 ) E ¯ η ( < p, 10 (¯ MC i,j GeV) / N ) the corresponding number of expected MC ¯ η reco log · E p, – 8 – ( (¯ 10 data i,j MC i,j N N − values for events which have been reconstructed by method ) ¯ 2 SL η χ ) and p, (¯ i, j MC i,j N  MC atmospheric muons MC atmospheric MC neutrinos ANTARES Data (2007-2016) i,j X 2 5 4 3 = 2 1 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0

, is shown. The distribution of the MC expectation assuming no neutrino

10

L evt 10 10 10 10 N GeV) have been considered, seven from 1.2 to 2.0, plus an additional underflow reco / 8 indicates the value of the applied cut on this parameter. The fitted functions used to θ . 2 log . Distribution of reduced reco − = 0 E ( The final fit has been performed on the 2-dimensional histograms shown in figure 2 SL 10 χ , as described in the next subsections. The dependency on oscillation parameters is taken ¯ events in the same bin. Thisunder number investigation, depends as on the well set as ofη on oscillation parameters, the set ¯ of parametersinto related acount to for systematic CC uncertainties, interactionssample. of The all second neutrino sum flavours runs which over contribute the to number the of final nuisance event parameters taken into account, where the first sum runs overthe the number histogram of bins events in of bin log ( log bin which accounts for all eventsfrom with 0.15 log to 1.0, the latter denoting vertically up-goingThe events. fit follows a log-likelihood approach, by minimising the function: estimate the background ofwith atmospheric their muons extrapolation are into thefor shown signal details). as region well left (solid to coloured the lines), cut value together (dashed colouredangle, lines, see cos text oscillation (left panel) is compared to what is observed in data (right panel). Eight bins in Figure 3 (SL). Data (black crosses)with with MC error neutrino bars events indicating (red theat line) statistical and uncertainty MC are atmospheric shown muons together (green line). The dashed black line JHEP06(2019)113

, , evt N ν n hor 350 300 250 200 150 100 50 /ν up ) ν 2.0

GeV 1.9 /

reco 1.8 (E

10 1.7

. The first energy bin Log 1.6 reco

θ 1.5

1.4

] have been parametrised by the 1.3 its uncertainty. The log-likelihood

30 k ANTARES η 1.2 σ

1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 reco ) cos( θ , and

k

evt N 2. – 9 – . 1 , as well as in the correction to the spectral index for bins with high statistics. For bins with a small 350 300 250 200 150 100 50 < ν 2 in the nominal neutrino flux spectral index has been n χ ) γ 2.0

GeV) / GeV 1.9 /

reco

reco 1.8 E (E

( ]. Such parameterisations compute a correction on the number

10 10 1.7 31

Log 1.6

GeV), and the reconstructed cosine of zenith, cos 1.5 /

1.4 reco

E ( 1.3 10

, and on the flux asymmetry between up-going and horizontal neutrinos, . Number (colour scale on the right side) of selected MC events assuming no oscillation ¯ ν ANTARES 1.2 ν/

being the assumed prior of the parameter An additional source of systematic uncertainty is the limited knowledge of the neutrino Since the treatment of the systematic uncertainties slightly differs between the stan-

. For what concerns the QE and RES processes, dedicated studies have been performed

