Measuring the atmospheric neutrino oscillation parameters and constraining the 3+1 neutrino model with ten years of ANTARES data

JHEP06(2019)113 Published for SISSA by Springer Received:December 21, 2018 Revised:April 24, 2019 Accepted:May 19, 2019 Published:June 21, 2019 Measuring the atmospheric neutrino oscillation parameters and constraining the 3+1 neutrino model with ten years of ANTARES data The ANTARES collaboration A. Albert,aM. Andr´e,bM. Anghinolfi,cG. Anton,dM. Ardid,eJ.-J. Aubert,f J. Aublin,gT. Avgitas,gB. Baret,gJ. Barrios-Mart´ı,hS. Basa,iB. Belhorma,j V. Bertin,fS. Biagi,kR. Bormuth,l,m J. Boumaaza,nS. Bourret,gM.C. Bouwhuis,l H. Brˆanza¸s,oR. Bruijn,l,p J. Brunner,fJ. Busto,fA. Capone,q,r L. Caramete,o J. Carr,fS. Celli,q,r,s M. Chabab,tR. Cherkaoui El Moursli,nT. Chiarusi,u M. Circella,vA. Coleiro,h,g M. Colomer,g,h R. Coniglione,kH. Costantini,fP. Coyle,f A. Creusot,gA.F. D´ıaz,wA. Deschamps,xC. Distefano,kI. Di Palma,q,r A. Domi,c,y R. Don`a,uC. Donzaud,g,z D. Dornic,fD. Drouhin,aT. Eberl,dI. El Bojaddaini,aa N. El Khayati,nD. Els¨asser,ab A. Enzenh¨ofer,d,f A. Ettahiri,nF. Fassi,nP. Fermani,q,r G. Ferrara,kL. Fusco,g,ac P. Gay,ad,g H. Glotin,ae R. Gozzini,hT. Gr´egoire,g R. Gracia Ruiz,aK. Graf,dS. Hallmann,dH. van Haren,af A.J. Heijboer,lY. Hello,x J.J. Hern´andez-Rey,hJ. H¨oßl,dJ. Hofest¨adt,dG. Illuminati,hC.W. James,ag,ah M. de Jong,l,m M. Jongen,lM. Kadler,ab O. Kalekin,dU. Katz,d N.R. Khan-Chowdhury,hA. Kouchner,g,ai M. Kreter,ab I. Kreykenbohm,aj V. Kulikovskiy,c,ak C. Lachaud,gR. Lahmann,dR. Le Breton,gD. Lef`evre,al E. Leonora,am G. Levi,u,ac M. Lincetto,fM. Lotze,hS. Loucatos,an,g G. Maggi,f M. Marcelin,iA. Margiotta,u,ac A. Marinelli,ao,ap J.A. Mart´ınez-Mora,eR. Mele,aq,ar K. Melis,l,p P. Migliozzi,aq A. Moussa,aa S. Navas,as E. Nezri,iC. Nielsen,g A. Nu˜nez,f,i M. Organokov,aG.E. P˘av˘ala¸s,oC. Pellegrino,u,ac M. Perrin-Terrin,f P. Piattelli,kV. Popa,oT. Pradier,aL. Quinn,fC. Racca,at N. Randazzo,am G. Riccobene,kA. S´anchez-Losa,vA. Salah-Eddine,tI. Salvadori, f,1 D.F.E. Samtleben,l,m M. Sanguineti,c,y P. Sapienza,kF. Sch¨ussler,an M. Spurio,u,ac 1Corresponding author. Open Access,c The Authors. Article funded by SCOAP3.https://doi.org/10.1007/JHEP06(2019)113 JHEP06(2019)113 Th. Stolarczyk,an M. Taiuti,c,y Y. Tayalati,nT. Thakore,hA. Trovato,kB. Vallage,an,g V. Van Elewyck,g,ai F. Versari,u,ac S. Viola,kD. Vivolo,aq,ar J. Wilms,aj D. Zaborov,f J.D. Zornozahand J. Z´u˜nigah aUniversit´e de Strasbourg, CNRS, IPHC UMR 7178, F-67000 Strasbourg, France bTechnical University of Catalonia, Laboratory of Applied Bioacoustics, Rambla Exposici´o, 08800 Vilanova i la Geltr´u, Barcelona, Spain cINFN — Sezione di Genova, Via Dodecaneso 33, 16146 Genova, Italy dFriedrich-Alexander-Universit¨at Erlangen-N¨urnberg, Erlangen Centre for Astroparticle Physics, Erwin-Rommel-Str. 1, 91058 Erlangen, Germany eInstitut 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 fAix Marseille University, CNRS/IN2P3, CPPM, Marseille, France gAPC, Universit´e Paris Diderot, CNRS/IN2P3, CEA/Irfu, Observatoire de Paris, Sorbonne Paris Cit´e, France hIFIC — Instituto de F´ısica Corpuscular (CSIC — Universitat de Val`encia), c/ Catedr´atico Jos´e Beltr´an 2, E-46980 Paterna, Valencia, Spain iLAM — Laboratoire d’Astrophysique de Marseille, Pˆole de l’ ´ Etoile Site de Chˆateau-Gombert, Rue Fr´ed´eric Joliot-Curie 38, 13388 Marseille Cedex 13, France jNational Center for Energy Sciences and Nuclear Techniques, B.P. 1382, R.P. 10001 Rabat, Morocco kINFN — Laboratori Nazionali del Sud (LNS), Via S. Sofia 62, 95123 Catania, Italy lNikhef, Science Park, Amsterdam, The Netherlands mHuygens-Kamerlingh Onnes Laboratorium, Universiteit Leiden, The Netherlands nUniversity Mohammed V in Rabat, Faculty of Sciences, 4 av. Ibn Battouta, B.P. 1014, R.P. 10000 Rabat, Morocco oInstitute of Space Science, RO-077125 Bucharest, M˘agurele, Romania pUniversiteit van Amsterdam, Instituut voor Hoge-Energie Fysica, Science Park 105, 1098 XG Amsterdam, The Netherlands qINFN — Sezione di Roma, P.le Aldo Moro 2, 00185 Roma, Italy rDipartimento di Fisica dell’Universit`a La Sapienza, P.le Aldo Moro 2, 00185 Roma, Italy sGran Sasso Science Institute, Viale Francesco Crispi 7, 00167 L’Aquila, Italy tLPHEA, Faculty of Science — Semlali, Cadi Ayyad University, P.O.B. 2390, Marrakech, Morocco uINFN — Sezione di Bologna, Viale Berti-Pichat 6/2, 40127 Bologna, Italy vINFN — Sezione