On $\theta_{23}$ Octant Measurement in $3+1$ Neutrino Oscillations in T2HKK

It has been pointed out that the mixing of an eV-scale sterile neutrino with active flavors can lead to loss of sensitivity to the $\theta_{23}$ octant (sign of $\sin^2\theta_{23}-1/2$) in long baseline experiments, because the main oscillation probability $P_0=4\sin^2\theta_{23}\sin^2\theta_{13}\sin^2\Delta_{31}$ can be degenerate with the sum of the interferences with the solar oscillation amplitude and an active-sterile oscillation amplitude in both neutrino and antineutrino oscillations, depending on CP phases. In this paper, we show that the above degeneracy is resolved by measuring the same beam at different baseline lengths. We demonstrate that Tokai-to-Hyper-Kamiokande-to-Korea (T2HKK) experiment (one 187~kton fiducial volume water Cerenkov detector is placed at Kamioka, $L=295$~km, and another detector is put in Korea, $L\sim1000$~km) exhibits a better sensitivity to the $\theta_{23}$ octant in those parameter regions where the experiment with two detectors at Kamioka is insensitive to it. Therefore, if a hint of sterile-active mixings is discovered in short baseline experiments, T2HKK is a better option than the plan of placing two detectors at Kamioka. We also consider an alternative case where one detector is placed at Kamioka and a different detector is at Oki Islands, $L=653$~km, and show that this configuration also leads to a better sensitivity to the $\theta_{23}$ octant.

In this paper, we pursue the possibility that a combination of neutrino-and antineutrinofocusing operations and measurements of the same beam at different baseline lengths resolves the degeneracy of P 0 and P 1 + P 2 , thereby resurrecting the sensitivity to the θ 23 octant in the 3 + 1 oscillations. For concreteness, we concentrate on Tokai-to-Hyper-Kamiokande-to-Korea (T2HKK) experiment [25] (for early proposals, see Refs. [26,27,28,29,30,5,31,32,33]), where one 187 kton fiducial volume water Cerenkov detector is placed at Kamioka (L = 295 km) and another 187 kton detector is in Korea (L ∼ 1000 km), which is a proposed extension of Tokaito-Hyper-Kamiokande (T2HK) experiment [34]. We conduct a comparative study of T2HKK experiment with the plan of placing two 187 kton detectors at Kamioka. We will demonstrate that for some values of CP phases φ 13 and φ 14 , the sensitivity to the θ 23 octant is lost in the latter experiment while the sensitivity is maintained in T2HKK, in spite of smaller statistics of T2HKK (as the baseline to Korea is longer than that to Kamioka). We further consider an alternative plan of placing a modest detector at Oki Islands [35,36,37,38] (L = 653 km) in addition to one 187 kton detector at Kamioka, and study how the sensitivity to the θ 23 octant changes in this case.
Previously, the sensitivity of T2HK and T2HKK to the θ 23 octant in the presence of mixings of an eV-scale sterile neutrino has been studied in Ref. [21]. Our study differs from it in that we consider both cases with θ 34 = 0 and θ 34 = 0, and further separately investigate the dependence of the sensitivity on θ 34 and that on θ 24 2 . In contrast, Ref. [21] only assumes substantially large values for θ 34 and varies θ 34 , θ 14 , θ 24 only simultaneously. As we show in the main text, non-zero values of θ 34 tend to increase the sensitivity to the θ 23 octant, and hence it is important to scrutinize the case with θ 34 = 0, which is the 'worst situation' for the θ 23 octant measurement. Also, since the sensitivity tends to increase with θ 34 and decrease with θ 24 , the dependences on θ 34 and θ 24 must be analyzed separately. This paper is organized as follows: In Section 2, we write the ν µ → ν e oscillation probability in the 3 + 1 oscillations, spot the terms that can lead to loss of sensitivity to θ 23 octant, and qualitatively state that this problem is mitigated in T2HKK experiment. In Section 3, we confirm the above qualitative statement through a numerical analysis. Section 4 summarizes the paper.