i

1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 reco ) cos( θ k γ η interaction model. At theby energy deep of inelastic interest scattering for (DIS),resonant with this (RES) a study, smaller scattering. the contribution cross Uncertainties fromthe section quasi in global is elastic the flux dominated (QE) DIS and normalisation cross∆ factor section can be incorporated in IceCube Collaboration [ of expected events as avalue function of of the the uncertainty neutrino on energy,been flavour, the chirality, found direction flux to and ratio. the bethe The strongly fit. two ratios correlated, considered thus in a this unique analysis nuisance have parameter is considered in which is left unconstrainedof during expected the events. fit, accounts Aused for variation as uncertainties ∆ additional on the nuisanceratio, total parameter. number Uncertainties onhave also the been neutrino/anti-neutrino taken flux into account. These uncertainties [ separately in the following subsections. 5.1 Treatment of systematicsThe standard for oscillation the analysis accounts standard forare six oscillation sources related analysis of to systematic uncertainties. the Three atmospheric neutrino flux. A global neutrino normalisation factor, h function converges to the standard number of entries the log-likelihood is more adequate. dard atmospheric oscillation analysis and the sterile neutrino analysis, they are described Figure 4 (left panel) and selected dataenergy, (right log panel), binned accordingcontains to all the events logarithm with of log the reconstructed JHEP06(2019)113 , , ◦ ν CP are n δ (5.2) 28) , the w . σ 0 A 1   , the other ), has been CC events, and 41 . 23 µ  θ ν nom A r = (8 ] which makes the and 13 22 θ 2 32 = 100 GeV defines the 10% from the nominal m 0  E 297. Different values of . K decays [ = 0 . The effects of , 40  1 12 θ B ) 2 0 E / are the two fitted parameters describing T  are covered by the uncertainty of the prior B and sin (E ]. By varying this parameter by , · 2  w 33  , contaminating the neutrino sample, is treated A B ] to model neutrino interactions. The dominant A µ – 10 – eV 5 N 33 − ) = and are listed in table ). 10 T  2 w as the corresponding fit parameters. The values of the 22 GeV [ ] as well as the values of the parameters, × B . (E B shows the distribution of the event ratios as a function  0 w 34 f 37 5 . B is fixed at zero.  and 12 = 7 CP . GeV, with a function of the form: w and δ (see table 3 2 21 ) and the one from the nominal MC simulation ( w γ ,A  m ]. The water absorption length had been measured several times at . Figure A ]. The different measurements, taken at two different wavelengths, var  r 17 35 ,B A  A ], which uses GENIE [ ) using 32 5.2 is the MC true neutrino energy, is left free in the fit but treated with a Gaussian prior at T 13 θ The effect of the modified water absorption length is described by the same functional The correction to the event rates, obtained by dividing the event rates from the mod- The number of atmospheric muons, Finally, detector and sea water related systematics have been studied as well. Dedi- Apart from the oscillation parameters under investigation, ∆ . Its default value is 1 A form of eq. ( fitted parameters taken into account inwhich the is minimisation left procedure unconstrained, byon while the the global spectral normalisation index, factor, ∆ where E the effect of thereference modified energy OM for photon detectionof efficiencies true and neutrino energy, together with its parameterisation. reconstructed as up-going. Whileof no the detector zenith-dependent is effect affectedin is by the seen, these energy variations. the range The energy 10–10 resulting response distributions have been fitted, the ANTARES site [ vary within about 10%. ified MC simulation ( computed as a function of the MC neutrino energy and zenith angle for and a modified watervalue, absorption but length, assuming keeping a theeasily variation same adjusted of wavelength to dependence. thechosen measured 10% The variations coincidence a overall rates conservative benchmark OMon from value, ANTARES in efficiency OMs line can [ with early be studies performed as an additional nuisance parameter.driven Its technique, value are and used uncertainty, as determined a with prior. the data- cated MC simulations have been generated with modified OM photon detection efficiencies oscillation parameters may play aular, role, but their effectwhich is is limited taken for from a thiswhich global study. are fit kept In fixed: [ partic- ∆ have been tested at thethe stage final of result. the Therefore MC sensitivity study and found to have no impact on systematic is found to beM related to the axialcorrection mass with for respect CC to resonance theof neutrino expected the production, number true of neutrino events energy has been and computed this as parameterisation a is function used in the final fit. with gSeaGen [ JHEP06(2019)113 14 δ 3 10 [GeV] T have been tested during E are left unconstrained as 24 +10% -10% . The choice of the neutrino δ and its associated phase 2 23 θ 02 14 w . 02 θ . 0 B 0 − channel. The fast oscillations due and e ν 2 32 w 92 16 . . m A 0 1 03  2 . 03 . 0 B 0 10 − – 11 – are expected to impact the result. Therefore both  CC events, as a function of true neutrino energy, due to 82 19 . . A 0 1 µ 24 ν δ 10% , the addition of a sterile neutrino in the model implies six new − +10% 1 do, these parameters are considered to be one of the sources of 23 ]. The other standard oscillation parameters are treated as previously are unobservable due to the limited energy resolution of the detector, ANTARES θ 10% (green) variation from the nominal value of the OM photon detection − 2 36 10 and 1 0 2 not measurable. It has been kept fixed at 0.5 eV 5 eV

.

