Revisiting T2KK and T2KO physics potential and $\nu_\mu$ - $\bar{\nu}_\mu$ beam ratio

We revisit the sensitivity study of the Tokai-to-Kamioka-and-Korea (T2KK) and Tokai-to-Kamioka-and-Oki (T2KO) proposals where a water Cerenkov detector with the 100 kton fiducial volume is placed in Korea ($L = 1000$ km) and Oki island ($L = 653$ km) in Japan, respectively, in addition to the Super-Kamiokande for determination of the neutrino mass hierarchy and leptonic CP phase ($\delta_{CP}$). We systematically study the running ratio of the $\nu_\mu$ and $\bar{\nu}_\mu$ focusing beams with dedicated background estimation for the $\nu_e$ appearance and $\nu_\mu$ disappearance signals, especially improving treatment of the neutral current $\pi^0$ backgrounds. Using a $\nu_\mu$ - $\bar{\nu}_\mu$ beam ratio between 3 : 2 and 2.5 : 2.5 (in unit of $10^{21}$POT with the proton energy of 40 GeV), the mass hierarchy determination with the median sensitivity of 3 - 5 $\sigma$ by the T2KK and 1 - 4 $\sigma$ by the T2KO experiment are expected when $\sin^2\theta_{23} = 0.5$, depending on the mass hierarchy pattern and CP phase. These sensitivities are enhanced (reduced) by $30\%$ - $40\%$ in $\Delta \chi^2$ when $\sin^2\theta_{23} = 0.6\, (0.4)$. The CP phase is measured with the uncertainty of $20^\circ$ - $50^\circ$ by the T2KK and T2KO using the $\nu_\mu$ - $\bar{\nu}_\mu$ focusing beam ratio between 3.5 : 1.5 and 1.5 : 3.5. These findings indicate that inclusion of the $\bar{\nu}_\mu$ focusing beam improves the sensitivities of the T2KK and T2KO experiments to both the mass hierarchy determination and leptonic CP phase measurement simultaneously with the preferred beam ratio being between 3 : 2 - 2.5 : 2.5 ($\times 10^{21}$POT).

Ideas of extending the Tokai-to-Kamioka (T2K) experiment with additional waterČerenkov detectors placed in Korea (Tokai-to-Kamioka-and-Korea, T2KK, experiment [15][16][17][18][19][20][21][22][23][24][25][26]) or in Oki island (Tokai-to-Kamioka-and-Oki, T2KO, experiment [25,27]) has been proposed to address those questions 1 . It has been shown that the T2KK experiment with a 100 kton fiducialvolume detector in Korea in addition to the SK detector is an appealing proposal if we can use the J-PARC neutrino beam with 0.64 MW beam power and the 2.5 • -3.0 • off-axis angle at the SK [17,18,20,22,24]. The authors of Ref. [17,18,20] investigated the sensitivities to the mass hierarchy and CP phase with the ν µ focusing beam in a simple manner, ignoring the effects of neutral current (NC) π 0 backgrounds, miss-identification of a muon as an electron, and smearing of reconstructed neutrino energy. Authors of Ref. [22] then re-evaluated the physics potential of the same T2KK setup with careful consideration on those effects.
Inclusion ofν µ focusing beams may improve the sensitivity of long-baseline oscillation experiments to the mass hierarchy since the matter effects, which enhance the mass-hierarchy difference in neutrino oscillation patterns, appear in the opposite way in ν µ andν µ oscillations. The impacts of including theν µ focusing beam in the T2KK experiment was studied in Ref. [24]. The authors considered the running ratio of the ν µ andν µ focusing beams of 5 : 0 and 2.5 : 2.5 in the unit of protons on target (POT) and argued that including theν µ focusing beam improves the sensitivity to the mass hierarchy determination significantly. The impacts of anti-neutrino beams was also studied in Ref. [16] for a different T2KK setup; two 270 kton detectors are each placed at Kamioka and Korea, receiving 2.5 • off-axis beams with the beam power of 4 MW and the total running time of eight years. Physics potential of the T2KO experiment was also investigated [25] with a similar analysis and conclusion as in Ref. [24]. However, those studies again did not consider the effects of the NC π 0 backgrounds, miss-identified muon, and events from other neutrino-nucleus interactions than the charged-current quasi-elastic (CCQE) one. Therefore, it is not very clear whether the ν µ -ν µ focusing beam ratio of 1 : 1 is the best for the mass hierarchy determination and CP phase measurement.
In this paper, we revisit the sensitivity study of the T2KK [22,24] and T2KO [25] experiments for the neutrino mass hierarchy and CP phase, studying the dependence of the sensitivities on the ν µ -ν µ focusing beam ratio systematically with dedicated estimation of backgrounds. Especially, the treatment of the NC π 0 backgrounds is improved in this analysis. The NC π 0 backgrounds is estimated using a realistic π 0 rejection probability based on the POLfit (Pattern Of Light fitter) algorithm [31], and the contribution form the coherent π 0 production process is taken into account, which is neglected in the previous analysis [22]. The uncertainty of the NC π 0 backgrounds is also reconsidered including the uncertainty from the axial masses in the models of neutrino-nucleus scattering cross sections [32,33].
The remaining part of this paper is organized as follows. After describing the T2KK and T2KO experimental setups in Section 2, our analysis details are discussed in Section 3. Results for the sensitivity of the T2KK and T2KO experiments to the mass hierarchy determination and CP phase measurements are presented in Sections 4 and 5, and our main conclusions are summarized in Section 6.

Simulation details of T2KK and T2KO experiments
In this section, we fix our notation and introduce useful approximated formulae for the ν µ → ν µ and ν µ → ν e oscillation probabilities. We then describe the experimental setups and discuss the simulation details of the expected signal event number in those experiments, taking into account of smearing of reconstructed neutrino energy due to the Fermi motions of target nuclei, detector resolution and contamination of events from non-CCQE neutrino-nucleus interactions. Simulation of the background events are also discussed: the NC single-π 0 background and its uncertainty, the secondary neutrino beam backgrounds, and miss-identified muon/electron backgrounds.

