Neutrino Oscillation Physics Potential of the T2K Experiment

The observation of the recent electron neutrino appearance in a muon neutrino beam and the high-precision measurement of the mixing angle $\theta_{13}$ have led to a re-evaluation of the physics potential of the T2K long-baseline neutrino oscillation experiment. Sensitivities are explored for CP violation in neutrinos, non-maximal $\sin^22\theta_{23}$, the octant of $\theta_{23}$, and the mass hierarchy, in addition to the measurements of $\delta_{CP}$, $\sin^2\theta_{23}$, and $\Delta m^2_{32}$, for various combinations of $\nu$-mode and \(\bar{\nu}\)-mode data-taking. With an exposure of $7.8\times10^{21}$~protons-on-target, T2K can achieve 1-$\sigma$ resolution of 0.050(0.054) on $\sin^2\theta_{23}$ and $0.040(0.045)\times10^{-3}~\rm{eV}^2$ on $\Delta m^2_{32}$ for 100\%(50\%) neutrino beam mode running assuming $\sin^2\theta_{23}=0.5$ and $\Delta m^2_{32} = 2.4\times10^{-3}$ eV$^2$. T2K will have sensitivity to the CP-violating phase $\delta_{\rm{CP}}$ at 90\% C.L. or better over a significant range. For example, if $\sin^22\theta_{23}$ is maximal (i.e $\theta_{23}$=$45^\circ$) the range is $-115^\circ<\delta_{\rm{CP}}<-60^\circ$ for normal hierarchy and $+50^\circ<\delta_{\rm{CP}}<+130^\circ$ for inverted hierarchy. When T2K data is combined with data from the NO$\nu$A experiment, the region of oscillation parameter space where there is sensitivity to observe a non-zero $\delta_{CP}$ is substantially increased compared to if each experiment is analyzed alone.

The observation of the recent electron neutrino appearance in a muon neutrino beam and the high-precision measurement of the mixing angle θ 13 have led to a re-evaluation of the physics potential of the T2K long-baseline neutrino oscillation experiment. Sensitivities are explored for CP violation in neutrinos, non-maximal sin 2 2θ 23 , the octant of θ 23 , and the mass hierarchy, in addition to the measurements of δ CP , sin 2 θ 23 , and ∆m 2 32 , for various combinations of ν-mode andν-mode data-taking. With an exposure of 7.8 × 10 21 protons-on-target, T2K can achieve 1-σ resolution of 0.050(0.054) on sin 2 θ 23 and 0.040(0.045) × 10 −3 eV 2 on ∆m 2 32 for 100%(50%) neutrino beam mode running assuming sin 2 θ 23 = 0.5 and ∆m 2 32 = 2.4 × 10 −3 eV 2 . T2K will have sensitivity to the CP-violating phase δ CP at 90% C.L. or better over a significant range. For example, if sin 2 2θ 23 is maximal (i.e θ 23 =45 • ) the range is −115 • < δ CP < −60 • for normal hierarchy and +50 • < δ CP < +130 • for inverted hierarchy. When T2K data is combined with data from the NOνA experiment, the region of oscillation parameter space where there is sensitivity to observe a non-zero δ CP is substantially increased compared to if each experiment is analyzed alone.

Introduction
The experimental confirmation of neutrino oscillations, where neutrinos of a particular flavor (ν e ,ν µ ,ν τ ) can transmute to another flavor, has profound implications for physics. The observation of a zenith-angle-dependent deficit in muon neutrinos produced by high-energy proton interactions in the atmosphere [1] confirmed the neutrino flavor oscillation hypothesis. The "anomalous" solar neutrino flux [2] problem was shown to be due to neutrino oscillation by more precise measurements [3,4,5,6]. Atmospheric neutrino measurements have provided further precision on the disappearance of muon neutrinos [7,8] and the appearance of tau neutrinos [9]. Taking advantage of nuclear reactors as intense sources, the disappearance of electron antineutrinos has been firmly established using both widely distributed multiple sources at an average distance of 180 km [6] and from specialized detectors placed within ∼ 2 km [10,11,12]. The development of high-intensity proton accelerators that can produce focused neutrino beams with mean energy from a few hundred MeV to tens of GeV have enabled measurements of the disappearance of muon-neutrinos (and muon antineutrinos) [8,13,14] and appearance of electron-neutrinos (and electron antineutrinos) [15,16,17,18] and tau-neutrinos [19] over distances of hundreds of kilometers.
While the early solar and atmospheric oscillation experiments could be described in a twoneutrino framework, recent experiments with diverse neutrino sources support a three-flavor oscillation framework. In this scenario, the three neutrino flavor eigenstates mix with three mass eigenstates (ν 1 , ν 2 , ν 3 ) through the Pontecorvo-Maki-Nakagawa-Sakata [20] (PMNS) matrix in terms of three mixing angles (θ 12 , θ 23 , θ 13 ) and one complex phase (δ CP ). The probability of neutrino oscillation depends on these parameters, as well as the difference of the squared masses of the mass states (∆m 2 21 , ∆m 2 31 , ∆m 2 32 ). Furthermore, there is an explicit dependence on the energy of the neutrino (E ν ) and the distance traveled (L) before detection. To date, all the experimental results are well-described within the neutrino oscillation framework as described in Sec. 2.