0.8 0.6 0.4 0.2 1.8 1.6 1.4 1.2

2 32 0 2 41 nom var /r r m m & . Expected event ratios for . Fitted values for the parameterisation of the event weight correction with a variation of 41 2 m As discussed in section Since the effect of a sterile neutrino would modify the oscillation pattern in a similar 10% from the nominal value of the OM photon detection efficiency and water absorption length. making ∆ mass hierarchy (NMH) asnormal well and as inverted hierarchy (NH/IH)the and fit. various values Furthermore, of contamination to has ensure been the fixed stability at of the the value fit found procedure, by the the atmospheric standard muon oscillation analysis. It recommended in [ discussed. mixing parameters to be accountedhave for. been fixed The mixing at angle zero,to since ∆ they mainly affect the For the sterile analysis, the fluxare as treated well in as the the cross same section way related as systematic described uncertainties inway the as previous subsection. ∆ systematic uncertainty for this analysis. Both ∆ Table 1  5.2 Treatment of systematics for the sterile oscillation analysis efficiency. Figure 5 a +10% (red) and JHEP06(2019)113 ]. 10 ]. A 31 ], applied , which is 2 is found to 37 but only on eV 23 13 3 θ θ − 10 × ) 4 3 . . 0 036 +0 − . 0 28 0 . . 56 09 64 10 3 4 0 ...... 24 1 0 0 0 0 11 +48 −  +0 − +0 − +0 − +12 −   0 . 0 10 81 45 . . . at (2 414 2 41 003 0 . 1 0 . Fit result 8 0 2 32 − ]. The mixing angle m 38 the best fit values and their errors are 0 28 05 0 . . . . A 1 0 120 0 1 M      none none none 0 0 Prior . . ). Results are presented in the following subsec- 41 00 . This ratio is affected by the oscillation phe- – 12 – 0 . . 0 mainly affects neutrinos with energies below the 740 and 8 0 6 A 5.1 13 ] M 02 is found. Concerning the spectral index correction, θ 2 . 0 . This parameter seems to compensate for the low value eV ] ] ¯ ν  3 ] ] σ σ ◦ ◦ − [ [ [ [ µ γ ν ν/ 04 . survival probability does not depend on sin ν n A N ∆ [10 13 23 θ θ µ ν/ M ν 2 32 = 1 )) whereas Parameter m ν 1.1 n ∆ , is found to be 18% lower. This value is within the atmospheric neutrino ν 99 (see eq. ( n . the complete list of all the fitted parameters for the standard oscillation analysis 2 . Priors and fitted values obtained from the minimisation for all the parameters considered have been fixed, to allow a more direct comparison with the result reported in [ = 0 ν : this has been derived from an alternative fit, for which all nuisance parameters 13 The distribution of the ratio between the reconstructed energy and the cosine of the ν n θ , no significant distortion from the nominal value is observed. The fitted value for the n γ can be understood ascos the detection threshold of ANTARES. reconstructed zenith is shown in figure but Under these conditions ∆ atmospheric muon contamination showsto a the strong MC pull expectations.found and at the For it corresponding both is prior, found which indicates incidentally no close sensitivity to these parameters. This be compatible with maximalneutrinos, mixing within its error.flux uncertainties The and global it normalisationnon-negligible is factor pull compatible for is with found on whatof was reported by other analyses [ In table is shown, togetherthreshold with of their ANTARES best-fit thisboth values analysis NH and is and their not IH.compatible sensitive priors. with The to best-fit the the Due value current NMH. to is world the found The best-fit for high results value ∆ hold [ energy for The minimisation procedure has beento done the using function the introduced ROOT in packagetions, equation Minuit2 for ( [ the standard oscillation analysis and6.1 the sterile oscillation analysis, Results respectively. for the standard oscillation analysis has been verified thatcase this of choice a does free muon not contamination. lead to better6 constraints with respect Results to the Table 2 in the standard oscillation analysis. JHEP06(2019)113 ◦ 4 . 24 θ are and = 0 = 4 23 23 θ 24 L θ δ 2 [GeV] and reco θ 2 32 2∆ log m /cos − . While compati- reco E 4 ANTARES . For comparison, also the and minimising the negative reco is found at a non-zero value. 23 θ θ 34 θ 50 100 150 200 cos and / MC world best fit MC best fit Data 2 32 ]. reco m . For IH instead the fit prefers 1 10 E 0

