A comprehensive study of the discovery potential of NOvA, T2K and T2HK experiments

With the recent measurement of reactor mixing angle $\theta_{13}$ the knowledge of neutrino oscillation parameters that describe PMNS matrix has improved significantly except the CP violating phase $\delta_{CP}$. The other unknown parameters in neutrino oscillation studies are mass hierarchy and the octant of the atmospheric mixing angle $\theta_{23}$. Many dedicated experiments are proposed to determine these parameters which may take at least 10 years from now to become operational. It is therefore very crucial to use the results from the existing experiments to see whether we can get even partial answers to these questions. In this paper we study the discovery potential of the ongoing NO$\nu$A and T2K experiments as well as the forthcoming T2HK experiment in addressing these questions. In particular, we evaluate the sensitivity of NO$\nu$A to determine neutrino mass hierarchy, octant degeneracy and to obtain CP violation phase after running for its scheduled period of 3 years in neutrino mode and 3 years in anti-neutrino mode. We then extend the analysis to understand the discovery potential if the experiments will run for (5$\nu$+5$\bar{\nu}$) years and (7$\nu$+3$\bar{\nu}$) years. We also show how the sensitivity improves when we combine the data from (3$\nu$+3$\bar{\nu}$) years of NO$\nu$A run with (3$\nu$+2$\bar{\nu}$) years of T2K and (3$\nu$+7$\bar{\nu}$) years of T2HK experiments. The CP violation sensitivity is marginal for T2K and NO$\nu$A experiments even for ten years data taking of NO$\nu$A. T2HK has a significance above 5$\sigma$ for a fraction of two-fifth values of the $\delta_{CP}$ space. We also find that $\delta_{CP}$ can be determined to be better than $35^\circ $, $21^\circ $ and $9^\circ $ for all values of $\delta_{CP}$ for T2K, NO$\nu$A and T2HK respectively.


I. INTRODUCTION
The discovery of neutrino oscillations has firmly established that neutrinos are massive.
It has marked the beginning of many neutrino oscillation experiments. The mixing of three neutrino flavors can be described by Pontecorvo-Maki-Nakagawa-Sakata matrix U P M N S [1,2] which is parameterized in terms of three rotation angles θ 12 , θ 23 , θ 13 and three CP-violating phases, one Dirac type δ CP and two Majorana types ρ and σ. The neutrino oscillation data accumulated over many years allows us to determine the solar and atmospheric neutrino oscillation parameters with very high precision. Recently, the reactor mixing angle θ 13 has been measured precisely [3][4][5][6][7] with a moderately large value, quite close to its previous upper bound.
The current results from recent neutrino oscillation experiments [8][9][10][11] and their global analysis [12][13][14][15] performed by several groups have implied that the minimal three neutrino framework is adequate to describe the observed oscillation phenomenology. The best-fit values and the 3σ ranges of the oscillation parameters from Ref. [15], are presented in Table-1.
Another important discovery in recent times is the precision measurement of sin 2 θ 23 by MINOS experiment [16] which is found to be non-maximal. Using the complete set of accelerator and atmospheric data they disfavored the maximal mixing by −2∆ log(L) = 1.54. They obtained the best-fit value for the mixing angle θ 23 as sin 2 θ 23 = 0.41 (LO) and sin 2 θ 23 = 0.61 (HO).
With these exciting discoveries of non-zero θ 13 and non-maximal θ 23 the focus of neutrino oscillation studies has now been shifted towards the determination of other unknown parameters. These include the determination of mass hierarchy, octant of the atmospheric mixing angle θ 23 , discovery of CP violation and the magnitude of the CP violating phase δ CP . In this paper we would like to investigate the prospects of addressing these issues with the off-axis long baseline experiments T2K, NOνA and T2HK.
The increasing number of long baseline neutrino oscillation experiments have intensified the quest for good theoretical models and precise experimental measurements of neutrinonucleus cross-sections. These oscillation experiments aim at measuring various neutrino mixing angles and most importantly determining CP-violation. From Eq. (1) it can be seen that to obtain the oscillation probabilities it is crucial to precisely reconstruct the  neutrino energy E ν . This in turn demands the requirement of well motivated theory and accurate measurements on neutrino-nucleus cross-sections. So far many experiments have measured the total cross-sections for neutrino ν µ N → µ − X and anti-neutrinoν µ N → µ + X scattering off nucleons [17] covering a wide range of energies. The knowledge of cross-section values for energies between 0.5 GeV to 4 GeV, which covers the range for NOνA, T2K and T2HK experiments, is limited (with an error of about 10%). Without considering a specific theoretical model, the errors on cross-sections are in the range of 20-50% [18].