The probability of ν µ → ν e oscillation, which is crucial for the θ 23 octant measurement, is given by where L is the baseline length, E is the neutrino energy, and ∆m 2 i1 = m 2 i − m 2 1 . V cc and V nc denote the potentials generated by charged and neutral current interactions, respectively, which are evaluated as V cc = −2V nc ≃ 0.193 × 10 −3 (ρ/(g/cm 3 ))/km with ρ being the matter density. The antineutrino oscillation probability P (ν µ →ν e ) is obtained by flipping the signs To gain physical insight, we expand Eq. (2) in the leading order of ∆m 2 21 L/E, θ 14 , θ 24 , θ 34 and the matter effect (approximated to be uniform), and further take an average over ∆m 2 41 L/(4E) oscillations. The ν µ → ν e andν µ →ν e oscillation probabilities are then written as

4E
, where A = 4 sin θ 13 sin θ 12 cos θ 12 sin(2θ 23 ) ∆m 2 21 L 4E , B = 4 sin θ 14 sin θ 24 sin θ 13 sin θ 23 , and the upper signs in ±, ∓ refer to the neutrino oscillation, and the lower signs to the antineutrino oscillation. First, we find that in the absence of sterile-active mixings, we have B = 0 and the impact of A on the θ 23 octant measurement can be reduced by a combination of neutrino-and antineutrino-focusing beams, since A term in Eq. (4) changes sign for neutrino and antineutrino.
Second, we observe that in the presence of sterile-active mixings (i.e. B = 0), a combination of neutrino-and antineutrino-focusing beams and two different baseline lengths revives the sensitivity to the θ 23 octant, because a different oscillating function, sin(∆m 2 31 L/(2E)), enters Eq. (5). In the rest of the paper, we will confirm this qualitative observation through simulations of T2HK, T2HKK, and T2HK+Oki experiments.
The term Eq. (6) represents the ordinary matter effect that is present in the standard oscillations, and does not affect the measurement of the θ 23 octant.
The last term Eq. (7) represents a synergy of the matter effect and sterile-active mixings. Here, F is a complicated function of L∆m 2 31 /(4E) and ±(φ 13 − φ 14 + φ 34 ), and is expected to be O(1). Since θ 34 is less constrained than θ 24 , this term can lead to interesting phenomenology in some long baseline experiments with large matter effect. We will study the impact of the term Eq. (7) numerically in the next section.

Oscillation Parameters
We make a simplifying assumption that θ 12 , θ 13 , ∆m 2 21 , |∆m 2 32 | and the true mass hierarchy have been measured precisely prior to a θ 23 measurement. Accordingly, we fix the values of θ 12 , θ 13 , ∆m 2 21 , |∆m 2 32 | at their Particle Data Group values [41] in the simulation, and fit the simulation results with the true values. Likewise, we perform the simulation for the normal hierarchy case (∆m 2 32 > 0) and fit the results by assuming the normal hierarchy, and do analogously for the inverted hierarchy case (∆m 2 32 < 0). For sterile-active mixing parameters, we consider a situation where sterile-active mixings have already been discovered and ∆m 2 41 , θ 14 , θ 24 , θ 34 have been measured precisely in short baseline experiments.
The full analysis is multi-dimensional, depending on benchmark values of θ 14 , θ 24 , θ 34 as well as unknown values of θ 23 and the three CP phases φ 13 , φ 14 , φ 34 . Such a complicated analysis does not provide clear physical insight. Therefore, first we restrict our study to the case with θ 34 = 0, and assume the largest values for θ 14 , θ 24 that are consistent with the current experimental bounds in order to assess the maximal impact of θ 14 , θ 24 mixings on the θ 23 octant measurement. The reason for taking θ 34 = 0 is that the θ 34 mixing affects the octant measurement only via the matter effect and so its impact should be studied separately. Here we fix ∆m 2 41 at a value around ∼ 1 eV 2 , since the results do not depend on the precise value of ∆m 2 41 . In contrast, we consider various values for φ 13 , φ 14 because the impact of θ 14 , θ 24 mixings on the θ 23 octant measurement depends keenly on φ 13 , φ 14 . We take two benchmark values for θ 23 , one in the lower octant and the other in the higher octant. The first benchmark parameter set, which is motivated by the above argument, is presented in Table 1.
In Table 1, the values of sin 2 θ 14 and ∆m 2 41 are marginally compatible with the 90% CL bound obtained by a reanalysis [42] of Bugey-3 [43] experiment, and the values of sin 2 θ 24 and ∆m 2 41 are marginally consistent with the 90% CL bound of the MINOS and MINOS+ results [44] and are outside the 90% CL bound from the IceCube [45]. The asymmetric benchmark values of sin 2 θ 23 are motivated by the 3σ range reported by NuFIT 4.1 [46,47], which is tilted towards the higher octant region. Note that φ 34 is an unphysical phase when θ 34 = 0.
