Complementarity Between Hyperkamiokande and DUNE in Determining Neutrino Oscillation Parameters

In this work we investigate the sensitivity to the neutrino mass hierarchy, the octant of the mixing angle $\theta_{23}$ and the CP phase $\delta_{CP}$ in the future long-baseline experiments T2HK and DUNE as well as in the atmospheric neutrino observation at Hyperkamiokande (HK). We show for the first time that the sensitivity is enhanced greatly if we combine these three experiments. Our results show that the hierarchy sensitivity of both T2HK and HK are limited due to the presence of parameter degeneracy. But this degeneracy is removed when T2HK and HK are added together. With T2HK+HK (DUNE), the neutrino mass hierarchy can be determined at least at $ 5 \sigma$ (8 $\sigma$) C.L. for any value of true $\delta_{CP}$. With T2HK+HK+DUNE the significance of the mass hierarchy increases to almost 15 $\sigma$ for the unfavorable value of $\delta_{CP}$. For these combined setup, octant can be resolved except $43.5^\circ<\theta_{23}<48^\circ$ at $5\sigma$ C.L for both the hierarchies irrespective of the value of $\delta_{CP}$. The significance of CP violation is around 10 $\sigma$ C.L. for $\delta_{CP} \sim \pm 90^\circ$. Apart from that these combined facility has the capability to discover CP violation for at least $68\%$ fraction of the true $\delta_{CP}$ values at $5 \sigma$ for any value of true $\theta_{23}$. We also find that, with combination of all these three, the precision of $\Delta m^2_{{\rm eff}}$, $\sin^2\theta_{23}$ and $\delta_{CP}$ becomes 0.3%, 2% and 20% respectively. We also clarify how the octant degeneracy occurs in the HK atmospheric neutrino experiment.


I. INTRODUCTION
With the discovery of the mixing angle θ 13 by the reactor experiments [1][2][3], the physics of neutrino oscillation has entered into an era of precision measurement. In the standard three flavor scenario, the phenomenon of neutrino oscillation can be parametrized by three mixing angles: θ 12 , θ 23 and θ 13 , two mass squared difference: ∆m 2 21 and ∆m 2 31 , and one Dirac type phase δ CP . Thanks to the neutrino experiments in the last two decades, the values of the three mixing angles and the values of the mass squared differences are now determined in the three flavor mixing framework to some precision [4][5][6]. The unknown quantities at present are: (i) the mass hierarchy or the sign of ∆m 2 31 (NH: normal hierarchy i.e., ∆m 2 31 > 0 or IH: inverted hierarchy i.e., ∆m 2 31 < 0 ), (ii) the octant of θ 23 (LO: lower octant i.e., θ 23 < 45 • or HO: higher octant i.e., θ 23 > 45 • ) and (iii) the CP phase δ CP . There are several experiments which are dedicated to determine these above mentioned unknowns.
The main difficulty in determining those unknowns is the presence of parameter degeneracy [7][8][9][10][11]. In parameter degeneracy, different sets of oscillation parameter leads to the same value of the oscillation probability. Due to this the true solutions can be mimicked by the false solutions and thus a unique determination of the parameters becomes difficult. One of the ways to overcome this degeneracy is to combine data from different experiments. As the degenerate parameter space is different for different experiments, combination of different experiments can lead to the removal of the fake solutions which may help in the unambiguous determination of the neutrino oscillation parameters. Recently, this strategy has been adopted by many to study the synergy between different on-going as well as future proposed experiments in determining the remaining unknowns of the neutrino oscillation parameter. In this regard, the well established method is to combine the data of the long-baseline and atmospheric neutrino oscillation experiments. The sensitivity to the remaining unknown parameters in long-baseline experiments comes from the appearance channel probabilities P (ν µ → ν e ) and P (ν µ →ν e ) while the atmospheric experiments can have sensitivity for both appearance and disappearance channel probabilities P (ν µ → ν µ ) and P (ν µ →ν µ ). In addition to that some of the authors also prefer to add the reactor data which has the sensitivity of the electron antineutrino disappearance channel (ν e →ν e ). Thus the combination of the long-baseline and atmospheric experiments along with the information of the precise value of θ 13 , is believed to remove the degenerate solutions and measure the current unknowns at a significant confidence level. In Refs. [12][13][14], the combined analysis on the CP sensitivity of the long-baseline exper-iments T2K [15], NOνA [16] and atmospheric experiment ICAL@INO [17] has been studied in detail. The combined analysis on the octant sensitivity of these experiments can be found in [18].