Thus, it is crucial to understand the role of this cross-section uncertainty in determination of CP violation for the above experiments. We do not emphasize on the actual size of these errors or the exact values of the resultant χ 2 , but on the relative effect of these errors on the determination of CP-violation in all these three experiments for various combinations of run periods.
NOνA [19] is an off-axis long baseline neutrino oscillation experiment designed to study ν µ → ν e appearance measurements using Fermilab NuMI muon neutrino beam (ν µ ). Its secondary aim is to precisely measure ν µ disappearance parameters. It uses a high intensity proton beam with a beam power of 0.7 MW with 6 × 10 20 POT/year. Its detector is a 14 kt totally active liquid scintillator detector (TASD) located at Ash River, 810 km from Fermilab. The detector is located slightly off the centerline (14 mrad) to the neutrino beam where one can find a large flux of neutrinos of 2 GeV energy. The oscillation from ν µ → ν e is expected to be maximum at this energy. It is scheduled to run 3 years in ν mode followed by 3 years inν mode. The detector properties of NOνA considered in our simulations are taken from Ref. [20] with the following characteristics :  [21][22][23]. The primary objective of this experiment is the discovery of CP asymmetry.
Since these experiments use ν µ beam and also will run in antineutrino mode their main focus is to study the appearance (ν µ → ν e ) and the disappearance channels (ν µ → ν µ ) along with their antineutrino counterparts. Since the leading term in the appearance channels ν µ → ν e (P µe ) and the corresponding antineutrino modeν µ →ν e (Pμē) is proportional to sin 2 2θ 13 sin 2 θ 23 and with the observation of moderately large value of θ 13 , these experiments are well-suited for the determination of mass hierarchy and the octant of θ 23 . Although, these ongoing NOνA and T2K experiments are not planned to measure δ CP or to explore CP violation in the neutrino sector, we would like to investigate whether it is possible to constrain the δ CP phase using the data from these two experiments. In other words, how much of the δ CP space can be ruled out by these experiments within the next 10 years. In particular, we would like to investigate • whether the combination of T2K (3+2) and NOνA (3+3) provide more quantitative answer on the above posed questions than each one of these experiments.
• how the sensitivity on δ CP , mass hierarchy and θ 23 octant will improve if NOνA runs for 10 years in the (5+5) and (7+3) combination of modes.
• the sensitivities of T2HK experiment for its scheduled run of 3 years in neutrino mode and 7 years in anti-neutrino beam.
• the effect of the uncertainty in ν µ N → µ − X andν µ N → µ + X cross-sections on the sensitivities of all these three experiments individually.
The paper is organized as follows. In section II we briefly describe the physics reach of these experiments. The prospect of octant resolution and mass hierarchy determination along with the effect of 10% error on cross-section uncertainty on their sensitivities are discussed in section III and IV. Section V contains the CP violation discovery potential, the effect of 10% error on cross-section uncertainty on CP violation discovery potential and the correlations between the CP violating phase δ CP and the mixing angles θ 13 and θ 23 . We summarize our results in Section VI.

II. PHYSICS REACH
As discussed before, the determination of the mass hierarchy, octant of the atmospheric mixing angle θ 23 and the search for CP violation in the neutrino sector are the important physics goals of the current and future oscillation experiments. A simple way to achieve the above three goals is to measure the oscillation probabilities P (ν µ → ν e ) and P (ν µ →ν e ).
Here we will try to see if the energy spectrum information will help us in resolving the octant degeneracy and mass hierarchy. We have used GLoBES package [27,28] for the simulation through out this paper. We consider the following true values for oscillation parameters as provided in Table-II, unless mentioned otherwise.  ( 0 ≤ δ CP ≤ 180 • ), P µe is much lower. The situations reverse for the antineutrino probability Pμē. Thus, LHP is the favorable half-plane for NH and UHP is for IH for neutrino mode.

III. OCTANT RESOLUTION AS A FUNCTION OF θ 23
In this section we present the results of our analysis on octant sensitivity of θ 23 for T2K, NOνA and T2HK experiments. We also show the results when the data from all the experiments are combined. Although the octant sensitivity of various long baseline experiments has been discussed extensively by many authors [29][30][31], here we would like to revisit the octant resolution potential of NOνA for its scheduled (3+3) years of run.
Furthermore, we also investigate the situation if it runs for next 10 years what would be the potential for resolving octant degeneracy for (5+5) as well as (7+3) years of running. We also see the synergy between T2K, NOνA and T2HK experiments for their scheduled runs.
