Physics potential of the combined sensitivity of T2K-II, NO$\nu$A extension, and JUNO

T2K-II: The ongoing Tokai-To-Kamioka (T2K) Abe et al. (2011) is the second generation of accelerator-based long-baseline (A-LBL) neutrino oscillation experiments located in Japan, and T2K-II Abe et al. (2016) is a proposal to extend the T2K run until 2026 before Hyper-Kamiokande Abe et al. (2018) 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 $\nu_{\mu}$ ($\overline{\nu}_{\mu}$) disappearance provide a precise measurement of the atmospheric neutrino parameters, $\sin^{2}2\theta_{23}$ and $\Delta m^{2}_{31}$, the $\nu_{e}$ ($\bar{\nu}_{e}$) appearance rates are driven by $\sin^{2}2\theta_{13}$ and are sensitive to $\delta_{\text{CP}}$ and the MH. The sensitivity of the A-LBL experiments such as T2K and NO$\nu$A to $\delta_{\text{CP}}$ and the MH can be understood via the following expression of the so-called CP asymmetry Suekane (2015), presenting a relative difference between $P_{(\nu_{\mu}\rightarrow\nu_{e})}$ and $P_{(\bar{\nu}_{\mu}\rightarrow\bar{\nu}_{e})}$ near the oscillation maximum, and corresponding to $\frac{|\Delta m^{2}_{31}|L}{4E_{\nu}}=\pi/2$.

NO$\nu$A extension or NO$\nu$A-II: Ongoing NuMI Off-axis $\nu_{e}$ Appearance (NO$\nu$A ) Ayres et al. (2007) 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 NO$\nu$A to explore the MH with high sensitivity via the matter effect Wolfenstein (1978) on the (anti-)neutrino interactions. From Eq. (II.1), it can be estimated that the matter effect in NO$\nu$A is $\sim 28.9\%$, which is slightly higher than the CP violation effect. However, these two effects, along with the ambiguity of the $\theta_{23}$ octant, are largely entangled. In other words, NO$\nu$A sensitivity on the neutrino MH depends on the value of $\delta_{\text{CP}}$. NO$\nu$A’s recent data Himmel (2020b) does not provide as much preference to the neutrino mass hierarchy as T2K Dunne (2020b) does since NO$\nu$A data shows no indication of the CP violation. Similar to T2K, NO$\nu$A 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. NO$\nu$A uses a near detector, located 1 km away from the production target, to characterize the unoscillated neutrino flux. The NO$\nu$A far detector is filled with liquid scintillator contained in PVC cells, totally weighted at 14 ktons with 63$\%$ active materials. NO$\nu$A takes advantage of machine learning for particle classification to enhance the event selection performance. In 2018 Acero et al. (2019), NO$\nu$A provided more than a 4$\sigma$ C.L. evidence of electron anti-neutrino appearance from a beam of muon anti-neutrinos. At the Neutrino 2020 conference, NO$\nu$A Himmel (2020b) reported a collected data sample from $2.6\times 10^{21}$ POT exposure. In Sanchez (2018), NO$\nu$A gives the prospect of extending the run through 2024, hereby called NO$\nu$A-II, in order to get a 3$\sigma$ C.L. or higher sensitivity to the MH in case the MH is normal and $\delta_{\text{CP}}$ is close to $-\pi/2$, and more than a 2$\sigma$ C.L. sensitivity to CPV.

JUNO: Jiangmen Underground Neutrino Observatory (JUNO) Djurcic et al. (2015) 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 ($\overline{\nu}_{e}$) 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 $\overline{\nu}_{e}$ spectrum, which uniquely displays the oscillation patterns driven by both solar and atmospheric neutrino mass-squared splittings Zhan et al. (2008). This feature can be understood via the $\bar{\nu}_{e}$ disappearance probability in the vacuum, which is expressed as follow:

Although T2K and NO$\nu$A 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 $\theta_{23}$ 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 $\delta_{\text{CP}}$ and the two scenarios of the neutrino mass hierarchy are still possible, and the mixing angle $\theta_{23}$ is explored in a range close to $45^{\circ}$.

