Feasibility of Determining Diffuse Ultra-High Energy Cosmic Neutrino Flavor Ratio through ARA Neutrino Observatory
aa r X i v : . [ a s t r o - ph . H E ] N ov Feasibility of Determining Diffuse Ultra-High Energy CosmicNeutrino Flavor Ratio through ARA Neutrino Observatory
Shi-Hao Wang,
1, 2, ∗ Pisin Chen,
1, 2, 3, 4, † Jiwoo Nam,
1, 2, ‡ and Melin Huang § Graduate Institute of Astrophysics,National Taiwan University, Taipei 10617, Taiwan, R.O.C. Leung Center for Cosmology and Particle Astrophysics,National Taiwan University, Taipei 10617, Taiwan, R.O.C. Department of Physics, National Taiwan University, Taipei 10617, Taiwan, R.O.C. Kavli Institute for Particle Astrophysics and Cosmology,SLAC National Accelerator Laboratory, Menlo Park, CA 94025, U.S.A.
Abstract
The flavor composition of ultra-high energy cosmic neutrinos (UHECN) carries precious informa-tion about the physical properties of their sources, the nature of neutrino oscillations and possibleexotic physics involved during the propagation. Since UHECN with different incoming directionswould propagate through different amounts of matter in Earth and since different flavors of chargedleptons produced in the neutrino-nucleon charged-current (CC) interaction would have differentenergy-loss behaviors in the medium, measurement of the angular distribution of incoming eventsby a neutrino observatory can in principle be employed to help determine the UHECN flavor ratio.In this paper we report on our investigation of the feasibility of such an attempt. Simulations wereperformed, where the detector configuration was based on the proposed Askaryan Radio Array(ARA) Observatory at the South Pole, to investigate the expected event-direction distribution foreach flavor. Assuming ν µ - ν τ symmetry and invoking the standard oscillation and the neutrinodecay scenarios, the probability distribution functions (PDF) of the event directions are utilized toextract the flavor ratio of cosmogenic neutrinos on Earth. The simulation results are summarizedin terms of the probability of flavor ratio extraction and resolution as functions of the number ofobserved events and the angular resolution of neutrino directions. We show that it is feasible toconstrain the UHECN flavor ratio using the proposed ARA Observatory. ∗ [email protected] † [email protected] ‡ [email protected] § [email protected] . INTRODUCTION The observed energy spectrum of comic rays has been extended to beyond 10 eV [1, 2],but little is known about their origins and acceleration mechanism, which are importantquestions in astrophysics [3]. Ultra-high energy cosmic rays (UHECRs) are thought to be ofextragalactic origin, such as being produced by active galactic nuclei (AGNs) or gamma-raybursts (GRBs) [4]. Such UHECRs can generate ultra-high energy ( E > eV) neutrinosvia photo-pion production or proton-proton interaction: p + γ → ∆ + → n + π + ,p + p → π + π − π , and the subsequent decays of charged pion and muon, e.g., π + → ν µ + µ + → ν µ + ¯ ν µ + ν e + e + , where the targets can be the intergalactic medium near the astrophysical sources [5, 6] (seeRef. [7] for a review), or the cosmic microwave background (CMB) photons (the Greisen-Zatsepin-Kuzmin process) [8, 9]. Neutrinos originating from the GZK process are knownas the cosmogenic neutrinos (or GZK neutrinos), which are guaranteed to exist based onthe fact that both initial-state particles, i.e., the UHECR and the CMB photon, have beenobserved and that the notion is consistent with the observed GZK cutoff in the cosmic rayspectrum [10]. Since the production of ultra high energy cosmic neutrinos (UHECNs) aretightly connected with UHECRs, such neutrino spectrum can help to resolve the puzzles ofcosmic rays such as their composition [11, 12], the energy spectrum at the sources, and thecosmological evolution of the sources [13, 14].Besides the overall spectrum, the relative flux ratio between different neutrino flavors, orbriefly, the flavor ratio, can also provide information about the physical properties of UHECRsources. For example, the transition of flavor ratio at the source from f Se : f Sµ : f Sτ = 1 : 2 : 0(pion source) to 0 : 1 : 0 (muon-damped source) with increasing energies due to synchrotronenergy loss of muons can be used to constrain the strength of cosmic magnetic field [15–17](neutrinos and antineutrinos are counted together because they are hard to be discriminatedin the UHE neutrino detection). Furthermore, during the propagation from the source to theEarth, the flavor composition of a neutrino would oscillate [18], and may even be altered by2ome new physics beyond the Standard Model, such as the neutrino decay [19–23], pseudo-Dirac states of neutrinos [24], sterile neutrinos with tiny mass differences [25], the violationof CPT or Lorentz invariance [20, 23, 26], and the quantum decoherence [23, 26, 27] (seeRef. [28] for a review). With extremely high energies and long traveling distances ( > To detect UHECNs, enormous amount of matter is required for the target due to their lowflux and tiny interaction cross section. There are four major detection strategies dependingon either neutrinos interact with nucleons via the neutral current (NC) interaction, ν l + N → ν l + X , or via the charged current (CC) interaction, ν l + N → l − + X , where l stands for lepton and X for hadronic debris that will develop into hadronic showers. Thefirst approach is to observe the optical Cherenkov lights emitted by secondary chargedparticles and showers by an array of optical sensors (e.g. photomultiplier tubes) deployeddeep in the medium, e.g. under-ice arrays such as AMANDA [35] and IceCube [36] at theSouth Pole, and Baikal, ANTARES, NESTOR, NEMO, KM3NET [37] underwater. Thesecond one is to detect horizontal or Earth-skimming neutrino-induced air showers, suchas Pierre Auger Observatory [38] and HiRes [39]. The third approach is to detect acousticwaves generated by the showers, which is still in the R&D stage [40]. The last and a verypromising one is to observe the radio Cherenkov emission from the neutrino-induced showersin dense media through the Askaryan effect [41]. Showers propagating in dense media woulddevelop about 20 % of excess negative charges and would emit Cherenkov radiation, whichis coherent in the radio frequencies up to a few GHz due to the compact shower size. Thiseffect has been verified in a series of beam experiments [42]. The radiated power in thecoherent regime is proportional to the square of net charges, which is roughly proportionalto the shower energy, making this technique especially sensitive to UHE showers and thusUHECNs. Another advantage of this approach is the long radio attenuation length in somenatural media, e.g. the Polar ice and salt, with lengths typically of order of 0.1 to 1 km, andtherefore detectors are able to monitor large target volume and achieve greater sensitivity.Observatories of this type are: the FORTE satellite [43] looking for neutrino signals fromthe Greenland ice; the balloon-borne antenna array ANITA [44] overlooking the Antarcticice; and radio telescopes looking for signals from the lunar regolith, e.g. GLUE [45] and In fact, most of these references consider neutrinos with energy > PeV. eV), e.g. under-ice antenna arrays RICE[47], ARA [48], and ARIANNA [49] in the Antarctic ice; and SalSA [50] in salt dome.It is impossible to distinguish between neutrino flavors from the NC interactions be-cause their only products are hadronic showers. On the other hand, the charged leptonsproduced in the CC interactions have different energy-loss characteristics in the medium,which provides an opportunity to identify the flavor. Optical Cherenkov neutrino telescope,e.g. IceCube, is able to identify the flavors by event topologies. The muon from the ν µ CCinteraction leaves a track; whereas showers from all flavor’s NC events and