Revisiting the sensitivity studies for leptonic CP violation and mass hierarchy with T2K, NOvA and LBNE experiments

Precision measurement of the neutrino mixing parameters and the determination of mass hierarchy are the primary goals of the present and upcoming neutrino experiments. In this work, we study the sensitivity of T2K,NO$\nu$A and LBNE experiments to discover leptonic CP violation and the determination of neutrino mass hierarchy. We obtain the correlation between the CP violating phase $\delta_{CP}$ and the mixing angles $\theta_{13}$, $\theta_{23}$ and the sensitivity to determine the octant of atmospheric mixing angle $\theta_{23}$. The entire analysis is done for a total 10 years (5$\nu$+ 5$\bar \nu$) of running of T2K, NO$\nu$A and LBNE experiments. Furthermore, we also consider the impact of cross section uncertainties on the CP violation sensitivity of LBNE experiment.


I. INTRODUCTION
The experimental endeavour in the past decades has firmly established the phenomenon of neutrino oscillations, i.e., the composition of neutrino flavors change as they propagate.
In the three neutrino framework, the three flavour eigenstates (ν e , ν µ , ν τ ) mix via the unitary Pontecorvo-Maki-Nakagawa-Sakata mixing matrix U P M N S [1,2], analogue of the CKM matrix V CKM that governs the mixing in the quark sector. This PMNS matrix can be parameterized in terms of three mixing angles (θ 12 , θ 13 and θ 23 ), which have all been measured experimentally and a CP-violating phase δ CP which is unknown. The probability for flavor oscillation also depends on the differences in the squared masses of the neutrinos, i.e., ∆m 2 21 and ∆m 2 31 , where ∆m 2 ij = m 2 i − m 2 j . The neutrino oscillation data accumulated over many years allow us to determine the solar and atmospheric neutrino oscillation parameters with very high precision. The mixing angles θ 12 and θ 23 as well as the mass square differences have been well constrained by various neutrino experiments. Recently, the reactor mixing angle θ 13 has been measured precisely [3][4][5][6] with a moderately large value. This provides a significant achievement in establishing the picture of three-flavor neutrino oscillations. The global analysis of the recent results of various neutrino oscillation experiments has been performed by several groups [7][8][9][10]. We have considered best-fit values and the 3σ ranges of the oscillation parameters from Ref.
[10] through out in our simulations.
There are however, still many open questions to be answered. These include : i.) The value of the CP violating phase δ CP is not yet constrained by any experiment. ii.) We still do not know the exact nature of neutrino mass hierarchy, i.e., whether the neutrino mass ordering is normal or inverted in nature. iii.) The possibility of observing CP violation in the neutrino sector due to the presence of the Dirac type CP violating phase in the neutrino mixing matrix. iv.) Another interesting and crucial development in recent times is the indication of non-maximal atmospheric mixing angle by the MINOS [11] and T2K [12] experiments. The global analyses of all the available neutrino oscillation data [7][8][9][10] also prefer the deviation of θ 23 value from maximal mixing i.e., sin 2 θ 23 = 0.5. Thus, for non-maximal value of θ 23 , one can have two possible solutions, one with θ 23 < 45 • for which (sin 2 θ 23 − 0.5) is negative and the other with θ 23 > 45 • for which (sin 2 θ 23 − 0.5) is positive. The former case is known as lower octant (LO) whereas the later one is known as higher octant (HO) solution. This corresponds to the problem of octant degeneracy of θ 23 . In this paper we would like study the sensitivity of the current and future long baseline experiments i.e., T2K, NOνA and LBNE in addressing some of these issues. Although some of these aspects have been studied in detail recently by various authors [13][14][15][16][17][18][19][20][21][22], in this paper we have attempted to do a complete analysis of all these issues in the context of the current generation and upcoming long baseline super-beam experiments. Another important difference is that in most of the previous analyses the LBNE flux files used are either atmospheric or NOνA (which is an off-axis experiment) flux files whereas we have considered the on-axis NuMI beam flux files for LBNE from [23]. In Ref. [18], the authors have studied the sensitivities to mass hierarchy, octant of θ 23 and CP violation for LBNE.
They have also included the data from T2K (5+0), NOνA (3+3) and also from atmospheric neutrinos. The difference between their and our work are: (i) we have not taken into account the effect atmospheric neutrinos (ii) we have considered 10 years of data for NoνA and T2K in the combinations (5+5) assuming that by the time LBNE will start data taking both NOνA and T2K would have completed 10 years of run (iii) we have also studied various correlations between δ CP and θ 23 /θ 13 , which will help us to constrain the value of δ CP .