di Bari, Via E. Orabona 4, 70126 Bari, Italy wDepartment of Computer Architecture and Technology/CITIC, University of Granada, 18071 Granada, Spain xG´eoazur, UCA, CNRS, IRD, Observatoire de la Cˆote d’Azur, Sophia Antipolis, France yDipartimento di Fisica dell’Universit`a, Via Dodecaneso 33, 16146 Genova, Italy zUniversit´e Paris-Sud, 91405 Orsay Cedex, France aaUniversity Mohammed I, Laboratory of Physics of Matter and Radiations, B.P. 717, Oujda 6000, Morocco abInstitut f¨ur Theoretische Physik und Astrophysik, Universit¨at W¨urzburg, Emil-Fischer Str. 31, 97074 W¨urzburg, Germany acDipartimento di Fisica e Astronomia dell’Universit`a, Viale Berti Pichat 6/2, 40127 Bologna, Italy JHEP06(2019)113 adLaboratoire de Physique Corpusculaire, Clermont Universit´e, Universit´e Blaise Pascal, CNRS/IN2P3, BP 10448, F-63000 Clermont-Ferrand, France aeLIS, UMR Universit´e de Toulon, Aix Marseille Universit´e, CNRS, 83041 Toulon, France af Royal Netherlands Institute for Sea Research (NIOZ) and Utrecht University, Landsdiep 4, 1797 SZ ’t Horntje (Texel), The Netherlands agInternational Centre for Radio Astronomy Research — Curtin University, Bentley, WA 6102, Australia ahARC Centre of Excellence for All-sky Astrophysics (CAASTRO), Australia aiInstitut Universitaire de France, 75005 Paris, France ajDr. Remeis-Sternwarte and ECAP, Friedrich-Alexander-Universit¨at Erlangen-N¨urnberg, Sternwartstr. 7, 96049 Bamberg, Germany akMoscow State University, Skobeltsyn Institute of Nuclear Physics, Leninskie gory, 119991 Moscow, Russia alMediterranean Institute of Oceanography (MIO), Aix-Marseille University, 13288, Marseille, Cedex 9, France; Universit´e du Sud Toulon-Var, CNRS-INSU/IRD UM 110, 83957, La Garde Cedex, France amINFN — Sezione di Catania, Via S. Sofia 64, 95123 Catania, Italy anIRFU, CEA, Universit´e Paris-Saclay, F-91191 Gif-sur-Yvette, France aoINFN — Sezione di Pisa, Largo B. Pontecorvo 3, 56127 Pisa, Italy apDipartimento di Fisica dell’Universit`a, Largo B. Pontecorvo 3, 56127 Pisa, Italy aqINFN — Sezione di Napoli, Via Cintia, 80126 Napoli, Italy arDipartimento di Fisica dell’Universit`a Federico II di Napoli, Via Cintia, 80126 Napoli, Italy asDepartamento de F´ısica Te´orica y del Cosmos & C.A.F.P.E., University of Granada, 18071 Granada, Spain atGRPHE — Universit´e de Haute Alsace — Institut Universitaire de Technologie de Colmar, Rue du Grillenbreit 34, BP 50568-68008 Colmar, France E-mail: [email protected] Abstract: The ANTARES neutrino telescope has an energy threshold of a few tens of GeV. This allows to study the phenomenon of atmospheric muon neutrino disappearance due to neutrino oscillations. In a similar way, constraints on the 3+1 neutrino model, which foresees the existence of one sterile neutrino, can be inferred. Using data collected by the ANTARES neutrino telescope from 2007 to 2016, a new measurement of ∆m2 32 and θ23 has been performed — which is consistent with world best-fit values — and constraints on the 3+1 neutrino model have been derived. Keywords: Neutrino Detectors and Telescopes (experiments), Oscillation ArXiv ePrint: 1812.08650 JHEP06(2019)113 Contents 1 Introduction 1 2 The ANTARES neutrino telescope 4 3 ANTARES data and Monte Carlo samples 4 4 Event reconstruction 5 5 Analysis 6 5.1 Treatment of systematics for the standard oscillation analysis 9 5.2 Treatment of systematics for the sterile oscillation analysis 11 6 Results 12 6.1 Results for the standard oscillation analysis 12 6.2 Results for the sterile oscillation analysis 13 7 Conclusions 16 A Sterile neutrinos and matter effects 16 1 Introduction Neutrino oscillations arise from the mixing between flavour (νe, νµ, ντ) and mass (ν1, ν2, ν3) eigenstates. The mixing parameters of the Pontecorvo-Maki-Nakagawa-Sakata matrix [1–3] (PMNS) and the differences between the mass eigenvalues regulate the oscillation probability. Neutrino oscillations have been detected by a variety of experiments, studying solar as well as atmospheric neutrinos, but also neutrinos produced from nuclear reactors and particle accelerators. For a comprehensive review see [4]. Atmospheric neutrinos are produced through the interaction of cosmic rays with nuclei in the Earth’s atmosphere. Their flux spans many orders of magnitude in energy, from GeV to hundreds of TeV. Being isotropic to first order, it allows to investigate a large range of baselines on the Earth’s surface, from ∼10 km of