Reaching the three above-mentioned unknowns depends on the ability to resolve the parameter degeneracies among $\delta_{\text{CP}}$, the sign of $\Delta m^{2}_{31}$, $\theta_{13}$, and $\theta_{23}$ Barger et al. (2002). Combining the data samples of the A-LBL experiments (T2K-II and NO$\nu$A-II) and JUNO would enhance the CPV search and the MH determination since the JUNO sensitivity to the MH has no ambiguity to $\delta_{\text{CP}}$. To further enhance the CPV search, one can break the $\delta_{\text{CP}}$-$\theta_{13}$ degeneracy by using the constraint of $\theta_{13}$ from reactor-based short-baseline (R-SBL) neutrino experiments such as Daya Bay Guo et al. (2007), Double Chooz Ardellier et al. (2006), and RENO Ahn et al. (2010). This combination also helps to solve the $\theta_{23}$ octant in the case of nonmaximal mixing.

The General Long-Baseline Experiment Simulator (GLoBES) Huber et al. (2005, 2007) is used for simulating the experiments and calculating their statistical significance. In this simulator, a number of expected events of $\nu_{j}$ from $\nu_{i}$ oscillation in the n-th energy bin of the detector in a given experiment is calculated as

where i, j are the charged lepton(s) associated with the initial and final flavor(s) of the neutrinos, $\Phi_{i}$ is the flux of the initial flavor at the source, $\sigma_{\nu_{j}}$ is the cross-section for the final flavor f, L is the baseline length, $E_{t}$ and $E_{r}$ are the incident and reconstructed neutrino energy, respectively, $\epsilon_{j}(E_{r})$ is the detection efficiency of the final flavor f, and N is the normalization factor for standard units in GLoBES. We describe the experiments using updated information of fluxes, signal and background efficiencies, and systematic errors. Remaining differences between the energy spectra of the simulated data sample at the reconstruction level obtained by GLoBES and the real experiment simulation can be due to the effects of the neutrino interaction model, the detector acceptance, detection efficiency variation as a function of energy, etc… These differences are then treated quantitatively using post-smearing efficiencies, consequently allowing us to match our simulation with the published spectra of each simulated sample from each experiment. Each experimental setup is validated at the event rate level and sensitivity level to ensure that physics reaches of the simulated data samples we obtain are in relatively good agreement with the real experimental setup.
For each T2K-II and NO$\nu$A-II experiment, four simulated data samples per each experiment are used: $\nu_{\mu}(\bar{\nu}_{\mu})$ disappearance and $\nu_{e}(\bar{\nu}_{e})$ appearance in both $\nu$-mode and $\bar{\nu}$-mode. The experimental specifications of these two experiments are shown in Table 2.

In T2K(-II), neutrino events are dominated by the charged current quasielastic (CCQE) interactions. Thus, for appearance (disappearance) in $\nu$-mode and $\bar{\nu}$-mode, the signal events are obtained from the $\nu_{\mu}\rightarrow\nu_{e}$ ($\nu_{\mu}\rightarrow\nu_{\mu}$) CCQE events and the $\bar{\nu}_{\mu}\rightarrow\bar{\nu}_{e}$ ($\bar{\nu}_{\mu}\rightarrow\bar{\nu}_{\mu}$) CCQE events, respectively. In the appearance samples, the intrinsic $\nu_{e}/\bar{\nu}_{e}$ contamination from the beam, the wrong-sign components, i.e., $\overline{\nu}_{\mu}\rightarrow\overline{\nu}_{e}$ ($\nu_{\mu}\rightarrow\nu_{e}$) in $\nu$-mode ($\bar{\nu}$-mode), respectively, and the neutral current (NC) events constitute the backgrounds. In the disappearance samples, the backgrounds come from $\nu_{\mu}$, $\overline{\nu}_{\mu}$ charged current (CC) interaction excluding CCQE, hereby called CC non-QE, and NC interactions. We use the updated T2K flux released along with Ref. Abe et al. (2015). In simulation, the cross section for low- and high-energy regions are taken from Refs. Messier (1999); Paschos and Yu (2002), respectively. In our T2K-II setup, an exposure of $20\times 10^{21}$ POT equally divided among the $\nu$-mode and the $\bar{\nu}$-mode is considered, along with a 50% effectively statistic improvement as presented in Ref. Abe et al. (2016). The signal and background efficiencies and the spectral information for T2K-II are obtained by scaling the T2K analysis reported in Ref. Abe et al. (2017) 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. Abe et al. (2016). A $3\%$ error is assigned for both the energy resolution and the normalization uncertainties of the signal and background in all simulated samples.

