Neutrino Oscillations with the MINOS, MINOS+, T2K, and Nova Experiments

Neutrino Oscillations with the MINOS, MINOS+, T2K, and NOvA Experiments.

Tsuyoshi Nakaya Kyoto University, Department of Physics Kyoto 606-8502 Japan E-mail: [email protected]

Robert K Plunkett Fermi National Accelerator Laboratory Batavia, Illinois 60510 USA E-mail: [email protected]

Abstract. This paper discusses recent results and near-term prospects of the long- baseline neutrino experiments MINOS, MINOS+, T2K and NOvA. The non-zero value of the third neutrino mixing angle θ13 allows experimental analysis in a manner which explicitly exhibits appearance and disappearance dependencies on additional parameters associated with mass-hierarchy, CP violation, and any non-maximal θ23. These current and near-future experiments begin the era of precision accelerator long- baseline measurements and lay the framework within which future experimental results will be interpreted. arXiv:1507.08134v4 [hep-ex] 20 Nov 2015 Neutrino Oscillations with the MINOS, MINOS+, T2K, and NOvA Experiments. 2

1. Introduction to Long-Baseline Accelerator Experiments

1.1. Motivation and 3-Flavor Model Beginning with the successful operation of the K2K experiment [24], the physics com- munity has seen a profound expansion of our knowledge of the mixing of neutrinos, driven by long-baseline accelerator experiments [7, 4, 54, 45, 25], experiments studying atmospheric neutrinos [26, 27, 28], solar neutrinos [9], and, most recently, high-precision experiments with reactor neutrinos [55]. In this article we describe the current generation of the running long-baseline neutrino experiments T2K and MINOS/MINOS+ , and the status of the NOvA experiment which was commissioned in 2014. Each of these experiments was designed with primary and secondary goals. For example, MINOS had

as its principle justiﬁcation the measurement, via disappearance of νµ, of the mixing 2 2 2 parameters sin θ23 and ∆m32, with particular emphasis on ∆m32. T2K and NOvA were primarily designed to elucidate the structure of the neutrino sector by studies of

νe appearance. However they are making and will continue to make very signiﬁcant contributions to the study of νµ disappearance as well. Similarly, MINOS has measured νe appearance and the angle θ13.

This situation leads us to a point we will emphasize throughout this article, namely that the traditional distinction between various modes of study of mixing, matter eﬀects, and CP violation is rapidly giving way to a more integrated approach which utilizes both major types of signals to gain maximal information about the somewhat complicated 3-neutrino sector. We will ﬁrst discuss the relevant formalism and the major measurements required to test it. Next we will provide a technical overview of the powerful neutrino beams required for the measurements, and then proceed to discuss the experiments together with their current measurements and expected sensitivities. Finally, we conclude the article with a discussion of the near-term future for these experiments.

It is our goal to familiarize the reader with the surprisingly rich information already available in studies of this sector of physics, the only one currently not well-handled by the Standard Model. In addition, we will provide context for future discussions of progress to be provided by these experiments and by the exciting future world of very large experiments, and very long baselines.

1.2. Core measurements In the three-neutrino model there is a close relationship among the disappearance and appearance modes of oscillation study, going back to their origin in the PMNS matrix. Following reference [36], it is possible to write the disappearance possibility for muon neutrinos in vacuum, as Neutrino Oscillations with the MINOS, MINOS+, T2K, and NOvA Experiments. 3

∆m2 L P (ν ν ) = 1 4 sin2 θ cos2 θ (1 sin2 θ cos2 θ ) sin2 eff (1) µ → µ − 23 13 − 23 13 4E 2 where ∆meff incorporates eﬀective leading dependences on the additional PMNS pa- 2 rameters ∆m21, θ13, and δCP as

2 2 2 2 2 ∆meff = ∆m32 + ∆m21 sin θ12 + ∆m21 cos δCP sin θ13 tan θ23 sin 2θ12 Equation 1 may be simply manipulated to yield a form appropriate for the baseline of the T2K experiment, namely

