Probing CP violation in neutrino oscillations with neutrino telescopes
aa r X i v : . [ h e p - ph ] J u l Probing CP violation in neutrino oscillationswith neutrino telescopes
Kfir Blum ∗ Department of Condensed Matter Physics,Weizmann Institute of Science, Rehovot 76100, Israel
Yosef Nir † Department of Particle Physics, Weizmann Institute of Science, Rehovot 76100, Israel
Eli Waxman
Department of Condensed Matter Physics,Weizmann Institute of Science, Rehovot 76100, Israel
Abstract
Measurements of flavor ratios of astrophysical neutrino fluxes are sensitive to the two yet un-known mixing parameters θ and δ through the combination sin θ cos δ . We extend previousstudies by considering the possibility that neutrino fluxes from more than a single type of sourceswill be measured. We point out that, if reactor experiments establish a lower bound on θ , thenneutrino telescopes might establish an upper bound on | cos δ | that is smaller than one, and by thatprove that CP is violated in neutrino oscillations. Such a measurement requires several favorableingredients to occur: (i) θ is not far below the present upper bound; (ii) The uncertainties in θ and θ are reduced by a factor of about two; (iii) Neutrino fluxes from muon-damped sourcesare identified, and their flavor ratios measured with accuracy of order 10% or better. For the lastcondition to be achieved with the planned km detectors, the neutrino flux should be close to theWaxman-Bahcall bound. It motivates neutrino telescopes that are effectively about 10 times largerthan IceCube for energies of O (100 T eV ), even at the expense of a higher energy threshold. † The Amos de-Shalit chair of theoretical physics ∗ Electronic address: kfir.blum,yosef.nir,[email protected] . INTRODUCTION One of the main goals of future neutrino experiments [1] is to observe CP violation inneutrino oscillations. The significance of such a measurement goes beyond the determinationof a fundamental parameter of Nature: it can give further qualitative support to leptogenesis,the idea that the observed baryon asymmetry of the Universe has its source in a leptonasymmetry generated in neutrino interactions. In some scenarios, it is even quantitativelyrelated to leptogenesis.Neutrino telescopes [2], such as the IceCube experiment, aim to observe neutrinos comingfrom astrophysical sources. The experiments will provide information on the direction,energy, and flavor of the incoming neutrinos. In particular, ratios between fluxes of differentflavors arriving to the detector can be measured. Ratios between these fluxes at the sourceare predicted by rather robust theoretical considerations.The modifications of the flavor ratios between source and detector originate from neutrinooscillations. This means that the relations between the fluxes at the source and the fluxes atthe detector depend on the neutrino parameters in a calculable way. Flavor measurementsin neutrino telescopes can thus provide information on the neutrino mixing parameters[3, 4, 5, 6, 7, 8, 9]. In particular, there is sensitivity to two yet unknown parameters: themixing angle θ and the CP violating phase δ .CP violation in neutrino oscillations can, in principle, be observed via interference terms.For neutrinos coming from astrophysical sources, such interference terms are washed out, andthe measured fluxes are therefore sensitive only to CP conserving parameters. Specifically,the measured flavor ratios are sensitive to the combination∆ ≡ sin θ cos δ. (1)Since θ is experimentally bounded from above and known to be small, it is convenient towrite the flavor ratios in the general form a + b ∆ , where a and b are known functions ofthe two measured parameters, θ and θ , but independent of θ and δ . The b ∆ termprovides a small correction to the zeroth order prediction a . If sin θ = 0, or if CP violationis maximal, i.e. δ = π/ π/
2, the correction term is absent.If sin θ is close to the present experimental upper bound, it is likely to be measured innear future reactor experiments [10]. In that case, if neutrino telescopes are able to exclude2 correction term as large as ± b sin θ , they will establish that cos δ = ± π . Our finalconclusion is that, with large θ and near-maximal CP violation, and under some favorablecircumstances, it may be possible for IceCube (or, more easily, for future, larger detectors)to establish CP violation in neutrino interactions. II. FLAVOR RATIOS AND MIXING PARAMETERS
