The need for an early anti-neutrino run of NOvA

The moderately large value of $\ty$, measured recently by reactor experiments, is very welcome news for the future neutrino experiments. In particular, the \nova experiment, with 3 years each of $\nu$ and $\anu$ runs, will be able to determine the mass hierarchy if one of the following two favourable combinations is true: normal hierarchy with $-180^\circ \leq \dcp \leq 0$ or inverted hierarchy with $0\leq \dcp \leq 180^\circ$. In this report, we study the hierarchy reach of the first 3 years of \nova data. Since $\sin^2 2 \tz$ is measured to be non-maximal, $\tz$ can be either in the lower or higher octant. Pure $\nu$ data is affected by $\ty$-hierarchy and octant-hierarchy degeneracies, which limit the hierarchy sensitivity of such data. A combination of $\nu$ and $\anu$ data is not subject to these degeneracies and hence has much better hierarchy discrimination capability. We find that, with a 3 year $\nu$ run, hierarchy determination is possible for only two of the four octant-hierarchy combinations. Equal 1.5 year runs in $\nu$ and $\anu$ modes give good hierarchy sensitivity for all the four combinations.

Introduction: Neutrino oscillations are the first and, so far, the only evidence we have for physics beyond standard model. During the past year and a half, a lot of excitement has been created after the reactor neutrino experiments measured the last neutrino mixing angle θ 13 to be non-zero [1][2][3]. In fact, its measured value is moderately large and is just below the upper limit established earlier. Daya Bay experiment gives the most precise value of sin 2 2θ 13 = 0.089±0.01 [1]. By the end of its run, the uncertainty is expected to be reduced from the present 10% to 5% [4]. Another important recent discovery is the precision measurement of sin 2 2θ 23 by MINOS, which found it to be non-maximal [5]. 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 all oscillation parameters reasonably well except δ CP , on which there is no information [6][7][8]. The observed energy dependence of the solar neutrino survival probability requires ∆ 21 to be positive. But, present data allow ∆ 31 to be either positive or negative. Given that |∆ 31 | ∆ 21 , we have two possibilities: m 3 m 2 > m 1 , called normal hierarchy (NH) or m 2 > m 1 m 3 , called inverted hierarchy (IH). 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.
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. 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 [9], the search 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 µe ) andν µ →ν e oscillation (Pμē). 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 . In particular, the NOνA experiment can determine the mass hierarchy [10,11] and octant of θ 23 [12,13] for about half the possible values of δ CP .
The leading term in P µe and in Pμē [14] are driven by the matter modified ∆ 31 , which depends on hierarchy. The second term, suppressed by the small parameter α = ∆ 21 /∆ 31 , depends on δ CP . For NH (IH), the first term in P µe becomes larger (smaller). For Pμē, the situation is reverse. The change in the first term can be canceled by changing the value of δ CP in second term, leading to hierarchy-δ CP degeneracy [15,16]. This is illustrated in fig. 1, where P µe and Pμē for NOνA are plotted. We see that, P µe and Pμē for NH and δ CP in the upper half plane (UHP) (0 ≤ δ CP ≤ 180 • ) are very close to or degenerate with those of IH and δ CP in the lower half plane (LHP) (−180 • ≤ δ CP ≤ 0). For these unfavorable combinations, NOνA has no hierarchy sensitivity [11]. Addition of T2K data gives rise to a small sensitivity [16,17]. On the other hand, for NH and δ CP in LHP the values of P µe (Pμē) are reasonably greater (lower) than the values of P µe (Pμē) for IH and any value of δ CP . Similarly, for IH and δ CP in UHP the values of P µe (Pμē) are reasonably lower (greater) than the values of P µe (Pμē) for NH and any value of δ CP . Hence, for these 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.
In this letter, we study the possible hierarchy reach of the first three years of NOνA data. We consider two options: (a) a 3 year ν run (3ν) and (b) equal ν andν runs of 1.5 years each (1.5ν+1.5ν). We find that the combined ν andν runs have a superior hierarchy determination capability compared to the pure neutrino run.