In these analyses, the reactor information has been taken into account in the form of prior on sin 2 2θ 13 . The hierarchy sensitivity of T2K, NOνA and ICAL@INO along with the combination of the reactor experiments has been done in [19]. The sensitivity of the long-baseline experiment LBNO [20] with the addition of T2K, NOνA and ICAL@INO has been studied in [21]. The combined analysis on the octant sensitivity of the atmospheric experiment PINGU [22] along with T2K, NOνA and reactor data has been studied in Ref. [23]. The synergistic study on the hierarchy sensitivity of PINGU by combining reactor experiment has been done in [24] whereas the same with combining beam based experiments and reactors is carried out in [25]. The synergy of the proposed long-baseline experiment DUNE [26], combining both the beam and atmospheric data along with T2K, NOνA is analyzed in [27,28]. The combined sensitivity of DUNE in conjunction with T2K, NOνA and ICAL@INO has been explored in [29]. All these studies shows that, the sensitivity of an individual experiment is significantly enhanced when added with the other experiments.
In this paper, for the first time we study the joint sensitivity of the long-baseline experiments T2HK [30], DUNE and the atmospheric experiment HK [31] in determining the remaining unknowns in neutrino oscillation sector. This is the so-called HK and LBNF complementarity which has been discussed in the literature [32]. We take this opportunity to work out the physics potential of these facilities in detail. The T2HK experiment is an upgrade of the ongoing T2K experiment which will use a detector to have a volume almost 25 times larger than the existing T2K detector. HK is the atmospheric counterpart of the T2HK experiment. On the other hand DUNE is a high statistics beam based experiment to use high beam power, large detector volume and longer baseline. Among all other existing facilities, these above mentioned experiments are the most promising future experiments in terms of both statistics and matter effect which have the maximum potential to reveal the true nature of the neutrino oscillation parameters. In this work we study: ( The ultimate goal of the research on neutrino oscillations in the standard three flavor mixing scheme is the determination of δ CP , since it is expected to be relevant to the baryon asymmetry of the universe. It has been known that the determination of δ CP is not easy because of so-called parameter degeneracy: Even if we know the values of the appearance oscillation probabilities P (ν µ → ν e ) and P (ν µ →ν e ) precisely, we cannot determine the oscillation parameters uniquely.
Earlier when θ 13 was not known precisely, there were three kinds of parameter degeneracy. The first parameter degeneracy is the intrinsic degeneracy [7], which occurs because the appearance oscillation probabilities are approximately quadratic in sin 2θ 13 for small θ 13 and we obtain the two solutions (θ 13 , δ CP ) and (θ ′ 13 , δ ′ CP ). This intrinsic degeneracy is not expected to cause a problem because we now know the value of θ 13 from the precise measurements of θ 13 by the reactor neutrino experiments [1][2][3]. The second one is the sign degeneracy [8]. At this moment we do not know whether the mass hierarchy is normal or inverted. Depending on whether the mass hierarchy is normal or inverted, the appearance oscillation probabilities vary, and ignorance of the mass hierarchy may give us a completely wrong region for δ CP . Thus the determination of the mass hierarchy is important for the measurement of δ CP . The third one is the octant degeneracy [9].