Our goal in this section is to derive analytical expressions for neutrino flavor fluxes thatcan be measured in neutrino telescopes and, in particular, in IceCube.Neutrino telescopes can identify the neutrino flavor ( α = e, µ, τ ) via its characteristicinteraction topology [11, 12]. IceCube has an energy threshold ∼ GeV for detectingmuon tracks, and ∼ T eV for detecting electron- and tau-related showers. Above an energythreshold ∼ P eV , it is possible to distinguish between the electron-related electromagneticshowers and the tau-related hadronic showers. Finally, around E ∼ . P eV , the Glashowresonance may allow the identification of ¯ ν e events [7, 13].We denote the flux of ν α + ¯ ν α measured at the detector by φ dα ; the flux of antineutrinos¯ ν α is denoted by ¯ φ dα . We consider the following flavor ratios: R ≡ φ dµ φ de + φ dτ , (2) S ≡ φ de φ dτ , (3) T ≡ ¯ φ de φ dµ . (4)Below E ∼ P eV , only R can be measured. At higher energies, S and perhaps T may becomeavailable. 3e denote the flux of ν α + ¯ ν α emitted from the source by φ sα . The relation between φ sα and φ dβ is given by φ dβ = P βα φ sα , (5)where P βα ≡ P ( ν α → ν β ) is the transition probability from a flavor ν α at the source to aflavor ν β at the detector.For propagation over astronomical distance scales, the distance-dependent oscillatoryterms average out, and P βα depends on mixing parameters only: P βα = X i | U αi | | U β i | . (6)Here U is the unitary transformation that relates the neutrino interaction eigenstates ν α ( α = e, µ, τ ) and mass eigenstates ν i ( i = 1 , , | ν α i = U ∗ αi | ν i i . (7)We parametrize the matrix U by three mixing angles, θ , θ and θ , and three CP violatingphases, δ , α and α : U = c c s c s e iδ − s c − c s s e iδ c c − s s s e iδ s c s s − c c s e iδ − c s − s c s e iδ c c e iα / e iα / , (8)where c ij ≡ cos θ ij , s ij ≡ sin θ ij . It is clear from Eq. (6) that P βα is independent of thephases α , . It depends on the three mixing angles θ ij and on δ .Since it is experimentally known that θ is small (see Table I), it is convenient to writedown the flavor transition probabilities to first order in ∆ (see Eq. (1)) [14, 15, 16]: P ee ≃ −
12 sin θ ,P eµ ≃
12 sin θ cos θ + 14 sin 2 θ sin 4 θ ∆ ,P µµ ≃ − (cid:0) cos θ sin θ + sin θ (cid:1) −
12 sin 2 θ cos θ sin 4 θ ∆ ,P eτ ≃
12 sin θ sin θ −
14 sin 2 θ sin 4 θ ∆ ,P µτ ≃
18 sin θ (cid:0) − sin θ (cid:1) + 18 sin 4 θ sin 4 θ ∆ . (9)The remaining probabilities can be derived from P αβ = P βα and P α P αβ = 1.4 II. ASTROPHYSICAL NEUTRINO SOURCES AND FLAVOR RATIOS
We consider two types of sources: • “Pion sources” (denoted by sub-index π ) provide the following flavor ratios: φ se : φ sµ : φ sτ = 1 : 2 : 0 . (10)As concerns the ν e − ¯ ν e decomposition of φ e , the situation depends on whether thepions are produced mainly by pp or pγ interactions:¯ φ sµ φ sµ = 12 , ¯ φ se φ se = / pp, pγ. (11) • “Muon-damped sources” (denoted by sub-index µ ) provide the following flavor ratios: φ se : φ sµ : φ sτ = 0 : 1 : 0 . (12)As concerns the ν µ − ¯ ν µ decomposition of φ µ , the situation depends on whether thepions are produced mainly by pp or pγ interactions:¯ φ sµ φ sµ = / pp, pγ. (13)The expectation is that all sources where the initial stage of neutrino production ischarged pion decays will undergo a transition from a “pion” to “muon-damped” flavor de-composition at high enough neutrino energies [17]. If energy losses are mainly due to syn-chrotron radiation and inverse compton emission, the transition region is expected to spanabout one decade in energy. The actual threshold energy cannot be determined model inde-pendently and, furthermore, is likely to differ from source to source. We assume here that,nevertheless, the transition is such that it will be possible to separate the neutrino events tolower-energy events from pion sources and higher-energy events from muon-damped sources.The dependence of the flavor ratios at the detector on the mixing parameters can beobtained as follows. One starts from the fluxes at the source (in arbitrary units), Eqs. (10),(11), (12) and (13). Then, the fluxes at the detector can be found by using Eq. (5) and theexpressions for the transition probabilities (6). Finally, the expressions are put in Eqs. (2),(3) and (4). 5 ABLE I: Experimental ranges of mixing angles [1]Parameter Best fit 1 σ range ‘Future’sin θ .