Neutrino oscillations in matter
We briefly review the neutrino oscillation probabilities in matter, presenting analytic approximations for the ν µ → ν µ (ν µ disappearance) and ν µ → ν e (ν e appearance) oscillation modes, which are useful for understanding the physics potential of the T2KK and T2KO experiments qualitatively.
We work in the three neutrino flavor scheme, where the neutrino flavor eigenstate |ν α (α = e, µ, τ ) are mixtures of the three mass eigenstates |ν i with their masses m i (i = 1, 2, 3) as Here U is the Maki-Nakagawa-Sakata (MNS) [14] matrix, which can be parameterized with the three mixing angles, θ 12 , θ 13 , θ 23 , and three phases, δ CP , φ 1 , φ 2 [34]. Among them, two phases can be eliminated in lepton number conserving processes, remaining one relevant phase, δ CP , to neutrino oscillation experiments. The definition regions of the four parameters are chosen as 0 ≤ θ 12 , θ 13 , θ 23 ≤ π/2 and −π ≤ δ CP ≤ π. The probability that an initial flavor eigenstate |ν α with energy E is observed as a flavor eigenstate |ν β after traveling a distance L in the matter of density ρ(x) (0 < x < L) is given by where the Hamiltonian inside matter is with δm ij ≡ m 2 i − m 2 j . a(x)/2E is the effective potential due to electrons in matter as where G F is the Fermi constant and n e (x) is the electron number density. In the translation from n e (x) to ρ(x), we assume that the number of the neutron in matter is same as that of proton. λ i (x)/2E andŨ (x) are the eigenvalues and the corresponding unitary matrix of the Hamiltonian at the distance x, respectively. To a good approximation [24,35,36], the matter density along the T2K, T2KO and T2KK baselines can be replaced by the averaged one, ρ(x) ≃ρ, and so as a(x) in Eq. (2.3), a(x) ≃ā. Then the oscillation probability, P να→ν β , can be expressed compactly by using x-independent eigenvalues, λ i , and a unitary matrix,Ũ , as Our numerical results are based on the above solution, Eq. (2.5a), and the main results are not affected significantly by the matter density profile as long as the mean matter density is chosen appropriately [24,25].
In this study we are mainly interested in the ν µ → ν e and ν µ → ν µ oscillation modes and their charge conjugated ones. It is useful to write them down in the approximated analytic forms as [17] where sin θ atm ≡ sin θ 23 cos θ 13 and ∆ ij ≡ δm 2 ij L/4E. Here A α , B α and C α (α = µ, e) are corrections due to the matter effect and smaller mass difference δm 2 21 : In these expressions we retain up to the sub-leading terms of ∆ 21 , sin 2 θ 13 and aL/4E. The corresponding probabilities for anti-neutrino oscillations can be obtained from the above expressions by reversing the sign of the matter effect term (a → −a) and the CP phase (δ CP → −δ CP ). These expressions are valid as long as those three parameters are negligibly smaller than unity; this is the case for T2K, T2KO and T2KK experiments, where typically L/E ∼ O(10 2 − The ν e appearance mode plays more important role in determining the mass hierarchy (i.e., the sign of ∆ 31 ) than the ν µ disappearance mode. This is because the appearance mode may have sensitivity to the mass hierarchy around oscillation peaks through the A e parameter, while the disappearance mode is lack of sensitivity around oscillation peaks since A µ ≃ 0. On the other hand, the disappearance mode is important in constraining the θ 23 mixing angle, which still has large uncertainty [34]. The ν e appearance mode also has sensitivity to the CP phase. It is sensitive to the sine of δ CP around the oscillation peaks, mainly through the A e parameter; on the other hand, it is sensitive to the the cosine of δ CP between oscillation maxima and minima, mainly through the B e parameter. Therefore, if we try to obtain the full information of the δ CP , it is not enough to observe just around the first oscillation peak, as we will see later.

Experimental setups
We use the ν µ andν µ focusing beam fluxes from the J-PARC with the proton energy of 40 GeV [37]. In Fig. 1, we show the fluxes corresponding to 10 21 POT (protons on target) at the SK. The ν µ (ν µ ) focusing beams include the primary, ν µ (ν µ ), and secondary,ν µ (ν µ ), ν e ,ν e , components, and we take them into account in our analyses. The baseline length from the J-PARC to the SK and Oki detectors are taken to be 295 km [38] and 653 km [25], respectively. The baseline length to a detector in Korea (Kr detector) can be taken from 1000 km to 1300 km in South Korea [16,17]. In this study, we place a Kr detector at the shortest baseline length, L = 1000 km, to receive the J-PARC neutrino beams with the smallest off-axis angle [17], which is preferred in terms of the sensitivity to the mass hierarchy determination [17,18,22,25]. For the nominal 2.5 • off-axis angle at the SK, a Kr detector receives the ∼ 1 • off-axis beam (OAB); the case of 3.0 • OAB at the SK is also investigated, corresponding to the 0.5 • OAB at a Kr detector [25]. On the other hand, variation of the off-axis angle does not affect sensitivities of the T2KO experiment to the mass hierarchy . L is the baseline length between the J-PARC and a detector, FV is the fiducial volume of a detector,ρ is the average matter density along a baseline, and OA is the off-axis angle of the J-PARC neutrino (anti-neutrino) beams at a detector. The first and second OA angles at the Oki and Kr detectors are related to the corresponding OA angles at the SK. These parameter values are used as default in our simulation unless otherwise mentioned.
and CP phase measurements significantly [25], and we only consider the 2.5 • off-axis angle at the SK for the T2KO experiment, corresponding to 0.9 • OAB at the Oki detector [25]. The averaged matter densities,ρ, along the baseline between J-PARC and SK, Oki or Kr detectors have been evaluated in Refs. [24,25] and taken as in Table 1 in this study. It is sufficient to use those averaged densities in those long-baseline neutrino experiments [24], and we neglect small effects from the variation of the matter density along those baselines.