◦

), which can be easily provided by a non-

evt no-osc 0.6 0.2 1.2 0.8 0.4 1.4 /N N 6 – 13 – [GeV] is found at 180 MC no osc MC world best fit MC best fit Data ). The non-sterile hypothesis is found at reco ] and MC assuming best-fit values of this analysis (green). The θ 24 1 δ 38 /cos reco E in our previous analysis [ distribution for data (black), MC without oscillation (red), MC assuming σ 3 (see figure reco . θ 34 ANTARES ] are shown. The latter two are calculated with all nuisance parameters at θ cos the 90% CL contour obtained in this work, in the plane of sin 38 / 7 reco E the complete list of all the fitted parameters for the sterile oscillation analysis 50 100 150 200 . 3 , is compared to those published by other experiments. The 1D projections, after , compared to 2 The non-oscillation hypothesis has been tested by performing the minimisation with In figure 2 32

0

σ evt N m 6 800 600 400 200 . 1000 oscillation dip (see discussionzero related value to of figure sin which corresponds to a 2-parameterslightly p-value different of but 11%. consistent with The respectanalysis. fitted to the values The of ones obtained ∆ complex in phase the standard oscillation 6.2 Results for theIn sterile table oscillation analysis for NH and IHis is found shown, to togetherThis be with can compatible their be best-fit with understood values zero, from and the the their slight best preference priors. of fit the While for ANTARES data for a shallower log-likelihood over all the other parameters. a fixed null value4 of the oscillation parameters, and it is discarded with a significance of ble with world data,shifted) ANTARES oscillation results minimum. seem to prefer a somewhat∆ shallower (or energy profiling over the othercomputed variable, by are looping over shown a as fine grid well. of values Confidence in ∆ level contours have been nomenon as can bedistribution seen of for MC the assuming lowest nobest-fit values neutrino values of oscillation, [ as well astheir the nominal one values. assuming Such the a world in 1D the distribution fit, does which not is carry the performed full on information the exploited 2D distribution shown in figure Figure 6 the world best-fit valuesleft (blue) plot [ shows event numbers whilewithout the oscillations. right plot illustrates the event ratio with respect to the MC JHEP06(2019)113 24 δ ]. The 42 ], Super- ], can be ◦ 31 = 0 the resulting 24 8 δ 036 . 8 6 28 2 1 . . . 97 55 09 93 63 10 . . 0 0 0 8 ...... 0 . 4 0 . The exclusion limit 0 0 0 5 72 +0 − +5 −   +0 − +0 − +0 − +2 − 24 0  9 .  5 . θ . 3 11 07 84 52 0 Fit IH . . . 2 ] or [IH, 1 41 011 25 − . 0 1 0 ◦ . obtained in this work (black 8 0 cos − 2 32 34 θ m = 180 2 ], and MINOS (light blue) [ 036 24 . 28 41 2 1 δ . 97 55 09 93 63 10 . . 6 0 8 0 0 8 71 ...... 4 0 0 0 0 0 5 and ∆ = sin +5 −    +0 − +0 − +0 − +0 − +2 −  9 0 5 2 . 23 . . | 11 07 84 52 θ . . . 3 1 4 Fit NH 41 011 25 180 . 0 1 0 τ . 2 8 0 U | − – 14 – and ], T2K (blue) [ 0 28 05 0 . . . . 40 24 1 0 0 1 θ     2 none none none none none none 0 0 Prior . . 41 00 0 . . 0 8 0 = sin 2 ] | 2 A (purple) [ 4 µ ν eV U ] ] 3 | ] ] ] ] ] σ σ ◦ ◦ ◦ ◦ ◦ − [ [ [ [ [ [ [ γ ν ], NO ν n A , which corresponds to both [NH, ∆ [10 13 23 34 24 24 39 δ θ θ θ θ 24 ν/ M 2 32 δ Parameter m ∆ . Contour at 90% CL in the plane of sin . Priors and fitted values obtained from the minimisation for all the parameters considered Exclusion contours are built by applying Wilks’ theorem. In figure analysis is observed. 90% and 99% CL exclusion limitsmatrix elements, have namely been computed onfor a 2D unconstrained grid in the plane of the two Table 3 in the sterile oscillation analysis. with otherwise identical results,(see as appendix). expected For the from other the parameters a degeneracy similar between behaviour as NMH for and the standard oscillation Figure 7 line) and comparedKamiokande to (green) the [ resultslateral by plots show other the experiments: 1D projections IceCube/DeepCore on the (red) plane [ of the two oscillation parameters under study. JHEP06(2019)113 ] ].