∆m2 L P (ν ν ) 1 (cos4 θ sin2 2θ + sin2 2θ sin2 θ ) sin2 31 (2) µ → µ ' − 13 23 13 23 4E Here we see the vital role of the mixing angle θ13, which couples in the PMNS matrix to the CP-violating phase δCP . It is now well-known that this angle is relatively large, approximately 9 degrees [9]. T2K, MINOS/MINOS+, and future NOvA measurements of this angle and its consequences are discussed in sections 3, 4 and 7. Because the earth between the beam creation point and the detector location forms an essential part of any long-baseline experiment, its effects on the measurements must be considered. This creates both problems and opportunities - problems because of the introduction of degeneracies between matter effects and CP violation, and opportunities because of the possibilities to exploit the differences between neutrino and antineutrino interactions, and from the two mass hierarchies.

2 1.2.1. Appearance measurements and sin 2θ13

A large value of θ13 is key to allowing an integrated approach to oscillation studies. MINOS, T2K, and soon NOvA use the appearance channel for νe with a muon neutrino beam to probe θ13 directly. In appropriate approximation for a muon neutrino with energy Eν of O(1) GeV traveling a distance of O(100) km, the leading order equation governing the appearance probability is: ∆m2 L P (ν ν ) sin2 θ sin2 2θ sin2 32 (3) µ → e ≈ 23 13 4E Equation 3 is applicable for both neutrino and anti-neutrino oscillations. The diﬀerence between neutrinos and anti-neutrinos in oscillation appears as a sub- 2 leading eﬀect, including the solar parameters θ12 and ∆m21 and CP violation phase δCP. The probability is expressed [32, 33] as P (ν ν ) sin2 2θ T α sin 2θ T + α sin 2θ T + α4T (4) µ → e ' 13 1 − 13 2 13 3 4 Neutrino Oscillations with the MINOS, MINOS+, T2K, and NOvA Experiments. 4

2 ∆m21 where α 2 is the small ( 1/30) ratio between the solar and atmospheric squared- ≡ ∆m31 ∼ mass splittings, and sin2[(1 A)∆] T = sin2 θ − , 1 23 (1 A2) − sin(A∆) sin[(1 A)∆] T = sin δ sin 2θ sin 2θ sin ∆ − , 2 CP 12 23 A 1 A sin(A∆) sin[(1− A)∆] T = cos δ sin 2θ sin 2θ cos ∆ − , 3 CP 12 23 A 1 A − sin2(A∆) T = cos2 θ sin2 2θ . 4 23 12 A2 2 ∆m32L 2 Here, ∆ = and A 2√2GF neEν/∆m , where Ne is the electron density of 4Eν ≡ 32 Earth’s crust. In Equation 4, the sign of the second term changes for anti-neutrinos, governing CP violation when all three mixing angles, including θ13, have nonzero values. With current best knowledge of oscillation parameters, the CP violation (sub-leading term) can be as large as 30% of the leading term. ∼ The A dependence arises from matter eﬀects (caused by additional terms in the Hamiltonian for the electron component of the neutrino eigenstate), which are coupled 2 2 with the sign of ∆m32. In this paper, we refer to ∆m32 > 0 as the normal mass hierarchy 2 and ∆m32 < 0 as the inverted one.

1.2.2. Hierarchy, Octants, and CP Violation (δCP )

The subleading terms shown in Equations 1 and 4 make it possible in principle for oscillation measurements to be sensitive to the octant of θ23 (θ23 < π/4 or θ23 > π/4), in the case it is not maximal (i.e. = π/4). Even for maximal mixing, the additional dependencies in Equation 4, and further terms added to Equation 2 enable the combination of disappearance and appearance results to begin to give clues to the CP and hierarchy puzzles.

2. Beamlines

2.1. NuMI The conceptual beginning of the NuMI beamline dates to the era of the construction of the Fermilab Main Injector. The beamline, together with its associated tunnels, experimental halls, surface buildings and infrastructure, was built between 1999 and 2004. Datataking with the beam began in March of 2005, and continues to this time. The complex currently consists of a primary beam transport, the NuMI target hall, a 675 meter He-filled decay pipe of one meter radius, a hadron absorber and muon flux monitor area, meters of rock shielding, and two experimental halls - the first housing the MINOS near detector and the MINERvA detector, and the second housing the NOvA near detector. Figure 1 shows the general configuration of the beamline. Neutrino Oscillations with the MINOS, MINOS+, T2K, and NOvA Experiments. 5

Figure 1. Layout of the NuMI beamline at Fermilab, showing main components of the target, focusing, and decay systems [2].

