Physics potentials with a combined sensitivity of T2K-II, NO$\nu$A extension and JUNO

Leptonic \textit{CP} violation search, neutrino mass hierarchy determination, and the precision measurement of oscillation parameters for a unitary test of the leptonic mixing matrix are among the major targets of the ongoing and future neutrino oscillation experiments. The work explores the physics reach for these targets by around 2027, when the third generation of the neutrino experiments starts operation, with a combined sensitivity of three experiments: T2K-II, NO$\nu$A extension, and JUNO. It is shown that a joint analysis of these three experiments can conclusively determine the neutrino mass hierarchy. Also, at certain values of \emph{true} \dcp, it provides closely around a $5\sigma$ confidence level (C.L.) to exclude \textit{CP}-conserving values and more than a $50\%$ fractional region of \emph{true} $\delta_{\text{CP}}$ values can be explored with a statistic significance of at least a $3\sigma$ C.L. Besides, the joint analysis can provide unprecedented precision measurements of the atmospheric neutrino oscillation parameters and a great offer to solve the $\theta_{23}$ octant degeneracy in the case of nonmaximal mixing.

It is well-established from the contribution of many neutrino experiments [6], using both the natural neutrino sources (solar and atmospheric neutrinos) and the man-made neutrino sources (reactor and accelerator neutrinos) that the two leptonic mixing angles, θ 12 and θ 23 , are large, θ 13 is relatively small but nonzero, and the mass-squared splitting |∆m 2 31 | is about 30 times larger than ∆m 2 21 . The global analysis of neutrino oscillation data is available, e.g., in Ref. [7,8], and is briefly summarized in Table I. Although a few percent precision measurements of three mixing angles and two mass-squared splittings have been achieved, a complete picture of neutrino oscillation has not been fulfilled yet. There are at least three unknowns, which the worldwide neutrino programs plan to address in the next decades.   [7,9] .
The first unknown is CP violation (CPV) in the neutrino oscillations. Despite a recent hint of maximal CPV from the δ CP measurement by the T2K experiment [10], whether CP is violated or not requires higher statistics to be established. The second unknown is the neutrino mass hierarchy (MH), which refers to the order of the three mass eigenvalues of neutrino mass eigenstates. Whether the MH is normal (m 1 < m 2 < m 3 ) or inverted (m 3 < m 1 < m 2 ) is still questionable. While the recent measurements from individual experiments [11][12][13] mildly favor the former, the efforts [8,14] for fitting jointly multiple neutrino data samples show that the preference to the normal MH becomes less significant.
Thus, more neutrino data is essential to shedding light on the neutrino MH. The third unknown on the list is about the mixing angle θ 23 . Its measured value is close to 45 • , which means the mass eigenstate ν 3 is comprised of an approximately equal amount of ν µ and ν τ , indicating some unknown symmetry between the second and third lepton generations.
In this paper, we show the prospect of reaching these unknowns in light of two acceleratorbased long-baseline neutrino experiments, T2K-II and NOνA extended program, and a reactor-based medium-baseline neutrino experiment, JUNO. The paper is organized as follows. Section II details the experimental specifications of these three experiments and elaborates on the simulation methodology. In Sec. III, we present our results on the MH determination, the CPV sensitivity, the resolution of the θ 23 octant, and the precise constraints of the oscillation parameters. We give the conclusion of the work in Sec. IV.

