The need for an early anti-neutrino run of NOνA

The moderately large value of θ13, measured recently by reactor experiments, is very welcome news for the future neutrino experiments. In particular, the NOνA experiment, with 3 years each of ν and v¯\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \overline{v} $$\end{document} runs, will be able to determine the mass hierarchy if one of the following two favourable combinations is true: normal hierarchy with −180° ≤ δCP ≤ 0 or inverted hierarchy with 0 ≤ δCP ≤ 180°. In this report, we study the hierarchy reach of the first 3 years of NOνA data. Since sin2 2θ23 is measured to be non-maximal, θ23 can be either in the lower or higher octant. Pure ν data is affected by θ13-hierarchy and octant-hierarchy degeneracies, which limit the hierarchy sensitivity of such data. A combination of ν and v¯\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \overline{v} $$\end{document} data is not subject to these degeneracies and hence has much better hierarchy discrimination capability. We find that, with a 3 year ν run, hierarchy determination is possible for only two of the four octant-hierarchy combinations. Equal 1.5 year runs in ν and v¯\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \overline{v} $$\end{document} modes give good hierarchy sensitivity for all the four combinations.


Introduction
Neutrino oscillations are one of the most significant evidences for physics beyond standard model. The discovery by the reactor neutrino experiments during the last two years, that θ 13 is non-zero, created a lot of excitement [1][2][3]. In fact, its measured value is moderately large and is just below the upper limit established earlier [4][5][6]. The Daya Bay experiment gives the most precise value: sin 2 2θ 13 = 0.089±0.01 [1]. By the end of Daya Bay's run, the uncertainty is expected to be reduced from the present 10% to 5% [7]. Another important recent discovery is the precision measurement of sin 2 2θ 23 by MINOS, which found it to be non-maximal [8]. This raises the problem of determining the true octant of θ 23 . Neutrino oscillations depend on two mass-squared differences, ∆ 21 = m 2 2 − m 2 1 and ∆ 31 = m 2 3 − m 2 1 , three mixing angles and a CP violating phase δ CP . Here m 1 , m 2 and m 3 are the masses of three mass eigenstates. The present oscillation data determine the mass-squared differences and mixing angles reasonably well [9][10][11]. The observed energy dependence of the solar neutrino survival probability requires ∆ 21 to be positive. But the present data allow ∆ 31 to be either positive or negative. The case of positive ∆ 31 is called normal hierarchy (NH) and that of negative ∆ 31 is called inverted hierarchy (IH). If the lightest neutrino mass is negligibly small, we have the following patterns: m 3 m 2 > m 1 for NH and m 2 > m 1 m 3 for IH. It is possible that all the three masses are nearly degenerate. In such a situation also the data allows either hierarchy. Determination of the neutrino mass hierarchy, the octant of θ 23 and the search for CP violation in neutrino sector are the important physics goals of current and future oscillation experiments.

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A number of models are proposed to explain the observed pattern of neutrino masses and mixing. Among these, the models predicting NH are qualitatively different from those predicting IH. Therefore, the determination of the neutrino mass hierarchy will enable us to distinguish between different types of models [12]. A large number of these models predict θ 13 to be zero and θ 23 to be maximal. A precise measurement of the deviations from these predictions will enable us to discern the pattern of symmetry breaking in the models. Ever since the possibility of generating baryon asymmetry via leptogenesis was raised [13], the search for leptonic CP violation has acquired great significance.
A simple way to achieve the above three goals is to measure the probabilities for ν µ → ν e oscillation (P (ν µ → ν e )) andν µ →ν e oscillation (P (ν µ →ν e )). The leading term in both these probabilities is proportional to sin 2 2θ 13 sin 2 θ 23 . Therefore, the moderately large value of θ 13 makes it possible for the current experiments to address the problems of both hierarchy and the octant of θ 23 . Appreciable matter effects in the NOνA experiment make it an excellent tool to determine the hierarchy for favourable values of parameters [14,15]. In addition, T2K and NOνA can determine octant of θ 23 at 2σ [16,17] for all values of δ CP .
2 Degeneracies in P (ν µ → ν e ) and P (ν µ →ν e ) Among the neutrino oscillation parameters, there are two small quantities: θ 13 and α = ∆ 21 /∆ 31 . By setting one or both to be zero, it was possible so far, to reduce all the measured survival probabilities to effective two flavour formulae. In the ν e appearance measurements at T2K and NOνA, the first non-trivial three flavour oscillation effects will be observed, which are proportional to the small quantities θ 13 and α. In the approximation of keeping only the terms which are second order in these small quantities, the ν µ → ν e oscillation probability is given by [18,19], +α cos θ 13 sin 2θ 12 sin 2θ 13 sin 2θ 23 cos(∆ + δ CP ) sin∆Â (2.1) [20]. The expression for P (ν µ →ν e ) is obtained by changing the signs ofÂ and δ CP in P (ν µ → ν e ). ∆ 31 is positive for NH and is negative for IH. From eq. (2.1), we see that the oscillation probability depends on unknowns, i.e. hierarchy, octant of θ 23 and δ CP , along with other parameters, such as θ 13 . A measurement of these probabilities, in general, gives rise to degenerate solutions.