Signal events
In this subsection we describe how to estimate the signal event numbers at the SK, Oki and Korea detectors. We consider charged-current quasi-elastic (CCQE) events, ν l n → l p orν l p → l n (l = µ or e), from the ν µ → ν µ and ν µ → ν e oscillation modes and their charge conjugated modes as signal events. The CCQE events are identified as events with only oneČerenkov ring from an electron or muon where the visible energy of the ring is required to be larger than 200 MeV. Since neutrino beam direction at a far detector is understood well in long-baseline experiments, we can reconstruct incoming neutrino energy for the CCQE events as [39] , (2.8) assuming that target nucleons are at rest. Here E ℓ , p ℓ and θ ℓ are the charged lepton's energy, magnitude of the three-momentum and polar angle about the neutrino beam direction; m p , m n and m l are the mass of a proton, neutron and charged-lepton, respectively, and E b n is the neutron binding energy in the target nucleus. For the anti-neutrino events, m p and m n should be exchanged and E b n should be replaced with the proton binding energy E b p in Eq. (2.8). It should be noted that in reality the reconstructed energy may be different from the incoming neutrino energy due to the Fermi motion of target nucleons inside nuclei and finite detector resolutions for lepton momenta and scattering angles.
The number of the signal events in the i th energy bin, E i rec < E < E i+1 rec , from the ν α → ν β oscillation mode at the waterČerenkov detector D (= SK, Oki, Kr) via the X-type neutrinonucleus interaction (X = CCQE, non-CCQE) are calculated as 3.0 • Table 2. The momentum and angular resolutions for muons and electrons at the SK detector [41].
where E i rec = 0.05GeV × i, E ν is an incoming neutrino energy, Φ D να is the flux of ν α at the detector D, P να→ν β is the neutrino oscillation probability including the matter effects with the mean matter densityρ D , and N Z is the number of the nucleus Z (hydrogen (H) or oxygen (O)) in the detector.σ X ν β Z is the cross section of the X-type ν β -Z interaction after imposing a CCQE selection cuts. The smearing function S X ν β Z (E ν , E rec ) returns the probability that the energy E rec is reconstructed from an event induced by an incoming neutrino with the energy E ν , taking into account the Fermi motion of the target nucleons and detector resolutions. The detection efficiency ofČerenkov rings and the electron/muon identification efficiencies will be discussed in Sections 2.4 and 3.
In order to estimate the cross sections of the CCQE signal,σ X ν β Z , we generate events induced by the neutrino and anti-neutrino charged-current interactions with the Monte-Carlo event generator Nuance v3.504 [40], imposing the CCQE selection criteria: Only one charged lepton (ℓ = µ ± or e ± ) with |p ℓ | > 200 MeV, No π 0 , K S , K L and K ± . (2.10d) The lower limit of the lepton momentum in the first criterion, (2.10a), is from the threshold of the waterČerenkov detector for muons. π ± with |p| > 200 MeV and γ with |p| > 30 MeV as well as π 0 , K S , K L and K ± (which are assumed to decay inside a detector) give rise to additional rings. Events with such additional rings are not selected as the CCQE events and are removed. The survived events after imposing the selection cuts consist of the genuine CCQE events and the other charged-current events (non-CCQE events). Some of the non-CCQE events arise from single soft π ± emission via the ∆ resonance [22]. We parameterize the CCQE and non-CCQE cross sections for target nuclei after imposing the selection criteria (2.10) and summarize them in Appendix A. The smearing effects due to the Fermi motion and detector resolution shown in Table 2 are taken into account by smearing functions. We made fitting formulae of the anti-neutrino smearing functions for numerical simulations and show them in Figs. 2 for incoming anti-neutrino energy of 1 and 2 GeV. The explicit expressions of the anti-neutrino smearing functions are found in Appendix B. Those for neutrinos are in Ref. [22]. The red circles show simulated distributions for genuine CCQE interactions, while the red histograms show the distributions based on the fitting formulae. The blue diamonds and histograms are for non-CCQE interactions. We see that the fitting formulae describe the simulated distributions well. Anti-neutrinos can interact with protons in hydrogens, in addition to Oxygens. Thus, the reconstructed energy distibutions show sharper peak than those for neutrinos since protons in hydrogens do not have the Fermi motion and are almost at rest for anti-neutrinos with O(1) GeV energy. (The smearing functions for neutrinos are shown in Fig.3 in Ref. [22].) (c)ν µ and (d)ν e with E ν = 2 GeV. Those events are generated by Nuance v3.504 [40], imposing the CCQE selection cuts (2.10) and applying the detector resolutions in Table 2 to the produced muons and electrons. The solid-red and dotted-blue histograms show the fitting formulae of the CCQE and non-CCQE smearing functions, respectively.

Background events
In this section we discuss the sources of background events taken into account in this study: neutral-current (NC) single-π 0 events, secondary neutrinos in the ν µ andν µ focusing beams and misidentified muon and electron events. NC single-π 0 events can be a substantial background source for the ν e andν e appearance modes, where one of the photons from a π 0 decay is lost, or the produced π 0 is so energetic that it decays to unresolved photons, mimicking an electron ring. NC neutrino-nucleus scatterings occurs through the quasi-elastic (NCQE), resonant π 0 production (NCRes), coherent π 0 production (NCCoh) or deep inelastic (NCDI) scatterings. The number of the NC single-π 0 events in the i th energy bin, E i rec < E < E i+1 rec , at the waterČerenkov detector D (= SK, Oki, Kr) via the Y -type neutrino-nucleus interaction (Y = NCQE, NCRes, NCCoh and NCDI) induced by the ν α component of the ν µ orν µ focusing beam are calculated as where E i rec = 0.05 GeV × i, andσ Y Z and S Y Z are the cross section and smearing function for the The probability of misidentifying a π 0 to an e ± as a function of π 0 momentum, based on the POLfit algorithm [31]. The black points show simulated data by the T2K collaboration [42], and the red curve show the fitted function to the data, Eq. (2.13).