31 0 . − .
33 0 . ± . θ .
47 0 . − .
55 0 . ± . θ . ≤ .
008 0 . ± . IV. DESCRIPTION OF ANALYSISA. Numerical input
The current best fit values and 1 σ ranges of the mixing angles are given in Table I [1].By the time that IceCube can carry out the measurements that we discuss in this work, itis likely that the knowledge – from other experiments – of the mixing angles will improve.Such progress is very significant for our purposes, as we see below. In particular, in orderthat the IceCube measurements will be able, even in principle, to show that δ = 0, it iscrucial that experiments establish that sin θ = 0. For the sake of our analysis, we assumethat reactor experiments will measure sin θ = 0 . ± .
013 [10]. (This value for θ corresponds to the current 2 σ allowed range [1].) For θ and θ we assume a factor of twoimprovement in the accuracy. The resulting ranges which we use to examine the questionof whether IceCube can discover CP violation are given in the column labelled ‘Future’ inTable I. As concerns the phase, we assume that it will remain unconstrained.To obtain an understanding of the dependence of the flavor ratios on the mixing parameter∆ , we use the central values for the two measured angles, θ and θ , and apply theapproximate relations (9). We obtain for the pion source R π = 0 . − . ,S π = 1 .
04 + 0 . ,T π = .
52 + 0 . pp, .
23 + 0 . pγ, (14)6nd for the muon-damped source R µ = 0 . − . ,S µ = 0 .
58 + 0 . ,T µ = .
30 + 0 . pp, pγ, (15)We emphasize, however, that in our calculations we use the full dependence on the mixingangles [see Eq. (6)], and not just the leading order (in ∆ ) expressions, Eqs. (9), (14) and(15). B. Experimental errors
It is not yet clear whether all of the flavor ratios defined in Section II will indeed beavailable at IceCube (or any future neutrino telescope). We assume that R π , R µ and S µ willbe measured, and consider cases where S π and T µ are available or not.The goal of this work is not to obtain a detailed realistic estimate of the accuracies thatare expected in the relevant measurements. Such an estimate depends on both features ofthe astrophysical neutrinos that are not yet known ( e.g. the actual total flux), and featuresof the detectors that will only become clear when these neutrinos are observed. The maingoal here is to find the accuracies that are required in order to establish that CP is violated.We thus consider the following experimental accuracies in the measurements of the variousflavor ratios:1. R π : we consider hypothetical accuracies of 5%, 10% or 20%. If the flux is close to theWaxman-Bahcall bound, then we expect O (100) events, and an error of order 10%seems realistic;2. S π : In the cases that it is available, we relate the accuracy to that of R π , byassuming a Poisson distribution of the number of events for each neutrino flavor.We neglect issues of efficiency in detecting tracks versus showers. This leads to∆ S π /S π = p S π (1 + S − π ) / (1 + R − π )(∆ R π /R π ). Using central values from Eq. (14),we obtain ∆ S π /S π = 1 . R π /R π ); 7 ABLE II: Scenarios for experimental accuraciesScenario ∆ R π /R π ∆ R µ /R µ ∆ S µ /S µ (∆ S π /S π ) (∆ T µ /T µ )(5 ,
5) 5 5 7 6 5(5 ,
10) 5 10 13 6 10(5 ,
20) 5 20 27 6 20(10 ,
10) 10 10 13 12 10(10 ,
20) 10 20 27 12 20(20 ,