The main contribution to the probabilities 1 − P (ν µ → ν µ ) and 1 − P (ν µ →ν µ ) is proportional to sin 2 2θ 23 , and we can only determine sin 2 2θ 23 from the disappearance channel. If θ 23 is not maximal, then we have two possibilities cos 2θ 23 > 0 and cos 2θ 23 < 0, so this ambiguity gives us two solutions (θ 23 , δ CP ) and (90 • − θ 23 , δ ′ CP ). The octant degeneracy can create a source of the uncertainty in θ 23 , so its resolution is important for precise measurement of θ 23 . These three types of degeneracies together created a eight fold degeneracy degeneracy [10] when the precise value of θ 13 was unknown. After the discovery of θ 13 , now the eight fold degeneracy breaks into a four fold degeneracy. At present the relevant degeneracy in the neutrino oscillation probability is known as the "generalized hierarchy-octant-δ CP degeneracy" [11]. This generalized degeneracy consists of hierarchy-δ CP degeneracy [33] and octant-δ CP degeneracy [34] which we discuss in the next section for the T2HK and DUNE baselines.

B. T2HK and DUNE
T2HK is the long baseline experiment which is planned in Japan, and its baseline length and peak energy is L=295 km, E ∼ 0.6 GeV, respectively. On the other hand, DUNE is another long baseline experiment which is planned in USA, and its baseline length and peak energy is L=1300 is too short for the matter effect so T2HK has poor sensitivity to the mass hierarchy. Hence the sign degeneracy can be in principle serious in determination of δ CP at T2HK. On the other hand, the baseline length of DUNE is comparable to the typical length which is estimated by the matter effect, so DUNE is expected to be sensitive to the mass hierarchy.
As was discussed in the case of T2K and NOνA in Ref. [33], we can see whether this so called hierarchy-δ CP degeneracy can be resolved or not by looking at the behaviors of the appearance channel probabilities. In Fig. 1, we plot the appearance channel probability spectrum for a fixed value of θ 23 = 45 • . The first and second panels are for neutrino and antineutrino probabilities of T2HK and the third and fourth panels are for neutrino and antineutrino probabilities of DUNE respectively. In each panel, the red band correspond to NH and the blue band corresponds to IH.
The width of the band is due to the variation of the phase δ CP . The overlap region between the red band and the blue band correspond to the hierarchy-δ CP degeneracy. From the Fig On the other hand, the situation of the octant degeneracy is different. As shown in [34], one can identify the octant-δ CP degenerate parameter space by looking at the appearance channel probabilities. In Fig. 2 we plot the same as that of Fig. 1 but for a fixed value of ∆m 2 31 . We assume the normal mass hierarchy for simplicity. In these plots the red band corresponds to LO The behaviors of the appearance probabilities P (ν µ → ν e ) (first and third panels) and P (ν µ →ν e ) (second and fourth panels) for the atmospheric neutrino measurements for the zenith angle cos Θ = −0.9 for the two different mass hierarchies (left two panels) and in the two different octants (right two panels). In the right two panels θ 23 = 42 • or θ 23 = 48 • is taken as a reference value, and Normal hierarchy is assumed.
applies to the case of δ CP = −90 • , and we conclude that both T2HK and DUNE has potential to resolve the octant degeneracy by themselves by combining the neutrino and antineutrino modes.
These conclusions also hold for the inverted hierarchy 1 .

C. Atmospheric neutrinos
The measurement of atmospheric neutrinos at Hyperkamiokande (HK) gives us information which is complimentary to long-baseline experiments. Its advantage is that we can get the information on the oscillation probabilities for wide ranges of the neutrino energy and the baseline length, particularly for more than 4000 km, so the measurement gives us information on the matter effect. On the other hand, its disadvantages are that there are uncertainties on the baseline length, that it is difficult to distinguish neutrinos and antineutrinos, and that the oscillation probability has to be deduced indirectly from the measured flux, because the measured flux is the sum of the each flux which are produced out of the original flux F 0 (ν α ) (α = e, µ) through neutrino oscillations.