ν e ’s CC eventslead to localized trigger patterns; and there are double-bang and lollipop events unique to ν τ induced by the τ decays. Beacom et al. [51] proposed a method to deduce the neutrinoflavor ratio from the measured events of different types, and its feasibility has been widelyinvestigated (e.g. [21, 29, 30, 32, 52]). However, the instrumented volume, currently of cubickilometer scale for the largest, limits the event rate for UHE neutrinos, which renders itchallenging to distinguish between ν µ and ν τ events at UHEs as the decay length of τ leptonexceeds the detector size [53].For radio Cherenkov telescopes such as ARA, the situation is a somewhat different.Though this approach cannot detect the track by a single charged particle, the ν µ and ν τ CC events can in principle be separated through the amount of energy deposited byleptons into electromagnetic and hadronic showers [53]. It has also been pointed out thatdifferent types of showers at UHEs can be distinguished according to their elongation by theLandau-Pomeranchuk-Migdal (LPM) effect [54, 55]. In addition, ν e CC events can generatemixed showers of both types and should have its own characteristic signal feature [56]. Thefeasibility of this method has been investigated in SalSA [57].Apart from the event signatures, the direction distribution of neutrino events also man-ifest themselves in the the energy-loss properties of leptons since neutrinos from differentdirections propagate through different amounts of matter in the Earth. For example, ν τ can undergo the regeneration process ( ν τ → τ → ν τ ) without being absorbed due to the τ decay [53, 58], and as a result can exhibit a higher flux in the up-going directions. Thereforethe neutrino distribution can be a useful tool for measuring the flavor ratio and should beapplicable to any detector with sufficient angular resolution of event direction. A similaridea that takes advantage of the event direction distribution has been proposed to constrain4eutrino-nucleon cross sections [59].In this paper we focus on the cosmogenic neutrinos and consider three expected flavorratios when they arrive at the Earth’s surface, that is, 1 : 1 : 1 expected in the standardoscillation scenario [18]; and 6:1:1 as well as 0:1:1 ratios predicted in the neutrino decaymodels [19] with normal and inverted neutrino mass hierarchy, respectively. The sum ofratios is normalized to unity in the following sections. With the detector configuration basedon ARA [48] currently under construction at the South Pole, we demonstrate the feasibilityof extracting the flavor ratio from the event direction distribution, while the inference onthe flavor ratio at the source is beyond our scope.In the next section we present the simulation setup, and derive the expected event direc-tion distribution for each flavor in Section III. The procedures for hypothetical experimentsand the extraction of flavor ratios from the direction distribution of pseudo-data are de-scribed in Section IV A and IV B. The successful probability of this method and the flavorratio resolution as functions of the number of observed events and angular resolution ofneutrino direction are reported in Section IV C. II. SIMULATION SETUP
Our simulation is built by integrating existing packages that consists of two parts. Thefirst part is the propagation of the neutrinos and the secondary charged leptons. Thesecond part is the simulation of event detection, including the conversion of particle energylosses to showers, the development of Cherenkov radiations from showers, the propagation ofCherenkov radiations to detectors, and the calculation of detector responses. The plannedconfiguration of ARA [48] and the ice properties at the South Pole are adopted as the settingin our simulations.
A. Neutrino Generation and Propagation Using MMC
The Muon Monte Carlo (MMC) package [60] is employed to generate neutrinos andpropagate all types of neutrinos and secondary charged leptons. In MMC, interaction crosssections of neutrinos are evaluated based on Ref. [61] with CTEQ6 parton distributionfunctions [62]. 5or charged leptons, energy losses via ionization, pair production, bremsstrahlung, pho-tonuclear interaction, and decay are taken into account. The Kelner-Kokoulin-Petrukhin(KKP) [63] and Bezrukov-Bugaev (BB) [64] parameterizations are chosen for cross sec-tion calculations of bremsstrahlung and photonuclear interaction, respectively. We do notpropagate secondary electrons and regard them as losing all of their energies to shower de-velopments within a short distance once they are generated. Taus decay into electron, muon,or hadrons are considered, and hence the ν τ → τ → ν τ regeneration in the propagation.Monoenergetic neutrinos are generated isotropically at the Earth’s surface and start theirpropagation to the detection volume. The Earth model provided in MMC simulation codeis used, where the density is calculated based on the Preliminary Reference Earth Model(PREM) [65] while the composition as well as the topography on the Earth’s crust are notconsidered. The detection volume, which is the South Polar ice sheet in the vicinity of ARA,is approximated by a cylindrical ice volume, centered at 1 km below the ice surface with aradius of 8 km and a height of 2 km. In the propagation, all neutrinos and charged leptonsare tracked until they either are absorbed by the Earth or exit the detection volume; energylosses greater than 1 PeV are treated stochastically. Only those neutrino events traversingthe detection volume with at least one stochastic energy loss are reserved for the eventdetection. The cutoff energy is chosen based on the consideration that the radio Cherenkovsignals emitted by showers below this energy would not be strong enough to trigger thedetector efficiently. B. Neutrino Event Detection Using SADE
The Simulation of Askaryan Detection and Events (SADE) package [66] is used for sim-ulating neutrino detection. Neutrino events recorded in the previous step, as describedin Section II A, are processed individually. Every energy loss exceeding 1 PeV within thedetection volume is converted into showers of corresponding types. Hadronic products ofneutrino CC and NC interactions, τ decay, and photonuclear interactions are turned intohadronic showers, while secondary electron, pair production, and bremsstrahlung are turnedinto electromagnetic showers. In general, a neutrino event can generate multiple showersin the detection volume. For example, a ν e event having CC interaction would generate ahadronic shower and an electromagnetic shower, whereas a ν µ or a ν τ event may have several6o even hundreds of showers produced by the secondary µ or τ lepton, respectively.In SADE, shower characteristics and the frequency spectrum of Cherenkov radiationare calculated with analytic formulae, where the latter is primarily based on Ref. [67] butwith a little bit modification of parametrization to account for the decoherence due to thelongitudinal and the lateral spreads of shower.A ray-tracing routine then finds the paths of both the direct and the reflected (due tothe ice-air interface at the surface) rays connecting each shower to each antenna in the icewhose index of refraction varies with depth. The flight time, the radiation spectrum takeninto account the frequency-dependent attenuation along the path as well as the polarizationat the antenna are calculated for each ray. The spectrum is then Fourier-transformed to theelectric field received by the antenna in the time domain accordingly.Ice properties such as the index of refraction [68], the temperature [69], and the radioattenuation length [48, 70] are based on the results of in situ measurements at the SouthPole. The index of refraction n and the ice temperature T (in ◦ C) depend only on the depth(in km), | z | , n ( z ) = 1 . − (1 . − .