24 10 δ 16

16 8

(6.1) (6.2) 6 to differ LogL

� 4 2 | -2

4 2 τ ] (blue). The

U 0 | 0 2 4 6 8 10 15 ° 24 ° θ = 0 = 0 2 ° 24 ° 24 δ δ = 0 = 0 24 24 δ δ sin 1 99% C.L. − This work This work NH, IceCube (2017) NH, IceCube (2017) IH, SK (2015) NH, 10 , . 2 − 10 ] (IH) limit. Also shown are limits for NH 16 ] (red) and Super-Kamiokande [ obtained in this work (black lines), and compared while the solid lines are for an unconstrained 16 ◦ 13) at 90% (99%) CL 24 3 . – 15 – − 68) at 90% (99%) CL θ ] is limited to events with reconstructed energy . 10 2 = 0 16 cos 24 24 ° 007 (0 40 (0 δ ° θ . . = 0 34 2 = 0 ° 24 ° 24 δ δ 0 0 θ = 0 = 0 24 24 δ δ sin 2 ]. All three experiments find the best fit for < < (IceCube/Deepcore) respectively. The coulored markers indicate 15 1 90% C.L. 2 2 − ◦ | | This work This work NH, IceCube (2017) NH, IceCube (2017) IH, SK (2015) NH, 4 4 10 = sin τ µ = 0 2 U | U | | 4 24 τ δ U | ) are left unconstrained in line with the IceCube/DeepCore analysis [ and 2 23 − θ 10 24 θ (2 which allow a direct comparison with the results from IceCube/DeepCore [ 2 2 56 GeV. In this work both of the standard atmospheric oscillation parameters ◦ < . 90% (left) and 99% (right) CL limits for the 3+1 neutrino model in the parameter plane = 0 = sin 2 reco and sin | 4 CP 3 E − µ After profiling over the other variable, the following limits on the two matrix elements The IceCube/DeepCore analysis [ δ 2 32