In our simulation of the atmospheric neutrino measurements at Hyperkamiokande, the data which is most sensitive to the mass hierarchy and the octant degeneracy is the e-like multi-GeV events 2 in the higher energy region for the zenith angle −1.0 < cos Θ < −0.8. The disappearance 1 Note that our discussion of degeneracy is valid only for a fixed value of energy E. In the realistic scenario the nature of these degeneracies can be different due to the energy dependence of the probability spectrum. 2 In the multi-GeV energy region (∼ 10 GeV), the ratio of the original e-like flux F 0 (ν e ) to the original µ-like probability P (ν e → ν e ) is insensitive to the matter effect, so what makes the difference between the two mass hierarchies is the appearance probability P (ν µ → ν e ). In Fig. 3, the appearance probabilities are plotted for cos Θ = −0.9. The first two panels are similar as that of Fig. 1 and the third and fourth panels are similar as that of Fig. 2. From the first two panels, it is clear that as far as the appearance probabilities are concerned, we should be able to determine the mass hierarchy on the condition that we can measure the appearance probabilities for the energy and the baseline length in the atmospheric neutrino measurements. As for the octant, third panel and fourth panels of Fig. 3 suggests that the atmospheric neutrino measurements may resolve the octant degeneracy as it combines both neutrino and antineutrino probabilities.

A. Simulation Details
Here we assume the following parameters for each experiment: 1) T2HK: We have taken the parameters from Ref. [30]. 3 Our χ 2 is a function of the two sets of the oscillation parameters p ex and p th , and is defined as: where {ξ j } are the pull variables, π j are the (1σ) systematic errors for the pull variable ξ j , with one F 0 (ν µ ) is approximately 1:6, so the relative contribution of the appearance probability in the total flux after oscillations is larger in the e-like flux Therefore the e-like events are more sensitive to the mass hierarchy than the µ-like ones. 3 In the recent design of the Hyperkamiokande project [35], two tanks with the fiducial volume 0.19 Mton each with the staging construction is planned. At this moment, the details are not known, so we will analyze T2HK and the atmospheric neutrino measurements at Hyperkamiokande with the parameters in the old design of Hyperkamiokande throughout this paper.
The 'experimental' data N i ( p ex ) are simulated using the 'true' oscillation parameters p ex , while the 'theoretical' events N i ( p th ) are generated using the 'test' oscillation parameters p th . The subscript i here runs over all the energy bins. The theoretical events get modified due to systematic errors as Systematics: an overall normalization error of 3.3% for both appearance and disappearance channel in neutrino mode and 6.2% (4.5%) for appearance (disappearance) channel in antineutrino mode. The normalization error is same for both signal and background. For tilt error we have taken 1% (5%) corresponding to signal (background) in appearance channel and 0.1% in disappearance channel for both signal and background. Tilt error for neutrinos and antineutrinos are taken to be same.
For T2HK we have reproduced the sensitivity as given in Ref. [30].
In the case of the DUNE experiment χ 2 is defined as in the same manner as T2HK. As in Ref. [38], we use the data set of the sub-GeV events with the two energy-bins, the multi-GeV events with the two energy-bins, and the combined stopping and through-going upward going µ events with the single energy-bin, where the number of the zenith angle bins is ten for all these channels.
Simulation of the atmospheric neutrino at Hyperkamiokande is done with the codes which were used in Refs. [38][39][40][41]. In our analysis, we compared our χ 2 and the value of χ 2 given by the HK collaboration in Ref. [30] and normalized our χ 2 for each analysis (mass hierarchy, octant, and CP violation) so that the two χ 2 approximately coincide with each other. Thus our sensitivity of the HK experiment matches with Ref. [30].
The analysis was performed using χ 2 -method. χ 2 depends on the two sets of the oscillation parameters p ex and p th , and is defined as where χ 2 sub−GeV ( p ex , p th ), χ 2 multi−GeV ( p ex , p th ) and χ 2 upward ( p ex , p th ) are χ 2 for the sub-GeV, muti-GeV, and upward µ events, and they are the same as χ 2 sub−GeV , χ 2 multi−GeV and χ 2 upward defined in Ref. [38], respectively. The way we introduce the systematic errors is an extension of that in the original analysis of atmospheric neutrinos. [42] In our code we have used the old atmospheric neutrino flux [43] instead of the new one [44], because our simulation results become closer to the ones by Superkamiokande collaboration with the old flux.