II. EXPERIMENTAL SPECIFICATIONS AND SIMULATION DETAILS
A. Experimental specifications of T2K-II, NOνA-II and JUNO T2K-II: The ongoing Tokai-To-Kamioka (T2K) [15] is the second generation of acceleratorbased long-baseline (A-LBL) neutrino oscillation experiments located in Japan, and T2K-II [16] is a proposal to extend the T2K run until 2026 before Hyper-Kamiokande [17] starts operation. The T2K far detector, SK, is located 295 km away from the neutrino production source, and receives the neutrino beam at an average angle of 2.5 o off-axis to achieve a narrow-band neutrino beam with a peak energy of 0.6 GeV. Being a gigantic Cherenkov detector with 50 ktons of pure water and approximately 13,000 photomultiplier tubes deployed, SK provides an excellent performance of reconstructing the neutrino energy and the neutrino flavor classification. This capability allows T2K(-II) to measure simultaneously the disappearance of muon (anti-)neutrinos and the appearance of electron (anti-)neutrinos from the flux of almost pure muon (anti-)neutrinos. While the data samples of the ν µ (ν µ ) disappearance provide a precise measurement of the atmospheric neutrino parameters, sin 2 2θ 23 and ∆m 2 31 , the ν e (ν e ) appearance rates are driven by sin 2 2θ 13 and are sensitive to δ CP and the MH. The sensitivity of the A-LBL experiments such as T2K and NOνA to δ CP and the MH can be understood via the following expression of the so-called CP asymmetry [18], presenting a relative difference between P (νµ→νe) and P (νµ→νe) near the oscillation maximum, and corresponding to ∼ − π sin 2θ 12 tan θ 23 sin 2θ 13 where the +(−) sign is taken for the normal (inverted ) MH, respectively. With the values listed in Table I [19] and presented an indication of CPV in the neutrino oscillation [10]. T2K originally planned to take data equivalent to 7.8 × 10 21 protons-on-target (POT) exposure. At the Neutrino 2020 conference, T2K [20] reported a collected data sample from 3.6 × 10 21 POT exposure. In Ref. [16], T2K proposes to extend the run until 2026 to collect 20 × 10 21 POT, allowing T2K to explore CPV with a confidence level (C.L.) of 3σ or higher if δ CP is close to −π/2 and to make precision measurements of θ 23 and |∆m 2 31 |. NOνA extension or NOνA-II: Ongoing NuMI Off-axis ν e Appearance (NOνA ) [21] is also the second generation of A-LBL neutrino experiments placed in the United States with a baseline of 810 km between the production source and the far detector. Such a long baseline allows NOνA to explore the MH with high sensitivity via the matter effect [22] on the (anti-)neutrino interactions. From Eq. (1), it can be estimated that the matter effect in NOνA is ∼ 28.9%, which is slightly higher than the CP violation effect. However, these two effects, along with the ambiguity of the θ 23 octant, are largely entangled. In other words, NOνA sensitivity on the neutrino MH depends on the value of δ CP . NOνA's recent data [23] does not provide as much preference to the neutrino mass hierarchy as T2K [20] does since NOνA data shows no indication of the CP violation. Similar to T2K, NOνA adopts the off-axis technique such that the far detector is placed at an angle of 14 mrad to the averaged direction of the neutrino beam. NOνA uses a near detector, located 1 km away from the production target, to characterize the unoscillated neutrino flux. The NOνA far detector is filled with liquid scintillator contained in PVC cells, totally weighted at 14 ktons with 63% active materials. NOνA takes advantage of machine learning for particle classification to enhance the event selection performance. In 2018 [24], NOνA provided more than a 4σ C.L. evidence of electron anti-neutrino appearance from a beam of muon anti-neutrinos. At the Neutrino 2020 conference, NOνA [23] reported a collected data sample from 2.6 × 10 21 POT exposure. In [25], NOνA gives the prospect of extending the run through 2024, hereby called NOνA-II, in order to get a 3σ C.L. or higher sensitivity to the MH in case the MH is normal and δ CP is close to −π/2, and more than a 2σ C.L. sensitivity to CPV.