For NO$\nu$A-II, we consider a total exposure of $72\times 10^{20}$ POT equally divided among $\nu$-mode and $\bar{\nu}$-mode Sanchez (2018). We predict the neutrino fluxes at the NO$\nu$A far detector by using the flux information from the near detector, given in Ref. Soplin and Cremonesi (2018), and normalizing it with the square of their baseline ratio. A 5% systematic error for all samples and 8–10% sample-dependent energy resolutions are assigned. Significant background events in the appearance samples stem from the intrinsic beam $\nu_{e}/\bar{\nu}_{e}$, NC components, and cosmic muons. In the appearance sample of the $\bar{\nu}$-mode, wrong-sign events from $\nu_{e}$ appearance events are included as the backgrounds in the simulation. We use the reconstructed energy spectra of the NO$\nu$A far detector simulated sample, reported in Ref. Acero et al. (2018), 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 $\nu$-mode and $\bar{\nu}$-mode, events from both CC $\nu_{\mu}$ and $\bar{\nu}_{\mu}$ interactions are considered to be signal events, which is tuned to match with the NO$\nu$A far detector simulated signal given an identical exposure. Background from the NC $\nu_{\mu}$ ($\bar{\nu}_{\mu}$) 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. Acero et al. (2018). Figure 2 shows the simulated NO$\nu$A-II event spectra as a function of reconstructed neutrino energy for $\nu_{e}$ appearance and $\nu_{\mu}$ disappearance channels in both $\nu$-mode and $\bar{\nu}$-mode, where normal MH is assumed, $\delta_{\text{CP}}$ is fixed at $0^{\circ}$, and other parameters are given in Table 1.

Tables 3 and 4 detail our calculated signal and background detection efficiencies for the electron (anti-)neutrino appearance and muon (anti-)neutrino disappearance, respectively, in T2K and NO$\nu$A. The two neutrino experiments reach a relatively similar performance for selecting the electron (anti-)neutrino appearance samples. While T2K gains due to the excellent separation of muons and electrons with the water Cherenkov detector, NO$\nu$A boosts the selection performance with the striking features of the liquid scintillator and the powerful deep learning. For selecting the disappearance samples, T2K outperforms since the T2K far detector is placed deep underground while the NOvA far detector is on the surface and suffers a much higher rate of cosmic ray muons.