After the analysis of the first benchmark Table 1, we vary sin 2 θ 34 , φ 34 and study the impact of a synergy of the matter effect and sterile-active mixings described by Eq. (7). Here we fix φ 13 , φ 14 at the values for which sterile-active mixings have most afflicted the θ 23 octant measurement in the first benchmark. The second benchmark parameter set, which is motivated by the above argument, is presented in Table 2.
In Table 2, the range of sin 2 θ 34 is chosen to satisfy the 90% CL bound on the sterile-tau neutrino mixing reported from Super-Kamiokande atmospheric neutrino measurement [48].
We will find in Section 3.5 that non-zero values of sin 2 θ 34 tend to mitigate the impact of sterile-active mixings on the θ 23 octant measurement, i.e. one is more likely to obtain the correct octant when the θ 34 mixing is present. Therefore, it is important to study how the impact of sterile-active mixings changes with θ 14 , θ 24 in the 'worst scenario' for the octant measurement where θ 34 = 0. To this end, we vary θ 24 while fixing θ 34 = 0. Here we fix sin 2 θ 14 = 0.04 because, as seen in Eqs. (3)-(7), the effects of sterile-active mixings on the θ 23 octant measurement appear only in the combination sin θ 14 sin θ 24 when θ 34 = 0, and thus varying sin 2 θ 14 is equivalent to varying sin 2 θ 24 . Here we take a specific combination of values of φ 13 , φ 14 for which sterile-active mixings have most afflicted the θ 23 octant measurement in the first benchmark. The third benchmark parameter set, which is motivated by the above argument, is presented in Table 3.

Experiments
We assume that J-PARC operates with 1.3 MW beam power, delivering 2.7 × 10 21 protonon-target (POT) flux per year. We adopt the values in Table 4 as the baseline and detector parameters, where the matter density is approximated to be uniform [49,50]. We consider three experiments, referred to as 'T2HK', 'T2HKK' and 'T2HK+Oki' in this paper. The configuration of each experiment is assumed as follows, and is summarized in Table 4.
• In 'T2HK', we assume that one 187 kton fiducial volume detector at Kamioka is exposed to a neutrino-focusing beam for 2.5 years and to an antineutrino-focusing beam for 2.5 years. Subsequently, two 187 kton detectors (374 kton in total) at Kamioka are exposed to a neutrino-focusing beam for 5 years and to an antineutrino-focusing beam for 5 years.
• In 'T2HKK', we assume that one 187 kton detector at Kamioka is exposed to a neutrinofocusing beam for 2.5 years and to an antineutrino-focusing beam for 2.5 years. Subsequently, one 187 kton detector at Kamioka and one 187 kton detector in Korea are exposed to a neutrino-focusing beam for 5 years and to an antineutrino-focusing beam for 5 years.
• In 'T2HK+Oki', we assume that one 187 kton detector at Kamioka is exposed to a neutrino-focusing beam for 2.5 years and to an antineutrino-focusing beam for 2.5 years. Subsequently, one 187 kton detector at Kamioka and a smaller 100 kton detector at Oki Islands are exposed to a neutrino-focusing beam for 5 years and to an antineutrinofocusing beam for 5 years.

Number of Events
The number of ν e +ν e events and the number of ν µ +ν µ events in the bin of 0.4+(i−1)·0.05 GeV < E < 0.4+i·0.05 GeV with a neutrino-focusing beam that are detected at a specific site, denoted by N e,i,site and N µ,i,site , and those with an antineutrino-focusing beam, denoted by N e,i,site and ) respectively denote ν µ andν µ fluxes per energy in a neutrino-focusing beam (in an antineutrino-focusing beam) at a specific detector. P site denotes a transition probability calculated with Eq. (2) for a specific site. N n,site and N p,site are respectively the number of neutrons and protons in the water Cerenkov detector at a specific site. σ denotes the cross sections for the ν ℓ n → ℓ − p andν ℓ p → ℓ + n processes.
We employ the result of Ref.
[51] for ν µ andν µ flux in neutrino-focusing and antineutrinofocusing beams emitted from J-PARC and detected at a water Cerenkov detector at Kamioka, Oki and in Korea if the neutrino oscillations were absent. For reference, we plot the flux in Fig. 1. The baseline length and beam off-axis angle of each detector are found in Table 4.
In the plots, the blue lines correspond to a neutrino-focusing beam and the red lines to an antineutrino-focusing beam. The solid lines indicate ν µ flux and the dashed linesν µ flux.  Table 4. The upper plot, the lower-left plot and the lower-right plot correspond to Kamioka, Oki and Korea, respectively. In the plots, the blue lines correspond to a neutrinofocusing beam and the red lines to an antineutrino-focusing beam. The solid lines indicate ν µ flux and the dashed linesν µ flux. ν e andν e components in the beams are neglected in our study.