35) exp( − . | z | ) , (1) T ( z ) = − . − . | z | + 5 . | z | . (2)The attenuation length in turn depends on the ice temperature as well as the radiationfrequency, and is plotted in Fig. 1. Note that the rising ice temperature with increasingdepth renders the attenuation length shorter and thus suppresses the detectability of signalsoriginating from the bottom part of ice. The birefringence of South Polar ice, which is thepolarization dependence of the wave speed and the attenuation due to the crystal anisotropyand orientation of ice, is not considered here. It is reported [71] that the birefringence isobserved at the bottom half of the ice sheet and will reduce about 5 % of the neutrinodetection volume.The detector based on the planned configuration of ARA [48] is a hexagonal array of37 antenna stations arranged in a triangular grid with 2 km spacing. The array covers atotal area of about 100 km . Each station is an autonomously operating cluster of eightvertically polarized (Vpol) and eight horizontally polarized (Hpol) antennas evenly deployedon four vertical strings with each string placed on one vertex of a square. Each string is ata maximum depth of 200 m and is loaded with two antenna pairs, where each antenna pair7 Frequency [GHz]0.2 0.4 0.6 0.8 1 1.2 A tt e nu a t i on L e ng t h [ k m ] Attenuation length v.s. Frequency in SADE C ° T= -50 C ° T= -45 C ° T= -40 C ° T= -30
Depth [km]0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 A tt e nu a t i on L e ng t h [ k m ] Attenuation length v.s. Depth in SADE f = 300 MHzf = 500 MHzf = 700 MHz
FIG. 1. Attenuation length used in SADE [66] as a function of the radiation frequency at differenttemperatures ( − ◦ C, − ◦ C, − ◦ C and − ◦ C, left panel) and as a function of the depth atdifferent frequencies (300 MHz, 500 MHz and 700 MHz, right panel), respectively. contains a Vpol antenna and an Hpol antenna. The values of parameters for the detectorsettings are listed in Table I.Given the incident electric field ~E at an antenna, the received signal voltage of the antenna V signal is V signal = 12 ~E · ~h eff , (3)with | ~h eff | = 2 r A eff Z ant nZ , (4) A eff = Gc πf , (5)where ~h eff and A eff are the effective height and the effective area of the antenna, respectively; Z ant the antenna impedance, Z ≃ n the index of refractionof surrounding medium, G the antenna gain, and c the speed of light in vacuum. Thedirection of effective height for Vpol antennas is in the vertical direction ˆ z , while it is in theazimuthal direction ˆ φ for the Hpol. In the simulation, the frequency response of the antennais assumed to be a single perfect passband, and the effective height is approximated by asingle value evaluated at the central frequency of the passband. For each antenna, signals8oming from different showers are summed in the time domain. We neglect the contributionfrom the reflected signals because they would have sufficient time delays and would suffermore attenuation than the direct ones due to longer path length and are more difficult tobe reconstructed after passing through the less compact snow near the surface (firn) wherethe index of refraction changes rapidly (see Eqn. 1).The root mean square (RMS) thermal noise voltage of an antenna V rms is defined as V rms = p k B T sys Z ant B, (6)where k B is Boltzmann’s constant, T sys the system noise temperature of the receiving antennasystem, f the radiation frequency and B the frequency bandwidth of the antenna. The valuesof antenna parameters used in the simulation are summarized in Table I.The trigger conditions for a detected event require that i ) the received voltage of atriggered antenna should exceed three times of its RMS noise voltage (i.e., V signal ≥ V rms ); ii ) at least five out of sixteen antennas in a station are triggered; and iii ) at least one station is triggered. III. ANGULAR DISTRIBUTION OF NEUTRINO EVENTS
In the simulation, monoenergetic neutrinos with initial energies log ( E ν / eV) = 17, 17 . .
5, 19 and 19 . E ν and flavor α , D α ( E ν , cos θ ), the followinginformation are needed: a ) the flux of isotropic cosmogenic neutrinos for all flavors, Φ ν ( E ν ), as well as the incidentflux ratio among three flavors at the Earth’s surface, f Ee : f Eµ : f Eτ , with P α f Eα = 1; b ) the interaction probability P int ,α ( E ν , cos θ ), which is defined as the probability thata neutrino or its secondary lepton traverses the Earth in the zenith direction cos θ with-out being stopped and interacts (with at least one energy loss exceeding 1 PeV) inside thedetection volume, to account for the propagation effect; and c ) the detection efficiency to the subsequent shower(s) generated by the neutrino eventinteracting inside the detection volume, ǫ det ,α ( E ν , cos θ ). That is, D α ( E ν , cos θ ) = 1 N f Eα Φ ν ( E ν ) P int ,α ( E ν , cos θ ) (7) × ǫ det ,α ( E ν , cos θ ) , ABLE I. Parameters for the detector configuration and the antennas, and their values used inthe simulation.Parameter (unit) ValueStation spacing (km) 2Radius of string (m) 10Number of strings per station 4Separation between paired antennas (m) 5Vertical spacing between antenna pairs (m) 20Maximum antenna depth (m) 200Number of Vpol antennas per station 8Number of Hpol antennas per station 8Vertical antenna configuration Vpol, Hpol above Vpol, HpolVpol frequency band: B V (MHz) 150-850Hpol frequency band: B H (MHz) 200-850Antenna impedance: Z ant (Ω) 50Antenna gain: G | ~h eff,V | (cm) 11.8Effective height of Hpol antenna: | ~h eff,H | (cm) 11.3System noise temperature of antenna: T sys (K) 300RMS noise voltage of Vpol: V rms,V (V) 1 . × RMS noise voltage of Hpol: V rms,H (V) 1 . × − Antenna trigger threshold ( V rms ) 3Station trigger threshold (antennas) 5 with the normalization factor N = X α Z dE ν Z − d cos θf Eα Φ ν ( E ν ) (8) × P int ,α ( E ν , cos θ ) ǫ det ,α ( E ν , cos θ ) , where the distribution has been normalized as a probability distribution independent of thetotal event rate, α = e, µ, τ , the superscript E indicates quantities on the Earth, and θ is10he angle between the local vertical axis of the detector (ˆ z ) and the direction of neutrinomomentum. Hence, events with negative cos θ are down-going, whereas those with positivevalues are up-going. The neutrino energy range considered in this article is log ( E ν / eV) =16 . . P int ( E ν , cos θ ) and detection efficiency ǫ det ( E ν , cos θ )has averaged over all events with energy loss above the threshold 1 PeV in the detectionvolume. In addition, in our simulation result, for more than about 95% of ν µ and ν τ eventsonly one shower can be detected, so we did not separate the detection efficiency and the an-gular distribution into single cascade channel from CC/NC interaction and multiple cascadechannel from µ / τ leptons. A. Flux and Flavor Ratio on Earth
The cosmogenic neutrino fluxes for all flavors, Φ ν ( E ν ), have been theoretically predictedin Refs. [14, 72, 73] (see Fig. 2). In the following analysis, the neutrino flux from Ref. [72](red solid curve; hereafter, ESS) is assumed. Note that it is the spectral shape that affectsthe event direction distribution instead of the overall flux normalization. Therefore oneexpects that neutrino fluxes predicted in [72], [73] (green dashed curve), and the optimisticscenario in [14] (purple dashed curve) should yield similar distributions. The flux predictedin the plausible scenario in [14] (blue dashed curve) has a steeper spectrum than others, andwe will present its results later in Sec. IV.The flavor ratio of neutrinos arriving at the Earth’s surface adopted in our analysis is f Ee : f Eµ : f Eτ = 1 / / /
3, as expected in the standard oscillation scenario [18];0 .
75 : 0 .
125 : 0 .