However the difference in χ 2 between the old flux and the new flux is small (∼ a few %), since our simulated experimental data are also evaluated with the same flux. So the choice of the atmospheric neutrino flux does not affect our conclusions very much. We have set the systematic errors to the same values as in Ref. [45] except several unimportant factors. In particular, we confirmed that taking a uncertainty in the E ν spectral index which is omitted in our analysis into consideration gives negligible contributions to χ 2 . See Refs. [38,46] for further details.
In our simulation we have fixed the value of sin 2 θ 12 = 0.3, sin 2 2θ 13 = 0.1, ∆m 2 21 = 7.5×10 −3 eV 2 and |∆m 2 eff | = 2.4 × 10 −3 eV 24 . This is a good approximation since these parameters are well constrained from the global analysis from the world neutrino data [4][5][6] and thus marginalization over these parameters will not affect the sensitivity much. We vary θ 23 from 40 • to 50 • and δ CP from −180 • to +180 • . For estimating the precision of θ 23 and ∆m 2 eff , we have kept δ CP fixed at −90 • for both true and test spectrum.

B. Sensitivity of HK
In this section we study the potential of the HK experiment to determine the neutrino mass hierarchy, octant and δ CP . Hierarchy sensitivity of an experiment is defined by taking the right hierarchy in the true spectrum and wrong hierarchy in the test spectrum. While calculating χ 2 we have marginalized over θ 23 and δ CP in the test spectrum. The purple band in the figure corresponds to sensitivity in NH and the blue band corresponds to sensitivity in IH. The width of the band is due to the variation of true θ 23 4 Following Ref. [47], we adopt the effective mass squared difference for the µ disappearance channel, and it is defined as ∆m 2 eff ≡ s 2 12 ∆m 2 31 + c 2 12 ∆m 2 32 + cos δ CP s 13 sin 2θ 12 tan θ 23 ∆m 2 21 . We consider this in our analysis because in vacuum, the hierarchy degeneracy do not occur for P µµ (∆m 2 31 ) = P µµ (−∆m 2 31 ) but it occurs for P µµ (∆m 2 eff ) = P µµ (−∆m 2 eff ). from 40 • to 50 • which is the current allowed values of θ 23 . From the figure we observe that, for conservative value of θ 23 , 5 the hierarchy χ 2 is close to 6 around δ CP = 0 • for both NH and IH and for optimistic value of θ 23 , the hierarchy sensitivity increases to χ 2 = 30 for NH around ±180 • and χ 2 = 18 for IH around δ CP = +60 • . Here it is interesting to see that the hierarchy sensitivity of IH is in general poorer as compared to NH. It is also important to note that for −180 • < δ CP < −60 • , the width of the IH band is very narrow. This is because for waterČerenkov detectors, it has been  shown that NH-LO is degenerate with IH-HO [25,48]. Thus because of the octant degeneracy, we obtain a very poor hierarchy sensitivity for HK in IH in the above mentioned parameter space.
Note that although NH is not free from this degeneracy but the degeneracy in NH seems less severe as compared to the degeneracy in IH. We will discuss this point in detail in the appendix.

C. Combined Sensitivity of T2HK, HK and DUNE
In this section we will study the synergy between T2HK, HK and DUNE in determining the unknowns of neutrino oscillation and also study up to what confidence level these unknowns can be determined.

Hierarchy
In the upper row of Fig. 5, we plot the significance χ 2 at which hierarchy can be determined as a function of true δ CP . The left panel is for true NH and the right panel is for true IH. These plots are similar as that of top left panel of Fig. 4. As it is explained in Subsect. II B, in the case of δ CP = −90 • with NH and δ CP = 90 • with IH, T2HK can resolve the mass hierarchy and the sensitivity of T2HK is good, whereas in the case of δ CP = 90 • with NH and δ CP = −90 • with IH, T2HK cannot resolve the mass hierarchy by itself, so the sensitivity of T2HK becomes poor. If we combine T2HK and the Hyperkamiokande atmospheric neutrino data, then T2HK+HK(atm) can resolve the sign degeneracy at 5 σ C.L. for any value of δ CP . Here it is important to note that for IH, in the region −180 • < δ CP < 0 • , both T2HK and HK has poor hierarchy sensitivity. In this parameter space, T2HK suffers from hierarchy degeneracy and HK suffers from octant degeneracy.