Beam from the Fermilab Booster is accelerated in the Main Injector to 120 GeV, and then extracted with a system of fast kickers. As part of an upgrade to beam power, commissioning is underway to stack beam in the Fermilab Recycler Ring before transfer to the Main Injector. At the NuMI target the proton beam consists of 6 batches, with a total extraction period of 10 µs. The time between extractions has varied from 2.2 s to the current 1.3 s. This low duty factor allows the MINOS and NOvA experiments to trigger on a simple timing window, which facilitates the surface location of the NOvA far detector. Overlaying multiple injections from the booster in the main injector (slip- stacking) has allowed the beam intensity to reach 375 kW. The NuMI beam for NOvA is anticipated to reach 700 kW, with a similar time structure. Secondary hadrons created in the interaction of the extracted proton beam with a 94 cm graphite target are focused by a system of two magnetic horns. Historically, the focused hadron momentum (which translates into neutrino energy) has been adjustable by moving the target w.r.t the first horn. This is more difficult in the NOvA beam configuration, which is optimized for the off-axis application. Figure 2 shows the focusing schema of the NuMI beam.The majority of the data samples used in MINOS physics analyses have used the NuMI beam line focused at its lowest practical energy configuration, with a peak neutrino energy at approximately 3 GeV. Additional samples at higher neutrino beam energy settings have been used to provide information on the intrinsic νe component of the beam, an important and irreducible background for the 2 measurement of sin 2θ13. Overall beam production for the MINOS running period 2005-2012, is shown in Figure 3. An important feature of horn-focused beamlines like the NuMI beamline is the ability to convert from focusing positive hadrons (primarily π+) to negative hadrons

(primarily π−), which creates a beam heavily enriched in antineutrinos. Using this

beam, MINOS has published special studies of the oscillation parameters ofν ¯µ [40]. Targetry which can withstand the repeated high power proton pulses needed for a neutrino beam represents a technical challenge. The NuMI targets used for MINOS data taking were constructed of 47 segmented graphite ﬁns. A total of 7 targets were used in the period 2005 to 2012, with exchanges usually due to failures in auxiliary Neutrino Oscillations with the MINOS, MINOS+, T2K, and NOvA Experiments. 6

Horn 1 Horn 2 baffle B target π+, K+

I I ⊗ π−

Figure 2. Detail of the magnetic focusing horn system for the NuMI beam line [41].

Figure 3. Total NuMI beam delivery during MINOS running. The total collected in the principle neutrino configuration was 10.7 x 1020 protons. For the antineutrino configuration the total was 3.4 x 1020 protons. cooling systems. In one case the target material experienced significant degradation, visible in the produced neutrino rates. Significant engineering changes have occurred for the targets to be used in the NOvA beam, which must withstand 700 kW operations. These include detailed changes to the graphite fins to allow for an increase in primary beam spot size from 1.1 mm to 1.3 mm, and, importantly, a significant relocation of the water cooling tubing to decrease its vulnerability. With these changes, it is expected that the NuMI targets in the NOvA era will survive a minimum of a year of high power operation before any replacement is needed.

2.2. Off axis neutrino beam An off axis neutrino beam (OAB) configuration [1] is a method to produce a narrow band energy neutrino beam. In the OAB configuration, the axis of the beam optics is Neutrino Oscillations with the MINOS, MINOS+, T2K, and NOvA Experiments. 7

intentionally shifted by a few degrees from the detector direction. With a ﬁnite decay angle, the neutrino energy becomes almost independent of the parent pion energy due to characteristics of the two body decay kinematics of the pion with Lorentz boost. The oﬀ-axis beam principle can be illustrated with a simple algebraic example. Let us model the beam as consisting of pions which are fully focused in the on-axis directions. The transverse and perpendicular components of the decay neutrino momentum obey the relations:

