# Revisiting T2KK and T2KO physics potential and \(\nu _\mu \)–\({\bar{\nu }}_\mu \) beam ratio

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## Abstract

We revisit the sensitivity study of the Tokai-to-Kamioka-and-Korea (T2KK) and Tokai-to-Kamioka-and-Oki (T2KO) proposals where a water \(\check{\mathrm{C}}\)erenkov 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 (\(\delta _{\scriptscriptstyle \mathrm{CP}}\)). 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 units 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 the leptonic CP phase measurement simultaneously with the preferred beam ratio being between 3:2–2.5:2.5 (\({\times } 10^{21}\)POT).

## 1 Introduction

After the accurate measurements of \(\sin ^2 2\theta _{13}\) by DayaBay [1, 2, 3, 4, 5, 6], Reno [7, 8] and Double Chooz [9, 10, 11, 12, 13] experiments, determination of the neutrino mass hierarchy and CP violating phase in the Maki–Nakagawa–Sakata (MNS) mixing matrix [14] has been the next targets in neutrino physics.

Ideas of extending the Tokai-to-Kamioka (T2K) experiment with additional water \(\check{\mathrm{C}}\)erenkov 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]) have been proposed to address those questions.^{1} It has been shown that the T2KK experiment with a 100 kton fiducial-volume 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 \(2.5^\circ \)–\(3.0^\circ \) 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 \(\nu _\mu \) focusing beam in a simple manner, ignoring the effects of neutral-current (NC) \(\pi ^0\) 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 of \({\bar{\nu }}_\mu \) 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 \(\nu _\mu \) and \({\bar{\nu }}_\mu \) oscillations. The impacts of including the \({\bar{\nu }}_\mu \) focusing beam in the T2KK experiment was studied in Ref. [24]. The authors considered the running ratio of the \(\nu _\mu \) and \({\bar{\nu }}_\mu \) focusing beams of 5:0 and 2.5:2.5 in the units of protons on target (POT) and argued that including the \({\bar{\nu }}_\mu \) focusing beam improves the sensitivity to the mass-hierarchy determination significantly. The impact 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 \(2.5^\circ \) off-axis beams with the beam power of 4 MW and the total running time of 8 years. The 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 \(\pi ^0\) 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 \(\nu _\mu \)–\({\bar{\nu }}_\mu \) 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 \(\nu _\mu \)–\({\bar{\nu }}_\mu \) focusing beam ratio systematically with dedicated estimation of backgrounds. Especially, the treatment of the NC \(\pi ^0\) backgrounds is improved in this analysis. The NC \(\pi ^0\) backgrounds is estimated using a realistic \(\pi ^0\) rejection probability based on the POLfit (Pattern Of Light fitter) algorithm [31], and the contribution form the coherent \(\pi ^0\) production process is taken into account, which is neglected in the previous analysis [22]. The uncertainty of the NC \(\pi ^0\) 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 Sect. 2, our analysis details are discussed in Sect. 3. Results for the sensitivity of the T2KK and T2KO experiments to the mass-hierarchy determination and CP phase measurements are presented in Sects. 4 and 5, and our main conclusions are summarized in Sect. 6.

## 2 Simulation details of T2KK and T2KO experiments

In this section, we fix our notation and introduce useful approximated formulas for the \(\nu _\mu \rightarrow \nu _\mu \) and \(\nu _\mu \rightarrow \nu _e\) 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-\(\pi ^0\) background and its uncertainty, the secondary neutrino beam backgrounds, and miss-identified muon/electron backgrounds.

### 2.1 Neutrino oscillations in matter

We briefly review the neutrino oscillation probabilities in matter, presenting analytic approximations for the \(\nu _\mu \rightarrow \nu _\mu \) (\(\nu _\mu \) disappearance) and \(\nu _\mu \rightarrow \nu _e\) (\(\nu _e\) appearance) oscillation modes, which are useful for understanding the physics potential of the T2KK and T2KO experiments qualitatively.