The T2K experiment began data taking in 2009 [22] and a major physics goal, the discovery of ν µ →ν e appearance, has been realized at 7.3 σ level of significance with just 8.4% of the total approved POT [17]. This is the first time an explicit flavor appearance has been observed from another neutrino flavor with significance larger than 5σ. This observation opens the door to study CP violation (CPV) in neutrinos as described in Sec. 2. Following this discovery, the primary physics goal for the neutrino physics community has become a detailed investigation of the three-flavor paradigm which requires determination of the CPviolating phase δ CP , resolution of the mass hierarchy (MH), and determination of the θ 23 octant, i.e., whether the mixing angle θ 23 is less than or greater than 45 • . T2K, along with the NOνA [23] experiment that recently began operation, will lead in the determination of these parameters for at least a decade.
This paper provides a comprehensive update of the anticipated sensitivity of the T2K experiment to the oscillation parameters as given in the original T2K proposal [21], and includes an investigation of the enhancements from performing combined fits including the projected NOνA sensitivity. It starts with a brief overview of the neutrino oscillation framework in Sec. 2, and a description of the T2K experiment in Sec. 3. Updated T2K sensitivities are given in Sec. 4, while sensitivities when results from T2K are combined with those from the NOνA experiment are given in Sec. 5. Finally, results of a study of the optimization of the ν andν running time for both T2K and NOνA are given in Sec. 6.

Neutrino Mixing and Oscillation Framework
Three-generation neutrino mixing can be described by a unitary matrix, often referred to as the PMNS matrix. The weak flavor eigenstates, ν e , ν µ , and ν τ are related to the mass eigenstates, ν 1 , ν 2 , and ν 3 , by the unitary mixing matrix U : where the matrix is commonly parameterized as with C ij (S ij ) representing cos θ ij (sin θ ij ), where θ ij is the mixing angle between the generations i and j. There is one irreducible phase, δ CP , allowed in a unitary 3×3 mixing matrix. 1 After neutrinos propagate through vacuum, the probability that they will interact via one of the three flavors will depend on the values of these mixing angles. As neutrinos propagate through matter, coherent forward scattering of electron-neutrinos causes a change in the effective neutrino mass that leads to a modification of the oscillation probability. This is the so-called matter effect. Interference between multiple terms in the transition probability can lead to CP violation in neutrino mixing if the phase δ CP is non-zero.
For T2K, the neutrino oscillation modes of interest are the ν µ → ν e appearance mode and the ν µ disappearance mode. The ν µ → ν e appearance oscillation probability (to first order approximation in the matter effect [24]) is given by ) are a consequence of the matter effect, where n e and ρ are the electron and matter densities, respectively. At the T2K peak energy of ∼ 0.6 GeV these terms account for nearly 10% of the oscillation probability. The equivalent expression for antineutrino appearance,ν µ →ν e , is obtained by reversing the signs of terms proportional to sin δ CP and a.
The ν µ disappearance oscillation probability is given by (where other matter effect and ∆m 2 21 terms can be neglected). The long-baseline ν µ → ν e appearance measurement is particularly sensitive to sin 2 2θ 13 (= 4C 2 13 S 2 13 ) and δ CP , while the ν µ disappearance measurement is more sensitive to sin 2 2θ 23 and ∆m 2 32 , as these parameters appear in the leading-order terms of the respective appearance and disappearance probability equations. Currently, the measured value of sin 2 2θ 23 is consistent with full mixing, but more data are required to know if that is the case. If the mixing is not maximal, the ν e appearance data have the potential to resolve the θ 23 octant degeneracy. The measurement of θ 13 from the reactor experiments is independent of the CP phase, and future measurements from Daya Bay [10], Double Chooz [11] and RENO [12] will reduce the θ 13 uncertainty such that the significance of the CP violating term will be enhanced for T2K. For reference, the third (CP-violating) term of Eq. 3 is 27% of the first term at the L/E corresponding to the oscillation maximum when sin 2 2θ 23 = 1, meaning that the CP-violating term makes a non-negligible contribution to the total ν e appearance probability. It is also important to recognize that since the sign of the CP-violating term is opposite for neutrino and antineutrino oscillations, data taken by T2K with an antineutrino beam for comparison to neutrino data may allow us to study CP violation effects.
The NOνA experiment is similar to T2K in the basic goals to measure ν µ disappearance and ν e appearance in an off-axis muon neutrino beam. The most important difference between the two experiments is the distance from the neutrino source to the far detector, 810 km for NOνA and 295 km for T2K, with a correspondingly higher peak neutrino beam energy for NOνA to maximize the appearance probability. NOνA is projected to have similar sensitivity compared to T2K for θ 23 , θ 13 , and δ CP , but better sensitivity to the sign of ∆m 2 32 since, as can be seen in the last term of Eq. 3, the size of the matter effect is proportional to the distance L. The combination of results from the two experiments at different baselines will further improve the sensitivity to the sign of ∆m 2 32 and to δ CP . 6 In this paper we present the updated T2K sensitivity to neutrino oscillation parameters using a large value of sin 2 2θ 13 similar to that measured by the reactor experiments, together with the sensitivity when projected T2K and NOνA results are combined.