P = P ∗ sin θ∗,P = γP ∗(1 + cos θ∗) E (5) T L ≈ ν

where P ∗ and θ∗ are the decay momentum and angle in the rest frame of the decaying particle. The ﬁxed oﬀ-axis angle condition is θlab = PT /PL. Near θ∗ = π/2, we have ∆P 0 for variations in θ∗, and therefore T ≈ ∆PT ∆PL = 0 (6) θlab ≈ Physically, the constraint on the angle means that the variation of neutrino energy

that normally occurs when θ∗ varies is greatly reduced, and parent particles of many energies contribute to a single peak in neutrino energy. As a consequence, the peak energy of the neutrino beam depends on the off-axis angle. Figure 5 illustrates this effect graphically for several off-axis angles in the NuMI configuration. By changing the off axis angle, it is possible to tune the neutrino beam energy to maximize the sensitivity of the oscillation parameters. As a reference, the off axis angle can be varied from 2.5 to 3.0 degree in the T2K beamline, which corresponds to a mean energy of neutrinos in the range from 0.5 to 0.9 GeV. The neutrino energy spectra at the far detector (Super-Kamiokande) with different off-axis angles in T2K are shown in Figure 4 [3]. In T2K, the off-axis angle is set to 2.5 degrees. The NOvA experiment is situated at an off-axis angle of 14 mrad (0.8 degrees). With a higher beam energy focusing than used for the MINOS program, this results in a large flux at the neutrino energy associated with oscillations. At the same time, it reduces backgrounds from neutral current (NC) interactions from higher energies and

from intrinsic beam νe, which have a wider energy distribution.

2.3. T2K neutrino beam J-PARC, the Japan Proton Accelerator Research Complex, is the accelerator complex supplying 30 GeV protons to the T2K experiment. An intense neutrino beam with a narrow-band energy spectrum is produced using the off-axis technique. The beam energy is tuned to the oscillation maximum ( 600 MeV for the T2K baseline of 295 km), which ∼ also suppresses the high energy component contributing to background generation. The left plot in Figure 6 shows the prediction of the T2K neutrino beam flux at the far T2K NEUTRINO FLUX PREDICTION PHYSICAL REVIEW D 87, 012001 (2013) 56Department of Physics, University of Winnipeg, Winnipeg, Manitoba, Canada 57Faculty of Physics and Astronomy, Wroclaw University, Wroclaw, Poland 58Department of Physics and Astronomy, York University, Toronto, Ontario, Canada (Received 2 November 2012; published 2 January 2013) The Tokai-to-Kamioka (T2K) experiment studies neutrino oscillations using an off-axis muon neutrino beam with a peak energy of about 0.6 GeV that originates at the Japan Proton Accelerator Research Complex accelerator facility. Interactions of the neutrinos are observed at near detectors placed at 280 m from the production target and at the far detector—Super-Kamiokande—located 295 km away. The flux prediction is an essential part of the successful prediction of neutrino interaction rates at the T2K detectors and is an important input to T2K neutrino oscillation and cross section measurements. A FLUKA and GEANT3-based simulation models the physical processes involved in the neutrino production, from the interaction of primary beam protons in the T2K target, to the decay of hadrons and muons that produce neutrinos. The simulation uses proton beam monitor measurements as inputs. The modeling of hadronic interactions is reweighted using thin target hadron production data, including recent charged pion and kaon measurements from the NA61/SHINE experiment. For the first T2K analyses the uncertainties on the flux prediction are evaluated to be below 15% near the flux peak. The uncertainty on the ratio of the flux predictions at the far and near detectors is less than 2% near the flux peak.