Simulations: NOνA experiment [18] 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 oscillation maximum energy of 1.4 GeV. It is scheduled to have equal ν andν runs of 3 years each, with a NuMI beam power of 700 kW, corresponding to 6 × 10 20 protons on target per year. In our simulations, we have used the re-tuned signal acceptance and background rejection factors taken from [17,19]. 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. The other parameters used are sin 2 2θ 13 = 0.089 and ∆m 2 eff = ±2.4 × 10 −3 eV 2 , where the positive (negative) sign is for NH (IH). ∆ 31 is derived from ∆m 2 eff from the expression given in [20]. In the case of 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 the lower octant (LO) and 0.59 for θ 23 in the higher octant (HO) [8]. The number of electron neutrino appearance events N e and the electron antineutrino appearance events Nē 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 numbers for the true and  the wrong hierarchies. The event number simulations and the ∆χ 2 calculation are done by using the software GLoBES [21,22]. The minimum ∆χ 2 is computed by doing a marginalization over the neutrino parameters. We took σ(∆m 2 eff ) = 3% [23] 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. 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 • ].
Effect of precision of sin 2 2θ 13 : In fig. 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 favorable 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 . The lower sensitivity of 3ν run is due to the marginalization over θ 13 . The 1.5ν + 1.5ν run is less sensitive to this marginalization. This occurs because of the following reason. Let us assume (NH and LHP) is true. Because δ CP values are in favorable half plane, for a given θ 13 , P µe (NH) is greater than P µe (IH) for any δ CP , implying a clean hierarchy separation. But, when we marginalize over θ 13 , it is possible to have P µe (IH, θ 13 ) close to P µe (NH, θ 13 ), where θ 13 > θ 13 . Hence, the ∆χ 2 for 3ν run will be moderate. However, forν case, Pμē(IH, θ 13 ) will be farther away from Pμē(NH, θ 13 ). Thus, if the value of θ 13 is chosen to minimize |N e (NH, θ 13 ) − N e (IH, θ 13 )|, then |Nē(NH, θ 13 ) − Nē(IH, θ 13 )| becomes larger. Varying θ 13 away from the true value of θ 13 can make ∆χ 2 ν very small but it will also make ∆χ 2 ν very large, leading to the total ∆χ 2 = ∆χ 2 ν + ∆χ 2 ν also being very large. Be-  cause of this, for 1.5ν +1.5ν run, the minimum of ∆χ 2 occurs when θ 13 very close to θ 13 even when the marginalization range of θ 13 is large. Therefore 1.5ν + 1.5ν run is less sensitive to marginalization in θ 13 and gives a better hierarchy reach compared to 3ν run, if the uncertainty in sin 2 2θ 13 is large. 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. Non-maximal θ 23 : 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 favorable hierarchy-δ CP combinations, (NH and LHP) and (IH and UHP). But, because of the octant degeneracy of θ 23 , we must consider four possible combinations of octant and hierarchy: LO-NH, HO-NH, LO-IH and HO-IH.
In fig. 3, we show the hierarchy capability assuming (NH and LHP). The left (right) panel corresponds to θ 23 in LO (HO). In fig. 4, we do the same for (IH and UHP), where again the left and the right panels have the same interpretation. 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 favorable 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. Addition of 5 year ν data from T2K leads only to a small improvement.
The very small values of ∆χ 2 , for the 3ν run, occur due to the marginalization over sin 2 θ 23 and δ CP . As mentioned before, the dominant term in P µe 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 µe (HO-NH) is significantly higher than P µe (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 favorable half-plane for 3ν (1.5ν + 1.5ν) run. Matter effects in IH make the leading term in P µe smaller and choosing LO makes it even smaller. So, for δ CP in UHP, P µe (LO-IH) is significantly smaller than P µe (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 favorable 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 µe whereas the choice of LO lowers it. Similarly, for the combination HO-IH, the choice of IH lowers P µe 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 that of the true hierarchy. Thus, it is possible to have P µe (LO-NH, δ CP ) mimic P µe (HO-IH, δ CP ), where 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 µe (LO-NH) ≈ P µe (HO-IH), the corresponding values of Pμē will be far apart. This is illustrated in fig. 5 for two cases, where the two left panels have δ CP = δ CP and the two right panels have δ CP = δ CP . The large separation in Pμē 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%.   Conclusions: NOνA experiment plans to have a 3 year ν run followed by a 3 yearν run. The hierarchy reach of the 3ν run depends on the uncertainty in sin 2 2θ 13 . 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. Equal ν andν runs of 1.5 years each have good hierarchy discrimination for all octant-hierarchy combinations, which is independent of the uncertainty in sin 2 2θ 13 . Therefore, it is imperative for NOνA to plan an earlyν run to get a first hint of hierarchy.