vertically down-going to ∼104km of vertically up-going neutrinos. In this paper the muon disappearance channel (Pνµ→νµ) is studied. The vacuum survival probability for a muon neutrino of energy Einteracting at a distance Lfrom its creation point is given by: Pνµ→νµ= 1 −4X j>i |Uµj|2|Uµi|2sin2∆m2 jiL 4E∼1−4|Uµ3|2(1 − |Uµ3|2) sin2∆m2 32L 4E, (1.1) – 1 – JHEP06(2019)113 where Uµi, Uµj are elements of the PMNS matrix U, and ∆m2 ji =m2 j−m2 iare the mass splittings between two mass eigenstates. The rightmost term describes the “single ∆m2 dominance” approximation, relevant in the energy domain considered for this analysis. Here the νµsurvival probability depends only on Uµ3= sin θ23 cos θ13 and ∆m2 32. For a vertically up-going atmospheric νµ, the first minimum of the survival probability described in equation (1.1) is reached at energies of ∼25 GeV. The formalism given in eq. (1.1) is further modified by matter effects [5–7] as the neutrinos propagate through the Earth. Throughout the paper, oscillation probabilities are calculated with the OscProb package [8] which treats matter effects for an arbitrary number of neutrino families numerically without approximations. The ANTARES neutrino telescope [9] has been designed and optimised for the exploration of the high-energy Universe by using neutrinos as cosmic probes. However, its energy threshold of about 20 GeV is sufficient, even if at the edge, to be sensitive to the first atmospheric oscillation minimum, making also the study of neutrino oscillations possible. As neutrinos and antineutrinos are indistinguishable on an event-by-event basis in neutrino telescopes, in the following muon (electron) neutrinos are refered to the sum of contributions from both neutrinos and antineutrinos. A previous analysis of ANTARES data, covering the data acquisition period from 2007 to 2010, represented the first study of this kind performed by a neutrino telescope, and measured the atmospheric neutrino oscillation parameters, ∆m2 32 and θ23 [10]. In the present work, data collected during 10 years have been studied with a new analysis chain that also includes a more comprehensive treatment of various systematic effects. Despite the fact that neutrino oscillation is a well established phenomenon, some observed experimental anomalies, such as the ones reported by the LSND [11] and MiniBooNE [12] collaborations, seem to indicate a deviation from the standard 3-flavour picture. These discrepancies could be partially explained by introducing in the model an additional neutrino state. However, since the number of weakly interacting families of light neutrinos is limited to three by the LEP results [13], the additional neutrinos have to be sterile, i.e., they do not undergo weak interactions. The 3+1 neutrino model foresees the existence of one sterile neutrino in addition to the three standard ones. A choice has to be made, how to extend the mixing matrix U from three to four families. In this analysis the convention from [14] (see “supplementary materials”) is adopted: U3+1 =R34R24R14R23R13R12 where Rij is the rotation matrix for angle θij. If j−i > 1, Rij also contains a CP-violating phase, δij . Six new real mixing parameters have to be accounted for: three new mixing angles, θ14,θ24 and θ34, a new mass splitting, ∆m2 41, and two new phases, δ14 and δ24. In line with other analyses of sterile neutrinos in the muon disappearance channel [14–16], θ14 = 0 is assumed, which also eliminates any dependency on δ14. Even though a sterile neutrino does not interact as the active flavours, its presence would still modify the oscillation pattern of the standard neutrinos, due to the fact that the standard neutrino flavours could oscillate into these additional sterile species. In particular, – 2 – JHEP06(2019)113 [GeV] ν E 10 2 10 µ ν -> µ ν P 0.0 0.2 0.4 0.6 0.8 1.0 No Sterile ° = 0 24 δ = 0.00, 34 θ 2 = 0.10, sin 24 θ 2 sin ° = 0 24 δ = 0.10, 34 θ 2 = 0.00, sin 24 θ 2 sin ° = 0 24 δ = 0.05, 34 θ 2 = 0.05, sin 24 θ 2 sin ° ,270 ° = 90 24 δ = 0.05, 34 θ 2 = 0.05, sin 24 θ 2 sin ° = 180 24 δ = 0.05, 34 θ 2 = 0.05, sin 24 θ 2 sin Figure 1. Survival probability of vertically up-going νµas a function of neutrino energy (calculated with [8]) for different values of mixing angles θ24,θ34 and δ24 with ∆m2 41 = 0.5 eV2, ∆m2 31 = 2.5·10−3eV2and sin22θ23 = 1. for up-going νµin the energy range of 20–100 GeV, non-zero values of Uµ4and Uτ4with Uµ4=e−iδ24 sin θ24 ,(1.2) Uτ4= sin θ34 cos θ24 .