Hierarchy-δ CP degeneracy
From the current measurements, we know that sin 2θ 13 ≈ 0.3 whereas |α| ≈ 0.03. Hence, the first term in P (ν µ → ν e ) (and in P (ν µ →ν e )) is much larger than second term and the third term is completely negligible. The largest amount of matter effect and hence hierarchy sensitivity, comes from the leading term. For NH (IH), the first term in P (ν µ → ν e ) becomes larger (smaller). For P (ν µ →ν e ), the situation is reverse. These changes in P (ν µ → ν e ) and in P (ν µ →ν e ) can be amplified or canceled by the second term, depending on the value of δ CP . This is illustrated in figure 1, where P (ν µ → ν e ) and P (ν µ →ν e ) are plotted for the NOνA experiment. For NH and δ CP in the lower half plane (LHP) (−180 • ≤ δ CP ≤ 0), the values of P (ν µ → ν e ) (P (ν µ →ν e )) are reasonably greater (lower) than the values of P (ν µ → ν e ) (P (ν µ →ν e )) for IH and any value of δ CP . Similarly, for IH and δ CP in the upper half plane (UHP) (0 ≤ δ CP ≤ 180 • ) the values of P (ν µ → ν e ) (P (ν µ →ν e )) are reasonably lower (greater) than the values of P (ν µ → ν e ) (P (ν µ →ν e )) for NH and any value of δ CP . Hence, for these favourable combinations, NOνA is capable of determining the hierarchy at a confidence level (C.L.) of 2σ or better, with 3 years each of ν andν runs. However, as mentioned above, the change in the first term can be canceled by the second term for unfavourable values of δ CP . This leads to hierarchy-δ CP degeneracy [21][22][23]. From figure 1, we see that, P (ν µ → ν e ) and P (ν µ →ν e ) for NH and δ CP in the UHP are very close to or degenerate with those of IH and δ CP in the LHP. For these unfavourable combinations, NOνA has no hierarchy sensitivity [15]. Addition of T2K data gives rise to a small sensitivity [23,24]. In this paper, we explore the further degeneracies in the case of the favourable hierarchy-δ CP combinations.

θ 13 -hierarchy degeneracy
Even if δ CP is in the favourable half-plane, there are further degeneracies which limit the hierarchy sensitivity of an experiment. For example, in eq. (2.1), the increase (reduction)

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in the first term for NH (IH) case, due to matter effect, can be canceled by choosing a lower (higher) value of θ 13 . This θ 13 -hierarchy degeneracy [21] can reduce the hierarchy sensitivity. However, a combination of ν andν data is not susceptible to this degeneracy. The reason is the following. In ν data, it is possible to have P (ν µ → ν e )(θ 13 , N H) ≈ P (ν µ → ν e )(θ 13 , IH) with θ 13 > θ 13 . However, for such a choice of θ 13 , we will have P (ν µ →ν e )(θ 13 , N H) significantly smaller than P (ν µ →ν e )(θ 13 , IH). Thus a degeneracy in the ν data is resolved by theν data (and vice-verse). If the allowed range of θ 13 is large, then a combination of ν andν data has better hierarchy sensitivity compared to pure ν data.