But when these experiments are combined there is a great enhancement in the sensitivity in that unfavorable values of δ CP . This reflects the synergy between these two experiments which are essential to achieve a hierarchy sensitivity around 5σ C.L. irrespective of the true value δ CP . On the other hand, for DUNE, the separation of the two mass hierarchies are good and DUNE itself has above 5σ C.L. sensitivity to the mass hierarchy for any value of δ CP . If we combine T2HK, the HK atmospheric neutrino data and DUNE, then the significance of the mass hierarchy becomes as large as 15 σ C.L., even for the unfavorable values of δ CP . This is a quite remarkable result which shows the potential of these experiments to discover neutrino mass hierarchy.

Octant
In the middle panels of Fig. 5, we plot the significance χ 2 at which the wrong octant can be excluded as a function of true θ 23 . The left panel is for true NH and the right panel is for true IH.
These figures are similar as that of the top right panel of Fig. 4 i.e., our results corresponds to the conservative values of δ CP . From the plots we see that the values of θ 23 for which octant can be resolved at 5σ C.L is almost the same for T2HK, T2HK+HK and DUNE. For these setups octant can be determined except 43 • < θ 23 < 49 • for both NH and IH at 5σ C.L. However when all the three experiments are combined we see that there is a significant amount of increase in the octant sensitivity for both the hierarchies. From the figure we see that T2HK+HK+DUNE can resolve octant except 43.5 • < θ 23 < 48 • for both the hierarchies at 5σ C.L.

CP Violation
In the lower panels of Fig. 5 we plot the fraction of true δ CP values for which CP violation can be discovered at a 5σ confidence level as function of true θ 23 . The left panel is for NH and the right panel is for IH. From the plots we see that as θ 23 increases the percentage of true δ CP decreases.
For T2HK+HK, the δ CP fraction varies from 60% to 70% and for T2HK+HK+DUNE, the δ CP fraction varies from 75% to 68%. From these results we clearly understand that the combination  archy, χ 2 is given. For octant, the region for θ 23 which is allowed at 5σ is given. For CP violation, the fraction of δ CP for which CPV can be discovered at χ 2 = 25 is given in %.
of these three experiments have the capability to discover CP violation at 5σ C.L. for at least 68% true values of δ CP . This is an incredible result since no other existing facilities can have such a reach to discover CP violation.
From these figures we also note that the sensitivities of DUNE and T2HK are similar. To understand this in Fig. 6, we plot the CP violation discovery χ 2 as a function of true δ CP for θ 23 = 48 • which is the present best-fit value of θ 23 . As was discussed in Subsect. II B, the sensitivity of T2HK to CP is poor for δ CP ∼ 90 • with NH and δ CP ∼ −90 • with IH. On the other hand due to large matter effect DUNE has almost equal sensitivity at ±90 • for both the hierarchies. But due to very high statistics, T2HK has very large CP sensitivity in the favorable values of δ CP and this gives almost comparable coverage of true δ CP values for which CP violation can be discovered at 5σ C.L. It is also important to note that if we combine the result of the Hyperkamiokande atmospheric neutrino data to T2HK, then sensitivity is good also for the unfavorable values of δ CP . This is because the hierarchy sensitivity of HK, removes the wrong hierarchy solutions of T2HK in the unfavorable parameter space. We note in passing that synergy between T2K and the atmospheric neutrino measurement at ICAL@INO for CP violation discovery was studied in Ref. [12]. Furthermore, if we combine T2HK, the Hyperkamiokande atmospheric neutrino and DUNE, then the sensitivity to CP violation reaches around 10 σ C.L. for δ CP ∼ −90 • for both NH and IH. In this section we study how much precisely the parameters δ CP , θ 23 and ∆m 2 eff can be measured by the set up under consideration. In Fig. 7  The left panel is for NH and the right panel is for IH. From the plot we see that, for T2HK there are wrong δ CP solutions mainly in the region 0 • < δ CP < 180 • in NH and −180 • < δ CP < 0 • in IH. But when the HK is added with it, the wrong solutions completely disappears. But in the case of DUNE there are no wrong δ CP solutions. It is also seen that the CP precision is excellent when all the three experiments are combined. Here it is interesting to note that the precision of δ CP is better at δ CP = 0 • as compared to δ CP = ±90 • .