In JUNO, the electron anti-neutrino $\bar{\nu}_{e}$ flux, which is produced mainly from four radioactive isotopes (${}^{235}\text{U}$, ${}^{238}\text{U}$, ${}^{239}\text{Pu}$, and ${}^{241}\text{Pu}$ Huber and Schwetz (2004)), is simulated with an assumed detection efficiency of $73\%$. The backgrounds, which have a marginal effect on the MH sensitivity, are not included in our simulation. In our setup, to speed up the calculation, we consider one core of 36 GW thermal power with an average baseline of 52.5 km instead of the true distribution of the reactor cores, baselines, and powers. The simulated JUNO specification is listed in Table 5 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 the $\bar{\nu}_{e}$ 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, NO$\nu$A-II, and JUNO, we implement a R-SBL neutrino experiment to constrain $\sin^{2}\theta_{13}$ at $3\%$ uncertainty, which is reachable as prospected in Ref. Cao and Luk (2016). This constraint is important to break the parameter degeneracy between $\delta_{\text{CP}}$-$\theta_{13}$, which is inherent from the measurement with the electron (anti-)neutrino appearance samples in the A-LBL experiments.
To calculate the sensitivity, a joint $\chi^{2}$ is formulated by summing over all individual experiments under consideration without taking any systematic correlation among experiments. For T2K-II and NO$\nu$A-II, we use a built-in $\chi^{2}$ function from GLoBES for taking the signal and background normalization systematics with the spectral distortion into account. For JUNO, a Gaussian formula for $\chi^{2}$ is implemented thanks to a high statistics sample in JUNO. For a given true value of the oscillation parameters, $\vec{\Uptheta}_{\text{truth}}=(\theta_{12},\theta_{13},\theta_{23},\delta_{\text{CP}};\Delta m^{2}_{21},\Delta m^{2}_{31})_{\text{truth}}$, at a test set of oscillation parameters, $\vec{\Uptheta}_{\text{test}}$, and systematic variations, $\vec{s}_{\text{syst.}}$, a measure $\chi^{2}(\vec{\Uptheta}_{\text{truth}}|\vec{\Uptheta}_{\text{test}},\vec{s}_{\text{syst.}})$ 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.

To estimate quantitatively the sensitivity of the experiment(s) to the MH determination, we calculate the statistical significance $\sqrt{\Delta\chi^{2}}$ 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 $\delta_{\text{CP}}$ 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 $\delta_{\text{CP}}$ with normal MH assumed, marginalized $\chi^{2}$ is calculated for each test value of $\delta_{\text{CP}}$ with the MH fixed to inverted. Then for each true value of $\delta_{\text{CP}}$, the minimum value of $\chi^{2}$, which is also equivalent to $\Delta\chi^{2}$ since the test value with normal MH assumed would give a minimum $\chi^{2}$ close to zero, is obtained. The results, in which we assume $\sin^{2}\theta_{23}=0.5$, are shown in the top plot of Fig. 4 for different experimental setups: (i) JUNO only; (ii) NO$\nu$A-II only; (iii) a joint of JUNO and NO$\nu$A-II; and (iv) a joint of JUNO, NO$\nu$A-II, T2K-II, and the R-SBL experiment. It is expected that the MH sensitivity of JUNO is more than a 3$\sigma$ C.L. and does not depend on $\delta_{\text{CP}}$. On the other hand, the NO$\nu$A-II sensitivity to the MH depends strongly on the true value of $\delta_{\text{CP}}$. A joint analysis of JUNO with the A-LBL experiments, NO$\nu$A-II and T2K-II, shows a great boost in the MH determination. This is expected since a joint analysis will break the parameter degeneracy between $\delta_{\text{CP}}$ and the sign of $\Delta m^{2}_{31}$. Due to the parameter degeneracy among $\delta_{\text{CP}}$ and the sign of $\Delta m^{2}_{31}$, $\theta_{13}$, and $\theta_{23}$ in the measurement with the A-LBL experiments, we also expect that the MH determination depends on the value of $\theta_{23}$. The combined sensitivity of all considered experiments at different values of $\theta_{23}$, (i) maximal mixing at $45^{\circ}$ ($\sin^{2}\theta_{23}=0.50$), (ii) LO at $41^{\circ}$ ($\sin^{2}\theta_{23}=0.43$), and (iii) HO at $51^{\circ}$ ($\sin^{2}\theta_{23}=0.60$), is shown in the center plot of Fig. 4. In the bottom plot of Fig. 4, we compare the MH sensitivity for two hypotheses: the MH is normal and the MH is inverted. The result reflects what we expect: (i) the MH resolving with JUNO is less sensitive to its truth since the dominating factor is the separation power between two oscillation frequencies driven by $|\Delta m^{2}_{31}|$ and $|\Delta m^{2}_{32}|$, shown in Eq. (II.1), and the relatively large mixing angle $\theta_{12}$; and (ii) for the A-LBL experiments like T2K and NO$\nu$A, the MH is determined through the MH-$\delta_{\text{CP}}$ degeneracy resolving as concisely described in Eq. (II.1). The $A_{\text{CP}}$ amplitude is almost unchanged when one switches from a normal MH to an inverted MH and simultaneously flips the sign of $\delta_{\text{CP}}$. Those results conclude that the wrong mass hierarchy can be excluded at a C.L. greater than $5\sigma$ for all the true values of $\delta_{\text{CP}}$ and for any value of $\theta_{23}$ in the range constrained by the experiments. In other words, the MH can be determined conclusively by a joint analysis of JUNO with the A-LBL experiments, NO$\nu$A-II and T2K-II.