Octant-hierarchy degeneracy
A more serious degeneracy, which limits the hierarchy sensitivity, is the octant-hierarchy degeneracy. MINOS experiment has measured sin 2 2θ 23 < 1 [8] and the global fits favour a non-maximal value of θ 23 [9][10][11]. There are two degenerate solutions, with θ 23 in the lower octant (LO) (sin 2 θ 23 < 0.5) and with θ 23 in the higher octant (HO) (sin 2 θ 23 > 0.5). Thus we have four possible octant-hierarchy combinations: LO-NH, HO-NH, LO-IH and HO-IH. As already stated, the first term in P (ν µ → ν e ) becomes larger (smaller) for NH (IH). The same term also becomes smaller (larger) for LO (HO). If the case HO-NH (LO-IH) is true, then the values of P (ν µ → ν e ) are significantly higher (smaller) than those for IH (NH) and any octant. For these two cases, pure ν data has good hierarchy determination capability. But the situation is very different for the two cases LO-NH and HO-IH. The increase (decrease) in the first term of P (ν µ → ν e ) due to NH (IH) is canceled (compensated) by the choice of LO (HO). Thus the two cases, LO-NH and HO-IH, have degenerate values for P (ν µ → ν e ). However, this degeneracy is not present in P (ν µ →ν e ), which receives a double boost (suppression) for the case of HO-IH (LO-NH). Thus the octant-hierarchy degeneracy in P (ν µ → ν e ) is broken by P (ν µ →ν e ) (and vice-verse) as in the case of θ 13 -hierarchy degeneracy. Therefore pure ν data has no hierarchy sensitivity if the cases LO-NH or HO-IH are true, but a combination of ν andν data will have a good sensitivity.

Simulation details
In this report, we study the possible hierarchy reach of the first three years of NOνA data. As shown in the previous section, a pure ν data is subject to θ 13 -hierarchy and octanthierarchy degeneracies, whereas a combination of ν andν data is not. Therefore, here we consider two options: (a) a 3 year ν run (labeled 3ν in the rest of the paper) and (b) equal ν andν runs of 1.5 years each (labeled 1.5ν+1.5ν).
NOνA experiment [25] consists of a 14 kiloton totally active scintillator detector (TASD), placed 810 km away from Fermilab, situated at a 0.8 • off-axis location from the NuMI beam. The ν flux peaks sharply at 2 GeV, close to the energy range 1.4-1.8 GeV, where the oscillation maxima occur for NH and for IH. It is scheduled to have equal ν and ν runs of 3 years each, with a NuMI beam power of 700 kW, corresponding to 6 × 10 20

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protons on target per year. In our simulations, we have used the re-tuned signal acceptance and background rejection factors taken from [24,26]. In the numerical simulations, we took the solar oscillation parameters to be sin 2 θ 12 = 0.30 and ∆ 21 = 7.5 × 10 −5 eV 2 , which have been kept fixed [11]. The other parameters used are sin 2 2θ 13 = 0.089 and ∆m 2 eff = ±2.4 × 10 −3 eV 2 [8], where the positive (negative) sign is for NH (IH). ∆ 31 is derived from ∆m 2 eff from the expression given in [27]. For θ 23 , we considered the cases of both maximal and non-maximal mixing. For maximal mixing (MM), sin 2 θ 23 = 0.5. For non-maximal mixing, we have used the two degenerate best-fit values of the global fits: 0.41 for θ 23 in LO and 0.59 for θ 23 in HO [11].
The spectrum of electron neutrino appearance events and that of the electron antineutrino appearance events are first computed for an assumed true hierarchy. The same quantities are calculated again for the wrong hierarchy and the ∆χ 2 is computed between the event spectra for the true and the wrong hierarchies. The event spectrum simulations and the ∆χ 2 calculation are done by using the software GLoBES [28,29]. The minimum ∆χ 2 is computed by doing a marginalization over the neutrino parameters. We took σ(∆m 2 eff ) = 3% [30] and σ(sin 2 2θ 13 ) = 10% in the preliminary calculations and 5% in later calculations. For both these parameters, the marginalization was done over 2σ range with Gaussian priors. The marginalization range for sin 2 θ 23 is its 3σ allowed range: [0.35, 0.65] and that of δ CP is the full range [−180 • , 180 • ]. 1 No priors were added for these two parameters.