Y -type ν-Z interaction after imposing the NC single-π 0 selection criteria: 12e) The first condition, Eq. (2.12a), selects NC events, while the other conditions eliminate multiring events. The π 0 momentum distributions from the ν µ focusing beams after imposing the above criteria are shown in Fig. 3 for various off-axis beam angles. The NC single-π 0 events are produced more with smaller off-axis beam angle because the fluxes of such neutrino beams are distributed in higher energy region as shown in Fig. 1. This is a disadvantage of using neutrino beams with smaller off-axis angles at a far detector, and a low π 0 misidentification probability is needed especially for the CP phase measurements. We parameterize the π 0 misidentification probability, P e/π 0 , used in our analysis as a function of π 0 momentum x [GeV] as based on the simulation [42] of the POLfit π 0 -rejection algorithm 2 [31]. The reference data and the fitted function are shown in Fig. 3(b). The misidentification probability is kept less than 0.2 for p π 0 < 0.6 GeV, where the π 0 backgrounds mostly distributes. The background events are then selected from the simulated NC single-π 0 events according to the misidentification probability, and the reconstructed energy of each background event is calculated with Eq. (2.8), assuming the misidentified π 0 as an electron.   03 GeV, respectively. The dashed-red and dashed-blue histograms show the 1σ uncertainty ranges of the resonant and coherent axial masses, respectively. The solid-green and purple histograms show the NC deep-inelastic (DI) and quasi-elastic (QE) single-π 0 components, and the black histogram is for the total NC single-π 0 backgrounds. Event numbers are calculated for a 100 kton detector using the 0.5 • off-axis beam with the ν µ (ν µ ) flux corresponding to 5 × 10 21 POT with the proton energy of 40 GeV.
The NC single-π 0 backgrounds significantly affect the sensitivity to the mass hierarchy and CP phase [22], and it is important to include their uncertainty properly in our analyses. One of the major uncertainty sources of the NC single-π 0 backgrounds is modeling of neutrino-nucleus interactions. In Fig. 4 we show the reconstructed energy distributions of the NC single-π 0 backgrounds calculated with Nuance for different neutrino-nucleus interactions. We see that the π 0 backgrounds mainly distribute in low energy region, where the contributions from resonant and coherent single-π 0 production processes dominate. These processes are implemented in Nuance based on the Rein-Sehgal's calculations [32,33]. Among the modeling parameters of the NC neutrino-nucleus interactions, axial form-factor masses (m A ) have not been measured accurately. Therefore, we vary the axial masses of the resonant and coherent single-pion production processes within their uncertainties: m Res A = 1.1±0.11 GeV [44] and m Coh A = 1.03±0.28 GeV [45]. As shown in Fig. 4, those uncertainties can be well approximated by 13% and 15% normalization uncertainties for the NC resonant and coherent single-π 0 backgrounds, respectively.
Another major source of uncertainty of the NC single-π 0 backgrounds arises from the π 0 misidentification probability, Eq. (2.13). The T2K collaboration estimated 10.8% uncertainty in the NC-π 0 background estimation due to the POLfit algorithm [42]. Since our modeling of the π 0 misidentification probability is based on the POLfit algorithm, we assign 11% uncertainty to the normalization of the NC single-π 0 backgrounds due to the π 0 misidentification. All in all, we include the 13% and 15% normalization uncertainties for the NC resonant and coherent single-π 0 backgrounds, respectively, and 11% normalization uncertainty for the total NC single-π 0 backgrounds. This treatment allows independent normalization corrections for the resonant and coherent NC single-π 0 backgrounds. The ν µ (ν µ ) focusing beams contain not only ν µ (ν µ ) but also other neutrino flavors, ν e ,ν e andν µ (ν µ ), secondary neutrino beams. Especially, for the ν µ → ν e andν µ →ν e oscillation modes, the ν e andν e secondary beams become major background sources. We simulate these secondary-neutrino events in the same way as the signal events described in Sec. 2.3.
There is also some probability of misidentifying a muon (electron)Čerenkov ring as an electron (muon), P e/µ (P µ/e ). Although these probabilities depend on the detector design and performance, we assume the same probabilities for the SK and a far detector in Oki and Korea as P e/µ = P µ/e = 1 ± 1%. (2.15) 3 χ 2 analysis Using the simulated signal and background events, we estimate the sensitivity of the T2KK and T2KO experiments to the mass hierarchy and CP phase (δ CP ), performing the χ 2 analysis. The χ 2 function used in this study can be written as The first two terms measure deviations of data from the theoretical predictions at the SK and a far detector in Oki or Korea, The µ-and e-like event numbers are calculated using the CC signal and NC single-π 0 background events (N i,X D and N i,Y π 0 ,D defined by Eqs. (2.9) and (2.11)) as where ε D µ (ε D e ) is the efficiency of detecting muon (electron)Čerenkov rings, and P D e/µ (P D µ/e ) is the probability of misidentifying the detected muon (electron)Čerenkov ring as an electron (muon). Detected neutrino flavors are already summed in N i,Y π 0 ,D . The anti-neutrino event   Table 3.   Figure 5. The reconstructed energy distributions at the SK detector with the ν µ (left panels) and ν µ (right panels) focusing beams. The former four panels show distributions for e-like events, and the latter four panels are for µ-like events. The 1st and 3rd rows are for the normal hierarchy case, while the 2nd and 4th rows are for the inverted hierarchy case. The dashed-blue, dotted-green, red and dash-dotted-purple histograms are for CCQE, non-CCQE, NC single-π 0 background and misidentified muon/electron background events, respectively. The black histogram shows the total of those contributions. The event numbers are calculated for T2K experiment with the 2.5 • OAB and the beam flux corresponding to 5 × 10 21 POT with the proton energy of 40 GeV.   As shown in those figures, T2KK and T2KO experiments can observe up to the second peak of the ν µ → ν e andν µ →ν e oscillations due to their long baseline length, while the T2K experiment only observe the first peak. Observing the several peaks of those oscillation modes has advantages especially for the accurate CP phase measurement because tails of the oscillation peaks have the information of both sin δ CP and cos δ CP .