The latest measured values of the neutrino mixing parameters (θ 12 , θ 23 , θ 13 , |∆m 2 32 |, ∆m 2 21 , δ CP ) are listed in Table 1 [25]. The CP-violating phase, δ CP , is not yet well constrained, nor is the sign of ∆m 2 32 ≡ m 2 3 − m 2 2 known. The sign of ∆m 2 32 is related to the ordering of the three mass eigenstates; the positive sign is referred to as the normal MH (NH) and the negative sign as the inverted MH (IH). Of the mixing angles, the angle θ 23 is measured with the least precision; the value of sin 2 2θ 23 in Table 1 corresponds to 0.4 < sin 2 (θ 23 ) < 0.6.

T2K Experiment
The T2K experiment [22] uses a 30-GeV proton beam accelerated by the J-PARC accelerator facility. This is composed of (1) the muon neutrino beamline, (2) the near detector complex, which is located 280 m downstream of the neutrino production target, monitors the beam, and constrains the neutrino flux parameterization and cross sections, and (3) the far detector, Super-Kamiokande (Super-K), which detects neutrinos at a baseline distance of 295 km from the target. The neutrino beam is directed 2.5 • away from Super-K, producing a narrow-band ν µ beam [26] at the far detector. The off-axis angle is chosen such that the energy peaks at E ν =∆m 2 32 L/2π ≈ 0.6 GeV, which corresponds to the first oscillation minimum of the ν µ survival probability at Super-K. This enhances the sensitivity to θ 13 and θ 23 and reduces backgrounds from higher-energy neutrino interactions at Super-K.
The J-PARC main ring accelerator provides a fast-extracted high-intensity proton beam to a graphite target located in the first of three consecutive electro-magnetic horns. Pions and kaons produced in the target are focused by the horns and decay in flight to muons and ν µ 's in the helium-filled 96-m-long decay tunnel. This is followed by a beam dump and a set of muon monitors, which are used to monitor the direction and stability of the neutrino beam.
The near detector complex contains an on-axis Interactive Neutrino Grid detector (INGRID) [27] and an off-axis magnetized detector, ND280. INGRID measures the neutrino interaction event rate at various positions from 0 • to ∼ 1 • around the beam axis, and provides monitoring of the intensity, direction, profile, and stability of the neutrino beam. The ND280 off-axis detector measures neutrino beam properties and neutrino interactions at approximately the same off-axis angle as Super-K. It is enclosed in a 0.2-T magnet that contains a subdetector optimized to measure π 0 s (PØD) [28], three time projection chambers (TPC1,2,3) [29] alternating with two one-ton fine-grained detectors (FGD1,2) [30], and an electromagnetic calorimeter (ECal) that surrounds the TPC, FGD, and PØD detectors. A side muon range detector (SMRD) [31] built into slots in the magnet return-yoke steel detects muons that exit or stop in the magnet steel. A schematic diagram of the detector layout has been published elsewhere [22].
The Super-K water Cherenkov far detector [32] has a fiducial mass of 22.5 kt contained within a cylindrical inner detector (ID) instrumented with 11,129 inward facing 20-inch phototubes. Surrounding the ID is a 2-meter wide outer detector (OD) with 1,885 outwardfacing 8-inch phototubes. A Global Positioning System receiver with <150 ns precision synchronizes the timing between reconstructed Super-K events and the J-PARC beam spill.
T2K employs various analysis methods to estimate oscillation parameters from the data, but in general it is done by comparing the observed and predicted ν e and ν µ interaction rates and energy spectra at the far detector. The rate and spectrum depend on the oscillation parameters, the incident neutrino flux, neutrino interaction cross sections, and the detector response. The initial estimate of the neutrino flux is determined from detailed simulations incorporating proton beam measurements, INGRID measurements, and pion and kaon production measurements from the NA61/SHINE [33,34] experiment. The ND280 detector measurement of ν µ charged current (CC) events is used to constrain the initial flux estimates and parameters of the neutrino interaction models that affect the predicted rate and spectrum of neutrino interactions at both ND280 and Super-K. At Super-K, ν e and ν µ charged current quasi-elastic (CCQE) events, for which the neutrino energy can be reconstructed using simple kinematics, are selected. Efficiencies and backgrounds are determined through detailed simulations tuned to control samples which account for final state interactions (FSI) inside the nucleus and secondary hadronic interactions (SI) in the detector material. These combined results are used in a fit to determine the oscillation parameters.
As of May 2013, T2K has accumulated 6.57 × 10 20 POT, which corresponds to about 8.4% of the total approved data. Results from this dataset on the measurement of θ 23 and |∆m 2 32 | by ν µ disappearance [14], and of θ 13 and δ CP by ν e appearance have been published [17]. It is reported in [17] that combining the T2K result with the world average value of θ 13 from reactor experiments leads to some values of δ CP being disfavored at 90% CL.

T2K Projected Sensitivities to Neutrino Oscillation Parameters
To demonstrate the T2K physics potential, we have performed sensitivity studies using combined fits to the reconstructed energy spectra of ν e (ν e ) and ν µ (ν µ ) events observed at Super-K with both ν-mode beam, andν-mode beam in the three-flavor mixing model. Results shown here generally use the systematic errors established for the 2012 oscillation analyses [35,16] as described below, although, in addition, we have studied cases with projected systematic errors as described in Sec. 4.5.