The statistical significance of $\sqrt{\Delta\chi^{2}}$ excluding the CP-conserving values ($\delta_{\text{CP}}$=0,$\pi$) or the sensitivity to CPV is evaluated for any true value of $\delta_{\text{CP}}$ with the normal MH assumed. For the minimization of $\chi^{2}$ 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 shows the CPV sensitivity as a function of the true value of $\delta_{\text{CP}}$ for both MH options obtained by different analyses: (i) T2K-II only; (ii) a joint of T2K-II and R-SBL experiments; (iii) a joint of T2K-II, NO$\nu$A-II, and R-SBL experiments; and (iv) a joint of T2K-II, NO$\nu$A-II, JUNO, and R-SBL experiments. The result shows that whether the MH is known or 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.

Table 6 shows the fractional region of all possible true $\delta_{\text{CP}}$ values for which we can exclude CP-conserving values of $\delta_{\text{CP}}$ to at least a $3\sigma$ C.L., obtained by the joint analysis of all considered experiments. Due to the fact that the MH is resolved completely with the joint analysis, the CPV sensitivities are quantitatively identical no matter whether the MH is assumed to be known or unknown.

We briefly discuss the implications that have arisen from our results in light of the recent updated results from T2K Dunne (2020b), NOvA Himmel (2020b), SK Nakajima (2020), IceCube DeepCore Blot (2020), and MINOS(+) Carroll (2020b) presented at the Neutrino 2020 conference. T2K prefers the normal MH with a Bayes factor of 3.4; SK disfavors the inverted MH at a 71.4–90.3$\%$ C.L.; both NO$\nu$A and MINOS(+) disfavor the inverted MH at a C.L. less than 1$\sigma$. The prospect of completely resolving the MH by combining T2K-II, NO$\nu$A-II, and JUNO by 2027 is thus very encouraging. On the leptonic CPV search, the leading measurement is from T2K where $35\%$ of $\delta_{\text{CP}}$ values are excluded at a $3\sigma$ C.L. Comparing this to Ref. Abe et al. (2020), 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, NO$\nu$A shows no indication of asymmetry of neutrino and antineutrino behaviors. With the combined analysis of T2K-II, NO$\nu$A-II, and JUNO by 2027, it is expected that more than half of the $\delta_{\text{CP}}$ values can be excluded with more than a $3\sigma$ C.L. If the true $\delta_{\text{CP}}$ is near $\delta_{\text{CP}}=\pm\frac{\pi}{2}$, discovery of the leptonic CPV with a $5\sigma$ C.L. is within reach. Regarding the octant of the $\theta_{23}$ mixing angles, T2K, NO$\nu$A, SK, and MINOS(+) data prefer nonmaximum with statistic significance between a 0.5$\sigma$ to 1.5$\sigma$ C.L. If the true value of $\theta_{23}$ is close to the best fit in the global data fit Esteban et al. (2020), $\theta_{23}$$=0.57$, a combined analysis of T2K-II, NO$\nu$A-II, and JUNO can exclude the wrong octant with a 3$\sigma$ 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 NO$\nu$A. Such activities are vital to realizing a grand framework for combining the special-but-statistically-limited neutrino data in the future.