JUNO: Jiangmen Underground Neutrino Observatory (JUNO) [26] is a reactor-based medium-baseline neutrino experiment located in China. JUNO houses a 20 kton large liquid scintillator detector for detecting the electron anti-neutrinos (ν e ) from the Yangjiang (YJ) and Taishan (TS) nuclear power plants (NPPs) with an average baseline of 52.5 km. Each of the six cores at the YJ nuclear plant will produce a power of 2.9 GW and the four cores at the TS NPP will generate 4.6 GW each. They are combined to give 36 GW of thermal power. JUNO primarily aims to determine the MH by measuring the surviving ν e spectrum, which uniquely displays the oscillation patterns driven by both solar and atmospheric neutrino mass-squared splittings [27]. This feature can be understood via thē ν e disappearance probability in the vacuum, which is expressed as follow: 4Eν . An averaged 52 km baseline of the JUNO experiment obtains the maximum oscillation corresponding to Φ 21 = π/2 around 3 MeV, and relatively enhances the oscillation patterns driven by the Φ 31 and Φ 32 terms. The relatively small difference between ∆m 2 31 and ∆m 2 32 make oscillation patterns in the normal and inverted MH scenarios distinguishable. To realize practically the capability of mass hierarchy resolution, JUNO must achieve a very good neutrino energy resolution, which has been demonstrated recently in Ref. [28], and collect a huge amount of data. With six years of operation, JUNO can reach a 3σ C.L. or higher sensitivity to the MH and achieve better than 1% precision on the solar neutrino parameters and the atmospheric neutrino mass-squared splitting |∆m 2 31 |. Although T2K and NOνA experiments have already collected 18% and 36% of the total proton exposure assumed in this study, respectively, we do not directly use their experimental data to estimate their final reaches. The main reason is that measurements of the CP violation, the mass hierarchy, and the mixing angle θ 23 are so far statistically limited except for a specific set of oscillation parameters. We thus carry out the study with the assumption that all values of δ CP and the two scenarios of the neutrino mass hierarchy are still possible, and the mixing angle θ 23 is explored in a range close to 45 • .
Reaching the three above-mentioned unknowns depends on the ability to resolve the parameter degeneracies among δ CP , the sign of ∆m 2 31 , θ 13 , and θ 23 [29]. Combining the data samples of the A-LBL experiments (T2K-II and NOνA-II) and JUNO would enhance the CPV search and the MH determination since the JUNO sensitivity to the MH has no ambiguity to δ CP . To further enhance the CPV search, one can break the δ CP -θ 13 degeneracy by using the constraint of θ 13 from reactor-based short-baseline (R-SBL) neutrino experiments   such as Daya Bay [30], Double Chooz [31], and RENO [32]. This combination also helps to solve the θ 23 octant in the case of nonmaximal mixing.

B. Simulation details
The General Long-Baseline Experiment Simulator (GLoBES) [35,36] is used for simulating the experiments and calculating their statistical significance. In this simulator, a number of expected events of ν j from ν i oscillation in the n-th energy bin of the detector in a given experiment is calculated as For each T2K-II and NOνA-II experiment, four simulated data samples per each experiment are used: ν µ (ν µ ) disappearance and ν e (ν e ) appearance in both ν-mode andν-mode. The experimental specifications of these two experiments are shown in Table II. In T2K(-II), neutrino events are dominated by the charged current quasielastic (CCQE) interactions.
Thus, for appearance (disappearance) in ν-mode andν-mode, the signal events are obtained from the ν µ → ν e (ν µ → ν µ ) CCQE events and theν µ →ν e (ν µ →ν µ ) CCQE events, respectively. In the appearance samples, the intrinsic ν e /ν e contamination from the beam, the wrong-sign components, i.e., ν µ → ν e (ν µ → ν e ) in ν-mode (ν-mode), respectively, and the neutral current (NC) events constitute the backgrounds. In the disappearance samples, the backgrounds come from ν µ , ν µ charged current (CC) interaction excluding CCQE, hereby called CC non-QE, and NC interactions. We use the updated T2K flux released along with Ref. [37]. In simulation, the cross section for low-and high-energy regions are taken from Refs. [38,39], respectively. In our T2K-II setup, an exposure of 20 × 10 21 POT equally divided among the ν-mode and theν-mode is considered, along with a 50% effectively statistic improvement as presented in Ref. [16]. The signal and background efficiencies and the spectral information for T2K-II are obtained by scaling the T2K analysis reported in Ref. [33] to the same exposure as the T2K-II proposal. In Fig. 1, the T2K-II expected spectra of the signal and background events as a function of reconstructed neutrino energy obtained with GLoBES are compared to those of the Monte-Carlo simulation scaled from Ref. [16]. A 3% error is assigned for both the energy resolution and the normalization uncertainties of the signal and background in all simulated samples.