Since the sensitivity depends on the true values of the oscillation parameters, a set of oscillation parameters (θ) is chosen as a test point for each study and is used to generate simulated 'observed' reconstructed energy spectra. Then, a hypothesis test for the set of parameters of interest (H 0 ) is applied using 8 The value of χ 2 (H 0 ) is calculated as −2 ln L(θ|H 0 ), where L(θ|H 0 ) is the likelihood to observe the spectrum generated at θ when the 'true' oscillation parameters are given by H 0 . The minimum value of χ 2 in the oscillation parameter space is given by χ 2 min . The oscillation parameter set which gives χ 2 min is equivalent to θ, since spectra are generated without statistical fluctuations in this analysis. When we test only one or two of the five varied oscillation parameters (sin 2 2θ 13 , δ CP , sin 2 θ 23 , ∆m 2 32 , and the MH), the tested parameters are fixed at a set of test points, and the remaining oscillation parameters are fit to give a minimized χ 2 (H 0 ).
In most cases, this ∆χ 2 closely resembles a χ 2 distribution for n degrees of freedom, where n corresponds to the number of tested oscillation parameters. Then, critical χ 2 values for Gaussian distributed variables can be used for determining confidence level (C.L.) regions [36]. Each simulated spectrum is generated at the MC sample statistical mean, and therefore the results of this test represent the median sensitivity. Thus the results of these studies indicate that half of experiments are expected to be able to reject H 0 at the reported C.L. This is accurate if two conditions are met: (1) the probability density function (pdf) for ∆χ 2 follows a true χ 2 distribution, and (2) the ∆χ 2 value calculated with the MC sample statistical mean spectra is equivalent to the median of the ∆χ 2 pdf. Then, ∆χ 2 can be used to construct C.L. contours. Studies using ensembles of toy MC experiments where statistical and systematic fluctuations are included have shown these assumptions to be fairly accurate except in the case of a mass hierarchy determination. This exception will be discussed in detail in Sec. 5.

Expected observables and summary of current systematic errors
Our sensitivity studies are based on the signal efficiency, background, and systematic errors established for the T2K 2012 oscillation analyses; however, we note that errors are lower in more recent published analyses. Since official T2K systematic errors are used, these errors have been reliably estimated based on data analysis, unlike previous sensitivity studies which use errors based only on simulation and estimations [21]. Systematic errors therefore include both normalization and shape errors, and are implemented as a covariance matrix for these studies, where full correlation between ν-andν-modes is generally assumed.
Fits are performed by calculating ∆χ 2 using a binned likelihood method for the appearance and disappearance reconstructed energy spectra in Super-K. Reconstructed appearance and disappearance energy spectra generated for the approved full T2K statistics, 7.8 × 10 21 POT, assuming a data-taking condition of either 100% ν-mode or 100%ν-mode, after the standard T2K cuts (equivalent to those used by previous analyses [35,16]) are given in Fig. 1. These spectra are generated assuming the nominal oscillation parameters given in Table 2. When performing fits, the oscillation parameters δ CP , sin 2 2θ 13 , sin 2 θ 23 , and ∆m 2 32 are considered unknown unless otherwise stated, while sin 2 2θ 12 and ∆m 2 21 are assumed fixed to the values given in this table. Tables 3 and 4 give the number of events expected with the T2K full statistics. Fig. 2 shows the dependence of the ν e appearance reconstructed energy spectrum on δ CP . Some of the sensitivities are enhanced by constraining the error on sin 2 2θ 13 based on the projected precision of reactor measurements. For this study, the uncertainty (referred to as the ultimate reactor error) on sin 2 2θ 13 is chosen to be 0.005, which corresponds to   Table 2 the 2012 systematic error only of the Daya Bay experiment [37] 2 . Although shape errors are used for this analysis, the total error on the number of events at Super-K is given in Table 5.

Expected 90% C.L. regions
In this section we show expected 90% C.L. intervals for the T2K full statistics of 7.8 × 10 21 POT. Contours showing both the T2K sensitivity for δ CP vs. sin 2 2θ 13 and for ∆m 2 32 vs. sin 2 θ 23 are provided, where the assumed true value of the oscillation parameters is 2 The statistical error is 0.010 for [37] Table 4: Expected numbers of ν µ orν µ disappearance events for 7.8 × 10 21 POT. The first two columns show the number of ν µ andν µ events, broken down into those that undergo charged-current quasi-elastic (CCQE) scattering at Super-K, and those that undergo other types of CC scattering (CC non-QE). The third column shows CC ν e andν events, both from intrinsic beam backgrounds and oscillations, while the fourth column shows NC events. indicated by a black cross. The oscillation parameters δ CP , sin 2 2θ 13 , sin 2 θ 23 , and ∆m 2 32 are considered unknown, as stated above. Both the NH and IH are considered, and ∆χ 2 values are calculated from the minimum χ 2 value for both MH assumptions. The blue curves are generated assuming the correct MH and the red curves are generated assuming the incorrect MH, such that if an experiment or combination of experiments from the global neutrino community were to determine the MH the red contour would be eliminated. A contour consisting of the outermost edge of all contours in each plot can be considered as the T2K sensitivity assuming an unknown MH. Figure 3 gives an example of the difference in the shape of the T2K sensitive region for νvs.ν-mode at true δ CP = −90 • (and the other oscillation parameters as given in Table 2) by comparing the ν-mode - Fig. 3 (a) -andν-mode - Fig. 3 (b) -C.L. contours without a reactor constraint at 50% of the full T2K POT. These two contours are then combined in Fig. 3 (c), which shows the 90% C.L. region for 50% ν-plus 50%ν-mode running to achieve  the full T2K POT. This demonstrates that δ CP can be constrained by combining ν-mode andν-mode data. Figures 4 and 5 show example 90% C.L. regions for δ CP vs. sin 2 2θ 13 at the full T2K statistics, both for T2K alone and including an extra constraint on the T2K predicted data fit based on the ultimate reactor error δ(sin 2 2θ 13 ) = 0.005 as discussed above, for true δ CP of 0 • and −90 • , respectively. In the case of δ CP = −90 • , we start to have sensitivity to resolve δ CP without degeneracies. Table 5: The systematic errors in percentage on the predicted number of events at Super-K (assuming the oscillation parameters given in Table 2 are the true values of the oscillation parameters) as used in the 2012 oscillation analyses.