Finally in Fig. 9 we plotted the 1σ, 2σ and 3σ contours in the ∆m 2 eff -θ 23 plane for various combinations of the experiments. The true value is taken as θ 23 = 45 • and |∆m 2 eff | = 0.0024 eV 2 . In generating these plots we have kept δ CP fixed at −90 • in both true and test spectrum. The upper panels are for NH and the lower panels are for IH. From the plots we see that T2HK and DUNE have similar sensitivity. If we combine T2HK, the HK atmospheric neutrino data and DUNE, then the errors in ∆m 2 eff , sin 2 θ 23 and δ CP become 0.3%, 2% and 20%, respectively.

E. Precision of θ 13
In this paper we have discussed only the measurements of ∆m 2 eff , θ 23 and δ CP . Let us discuss briefly whether the combination of T2HK+HK+DUNE can improve the current precision on θ 13 which is obtained by the reactor experiments. It is known [49] that if we fix the values of the both appearance probabilities P (ν µ → ν e ) and P (ν µ →ν e ), then it gives us a quadratic curve in the (sin 2 2θ 13 , 1/ sin 2 θ 23 ) plane, where a point sweeps the quadratic curve as δ CP varies from 0 to 2π.
The region of the quadratic curve in the (sin 2 2θ 13 , 1/ sin 2 θ 23 ) plane is the necessary and sufficient condition which can be obtained from the appearance probabilities P (ν µ → ν e ) and P (ν µ →ν e ) only. To get information on θ 13 from this quadratic curve without the reactor data, we need to combine the disappearance probabilities P (ν µ → ν µ ) and P (ν µ →ν µ ) with the appearance probabilities. Because the error δ(sin 2 θ 23 ) =(1/4)δ(sin 2 2θ 23 )/(1−sin 2 2θ 23 ) 1/2 is enhanced due to the singular Jacobian factor at the maximal mixing [50], the uncertainty in the vertical coordinate in the (sin 2 2θ 13 , 1/ sin 2 θ 23 ) plane is expected to be large. Notice that the uncertainty in the horizontal coordinate sin 2 2θ 13 is not enhanced in the case of the reactor measurements, because sin 2 2θ 13 appears linearly in the disappearance probability in the reactor experiments. From this discussion, we expect that it is difficult for the combination of T2HK+HK+DUNE to give a precision on θ 13 which is competitive with the current one from the reactor experiments.

IV. CONCLUSION
In this paper we have studied the sensitivity of T2HK, HK and DUNE to mass hierarchy, octant of the mixing angle θ 23 and δ CP . The main results of our analysis are summarized in Table I.
Although it is difficult for T2HK to resolve the sign degeneracy for unfavorable region of the CP phase, when we combine it with the atmospheric neutrino measurement at Hyperkamiokande, we can determine the mass hierarchy at 5 σ C.L. for any value of δ CP . We have also clarified how the octant degeneracy occurs and why its behavior depends on the mass hierarchy in the HK atmospheric neutrino measurements. On the other hand, DUNE can determine the mass hierarchy at least at 8 σ C.L. by itself. Furthermore, if we combined all of them, then the significance to mass hierarchy is at least 15 σ C.L. In our analysis we found out that the octant sensitivity of T2HK, T2HK+HK and DUNE are quite similar in ruling out the wrong octant at 5σ C.L. But for T2HK+HK+DUNE the increase in the octant sensitivity is significant. For CP violation discovery we find that the combination T2HK+HK can measure CP violation at 8 σ C.L. for δ CP = ±90 • and for T2HK+HK+DUNE the significance for CP violation is around 10 σ C.L. for δ CP = ±90 • .