*U*is the Maki–Nakagawa–Sakata (MNS) [14] matrix, which can be parameterized with the three mixing angles, \(\theta _{12}, \theta _{13}, \theta _{23}\), and three phases, \(\delta _{\scriptscriptstyle \mathrm{CP}}\), \(\phi _1\), \(\phi _2\) [34]. Among them, two phases can be eliminated in lepton number conserving processes, remaining one relevant phase, \(\delta _{\scriptscriptstyle \mathrm{CP}}\), to neutrino oscillation experiments. The definition regions of the four parameters are chosen as \(0\le \theta _{12},\theta _{13},\theta _{23}\le \pi /2\), and \(-\pi \le \delta _{\scriptscriptstyle \mathrm{CP}}\le \pi \).

*E*is observed as a flavor eigenstate \(|\nu _\beta \rangle \) after traveling a distance

*L*in the matter of density \(\rho (x)\) \((0<x<L)\) is given by

*a*(

*x*) / 2

*E*is the effective potential due to electrons in matter as

*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, \(\rho (x) \simeq \bar{\rho }\), and so as

*a*(

*x*) in Eq. (2.3), \(a(x) \simeq \bar{a}\). Then the oscillation probability, \( P_{\nu _\alpha \rightarrow \nu _\beta }\), can be expressed compactly by using

*x*-independent eigenvalues, \(\lambda _i\), and a unitary matrix, \(\tilde{U}\), as

*aL*/ 4

*E*. The corresponding probabilities for anti-neutrino oscillations can be obtained from the above expressions by reversing the sign of the matter effect term (\(a \rightarrow -a\)) and the CP phase (\(\delta _{\scriptscriptstyle \mathrm{CP}}\rightarrow -\delta _{\scriptscriptstyle \mathrm{CP}}\)). 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 \(L/E \sim \mathcal{O}(10^2{-}10^3) [1/\mathrm{eV}^2]\).

The \(\nu _e\) appearance mode plays a more important role in determining the mass hierarchy (i.e., the sign of \(\Delta _{31}\)) than the \(\nu _\mu \) 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 \(A^\mu \simeq 0\). On the other hand, the disappearance mode is important in constraining the \(\theta _\mathrm{23}\) mixing angle, which still has large uncertainty [34]. The \(\nu _e\) appearance mode also has sensitivity to the CP phase. It is sensitive to the sine of \(\delta _{\scriptscriptstyle \mathrm{CP}}\) around the oscillation peaks, mainly through the \(A^e\) parameter; on the other hand, it is sensitive to the cosine of \(\delta _{\scriptscriptstyle \mathrm{CP}}\) between oscillation maxima and minima, mainly through the \(B^e\) parameter. Therefore, if we try to obtain the full information of the \(\delta _{\scriptscriptstyle \mathrm{CP}}\), it is not enough to observe just around the first oscillation peak, as we will see later.

### 2.2 Experimental setups

We use the \(\nu _{\mu }\) and \({\bar{\nu }}_{\mu }\) 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 \(\nu _{\mu } \,({\bar{\nu }}_{\mu })\) focusing beams include the primary, \(\nu _{\mu } \,({\bar{\nu }}_{\mu })\), and secondary, \({\bar{\nu }}_{\mu } \,(\nu _{\mu })\), \(\nu _e\), \({\bar{\nu }}_e\), 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 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\(^\circ \) off-axis angle at the SK, a Kr detector receives the \({\sim } 1^\circ \) off-axis beam (OAB); the case of \(3.0^\circ \) OAB at the SK is also investigated, corresponding to the \(0.5^\circ \) 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 and CP phase measurements significantly [25], and we only consider the \(2.5^\circ \) off-axis angle at the SK for the T2KO experiment, corresponding to \(0.9^\circ \) OAB at the Oki detector [25].