Furthermore, as discussed in Ref. [24], the uncertainties in cross-sections play a crucial role in the determination of CP violation sensitivities of various long baseline super-beam experiments. Without considering any specific theoretical model, the errors on cross-sections are expected to be in the range of (20-50)%. In this paper, we have studied the impact of these cross-section uncertainties on the CP violation sensitivity of LBNE experiment.
The paper is organized as follows. In Section 2, we discuss the δ CP dependence of neutrino oscillation probabilities and also show how it is correlated with the octant of θ 23 and netrino mass ordering. The experimental details of the long-baseline experiments (NOνA, T2K and LBNE) are briefly discussed in Section 3. The CP violation sensitivity and the determination of mass hierarchy are outlined in sections 4 and 5. Section 6 contains the results on octant sensitivity determination of these experiments. The correlations between the CP violating phase δ CP and the mixing angles θ 12 and θ 23 are presented in Section 7. Section 8 contains the summary and conclusion.

II. EFFECT OF MASS HIERARCHY AND θ 23 OCTANT ON δ CP SENSITIVITY
The three-flavor neutrino oscillation effects can be systematically demonstrated by considering oscillation channels ν µ → ν e andν µ →ν e . The detailed study of these channels at the long-baseline experiments is capable of addressing almost all the four major issues discussed in the previous section. In particular, appearance channel, i.e., ν µ → ν e is very sensitive to explore the CP violation effect in neutrino oscillation experiments which can be understood as follows. In matter of constant density, the appearance probability P µe , which depends on δ CP in its sub-leading term can be expressed as [25][26][27] P µe ≃ sin 2 θ 23 sin 2 2θ 13 (1 −Â) 2 + α sin 2θ 13 sin 2θ 12 sin 2θ 23 cos θ 13 cos(∆ + δ CP ) sin(Â∆) where 2 31 . All six parameters governing neutrino oscillations (θ 12 , θ 23 , θ 13 , ∆m 2 21 , ∆m 2 31 and δ CP ) appear in this equation. It should be noted that the parameters α, ∆ andÂ are sensitive to the neutrino mass ordering i.e., to the sign of ∆m 2 31 . Furthermore, the sign ofÂ changes with the sign of ∆m 2 31 , which implies that the matter effect can be used to determine the mass hierarchy. AlsoÂ changes sign while going from neutrino to antineutrino mode, which indicates that it can mimic CP violation and hence complicates the extraction of δ CP by comparing the data from neutrino and antineutrino modes. Thus, for large θ 13 from the dominant first term of Eq. (1), one can determine sin 2 θ 23 or in other words the octant of θ 23 . Secondly, as this term contains large matter effect, the nature of mass ordering can also be extracted from it. The second sub-dominant term is sensitive for the determination of CP violation as it contains both sin δ CP and cos δ CP terms. As discussed in detail in Ref. [15], the following points can be inferred from Eq. (1).
• The CP violation phase δ CP appears in combination with the atmospheric masssquared difference as cos(∆ + δ CP ) and hence, it suffers from the hierarchy-δ CP degeneracy. This in turn limits the CP violation sensitivity which can be clearly understood from baseline of 1300 km. In our analysis, we have used the relation between the atmospheric parameters (∆m 2 atm ) and θ µµ measured and standard oscillation parameter in nature as [28][29][30] sin θ 23 = sin θ µµ cos θ 13 (2) ∆m 2 31 = ∆m 2 atm + ∆m 2 21 (cos 2 θ 12 − cos δ CP sin θ 13 sin 2θ 12 tan θ 23 ) where ∆m 2 atm is taken to be positive (negative) for Normal Hierarchy (Inverted Hierarchy). We consider the true curves of δ CP = ±90 • and true hierarchy to be normal for both the panels. The test values for δ CP = 0 and 180 • and test NH is shown in left panel and the same for test hierarchy as inverted is shown on the right panel. Thus, the left panel represents the separation between the CP conserving test (δ test CP = 0, π) and maximally CP violation true (δ true CP = −π/2 or π/2) when the hierarchy is known while the right panel represents the same when the hierarchy is unknown. Hence, one can see that the separation between the true cases i.e., (NH, δ CP = ±π/2) from the corresponding test CP conserving cases (NH/IH, δ CP =0 or π) is hierarchy dependent, which will effectively introduce hindrance in the CP sensitivity measurements.
• The probability for neutrinos P µe is higher for NH than for IH due to matter effects as seen from the first term in Eq. (1).

III. EXPERIMENTAL SPECIFICATIONS FOR THE SIMULATION STUDIES
To determine the sensitivity of various observables in the currently running and upcoming long-baseline experiments, the simulation is done using the GLoBES package [31,32]. First we briefly describe the procedure that we have adopted for obtaining the numerical results.