Appearance Disappearance
Flux and cross section constrained by the near detector 5.0 % 4.2 % Cross section not constrained by the near detector 7.4 % 6.2 % Super-K detector and FSI 3.9 % 11.0 % Total 9.7 % 13.3 % Figure 6 shows example 90% C.L. regions for ∆m 2 32 vs. sin 2 θ 23 at the full T2K statistics for sin 2 θ 23 = 0.4. The θ 23 octant can be resolved in this case by combining both ν-mode and ν-mode data and also including a reactor constraint on θ 13 , where this combination of inputs is required to resolve degeneracies between the oscillation parameters sin 2 θ 23 , sin 2 2θ 13 , and δ CP , demonstrating the importance of the reactor constraint in this case.

Sensitivities for CP-violating term, non-maximal θ 23 , and θ 23 octant
The sensitivities for CP violation, non-maximal θ 23 , and the octant of θ 23 depend on the true oscillation parameter values. Fig. 7 shows the expected ∆χ 2 for the sin δ CP = 0 hypothesis, for various true values of δ CP and sin 2 θ 23 . To see the dependence more clearly, ∆χ 2 is plotted as a function of δ CP for various values of sin 2 θ 23 in Fig. 8 (normal MH case) and Fig. 9 (inverted MH case). For favorable sets of the oscillation parameters and mass hierarchy, T2K will have greater than 90% C.L. sensitivity to non-zero sin δ CP . Figures 10 and 11 show the sin 2 θ 23 vs. δ CP regions where T2K has more than a 90% C.L. sensitivity to reject maximal mixing or reject one octant of θ 23 . In each of these figures, the oscillation parameters δ CP , sin 2 2θ 13 , sin 2 θ 23 , ∆m 2 32 , and the MH are considered unknown and a constraint based on the ultimate reactor error is used. Note that the T2K sensitivity to reject maximal mixing is roughly independent of ν −ν running ratio, while the sensitivity to reject one octant is better when ν-andν-modes are combined. Again the combination of ν-andν-modes, as well as the tight constraint on θ 13 from the reactor measurement, are all required to resolve degeneracies between the oscillation parameters sin 2 θ 23 , sin 2 2θ 13 , and δ CP in order to resolve the θ 23 octant.
These figures show that by running with a significant amount ofν-mode, T2K has sensitivity to the CP-violating term and octant of θ 23 for a wider region of oscillation parameters (δ CP , θ 23 ) and for both mass hierarchies, particularly when systematic errors are taken into account. The optimal running ratio is discussed in more detail in Sec. 6.

Precision or sensitivity vs. POT
The T2K uncertainty (i.e. precision) vs. POT for sin 2 θ 23 and ∆m 2 32 is given in Fig. 12 for the 100% ν-mode running case and the 50% plus 50% ν −ν-mode running case. The precision includes either statistical errors only, statistical errors combined with the 2012 systematic errors, or statistical errors combined with conservatively-projected systematic errors for the full POT. See Sec. 4.5 for details about the projected systematic errors used. (c) 50% ν-, 50%ν-mode. (a) 100% ν-mode. (b) 50% ν-, 50%ν-mode. (c) 100% ν-mode, with ultimate reactor constraint. (d) 50% ν-, 50%ν-mode, with ultimate reactor constraint. Generally, the effect of the systematic errors is reduced by running with combined ν-mode andν-mode. When running 50% in ν-mode and 50% inν-mode, the statistical 1σ uncertainty of sin 2 θ 23 and ∆m 2 32 is 0.045 and 0.04 × 10 −3 eV 2 , respectively, at the T2K full statistics. It should be noted that the sensitivity to sin 2 θ 23 shown here for the current exposure (6.57 × 10 20 POT) is significantly worse than the most recent T2K result [14], and in fact the recent result is quite close to the final sensitivity (at 7.8 × 10 21 POT) shown. This apparent discrepancy comes from three factors. About half of the difference between the expected (c) 100% ν-mode, with ultimate reactor constraint. (d) 50% ν-, 50%ν-mode, with ultimate reactor constraint.   disappearance, the larger the error on sin 2 θ 23 becomes (where the studies here assume a true value of sin 2 θ 23 slightly lower than the point of maximal disappearance -sin 2 θ 23 = 0.5). Therefore, if results from future running continue to favor maximal disappearance we expect modest improvements in our current constraints, eventually approaching a value close to, and possibly slightly better than, the predicted final sensitivity shown here. Figure 13 shows the sin 2 θ 23 region where maximal mixing or one of the θ 23 octants can be rejected, as a function of POT in the case of 50% ν-plus 50%ν-mode running. Although these  plots are made under the condition that the true mass hierarchy is normal and δ CP = 0 • , dependence on these conditions is moderate in the case of 50% ν-plus 50%ν-mode running.