(1.3) can lead to distortions in their survival probability. This is illustrated in figure 1which shows the νµsurvival probability for maximal mixing of θ23 and different combinations of the mixing parameters θ24, θ34 and δ24. If only θ34 is non-zero, the survival probability of νµwith respect to the non-sterile hypothesis is only modified close to the first oscillation minimum. The case of both θ24 and θ34 being non-zero leads instead to a significant shift of the first oscillation minimum in energy (depending on δ24) and modifies the event rate up to energies of few hundred GeV, easily accessible with ANTARES. The fast wiggles due to ∆m2 41 = 0.5 eV2will be smeared out by detector resolution effects, therefore no sensitivity to this parameter is expected. The surprisingly strong effect of δ24 on the νµsurvival probability, neglected in all similar analyses so far, is further detailed in the appendix. Since the effect of an additional sterile neutrino would be visible in the same energy and zenith range as the νµdisappearance, the same analysis chain and data sample can be exploited to constrain the 3+1 neutrino model parameters. In this paper, the results of an investigation aiming to constrain the mixing angles θ24 and θ34 of the 3+1 neutrino model are also reported. The paper is organised as follows: in section 2the ANTARES neutrino telescope is briefly described and its detection principle is illustrated; the ANTARES data sample as well as the Monte Carlo (MC) chain are presented in section 3, while the event reconstruction is discussed in section 4. Section 5is dedicated to the event selection and the – 3 – JHEP06(2019)113 minimisation procedure. The results are presented in section 6, while conclusions are given in section 7. 2 The ANTARES neutrino telescope The ANTARES neutrino telescope is located in the Mediterranean Sea, 40 km off the coast of Toulon, France, at a mooring depth of about 2475 m. The detector was completed in 2008. ANTARES is composed of 12 detection lines, each one equipped with 25 storeys of 3 optical modules (OMs), except line 12 with only 20 storeys of OMs, for a total of 885 OMs. The horizontal spacing among the lines is ∼60 m, while the vertical spacing between the storeys is 14.5 m. Each OM hosts a 10-inch photomultiplier tube (PMT) from Hamamatsu [17], whose axis points 45◦downwards. All signals from the PMTs that pass a threshold of 0.3 single photoelectrons (hits) are digitised and sent to the shore station [18,19]. The on-shore trigger system [20] performs a hit selection based on causality relations and builds events under the hypothesis that the selected hits originate from Cherenkov radiation induced by relativistic charged particles as they are produced in neutrino interactions close to the ANTARES instrumented volume. The main sources of optical background registered by the ANTARES PMTs are represented by Cherenkov light from decay products of the radioactive isotope 40K, naturally present in sea-water, by light emitted through bioluminescence by living organisms, and by energetic atmospheric muons, which can penetrate deeply under the sea and reach the detector from above. 3 ANTARES data and Monte Carlo samples ANTARES data collected from 2007 to 2016 have been considered in the analysis. After excluding data acquired under adverse conditions, a total of 2830 days of live time has been evaluated. The aim of the MC production is to reproduce in the most realistic way the events expected at the detector, as well as the response of the apparatus when recording these events. In order to account for changes of the environmental conditions, as well as for the different operational status of the detector and its components over time, a run-byrun MC approach is applied [21]. A typical run lasts few hours. Several time dependent conditions are taken from real data and applied to the run-by-run MC. First, temporarily or permanently non-operational OMs are masked in the simulation. Secondly, background light conditions, which might vary due to bioluminescence, are measured every 104 ms for each individual OM. These samples are directly used as input for the background light simulation. Thirdly, individual OM efficiencies are considered, as calculated on an approximately weekly basis from 40K coincidence rates [22]. Finally, the acoustics based position