For NOνA-II, we consider a total exposure of 72 × 10 20 POT equally divided among ν-mode andν-mode [25]. We predict the neutrino fluxes at the NOνA far detector by using the flux information from the near detector, given in Ref. [40], and normalizing it with the square of their baseline ratio. A 5% systematic error for all samples and 8-10% sampledependent energy resolutions are assigned. Significant background events in the appearance samples stem from the intrinsic beam ν e /ν e , NC components, and cosmic muons. In the appearance sample of theν-mode, wrong-sign events from ν e appearance events are included as the backgrounds in the simulation. We use the reconstructed energy spectra of the NOνA far detector simulated sample, reported in Ref. [41], to tune our GLoBES simulation. The low-and high-particle identification score samples are used, but the peripheral sample is not since the reconstructed energy information is not available. In the disappearance samples of both ν-mode andν-mode, events from both CC ν µ andν µ interactions are considered to be signal events, which is tuned to match with the NOνA far detector simulated signal given an identical exposure. Background from the NC ν µ (ν µ ) interactions is taken into consideration and weighted such that the rate at a predefined exposure is matched to a combination of the reported NC and cosmic muon backgrounds in Ref. [41]. Figure 2 shows the simulated NOνA-II event spectra as a function of reconstructed neutrino energy for ν e appearance and ν µ disappearance channels in both ν-mode andν-mode, where normal MH is assumed, δ CP is fixed at 0 • , and other parameters are given in Table I.   Tables III and IV detail Table I are  In JUNO, the electron anti-neutrinoν e flux, which is produced mainly from four radioactive isotopes ( 235 U, 238 U, 239 Pu, and 241 Pu [42]), is simulated with an assumed detection efficiency of 73%. The backgrounds, which have a marginal effect on the MH sensitivity,    Table V and the event rate distribution as a function of the neutrino energy is shown in Fig. 3. For systematic errors, we commonly use 1% for the errors associated with the uncertainties of the normalization of theν e flux produced from the reactor core, the normalization of the detector mass, the spectral normalization of the signal, the detector response to the energy scale, the isotopic abundance, and the bin-to-bin reconstructed energy shape.
Besides T2K-II, NOνA-II, and JUNO, we implement a R-SBL neutrino experiment to constrain sin 2 θ 13 at 3% uncertainty, which is reachable as prospected in Ref. [44]. This constraint is important to break the parameter degeneracy between δ CP -θ 13 , which is inherent from the measurement with the electron (anti-)neutrino appearance samples in the A-LBL experiments.
To calculate the sensitivity, a joint χ 2 is formulated by summing over all individual experiments under consideration without taking any systematic correlation among experiments. For T2K-II and NOνA-II, we use a built-in χ 2 function from GLoBES for  [26] taking the signal and background normalization systematics with the spectral distortion into account. For JUNO, a Gaussian formula for χ 2 is implemented thanks to a high statistics sample in JUNO. For a given true value of the oscillation parameters, Θ truth = (θ 12 , θ 13 , θ 23 , δ CP ; ∆m 2 21 , ∆m 2 31 ) truth , at a test set of oscillation parameters, Θ test , and systematic variations, s syst. , a measure χ 2 ( Θ truth | Θ test , s syst. ) is calculated. It is then minimized over the nuisance parameters (both systematic parameters and marginalized oscillation parameters) to obtain the statistical significance on the hyperplane of parameters of interest.