The sensitivity to reject the null hypothesis sin δ CP = 0 depends on the true oscillation parameters and is expected to be greatest for the case δ CP = +90 • and inverted MH. Figure 14 shows how the expected ∆χ 2 evolves as a function of POT in this case, as well as for δ CP = −90 • and normal MH, another case in which the sensitivity is high. These plots indicate the earliest case for T2K to observe CP violation. If the systematic error size is negligibly small, T2K may reach a higher sensitivity at an earlier stage by running in 100%  In Sec. 4.4 we showed the T2K sensitivity with projected systematic errors which are estimated based on a conservative expectation of T2K systematic error reduction. In this case the systematic error on the predicted number of events in Super-K is about 7% for the ν µ and ν e samples and about 14% for theν µ andν e samples. These errors were calculated by reducing the 2012 oscillation analysis errors by removing certain interaction model and cross section uncertainties from both the ν e -and ν µ -mode errors, and by additionally scaling all ν µ -mode errors down by a factor of two. Errors for theν µ -andν e -modes were estimated to be twice those of the ν µ -and ν e -modes, respectively. These  where the T2K oscillation analysis errors have similarly been reduced by improvements in understanding the relevant interactions and cross sections.
For the measurement of δ CP , studies have shown that it is desirable to reduce this to 5∼8% for the ν e sample and ∼10% for theν e sample to maximize the T2K sensitivity with full statistics. The measurement of δ CP is nearly independent of the size of the error on the ν µ andν µ samples as long as we can achieve uncertainty onν µ similar to the current uncertainty on ν µ . For the measurement of θ 23 and ∆m 2 32 , the systematic error sizes are 21  significant compared to the statistical error, and the result would benefit from systematic error reduction even for uncertainties as small as 5%.
These error reductions may also be achievable with the implementation of further T2K and external cross section and hadron production measurements, which continue to be made with improved precision.      Table 2. The solid curves include statistical errors only, while the dashed curves assume the 2012 systematic errors (black) or the projected systematic errors (red). A constraint based on the ultimate reactor precision is included.

T2K and NOνA Combined Sensitivities
The ability of T2K to measure the value of δ CP (or determine if CPV exists in the lepton sector) is greatly enhanced by the determination of the MH. This enhancement results from the nearly degenerate ν e appearance event rate predictions at Super-K in the normal hierarchy with positive values of δ CP compared to the inverted hierarchy with negative values of δ CP . Determination of the MH thus breaks the degeneracy, enhancing the δ CP resolution for ∼50% of δ CP values. T2K does not have sufficient sensitivity to determine the mass hierarchy by itself. The NOνA experiment [23], which started operating in 2014,  has a longer baseline (810 km) and higher peak neutrino energy (∼ 2 GeV) than T2K. Accordingly, the impact of the matter effect on the predicted far detector event spectra is larger in NOνA than in T2K, leading to a greater sensitivity to the mass hierarchy. Because of the complementary nature of these two experiments, better constraints on the oscillation parameters, δ CP , sin 2 θ 23 and the MH can be obtained by comparing the ν µ → ν e oscillation probability of the two experiments. To evaluate the benefit of combining the two experiments, we have developed a code based on GLoBES [39,40]. The studies using projected T2K and NOνA data samples show the full physics reach for the two experiments, individually and combined, along with studies aimed at optimization of the ν-mode toν-mode running ratios of the two experiments. Figure 15 shows the relation between the expected number of events of T2K and NOνA for various values of δ CP , sin 2 θ 23 and mass hierarchies. The NH and IH predictions occupy distinct regions in the plot suggesting how a combined analysis T2K-NOνA fit leads to increased sensitivity. However, this plot does not include the (statistical + systematic) uncertainties on measurements of these event rates. This would result in regions of overlap where the MH can not be determined, and the sensitivity to δ CP is degraded.
In order to evaluate the effect of combining the results from T2K and NOνA quantitatively, we have conducted a T2K-NOνA combined sensitivity study. The GLoBES [39,40] software package was used to fit oscillation parameters based on the reconstructed neutrino energy spectra of the two experiments. The fits were conducted by minimizing ∆χ 2 which is calculated from spectra generated with different sets of oscillation parameters, and includes penalty terms for deviations of the signal and background normalizations from nominal. The parameter values and parameters accounting for systematic uncertainties within their uncertainties). The oscillation parameters, unless otherwise stated, are those shown in Table 2. The GLoBES three-flavor analysis package works very similarly to fitter used for the studies presented in Section 4. Several validation studies were done to ensure that the two methods produced the same results when given the same inputs.