calibration, performed every few minutes, is applied. All these detailed inputs assure an authentic description of the detector response for each individual run. Remaining uncertainties are small and can be handled as global parameters which are discussed below. They are included in the analysis as systematic uncertainties. – 4 – JHEP06(2019)113 Neutrino interactions of all flavours have been simulated with the GENHEN [23] package, developed inside the ANTARES Collaboration. It allows to reproduce neutrino interactions in the GeV to multi-PeV energy range. MC neutrino events can be weighted to reproduce different physical expectations. For atmospheric neutrinos with Eν∈[20– 100] GeV, a MC sample almost three hundreds times larger than the data sample is available. The model by Honda et al. [24] for the Fr´ejus site is used in this work. Even though the sub-marine location of ANTARES provides a good shielding against atmospheric muons, still a large amount of them will reach the detector. The event generator used in ANTARES to simulate atmospheric muons is MUPAGE [25]; the energy and angular distributions, as well as the multiplicity of muons propagating in sea water are parameterised. The contribution from this background is also evaluated from the data itself. Particle propagation and Cherenkov light production are simulated using a GEANTbased [26] package [23], which takes into account all relevant physics processes and computes the probability that photons emitted by a particle reach the OM surface, producing a hit. Finally the detector response is simulated, including the digitisation and filtering of hits. At this stage a realistic optical noise is added on each OM for each data acquisition run of the detector, and the time evolution of the detector configuration is accounted for as described above. 4 Event reconstruction Charged-current (CC) interactions of muon neutrinos produce a muon propagating through the detector and inducing Cherenkov light. They are identified as track-like events. The event reconstruction and selection used in the analysis have been optimised to select such events. On the other hand, νeCC interactions, as well as neutral-current interactions (NC) of all flavours produce hadronic showers. In the case of νeCC interactions an electromagnetic shower is produced as well. Moreover, ντCC events can be produced as the result of νµ→ντoscillations with and without muons in the final state. All these events constitute an additional source of background for this study. Events have been reconstructed using two different algorithms, described in detail in [27,28]. In the following discussion these algorithms will be referred to as method A and method B, respectively. Both are optimised for events induced by GeV-scale νµCC interactions. In method Aa hit selection, based on time and spatial coincidences of hits, is applied and a χ2-fit is performed in order to find the best track. Events can have a singleline topology (SL), if all the selected hits have been recorded in the same detector line, or a multi-line topology (ML), when hits belong to OMs of different lines. Method Bconsists of a chain of fits, aimed to improve at each step the track estimation. Starting from a hit selection, a first prefit, based on a directional scan with a large number of isotropically distributed directions, is performed. The best 9 directions are used as starting points for the final likelihood (log L) fit. Once the muon track has been reconstructed, its length, Lµ, is computed. This is done, for ML events, by projecting back to the track the first and last selected hit. For SL events, – 5 – JHEP06(2019)113 since a vertex estimation is not possible due to the lack of azimuth information, the track length is estimated from the z-coordinates of the uppermost and lowermost storey which have recorded the selected hits and taking into account the reconstructed zenith angle. The muon energy estimation is based on the fact that muons in the few-GeV energy range can be treated as minimum ionising particles, and their energy can be estimated from their track length Lµ: Ereco =Lµ×0.24 GeV/m ,(4.1) where the factor 0.24 GeV/m represents the energy loss of muons in sea water in the energy range of 10–100 GeV [29]. This quantity is used in the following as estimator for the neutrino energy. The energy resolution of fully contained muons is dominated by the spacing of the detector elements and is found to be around 5 GeV. For muons leaving the detector only a lower limit for their energy can be derived, corresponding to their visible length inside the instrumented volume. More details on the muon energy resolution can be found in [10]. 