Discrimination of mass hierarchy with atmospheric neutrinos at a magnetized muon detector
aa r X i v : . [ h e p - ph ] F e b The discrimination of mass hierarchy with atmospheric neutrinos at a magnetizedmuon detector
Abhijit Samanta ∗ Harish-Chandra Research Institute, Chhatnag Road, Jhusi, Allahabad 211 019, India (Dated: October 30, 2018)We have studied the mass hierarchy with atmospheric neutrinos considering the muon energy andzenith angle of the event at the magnetized iron calorimeter detector. For χ analysis we havemigrated the number of events from neutrino energy and zenith angle bins to muon energy andzenith angle bins using the two-dimensional energy-angle correlated resolution functions. The bin-ning of data is made in two-dimensional grids of log E − L . plane to get a better reflection of theoscillation pattern in the χ analysis. Then the χ is marginalized considering all possible system-atic uncertainties of the atmospheric neutrino flux and cross section. The effects of the ranges ofoscillation parameters on the marginalization are also studied. The lower limit of the range of θ for marginalization is found to be very crucial in determining the sensitivity of hierarchy for a given θ . Finally, we show that one can discriminate atmospheric neutrino mass hierarchy at >
90% C.L.if the lower limit of θ ≥ ◦ . PACS numbers: 14.60.Pq
I. INTRODUCTION
The origin of masses of the particles and their inter-actions have been successfully described by the stan-dard model (SM). The recent discovery of neutrinomasses and their mixing through neutrino oscillation[1, 2] opens a new window beyond the SM. This providesthe measurements of mass squared differences ∆ m ji = m j − m i and mixing angles θ ij . At present 1(3) σ ranges are [3]: ∆ m = 7 . +0 . − . ( +0 . − . ) × − eV , | ∆ m | = 2 . +0 . − . ( +0 . − . ) × − eV , sin θ =0 . +0 . − . ( +0 . − . ), sin θ = 0 . +0 . − . ( +0 . − . ), andsin θ = 0 . +0 . − . ( < . θ ij are the mixingangles in Pontecorvo, Maki, Nakagawa, Sakata mixingmatrix [4]. Currently, there is no constraint on the CPviolating phase δ CP or on the sign of ∆ m . The neu-trino mass ordering is one of the key problems to build upthe fundamental theory of the particles beyond the SM.This can be probed through neutrino oscillation. In thispaper we have concentrated on the prospects to resolvethe mass hierarchy using atmospheric neutrinos.The mass hierarchy with atmospheric neutrinos hasbeen studies in [5–7] with a large magnetized IronCALorimeter (ICAL) detector, which is being strong con-sidered for the India-based Neutrino Observatory [8]. In[9], the authors estimated the sensitivity of a large liq-uid argon detector. This is nonmagnetized, but can de-tect both electron and muon. So, here the statistics arehigh compared to ICAL. However, these studies have ∗ E-mail address: [email protected] dealt with neutrino energy and zenith angle and assumedfixed Gaussian resolutions for them separately. It is alsoshown that the confidence level of determining hierarchychanges drastically with the change in width of resolu-tions. Moreover, since all of the particles produced inthe neutrino interactions can not be reconstructed, theresolutions are not Gaussian in nature. There are manyneutral particles in the deep inelastic processes, and thenon-Gaussian nature of the resolutions increases as onegoes to higher energies. Again, the resolutions are dif-ferent for neutrino and antineutrino (see Fig. 4 of [10])since the different quantum numbers are involved in thescattering matrix elements for neutrino and antineutrino.At energies around 1 GeV, most of the events are quasi-elastic, and the muon carries almost all of the energy ofthe neutrino. Here, the energy resolution is very good.As the energy increases, the deep inelastic process dom-inates and the energy resolution begins to worsen by de-veloping a more prominent non-Gaussian nature. How-ever, the trend