125 and 0 : 0 . . E/eV) log15 16 17 18 19 20 21 ] - s r - s - ( E )) [ c m Φ ( E l og -21-20-19-18-17-16-15-14-13 ESS2001Ahlers2011Kotera2010, the "optimistic"Kotera2010, the "plausible"
FIG. 2. Differential cosmogenic neutrino fluxes predicted by [72] (ESS, red solid curve), [73] (greendotted), the optimistic (blue dashed) and the plausible (purple dot-dashed) scenarios in [14]. Theshaded region indicates the neutrino energy range considered in the analysis, log ( E ν / eV) = 16 . . B. Interaction Probability
To obtain the interaction probability P int from the simulation results, we first divide thezenith angle of neutrinos cos θ into 100 bins with a width of 0 .
02. The probability at eachbin is defined as the ratio of the number of survival events inside the detection volumeto the number of initial incoming events at the Earth’s surface. The probability does notdepend on the azimuthal angle of the neutrino due to the axial symmetry of our Earth modeland detection volume. The probabilities P int ,α ( E ν , cos θ ) for initial neutrino energies E ν =10 eV, 10 eV and 10 eV are shown in Fig. 3.For different flavors of neutrinos with the same initial energy, the interaction probabilitiesare about the same as cos θ approaches − θ and reach a maximum near the horizontaldirection (cos θ ≃ µ and τ leptons can on the average propagate distance of order of 10 km before come to astop [75]. Muons lose their energy mostly via pair production, while τ leptons via pairproduction as well as photonuclear interaction. So the probabilities of finding ν µ and ν τ arehigher than that for ν e . The interaction probability in these directions also increases withneutrino energies because of the decrease of the neutrino interaction length and the leptonpropagation range.The probability for up-going neutrinos (cos θ >
0) is suppressed as the neutrino travelingdistance becomes longer than the interaction length. The higher the initial neutrino energy,the shorter the interaction length, and hence the distribution terminates at smaller cos θ .The Earth attenuates the neutrino both in energy through NC and CC interactions and innumber through the stoppage of the charged lepton produced in the CC interaction. As aspecial case, τ leptons, having a decay length of about 50 × ( E τ / PeV) m, can transform to ν τ through decay before losing too much energy [53, 58]. Therefore ν τ coming from belowthe horizon would have an apparent larger probability than other two flavors. This is acritical feature for the flavor ratio determination proposed in this article, as we will furtherellaborate below. C. Detection Efficiency
The detection efficiency ǫ det defined here depends not only on the nature of Cherenkovradiation, the ice properties and the detector configuration, but also on how neutrinos andsecondary leptons deposit their energies in the shower. But for the simplicity of compu-13 cos-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 i n t P /eV)=17 ν (E log e ν µ ν τ ν θ cos-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 i n t P /eV)=18 ν (E log e ν µ ν τ ν θ cos-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 i n t P /eV)=19 ν (E log e ν µ ν τ ν FIG. 3. Interaction probability P int (see text for definition) as a function of the neutrino zenithdirection cos θ for ν e (red dash-dotted), ν µ (green solid), and ν τ (blue dashed), and for initialenergies E ν = 10 eV, 10 eV and 10 eV (from top to bottom panel). θ ≃ − . ◦ in ice, so it is less possi-ble to cover the area where the antennas are deployed. This leads to the common cutoff atcos θ ≃ − . ν e (top panel) and ν µ (middle panel) at cos θ & .
1. This is due to the smallness of thenumber of events arriving at the detection volume. Such results are therefore not reliable.However one generic feature remains valid; that is, the up-going neutrino events diminishsince their energies are severely damped by the Earth.The detection efficiency for ν e is the highest among the three flavors, because once CCinteraction occurs all the neutrino energy is released into showers and strong signals areemitted. For ν µ , although there are plenty of electromagnetic showers produced by muon,these showers tend to have lower energies so that they are less likely to be detected by thesparse antenna array. This leads to lower detection efficiency of ν µ , where the NC-inducedhadronic showers account for about 80 % of detected events for 1 EeV ν µ . The situation issimilar for ν τ , where most hadronic showers from photonuclear interaction and electromag-netic showers from pair production do not trigger detector efficiently while hadronic showersinduced by NC interaction and τ decay ( τ → ν τ + hadrons) account for the most detectedevents. The convergence of efficiency for up-going ν τ s of different initial energies resultsfrom the degradation of neutrino energy to few PeV by the regeneration process and NCinteraction [53]. 15 cos-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 d e t ∈ e ν Flavor: /eV)= 17 ν (E log /eV)= 17.5 ν (E log /eV)= 18 ν (E log /eV)= 18.5 ν (E log /eV)= 19 ν (E log /eV)= 19.5 ν (E log θ cos-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 d e t ∈ µ ν Flavor: /eV)= 17 ν (E log /eV)= 17.5 ν (E log /eV)= 18 ν (E log /eV)= 18.5 ν (E log /eV)= 19 ν (E log /eV)= 19.5 ν (E log θ cos-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 d e t ∈ τ ν Flavor: /eV)= 17 ν (E log /eV)= 17.5 ν (E log /eV)= 18 ν (E log /eV)= 18.5 ν (E log /eV)= 19 ν (E log /eV)= 19.5 ν (E log FIG. 4. The detection efficiency ǫ det (see text for definition) as a function of neutrino zenithaldirection cos θ for initial energies log ( E ν / eV) = 17 (red solid line), 17.5 (red dashed), 18 (greensolid), 18.5 (green dashed), 19 (blue solid), 19.5 (blue dashed), and for flavor ν e , ν µ , and ν τ (fromtop to bottom panel, respectively). . Event Direction Distribution and All-Sky Flavor Ratios of Events Because the initial neutrino energy is sampled only with discrete values of equal logarith-mic interval ∆ ≡ ∆ log E ν = 0 . P int and ǫ det , the direction distribu-tion integrated over the j -th energy bin ranging from log E j − ∆ / E j + ∆ /
2, isapproximated by Z D α ( E ν , cos θ ) dE ν ≃ N f Eα [ Z E j × ∆ / E j × − ∆ / Φ ν ( E ν ) dE ν ] (9) × P int ,α ( E j , cos θ ) ǫ det ,α ( E j , cos θ ) , with log ( E j / eV) = 17, 17 .
5, 18, 18 .
5, 19 and 19 .
5. The results assuming the ESS neutrinoflux are plotted in Fig. 5, where the total area under the distributions for each flavor hasbeen normalized to unity and the relative fraction contributed by each bin is also shown.It appears that EeV neutrinos contribute the most to the detected events for every flavor,because of the compromise between two competing effects: the decrease in the neutrino fluxversus the increase in the detection efficiency with neutrino energy.Finally, after summing over all distributions of different energy bins, the expected eventdirection distribution for each flavor D α (cos θ ) is obtained, which is shown in Fig. 6. Wedefine the “all-sky” flavor ratio of detected events, f e : f µ : f τ , as the event ratios amongflavors after integrating over all zenith directions. If the ESS neutrino flux and incidentflavor ratios at the Earth’s surface, f Ee : f Eµ : f Eτ = 1 / / /
3, are assumed, we findthat f e : f µ : f τ = 0 .
584 : 0 .