Effect of precision of sin 2 2θ 13 on hierarchy determination
In figure 2 we have shown the hierarchy determination potential of NOνA assuming a 10% uncertainty in sin 2 2θ 13 . The plots show ∆χ 2 vs. δ CP (true) for θ 23 = 45 • , for both 3ν and 1.5ν + 1.5ν runs. The left panel is for NH and LHP and the right panel is for IH and UHP. We see from these plots that a 2σ hierarchy determination is possible for about 50% of the favourable half plane for 1.5ν + 1.5ν run, whereas a 3ν run can determine hierarchy for only a smaller range. In particular, if IH and UHP is true, a 2σ hierarchy determination is not possible for any δ CP . Here the number of σ is taken to be ∆χ 2 .
The lower sensitivity of 3ν run is due to the marginalization over θ 13 . Because of the relatively large range of variation for θ 13 , it is possible for P (ν µ → ν e )(θ 13 , IH) to come reasonably close to P (ν µ → ν e )(θ 13 , N H), thus reducing the ∆χ 2 . As explained in the previous section, the 1.5ν + 1.5ν run is less sensitive to this marginalization and gives a larger ∆χ 2 . If the uncertainty in sin 2 2θ 13 is reduced to 5%. the hierarchy reach for 3ν does improve and becomes equal to that of 1.5ν + 1.5ν run.

Resolving the octant-hierarchy degeneracy
We now assume that σ(sin 2 2θ 13 ) = 5% and take θ 23 to be non-maximal. Once again we limit ourselves to the favourable hierarchy-δ CP combinations, NH and LHP and IH and UHP. But   In figure 3, we show the hierarchy capability assuming NH and LHP. The left (right) panel corresponds to θ 23 in LO (HO). In figure 4, we do the same for IH and UHP. From these figures, we see that for HO-NH and LO-IH, 3ν run does have a better hierarchy reach compared to 1.5ν + 1.5ν run and is capable of giving a better than 2σ hierarchy discrimination for more than half of the favourable half plane. But, for the other two possibilities, LO-NH and HO-IH, 3ν run has no hierarchy sensitivity whereas 1.5ν + 1.5ν  run has reasonable hierarchy sensitivity. The very small values of ∆χ 2 , for the 3ν run, occur due to the marginalization over sin 2 θ 23 and δ CP . Addition of 5 year ν data from T2K leads only to a small improvement.
As mentioned before, the dominant term in P (ν µ → ν e ) is proportional to sin 2 2θ 13 sin 2 θ 23 . Matter effects in NH make this term larger and choosing HO makes it even larger. Hence, for δ CP in LHP, P (ν µ → ν e )(HO-NH) is significantly higher than P (ν µ → ν e )(IH) for any values of neutrino parameters. Because of the double increase in the probability, the statistics for HO-NH will be quite large. Hence, this combination has 2σ hierarchy discrimination for 87% (68%) of the favourable half-plane for 3ν (1.5ν + 1.5ν) run. Matter effects in IH make the leading term in P (ν µ → ν e ) smaller and choosing LO makes it even smaller. So, for δ CP in UHP, P (ν µ → ν e )(LO-IH) is significantly smaller than P (ν µ → ν e )(NH) for any values of neutrino parameters. This double decrease in probability, leads to the lowest statistics for LO-IH. Here, 3ν (1.5ν + 1.5ν) run can determine hierarchy at 2σ for 35% (20%) of favourable half-plane. However, it must be emphasized that, in these two cases HO-NH and LO-IH, the hierarchy reach of 1.5ν + 1.5ν is only slightly worse than that of 3ν.
But, for the combination of LO-NH, the choice of NH increases P (ν µ → ν e ) whereas the choice of LO lowers it. Similarly, for the combination HO-IH, the choice of IH lowers P (ν µ → ν e ) and the choice of HO increases it. The marginalization over θ 23 and δ CP leads to a wrong hierarchy probability being very close to the true hierarchy probability. Thus, it is possible to have P (ν µ → ν e )(NH, θ 23 < 45 • , δ CP ) mimic P (ν µ → ν e )(IH, θ 23 > 45 • , δ CP ), where θ 23 and θ 23 may or may not be complementary and δ CP and δ CP may or may not be equal. But, in the case ofν, both the choices LO and NH lead to a reduction in the probability and both the choices HO and IH increase the probability. Whenever it is possible to have P (ν µ → ν e )(NH, θ 23 , δ CP ) ≈ P (ν µ → ν e )(IH, θ 23 , δ CP ), the corresponding values of P (ν µ →ν e ) will be far apart. This is illustrated in figure 5 for two cases, where θ 23 and θ 23 are complementary. For the two left panels δ CP = δ CP and for the two right panels δ CP = δ CP . The large separation in P (ν µ →ν e ) leads to a far better hierarchy discrimination for 1.5ν + 1.5ν run compared to 3ν run. All the results discussed above are neatly summarized in the table I. In all cases, the 1.5ν + 1.5ν data is insensitive to the uncertainty in sin 2 2θ 13 . Except for the no-sensitivity combinations, LO-NH and HO-IH, the 3ν data shows noticeable improvement when the uncertainty is reduced to 5% but none with further reduction to 2%.
In the most recent global fits of the neutrino oscillation data [31], the best-fit value of sin 2 θ 23 in LO is 0.45, (i. e. closer to the maximal mixing value), though the best-fit value in HO remains at 0.59. We have redone our calculations and compared the hierarchy discrimination ability of 3ν vs 1.5ν + 1.5ν data of NOνA, for these new values of sin 2 θ 23 . These results are shown in figures 6 and 7. As we see from these figures, even with the smaller deviation of θ 23 from maximality, the 3ν run of NOνA has no hierarchy sensitivity for the two combinations LO-NH and HO-IH, whereas the 1.5ν+1.5ν run has good hierarchy determination capability for all four combinations.
The most recent results of the T2K experiment [33] give sin 2 θ 23 = 0.514 +0.055 −0.055 (0.511 ± 0.055) for NH (IH). These values seem to favour maximal mixing but a deviation from maximality is also very likely. The parameters we have chosen here fall within the 2σ range of these measurements. Even if the deviation of θ 23 from maximality is very small (| sin 2 θ 23 − 0.5| = 0.02), the hierarchy sensitivity of 1.5ν + 1.5ν run is better than that of 3ν run for the two combinations LO-NH and HO-IH. This is illustrated in figures 8 and 9.  figure 10. From this figure, we see that there is reasonable hierarchy sensitivity for the combination HO-NH, even from 1.5 years of ν data, but not for the combination LO-IH. This is expected because P (ν µ → ν e ) receives a double boost in the case of HO-NH and hence there will be a large number of signal events. For LO-IH, P (ν µ → ν e ) gets a double suppression and hence the statistics in the 1.5ν run are not sufficient to rule out the wrong hierarchy. Addition of 2 years of ν data from T2K leads to no significant change. This leads us to a very interesting conclusion: the physics capabilities of NOνA are enhanced if it has 1.5ν + 1.5ν runs during the first three years. This statement is true for any octant-hierarchy combination. We see above that, for the combination of HO-NH, a 2σ hint of hierarchy is possible for half of LHP, even with 1.5 years of ν run. If the hierarchy is known after such a run, then a run plan, which has the best CP sensitivity, is preferable. To maximize the CP sensitivity, it is desirable to have equal number of ν andν events [34]. This requires a longerν run because theν cross sections are smaller. Hence, if HO-NH is true, a hierarchy hint can be obtained with a 1.5ν run, after which it is preferable to run NOνA inν mode only. For the other three octant-hierarchy combinations, 1.5ν run does not give a hint of hierarchy. In such a situation, a switch toν run will guarantee a 2σ hierarchy discrimination for a reasonable fraction of the favourable half plane of δ CP .