DOI: 10.1103/PhysRevD.87.012001 PACS numbers: 24.10.Lx, 14.60.Lm

I. INTRODUCTION neutrino beam. By positioning a detector at an angle relative to the focusing axis, one will, therefore, see neutrinos with a Predicting the neutrino flux and energy spectrum is an narrow spread in energy. The peak energy of the neutrino important component of analyses in accelerator neutrino beam can be varied by changing the off-axis angle as illus- experiments [1–4]. However, it is difficult to simulate the trated in the lower panel of Fig. 1. In the case of T2K, the off- flux precisely due to uncertainties in the underlying physi- axis angle is set at 2.5 so that the neutrino beam at SK has a cal processes, particularly hadron production in proton- peak energy at about 0.6 GeV, near the expected first oscil- nucleus interactions. To reduce flux-related uncertainties, lation maximum (Fig. 1). This maximizes the effect of the neutrino oscillation experiments are sometimes conducted neutrino oscillations at 295 km as well as reduces back- by comparing measurements between a near detector site ground events. Since the energy spectrum changes depending and a far detector site, allowing for cancellation of corre- on the off-axis angle, the neutrino beam direction has to be lated uncertainties. Therefore, it is important to correctly precisely monitored. predict the relationship between the fluxes at the two detector sites, described below as the far-to-nearNeutrino ratio. Oscillations with the MINOS, MINOS+, T2K, and NOvA Experiments. 8 Tokai-to-Kamioka (T2K) [5,6] is a long-baseline neutrino oscillation experiment that uses an intense muon 1

neutrino beam to measure the mixing angle 13 via the ) µ ν e appearance [7] and the mixing angle 23 and mass 2

→ sin 2 = 1.0 2 0.5 θ23 difference Ám via the disappearance [8]. The muon µ 32 ν ∆m2 = 2.4 × 10-3 eV2 neutrino beam is produced as the decay products of pions P( 32 and kaons generated by the interaction of the 30 GeV proton beam from the Japan Proton Accelerator Research Complex (J-PARC) with a graphite target. The properties 1 OA 0.0° of the generated neutrinos are measured at near detectors OA 2.0° placed 280 m from the target and at the far detector, OA 2.5° Super-Kamiokande (SK) [9], which is located 295 km

away. The effect of oscillation is expected to be negligible (A.U.) 0.5 at the near detectors and signiﬁcant at SK. µ ν 295km

The T2K experiment employs the off-axis method [10] to Φ generate a narrow-band neutrino beam and this is the first time this technique has been used in a search for neutrino oscillations. The method utilizes the fact that the energy of 0 a neutrino emitted in the two-body pion (kaon) decay, the 0123 dominant mode for the neutrino production, at an angle E (GeV) relative to the parent meson direction is only weakly depen- ν dent on the momentum of the parent. The parent þðÀÞ’s are FIG. 1 (color online). Muon neutrino survival probability at Figure 4. The neutrino oscillation probability of νµ νµ and the neutrino energy focused parallel to the proton beam axis to produce the (anti-) 295 km and neutrino fluxes for different off-axis angles.→ spectrum with different off-axis angles in T2K from [3]

012001-3

Medium Energy Tune

80 on-axis 7 mrad off-axis 14 mrad off-axis 21 mrad off-axis

20 CC events / kt 1E21 POT 0.2 GeV μ ν

0 0 2 4 6 8 10 Eν (GeV)

Figure 5. The neutrino spectra (ﬂux times cross-section) for various angles in a medium-energy NuMI beam [39]. Neutrino Oscillations with the MINOS, MINOS+, T2K, and NOvA Experiments. 9 detector, Super-Kamiokande (Super-K). The ﬂux is dominated by muon neutrinos with a small fraction (at the level of a few %) of intrinsic electron neutrinos, referred to as

”beam νe”. The beam νe component is a major background when searching for electron neutrino appearance.

run1-4 at SK run5c at SK

6 6 10

p.o.t) 10 p.o.t) 21 21 νµ νe νµ νe 5 νµ νe 10 νµ νe 105 /50MeV/10 /50MeV/10 2 2

104

Flux (/cm 104 Flux (/cm 103

103 102

0 2 4 6 8 10 0 2 4 6 8 10 Eν (GeV) Eν (GeV)

Figure 6. Prediction of the T2K beam flux for neutrinos and antineutrinos at Super- K. The left plot is for the neutrino beam mode made by focusing the positively charged particles, and the right is for the anti-neutrino mode made by focusing the negative ones. The flux above Eν = 10 GeV is not shown although the flux is simulated up to Eν = 30 GeV.