The T2K, NOνA, and combined sensitivities were generated using a modified version of GLoBES that allowed for use of inputs generated from Monte Carlo simulations of T2K neutrino interactions in the Super-Kamiokande detector. The inputs describing the NOνA experiment were developed in conjunction with NOνA collaborators, and validated against official NOνA sensitivity plots [41,42,43]. We assume the same run plan as presented in NOνA's TDR: 1.8 × 10 21 POT for ν and 1.8 × 10 21 POT forν modes. Table 6 summarizes the expected number of events for NOνA. The GLoBES inputs defining the analysis sample acceptances for the signal, the NC background, the ν µ CC background, and the ν e CC background were tuned to match the official event rate prediction from NOνA.
Since NOνA has only recently began taking data, detailed evaluation of systematic uncertainties is not yet published. Therefore, the combined sensitivity studies used a simplified systematics treatment for both T2K and NOνA: a 5% normalization uncertainty on signal events and a 10% normalization uncertainty on background events for both appearance and disappearance spectra. Uncertainties that impact the spectral shape are not considered. This is a reasonable choice since both experiments use a narrow band beam and much of the oscillation sensitivity comes from the measured event rates. The uncertainties are assumed to be uncorrelated for ν e appearance,ν e appearance, ν µ disappearance, andν µ disappearance. This simple systematics implementation is the same as the one adopted in the NOνA TDR and is also a reasonable representation of the projected uncertainties at T2K. The sensitivities shown here are obtained assuming sin 2 2θ 13 = 0.1 with the projected reactor constraint of 5%. When determining the MH, ∆χ 2 is not distributed according to a χ 2 distribution because the MH is a discrete, rather than a continuous, variable. Toy MC studies, where many pseudo-experiments are generated with statistical and systematic fluctuations, were used to evaluate the validity of applying a ∆χ 2 test statistic, as given in Eq. 5, for the MH determination.
The left column of Fig. 16 shows distributions for a test static for H 0 = IH: where χ 2 IH and χ 2 N H are the minimum χ 2 values obtained by fitting the oscillation parameters while fixing the MH to the inverted or normal mass hierarchy, respectively. This T is plotted here instead of ∆χ 2 for easier interpretation. In the figure, the blue (red) distributions are for the case where test or 'observed' spectra were generated for the inverted (normal) mass hierarchy with statistical and systematic fluctuations. Except for δ CP , the test oscillation parameters were fixed to the nominal values given in Table 2. The value of δ CP was fixed to that given in each caption for the NH, while it was thrown over all values of δ CP for the IH. This is done in order to calculate the p-value for H 0 = IH with unknown δ CP when the test point is in the NH [44]. The right column of Figure 16 is the same, but with the opposite MH hypothesis test (H 0 = NH): with a test point in the IH. The T -value calculated using the spectrum generated from the MC sample statistical mean (T M C ), which is generally used in this paper, is compared with the median T -value for the ensemble of toy MC experiments (T median ) in Table 7 for different oscillation parameter sets. The p-values calculated for T M C , assuming that ∆χ 2 follows a true χ 2 distribution, compared with the p-values calculated as the fraction of the T distribution for H 0 = (correct MH) above T median are also given. Figures 17 through 19 show plots of expected C.L. contours for T2K, NOνA and a T2K-NOνA combined fits as functions of sin 2 θ 23 vs. δ CP . Regions where sin δ CP = 0, one MH and one θ 23 octant are expected to be ruled out at the 90% C.L are shown. Significantly wider regions are covered by combining the results from T2K and NOνA.
In Figures 20 and 21 the ∆χ 2 for for sin δ CP = 0 and for each MH is plotted as a function of 'true' δ CP in case of sin 2 (θ 23 ) = 0.5. The 'true' value of sin 2 (θ 23 ) = 0.5 was chosen to   present a simplified view of the sensitivities for maximal mixing. The complex structure for positive (negative) values of δ CP with a true NH (IH) is due to the fact that ∆χ 2 calculation profiles over MH, and the expected number of ν e appearance events is nearly degenerate in these regions. These figures demonstrate the sensitivity of the two experiments, as well as the benefit of combined analysis of the two data sets on the ability to determine MH and CPV.

Neutrino Mode and Antineutrino Mode Running Time Optimization
As previously shown in Sec. 4, a significant fraction ofν-mode running improves the sensitivity to CP violation, especially when systematic uncertainties are taken into account. In this section studies of the ν:ν running ratios are shown for T2K, NOνA, and combined fits of T2K+NOνA simulated data using the tools developed Sec. 5. A set of metrics are defined that characterize the ability of each experiment or a combined fit of both experiments to constrain δ CP , reject δ CP = 0, or determine the MH. The following metrics are used in these studies: • δ CP half-width: The 1σ half-width is defined as half of the 1σ Confidence Interval (C.I.) about the true value of δ CP . In some cases there are degenerate 1σ C.I. regions in δ CP that are disconnected from the central value. In this case half of the width of the degenerate region is added to this metric. This is a measure of the precision that can be acheived in measurment of δ CP . • Median ∆χ 2 for δ CP = 0: This metric defines the ∆χ 2 value for which 50% of true δ CP values can be distinguished from δ CP = [0, π]. This is a measure of sensitivity to CPV. • Lowest ∆χ 2 for mass hierarchy determination: This metric defines the ∆χ 2 value at which the mass hierarchies can be distinguished for 100% of true δ CP values.