On the other hand, the theoretical predictions of the µ-and e-like event numbers, (N i µ,D ) fit where f D V and f D να are the normalization factors for the fiducial volume of and the ν α flux at a detector D (= SK, Oki, Kr), respectively. f X ν β is the normalization factor for the CC cross section of a neutrino flavor ν β via a X(= CCQE or non-CCQE) interaction. f NCRes π 0 and f NCCoh π 0 are the normalization factors for the NC cross sections of resonant and coherent singleπ 0 production processes, respectively, while f NC π 0 is the overall normalization factor for the NC single-π 0 backgrounds, mainly reflecting the uncertainty of the π 0 misidentification probability, Eq. (2.13). These factors are varied in the minimization of the χ 2 function, and their deviation from unity measures systematic uncertainties.
Using the above normalization factors, detection efficiencies (ε e , ε µ ) and misidentification probabilities (P e/µ , P µ/e ), we take into account effects of the systematic uncertainty in the χ 2 function as where S fit is the systematic parameter value used to calculate the theoretical predictions, S input is the one used to generate the data, and δS is the uncertainty of the parameter. We summarize the systematic parameters used in our analysis in Table 3, where the uncertainties related to the NC π 0 backgrounds are assigned based on the discussions in Sec. 2.4, while the other uncertainties are taken as in the previous study [22]. Finally, χ 2 para accounts for external constraints on the physical parameters as where P fit is the parameter value used to calculate the theoretical predictions, P input is the one used to generate the data, and δP is the uncertainty of the parameter. We summarize the physical parameters used in our analysis in Table 3 as well, where the parameter values are based on Ref. [34], except the uncertainty of sin 2 2θ 13 , for which we use the uncertainty achieved by DayaBay collaboration [46], and the matter densities, which are taken from the Ref. [24,25]. The sensitivities to the mass hierarchy is then estimated using the test statistic defined as where χ 2 min | NH(IH) is the minimum of the χ 2 function under the assumption of the normal (inverted) hierarchy. The distribution of the ∆χ 2 MH due to the fluctuation of data can be approximated as [47][48][49] 8) where N (µ, σ) denotes the normal distribution with mean µ and standard deviation σ; ∆χ 2 MH is the ∆χ 2 MH obtained with the average experiment (the Asimov data set [50]). It has been shown with explicit Monte Carlo studies that this approximation holds with good accuracy for long baseline experiments which give |∆χ 2 MH | ≫ 1 [49]. With this approximation, we may calculate the probability that an experiment rejects the wrong mass hierarchy hypothesis with a given confidence level. For example, ∼ 50% is the probability for an experiment to reject the wrong mass hierarchy with the |∆χ 2 MH |-σ confidence level. The sensitivity to the CP phase measurement is estimated in terms of (∆χ 2 ) min defined as where the minimum of the χ 2 function in the first term is found by fixing some model parameters θ and marginalizing the other parameters θ ′ , assuming that the true mass hierarchy is known, while the minimum of the χ 2 function in the second term is found by marginalizing the whole parameters, θ and θ ′ . Under certain conditions (especially linear dependence of the theoretical prediction on the parameters θ), the (∆χ 2 ) min (θ true ) is known to be approximately distributed as the χ 2 distribution of N θ degrees of freedom (d.o.f) when the data size is large, where θ true is the true values of the parameters θ, and N θ is the number of the fixed parameters [51].
Since the CP phase is a cyclic parameter in the oscillation probabilities, the linearity condition is not satisfied in general, and deviation of the distribution of (∆χ 2 ) min (δ CPtrue ) form the χ 2 distribution of 1 d.o.f would be expected [52,53]. However, this deviation is not so significant for experiments with sufficiently high sensitivity such that the 1-σ uncertainty of δ CP measurement is less than ∼ 20 • for δ CP = 0 • [52]. This is the case for the T2KK and T2KO experiments as we will see later, and we estimate sensitivity to the CP phase of those experiments based on the χ 2 distribution approximation of the (∆χ 2 ) min (δ CPtrue ) distribution 3 . The n-σ confidence interval of the CP phase measurement, [δ a CP , δ b CP ] nσ (δ a CP < δ b CP ), is then estimated such that (3.10)

Sensitivity to the mass hierarchy determination
In this section, we present the results for the sensitivity studies on the mass hierarchy determination by the T2KK and T2KO experiments, discussing the sensitivity dependence on the ν µ -ν µ focusing beam ratio and sin 2 θ 23 . In Fig. 8, the sensitivity of the T2KK experiments to the mass hierarchy determination is shown. The left and right panels are for the normal and inverted hierarchy cases, while the upper and lower panels are for the 3.0 • (0.5 • ) and 2.5 • (1.0 • ) off-axis beams, OAB, at the SK (Kr) detector, respectively. The true value of sin 2 θ 23 is assumed to be 0.5. The blue (dashedblue), green (dashed-green), orange (dashed-orange) and red curves show the absolute value of the ∆χ 2 MH , Eq. (3.8), for rejecting the wrong mass hierarchy when the ratio of the ν µ andν µ focusing beams is 5 : 0 (0 : 5), 4 : 1 (1 : 4), 3 : 2 (2 : 3) and 2.5 : 2.5 (×10 21 POT with the proton energy of 40 GeV), respectively. It is shown that includingν µ focusing beam can improve the sensitivity, especially in high sensitivity regions. Although inclusion of theν µ focusing beam causes reduction of sensitivities for some δ CP , this can be alleviated by adjusting the beam ratio appropriately. In order to minimize the reduction of the sensitivities, ν µ :ν µ = 4 : 1 is the best ratio for both OAB cases. Comparing the lowest |∆χ 2 MH | in the whole range of the CP phase, ν µ :ν µ = 4 : 1 is the best ratio for the 3.0 • OAB at the SK, and 3 : 2 -2 : 3 are the best for the 2.5 • OAB at the SK. In terms of the highest sensitivity, 4 : 1, 3 : 2 and 2.5 : 2.5 beam ratios give comparable sensitivity for the normal hierarchy, but 3 : 2 -2.5 : 2.5 are significantly better than 4 : 1 for the inverted hierarchy case. Thus, around 3 : 2 -2.5 : 2.5 would be a preferred choice for 3.0 • OAB at the SK. For the 2.5 • OAB at the SK, the beam ratio of 3 : 2 -2.5 : 2.5 would be a preferred choice. Although there is not such a ν µ -ν µ focusing beam ratio that gives the best sensitivity for any δ CP values and mass hierarchies, the beam ratio between 4 : 1 and 2.5 : 2.5 would be a reasonable choice for both 2.5 • and 3.0 • OAB at the SK.