154 : 0 . ν e events account for the most portion becauseof their higher detection efficiency. The ν τ events have a different shape compared to theother two flavors especially in the horizontal and up-going directions, primarily due to itsspecial interaction probability (see the graph in the right panel, Fig. 6). These will be usedto extract the flavor composition at the Earth’s surface ( f E s) in the next section. However,the resemblance between ν e and ν µ distributions will lead to the degeneracy in the flavorratio extraction, and an extra constraint is required, for example, the ν µ - ν τ symmetry.The event ratio f for other initial flavor ratio f E can be derived from the result above,which is denoted by f and f E . The ν e event ratio is f e = f e, f Ee f Eµ, f Eτ, f e, f Ee f Eµ, f Eτ, + f µ, f Ee, f Eµ f Eτ, + f τ, f Ee, f Eµ, f Eτ , (10)and similarly for f µ and f τ . The conversion from f s to f E s can be done by just interchanging17 with f E . The relation between ν e event ratio f e and initial flavor ratio f Ee is plotted inFig. 7, assuming ESS neutrino flux. IV. PSEUDO-OBSERVATION AND FLAVOR RATIO EXTRACTION
To investigate the discriminating power of flavor ratio reconstruction using event directiondistribution, we generate pseudo-observation data from simulated events and fit the directiondistribution to extract the flavor ratios, and repeat the processes to determine the statisticaluncertainty of the extracted ratio. Results assuming different incident flux ratios, numbersof observed events, and angular resolution of detector are then presented.
A. Pseudo-Data Samples
The pseudo-data sample is prepared in three steps. First, neutrino events with differentflavors and directions are randomly generated according to the expected angular distribu-tions, D α (cos θ ). Secondly, the zenith angle, θ , of each pick-up event is smeared by adding aGaussian distributed random number with zero mean and the standard deviation ∆ θ equalto an assigned experimental angular error in reconstructed neutrino direction. Althoughmultiple cascade events in principle have different angular resolution than single cascadeones, but for their rareness we just assigned the same resolution for both types of events.Finally, this event collection is evenly divided into subsets, where each data set representsa hypothetical experimental data sample with a total number of detected events equal to N obs and these data sets constitute a statistical ensemble.In the following, N obs varies from 50 to 500 with an increment of 50, while ∆ θ from 0 ◦ to 6 ◦ . B. Fitting Pseudo-Data
To construct the fitting function for the hypothetical experimental data samples withangular resolution ∆ θ , the expected direction distribution for each flavor α is convolvedwith a Gaussian resolution function of standard deviation ∆ θ in θ space, G ( θ ′ ; θ, (∆ θ ) ).18 cos-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 P r ob a b ili t y e ν logE(eV)= 17 : 0.045logE(eV)= 17.5 : 0.208logE(eV)= 18 : 0.352logE(eV)= 18.5 : 0.255logE(eV)= 19 : 0.108logE(eV)= 19.5 : 0.032 θ cos-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 P r ob a b ili t y µ ν logE(eV)= 17 : 0.030logE(eV)= 17.5 : 0.146logE(eV)= 18 : 0.309logE(eV)= 18.5 : 0.285logE(eV)= 19 : 0.166logE(eV)= 19.5 : 0.064 θ cos-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 P r ob a b ili t y τ ν logE(eV)= 17 : 0.045logE(eV)= 17.5 : 0.228logE(eV)= 18 : 0.362logE(eV)= 18.5 : 0.228logE(eV)= 19 : 0.102logE(eV)= 19.5 : 0.035 FIG. 5. The expected direction distribution integrated over the energy bin, log ( E j ) ± . ( E j / eV) = 17 (red solid), 17.5 (red dashed), 18 (green solid),18.5 (green dashed), 19 (blue solid), 19.5 (blue dashed), for flavor ν e , ν µ , and ν τ (from top tobottom panel, respectively). The distributions have been normalized so that each one representsits fractional contribution to the corresponding flavor, and the relative fraction of each energy binis listed in the legend. The ESS neutrino spectrum [72] is assumed. cos-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 P r ob a b ili t y ESS2001: 0.584 e ν : 0.154 µ ν : 0.262 τ ν θ cos-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 P r ob a b ili t y ESS2001: 0.333 e ν : 0.333 µ ν : 0.333 τ ν FIG. 6. Top panel: the expected direction distribution of detected neutrino events for each flavor α , D α (cos θ ), assuming the ESS flux [72] and the incident flux ratios f Ee : f Eµ : f Eτ = 1 / / / f e : f µ : f τ is 0 .
584 : 0 .
153 : 0 . / The convolved distribution, M α ( θ ; ∆ θ ) ∝ Z D α ( θ ) G ( θ ′ ; θ, (∆ θ ) ) dθ ′ , (11)is then normalized so that Z − M α (cos θ ; ∆ θ ) d cos θ = 1 , e fraction on the Earth f e ν Initial 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 e eve n t s f e ν F r ac t i on o f d e t ec t e d FIG. 7. The relation of the fraction of detected ν e events, f e versus the ν e fraction of incidentneutrino flux at the Earth’s surface, f Ee , where the ESS flux model [72] is assumed. The values forthe standard case ( f Ee : f Eµ : f Eτ = 1 / / /
3) and for the neutrino decay case (0 .
75 : 0 .
125 :0 . and the probability density function of event direction can be expressed as P (cos θ ; f , f , ∆ θ ) = f M e (cos θ ) + (1 − f ) (12) × [ f M µ (cos θ ) + (1 − f ) M τ (cos θ )] , where f and f are unknown fraction coefficients with values between zero and one. Thisexpression ensures f and f are independent of each other, and they are simply related tothe event flavor ratios by f e = f , f µ = (1 − f ) f , and f τ = (1 − f )(1 − f ).The maximum likelihood estimation is applied for flavor ratio extraction. For an experi-mental data set with total events of N obs , the likelihood function is defined as L (cos θ i ; f , f , ∆ θ ) = N obs Y i =1 P (cos θ i ; f , f , ∆ θ ) , (13)where the subscript i = 1 , , . . . , N obs stands for the i -th event. The true values of f and f are estimated by maximizing L , or equivalently minimize the negative log-likelihood (NLL), − ln L = − N obs X i =1 ln P (cos θ i ; f , f , ∆ θ ) . (14)21e perform grid search in the parameter space to find the minimum of NLL and the associ-ated best fit values of f and f , denoted as ˆ f (or ˆ f e ) and ˆ f . A reconstruction is defined asfailed if ˆ f or ˆ f is outside either boundary (zero or one). It may result from the insufficientstatistics of events, or that the true value is near the boundary. However, an extra constraint is required to avoid multiple solutions arising from thesimilarity in distribution shape between ν e and ν µ , as pointed out in Section III D. The ν µ - ν τ symmetry, i.e. the incident ν µ and ν τ fluxes on the Earth are of equal amount ( f Eµ = f Eτ ),is imposed for this purpose. As a consequence, the value of f is known and fixed, and thereis only one parameter f , the fraction of ν e events, left to be fitted. In addition, for simplicity,we assume the observer knows the exact shape of neutrino spectrum; that is, the neutrinofluxes assumed for generating pseudo-data samples and fitting function are identical. Thelatter constraint will be relaxed in the next section and the mismatch in shape between realand expected spectra will introduce a systematic bias to the extracted ratio. C. Successful Probability and Flavor Ratio Resolution
After every data set in the ensemble has been fitted for a given N obs and ∆ θ , the successfulprobability of flavor ratio extraction is calculated as the number of successful data setsdivided by the total number of data sets. The result is shown in Fig. 8.In general, the successful probability increases with the number of observed events anda better angular resolution, as expected. The probability for the standard scenario (toppanel) is greater than those for the decay scenarios (the one with normal hierarchy is shownin the middle panel) since the latter have true ν e ratios so closed to the boundaries thateven small statistical fluctuation may result in failure. But the probability for initial ratio of0 : 0 . . ν e ratio is on the boundary.For the standard scenario, the successful probability is greater than 70 % for N obs ≥ N obs ≥ ◦ . On theother hand, for the decay scenario with normal hierarchy, the probability is over 50 % once N obs ≥ When doing the fitting, ˆ f and ˆ f are permitted to have unphysical values. ν e event fraction as the spread of all fitted values ˆ f e s in theensemble with respect to the expected value f e, exp , R ± ( N obs , ∆ θ ) = vuut N s − N s X i =1 ( ˆ f e,i − f e, exp ) , (15)where N s is the number of hypothetical experimental data sets. If the fitting is not disturbedby the boundary cutoff, then the distribution of ˆ f e s asymptotically approaches a Gaussiandistribution as the number of observed events N obs increases, and therefore the resolutiondefined here is approximately 1 σ uncertainty at 68 % confidence level. In order to reducethe effect induced by boundary cutoff, the resolution is calculated separately at both sidesof the expected value and denoted as R + and R − .The upper and lower bounds of ν e event fraction, f e, exp ± R ± , are then transformed intothe corresponding flavor ratios at the Earth’s surface, f Ee, true ± R E ± , according to Eqn. 10(with f interchanging with f E ; see also Fig. 7). The resolution of ν e ratio on the Earth R E for given N obs and ∆ θ is plotted in Fig. 9. The lower (upper) resolution is taken in thedecay scenario with normal (inverted) hierarchy. The resolution in the standard scenariois asymmetric on both sides, with the upper resolution worse than the lower one typicallyby about 0 .