III. RESULTS
Throughout this work, unless otherwise mentioned, we consider the true mass hierarchy to be normal and oscillation parameters to be as given in Table I. Dependence of the MH resolving on the θ 13 mixing angle is compensated for in Appendix A. In Appendix B, as a message to emphasize the vitality of statistics in neutrino experiments, we provide a study on how the total of the T2K-II POT exposure can have a significant impact on the sensitivity results.

A. Determining the neutrino mass hierarchy
To estimate quantitatively the sensitivity of the experiment(s) to the MH determination, we calculate the statistical significance ∆χ 2 to exclude the inverted MH given that the null hypothesis is a normal MH, which is indicated by the recent neutrino experiment results.
The sensitivity is calculated as a function of true δ CP since for the A-LBL experiments, the capability to determine the MH depends on the values of the CP -violating phase. Technically, for each true value of δ CP with normal MH assumed, marginalized χ 2 is calculated for each test value of δ CP with the MH fixed to inverted. Then for each true value of δ CP , the minimum value of χ 2 , which is also equivalent to ∆χ 2 since the test value with normal MH assumed would give a minimum χ 2 close to zero, is obtained. The results, in which we assume sin 2 θ 23 = 0.5, are shown in the top plot of Fig. 4 for different experimental setups:  In other words, the MH can be determined conclusively by a joint analysis of JUNO with the A-LBL experiments, NOνA-II and T2K-II. We find out that in Ref. [45] the authors address a similar objective and come to a quite similar conclusion even though a different calculation method and assumption of the experimental setup are used.

B. Unravelling leptonic CP violation
The statistical significance of ∆χ 2 excluding the CP -conserving values (δ CP =0,π) or the sensitivity to CPV is evaluated for any true value of δ CP with the normal MH assumed.
For the minimization of χ 2 over the MH options, we consider two cases: (i) MH is known and normal, the same as the truth value, or (ii) MH is unknown. Figure 5  unknown affects the first three analyses, but not the fourth. This is because, as concluded in the above section, the MH can be determined conclusively with a joint analysis of all considered experiments. It can be seen that the sensitivity to CP violation is driven by T2K-II and NOνA-II. Contribution of the R-SBL neutrino experiment is significant only at the region where δ CP is between 0 and π and when the MH is not determined conclusively.
JUNO further enhances the CPV sensitivity by lifting up the overall MH sensitivity and consequently breaking the MH-δ CP degeneracy. At δ CP close to −π/2, which is indicated by recent T2K data [10], the sensitivity of the joint analysis with all considered experiments can approximately reach a 5σ C.L. We also calculate the statistical significance of the CPV sensitivity as a function of true δ CP at different values of θ 23 , as shown in Fig. 6. When an inverted MH is assumed, although A CP amplitude fluctuates in the same range as with a normal MH, the probability and rate of ν e appearance becomes smaller to make the statistic error, σ stat. νe , lower. In sum, sensitivity to CP violation, which is proportional to A CP /σ stat. νe , is slightly higher if the inverted MH is assumed to be true as shown at the bottom of the    from Ref. [7]. Figure 7a shows a 3σ C.L. allowed region of sin 2 θ 13 -δ CP obtained with a joint analysis of the T2K-II and NOνA-II experiments. The precision of sin 2 θ 13 can be achieved between 6.5% and 10.7% depending on the true value of δ CP . It will be interesting to compare the measurements of θ 13 from R-SBL experiments and from the A-LBL experiments with such high precision.
As shown in Fig. 7b, both JUNO alone and a combined sensitivity of T2K-II and NOνA-II experiments can reach a sub-percent-level precision on the atmospheric mass-squared splitting ∆m 2 31 . A comparison at such precision may provide a very good test for the PMNS framework. Besides, assuming a maximal mixing sin 2 θ 23 = 0.5, a combined sensitivity of T2K-II and NOνA-II can achieve approximately 6% and 3% precision for the upper and lower limit on sin 2 θ 23 . A capability to solve the θ 23 octant in case the mixing angle θ 23 is not maximal is discussed below.