The proton beam is directed onto a graphite target which is designed to accept 750 kW beam power. The target is a graphite rod of 91.4 cm long and 2.6 diameter with 1.8g/cm3 density. The target is helium-cooled. Since the current beam power is still around 350 kW, there is still a margin for safety. The details of the T2K target are found in ref. [13]. The positively charged particles (mainly pions) produced are focused by three magnetic horns, typically operated at 250 kA. The decay of the charged particles in a 100 m decay volume produces the neutrino beam. By reversing the direction of the horn current, negatively charged particles are focused to produce the anti-neutrino beam. The prediction of the anti-neutrino beam flux is shown at the right plot in Figure 6. Thanks to the off-axis technique, the signal to noise ratio of the anti-neutrino beam flux is as good as 30 at the flux peak, while the wrong sign component of ‡ neutrinos is broadly distributed in energy. T2K started physics data taking in January 2010. Although the data taking was interrupted on March 11, 2011 by the Great East Japan Earthquake, the experiment collected 6.57 1020 POT (protons on target) for analysis before May 2013 . The × § The signal to noise ratio of anti-neutrinos to neutrinos is typically much worse than that of neutrinos ‡ to anti-neutrinos. J-PARC stopped operation in May 2013 because of the hadron hall accident. § Neutrino Oscillations with the MINOS, MINOS+, T2K, and NOvA Experiments. 10 history of data taking is shown in Figure 7. In 2014, anti-neutrino beam running began. Today, a maximum beam power of 370 kW has been recorded in J-PARC.

Figure 7. The history of the delivered protons to the T2K experiment for analysis. The dots show the number of protons per pulse, and the lines show the integrated number of protons. The red dots are for the neutrino beam running, and the purple dots are for the anti-neutrinos.

3. Electron Appearance Analysis

3.1. MINOS The MINOS experiment, with its magnetized steel calorimeters, was principally designed to detect and classify the charged current reactions νµ + N µ− + X andν ¯µ + N + → → µ + X. In order to detect and measure the appearance of νe andν ¯e, which indicate a non-zero θ13, sophisticated statistical techniques must be used to disentangle the relative contributions of this signal from the similar neutral-current background. To do this, MINOS uses the LEM (Library Event Matching) technique [44]. This procedure uses large (> 107) simulated samples of signal and background events to form event- by-event comparisons of the observed deposited charge in detector channels with the equivalent simulated deposited charge in the library events. The LEM procedure gives a set of output variables which are used as input to a simple artiﬁcial neural net, giving a statistical discriminant ( αLEM ), which can be used to identify signal and background components of the data.

The discriminant αLEM is formed by the output of a neural net which has been given as inputs the reconstructed event energy and characteristics of the 50 best-matched Neutrino Oscillations with the MINOS, MINOS+, T2K, and NOvA Experiments. 11

library events, namely i) the fraction of these library events that are nue CC events, (ii) their average inelasticity (y) and (iii) the average overlapping fraction of charge on

strips between the data and the 50 library events. Events with αLEM > 0.6 are selected for further analysis.

Next, to search for appearance of νe due to the oscillation phenomenon, the spectra of the varying background components present in the beam (νµ-CC, NC, and residual beam νe-CC) need to be estimated using data from the near detector. This is done in MINOS by comparing samples obtained from diﬀerent beam focusing conﬁgurations for decaying secondaries, as discussed in ref. [43]. Figure 8, from ref. [45] shows the far

detector MINOS data and the expected backgrounds, for various bins of αLEM .

The final elements required to produce an appearance measurement are extrapolation of the background (and oscillated signal) estimates between near and far detectors, and an estimate of the signal efficiency. The first is done by comparison of the background measured in the ND with its simulated value, giving a correction factor that can be applied to the equivalent simulation of the far detector. The technique is simpler than that used for MINOS CC appearance measurement; however the essential equivalence of the methods has been demonstrated in ref. [41]. In order to estimate the signal efficiency, hybrid events were created by substitution of a simulated electron shower in shower-subtracted, well-identified CC events. The efficiencies obtained were > 55% in both beam configurations.

The principal systematic errors affect the result are uncertainties in the background estimation and in the signal efficiency. They are 3.8% (4.8%) and 2.8% (3.1%), respectively, for the ν (¯ν) modes. The measurement is dominated by statistical errors, affecting both the signal and the background estimation. MINOS systematic errors are discussed further in section 5.3.