20) 20 20 27 24 20 R µ : we consider hypothetical accuracies which are at best the same as the error on R π and at worst 20%;4. S µ : Following the same line of thought as for S π , we use ∆ S µ /S µ = q S µ (1 + S − µ ) / (1 + R − µ )(∆ R µ /R µ ). Using central values from Eq. (15), we obtain∆ S µ /S µ = 1 . R µ /R µ );5. T µ : In the cases that it is available, we assume ∆ T µ /T µ = ∆ R µ /R µ .The various scenarios can be defined by the assumed accuracies in R π and R µ : Wedenote by ( a, b ) a scenario where the errors are ∆ R π /R π = a % and ∆ R µ /R µ = b %. The sixscenarios that we consider are presented in Table II.We thus consider a hypothetical set of measurements – R , S , T and sin θ ij – whichprovide information on θ ij and δ . The statistical procedure by which this information isextracted is described in the following section. C. Statistical procedure
Given a measurement of an observable Y meas = h Y i ± σ Y , we construct χ ( θ ij , δ ) = P Y h h Y i− Y ( θ ij ,δ ) σ Y i , where Y ( θ ij , δ ) represents the theoretical description of the Y observable.The uncertainty σ Y is given in Table I for sin θ ij and in Table II for R , S and T . A8tatistical handling of the parameters is performed by analyzing the quantity ∆ χ ( θ ij , δ ) = χ − min θ ij ,δ { χ } .We define the N -dimensional “ α % CL acceptance region”, for a subset of N out ofthe four mixing parameters ( θ ij , δ ), by the region in the N parameter space for which∆ χ < C − ( α, N ). Here ∆ χ is obtained by marginalizing ∆ χ with respect to the4 − N redundant parameters and C − ( α, N ) is the inverse chi-square CDF with N degreesof freedom, evaluated at the point α . We have compared this procedure to the more compu-tationally demanding FC construction, (as described in [18] and demonstrated, for example,in [19]) under the assumption of gaussian measurement errors, for several sample configura-tions. We have found a reasonable agreement between our simplified method and the full FCroutine, with the former tending in general to supply slightly more conservative acceptanceregions.We define the “ α % CL acceptance interval”, for a specific parameter, by the set ofparameter values for which the condition ∆ χ < C − ( α,
1) is satisfied, with ∆ χ givenby marginalizing ∆ χ with respect to all of the other parameters.An “ α % CL fraction of coverage” is further defined for a specific parameter as the per-centage of the parameter range that is included in the α % CL acceptance interval. Thelower is this fraction, the stronger is the exclusion power of the experiment with respect tothe relevant parameter.We say that a specific value of a parameter is excluded with α % confidence, if this value isnot contained in the corresponding α % acceptance interval. This notion will be used below,when we discuss the prospects of various measurement scenarios do exclude CP conservationin neutrino oscillations. V. RESULTSA. Neglecting uncertainties in θ and θ To understand the abilities and difficulties that are intrinsic to the measurements byneutrino telescopes, we first carry out an analysis where θ and θ are held fixed at theircurrent best fit values. In the next section, we will study the implications of the uncertaintiesin these angles. 9e begin by choosing specific values for the parameters θ and δ , which we call “trueparameters”. Concretely, we assume a true value θ = 0 .
15, and consider mainly threepossibilities for the true value of δ : the two CP conserving ones ( δ = 0 , π ) and the maximallyCP violating one ( δ = π/ θ and δ (obtaining, of course, the “truevalues” as the best-fit parameters, but with acceptance regions that are different betweenthe various scenarios).The resulting 90% CL acceptance regions in the θ − δ plane are presented, for the sixscenarios, in Figs. 1, 2 and 3. As can be seen in the figures, for some cases, the neutrinotelescope measurements can mildly improve our knowledge of θ compared to the reactorconstraint.As concerns δ , the 90% CL fraction of coverage in case that all the relevant observableswill be measured is shown in Fig. 4, for true θ = 0 .