D. Discussion
We briefly discuss the implications that have arisen from our results in light of the recent updated results from T2K [20], NOvA [23], SK [46], IceCube DeepCore [47], and MI-NOS(+) [48]  values are excluded at a 3σ C.L. Comparing this to Ref. [10], although the statistic significance of excluding CP conservation is reduced from a 95% C.L. to a 90% C.L., the updated data looks more consistent with the PMNS prediction than before. While SK also favors the maximum CP violation, NOνA shows no indication of asymmetry of neutrino and antineu-trino behaviors. With the combined analysis of T2K-II, NOνA-II, and JUNO by 2027, it is expected that more than half of the δ CP values can be excluded with more than a 3σ C.L. If the true δ CP is near δ CP = ± π 2 , discovery of the leptonic CPV with a 5σ C.L. is within reach. Regarding the octant of the θ 23 mixing angles, T2K, NOνA, SK, and MINOS(+) data prefer nonmaximum with statistic significance between a 0.5σ to 1.5σ C.L. If the true value of θ 23 is close to the best fit in the global data fit [8], θ 23 = 0.57, a combined analysis of T2K-II, NOνA-II, and JUNO can exclude the wrong octant with a 3σ C.L. There is a room for improvement in the above-mentioned physic potentials, for example, by adding an atmospheric neutrino data sample from the SK experiment. There are ongoing efforts to combine data from T2K and SK along with a joint analysis of T2K and NOνA. Such activities are vital to realizing a grand framework for combining the special-but-statistically-limited neutrino data in the future.

IV. CONCLUSION
We have studied the physics potentials of a combined analysis of the two accelerator-based long-baseline experiments, T2K-II and NOνA-II, and a reactor-based medium-baseline experiment, JUNO. We have shown that the combined analysis will unambiguously determine the neutrino mass hierarchy given any true values of δ CP and θ 23 within the present allowed range. The combined analysis provides a very appealing sensitivity for the leptonic CP violation search. Particularly, CP -conserving values of δ CP can be excluded with at least a 3σ C.L. for 50% of all the possible true values of δ CP . At CP violation phase values close to δ CP = ± π 2 , a discovery of CP violation in the leptonic sector at the ∼ 5σ C.L. becomes possible. Besides, a combined analysis of T2K-II and NOνA-II can reach a few percent precision on the θ 13 mixing angle and sub-percent-level precision on the ∆m 2 31 mass-squared splitting, which can provide interesting tests of the standard PMNS framework by comparing the results to measurements from reactor-based short-baseline neutrino experiments and JUNO, respectively. Also, a combined analysis of T2K-II and NOνA-II offers a great sensitivity to determine the octant of the θ 23 mixing angle.
Finally, we would like to emphasize that the joint analysis in reality is foreseen to be more complicated than what we have done. Many systematic sources must be taken into account for each experiment and for a joint analysis; the correlation of systematic errors among experiments are important for extracting precisely the oscillation parameters. However, we affirm that the above conclusions are still valid since the measurement uncertainties, particularly for CP violation and the neutrino mass hierarchy, are still dominated by statistical errors.  [7], sin 2 θ 13 = 0.02221 is with NuFIT 5.0 [8], and sin 2 θ 13 = 0.02034 is a 3σ C.L. lower limit. Normal MH and sin 2 θ 23 = 0.5 are assumed to be true.
above a 5σ C.L., the CPV sensitivity depends significantly on the POT exposure as shown in Fig. 10. However there is still a large fraction of δ CP value excluded with a 3σ C.L. The study emphasizes the importance of providing as many proton beams as possible to the T2K experiment for reaching the highest capability of CPV search.