5 Analysis To achieve the best sensitivity to the measurement of the oscillation parameters, a set of quality criteria has been applied. The selection of νµCC events has been optimised by performing a preliminary Monte Carlo (MC) sensitivity study, before applying the whole analysis chain to data. The main parameter on which the selection is based is the reduced χ2for method Aand the log Lfor method B. Events reconstructed by method Aand passing the corresponding event selection are kept. The events discarded by this procedure are further reconstructed by method B; they are kept in the analysed sample if the corresponding selection criteria are passed. Only events which are reconstructed as up-going are used in the following. A minimum number of five storeys with selected hits is required, in order to minimise the background induced by atmospheric muons. In figure 2the distribution of the MC true neutrino energy, ET, for selected νµCC events is shown. For the histogram with the solid line no neutrino oscillations are assumed, while the dashed one refers to a 2-flavour oscillation scenario with maximal mixing and ∆m2 32 = 2.46 ×10−3eV2. As can be seen, atmospheric neutrino oscillations affect the expected event distribution for ET.100 GeV. About 7590 well-reconstructed νµCC events are expected in a live time of 2830 days when oscillations are neglected. Roughly one half of these events are reconstructed with method A(ML), while methods A(SL) and Bboth contribute with approximately 25% to this event sample. Further, ∼40 νe CC events are selected. Oscillations reduce the number of expected events by ∼720 events. This reduction is dominantly seen in the A(SL) sample (∼60%) which contains the lowest energetic and most vertical events, while the other two reconstruction methods contribute each about 20%. ντCC events reduce this oscillation signal by ∼20 events, taking into account the energy-dependent cross section ratio σ(ντCC)/σ(νµCC) (about – 6 – JHEP06(2019)113 [GeV] reco θ/cos reco E 50 100 150 200 evt N 0 200 400 600 800 1000 MC no osc MC world best fit MC best fit Data ANTARES [GeV] reco θ/cos reco E 50 100 150 200 no-osc /N evt N 0 0.2 0.4 0.6 0.8 1 1.2 1.4 MC world best fit MC best fit Data ANTARES Figure 6.Ereco/cos θreco distribution for data (black), MC without oscillation (red), MC assuming the world best-fit values (blue) [38] and MC assuming best-fit values of this analysis (green). The left plot shows event numbers while the right plot illustrates the event ratio with respect to the MC without oscillations. nomenon as can be seen for the lowest values of Ereco/cos θreco. For comparison, also the distribution of MC assuming no neutrino oscillation, as well as the one assuming the world best-fit values [38] are shown. The latter two are calculated with all nuisance parameters at their nominal values. Such a 1D distribution does not carry the full information exploited in the fit, which is performed on the 2D distribution shown in figure 4. While compatible with world data, ANTARES results seem to prefer a somewhat shallower (or energy shifted) oscillation minimum. In figure 7the 90% CL contour obtained in this work, in the plane of sin2θ23 and ∆m2 32, is compared to those published by other experiments. The 1D projections, after profiling over the other variable, are shown as well. Confidence level contours have been computed by looping over a fine grid of values in ∆m2 32 and θ23 and minimising the negative log-likelihood over all the other parameters. The non-oscillation hypothesis has been tested by performing the minimisation with a fixed null value of the oscillation parameters, and it is discarded with a significance of 4.6σ, compared to 2.3σin our previous analysis [10]. 