Summary of the parameters related to detectors at Kamioka (SK), Oki island (Oki) and Korea (Kr). *L* is the baseline length between the J-PARC and a detector, FV is the fiducial volume of a detector, \(\bar{\rho }\) 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

### 2.3 Signal events

*e*), from the \(\nu _\mu \rightarrow \nu _\mu \) and \(\nu _\mu \rightarrow \nu _e\) oscillation modes and their charge conjugated modes as signal events. The CCQE events are identified as events with only one \(\check{\mathrm{C}}\)erenkov ring from an electron or muon where the visible energy of the ring is required to be larger than 200 MeV. Since the neutrino beam direction at the far detector is understood well in long-baseline experiments, we can reconstruct the incoming neutrino energy for the CCQE events by [39]

*X*-type neutrino–nucleus interaction (\(X=\) CCQE, non-CCQE) are calculated as

*D*, \(P_{\nu _{\alpha }\rightarrow \nu _{\beta }}\) is the neutrino oscillation probability including the matter effects with the mean matter density \(\bar{\rho }^D\), and \(N_Z\) is the number of the

*Z*nuclei (hydrogen (

*H*) or oxygen (

*O*)) in the detector. \(\hat{\sigma }^{X}_{\nu _\beta Z}\) is the cross section of the

*X*-type \(\nu _\beta \)–

*Z*interaction after imposing a CCQE selection cuts. The smearing function \(S^X_{\nu _\beta Z}(E_\nu , E_\mathrm{rec})\) returns the probability that the energy \(E_\mathrm{rec}\) is reconstructed from an event induced by an incoming neutrino with the energy \(E_\nu \), taking into account the Fermi motion of the target nucleons and detector resolutions. The detection efficiency of \(\check{\mathrm{C}}\)erenkov rings and the electron/muon identification efficiencies will be discussed in Sects. 2.4 and 3.

The momentum and angular resolutions for muons and electrons at the SK detector [41]

\({\delta p}/{p}~~(\%)^{}_{}\) | \(\delta \theta \) (\(^\circ \)) | |
---|---|---|

\(\mu \) | \(1.7 +0.7/{\sqrt{p\hbox {[GeV]}}}\) | \(1.8^\circ \) |

| \(0.6+2.6/{\sqrt{p\hbox {[GeV]}}}\) | \(3.0^\circ \) |

### 2.4 Background events

In this section we discuss the sources of background events taken into account in this study: neutral-current (NC) single-\(\pi ^0\) events, secondary neutrinos in the \(\nu _\mu \) and \({\bar{\nu }}_\mu \) focusing beams and misidentified muon and electron events.

*D*(=SK, Oki, Kr) via the

*Y*-type neutrino–nucleus interaction (\(Y=\) NCQE, NCRes, NCCoh and NCDI) induced by the \(\nu _\alpha \) component of the \(\nu _\mu \) or \({\bar{\nu }}_\mu \) focusing beam are calculated as

*Y*-type \(\nu \)–

*Z*interaction after imposing the NC single-\(\pi ^0\) selection criteria:

*x*[GeV], as

^{2}[31]. The reference data and the fitted function are shown in Fig. 3b. The misidentification probability is kept less than 0.2 for \(p_{\pi ^0} < 0.6\) GeV, where the \(\pi ^0\) backgrounds mostly distributes. The background events are then selected from the simulated NC single-\(\pi ^0\) events according to the misidentification probability, and the reconstructed energy of each background event is calculated with Eq. (2.8), assuming the misidentified \(\pi ^0\) to be an electron.

The NC single-\(\pi ^0\) 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-\(\pi ^0\) backgrounds is modeling of neutrino–nucleus interactions. In Fig. 4 we show the reconstructed energy distributions of the NC single-\(\pi ^0\) backgrounds calculated with Nuance for different neutrino–nucleus interactions. We see that the \(\pi ^0\) backgrounds mainly distribute in low-energy region, where the contributions from resonant and coherent single-\(\pi ^0\) production processes dominate. These processes are implemented in Nuance based on the Rein–Sehgal 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: \(m_A^\mathrm{Res} = 1.1 \pm 0.11\) GeV [44] and \(m_A^\mathrm{Coh} = 1.03 \pm 0.28\) 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-\(\pi ^0\) backgrounds, respectively.