15 and scanning values of true δ between 0 and π . Since only CP-even quantities are considered, the results for δ = π + θ are equal to those for δ = π − θ . We can make the following statements:1. If the neutrino telescope measurements reach the accuracy assumed in this work, theyare likely to exclude a certain range of δ .2. If the Dirac phase is small (that is close to 0 or π ), the excluded range will be quitesignificant.3. The combination of all available observables is usually significantly more efficient thanpartial combinations.4. The power of combining measurements is particularly significant as resolutions getworse and in the large phase ( δ ∼ π/
2) case.5. If only R π is measured, no range of δ will be excluded.The main question that we are asking is the following: Given a hypothetical situationwhere δ ∼ π/
2, will IceCube be able to establish CP violation, that is, exclude 0 and π from the acceptance interval in δ ? The answer depends of course on which of the various10cenarios described in Table II, if any, will indeed be achieved in the experiment. The mainlessons that we draw from our calculations are the following: (5,5): Measuring R µ and S µ with an accuracy that is significantly better than 10 percentwill enable a discovery of CP violation in neutrino oscillations. (5,10): With this scenario, the sensitivity to CP violation is only marginally affected ifeither T µ or S π are removed from the analysis. Studying the acceptance interval for δ , onefinds that CP violation may be established even without either T µ or S π . This result will befurther qualified when we elaborate on the scenario, below. (10,10): If both T µ and S π are measured, with an accuracy ∼ R π can be somewhat relaxed. ((5,10,20),20): If the flavor ratios from muon-damped sources cannot be measured withan accuracy significantly better than 20%, then even an excellent measurement of flavorratios from pion sources will not exclude CP conservation.We learn that the (5 ,
10) scenario gives a reasonable sense of the minimal required setof measurements and accuracies in order that a discovery that CP is violated in neutrinooscillations will become possible. Further insight into the role of each of the five observablesin achieving this goal is given in Fig. 5, depicting the flavor ratios as a function of δ andthe χ composition for true δ = π/
2. While measurements of R π and R µ at the assumedaccuracies suffice to exclude δ = π , at least one of S π or T µ needs to be added in order toexclude δ = 0.The probability that CP conserving values of δ will be excluded as a function of thetrue δ , within the four scenarios (5,5), (5,10), (5,20) and (10,10), is shown in Fig. 6. Toproduce this plot, we generated a large sample (1000) of random sets of observables withthe prescribed statistics, then checked for each realization whether δ = 0 or π is containedin the resulting acceptance interval. For example, with zero uncertainties in θ and θ , theconditional probability to exclude CP conservation in the (10,10) scenario given maximalphase is about 50%. Note that statistical fluctuations may lead to erroneous exclusion ofCP conservation even with sin δ = 0. The fact that the (10,10) scenario is more likely than(5,20) to establish CP violation is suggestive for future detector optimizations: If the errorson θ and θ at the time of analysis are significantly reduced, then it may be preferable toimprove the detection efficiency at the higher range of the spectrum, E >
T eV , even atthe cost of somewhat weaker efficiency at lower energies.11 . Taking into account uncertainties in θ and θ As a first step in this analysis, we considered the present ranges for θ and θ (see TableI). The potential of neutrino telescopes to exclude a range of δ can be seen from Fig. 4(upper right panel). The impact of the uncertainties in θ and θ can be understood bycomparing it to the upper left panel. We learn that, with present accuracies, the excludedranges are weaker by 30-50% compared to the idealized case of zero uncertainties. (Theimportance of this ingredient in the analysis was noted in [20].)As a second step, we assumed experimental errors on sin θ and sin θ that are reducedby a factor of two compared to the present (see Table I). The results are shown in Fig. 4(lower panel). By comparing to the upper right panel, we learn that such an improvementwill entail an exclusion power stronger by about 20% compared to the situation that presentuncertainties remain.Concerning the probability that CP violation will be established, we repeat the analysiswith the present and with the assumed future uncertainties for the four leading scenarios.The results are shown in Fig. 6. Without an improvement in the determination of θ and θ ,only the very optimistic scenario (5,5) allows a discovery. With the assumed improvements,the more realistic (5,10) scenario also has over 30% probability to make such a discovery.The (5,20) and (10,10) scenarios are not powerful enough to do so. C. Discussion
A related analysis has been performed previously in Refs. [6, 8], which highlighted thesynergy between neutrino telescopes and terrestrial experiments. The conclusion in Refs.[6, 8] regarding the impact of neutrino telescopes on the issue of CP violation is morepessimistic than ours. The main difference lies in the fact that Refs. [6, 8] consider theinformation of one type of sources at a time, and indeed we agree with the pessimisticconclusion in this case. What we show, however, is that by combining the two types ofsources that we considered, the ability to exclude CP conservation improves considerably.Actually, if this combination of sources is indeed available (and the experimental accuracy issimilar to or better than our (10,10) scenario), the exclusion power that neutrino telescopeshave on δ will be comparable to the proposed superbeams [21]. (This situation actually12einforces the point made in [6]: since the δ -dependencies of the IceCube and the superbeammeasurements are different, the information from the two will be complimentary.)Ref. [22] points out that variations in the flavor ratios between sources can reach theten percent level and consequently play an important role in the investigation of the mixingparameters from astrophysical neutrinos. In particular, the resulting uncertainties maywash-out the effects of the ∆ terms, especially in the case of low θ . We agree thatflavor composition uncertainties at the source would tighten greatly the requirements onthe experimental precision. There are two reasons, however, why we think that this issuemay have only limited consequences for our purposes. First, by the time that this analysiscan be carried out in IceCube, the theoretical analysis of neutrino spectra, which is onlyat its beginning [17, 22], is likely to improve considerably. In particular, higher qualityelectromagnetic data, from radio to TeV photon energies, will become available. Second,our study is relevant only for the case of large θ where, as we have argued, 10% accuracymight be just enough for our purposes if a global analysis of flavor-dependent spectrum willbe possible.The general trends reflected in our results can be simply understood, based on Eqs. (14)and (15). We rewrite them as follows: R π = 0 .