After establishment of these techniques and their systematic errors, MINOS can now use the near detector spectrum to extrapolate the expectation of signal and background

for hypothesized values of the the physical parameters θ13, δCP , and mass hierarchy, to determine statistically allowed and disallowed regions. The overall background estimation for the ν beam configuration is 127.7 background events. For parameter values of sin2 2θ = 0.1, δ = 0, θ = π/4 and normal hierarchy, 33.7 1.9 appearance events 13 CP 23 ± are expected, giving a signal/background ratio of S/B = 0.26. A total of 152 events are observed. Contributions to the analysis from theν ¯ beam configuration are small, totaling only 21.4 expected and 20 observed events. The result from ref. [45] is shown in Figure 9 for the full MINOS dataset, consisting of 10.6 1020 POT for the ν beam × configuration and 3.3 1020 POT for theν ¯ beam configuration. This analysis does not × distinguish between neutrinos and antineutrinos. The result shows the characteristic

features of periodic variation with δCP , as well as shape inversion with hierarchy choice. Neutrino Oscillations with the MINOS, MINOS+, T2K, and NOvA Experiments. 12

Figure 8. MINOS far detector data with statistical errors [45] used for νe appearance 2 2 analysis, compared with expectations for sin (2θ13) = 0.051, ∆m32 > 0, δ = 0, and θ23 = π/4 .

2 2 +0.038 MINOS cites best-fit values for 2 sin 2θ13 sin θ23 of 0.051 0.030 under the normal hierar- +0.052 − chy assumption, and 0.093 0.049 for the inverted hierarchy, with 90% confidence ranges − of 0.01 - 0.12 and 0.03 - 0.18 , respectively. The best fits are all computed for δCP = 0, and θ23 < π/4.

It is of interest to examine the parameter space probed by this appearance analysis more closely. The MINOS collaboration has computed the change in likelihood for ex- cursions of the CP-violating phase δCP for four combinations of the hierarchy and θ23 octant parameters. This result from ref. [45] is shown in Figure 10. The experiment disfavors 31% of the total 3-parameter space (δCP , hierarchy, octant) at 68%C.L. and shows a suggestive, but statistically limited preference for the inverted mass hierarchy scenario. Neutrino Oscillations with the MINOS, MINOS+, T2K, and NOvA Experiments. 13

Figure 9. MINOS allowed contours at 68% and 90% conﬁdence for the measured 2 2 quantity 2 sin 2θ13 sin θ23, as a function of δCP . Systematic and statistical uncertainties are included, and results for both assumed hierarchies are displayed.

3.2. T2K The first evidence of non-zero θ was reported [5, 44] in the ν ν appearance 13 µ → e channel in 2011. Today, with more data collected in T2K, the ν ν transition is well µ → e established [7]. Twenty-eight electron candidate events in T2K have been observed in the T2K far detector (Super-K) by requiring one Cherenkov ”ring”, identified as an electron type with visible energy greater than 100 MeV. In addition, a newly developed algorithm was applied to suppress background events with a π0 2γ, where one of the photons is → missed in reconstruction. The details of event selection are found in [6, 7]. The number of observed events, compared with the expectations is shown in Table 1. The observed number of events, 28, is significantly larger than the expected number, 4.92 0.55, with 2 ± θ13 = 0, but is consistent with the expectation of 21.6 with sin 2θ13 = 0.10 and δCP = 0. The best fit value of θ has been evaluated to be sin2 2θ = 0.140 0.038(0.170 13 13 ± ± 0.045) with a 68 % confidence level (C.L.), by fixing the other oscillation parameters: Neutrino Oscillations with the MINOS, MINOS+, T2K, and NOvA Experiments. 14

Figure 10. Variation of likelihood (compared to best ﬁts) with δCP , plotted as 2∆ln(L), for the observed νe appearance in MINOS, shown for varying combinations − of other oscillation parameters. Values above the horizontal dashed lines are disfavored at either 68% or 90% C.L.

