Towards extracting the best possible results from NOvA

The NuMI Off-Axis $\nu_{e}$ Appearance (NO$\nu$A) is the currently running leading long-baseline neutrino oscillation experiment, whose main physics goal is to explore the current issues in the neutrino sector, such as determination of the neutrino mass ordering, resolution of the octant of atmospheric mixing angle and to constrain the Dirac-type CP violating phase $\delta_{CP}$. In this paper, we would like to investigate whether it is possible to extract the best possible results from NO$\nu$A with a shorter time-span than its scheduled run period by analyzing its capability to discriminate the degeneracy among various neutrino oscillation parameters within four years of run time, with two years in each neutrino and antineutrino modes. Further, we study the same by adding the data from T2K experiment for a total of five years run with 3.5 years in neutrino mode and 1.5 years in antineutrino mode. We find that NO$\nu$A (2+2) has a better oscillation parameter degeneracy discrimination capability compared to its scheduled run period for four years, i.e, NO$\nu$A (3+1).


I. INTRODUCTION
The results from various neutrino oscillation experiments [1][2][3][4][5][6][7][8] confirm that neutrino flavours mix with each other and neutrinos do possess tiny but non-zero masses. This mixing of neutrino can be described by a unitary matrix, so called Pontecorvo-Maki-Nagakawa-Sakata (PMNS) matrix, which is parameterized by three mixing angles, often referred to as the solar mixing angle (θ 12 ), atmospheric mixing angle (θ 23 ), reactor mixing angle (θ 13 ) and a Dirac-type CP violating phase (δ CP ) [9,10]. The probability of neutrino oscillation depends on these parameters as well as on two mass squared differences namely, the solar mass squared difference (∆m 2 21 ) and the atmospheric mass squared difference (∆m 2 31 ). All these parameters are determined through various neutrino experiments except the Dirac CP phase. However, we do not know the mass ordering of neutrinos, i.e, the sign of ∆m 2 31 and this left us with two choices like normal ordering/hierarchy (NH) with ∆m 2 31 > 0 and inverted ordering/hierarchy with ∆m 2 31 < 0. Furthermore, recent experimental result from MINOS [11] shows that θ 23 is non-maximal. Therefore, the octant of the mixing angle NOνA [12] experiment is currently running long-baseline neutrino oscillation experiment, which uses an upgraded NuMI beam power of 0.7 MW at Fermilab. It has a 14 kton totally active scintillator detector (TASD) placed 0.8 • off-axis from the NuMI beam near the Ash River, situated 810 km far away from Fermilab. It also has a 0.3 kton near detector located at the Fermilab site to monitor the un-oscillated neutrino or anti-neutrino flux. This experiment is designed to observe both ν e (ν e ) appearance events and ν µ (ν µ ) disappearance events. The main physics goals of this experiment are • Appearance events: To determine the value of θ 13 , determination of the octant of θ 23 , mass ordering and constrain the Dirac CP phase.
The determination of these parameters by an oscillation experiment like NOνA, which is mainly rely on the oscillation probability, is extremely difficult due to the parameter degeneracies, since various combination of these parameters give the same probability. A lot of work has been done in the literature to resolve these degeneracies among oscillation parameters [13][14][15]. Moreover, there was a suggestion for the need of an early anti-neutrino run to get a first hint of mass ordering in NOνA [16]. There it has been shown that the sensitivity for the determination of mass hierarchy is above 2σ (i.e., χ 2 > 4) only for δ CP value around ∓90 • for true hierarchy and octant as NH-LO or HO-IH, where the scheduled run time, i.e., (3 yrs in ν mode + 0 yr inν mode) gives almost null sensitivity.
The scheduled run period of NOνA is for a total of six years with first three years in neutrino mode followed by the next three years in antineutrino mode. Therefore, it is of great importance to study the ability to discriminate the degeneracies between different oscillation parameters of this experiment within a minimal time-span, since it leads to an early understanding of neutrino oscillation parameter space. In this context, we investigate in this paper how to extract the best possible results from NOνA with shortest time-span by analyzing its physics potential and degeneracy discrimination capability for a total of four years of runs, with two years in each neutrino and antineutrino modes. We have shown that the (2 + 2) years of run will provide much better sensitivity for the mass hierarchy determination in comparison to its scheduled run for four years i.e, 3 years in neutrino mode followed by one year in anti neutrino mode. Furthermore, we study the same by adding data from T2K experiment for a total of five years run with 3.5 years in neutrino mode and 1.5 years in antineutrino mode.
The outline of this paper is as follows. In section II, we present the details of simulation of T2K and NOνA experiments that we have considered in this work. The neutrino oscillation parameter degeneracies are discussed in section section III. Section IV contains the discussion about the mass ordering and octant determination. Finally, we summarize our results in section V.