Another major source of uncertainty of the NC single-\(\pi ^0\) backgrounds arises from the \(\pi ^0\) misidentification probability, Eq. (2.13). The T2K collaboration estimated 10.8% uncertainty in the NC-\(\pi ^0\) background estimation due to the POLfit algorithm [42]. Since our modeling of the \(\pi ^0\) misidentification probability is based on the POLfit algorithm, we assign \(11\%\) uncertainty to the normalization of the NC single-\(\pi ^0\) backgrounds due to the \(\pi ^0\) misidentification.

All in all, we include the 13 and 15% normalization uncertainties for the NC resonant and coherent single-\(\pi ^0\) backgrounds, respectively, and 11% normalization uncertainty for the total NC single-\(\pi ^0\) backgrounds. This treatment allows independent normalization corrections for the resonant and coherent NC single-\(\pi ^0\) backgrounds.

The \(\nu _\mu \,({\bar{\nu }}_\mu )\) focusing beams contain not only \(\nu _{\mu }\,({\bar{\nu }}_{\mu })\) but also other neutrino flavors, \(\nu _e\), \({\bar{\nu }}_e\) and \({\bar{\nu }}_\mu \, (\nu _\mu )\), secondary neutrino beams. Especially, for the \(\nu _\mu \rightarrow \nu _e\) and \({\bar{\nu }}_\mu \rightarrow {\bar{\nu }}_e\) oscillation modes, the \(\nu _e\) and \({\bar{\nu }}_e\) secondary beams become major background sources. We simulate these secondary-neutrino events in the same way as the signal events described in Sect. 2.3.

## 3 \(\chi ^2\) Analysis

*e*-like event numbers, respectively, in the

*i*th bin of the \(E_\mathrm{rec}\) distributions measured at a detector \(D\,(=\)SK, Oki, Kr) from the \(\nu _{\mu }\) focusing beam, and \((\overline{N}_{\mu ,D}^{i})^\mathrm{input}\) and \((\overline{N}_{e,D}^{i})^\mathrm{input}\) are those from the \({\bar{\nu }}_{\mu }\) focusing beam. The summation runs over all the \(E_\mathrm{rec}\) bins from 0.4 to 5.0 GeV at both the SK and the far (Oki or Kr) detectors.

*e*-like event numbers are calculated using the CC signal and NC single-\(\pi ^0\) background events (\(N^{i,X}_D\) and \(N^{i,Y}_{\pi ^0,D}\) defined by Eqs. (2.9) and (2.11)) as

*e*-like events are shown in Figs. 5, 6 and 7, which are calculated using the input parameter values in Table 3.

The systematic and physical parameters in the \(\chi ^2\) function, Eq. (3.1), where *D* stands for the detector site (SK, Oki and Kr), and \(\nu _\alpha \) or \(\nu _\beta \) denotes neutrino species (\(\nu _\mu ,{\bar{\nu }}_\mu ,\nu _e\) and \({\bar{\nu }}_e\)). These input values and uncertainties are used in the sensitivity study otherwise mentioned

Systematic parameters ( | Input value (\(S_\mathrm{input}\)) | Uncertainty (\(\delta S\)) |
---|---|---|

Fiducial volume of detectors (\(f_V^D\)) | 1.00 | 0.03 [22] |

Neutrino flux at a detector (\(f_{\nu _{\alpha }}^D)\) | 1.00 | 0.03 [22] |

CCQE cross sections (\(f^\mathrm{CCQE}_{\nu _\beta }\)) | 1.00 | 0.03 [22] |

Non-CCQE cross sections (\(f^\mathrm{nonCCQE}_{\nu _\beta }\)) | 1.00 | 0.20 [22] |

Misidentified NC \(\pi ^0\) events (\(f_{\pi ^0}^\mathrm{NC}\)) | 1.00 | 0.11 |

Misidentified NC resonant \(\pi ^0\) events (\(f^\mathrm{NCRes}_{\pi ^0}\)) | 1.00 | 0.13 |

Misidentified NC coherent \(\pi ^0\) events (\(f^\mathrm{NCCoh}_{\pi ^0}\)) | 1.00 | 0.15 |

Detection efficiency of electron \(\check{\mathrm{C}}\)erenkov rings (\(\epsilon _{e}^D)\) | 0.90 | 0.05 [22] |