Conclusions
NOνA experiment is about to start taking data. Among its physics goals are (a) the determination of neutrino mass hierarchy, (b) the determination of the octant of θ 23    (c) the discovery of leptonic CP violation. The hierarchy reach of pure ν data is subject to θ 13 -hierarchy and octant-hierarchy degeneracies, whereas equal ν-ν runs are free from them. If the uncertainty in sin 2 2θ 13 remains at the present 10% level, then the combination 1.5ν + 1.5ν run has better hierarchy sensitivity compared to pure 3ν run. Even when this uncertainty is reduced to 5%, the 3ν run fails to give any hierarchy discrimination, if the true combinations are LO-NH or HO-IH, whereas the combined 1.5ν + 1.5ν run has good hierarchy discrimination for all four octant-hierarchy combinations. We argue that it is advantageous for NOνA to have equal 1.5 years of ν andν runs during the first three years. We find that 1.5ν run gives a 2σ hierarchy hint if the combination HO-NH is true and δ CP is in LHP. In such a situation, it is better to switch toν to maximize the CP sensitivity. For the other three octant-hierarchy combinations, 1.5ν run has poor or no hierarchy sensitivity. Following this up with a 1.5 yearν run will give a better chance of hierarchy discrimination, if δ CP is in the favourable half plane.
Finally, what should happen after 1.5ν + 1.5ν run? If no hint of hierarchy is obtained, then a farther 1.5ν + 1.5ν run seems preferable. Then, the full hierarchy discrimination capability of 3ν + 3ν run of NOνA will be realised. If a hint of hierarchy is found, then having the additional run inν mode is likely to give the best CP sensitivity.