of angular resolution is quite opposite, itswidth is very wide at low energy, and it improves with anincrease in energy. So, it is very important to study thesensitivity of a detector in terms of directly measurablequantities.We have studied the neutrino mass hierarchy for themagnetized ICAL detector with atmospheric neutrinosgenerating events by NUANCE-v3 [11] and consider-ing the muon energy and direction (directly measurablequantities) of the event. Because of heavy mass of themuon, it looses energy mostly via ionization and atomicexcitation during its propagation through a medium.Since ICAL is a tracking detector, it gives a clean sin-gle track in the detector. The muon energy can be mea-sured from the bending of the track in the magnetic fieldor from the track length in the case of a fully containedevent. The direction can be measured from the tangentof the track at the vertex. From the GEANT [12] simula-tion of the ICAL detector it is found that the energy andangular resolutions of the muons are very high (4-10% forenergy and 4-12% for zenith angle) and negligible com-pared to the resolutions obtained from the kinematics ofthe scattering processes. A new method for migrationfrom true neutrino energy and zenith angle to muon en-ergy and zenith angle has been introduced in [13] andsubsequently used in [10, 14, 15].On the other hand, the binning of the data is alsoan important issue when one considers the binning ofthe events in reconstructed energy and zenith angle bins.The reasons are the following: The atmospheric neutrinoflux changes very rapidly with energy following a powerlaw (roughly ∼ E − . ). Again, the behavior of the os-cillation probability changes with the change in zones of E − L plane. This has been discussed in [10, 14]. Incase of proper binning of the data [10], the precision ofoscillation parameters improves. Then it helps in hierar-chy discrimination by reducing the effect of the ranges ofthe parameters, over which the marginalization is carriedout.In our previous work [13], the χ analysis has been car-ried out considering the ratio of total up and total downgoing events for each resonance zone. Since the ratioup/down cancels all overall uncertainties, we consideredonly the energy dependent one. However, it should benoted here that the ratio of two Gaussian observables isnot an exact Gaussian function. Our previous χ studyassuming the up/down ratio as a Gaussian function wasmotivated by the cancellations of all overall uncertainties.So, the result was an approximated one.It is important to estimate the sensitivity of hierarchydetermination with realistically measurable parametersof the experiments through a detailed analysis. Here,we have performed the χ analysis following Poissoniandistribution and considering the number of events in thegrids of the log E − L . plane of the muon. This followsPoissonian (Gaussian) distribution for a less (large) num-ber of events. For a large number of events Poissoniandistribution tends to a Gaussian one. We have takeninto account all possible systematic uncertainties usingthe pull method of the χ analysis. Expecting the lowerbound of θ to be known from other experiments likeDouble Chooz [16] or NO ν A [17], we have also estimatedthe improved sensitivity considering the lower bound of θ for marginalization as the input value of θ for gen-erating the experimental data for the χ analysis. II. OSCILLATION FORMALISM
To understand the analytical solution of time evolu-tion of neutrino propagation through matter, we adoptthe so-called “one mass scale dominance” frame work: | ∆ m | << | m − m , | [18, 19]. With this one mass scale dominance approximation,the survival probability of ν µ can be expressed as P mµµ = 1 − cos θ m sin θ × sin (cid:20) . (cid:18) (∆ m ) + A + (∆ m ) m (cid:19) LE (cid:21) − sin θ m sin θ × sin (cid:20) . (cid:18) (∆ m ) + A − (∆ m ) m (cid:19) LE (cid:21) − sin θ sin θ m sin (cid:20) .