In Fig. 9, we show the sensitivities of the T2KO experiment. Improvement of the sensitivities by including theν µ focusing beam is significant in the high sensitivity region, preferring the running ratio of 3 : 2 -2 : 3, while improvement in the low sensitivity region is not so evident. Comparing to the T2KK experiments, the sensitivity is lower by 20% -80% in |∆χ 2 MH |. The lower sensitivity in the T2KO experiment is basically due to the smaller matter effects. The difference between the mass hierarchies mainly shows up in A e and B e in Eq. (2.6b) through the matter effects, and enhanced by the baseline length. Because of the shorter baseline length to the Oki detector than the detector in Korea, this enhancement is reduced more easily by Figure 9. Same as Fig. 8, but for the T2KO experiment with the 2.5 • (0.9 • ) off-axis beam at the SK (Oki). Figure 10. The dependence of the |∆χ 2 MH | on sin 2 θ 23 for the T2KK and T2KO experiments to reject the wrong mass hierarchy as functions of the CP phase. The left and right plots are for the T2KK and T2KO experiments with the 2.5 • OAB at the SK, respectively. The red and dashed-blue curves are for the normal and inverted hierarchy cases. sin 2 θ 23 is assumed to be 0.6, 0.5 and 0.4 from the top to the bottom curves. The ν µ -ν µ focusing beam ratio is fixed at ν µ :ν µ = 2.5 : 2.5 ×10 21 POT with the proton energy of 40 GeV.
adjusting the CP phase, resulting in the lower sensitivity to the mass hierarchy in the T2KO experiment.
We show the sin 2 θ 23 dependence of the sensitivity to the mass hierarchy determination in Fig. 10. The left and right plots are for the T2KK and the T2KO experiments with the 2.5 • OAB at the SK, respectively. The red and dashed-blue curves are for the normal and inverted hierarchy cases. sin 2 θ 23 is assumed to be 0.6, 0.5 and 0.4 from the top to the bottom curves for both mass hierarchy cases. We fix the ν µ -ν µ beam ratio at 2.5 : 2.5, but dependence on sin 2 θ 23 are similar in the other beam ratios. The |∆χ 2 MH | is reduced by up to 30% -40% when sin 2 θ 23 decreases by 0.1 since the number of the ν e appearance signal decreases. We will reject the wrong mass hierarchy with |∆χ 2 MH | > 8 (T2KK) and > 3 (T2KO) for any CP phases and sin 2 θ 23 > 0.4. In the most sensitive region around δ CP = −90 • for the normal hierarchy case, we may reject the wrong mass hierarchy with |∆χ 2 MH | > 20 in the T2KK experiment and > 14 in the T2KO experiment with sin 2 θ 23 > 0.4. For the inverted hierarchy case, the most sensitive region is around δ CP = 90 • , and we may reject the wrong mass hierarchy with |∆χ 2 MH | > 18 in the T2KK experiment and > 14 in the T2KO experiment with sin 2 θ 23 > 0.4. Figure 11. The uncertainty of CP phase measurements as functions of the CP phase when sin 2 θ 23 = 0.5, and mass hierarchy is known to be the normal hierarchy. The solid-red, solid-blue, solid-green, dashed-blue and dashed-red curves are for the ν µ -ν µ focusing beam ratio of 4.5 : 0.5, 3.5 : 1.5, 2.5 : 2.5, 1.5 : 3.5 and 0.5 : 4.5 ×10 21 POT with the proton energy of 40 GeV, respectively.

Sensitivity to the CP phase measurement
In this section, we discuss the sensitivities of the T2KK and T2KO experiments to the CP phase measurement, comparing to an experiment where a 100 kton detector is placed at the Kamioka site in addition to the 22.5 kton SK detector, which is called as the T2K 122 experiment in this study. Comparison with this experiment will clearly show the dependence of the CP phase sensitivity on the baseline length. We put emphasis on the effects of including theν µ focusing beam. In Fig. 11, we show the uncertainties of the CP phase measurements as functions of the true CP phase, δ true CP , for the four experiments: (a) T2KK with 3.0 • OAB at the SK and 0.5 • OAB at the Kr, (b) T2KK with 2.5 • OAB at the SK and 1.0 • OAB at the Kr, (c) T2KO and (d) T2K 122 . The uncertainty is defined by the deviation of the test δ CP from the true δ CP which gives (∆χ 2 ) min = 1. The curves correspond to the different ν µ -ν µ focusing beam ratios: 4.5 : 0.5 (solid-red), 3.5 : 1.5 (solid-blue), 2.5 : 2.5 (solid-green), 1.5 : 3.5 (dashed-blue) and 0.5 : 4.5 (dashed-red) in the unit of 10 21 POT. The uncertainty of the CP phase measurements is smallest around δ CP = 0 • and 180 • . This is because the uncertainty mainly reflects the sin δ CP dependence of the signal event number since the magnitude of the sin δ CP term is larger than that of the cos δ CP term in Eq. (2.6b) on average.