6.2 Results for the sterile oscillation analysis In table 3the complete list of all the fitted parameters for the sterile oscillation analysis for NH and IH is shown, together with their best-fit values and their priors. While θ24 is found to be compatible with zero, the best fit for θ34 is found at a non-zero value. This can be understood from the slight preference of the ANTARES data for a shallower oscillation dip (see discussion related to figure 6), which can be easily provided by a nonzero value of sin θ34 (see figure 1). The non-sterile hypothesis is found at −2∆ log L= 4.4 which corresponds to a 2-parameter p-value of 11%. The fitted values of ∆m2 32 and θ23 are slightly different but consistent with respect to the ones obtained in the standard oscillation analysis. The complex phase δ24 is found at 180◦. For IH instead the fit prefers δ24 = 0◦ – 13 – JHEP06(2019)113 Figure 7. Contour at 90% CL in the plane of sin2θ23 and ∆m2 32 obtained in this work (black line) and compared to the results by other experiments: IceCube/DeepCore (red) [31], SuperKamiokande (green) [39], NOνA (purple) [40], T2K (blue) [41], and MINOS (light blue) [42]. The lateral plots show the 1D projections on the plane of the two oscillation parameters under study. Parameter Prior Fit NH Fit IH θ24 [◦] none 1.5+2.0 −5.01.5+2.0 −5.0 θ34 [◦] none 25.9+5.1 −4.225.9+5.1 −4.2 δ24 [◦] none 180 ±71 0 ±72 nνnone 0.84+0.10 −0.09 0.84+0.10 −0.09 ν/ν [σ] 0.0±1.0 1.07+0.63 −0.55 1.07+0.63 −0.55 ∆γ0.00 ±0.05 −0.011 ±0.036 −0.011 ±0.036 ∆m2 32 [10−3eV2] none 3.0+0.8 −0.6−3.0+0.6 −0.8 θ23 [◦] none 52 ±8 52 ±8 θ13 [◦] 8.41 ±0.28 8.41 ±0.28 8.41 ±0.28 MA[σ] 0.0±1.0 0.11+0.93 −0.97 0.11+0.93 −0.97 Table 3. Priors and fitted values obtained from the minimisation for all the parameters considered in the sterile oscillation analysis. with otherwise identical results, as expected from the degeneracy between NMH and δ24 (see appendix). For the other parameters a similar behaviour as for the standard oscillation analysis is observed. Exclusion contours are built by applying Wilks’ theorem. In figure 8the resulting 90% and 99% CL exclusion limits have been computed on a 2D grid in the plane of the two matrix elements, namely |Uµ4|2= sin2θ24 and |Uτ4|2= sin2θ34 cos2θ24. The exclusion limit for unconstrained δ24, which corresponds to both [NH,δ24 = 180◦] or [IH,δ24 = 0◦], can be – 14 – JHEP06(2019)113 8 0 2 4 6 8 10 0 2 4 6 8 10 -2�LogL 24 θ 2 sin 3− 10 2− 10 1− 10 24 θ 2 cos 34 θ 2 sin 2− 10 1− 10 90% C.L. This work ° = 0 24 δThis work NH, ° = 0 24 δIceCube (2017) NH, ° = 0 24 δIceCube (2017) IH, ° = 0 24 δSK (2015) NH, 24 θ 2 sin 3− 10 2− 10 1− 10 99% C.L. This work ° = 0 24 δThis work NH, ° = 0 24 δIceCube (2017) NH, ° = 0 24 δIceCube (2017) IH, ° = 0 24 δSK (2015) NH, 0 2 4 6 8 10 0 2 4 6 8 10 -2�LogL Figure 8. 90% (left) and 99% (right) CL limits for the 3+1 neutrino model in the parameter plane of |Uµ4|2= sin2θ24 and |Uτ4|2= sin2θ34 cos2θ24 obtained in this work (black lines), and compared to the ones published by IceCube/DeepCore [16] (red) and Super-Kamiokande [15] (blue). The dashed lines are obtained for NH and δ24 = 0◦while the solid lines are for an unconstrained δ24 (this work) or for IH and δ24 = 0◦(IceCube/Deepcore) respectively. The coulored markers indicate the best-fit values for each experiment. The 1D projections after profiling over the other variable are also shown for the result of this work. directly compared to the IceCube/DeepCore [16] (IH) limit. Also shown are limits for NH and δCP = 0◦which allow a direct comparison with the results from IceCube/DeepCore [16] (NH) and Super-Kamiokande [15]. All three experiments find the best fit for |Uτ4|2to differ from zero. Our results exclude regions of the parameter space not yet excluded by other experiments. The IceCube/DeepCore analysis [16] is limited to events with reconstructed energy lower than 56 GeV, while the distortion on the oscillation pattern possibly produced by the presence of a sterile neutrino would be evident also at higher reconstructed energies. The present analysis includes events with reconstructed energy up to 100 GeV. It has been verified that the ANTARES limits degrade when restricting the analysis to events with Ereco <56 GeV. In this work both of the standard atmospheric oscillation parameters ∆m2 32 and sin2(2θ23) are left unconstrained in line with the IceCube/DeepCore analysis [16]. After profiling over the other variable, the following limits on the two matrix elements can be derived: |Uµ4|2<0.007 (0.13) at 90% (99%) CL ,(6.1) |Uτ4|2<0.40 (0.68) at 90% (99%) CL .