49 [1 − . s / .
15) cos δ ] ,S π = 1 .
04 [1 + 0 . s / .
15) cos δ ] ,R µ = 0 .
62 [1 − . s / .
15) cos δ ] ,S µ = 0 .
58 [1 + 0 . s / .
15) cos δ ] ,T µ = 0 .
30 [1 + 0 . s / .
15) cos δ ] . (16)We learn the following: • The ratios related to muon-damped sources are more sensitive to the cos δ -dependentterms than those related to pion sources; • To be sensitive to the cos δ -dependent terms, the accuracy should be of order 10% orbetter; • The required accuracy scales with s . If, for example, s ∼ .
05, sensitivity to cos δ will be achieved only with accuracy better than 5%, which seems out of reach forIceCube. 13 I. CONCLUSIONS
We have studied the potential of combining measurements of flavor ratios in neutrinotelescopes with observation of θ = 0 by reactor experiments in constraining δ , the CPviolating phase in the lepton mixing matrix. We reached the following conclusions: • Since the neutrino telescopes are sensitive only to the combination ∆ ≡ sin θ cos δ ,they can constrain δ only if sin θ is not too small [6]. • Neutrino telescope may exclude at 90% CL up to 30% of the a-priori allowed rangefor δ , even with present accuracies in θ and θ . • Since the ∆ -term is maximized in size for cos δ = ±
1, the exclusion region is largestif CP is nearly conserved [6]. • Reduced uncertainties in θ and θ can enlarge the excluded region to about 50% ofthe a-priori allowed range, and give sensitivity even for cos δ ∼ • Measuring flavor ratios of fluxes from muon-damped sources will further strengthenthe exclusion power (compared to measurements based on solely pion sources). Theirsignificance is particularly important for cos δ ∼ δ is large ( ∼ π/ π , and by that prove thatCP is violated in neutrino interactions. Our conclusions regarding this question are thefollowing: • sin θ must be large, between current 1 − σ upper bounds. • The neutrino flux must not be lower than the Waxman-Bahcall bound. If the fluxis smaller, a larger neutrino telescope may still achieve this goal, within a reasonabletime scale ( < ∼
10 years). • Neutrino flux from muon-damped sources must be identified, and the related flavorratios measured with accuracy better than 10%.14
The uncertainties on θ and θ must be reduced by other experiments by a factor ofabout two.Even if all these conditions are met, the probability of excluding CP conservation in neutrinooscillations is at best 60%.The strongest sensitivity to cos δ arises in flavor ratios related to muon-damped sources.On the theoretical side, a more careful study of the transition at high energy from pion-sourceto muon-damped source is important for better understanding of this crucial ingredient inour analysis [22]. On Nature’s side, the lower the transition energy, and the sharper the tran-sition, the higher statistics of events from muon-damped source that will become availableand, consequently, the better chances are that a neutrino telescope will contribute signifi-cantly to understanding CP violation in neutrino oscillations. Finally, on the experimentalside, a neutrino telescope that is effectively ten times bigger than IceCube, for neutrinoenergy ∼ T eV (see Section V A), is well motivated by our arguments.The fact that establishing CP violation in IceCube, an experiment under construction, isnot manifestly impossible is exciting. While a combination of several favorable circumstancesis required to achieve such a goal, it is worth to refine this analysis, to prepare for a fortunatecase that these circumstances are fulfilled by the parameters of Nature and by the capabilitiesof neutrino telescopes.