The cross sections for charged current quasi-elastic scattering between a neutrino and a proton, ν ℓ n → ℓ − p, and that between an antineutrino and a neutron,ν ℓ p → ℓ + n, (p and n denote proton and neutron, respectively, and ℓ denotes e or µ) are obtained from Ref. [52]. ε e,site , ε µ,site respectively denote the efficiencies for electrons and muons of the far detector at a specific site, in a neutrino-focusing run. f Φν , f Φν account for the uncertainty of neutrino and antineutrino fluxes, respectively, in a neutrino-focusing beam. f σe (f σµ ) accounts for the uncertainty of the weighted sum of charged current quasi-elastic scattering cross sections of ν e andν e (ν µ andν µ ) that is estimated from near detector data and theory, in a neutrino-focusing run. ε e,site , ε µ,site , f Φν , f Φν , f σe , f σµ are the corresponding quantities in an antineutrino-focusing run. In this paper, we neglect energy dependence of the above systematic parameters. For the efficiencies of the far detectors, based on Table 9 of Ref. [53], we assume 3 ε e,site = 1 ± 0.007, ε µ,site = 1 ± 0.010, ε e,site = 1 ± 0.017, ε µ,site = 1 ± 0.011. (11) ε e,site and ε e,site at each site are maximally correlated.
ε µ,site and ε µ,site at each site are maximally correlated.
For the flux uncertainty, based on Fig. 9 of Ref. [53], we approximate For the cross section uncertainty, based on Table 9 of Ref. [53], we assume All uncertainties above are assumed to be uncorrelated unless otherwise stated.
Additionally, we fit the normal hierarchy events by assuming the normal hierarchy, and the inverted hierarchy events by assuming the inverted hierarchy, because the mass hierarchy may have been determined before T2HK starts operation.
In this paper, we are interested in the danger of wrong measurement of the θ 23 octant. Thus, we evaluate the significance of rejecting the wrong octant, ∆χ 2 , which is the square root of the difference between the minimum of χ 2 for sin 2 θ 23 > 1/2 and that for sin 2 θ 23 < 1/2.

Results
Before going to the main results, we show in Table 5 the total numbers of ν µ +ν µ events at the Kamioka, Oki and Korea detectors in T2HK, T2HKK and T2HK+Oki experiments when there were no oscillations, corresponding to the configurations of Table 4 and the assumption that J-PARC operates with 1.3 MW beam power.  Table 1. The results are presented as a function of (φ 13 , φ 14 ) and shown separately for the normal and inverted mass hierarchies. Fig. 2 is the result when the mass hierarchy is normal, and Fig. 3 is the result when the mass hierarchy is inverted. The red, orange, green, cyan, blue and purple contours correspond to ∆χ 2 = 1 2 , 1.5 2 , 2 2 , 2.5 2 , 3 2 , 3.5 2 , respectively.  Table 1. The mass hierarchy is normal. The red, orange, green, cyan, blue and purple contours correspond to ∆χ 2 = 1 2 , 1.5 2 , 2 2 , 2.5 2 , 3 2 , 3.5 2 , respectively. The following observations are made from Figs. 2,3.
• T2HKK and T2HK+Oki give a larger significance of rejecting the wrong θ 23 octant than T2HK in all cases, in spite of their smaller statistics than T2HK (see Table 5). This confirms the qualitative argument of Section 2 of the present paper.
• Comparing T2HKK and T2HK+Oki, we observe that T2HKK gives a larger significance, in spite of the fact that T2HKK has smaller statistics than T2HK+Oki (see Table 5).
The reason that T2HK+Oki, in spite of its larger statistics, shows a smaller improvement of the octant sensitivity than T2HKK is understood as follows: As shown in Fig. 1, under our assumption that the beam off-axis angle at Oki is 1.0 • , the flux at Oki has a (broad) peak around E ≃ 1.4 GeV. Thus, the value of L/E at the flux peak is 653 km/1.4 GeV for Oki, and it is 295 km/0.6 GeV for Kamioka. Since 653 km/1.4 GeV and 295 km/0.6 GeV are close values, a major portion of (anti)neutrinos oscillate in a similar manner at Oki and Kamioka, which spoils our attempt to resolve the degeneracy between Eq. (3) and Eqs. (4),(5) by combining different baselines.
Next, we present ∆χ 2 with sin 2 θ 23 =0.44 Eq. (16) and that with sin 2 θ 23 =0.60 Eq. (17) in the second benchmark Table 2. The results are presented as a function of (sin 2 θ 34 , φ 34 ) and shown separately for the normal and inverted mass hierarchies. Fig. 4 is the result when the mass hierarchy is normal, and Fig. 5 is the result when the mass hierarchy is inverted.  Table 2. The mass hierarchy is normal. The orange, green, cyan, blue and purple contours correspond to ∆χ 2 = 1.5 2 , 2 2 , 2.5 2 , 3 2 , 3.5 2 , respectively.
The following observations are made from Figs. 4,5.
• T2HKK and T2HK+Oki give a larger significance of rejecting the wrong θ 23 octant than T2HK in all cases. Also, T2HKK shows a larger significance than T2HK+Oki in all cases.
• Non-zero values of θ 34 tend to increase the significance of rejecting the wrong θ 23 octant in both cases with sin 2 θ 23 = 0.44 and 0.60, for both mass hierarchies, in all the experiments. Interestingly, even though the matter effect is small in T2HK, we still obtain an increase in the significance in T2HK. The above results are because the term Eq. (7) is proportional to sin θ 23 and flips sign for neutrinos and antineutrinos. Hence, a measurement of this term using the combination of neutrino-focusing and antineutrino-focusing operations offers a new probe for the θ 23 octant.
• In the case with sin 2 θ 23 = 0.60 and for small sin 2 θ 34 , there is a tiny parameter region where the significance of rejecting the wrong θ 23 octant decreases with θ 34 . The presence of such a region is probably because the term Eq. (7) mimics the first part of Eq. (4) (remember that φ 13 = π in the second benchmark and so the true value of the first part of Eq. (4) is 0), which makes it harder to distinguish the term Eq. (4) from the term Eq. (3).
The following observations are made from Figs. 6,7.
• T2HKK and T2HK+Oki give a larger significance of rejecting the wrong θ 23 octant than T2HK in all cases. Also, T2HKK shows a larger significance than T2HK+Oki in all cases.
• When sin 2 θ 23 = 0.44, for φ 13 ≃ 3π/2 and for large sin 2 θ 24 , there is a tiny parameter region where the significance increases with sin 2 θ 24 . The presence of such a region is probably because for larger sin 2 θ 24 , the first part of Eq. (4) is less likely to mimic the second part of Eq. (5) (remember that φ 13 − φ 14 = 0, π in the third benchmark and so the true value of the second part of Eq. (5) is zero), which makes it easier to distinguish the first part of Eq. (4) from the other terms. Then the combination of neutrino-and antineutrino-focusing operations can more easily resolve the degeneracy between Eq. (3) and Eq. (4).
We note in passing that in the χ 2 minimum in the wrong octant region, the fitted value of φ 13 is close to its true value. So, even if the sensitivity to the θ 23 octant is lost, the measurement of the standard CP phase is relatively unaffected.

Summary
We have confirmed that in the presence of sterile-active mixings with non-zero θ 14 and θ 24 , T2HK experiment with two 187 kton detectors at Kamioka can lose sensitivity to the θ 23 octant, depending on values of CP phases. We have revealed that T2HKK exhibits a better sensitivity to the θ 23 octant in all parameter regions including those where the experiment with two detectors at Kamioka loses its sensitivity, in spite of smaller statistics of T2HKK. The better sensitivity of T2HKK is because measurements of the same beam at Kamioka and Korea, which involve distinctively different baseline lengths, resolve the degeneracy between the atmospheric oscillation probability and the sum of the interferences with the solar oscillation amplitude and active-sterile oscillation amplitude. We have further studied the impact of nonzero θ 34 mixing and found that the sensitivity to the θ 23 octant tends to increase with θ 34 in all parameter regions in all the experiments.
Our results suggest that if a hint of sterile-active mixings with θ 14 = 0 and θ 24 = 0 is discovered but θ 34 is 0 or substantially smaller than the current experimental bound, T2HKK is a preferable option compared to the plan of placing the two detectors at Kamioka, for the determination of the θ 23 octant.
Additionally, we have studied a case where one 187 kton detector is at Kamioka and a 100 kton detector is at Oki Islands. This case also shows a better sensitivity to the θ 23 octant in those parameter regions where the plan of two 187 kton detectors loses its sensitivity, but the improvement is quite mild compared to T2HKK, in spite of its larger statistics than T2HKK.