II. SIMULATION DETAILS
We simulate the neutrino oscillation events for T2K (Tokai-to Kamioka) as well as NOνA experiments by using GLoBES package [17,18]. T2K is also a currently running off-axis long-baseline experiment, which has been designed to study the phenomenon of neutrino oscillation. It uses an upgraded beam power of 0.77 MW and has the water cherenkov detector of mass 22.5 kton placed about 295 km away from Tokai. We simulate T2K experiment with updated experimental description as given in [22]. As we mentioned earlier, NOνA experiment is an off-axis experiment with a baseline of 810 km, which uses a beam power of 0.7 MW and a detector of mass 14 kton. The experimental specifications of NOνA are taken from [23] with the following characteristics: Signal efficiency: 45% for ν e andν e signal; 100% ν µ CC andν µ CC.
Background efficiency: a) Mis-ID muons acceptance: 0.83% for ν µ CC, 0.22% forν µ CC ; b) NC background acceptance: 2% for ν µ NC, 3% forν µ NC ; c) Intrinsic beam contamination: 26% for ν e , 18% forν e , and we consider 5% uncertainty on signal normalization and 10% on background normalization. The migration matrices for NC background smearing are taken from [23]. The true values of oscillation parameters that we use in our simulation are listed in the Table-I [24].  FIG. 1: Neutrino and antineutrino appearance events for the ν µ → ν e versusν µ →ν e channels by assuming both IH and NH and for lower and higher octants of θ 23 .

III. NEUTRINO OSCILLATION PARAMETER DEGENERACIES
The parameter degeneracies in neutrino oscillation sector are of mainly three types and they are: (δ CP , θ 13 ), sign of ∆m 2 31 and (θ 23 , π/2 − θ 23 ). Recently, the reactor experiments such as Daya Bay [25,26], Double Chooz [27] and RENO [28] have precisely measured the value of the reactor angle as sin 2 2θ 13 ≈ 0.089 ± 0.01. Therefore, the eight fold degeneracy is reduced to four-fold degeneracy. Out of these degeneracies, the degeneracy in which θ 23 can't be distinguished from (π/2 − θ 23 ) is called octant degeneracy and the degeneracy in the sign of ∆m 2 31 is called hierarchy ambiguity. So far, we are left with four degeneracies and they are represented as NH-HO, NH-LO, IH-HO and IH-LO, where NH/IH (HO/LO) stands for Normal/Inverted ordering (Higher/Lower Octant). Resolution of these degeneracies are the main challenges of the present and future long-baseline neutrino oscillation experiments, which are mainly looking for oscillation from ν µ (ν µ ) → ν e (ν e ). The expression for the oscillation probability, which is up to first order in sin θ 13 and α ≡ ∆ 21 /∆ 31 is given as [29][30][31] P (ν µ → ν e ) ≈ sin 2 2θ 13 sin 2 θ 23 sin 2 (Â − 1)∆ (Â − 1) 2 + α cos θ 13 sin 2θ 12 sin 2θ 13 sin 2θ 23 sinÂ∆ where The best way to express the degeneracies without any mathematical expression is simply by using bi-events curves. The bi-events plots for various octant-hierarchy combinations of T2K and NOνA are depicted in Fig. 1

IV. MASS HIERARCHY AND OCTANT DETERMINATION
In this section, we obtain the potential of NOνA experiment to determine the mass hierarchy and octant of atmospheric mixing angle and discuss the role of mass hierarchyoctant parameter degeneracy in the determination of these parameters.