Acknowledgments
We are grateful to Concha Gonzalez-Garcia, Francis Halzen, and Walter Winter for usefuldiscussions. The research of Y.N. is supported by the Israel Science Foundation founded bythe Israel Academy of Sciences and Humanities, the United States-Israel Binational ScienceFoundation (BSF), Jerusalem, Israel, the German-Israeli foundation for scientific researchand development (GIF), and the Minerva Foundation. E.W.’s research is partly supportedby the ISF, AEC and Minerva grants. [1] For a review, see e.g.
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Meloni, arXiv:0704.0718 [astro-ph]. π /4 π /23 π /4 π π /43 π /27 π /42 π δ C P θ e3 π , sin θ R π , S π , sin θ R µ , S µ , sin θ R π , R µ , S µ , sin θ R π , R µ , S π , S µ , sin θ R π , R µ , S π , S µ , T µ , sin θ (5,5) π /4 π /23 π /4 π π /43 π /27 π /42 π δ C P θ e3 π , sin θ R π , S π , sin θ R µ , S µ , sin θ R π , R µ , S µ , sin θ R π , R µ , S π , S µ , sin θ R π , R µ , S π , S µ , T µ , sin θ (5,10) π /4 π /23 π /4 π π /43 π /27 π /42 π δ C P θ e3 π , sin θ R π , S π , sin θ R µ , S µ , sin θ R π , R µ , S µ , sin θ R π , R µ , S π , S µ , sin θ R π , R µ , S π , S µ , T µ , sin θ (5,20) π /4 π /23 π /4 π π /43 π /27 π /42 π δ C P θ e3 π , sin θ R π , S π , sin θ R µ , S µ , sin θ R π , R µ , S µ , sin θ R π , R µ , S π , S µ , sin θ R π , R µ , S π , S µ , T µ , sin θ (10,10) π /4 π /23 π /4 π π /43 π /27 π /42 π δ C P θ e3 π , sin θ R π , S π , sin θ R µ , S µ , sin θ R π , R µ , S µ , sin θ R π , R µ , S π , S µ , sin θ R π , R µ , S π , S µ , T µ , sin θ (10,20) π /4 π /23 π /4 π π /43 π /27 π /42 π δ C P θ e3 π , sin θ R π , S π , sin θ R µ , S µ , sin θ R π , R µ , S µ , sin θ R π , R µ , S π , S µ , sin θ R π , R µ , S π , S µ , T µ , sin θ (20,20) FIG. 1: 90%CL (2 d.o.f.) allowed regions for true δ CP = 0. π /4 π /23 π /4 π π /43 π /27 π /42 π δ C P θ e3 π , sin θ R π , S π , sin θ R µ , S µ , sin θ R π , R µ , S µ , sin θ R π , R µ , S π , S µ , sin θ R π , R µ , S π , S µ , T µ , sin θ (5,5) π /4 π /23 π /4 π π /43 π /27 π /42 π δ C P θ e3 π , sin θ R π , S π , sin θ R µ , S µ , sin θ R π , R µ , S µ , sin θ R π , R µ , S π , S µ , sin θ R π , R µ , S π , S µ , T µ , sin θ (5,10) π /4 π /23 π /4 π π /43 π /27 π /42 π δ C P θ e3 π , sin θ R π , S π , sin θ R µ , S µ , sin θ R π , R µ , S µ , sin θ R π , R µ , S π , S µ , sin θ R π , R µ , S π , S µ , T µ , sin θ (5,20) π /4 π /23 π /4 π π /43 π /27 π /42 π δ C P θ e3 π , sin θ R π , S π , sin θ R µ , S µ , sin θ R π , R µ , S µ , sin θ R π , R µ , S π , S µ , sin θ R π , R µ , S π , S µ , T µ , sin θ (10,10) π /4 π /23 π /4 π π /43 π /27 π /42 π δ C P θ e3 π , sin θ R π , S π , sin θ R µ , S µ , sin θ R π , R µ , S µ , sin θ R π , R µ , S π , S µ , sin θ R π , R µ , S π , S µ , T µ , sin θ (10,20) π /4 π /23 π /4 π π /43 π /27 π /42 π δ C P θ e3 π , sin θ R π , S π , sin θ R µ , S µ , sin θ R π , R µ , S µ , sin θ R π , R µ , S π , S µ , sin θ R π , R µ , S π , S µ , T µ , sin θ (20,20) FIG. 2: 90%CL (2 d.o.f.) allowed regions for true δ CP = π . π /4 π /23 π /4 π π /43 π /27 π /42 π δ C P θ e3 π , sin θ R π , S π , sin θ R µ , S µ , sin θ R π , R µ , S µ , sin θ R π , R µ , S π , S µ , sin θ R π , R µ , S π , S µ , T µ , sin θ (5,5) π /4 π /23 π /4 π π /43 π /27 π /42 π δ C P θ e3 π , sin θ R π , S π , sin θ R µ , S µ , sin θ R π , R µ , S µ , sin θ R π , R µ , S π , S µ , sin θ R π , R µ , S π , S µ , T µ , sin θ (5,10) π /4 π /23 π /4 π π /43 π /27 π /42 π δ C P θ e3 π , sin θ R π , S π , sin θ R µ , S µ , sin θ R π , R µ , S µ , sin θ R π , R µ , S π , S µ , sin θ R π , R µ , S π , S µ , T µ , sin θ (5,20) π /4 π /23 π /4 π π /43 π /27 π /42 π δ C P θ e3 π , sin θ R π , S π , sin θ R µ , S µ , sin θ R π , R µ , S µ , sin θ R π , R µ , S π , S µ , sin θ R π , R µ , S π , S µ , T µ , sin θ (10,10) π /4 π /23 π /4 π π /43 π /27 π /42 π δ C P θ e3 π , sin θ R π , S π , sin θ R µ , S µ , sin θ R π , R µ , S µ , sin θ R π , R µ , S π , S µ , sin θ R π , R µ , S π , S µ , T µ , sin θ (10,20) π /4 π /23 π /4 π π /43 π /27 π /42 π δ C P θ e3 π , sin θ R π , S π , sin θ R µ , S µ , sin θ R π , R µ , S µ , sin θ R π , R µ , S π , S µ , sin θ R π , R µ , S π , S µ , T µ , sin θ (20,20) FIG. 3: 90%CL (2 d.o.f.) allowed regions for true δ CP = π/ δ CP δ C P c o v e r age [ % ] π /4 π /2 3 π /4 π (5,5)(5,10)(5,20)(10,10)(10,20)(20,20) θ & θ fixed 60657075808590951000 π /4 π /2 3 π /4 π true δ CP δ C P c o v e r age [ % ] (5,5)(5,10)(5,20)(10,10)(10,20)(20,20) π /4 π /2 3 π /4 π δ C P c o v e r age [ % ] true δ CP (5,5)(5,10)(5,20)(10,10)(10,20)(20,20)reduced θ & θ uncertainties FIG. 4: 90% CL fraction of coverage for δ in the six scenarios defined in Table II. The three panelsdiffer in the uncertainties attributed to θ and θ (see Table I): (upper left) Zero uncertainties;(upper right) Present uncertainties; (bottom) ‘Future’ uncertainties. .30.40.50.60.70.80.911.1 δ CP X π /4 π /2 3 π /4 π π /4 3 π /2 7 π /4 2 π R µ S µ R π S π T µ δ CP ∆ χ π /4 π /2 3 π /4 π π /4 3 π /2 7 π /4 2 π R π S π R µ S µ T µ FIG. 5: The (5,10) scenario: (left) The flavor ratios as a function of δ and their one-sigma range(arrows mark the central values corresponding to δ = π/ χ composition for true δ = π/
2. Both panels correspond to θ and θ fixed at their best-fit values, θ = 0 . π /4 π /2 3 π /4 π true δ CP P r ob ’ [ % ] (5,5)(5,10)(5,20)(10,10) θ e2 & θ µ fixed 510152025303540450 π /4 π /2 3 π /4 π P r ob ’ [ % ] true δ CP (5,5)(5,10)(5,20)(10,10) δ CP P r ob ’ [ % ] π /4 π /2 3 π /4 π (5,5)(5,10)(5,20)(10,10)reduced θ e2 & θ µ uncertainties FIG. 6: Probability to exclude CP conservation with 90%CL. The three panels differ in the un-certainties attributed to θ and θ (see Table I): (upper left) Zero uncertainties; (upper right)Present uncertainties; (bottom) ‘Future’ uncertainties.(see Table I): (upper left) Zero uncertainties; (upper right)Present uncertainties; (bottom) ‘Future’ uncertainties.