8 6 4 2 0 1 2 10

10 −

2 8 6 4 −

0 U

10 � 34

24 | 8 θ θ

LogL -2

sin cos m

10 10 2 2 can be derived: the presence of aThe sterile present neutrino analysis would bebeen includes evident verified events also that with at the reconstructedwith higher ANTARES reconstructed energy limits energies. up degrade∆ to when 100 restricting GeV. the analysis It to has events from zero. Our resultsexperiments. exclude regions of the parameter space not yetlower excluded than by 56 other GeV, while the distortion on the oscillation pattern possibly produced by are also shown for the result of this work. directly compared to the IceCube/DeepCoreand [ (NH) and Super-Kamiokande [ Figure 8 of to the ones publisheddashed lines by are IceCube/DeepCore obtained [ (this for work) or NH for and IHthe and best-fit values for each experiment. The 1D projections after profiling over the other variable JHEP06(2019)113 µ ν Ile-de- ˆ , are consistent with what has ◦ ) 11 +12 − = (45 23 θ – 16 – and 2 eV 3 − 10 ]. However, in this appendix some common approximations 8 × ) 4 3 . . 0 +0 − 0 . . σ 6 . = (2 2 32 m Exploiting the same analysis chain and the same data set, a further study has allowed A Sterile neutrinos andFor matter the effects analysis presentedsoftware in package this OscProb paper, [ are oscillation applied, probabilities to are derive analyticalallow evaluated to formulae. with get These the a are better NOT understanding used of for the the interplay analysis between different itself parameters. but The Morocco. We also acknowledge thefor technical the support sea of Ifremer, operationthank AIM J. and and Coelho the Foselev for Marine CC-IN2P3 enlightening for discussions on the matter computing effects facilities. for sterile We neutrinos. would like to tific schools supporting grants,Research, Development Russia; and Executive Innovation (UEFISCDI), UnitCompetitividad Romania; for Ministerio (MINECO): de Financing Plan Econom´ıay Higher Estatal-2-P Education, de and Investigaci´on(refs. FPA2015-65150-C3-1-P, -3-P, (MINECO/FEDER)),solider Severo MultiDark Ochoa (MINECO), Centre andciana), of Prometeo Spain; Ministry Excellence and of Grisol´ıaprograms and Higher (Generalitat Education, Red Scientific Valen- Con- Research and Professional Training, France (DIM-ACAV), (contrat R´egionAlsace CPER), R´egionProvence-Alpes-Cˆoted’Azur, duD´epartement Var and Ville dedung La und Seyne-sur-Mer, Forschung France; (BMBF), BundesministeriumItaly; Germany; Nederlandse f¨urBil- organisatie Istituto voor Wetenschappelijk Nazionale OnderzoekCouncil (NWO), di of the Fisica the Netherlands; President Nucleare of (INFN), the Russian Federation for young scientists and leading scien- de la Recherche Scientifiquealternatives (CNRS), (CEA), Commissariat Commission `al’´energieatomique et (FEDER Europ´eenne Institut aux fund Universitaire ´energies and de Marie France (IUF),Sorbonne Curie IdEx Paris Program), program Cit´e(ANR-10-LABX-0023 and and UnivEarthS(ANR-11-LABX-0060) ANR-11-IDEX-0005-02), and Labex the Labex program A*MIDEX project at OCEVU (ANR-11-IDEX-0001-02), R´egion the parameter space not yet excluded by other experiments. Acknowledgments The authors acknowledge the financial support of the funding agencies: Centre National been published by othersignificance experiments. of 4 The non-oscillation hypothesis is discarded withto a constrain, for themodel, first which time foresees with the ANTARES, the existence parameter of space one of sterile the neutrino. 