27 (∆ m ) m LE (cid:21) (1)The mass squared difference (∆ m ) m and mixing an-gle sin θ m in matter are related to their vacuum valuesby(∆ m ) m = q ((∆ m ) cos 2 θ − A ) + ((∆ m ) sin 2 θ ) sin θ m = (∆ m ) sin 2 θ p ((∆ m ) cos 2 θ − A ) + ((∆ m ) sin 2 θ ) . (2)where, the matter term A = 2 √ G F n e E = 7 . × − eV ρ (gm / cc) E (GeV) eV . Here, G F and n e arethe Fermi constant and the electron number density inmatter and ρ is the matter density. The evolution equa-tion for antineutrinos has the sign of A reversed.From Eqs. 1 and 2 it is seen that a resonance in P mµµ will occur for neutrinos (antineutrinos) with normal hi-erarchy (inverted hierarchy) whensin θ m → , A = ∆ m cos 2 θ . (3)Then resonance energy can be expressed as E = (cid:20) × . × − Y e (cid:21) (cid:20) | ∆ m | eV cos 2 θ (cid:21) (cid:20) gm / cc ρ (cid:21) . (4)The difference ∆ P between P µµ (∆ m ) and P µµ ( − ∆ m ) has been plotted in Fig. 1 for ν µ and ¯ ν µ with θ = 2 ◦ and θ = 10 ◦ , respectively, as anoscillogram in a two-dimensional plane of E − cos θ nadir .We see that there exists a difference in all E − cos θ nadir space. This is not due to the matter effect, but due tothe nonzero value of ∆ m . So this exists for θ = 0 ◦ also. It is seen that the resonance zones are different for ν and ¯ ν and both their areas and amplitudes squeezewith a decrease in θ values. III. THE χ ANALYSIS
The χ is calculated according to the Poisson proba-bility distribution. The binning the data is made in two-dimensional grids in the plane of log E - L . of themuon. The method for migration of number of eventsfrom neutrino to muon energy and zenith angle bins, thenumber of bins, the systematic uncertainties, and thecuts at the near horizons are described in [14]. -0.8-0.4 0 0.4 0.8Neutrino, θ =10 o -1 -0.8 -0.6 -0.4 -0.2 0 0.2cos θ nadir E ( G e V ) -0.8-0.4 0 0.4 0.8Anti-neutrino, θ =10 o -1 -0.8 -0.6 -0.4 -0.2 0 0.2cos θ nadir -0.8-0.4 0 0.4 0.8Neutrino, θ =2 o E ( G e V ) -0.8-0.4 0 0.4 0.8P µµ (IH) - P µµ (NH) Anti-neutrino, θ =2 o FIG. 1:
The oscillogram of the difference P µµ ( − ∆ m ) − P µµ (∆ m ) in the E − cos θ nadir plane for θ = 2 ◦ (top) and θ = 10 ◦ (bottom) with ν µ (left) and ¯ ν µ (right), respectively. We set | ∆ m | = 2 . × − eV , θ = 45 ◦ , δ CP = 180 ◦ , and other oscillationparameters at their best-fit values. The fact is that neutrino cross sections have not beenprecisely measured at all energies, and the neutrino fluxis also not precisely known at every energy. There mayarise an energy dependent systematic uncertainty. Wemay not always realize this. The program NUANCEconsiders difference processes of interactions at differentenergies using the Monte Carlo method. This may notbe fully captured when we generate theoretical data byfolding the flux with total cross section and smearingwith the resolution functions[10]. The similar situationmay happen in real experiments also. The mass hierar-chy is determined considering the difference in numberof events between normal hierarchy (NH) and invertedhierarchy (IH). This arises for the resonance in neutrinopropagation through matter. It happens for some partic-ular zones of energy and the baseline. So, there may arisea large difference in estimated hierarchy sensitivities withand without consideration of this systematic uncertainty.One can generate the experimental data for χ analysisin two ways: I) directly from NUANCE simulation fora given set of oscillation parameters with 1 Mton.yearexposure and then binning the events in muon energyand zenith angle bins; II) considering the oscillated at-mospheric neutrino flux for a given set of parameters andthen folding with time of exposure, total cross section, de-tector mass and finally smearing it with the energy-anglecorrelated resolution functions. This is similar with themethod of generating the theoretical data for χ analysis.One can generate the theoretical data directly from ahuge data set, say, 500 Mton.year data (to ensure the statistical error negligible) for each set of oscillation pa-rameters and then reducing it to 1 Mton.year equivalentdata from it, which would be the more straightforwardway for method I of generating experimental data. Inthis case, the effect of the above systematic uncertaintywill not come into the play. The marginalization studywith this method is almost an undoable job in a normalCPU. When we adopt different methods for generatingtheoretical and experimental data, the significant effectof the above systematic uncertainties come into the re-sults. So, we adopt the same method for them and herewe consider method II. IV. RESULTS