The indistinguishability of θ 23 and (π/2 − θ 23 ) is known as octant degeneracy. The P v µe = sin 2 θ 23 sin 2 2θ 13 sin 2 1.27 The leading order term in the ν µ survival probability (P v µµ ) depends on sin 2 2θ 23 and one cannot distinguish between P v µµ (θ 23 ) and P v µµ (π/2 − θ 23 ). This kind of degeneracy that comes from the inherent structure of neutrino oscillation probability is called intrinsic octant degeneracy. Where as in the case of P v µe the degeneracy of the octant with the parameter θ 13 comes into play, since it depends on the parameter combination sin 2 θ 23 sin 2 2θ 13 . The values of θ 23 in opposite octant for different values of θ 13 and δ CP can have the same probabilities, i.e, P v µe (θ 23 , θ 13 , δ CP ) = P v µe (π/2 − θ 23 , θ l 13 , δ l CP ). This also gives rise to octant degeneracy. Before doing the simulation, here we would like to emphasize that the relation between atmospheric parameters (∆m 2 atm ) and θ µµ , measured in MINOS, and the standard oscillation parameters in nature are given as [32,33] sin θ 23 = sin θ µµ cos θ 13 , ∆m 2 31 = ∆m 2 atm + (cos 2 θ 12 − cos δ CP sin θ 13 sin 2θ 12 tan θ 23 )∆m 2 21 .
It is clear from the above relations that the observed value of moderately large θ 13 significantly affects the oscillation parameters. So here we use corrected definitions of these parameters to analyze octant sensitivity. We allocate measured values ∆m 2 atm and θ µµ and calculate oscillation probabilities in terms of ∆m 2 31 and θ 23 . We have chosen the true values of oscillation parameters given in Table-II We simulate the long baseline experiments T2K, NOνA, T2HK using the GLoBES package. For T2K we assume 3 years of running in neutrino mode and 2 years in antineutrino mode. For NOνA, we consider 3 years of neutrino running followed by 3 years of antineutrino running. We consider 3 years of neutrino running followed by 7 years of antineutrino running for T2HK. Furthermore, we also consider the case if NOνA continues the data taking for ten years beyond its scheduled (3+3) years and perform the analysis for two possible combinations (5+5) and (7+3) years of running.  We have studied the effect of cross-section uncertainty in the octant discovery potential of T2K and NOνA experiments by considering an optimistic error of 10% on the individual cross-sections of ν µ and ν e . In Fig. 4 shows a narrow band around the values of sin 2 θ 23 due to uncertainty in cross-section for T2K and NOνA experiments.

IV. MASS HIERARCHY DETERMINATION
Determination of neutrino mass hierarchy is one of the outstanding issues in neutrino oscillation physics. The conventional method to achieve this is by using matter effects in very long baseline neutrino oscillation experiments as the matter effects enhance the separation between oscillation spectra, and therefore the event spectra between the normal and inverted hierarchy. In this section we describe the capabilities of T2K, NOνA and T2HK experiments for the determination of mass hierarchy.
To obtain the χ 2 as a function of true value of δ CP , we have kept true parameters as in Table-II except for sin 2 θ 23 = 0.5 and varying the true value of δ CP in its full range (−π, π).
We have obtained the test values by varying ∆m 2 atm in the IH (NH) range for true NH (IH). We have marginalized over ∆m 2 31 , sin 2 2θ 13 and sin 2 θ 23 in their 3σ ranges and added prior to sin 2 2θ 13 with σ(sin 2 2θ 13 ) = 0.01. The mass hierarchy significance has a δ CP coverage of 75% for T2HK experiment alone and 90% for combined data of T2K, NOνA and T2HK experiments above 3σ.
To study the effect of cross-section uncertainty in the determination of mass hierarchy for the above experiments, we have assumed an optimistic error of 10% on the individual cross-sections of ν µ and ν e for NOνA and T2HK experiments. where as for T2HK experiment (from the bottom panel) it is observed to be ∼ 14% (∼ 10%) for significance above 5σ.

V. CP VIOLATION DISCOVERY POTENTIAL
Accelerator based long baseline neutrino oscillation experiments can address CPV problem through the appearance channels of ν µ → ν e andν µ →ν e . From Eq. (1) we can see that the CP violating effects of δ CP are modified by all the three mixing angles and their combinations thus resulting in an eight fold parameter degeneracy. In order to obtain the significance of CP violation sensitivity we simulated the true event spectrum by keeping the true values of oscillation parameters as in Table-II except for sin 2 θ 23 = 0.5 and varying the true value of δ CP in the range (−π, π) and compared those with test event spectrum for δ CP =0 or π and thus obtained the minimum χ 2 . We have considered the sign degeneracy of ∆m 2 31 by marginalizing over it, in both NH and IH 3σ ranges, sin 2 2θ 13 and sin 2 θ 23 in their 3σ and added prior to sin 2 2θ 13 . In Fig. 7, we plot the sensitivity to rule out the CP conserving scenarios, as a function of true δ CP assuming NH as the true hierarchy. From the figure one can notice that T2K by itself has no CP violation sensitivity at 2σ C.L.. For NOνA with (3+3) years of running there will be CP violation sensitivity above 1.5σ level for about one-third of the CP violating phase δ CP space. Furthermore, the synergistic combination of NOνA and T2K leads to much better CP violation sensitivity compared to the individual capabilities. Even the combination of NOνA (3+3) and T2K (3+2) has comparable sensitivity as for 10 years running of NOνA.