Each metric is calculated for a T2K+NOνA combined analysis for various ν:ν run ratios. Figure 22 gives the lowest ∆χ 2 values for mass hierarchy determination for ν:ν variations in a combined T2K+NOνA fit. They are computed from the results of studies like the one shown in Fig. 21 and conservatively summarize the content of the plot in one data point. For example, the lowest ∆χ 2 value for mass hierarchy determination at 1:0 (100% ν  where the incorrect mass hierarchy is predicted to be rejected at 90% C.L. for T2K+NOνA, assuming simple normalization systematics as described in the text. Figure 24 summarizes the data in Fig. 22 and compares it with the metric calculated for T2K only running. The black curve gives the lowest ∆χ 2 for MH determination in a combined, T2K+NOνA, fit as a function of T2K ν:ν running ratio with the NOνA running fixed at 1:1. As shown previously, the T2K data set alone has almost no sensitivity to the MH determination. The curves for 5:5 NOνA running with systematics (black dashed) shows an optimal T2K running ratio of around 6:4 for a combined fit. However, the metric is very flat with respect to the T2K ν:ν run ratio for ν running greater than 50%. Figure 25 shows  the summary for median ∆χ 2 for sin δ CP = 0. T2K run ratios between 1:0 and 5:5 produce relatively similar values of median ∆χ 2 for the combined fit. This is also true for combined T2K+NOνA running independent of the NOνA run plan optimization. There is a slight preference for all neutrino running in T2K in the combined fit. Figure 26 summarizes the δ CP 1σ width at various values of δ CP . Again, relatively similar values of δ CP 1σ width are expected for the T2K run ratios between 1:0 and 1:9. The predicted ∆χ 2 for rejecting sin δ CP = 0 hypothesis, as a function of δ CP for T2K (red), NOνA (blue), and T2K+NOνA (black). Dashed (solid) curves indicate studies where normalization systematics are (not) considered. The 'true' value of sin 2 (θ 23 ) is assumed to be 0.5, and the 'true' MH is assumed to be the NH (top) or the IH (bottom). The 'test' MH is unconstrained.
All of the metrics demonstrate a relatively flat response between approximately 7:3 and 3:7 for T2K and for T2K+NOνA (5:5) with systematics, with a worse response outside that range. These results are consistent with several other studies not shown in this paper (e.g. the measures of the precision on sin 2 θ 13 in ν-mode and inν-mode). The results are also robust with respect to reasonable variations in sin 2 θ 23 , δ CP and the MH. Thus, the results suggest that T2K run with a ν-mode toν-mode at ratio of 1:1 with an allowed variation is assumed to be 0.5, and the 'true' MH is assumed to be the NH (top) or the IH (bottom). The 'test' MH is unconstrained.
of ±20% of the total exposure. The variation can be used to optimize the experiment to any one analysis without significant degradation of the sensitivity to any other analysis. A more detailed optimization of the ν:ν run ratio will require tighter constraints on oscillation parameters from future analyses, a more detailed treatment of systematic uncertainties from both T2K and NOνA, and a clear prioritization of analysis goals from the T2K and NOνA collaborations.

Summary
In this paper we have presented studies of the T2K experiment sensitivity to oscillation parameters by performing a three-flavor analysis combining appearance and disappearance, for both ν-mode, andν-mode assuming the expected full statistics of 7.8 × 10 21 POT. The T2K precision study includes either statistical errors only, systematic errors established for the 2012 oscillation analyses, or conservatively projected systematic errors, and takes into consideration signal efficiency and background. We have derived the sensitivity to the oscillation parameters sin 2 2θ 13 , δ CP , sin 2 2θ 23 , and ∆m 2 32 for a range of the true parameter 35   values and using constraints from other experiments. For example, with equal exposure of ν-mode andν-mode and using signal efficiency from the 2012 analysis we project a dataset of approximately 100 ν e and 25ν e appearance events and 390 (270) ν µ and 130 (70)ν µ CCQE (CC non-QE) events. From these data, with the projected systematic uncertainties we would achieve a 1-σ resolution of 0.050(0.054) on sin 2 θ 23 and 0.040(0.045) × 10 −3 eV 2 on ∆m 2 32 for 100%(50%) neutrino beam mode running. T2K will also have sensitivity to the CP-violating phase δ CP at 90% C.L. or higher over a significant range. For example, if sin 2 θ 23 is maximal (i.e θ 23 =45 • ) the range is −115 • < δ CP < −60 • for normal hierarchy and +50 • < δ CP < +130 • for inverted hierarchy.
Since the ability of T2K to measure the value of δ CP is greatly enhanced by the knowledge of the mass hierarchy we have also incorporated the expected data from the NOνA experiment into our projections using the GLoBES tools. With the same normalization uncertainties of 5% on the signal and 10% on the background for both experiments we find, for example, that the predicted ∆χ 2 for rejecting the δ CP = 0 hypothesis for δ CP = +90 • , IH and sin 2 θ 23 = 0.5 from the combined experiment fit is 8.2 compared to 4.3 and 3.2 for T2K and NOνA alone, 38 respectively. The region of oscillation parameter space where there is sensitivity to observe a non-zero δ CP is substantially increased compared to if each experiment is analyzed alone.
From the investigation of dividing the running time between ν-andν-modes we found that an even split gives the best sensitivity for a wider region of the oscillation parameter space for both T2K data alone, and for T2K data in combination with NOνA, though the dependence on the ratio is not strong.
It is anticipated that the results of these studies will help to guide the optimization of the future run plan for T2K.