We marginalize the χ over all the oscillation parame-ters ∆ m , θ , θ , and δ CP along with solar oscillationparameters ∆ m and θ for both NH and IH with ν sand ¯ ν s separately for a given set of input data. Then wefind the total χ [= χ ν + χ ν ].We have chosen the range of ∆ m = 2 . − . × − eV , θ = 37 ◦ − ◦ , θ = 0 ◦ − . ◦ and δ CP = 0 ◦ − ◦ . We set the range of ∆ m = 7 . − . × − eV and θ = 30 . ◦ − ◦ . However, the effect of ∆ m comesin the subleading order in the oscillation probability when E ∼ GeV and it is marginal. We set the input of atmo-spheric oscillation parameters | ∆ m | = 2 . × − eV , θ = 45 ◦ , and δ CP = 180 ◦ and the solar parameters∆ m = 7 . × − eV and θ = 33 ◦ . We set IH as in- χ Input θ ( o ) FIG. 2:
The variation of marginalized χ (false) with different input values of θ . χ Input θ ( o )Input IH, θ =7.5 o true hierarchyfalse hierarchy 0246810121416 χ Input θ ( o )Input IH, θ =7.5 o true hierarchyfalse hierarchy 05101520 . . . . . . . . χ Input ∆ m (eV )Input IH, θ =7.5 o true hierarchyfalse hierarchy FIG. 3:
The variation of χ with θ , θ , and ∆ m , respectively for both true and false hierarchy. The χ is marginalized withrespect to all oscillation parameters except with which it varies. put. The variation of marginalized χ for different inputvalues of θ is shown in Fig. 2.To see the effect of marginalization over the ranges ofparameters, we have shown the variation of χ for bothtrue and false hierarchy as a function of θ , θ and | ∆ m | in Fig. 3. Here, we have marginalized the χ over all oscillation parameters except one with which itvaries. We find that significant improvement will comeif the lower bound of θ improves from zero. This isdemonstrated in the first plot in Fig. 3. We also seefrom this figure that there is no significant effect of otherparameters on marginalization since they are well deter-mined in this experiment. We find from Fig. 4 that if thelower limit is 5 ◦ , the mass hierarchy can be determinedat a confidence level > θ for marginalization is the in-put value for the experimental data set. Considering thisconstraint we have plotted the variation of χ with inputvalues of θ in Fig. 5. This will give the lower limitof the sensitivity of the mass hierarchy in this detectorwhen the lower limit of θ is known from other experi-ments like Double Chooz [16] or NO ν A [17]. A significantimprovement is observed comparing Figs. 2 and 5.
Acknowledgments:
This research has been supportedby funds from Neutrino Physics projects at HRI. The useof excellent cluster computational facility installed by thefunds of this project is also gratefully acknowledged. Thegeneral cluster facility of HRI has also been used at theinitial stages of the work. [1] B. Pontecorvo, J. Exp. Theor. Phys. , 549 (1957); Sov.Phys. JETP, , 429 (1958).[2] Y. Fukuda et al. [Super-Kamiokande Collaboration],Phys. Rev. Lett. , 1562 (1998) [arXiv:hep-ex/9807003].[3] G. L. Fogli et al. , Phys. Rev. D , 033010 (2008)[arXiv:0805.2517 [hep-ph]].[4] Z. Maki, M. Nakagawa and S. Sakata, Prog. Theor. Phys. , 870 (1962). B. Pontecorvo, JETP (1958) 429; (1958) 172;[5] D. Indumathi and M. V. N. Murthy, Phys. Rev. D ,013001 (2005) [arXiv:hep-ph/0407336]. [6] R. Gandhi, P. Ghoshal, S. Goswami, P. Mehta,S. U. Sankar and S. Shalgar, Phys. Rev. D , 073012(2007) [arXiv:0707.1723 [hep-ph]].[7] S. T. Petcov and T. Schwetz, Nucl. Phys. B , 1 (2006)[arXiv:hep-ph/0511277].[8] See “India-based Neutrino Observatory: Project Report.Volume I,” available at .[9] R. Gandhi, P. Ghoshal, S. Goswami and S. Uma Sankar,Phys. Rev. D , 073001 (2008) [arXiv:0807.2759 [hep- χ Input θ ( o )Input IH, θ =5 o true hierarchyfalse hierarchy FIG. 4:
The same as Fig. 3, but with θ only and with input θ = 5 ◦ . χ Input θ ( o ) FIG. 5:
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