3+1 ANTARES neutrino excludes values of Ten years of ANTARES dataspheric have neutrino oscillation been parameters. analysed The toto analysis provide chain our has a previously been measurement optimised published of withintroducing study, respect the a by atmo- more combining elaborate two treatment trackresults, of reconstruction various ∆ sources algorithms of and systematic uncertainties. The 7 Conclusions JHEP06(2019)113 if is are and π 32 4 (A.4) (A.5) (A.6) (A.1) (A.2) (A.3) ν A + m δ /E 2 32 disappears → or from the δ δ . m δ ]. 23 cos 45 θ , | = ∆ , ) s 4 s | θ θ 4 32 , µ the neutron density) A iδ U | n sin 2 − cos 2 ). | e . N + . This yields ) iδ 23 23 i ) 2  θ θ | 2 , ) − | = 0. Further, 4 4 4 | L τ ) ) s τ 4 2 2 U m θ | µ | sin 2 U 4 4 E U s τ | τ ( − | + cos 2 − | A 2 U 2 for all U | 2 | δ + / | 2 32 | 4 sin + 4 + µ A (3) µ µµ 2 cos m 2 U | breaks this degeneracy. The impact of | | U P θ 4 ) = 1 4 s 2 + 2 2 µ s . The impact of the CP-phase µ θ ) − | E θ 2 2 − | 2 U 1 U | | | 4 | s 4 (1 θ (1 )( sin µ 2 )( ] are reproduced. With sin 2 L/ | 2 2 2 | | i | U 4 | − – 17 – 2 4 4 4 15 τ 4 23 µ µ the Fermi constant and 1 µ sin 2 U m − | θ | U U | U can be probed in this scheme, which is applied in survival probability in matter had been pointed out defining 2 2 2 s F | ) (∆ µ 4 2 2 A − | − | G 2 − | | 24 | ν µ 4 4 = (1 ]): 2 θ (sin 2 + | U = 0, which leads to sin 2 τ µ (1 s | 2 (1 2 ( 4 µ 15 U U ), which permits any use, distribution and reproduction in τ ν 2 / A 23 | | ) = 0. Deviation from maximal mixing in p θ U 4 → 2 32 2 | | 2 τ − | µ s 4 ν A 2 θ U = sin τ | = = = 0; second, fast wiggles due to oscillations involving and disappearance analyses. However, when analysing atmospheric P U 2 s s | sin | 2 µ + 2 θ θ ]. In eq. (4.13), a complex phase is present in the non-diagonal | = (1 13 4 ν 4 2 s θ µ 2 32 + µ µ 15 ν in eq. (4.16) acquires an extra term exp( A U 2 A CC-BY 4.0 | U | = sin 2 | cos 2 s → 4 = 0 or + q θ µ µ ]: first, it is assumed, that the first generation decouples completely, i.e. This article is distributed under the terms of the Creative Commons ν 2 12 survival probability in the 3-flavour scheme, i.e. without additional ster- | m U 2 32 1 P | θ 4 ( = 0 or cos 2 µ E 43 A µ , n ν ], while the influence of complex phases is also discussed in [ U | = = N 23 15 44 θ F 2 m m the θ G E ) (see also eq. (4.23) of [ 2 survival probability in matter is fully defined and can be written equivalently to 2 (3) = 0 and µµ √ = 0 the original expressions from [ µ P sin 2 A.1 δ ν 2 21 = m s already in [ Open Access. Attribution License ( any medium, provided the original author(s) and source are credited. which describes well all features shown inwhen figure either completely degenerate with changingeither the cos mass 2 hierarchy, i.e.symmetry swapping between the sign of the neutrino mass hierarchy on the the eq. ( For A This in turn modifies eqs. (4.18) and eq. (4.19): term of the matrix, whichphase is is neglected, kept, i.e. sin 2 set to zero, in subsequent steps. If instead this with ile neutrinos. Only most accelerator based neutrinos, matter effects cannoteqs. be (4.13)–(4.25) neglected. of [ An analytical formalism is developped in hypotheses [ ∆ assumed to be unobservable, i.e. sin survival probability in vacuum in the 3 + 1 model can be simplified with the following two JHEP06(2019)113 , ]. , ]. ]. ]. ] SPIRE ]. IN SLD e SPIRE ]. SPIRE , ¯ ν ][ IN SPIRE IN ]. IN [ ][ Precision SPIRE , ][ ]. SPIRE IN IN ][ SPIRE , ]. ][ IN ]. SPIRE ]. ][ IN arXiv:1410.2008 (1978) 2369 , [ . SPIRE (2006) 257 SPIRE IN SPIRE arXiv:1206.0645 (2012) 052009 collaborations, IN [ ][ IN [ D 17 427 arXiv:1007.2549 hep-ex/0104049 ][ [ , [ D 86 (1985) 913 [ arXiv:1710.06488 42 [ astro-ph/0610029 (2015) 052019 (1986) 17 (2012) 224 [ arXiv:1702.05160 (2010) 59 Phys. Rev. [ Phys. Rept. , Dual baseline search for muon antineutrino , C 9 D 91 Phys. Rev. (2001) 112007 ]. , B 714 2 ]. astro-ph/0112172 ]. 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