Detection efficiency of muon \(\check{\mathrm{C}}\)erenkov rings (\(\epsilon _{\mu }^D)\) | 1.00 | 0.01 [22] |

\(\mu \)-to- | 0.01 | 0.01 [22] |

| 0.01 | 0.01 [22] |

Physical parameters ( | Input value (\(P_\mathrm{input}\)) | Uncertainty (\(\delta P\)) |
---|---|---|

\(\sin ^2 2\theta _{12}\) | 0.875 | 0.024 [34] |

\(\sin ^2 2\theta _{13}\) | 0.095 [34] | 0.005 [46] |

\(\sin ^2 \theta _{23}\) | 0.5 | 0.1 [34] |

\(\delta m^2_{21}\mathrm{[eV]}^2\) | \(7.50\times 10^{-5}\) | \(0.20\times 10^{-5}\) [34] |

\(|\delta m^2_{32}|\mathrm{[eV]}^2\) | \(2.32\times 10^{-3}\) | \(0.10\times 10^{-3}\) [34] |

\(\delta _{\scriptscriptstyle \mathrm{CP}}\) | \(0^\circ \) | – |

\(\bar{\rho }^\mathrm{SK}[\mathrm{g}/\mathrm{cm}^3]\) | 2.60 | 6% [25] |

\(\bar{\rho }^\mathrm{Oki}[\mathrm{g}/\mathrm{cm}^3]\) | 2.75 | 6% [25] |

\(\bar{\rho }^\mathrm{Kr}[\mathrm{g}/\mathrm{cm}^3]\) | 2.9 | 6% [25] |

As shown in those figures, T2KK and T2KO experiments can observe up to the second peak of the \(\nu _\mu \rightarrow \nu _e\) and \({\bar{\nu }}_\mu \rightarrow {\bar{\nu }}_e\) oscillations due to their long-baseline length, while the T2K experiment only observes 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 \(\sin \delta _{\scriptscriptstyle \mathrm{CP}}\) and \(\cos \delta _{\scriptscriptstyle \mathrm{CP}}\).

*e*-like event numbers, \((N_{\mu ,D}^{i})^\mathrm{fit}\) and \((N_{e,D}^{i})^\mathrm{fit}\), are calculated as

*D*(\(=\)SK, Oki, Kr), respectively. \(f_{\nu _\beta }^X\) is the normalization factor for the CC cross section of a neutrino flavor \(\nu _\beta \) via a \(X (=\)CCQE or non-CCQE) interaction. \(f_{\pi ^0}^\mathrm{NCRes}\) and \(f_{\pi ^0}^\mathrm{NCCoh}\) are the normalization factors for the NC cross sections of resonant and coherent single-\(\pi ^0\) production processes, respectively, while \(f_{\pi ^0}^\mathrm{NC}\) is the overall normalization factor for the NC single-\(\pi ^0\) backgrounds, mainly reflecting the uncertainty of the \(\pi ^0\) misidentification probability, Eq. (2.13). These factors are varied in the minimization of the \(\chi ^2\) function, and their deviation from unity measures systematic uncertainties.

^{3}The

*n*-\(\sigma \) confidence interval of the CP phase measurement, \([\delta _\mathrm{CP}^a,\delta _\mathrm{CP}^b]_{n\sigma } \,\,(\delta _\mathrm{CP}^{a} < \delta _\mathrm{CP}^{b})\), is then estimated such that

## 4 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 \(\nu _\mu \)–\({\bar{\nu }}_\mu \) focusing beam ratio and \(\sin ^2 \theta _{23}\).