06, due to the nonlinear conversion relation between event ratios and flux ratios(Fig. 7), and hence the average value is taken.To measure the separability of a flavor ratio predicted in a “fake” scenario from thatextracted from a given “true” scenario, we define the discriminating power of flavor ratiosas discriminating power ≡ | f e, model − f e, exp | R , (16)where f e, model is the ν e event ratio expected by the scenario to be examined, and f e, exp and R are the expected ν e event ratio and its resolution in the assumed true scenario, respectively.Note it is the ν e “event” ratio and its resolution that are used in the definition because ofthe asymptotic normality of its distribution, whereas the distribution of ν e flux ratio f Ee isskew due to the nonlinear conversion relation. The discriminating powers assuming differenttrue scenarios are shown in Fig. 10.In Ref. [48], the ARA simulation result reports an angular resolution of neutrino direc-tion about 6 ◦ . Under this circumstances, given 100 observed neutrino events, the successful23 bs Number of observed events N0 100 200 300 400 500 S u ccess f u l p r ob a b ili t y Ratio: 1:1:1
ESS2001 ° = 0.0 θ ∆ ° = 2.0 θ ∆° = 4.0 θ ∆ ° = 6.0 θ ∆ obs Number of observed events N0 100 200 300 400 500 S u ccess f u l p r ob a b ili t y Ratio: 6:1:1
ESS2001 ° = 0.0 θ ∆ ° = 2.0 θ ∆° = 4.0 θ ∆ ° = 6.0 θ ∆ obs Number of observed events N0 100 200 300 400 500 S u ccess f u l p r ob a b ili t y Ratio: 0:1:1
ESS2001 ° = 0.0 θ ∆ ° = 2.0 θ ∆° = 4.0 θ ∆ ° = 6.0 θ ∆ FIG. 8. The successful probability of flavor ratio extraction as a function of the number ofobserved events N obs , assuming initial ratio of 1 / / / .
75 : 0 .
125 : 0 . . . N obs ranges from 50 to 500 with an intervalof 50. Results for different angular resolutions are plotted: ∆ θ = 0 ◦ (red), 2 ◦ (green), 4 ◦ (blue), 6 ◦ (black). The ESS neutrino flux [72] is assumed in both data generating and fitting, and the ν µ - ν τ symmetry is assumed in fitting so that the relative ratio between them is known and fixed. bs Number of observed events N0 100 200 300 400 500 r a t i o on E a r t h e ν R es o l u t i on o f Ratio: 1:1:1
ESS2001 ° = 0.0 θ ∆ ° = 2.0 θ ∆° = 4.0 θ ∆ ° = 6.0 θ ∆ obs Number of observed events N0 100 200 300 400 500 r a t i o on E a r t h e ν R es o l u t i on o f Ratio: 6:1:1
ESS2001 ° = 0.0 θ ∆ ° = 2.0 θ ∆° = 4.0 θ ∆ ° = 6.0 θ ∆ obs Number of observed events N0 100 200 300 400 500 r a t i o on E a r t h e ν R es o l u t i on o f Ratio: 0:1:1
ESS2001 ° = 0.0 θ ∆ ° = 2.0 θ ∆° = 4.0 θ ∆ ° = 6.0 θ ∆ FIG. 9. The resolution of ν e ratio at the Earth’s surface R E (see text for definition) versus thenumber of observed events N obs for angular resolution of neutrino direction ∆ θ = 0 ◦ (the perfectcase, red solid line and open square), 2 ◦ (green dashed line and full square), 4 ◦ (blue dash-dottedline and open circle), and 6 ◦ (black dotted line and full circle). Results assuming initial flavorratios of : : (top panel), 0 .
75 : 0 .
125 : 0 .