Owing to the fact that main goal of T2HK experiment is to determine CP violation, we can observe that it has a significance of above 5σ C.L for a fraction of two-fifth values of the CP violating phase δ CP space. This in turn boosts up the sensitivity when its data is added to NOνA (3+3) yrs and T2K (3+2) yrs. In the lower panel one can observe that the sensitivity of NOνA increases slightly for 10 years of run time, with (5ν + 5ν) combination has better sensitivity than that of (7ν + 3ν) combination. The drop in the half planes of The top most panel of Fig. 8 shows a very thin band induced, when compared to that in the next two panels, due to a 10% error on the individual cross-sections of ν µ and ν e for (3ν+2ν) years run T2K experiment. Clearly, here the sensitivity to CP-violation is not effected much by this cross-section uncertainty. For NOνA experiment the induced error in the values of δ CP for which significance is above 1.5 σ, when the true hierarchy is NH (IH), is ∼ 12% (∼ 10%), where as for T2HK experiment (from the last panel) it is observed to be ∼ 7% (∼ 5%) for significance above 5 σ.
A. Correlation between δ CP and θ 13 The knowledge of reactor mixing angle θ 13 plays a crucial role in the discovery potential of δ CP . The recent discovery of large value of θ 13 has established the need to study and understand the dependency between δ CP and θ 13 . In this subsection, we discuss the correlation between the oscillation parameters θ 13 and δ CP . In obtaining the confidence region, we have kept true values as in Table-II  the test value of sin 2 2θ 13 and δ CP in their 3σ ranges. We have marginalized over sin 2 θ 23 in the LO(HO) range for true value of LO(HO) and ∆m 2 31 in NH (IH) for true NH (IH). In this analysis we have kept both true and test hierarchy as normal hierarchy. We also added prior to sin 2 2θ 13 . is to provide answers to some of these unsolved questions.
In this paper we have investigated the prospects of the determination of mass hierarchy, the octant of θ 23 and the observation of CP violation in the neutrino sector due to δ CP with the currently running accelerator based neutrino experiments NOνA and T2K and the forthcoming T2HK experiment. As the reactor mixing angle θ 13 is now known to be significantly large the oscillation probability P (ν µ → ν e ) and its corresponding antineutrino counterpart are sensitive for the determination of mass hierarchy and θ 23 octant. We found that T2K experiment with (3ν +2ν) years of running can resolve the octant degeneracy with nearly 2σ C.L. if the true value of θ 23 to be around sin 2 θ 23 = 0.41 (LO) or sin 2 θ 23 = 0.59 (HO). The sensitivity increases to nearly 3σ with (3ν +3ν) years running of NOνA. However, if we combine the data from these two experiments the sensitivity increases significantly than the sensitivities of individual experiments. Furthermore, if we assume that NOνA continues data taking for 10 years then octant degeneracy can be resolved with NOνA experiment alone with more than 3σ significance. For the determination of mass hierarchy also it is possible to rule out nearly one-third of the δ CP space at 3σ C.L. if we use the synergy between NOνA and T2K experiments. In this case the sensitivity increases significantly for ten years of running of NOνA with (5 + 5) combination is found to be more suitable than the combination of (7+3) years.
Measuring CP violation in the lepton sector is another important challenging problems today. We have also performed a systematic study of the CP sensitivity of the current long-baseline experiments T2K and NOνA. Although these experiments are not planned to study leptonic CP violation, we analyze the synergies between these set-ups which may aid in CP violation discovery by constraining the value of δ CP . Although dedicated long baseline experiments like LBNE, LBNO are planned to study CP violation in neutrino sector, we may have the the first hand information on δ CP from these experiments much before those dedicated facilities are operational. We found that T2K by itself has marginal CP violation sensitivity at 1σ C.L.. For NOνA with (3+3) years of running there will be CP violation sensitivity above 1.5 σ level for about one-third of the CP violating phase δ CP space. The sensitivity increases slightly for 10 years of run time, with (5ν + 5ν) combination has better sensitivity than that of (7ν + 3ν) combination. The data from T2HK experiment will improve the CPV sensitivity significantly. We have also obtained the Confidence regions in the δ CP − θ 13 (θ 23 ) plane for both T2K and NOνA experiments.