We calculate ∆χ 2 using the default definition in GLoBES. We then minimize the ∆χ 2 to compute the sensitivities on various parameters. The following are the experimental specifications for T2K, NOνA and LBNE setups that have been used in our analysis.
T2K: In the T2K experiment, a ν µ beam from J-PARK is directed towards Super-Kamiokande detector which is 22.5 kt (Water Cerenkov detector), 295 km away. It uses a 0.77 MW beam planned to run effectively for 5(ν) + 0(ν) or 3(ν) + 2(ν) years. The initial plan of T2K experiment was to run for five years with 10 21 proton on target per year. In this paper we consider the option of T2K running for 5(ν) + 5(ν) years and incorporate those results with NOνA and LBNE 10 years of run. The details of T2K experiment can be found from [34]. We have considered input files for T2K from GLoBES package [33][34][35] NOνA: NOνA is a 14 kt totally active scintillator detector (TASD) located at Ash River, a distance of 810 km from Fermilab [36,37]. The beam power is assumed to be 0.7 MW NuMI beam with 6.0 × 10 20 pot/year. This experiment is scheduled to have three years in neutrino mode first and after that three years run in anti-neutrino mode. However, in our analysis we consider the running for 5(ν) + 5(ν) years by 2024.
The following are the signal and background efficiencies considered in our simulation: Signal efficiency : 45% for ν e andν e signal; 100% ν µ CC andν µ CC.
We consider 5% uncertainty on signal normalization and 10% on background normalization.
The migration matrices for NC background smearing are taken from [38].
LBNE: For LBNE, we consider 35 kt LAr detector at 1300 km baseline length [39]. The neutrino beam (0.5 -8 GeV) is obtained from a proton beam of 700 KW beam power and 120 GeV beam energy resulting in 6 × 10 20 protons on target (POT) per year. We consider 5 years of data taken by detector in ν beam mode and 5 years inν beam mode. The GLoBES files and the detector parameter assumptions are taken from [40]. We consider 5% uncertainty on signal normalization and 10% on background normalization. Furthermore, we have not considered the effect of near detector (ND) in our analysis. As discussed in Ref. [18], the presence of ND will reduce the systematic uncertainties of ν e signal (background) from 5% (10%) to 1% (1%) and this in turn will enhance the various sensitivities a bit more.
Our primary objective is to perform the sensitivity studies with LBNE setup. However, by the time LBNE will start data taking, which is expected to be around 2022, both T2K and NOνA will have nearly 10 years of run. Therefore, we would incorporate the T2K and NOνA data to the LBNE data set to perform the simulation. For all the three experiments we consider two cases of runs in neutrino and anti-neutrino modes: i) 5 yrs in neutrino mode and 5 yrs in anti-neutrino mode.

IV. CP VIOLATION SENSITIVITY WITH T2K, NOνA AND LBNE
The determination of the CP violating phase δ CP is one of the most challenging problems in neutrino physics today. Since δ CP is associated with the mixing angle θ 13 in the PMNS matrix, the recent measurement of a non-zero and moderately large value of this angle by reactor and accelerator experiments is expected to be conducive for the measurement of δ CP .
Since θ 13 is found to be moderately large it is possible for NOνA and T2K to provide some hint on δ CP . In this section, we discuss the detection of CP violation, i.e., the ability of an experiment to exclude the cases δ CP = 0 or 180 • The sensitivity of the experiment to observe CP violation is evaluated at a given value of δ CP is done by minimizing the χ 2 at the fixed test values of 0 and π. Thus, we determine two quantities: and then take The significance of CP violation is obtained by using σ = ∆χ 2 . Furthermore, we have marginalised over ∆m 2 31 , sin 2 θ 23 , sin 2 2θ 13 over their 3σ ranges. We also added prior for sin 2 2θ 13 with σ(sin 2 2θ 13 ) = 0.01. We present our results as a function of δ CP in Fig.- uncertainties. Furthermore, it should also be noted that the region close to maximal CP violation, (i.e., δ CP = π/2) affected much due to these uncertainties. Also, as generally anticipated, there is an enhancement in these uncertainties with the increase in detector volume. Long-baseline experiments such as NOνA, T2K and LBNE primarily use the ν µ → ν e and the corresponding anti-neutrino oscillation channels to determine the neutrino mass hierarchy (MH). Using the approximate perturbative formula for the probability P µe , it can be seen that there is a hierarchy-δ CP degeneracy as discussed in section 2. As a result, the hierarchy sensitivity of these experiments is a strong function of the value of the CP violating phase δ CP .