On the other hand, the uncertainty is largest around δ CP = ±60 • and ±120 • as clearly shown in the T2K 122 experiment, Fig. 11(d); for the T2KK and T2KO experiments, the low sensitivity regions slightly shift from ±60 • and ±120 • due to the matter effects [54]. This low sensitivity reflects the degeneracy between δ CP and π −δ CP in sin δ CP . To resolve the degeneracy, we need information of the cos δ CP term, which becomes large around tails of oscillation peaks. The T2KK and T2KO experiments observe up to the second peak of the ν µ → ν e andν µ →ν e oscillations, while the T2K 122 experiment only observes the first peak (see Figs. 5 -7). Therefore, the former experiments are more sensitive to the cos δ CP term and can measure the CP phase more accurately around those low sensitive regions.
The effects of including theν µ focusing beam can be understood in terms of correlations of the oscillation parameters between the ν µ andν µ focusing beams. We illustrate this point taking the T2K 122 experiment as an example. In Fig. 12, we show the ∆χ 2 minimums and pull factors of the oscillation parameters for the different ν µ -ν µ focusing beam ratios as functions of the test δ CP . The pull factor of a fitting parameter X is defined as (X fit − X input )/δX, where δX is the uncertainty of the parameter. In the upper-left panel (a), solid-red, dashed-red and dash-dotted blue curves show the ∆χ 2 minimums for the ν µ -ν µ focusing beam ratios of 5 : 0, 0 : 5 and 2.5 : 2.5 (×10 21 POT with the proton energy of 40 GeV), respectively. Here, the true CP phase is assumed to be 0 • , and the mass hierarchy is known to be the normal hierarchy. In the lower plot (c), we show the corresponding pull factors of the oscillation parameters for ν µ :ν µ = 5 : 0 (upper-half panel) and 0 : 5 (lower-half panel). We see that each pull factor of sin 2 2θ 13 and sin 2 θ 23 shows clear anti-correlation between the ν µ andν µ focusing beams, i.e., the sign of the each pull factor is opposite between those focusing beams. This is because the sign of those pull factors is mainly related to the sign of the sin δ CP term in the ν µ → ν e oscillation probability (Eq. (2.6b)), which is inverted for the anti-neutrino beam case. Thus, inclusion of theν µ focusing beam would restrict the deviations of sin 2 2θ 13 and sin 2 θ 23 , resulting in the larger ∆χ 2 minimum for the 2.5 : 2.5 beam ratio than 5 : 0 in the upper panel (a). For the δ true CP = 60 • case (right panels in Fig. 12), on the other hand, the anti-correlation of the pull-factors is not so evident for 60 • δ test CP 120 • resulting in the rather reduction of the accuracy of the CP phase measurement when includingν µ focusing beams. Similar situation occurs for δ true CP ∼ −60 • and ±120 • as well. In Fig. 13, we show the uncertainties of the CP phase measurements for the inverted hierarchy case. The uncertainties show similar dependences on the ν µ -ν µ focusing beam ratio as in the normal hierarchy case, showing that the 3.5 : 1.5 -1.5 : 3.5 beam ratio give the smallest uncertainty except for δ CP ∼ ±60 • and ±120 • . Using the 2. We also show the sensitivities to the CP phase measurements in the test-δ CP vs. true-δ CP plane in Fig. 14 when sin 2 θ 23 = 0.5 and the mass hierarchy is known to be the normal hierarchy. The ν µ -ν µ focusing beam ratio is fixed at 2.5 : 2.5. The solid-red, dashed-blue and dash-dottedgreen contours show (∆χ 2 ) min = 1, 4, 9, respectively, for (a) T2KK with 3.0 • OAB at the SK and 0.5 • OAB at the Kr, (b) T2KK with 2.5 • OAB at the SK and 1.0 • OAB at the Kr, (c) T2KO and (d) T2K 122 experiments. We see that both δ test CP = 0 • and 180 • are rejected with the significance of (∆χ 2 ) min > 9 for −100 • < δ CP < −60 • and 70 • < δ CP < 120 • by the T2KO experiment and for −120 • < δ CP < −60 • and 60 • < δ CP < 120 • by the T2K 122 experiment. The T2KK experiment has less sensitivity to the CP violation than the T2KO and T2K 122 experiments and rejects both δ test CP = 0 • and 180 • with the significance of 9 > (∆χ 2 ) min > 4 for −120 • (−135 • ) < δ CP < −45 • and 45 • < δ CP < 140 • for the normal (inverted) hierarchy case. The low sensitivity regions due to the δ CP and π − δ CP degeneracy of sin δ CP can be seen around δ test CP = π − δ CP . We also show the sensitivity plots for the inverted hierarchy case in Fig. 15. The sensitivity is similar to that of the normal hierarchy case with slight difference due to the relative sign change between the sin δ CP and cos δ CP terms in the ν µ → ν e andν µ →ν e oscillation probability, Eq (2.6b).
The sin 2 θ 23 dependence of the sensitivity to the CP phase measurements is shown in Fig. 16. These are the same sensitivity plots as Fig. 14(a) but for sin 2 θ 23 = 0.4 (left plot) and 0.6 (right plot) for the T2KK experiment with 3.0 • OAB at the SK and 0.5 • OAB at the Kr. It is assumed that the mass hierarchy is known to be the normal hierarchy. The sensitivity is better for smaller sin 2 θ 23 as shown in the figure. This is because the coefficients of the sin δ CP and cos δ CP in the ν µ → ν e oscillation probability, Eq. (2.6b), is proportional to 1/ tan θ 23 . The percentages of the Figure 13. Same as Fig. 11 but for the inverted hierarchy case.