A. Mass hierarchy determination
For the mass hierarchy determination, we obtain the sensitivity by calculating the χ 2 with which one can rule out the wrong hierarchy from the true hierarchy. We express this sensitivity as a function of true value of δ CP , since it can be seen from Eq. (1) that there exists a degeneracy between hierarchy and δ CP . Therefore, we simulate true events by taking NH (IH) as true hierarchy and test events by taking IH (NH) as test hierarchy for each true value of δ CP . We obtain the χ 2 by using GLoBES and compare both event rates for full range of δ CP . We do marginalization over all other parameters in order to get minimum χ 2 .
We also add a prior on sin 2 2θ 13 . We obtain this χ 2 for various true values of sin 2 θ 23 (i.e, In Fig. 2, we plot the value of χ 2 , obtained for maximal mixing of atmospheric angle, as a function of δ CP . The left panel corresponds to true NH and the right panel is for true IH. From these figures, we can see that the potential to determine mass hierarchy for NOνA is above 2σ for less than half of parameter space of δ CP and it also depends on the neutrino mass ordering. The mass hierarchy sensitivity of NOνA (2+2) is lower (higher) than that Hence, NOνA (2+2) has a good mass hierarchy discrimination capability compared to the scheduled run of NOνA for four years. Thus, we can have an early information about the nature of mass ordering if NOνA runs in (2ν + 2ν) mode rather than its scheduled run of (3ν + 1ν) years. Furthermore, if nature would be kind enough in the sense that the real  In Fig. 4, the obtained χ 2 is plotted as a function of sin 2 θ 23 . Left panel corresponds to NH and the right panel corresponds to IH as true hierarchies. From the plots, it is clear that the potential to determine the octant of atmospheric angle is better for NOνA (2+2), when compared with NOνA (3+1). We can also see that a combined analysis of NOνA (2+2) and T2K (3.5+1.5) has good octant resolution sensitivity.

C. Correlation between θ 23 and ∆m 2 32
The discovery reach of mass hierarchy and octant of atmospheric mixing angles are crucial because of the degeneracies between the oscillation parameters. Therefore, resolution of these degeneracies is very important to have a clear understanding of the neutrino mixing phenomenon.
In this section, we focus on the θ 23 and ∆m 2 32 degeneracy. First of all, we would like to see how does the hierarchy ambiguity affect sin 2 θ 23 -∆m 2 32 parameter space. Therefore, we simulate the true events for maximal value of sin 2 θ 23 (sin 2 θ 23 = 0.5) and test events for . We obtain the χ 2 by comparing true events and test events. We also do marginalization for both sin 2 2θ 13 and δ CP and add a prior on sin 2 2θ 13 . In Fig. 5, the obtained χ 2 is plotted as a function of sin 2 θ 23 and ∆m 2 32 . From the plots, we can see that there is small difference in the allowed parameter space for NH and IH. However, there is no difference in the allowed parameter space for (2+2) and (3+1) years of NOνA running as far as the determination of ∆m 2 32 is concerned. We also get similar results when we compare the parameter space for both NOνA (2+2) and NOνA (3+1), and as expected such parameter spaces are significantly reduced when compared with NOνA (2+1) and NOνA (3+0). It should also be noted from the figure that the parameter space is substantially reduced for a combined analysis of T2K and NOνA.
Therefore, if we combine the (2+2) years of NOνA results with (3.5+1.5) T2K results, the significance of the atmospheric mass square determination will improve significantly.

V. SUMMARY AND CONCLUSIONS
At this point of time, where NOνA experiment already started taking data, it is crucial to analyze how to extract the best results from this experiment with shortest time span for a complete understanding of oscillation parameters. In this paper, we discussed the physics potential as well as the role of parameter degeneracies in the determination of oscillation parameters of NOνA experiment with a total of four years of runs with (2ν+2ν) mode.
We find that the parameter degeneracy discrimination capability of NOνA (2+2) is quite good when compared with NOνA (3+1). Looking all these results from our analysis, it is strongly urged that after two years of neutrino running, NOνA should run for two years in antineutrino mode to provide better information about the determination of neutrino mass ordering and the octant of atmospheric mixing angle.