125 (middle panel), and 0 : 0 . . ν µ - ν τ symmetry is assumed in fitting so that the relative ratio between them is known and fixed. . σ (1 . σ ) level. If the decay scenario with normal (inverted) hierarchy is real, then the dis-criminating power with respect to the standard scenario will be at about 0 . σ (1 . σ ) level,while that to its inverted (normal) counterpart will be at about 2 . σ (1 . σ ) level. Hencea preliminary constraint on the neutrino decay can be set by the method proposed in thisarticle.If the number of the observed events is doubled with the angular resolution fixed ( N obs =200, ∆ θ = 6 ◦ ), the successful probability rises to 85 % for the standard scenario while stillaround 50 % for the decay ones; the discriminating power between the standard scenarioand either of the decay scenario increases by about 0 . . σ while by about 0 . σ betweenthe decay scenarios. If the angular resolution is improved by 4 ◦ with the number of observedevents fixed ( N obs = 100, ∆ θ = 2 ◦ ), the successful probability becomes about 80 % for thestandard scenario while still around 50 % for both decay ones; and the discriminating powerbetween the standard scenario and either of decay one increases by about 0 . . σ , andby about 0 . σ between the decay scenarios. The limit of our method with perfect angularresolution is also shown in Fig. 8, Fig. 9 and Fig. 10 (red curves).To accumulate more neutrino events, apart from waiting for more neutrinos to come, onecan increase the event rate by extending the antenna array. To achieve a higher angularresolution of neutrino direction, on the other hand, an antenna array with denser grid isrequired for a better imaging of the Cherenkov cone. Therefore the results presented herecan serve as a reference for optimizing the future detector configuration of ARA or othersimilar observatories. D. Systematic Uncertainty from Neutrino Spectrum
Fig. 11, Fig. 12 and Fig. 13 show the results assuming the neutrino flux predicted in the“plausible” scenario in Ref. [14] (blue dashed curve in Fig. 2). The successful probabilityis lower than that assuming the ESS flux and the resolution is worse. Recall that the fluxin Ref. [14] is steeper than the ESS one, i.e. there are more neutrino events with lowerenergies. Since the interaction probabilities between flavors are less distinct from each other26t lower energies (see Fig. 3), the event direction distributions between flavors are harder tobe distinguished.If the observer does not exactly know the “real” neutrino spectrum (i.e., the one usedto generate pseudo-data), then the mismatch in shape between the real and the expected(i.e., the one used to fit the data) spectra will introduce a systematic bias to the extractedratio. We found that if the real spectrum is the plausible scenario in [14] while the spectrumexpected by the observer is the ESS model, the bias is about 10 %. More specifically, theaverage fitted value drops from 0.58 to 0.53, because a steeper neutrino spectrum leads tomore events coming from below the horizon due to the longer interaction length of neutrinoswith lower energies. In order to fit the event distribution from the steeper spectrum withthe flatter one, the ν τ fraction has to be increased while the ν e fraction decreased. E. Systematic Uncertainty from Neutrino Cross Sections
The calculation of neutrino cross section in Ref. [61] reports an uncertainty of factor of2 ± at around 10 eV. Therefore we vary the neutrino cross section by a factor of two, andthe results are shown in Fig. 14, Fig. 15 and Fig. 16. A larger neutrino cross section leadsto lower successful probability and resolution of flavor ratio, because the shorter interactionlength increase the fraction of events in the down-going directions where the interactionprobabilities for different flavors are more or less the same, while makes the more distinct,up-going parts slightly decrease. V. CONCLUSION AND FUTURE WORKS
In summary, measuring the flavor ratio of UHE cosmic neutrinos can not only reveal thephysical conditions at UHECR sources but also probe neutrino oscillation parameters andnon-standard physics that might involve during the propagation. Neutrino observatoriesusing the radio Cherenkov technique such as ARA [48], ARIANNA [49], and SalSA [50], aresensitive in the UHE regime and expected to accumulate of order of 10 to 100 cosmogenicneutrinos per year in the near future, and hence provide sufficient statistics for the flavorratio identification.In this work, the direction distribution of neutrino events is proposed to determine the27avor ratio. In order to investigate its feasibility, a simulation is constructed and the ex-pected distribution for ARA is derived. It is found that ν τ distribution has a different shapefrom the other two, but the distributions for ν e and ν µ resemble each other. Therefore anadditional constraint, the ν µ - ν τ symmetry, is imposed on the fitting of data distribution toavoid multiple solutions.This method is proved to be feasible for ARA, e.g., for 100 events and an angular reso-lution of 6 ◦ , the successful probability is about 70 % in the standard scenario and over 50 %for the neutrino decay models considered here. This method is also able to set a preliminaryconstraint on the neutrino decay, e.g., given an angular resolution of 6 ◦ , it requires about250 events to separate the standard scenario from the decay model with inverted hierarchyby about 2 σ level. Therefore, the flavor resolution as a function of the number of observedevents and the angular resolution of neutrino direction presented here can serve as a ref-erence for optimizing the future configuration of ARA. Similar procedures can be done forother observatories.However, we have not fully taken advantage of all the information possessed by neutrinoevents. For example, the spatial distribution and the energies of showers, which can beretrieved by incorporating the vertex reconstruction, can help to distinguish electron flavorfrom the other two; and the characteristics of radio signals may allow the classification ofshower types and further the identification of the neutrino flavor event by event. In addition,the near-field effect of Cherenkov radiation has to be considered, since in some cases theobservation distances are comparable to the shower lengths and the far-field approximationapplied here would underestimates both the signal strength and the angular width of theCherenkov cone, as pointed out in Ref. [76]. The neutrino spectrum and the cross sectionsshould also be incorporated into the fitting in order to reduce the systematic uncertainties.These will be included in the future simulation, and we expect to get a higher resolution ofthe neutrino flavor ratio. ACKNOWLEDGMENTS
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ESS2001 ° = 0.0 θ ∆ ° = 2.0 θ ∆° = 4.0 θ ∆ ° = 6.0 θ ∆ obs Number of observed events N0 100 200 300 400 500 D i sc r i m i n a t i ng po w e r Discriminating Power for 1:1:1 to 6:1:1
ESS2001 ° = 0.0 θ ∆ ° = 2.0 θ ∆° = 4.0 θ ∆ ° = 6.0 θ ∆ obs Number of observed events N0 100 200 300 400 500 D i sc r i m i n a t i ng po w e r Discriminating Power for 6:1:1 to 0:1:1
ESS2001 ° = 0.0 θ ∆ ° = 2.0 θ ∆° = 4.0 θ ∆ ° = 6.0 θ ∆ obs Number of observed events N0 100 200 300 400 500 D i sc r i m i n a t i ng po w e r Discriminating Power for 6:1:1 to 1:1:1
ESS2001 ° = 0.0 θ ∆ ° = 2.0 θ ∆° = 4.0 θ ∆ ° = 6.0 θ ∆ obs Number of observed events N0 100 200 300 400 500 D i sc r i m i n a t i ng po w e r Discriminating Power for 0:1:1 to 1:1:1
ESS2001 ° = 0.0 θ ∆ ° = 2.0 θ ∆° = 4.0 θ ∆ ° = 6.0 θ ∆ obs Number of observed events N0 100 200 300 400 500 D i sc r i m i n a t i ng po w e r Discriminating Power for 0:1:1 to 6:1:1
ESS2001 ° = 0.0 θ ∆ ° = 2.0 θ ∆° = 4.0 θ ∆ ° = 6.0 θ ∆ FIG. 10. The discriminating power of ν e flavor ratio versus the number of observed events N obs forangular resolution of neutrino direction ∆ θ = 0 ◦ (the perfect case, red solid line and open square),2 ◦ (green dashed line and full square), 4 ◦ (blue dash-dotted line and open circle), and 6 ◦ (blackdotted line and full circle). Results assuming initial flavor ratios of 1 / / / .
75 : 0 .
125 : 0 .