Summary and conclusion
In this paper, we have revisited the previous analysis of Ref. [22,24,25] on the sensitivities to the mass hierarchy determination and leptonic CP phase measurements of the Tokai-to-Kamioka-and-Korea (T2KK) [15][16][17][18][19][20][21][22][23][24][25][26] and Tokai-to-Kamioka-and-Oki (T2KO) experiments [25,27], putting emphasis on the ν µ andν µ focusing beam ratio with dedicated estimation of backgrounds. We place a Super-Kamiokande (SK) type waterČerenkov detector of 100 kton fiducial volume in Korea (T2KK) or Oki island (T2KO) at 1000 km and 653 km away from the J-PARC neutrino facility, respectively. The neutral current (NC) single-π 0 background and its uncertainty are estimated by using the realistic π 0 rejection probability based on the POLfit algorithm [42], taking into account the coherent π 0 production processes, which is neglected in the previous analysis [22], and including the uncertainty of axial masses in the neutrino-nucleus interaction model [32,33]. The sensitivities are then evaluated using the standard χ 2 analysis. We found that the wrong mass hierarchy is rejected with |∆χ 2 MH | > 10 in the T2KK and |∆χ 2 MH | > 3 in the T2KO experiment for any CP phases when sin 2 θ 23 = 0.5, using the ν µ andν µ focusing beam ratio between 3 : 2 and 2.5 : 2.5 (in unit of 10 21 POT with the proton energy of 40 GeV). It should be noted that the |∆χ 2 MH | quoted in this study is regarded as the average sensitivity expected for an experiment because we neglect the statistical fluctuations in input data set. Although a rigorous interpretation of the |∆χ 2 MH | needs dedicated statistical Figure 14. The sensitivities to the CP phase measurements in the test-δ CP vs. true-δ CP plane when sin 2 θ 23 = 0.5, and mass hierarchy is known to be the normal hierarchy. (a) T2KK with 3.0 • OAB at the SK, (b) T2KK with 2.5 • OAB at the SK, (c) T2KO and (d) T2K 122 . The ν µ -ν µ focusing beam ratio is fixed at 2.5 : 2.5 ×10 21 POT with the proton energy of 40 GeV. The solid-red, dashed-blue and dash-dotted-green contours show (∆χ 2 ) min = 1, 4, 9, respectively.
consideration, above |∆χ 2 MH | may be roughly interpreted as the 80% probabilities of determining the mass hierarchy with > 2.6 σ for T2KK and > 1.3 σ for T2KO, respectively, assuming the Gaussian distribution for the ∆χ 2 MH in the T2KK and T2KO experiments (See Fig.2 in Ref. [49] for the interpretation).
In the most sensitive region around δ CP ∼ −90 • for the normal hierarchy case, we reject the wrong mass hierarchy with |∆χ 2 MH | ∼ 32 in the T2KK experiment (3.0 • OAB at the SK) and with |∆χ 2 MH | ∼ 20 in the T2KO experiment, using the 3 : 2 -2.5 : 2.5 beam ratio. These |∆χ 2 MH | correspond to the 80% probabilities of the mass hierarchy determination with > 4.9 σ for the T2KK experiment and with > 3.8 σ for the T2KO experiment. On the other hand, for the inverted hierarchy case, we reject the wrong mass hierarchy with |∆χ 2 MH | ∼ 30 in the T2KK experiment (3.0 • OAB at the SK) and with |∆χ 2 MH | ∼ 18 in the T2KO experiment around δ CP ∼ 90 • , using the same beam ratio as the normal hierarchy case. These |∆χ 2 MH | correspond to the 80% probabilities of the mass hierarchy determination with > 4.7 σ for the Figure 15. Same as Fig. 14 but for the inverted hierarchy case. Figure 16. Same as Fig. 14 (a) but for sin 2 θ 23 = 0.4 (left) and sin 2 θ 23 = 0.6 (right).
T2KK experiment and with > 3.6 σ for the T2KO experiment. These sensitivities are obtained for sin 2 θ 23 = 0.5 and enhanced (reduced) by 30% -40% in |∆χ 2 MH | for sin 2 θ 23 = 0.6 (0.4). We also examined the sensitivity to the CP phase measurements. The ν µ andν µ focusing beam ratio between 3.5 : 1.5 and 1.5 : 3.5 give the smallest uncertainty for most of the CP phases. Employing the 2.5 : 2.5 beam ratio, the T2KK and T2KO experiments measure the CP phase with the uncertainty of ∼ 20 • -50 • (T2KK with 3.0 • OAB at the SK), ∼ 20 • -45 • (T2KK with 2.5 • OAB at the SK and T2KO) , depending on the CP phase. We can measure the CP phase most accurately around δ CP ∼ 0 • and ∼ 180 • , while the uncertainty is largest around δ CP ∼ ±60 • and ∼ ±120 • . A long baseline is helpful to improve the CP phase measurements around those large uncertainty regions. The mass hierarchy and sin 2 θ 23 dependence of the CP phase measurements are not so large. The CP violation in the lepton sector is detected with (∆χ 2 ) min > 9 for −100 • < δ CP < −60 • and 70 • < δ CP < 120 • by the T2KO experiment, while the T2KK experiment detects the CP violation only with (∆χ 2 ) min > 4. In either experiments, we need larger statistics to establish the CP violation in wide range of the CP phases.
As discussed in this paper, the T2KK and T2KO experiments can improve their sensitivity to both the mass hierarchy determination and leptonic CP phase measurement using ν µ and ν µ focusing beams with 3 : 2 -2.5 : 2.5 beam ratio. This improvement is significant especially for the mass hierarchy determination, lifting the highest sensitivities in the T2KK (both 2.5 • and 3.0 • OAB at the SK) and T2KO experiments. The lowest sensitivities are improved in the T2KK experiment with 2.5 • OAB at the SK, while the improvement is not so evident in the other experiments. The T2KK experiment allows us to determine the mass hierarchy and measure the leptonic CP phase simultaneously. The T2KO experiment also has the sensitivity to the CP phase measurement, while its physics potential for the mass hierarchy determination is not as good as that of the T2KK experiment.

B.1.2 proton interaction
The E rec distributions of theν µ induced CCQE events via interactions with protons can be parameterized by up to two Gaussians, where Θ is a step function. The weight factors r α n , variances σ α n (MeV) and energy shifts δE α n (MeV) are expressed as