It is also quite impressive that with the combination of all the three experiment CP violation can be established at 5σ C.L for at least 68% true values of δ CP . In the combination of all these experiments above, the precision in ∆m 2 eff , θ 23 and δ CP is 0.3%, 2% and 20%. The precision in the first two parameters is improved by one order of magnitude compared with the current data. We will be in the era of precision measurements of neutrino oscillation parameters, and combination of Hyperkamiokande and DUNE will play an important role in determination of δ CP as well as  NH42  NH44  NH46  NH48  NH50  IH40  IH42  IH44  IH46  IH48  IH50   620 NH42  NH44  NH46  NH48  NH50  IH40  IH42  IH44  IH46  IH48  IH50   1850     where we have introduced the simplified notations Here F 0 (ν α ) is the original flux of ν α before oscillations, P (ν α → ν β , MH) is the oscillation probability with a given Mass Hierarchy (either the True Hierarchy MH = T H or the Wrong Hierarchy MH = W H), σ(ν β ) (σ(ν β )) is the cross section for ν β (ν β ) to produce the charged lepton ℓ − β (ℓ + β ), and j stands for the index for the zenith angle bin ((j − 6)/5 < cos Θ < (j − 5)/5). To see how degeneracy occurs, let us take a look, for simplicity, at the number of events for the zenith bin j = 1 (−1.0 < cos Θ < −0.8), in which the matter effect is expected to be important. Assuming that data is taken at Hyperkamiokande for 2000 days with 0.56 Mton fiducial volume, the numbers of events for each channel are estimated and are given in Table II. From these numbers of events we can understand that the main contribution for the µ-like events comes from the ν µ → ν µ channel, and that the main contribution for the e-like events comes from the ν µ → ν e channel. For NH (IH) we have more events in the neutrino (antineutrino) mode. To understand these results qualitatively, we have to take into account a few facts. First of all, in the case of the normal (inverted) hierarchy, the oscillation probability P (ν µ → ν e ) (P (ν µ →ν e )) is enhanced due to the matter effect. Secondly, around the neutrino energy region E ∼ 10GeV, the ratio of the atmospheric neutrinos for the zenith bin j = 1 (−1.0 < cos Θ < −0.8) is [44] F (ν µ ):F (ν µ ):F (ν e ):F (ν e ) ≃ 40:40:10:7. The reason that the µ/e ratio is high at E ∼ 10GeV is because muons do not have enough time to decay to produce e ± and ν e orν e at such a high energy.
In Fig. 10 we show the numbers of the multi-GeV events (e-like, µ-like) for the zenith angle −1 < cos Θ < −0.8, assuming that we can separate the neutrino and antineutrino modes. The left (right) panels are for the neutrino (antineutrino) modes. Because of the enhancement due to the matter effect, a remarkable dependence on θ 23 as well as separation between the two mass hierarchies can be seen for NH (IH) in the neutrino (antineutrino) mode. These features can be seen for the zenith angle region −1 < cos Θ −0.4. For the zenith angle region cos Θ −0.4, the matter effect is not so dramatic, and we do not have the distinction between the two mass hierarchies.
In Fig. 11 we show the numbers of the multi-GeV events (e-like, µ-like), in which the neutrino and antineutrino events are combined together, as is done in most of the data of waterČerenkov detectors. From this plot it is easy to see that the NH -LO solution is confused with the IH -HO solution for both the e-like and µ-like events. Since the variation of the numbers of events with respect to θ 23 is smaller in IH than in NH, the minimum value of hierarchy χ 2 is expected to be larger in the case of T H = NH than in the case of T H = IH. This implies that the significance of the mass hierarchy is larger in the case of T H = NH than in the case of T H = IH.