125 (middle panels), and 0 : 0 . . ν µ - ν τ symmetry is assumed infitting so that the relative ratio between them is known and fixed. bs Number of observed events N0 100 200 300 400 500 S u ccess f u l p r ob a b ili t y Ratio: 1:1:1
Kotera2010, the "plausible" ° = 0.0 θ ∆ ° = 2.0 θ ∆° = 4.0 θ ∆ ° = 6.0 θ ∆ obs Number of observed events N0 100 200 300 400 500 S u ccess f u l p r ob a b ili t y Ratio: 6:1:1
Kotera2010, the "plausible" ° = 0.0 θ ∆ ° = 2.0 θ ∆° = 4.0 θ ∆ ° = 6.0 θ ∆ obs Number of observed events N0 100 200 300 400 500 S u ccess f u l p r ob a b ili t y Ratio: 0:1:1
Kotera2010, the "plausible" ° = 0.0 θ ∆ ° = 2.0 θ ∆° = 4.0 θ ∆ ° = 6.0 θ ∆ FIG. 11. The successful probability of flavor ratio extraction. Similar to Fig. 8 except that theassumed neutrino flux is the plausible scenario in Ref. [14] (blue dashed curve in Fig. 2). bs Number of observed events N0 100 200 300 400 500 r a t i o on E a r t h e ν R es o l u t i on o f Ratio: 1:1:1
Kotera2010, the "plausible" ° = 0.0 θ ∆ ° = 2.0 θ ∆° = 4.0 θ ∆ ° = 6.0 θ ∆ obs Number of observed events N0 100 200 300 400 500 r a t i o on E a r t h e ν R es o l u t i on o f Ratio: 6:1:1
Kotera2010, the "plausible" ° = 0.0 θ ∆ ° = 2.0 θ ∆° = 4.0 θ ∆ ° = 6.0 θ ∆ obs Number of observed events N0 100 200 300 400 500 r a t i o on E a r t h e ν R es o l u t i on o f Ratio: 0:1:1
Kotera2010, the "plausible" ° = 0.0 θ ∆ ° = 2.0 θ ∆° = 4.0 θ ∆ ° = 6.0 θ ∆ FIG. 12. The resolution of ν e ratio at the Earth’s surface versus the number of observed events.Similar to Fig. 9 except that the assumed neutrino flux is the plausible scenario in Ref. [14] (bluedashed curve in Fig. 2). bs Number of observed events N0 100 200 300 400 500 D i sc r i m i n a t i ng po w e r Discriminating Power for 1:1:1 to 0:1:1
Kotera2010, the "plausible" ° = 0.0 θ ∆ ° = 2.0 θ ∆° = 4.0 θ ∆ ° = 6.0 θ ∆ obs Number of observed events N0 100 200 300 400 500 D i sc r i m i n a t i ng po w e r Discriminating Power for 1:1:1 to 6:1:1
Kotera2010, the "plausible" ° = 0.0 θ ∆ ° = 2.0 θ ∆° = 4.0 θ ∆ ° = 6.0 θ ∆ obs Number of observed events N0 100 200 300 400 500 D i sc r i m i n a t i ng po w e r Discriminating Power for 6:1:1 to 0:1:1
Kotera2010, the "plausible" ° = 0.0 θ ∆ ° = 2.0 θ ∆° = 4.0 θ ∆ ° = 6.0 θ ∆ obs Number of observed events N0 100 200 300 400 500 D i sc r i m i n a t i ng po w e r Discriminating Power for 6:1:1 to 1:1:1
Kotera2010, the "plausible" ° = 0.0 θ ∆ ° = 2.0 θ ∆° = 4.0 θ ∆ ° = 6.0 θ ∆ obs Number of observed events N0 100 200 300 400 500 D i sc r i m i n a t i ng po w e r Discriminating Power for 0:1:1 to 1:1:1
Kotera2010, the "plausible" ° = 0.0 θ ∆ ° = 2.0 θ ∆° = 4.0 θ ∆ ° = 6.0 θ ∆ obs Number of observed events N0 100 200 300 400 500 D i sc r i m i n a t i ng po w e r Discriminating Power for 0:1:1 to 6:1:1
Kotera2010, the "plausible" ° = 0.0 θ ∆ ° = 2.0 θ ∆° = 4.0 θ ∆ ° = 6.0 θ ∆ FIG. 13. The discriminating power of ν e flavor ratio versus the number of observed events. Similarto Fig. 10 except that the assumed neutrino flux is the plausible scenario in Ref. [14] (blue dashedcurve in Fig. 2). bs Number of observed events N0 100 200 300 400 500 S u ccess f u l p r ob a b ili t y Ratio: 1:1:1
ESS2001 ° = 0.0 θ ∆ ° = 2.0 θ ∆° = 4.0 θ ∆ ° = 6.0 θ ∆ obs Number of observed events N0 100 200 300 400 500 S u ccess f u l p r ob a b ili t y Ratio: 6:1:1
ESS2001 ° = 0.0 θ ∆ ° = 2.0 θ ∆° = 4.0 θ ∆ ° = 6.0 θ ∆ obs Number of observed events N0 100 200 300 400 500 S u ccess f u l p r ob a b ili t y Ratio: 0:1:1
ESS2001 ° = 0.0 θ ∆ ° = 2.0 θ ∆° = 4.0 θ ∆ ° = 6.0 θ ∆ FIG. 14. The probability of successful flavor ratio reconstruction. Same as Fig. 8 except that theneutrino cross section is multiplied by a factor of two. bs Number of observed events N0 100 200 300 400 500 r a t i o on E a r t h e ν R es o l u t i on o f Ratio: 1:1:1
ESS2001 ° = 0.0 θ ∆ ° = 2.0 θ ∆° = 4.0 θ ∆ ° = 6.0 θ ∆ obs Number of observed events N0 100 200 300 400 500 r a t i o on E a r t h e ν R es o l u t i on o f Ratio: 6:1:1
ESS2001 ° = 0.0 θ ∆ ° = 2.0 θ ∆° = 4.0 θ ∆ ° = 6.0 θ ∆ obs Number of observed events N0 100 200 300 400 500 r a t i o on E a r t h e ν R es o l u t i on o f Ratio: 0:1:1
ESS2001 ° = 0.0 θ ∆ ° = 2.0 θ ∆° = 4.0 θ ∆ ° = 6.0 θ ∆ FIG. 15. The resolution of ν e ratio on the Earth. Same as Fig. 9 except that the neutrino crosssection is multiplied by a factor of two. bs Number of observed events N0 100 200 300 400 500 D i sc r i m i n a t i ng po w e r Discriminating Power for 1:1:1 to 0:1:1
ESS2001 ° = 0.0 θ ∆ ° = 2.0 θ ∆° = 4.0 θ ∆ ° = 6.0 θ ∆ obs Number of observed events N0 100 200 300 400 500 D i sc r i m i n a t i ng po w e r Discriminating Power for 1:1:1 to 6:1:1
ESS2001 ° = 0.0 θ ∆ ° = 2.0 θ ∆° = 4.0 θ ∆ ° = 6.0 θ ∆ obs Number of observed events N0 100 200 300 400 500 D i sc r i m i n a t i ng po w e r Discriminating Power for 6:1:1 to 0:1:1
ESS2001 ° = 0.0 θ ∆ ° = 2.0 θ ∆° = 4.0 θ ∆ ° = 6.0 θ ∆ obs Number of observed events N0 100 200 300 400 500 D i sc r i m i n a t i ng po w e r Discriminating Power for 6:1:1 to 1:1:1
ESS2001 ° = 0.0 θ ∆ ° = 2.0 θ ∆° = 4.0 θ ∆ ° = 6.0 θ ∆ obs Number of observed events N0 100 200 300 400 500 D i sc r i m i n a t i ng po w e r Discriminating Power for 0:1:1 to 1:1:1
ESS2001 ° = 0.0 θ ∆ ° = 2.0 θ ∆° = 4.0 θ ∆ ° = 6.0 θ ∆ obs Number of observed events N0 100 200 300 400 500 D i sc r i m i n a t i ng po w e r Discriminating Power for 0:1:1 to 6:1:1
ESS2001 ° = 0.0 θ ∆ ° = 2.0 θ ∆° = 4.0 θ ∆ ° = 6.0 θ ∆ FIG. 16. The discriminating power of ν e flavor ratio. Same as Fig. 10 except that the neutrinocross section is multiplied by a factor of two.flavor ratio. Same as Fig. 10 except that the neutrinocross section is multiplied by a factor of two.