In order to minimize the reduction of the sensitivities, \(\nu _\mu : {\bar{\nu }}_\mu = 4{:}1\) is the best ratio for both OAB cases. Comparing the lowest \(|\overline{\Delta \chi ^2_\mathrm{MH}}|\) in the whole range of the CP phase, \(\nu _\mu : {\bar{\nu }}_\mu = 4{:}1\) is the best ratio for the \(3.0^\circ \) OAB at the SK, and 3:2–2:3 are the best for the \(2.5^\circ \) 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\(^\circ \) OAB at the SK. For the \(2.5^\circ \) 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 \(\nu _\mu \)–\({\bar{\nu }}_\mu \) focusing beam ratio that gives the best sensitivity for any \(\delta _{\scriptscriptstyle \mathrm{CP}}\) values and mass hierarchies, the beam ratio between 4:1 and 2.5:2.5 would be a reasonable choice for both \(2.5^\circ \) and \(3.0^\circ \) OAB at the SK.

## 5 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 the \(\mathrm{T2K}_{122}\) 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 the \({\bar{\nu }}_\mu \) focusing beam.

*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 units of \(10^{21}\) POT. The uncertainty of the CP phase measurements is smallest around \(\delta _{\scriptscriptstyle \mathrm{CP}}= 0^\circ \) and \(180^\circ \). This is because the uncertainty mainly reflects the \(\sin \delta _{\scriptscriptstyle \mathrm{CP}}\) dependence of the signal event number since the magnitude of the \(\sin \delta _{\scriptscriptstyle \mathrm{CP}}\) term is larger than that of the \(\cos \delta _{\scriptscriptstyle \mathrm{CP}}\) term in Eq. (2.6b) on average.

On the other hand, the uncertainty is largest around \(\delta _{\scriptscriptstyle \mathrm{CP}}= \pm 60^\circ \) and \(\pm 120^\circ \) as clearly shown in the \(\mathrm{T2K}_{122}\) experiment, Fig. 11d; for the T2KK and T2KO experiments, the low sensitivity regions slightly shift from \({\pm } 60^\circ \) and \(\pm 120^\circ \) due to the matter effects [54]. This low sensitivity reflects the degeneracy between \(\delta _{\scriptscriptstyle \mathrm{CP}}\) and \(\pi -\delta _{\scriptscriptstyle \mathrm{CP}}\) in \(\sin \delta _{\scriptscriptstyle \mathrm{CP}}\). To resolve the degeneracy, we need information of the \(\cos \delta _{\scriptscriptstyle \mathrm{CP}}\) term, which becomes large around tails of oscillation peaks. The T2KK and T2KO experiments observe up to the second peak of the \(\nu _\mu \rightarrow \nu _e\) and \({\bar{\nu }}_\mu \rightarrow {\bar{\nu }}_e\) oscillations, while the \(\mathrm{T2K}_{122}\) experiment only observes the first peak (see Figs. 5, 6, 7). Therefore, the former experiments are more sensitive to the \(\cos \delta _{\scriptscriptstyle \mathrm{CP}}\) term and can measure the CP phase more accurately around those low sensitive regions.

As for the \(\nu _\mu \)–\({\bar{\nu }}_\mu \) focusing beam ratio, \(\nu _\mu {:}{\bar{\nu }}_\mu = 3.5{:}1.5{-}1.5{:}3.5\) give the smallest uncertainty for most of the CP phases, except for the low sensitivity region, where the ratio of 4.5:0.5 gives the best accuracy. Using the 2.5:2.5 beam ratio, for example, the T2KK and T2KO experiments measure the CP phase with the uncertainty of \({\sim } 20^\circ \)–\(50^\circ \) (T2KK with \(3.0^\circ \) OAB at the SK), \({\sim } 20^\circ \)–\(45^\circ \) (T2KK with \(2.5^\circ \) OAB at the SK and T2KO) and \({\sim } 15^\circ \)–\(70^\circ \) (\(\mathrm{T2K}_{122}\)), depending on the CP phase.