(6.2) – 15 – JHEP06(2019)113 7 Conclusions Ten years of ANTARES data have been analysed to provide a measurement of the atmospheric neutrino oscillation parameters. The analysis chain has been optimised with respect to our previously published study, by combining two track reconstruction algorithms and introducing a more elaborate treatment of various sources of systematic uncertainties. The results, ∆m2 32 = (2.0+0.4 −0.3)×10−3eV2and θ23 = (45+12 −11)◦, are consistent with what has been published by other experiments. The non-oscillation hypothesis is discarded with a significance of 4.6σ. Exploiting the same analysis chain and the same data set, a further study has allowed to constrain, for the first time with ANTARES, the parameter space of the 3+1 neutrino model, which foresees the existence of one sterile neutrino. ANTARES excludes values of the parameter space not yet excluded by other experiments. Acknowledgments The authors acknowledge the financial support of the funding agencies: Centre National 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), IdEx program and UnivEarthS Labex program at Sorbonne Paris Cit´e (ANR-10-LABX-0023 and ANR-11-IDEX-0005-02), Labex OCEVU (ANR-11-LABX-0060) and the A*MIDEX project (ANR-11-IDEX-0001-02), R´egion ˆ Ile-deFrance (DIM-ACAV), R´egion Alsace (contrat CPER), R´egion Provence-Alpes-Cˆote d’Azur, D´epartement du Var and Ville de La Seyne-sur-Mer, France; Bundesministerium f¨ur Bildung und Forschung (BMBF), Germany; Istituto 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 leading scientific schools supporting grants, Russia; Executive Unit for Financing Higher Education, Research, Development and Innovation (UEFISCDI), Romania; Ministerio de Econom´ıa y Competitividad (MINECO): Plan Estatal de Investigaci´on (refs. FPA2015-65150-C3-1-P, -2-P and -3-P, (MINECO/FEDER)), Severo Ochoa Centre of Excellence and Red Consolider MultiDark (MINECO), and Prometeo and Grisol´ıa programs (Generalitat Valenciana), Spain; Ministry of Higher Education, Scientific Research and Professional Training, Morocco. We also acknowledge the technical support of Ifremer, AIM and Foselev Marine for the sea operation and the CC-IN2P3 for the computing facilities. We would like to thank J. Coelho for enlightening discussions on matter effects for sterile neutrinos. A Sterile neutrinos and matter effects For the analysis presented in this paper, oscillation probabilities are evaluated with the software package OscProb [8]. However, in this appendix some common approximations are applied, to derive analytical formulae. These are NOT used for the analysis itself but allow to get a better understanding of the interplay between different parameters. The νµ – 16 – JHEP06(2019)113 survival probability in vacuum in the 3 + 1 model can be simplified with the following two hypotheses [15,43]: first, it is assumed, that the first generation decouples completely, i.e. ∆m2 21 = 0 and θ12 =θ13 = 0; second, fast wiggles due to oscillations involving m4are assumed to be unobservable, i.e. sin2(∆m2 4iL/4E) = 1/2 for all i. This yields Pνµ→νµ= (1 − |Uµ4|2)2P(3) µµ +|Uµ4|4,(A.1) with P(3) µµ the νµsurvival probability in the 3-flavour scheme, i.e. without additional sterile neutrinos. Only |Uµ4|2= sin2θ24 can be probed in this scheme, which is applied in most accelerator based νµdisappearance analyses. However, when analysing atmospheric neutrinos, matter effects cannot be neglected. An analytical formalism is developped in eqs. (4.13)–(4.25) of [15]. In eq. (4.13), a complex phase is present in the non-diagonal term of the matrix, which is neglected, i.e. set to zero, in subsequent steps. If instead this phase is kept, sin 2θsin eq. (4.16) acquires an extra term exp(−iδ). sin 2θs=2p|Uµ4|2|Uτ4|2(1 − |Uµ4|2− |Uτ4|2) (1 − |Uµ4|2)(|Uµ4|2+|Uτ4|2)e−iδ,(A.2) cos 2θs=|Uτ4|2− |Uµ4|2(1 − |Uµ4|2− |Uτ4|2) (1 − |Uµ4|2)(|Uµ4|2+|Uτ4|2).(A.3) This in turn modifies eqs. (4.18) and eq. (4.19): E2 m=A2 32 +A2 s+ 2A32As(sin 2θ23|sin 2θs|cos δ+ cos 2θ23 cos 2θs),(A.4) sin 2θm=1 EmqA2 32 sin22θ23 +A2 s|sin 2θs|2+ 2A32Assin 2θ23|sin 2θs|cos δ . (A.5) For δ= 0 the original expressions from [15] are reproduced. With A32 = ∆m2 32/Eνand As=√2 2GFNn(|Uµ4|2+|Uτ4|2)/2 (GFthe Fermi constant and Nnthe neutron density) the νµsurvival probability in matter is fully defined and can be written equivalently to eq. (A.1) (see also eq. (4.23) of [15]): Pνµ→νµ= (1 − |Uµ4|2)21−sin22θmsin2(EmL)+|Uµ4|4,(A.6) which describes well all features shown in figure 1. The impact of the CP-phase δdisappears when either |Uµ4|2= 0 or |Uτ4|2= 0, which leads to sin 2θs= 0. 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