To obtain the mass hierarchy significance we consider two cases. In the former we consider true hierarchy to be normal hierarchy (NH) and obtain the test values of ∆χ 2 by assuming inverted hierarchy as test hierarchy. In the later we consider true hierarchy to be inverted hierarchy (IH) and assume normal hierarchy to be the test hierarchy while obtaining the ∆χ 2 value. We marginalized the test values of ∆m 2 31 , sin 2 θ 23 , sin 2 2θ 13 over their 3σ ranges in both the cases. We also added prior for sin 2 2θ 13 with σ(sin 2 2θ 13 ) = 0.01. In Fig. 5 The leading order term in the above equation has its entire dependency on sin 2 2θ 23 giving rise to intrinsic octant degeneracy.
We first look into the bi-probability plots for LBNE experiment with (5+5) years of run, to estimate its capabilities in determining mass hierarchy and resolving octant degeneracy. The For the analysis of octant determination of θ 23 , we have used GLoBES to evaluate ∆χ 2 .
In the right panel of Fig. 6, we illustrate the ability of NOνA+T2K+LBNE to determine the octant as a function of the true value of θ 23 . We see that with LBNE and NOνA+T2K+LBNE, the octant can be determined at >5σ C.L. when sin 2 θ 23 = 0.41. For values closer to 45 • (sin 2 θ 23 = 0.5), the combined data from T2K+NOνA+LBNE is sensitive enough (more than 5σ C.L.) to determine the mass hierarchy.
A. Allowed regions in test δ CP and test sin 2 θ 23 plane Next, we would like to study the correlation between δ CP and sin 2 θ 23 for different combi-  Here also we have marginalized over θ 23 and ∆m 2 31 . We have obtained 1σ, 2σ and 3σ contours by considering three true values for δ CP = 0, −π/2, +π/2.
We have set 10% error on each of the solar parameters and a 5% error for the matter density and assumed the hierarchy to be normal. The result is presented on the left panel of Fig.   8, where the blue/green/red curves correspond to 1/2/3σ measurement contours for a total (5+5) yrs running of T2K+NOνA+LBNE.
The analogous plot between δ CP and θ 23 has been obtained following similar procedure running of (5+5) yrs for T2K+NOνA+ LBNE (right panel) in ν +ν mode and the corresponding result is shown in the right panel of Fig. 7. The allowed region is very tightly constrained which indicates that with 10 years of LBNE data taking it is possible to obtain the value of δ CP . The overlap in the 3σ contours in both the plots can be accounted by the fact that we are considering three different true values for δ CP = 0, −π/2, +π/2.

VIII. SUMMARY AND CONCLUSION
In this paper we have explored the possibility of determining the mass hierarchy, octant of the atmospheric mixing angle θ 23 and the CP violation discovery potential in the current generation and upcoming long baseline experiments T2K, NOνA and LBNE and our findings are summarized below.
• For long-baseline experiments, it is well known that the measurement of the mass hierarchy is easier than a measurement of δ CP because matter effects enhance the separation between the oscillation spectra, and hence the event rates between normal and inverted hierarchies. The determination of mass hierarchy is defined as the ability to exclude any degenerate solution for the wrong (fit) hierarchy at a given confidence level. From our analysis, we find that if we combine the results from all the three experiments for (5ν + 5ν) years of run we can determine the mass hierarchy of neutrinos above 5σ.
• The octant sensitivity of θ 23 also increases noticeably if we combine the results of the three experiments T2K, NOνA and LBNE. Even if the true value of θ 23 lies around 40 • , it is possible to disentangle between the different θ 23 octants at 5σ CL.
• The CP violation discovery potential in the long baseline experiments is also quite promising. A discovery of CP violation, if it exists, basically means being able to exclude the CP-conserving values i.e., δ CP = 0 • or 180 • at a given confidence level. From our analysis, we found that it is possible to measure the CP violation phase above 3σ C.L. for about 50% of the true δ CP range if we combine the data from all three experiments. Furthermore, it should also be noted that the CP violation measurement becomes very difficult for the δ CP values which are closer to 0 • or 180 • . Therefore, whilst it is possible to discover the mass hierarchy for all possible values of δ CP , the same is not true for CP violation.
• The cross-section uncertainties play a crucial role in determining the CP violation sensitivity. These uncertainties affect significantly the region of maximal-CP violation.
• From the correlation plots between δ CP and sin 2 θ 23 as well as from δ CP and θ 13 /θ 23 (Figs. 6 and 7), one can see that δ CP is severely constrained implying a definitive measurement on δ CP could be possible with 10 years of LBNE data taking.
In conclusion, we find that combining the data of (5ν + 5ν) years of running T2K, NOνA and LBNE will help us to resolve most of the ambiguities associated with the neutrino sector.