*X*is defined as \((X^\mathrm{fit} -X^\mathrm{input})/\delta X\), where \(\delta X\) is the uncertainty of the parameter. In the upper-left panel (a), solid-red, dashed-red and dash-dotted blue curves show the \(\Delta \chi ^2\) minimums for the \(\nu _\mu \)–\({\bar{\nu }}_\mu \) focusing beam ratios of 5:0, 0:5 and 2.5:2.5 \((\times 10^{21}\) POT with the proton energy of 40 GeV), respectively. Here, the true CP phase is assumed to be \(0^\circ \), 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 \(\nu _\mu {:} {\bar{\nu }}_\mu = 5{:}0\) (upper-half panel) and 0:5 (lower-half panel). We see that each pull factor of \(\sin ^2 2\theta _{13}\) and \(\sin ^2 \theta _{23}\) shows clear anti-correlation between the \(\nu _\mu \) and \({\bar{\nu }}_\mu \) 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 \(\sin \delta _{\scriptscriptstyle \mathrm{CP}}\) term in the \(\nu _\mu \rightarrow \nu _e\) oscillation probability (Eq. (2.6b)), which is inverted for the anti-neutrino beam case. Thus, inclusion of the \({\bar{\nu }}_\mu \) focusing beam would restrict the deviations of \(\sin ^2 2\theta _{13}\) and \(\sin ^2 \theta _{23}\), resulting in the larger \(\Delta \chi ^2\) minimum for the 2.5:2.5 beam ratio than 5:0 in the upper panel (a). For the \(\delta _{\scriptscriptstyle \mathrm{CP}}^\mathrm{true} = 60^\circ \) case (right panels in Fig. 12), on the other hand, the anti-correlation of the pull factors is not so evident for \(60^\circ \lesssim \delta _\mathrm{CP}^\mathrm{test} \lesssim 120^\circ \) resulting in the rather reduction of the accuracy of the CP phase measurement when including \({\bar{\nu }}_\mu \) focusing beams. A similar situation occurs for \(\delta _{\scriptscriptstyle \mathrm{CP}}^\mathrm{true} \sim -60^\circ \) and \(\pm 120^\circ \).

## 6 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 \(\nu _\mu \) and \({\bar{\nu }}_\mu \) focusing beam ratio with dedicated estimation of backgrounds. We place a Super-Kamiokande (SK) type water \(\check{\mathrm{C}}\)erenkov 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-\(\pi ^0\) background and its uncertainty are estimated by using the realistic \(\pi ^0\) rejection probability based on the POLfit algorithm [42], taking into account the coherent \(\pi ^0\) 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 \(\chi ^2\) analysis.

We found that the wrong mass hierarchy is rejected with \(|\overline{\Delta \chi ^2_\mathrm{MH}}| > 10\) in the T2KK and \(|\overline{\Delta \chi ^2_\mathrm{MH}}| > 3\) in the T2KO experiment for any CP phases when \(\sin ^2 \theta _{23} = 0.5\), using the \(\nu _\mu \) and \({\bar{\nu }}_\mu \) focusing beam ratio between 3:2 and 2.5:2.5 (in units of \(10^{21}\)POT with the proton energy of 40 GeV). It should be noted that the \(|\overline{\Delta \chi ^2_\mathrm{MH}}|\) 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 \(|\overline{\Delta \chi ^2_\mathrm{MH}}|\) needs dedicated statistical consideration, above \(|\overline{\Delta \chi ^2_\mathrm{MH}}|\) may be roughly interpreted as the \(80\%\) probabilities of determining the mass hierarchy with \({>} 2.6\,\sigma \) for T2KK and \({>} 1.3\,\sigma \) for T2KO, respectively, assuming the Gaussian distribution for the \(\Delta \chi ^2_\mathrm{MH}\) in the T2KK and T2KO experiments (see Fig. 2 in Ref. [49] for the interpretation).

In the most sensitive region around \(\delta _{\scriptscriptstyle \mathrm{CP}}\sim -90^\circ \) for the normal hierarchy case, we reject the wrong mass hierarchy with \(|\overline{\Delta \chi ^2_\mathrm{MH}}| \sim 32\) in the T2KK experiment (\(3.0^\circ \) OAB at the SK) and with \(|\overline{\Delta \chi ^2_\mathrm{MH}}| \sim 20\) in the T2KO experiment, using the 3:2–2.5:2.5 beam ratio. These \(|\overline{\Delta \chi ^2_\mathrm{MH}}|\) correspond to the \(80\%\) probabilities of the mass-hierarchy determination with \({>} 4.9\,\sigma \) for the T2KK experiment and with \({>} 3.8\, \sigma \) for the T2KO experiment. On the other hand, for the inverted hierarchy case, we reject the wrong mass hierarchy with \(|\overline{\Delta \chi ^2_\mathrm{MH}}| \sim 30\) in the T2KK experiment (\(3.0^\circ \) OAB at the SK) and with \(|\overline{\Delta \chi ^2_\mathrm{MH}}| \sim 18\) in the T2KO experiment around \(\delta _{\scriptscriptstyle \mathrm{CP}}\sim 90^\circ \), using the same beam ratio as the normal hierarchy case. These \(|\overline{\Delta \chi ^2_\mathrm{MH}}|\) correspond to the \(80\%\) probabilities of the mass-hierarchy determination with \({>} 4.7\,\sigma \) for the T2KK experiment and with \({>} 3.6\, \sigma \) for the T2KO experiment. These sensitivities are obtained for \(\sin ^2 \theta _{23} = 0.5\) and enhanced (reduced) by 30–40% in \(|\overline{\Delta \chi ^2_\mathrm{MH}}|\) for \(\sin ^2 \theta _{23} = 0.6\, (0.4)\).

We also examined the sensitivity to the CP phase measurements. The \(\nu _\mu \) and \({\bar{\nu }}_\mu \) 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 \({\sim } 20^\circ \)–\(50^\circ \) (T2KK with \(3.0^\circ \) OAB at the SK), \({\sim } 20^\circ \)–\(45^\circ \) (T2KK with \(2.5^\circ \) OAB at the SK and T2KO), depending on the CP phase. We can measure the CP phase most accurately around \(\delta _{\scriptscriptstyle \mathrm{CP}}\sim 0^\circ \) and \({\sim } 180^\circ \), while the uncertainty is largest around \(\delta _{\scriptscriptstyle \mathrm{CP}}\sim \pm 60^\circ \) and \({\sim } \pm 120^\circ \). A long baseline is helpful to improve the CP phase measurements around those large uncertainty regions. The mass hierarchy and \(\sin ^2 \theta _{23}\) dependence of the CP phase measurements are not so large. The CP violation in the lepton sector is detected with \((\Delta \chi ^2)_\mathrm{min}> 9\) for \(-100^\circ< \delta _{\scriptscriptstyle \mathrm{CP}}< -60^\circ \) and \(70^\circ< \delta _{\scriptscriptstyle \mathrm{CP}}< 120^\circ \) by the T2KO experiment, while the T2KK experiment detects the CP violation only with \((\Delta \chi ^2)_\mathrm{min}> 4\). In either experiment, we need larger statistics to establish the CP violation in a 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 the leptonic CP phase measurement using \(\nu _\mu \) and \({\bar{\nu }}_\mu \) 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\(^\circ \) and 3.0\(^\circ \) OAB at the SK) and T2KO experiments. The lowest sensitivities are improved in the T2KK experiment with \(2.5^\circ \) 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.

## Footnotes

- 1.
- 2.
Recently, more efficient \(\pi ^0\) rejection algorithm has been developed by the T2K collaboration [43], and our NC \(\pi ^0\) background estimation may be regarded as a conservative one.

- 3.
In this approximation, the resultant sensitivities would be slightly overestimated, as discussed in Ref. [52].

## Notes

### Acknowledgements

We would like to thank T. Nakaya and M. Yokoyama for useful discussions and comments on the CP phase sensitivity study of \(\mathrm{T2K}_{122}\) setup. Y.T. wishes to thank the Korea Neutrino Research Center and KIAS, where part of this work was done. This work is in part supported by the Grant in Aid for Scientific Research No. 25400287 (K.H.) and No. 26400254 (N.O. and Y.T.) from MEXT, Japan, by National Research Foundation of Korea (NRF) Research Grant NRF-2015R1A2A1A05001869 (P.K.), and by the NRF grant funded by the Korea government (MSIP) (No. 2009-0083526) through Korea Neutrino Research Center at Seoul National University (P.K. and Y.T.).

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