Table 1. The number of observed events with the MC expectations and eﬃciencies. 2 2 2 The oscillation parameters are assumed to be sin 2θ13 = 0.1, sin θ23 = 0.5, ∆m32 = −3 2 2 | | 2.4 10 eV , δCP = 0, and ∆m > 0 × 32 Total Signal ν ν ν + ν Beam ν + ν Data µ → e µ µ e e NC MC CC CC CC

νe events 28 21.6 17.3 0.1 3.2 1.0 Eﬃciency [%] - - 61.2 0.0 19.1 0.4

2 5 2 2 2 3 2 sin 2θ = 0.306, ∆m = 7.6 10− eV , sin θ = 0.5, ∆m = 2.4 10− eV , and 12 21 × 23 | 32| × δCP = 0. Figure 11 shows the electron momentum versus angle distribution (sensitive 2 to the oscillation), which is used to extract the oscillation parameters sin 2θ13 and δCP which give the best fit values. The significance for a nonzero θ13 is calculated to be 7.3 σ. 2 Allowed regions for sin 2θ13 as a function of δCP are evaluated as shown in Figure 12, 2 2 where the values of sin θ23 and ∆m32 are varied in the fit with additional constraints PHYSICAL REVIEW LETTERS week ending PRL 112, 061802 (2014) 14 FEBRUARY 2014 uncertainty for both the FC and the FV selection is 1%. The TABLE II. The uncertainty (rms/mean in %) on the predicted decay-electron rejection cut has errors of 0.2%–0.4%, number of signal νe events for each group of systematic uncertainties for sin22θ 0.1 and 0. The uncorrelated ν depending on neutrino flavor and interaction type. The 13 ¼ uncertainties for the single electronlike ring selection and interaction uncertainties are those coming from parts of the π0 rejection are estimated by using the SK atmospheric neutrino interaction model that cannot be constrained with ND280. neutrino data and SK cosmic-ray muons. Electron-neutrino CC-enriched control samples based on these cuts were Error source [%] sin22θ 0.1 sin22θ 0 prepared, and the differences between MC predictions and 13 ¼ 13 ¼ data are used to extract the systematic uncertainty. The Beam flux and near detector 2.9 4.8 uncertainty associated with the π0 background is deter- (without ND280 constraint) (25.9) (21.7) Uncorrelated ν interaction 7.5 6.8 mined by constructing a hybrid sample with either an Far detector and FSI SI PN 3.5 7.3 electronlike ring taken from the atmospheric data sample or þ þ from decay-electrons selected in the stopping muon data Total 8.8 11.1 sample, and a MC-generated gamma ray assuming π0 kinematics. The selection cut systematic uncertainty is calculated to be 1.6% for signal events and 7.3% for presented in this Letter with other measurements to 2 background events. The total SK selection uncertainty is better constrain sin 2θ13 and δCP, the Lconst term can also 2 2 2 2.1% for the νe candidate events assuming sin 2θ13 0.1. be used to apply additional constraints on sin 2θ13, sin θ23, ¼ 2 Additional SK systematic uncertainties are due to final- and Δm32. state interactions (FSI) of pions that occur inside the target The following oscillation parameters are fixed in the 2 2 5 2 nucleus, as well as secondary interactions (SI) of pions and analysis: sin θ12 0.306, Δm21 7.6 × 10− eV [29], 2 ¼ 2 ¼ 3 2 photonuclear (PN) interactions of photons that occur out- sin θ23 0.5, Δm 2.4 × 10− eV [30], and ¼ j 32j ¼ side of the target nucleus. The treatment of the FSI and SI δ 0. For the normal (inverted) hierarchy case, the CP ¼ uncertainties is the same as in the previous analysis [28]. best-fit value with a 68% confidence level (C.L.) is 2 0.038 0.045 For this analysis, a new simulation of PN interactions has sin 2θ13 0.140þ0.032 (0.170þ0.037). Figure 3 shows the ¼ − − been added to the SK MC. In the final νe event sample, 15% best-fit result, with the 28 observed νe events. The alter- 0 rec of the remaining π background is due to events where native analysis using Eν and a profile likelihood method one of the π0 decay photons is absorbed in a PN interaction. produces consistent best-fit values and nearly identical rec A systematic uncertainty of 100% is assumed for the confidence regions. Figure 4 shows the Eν distribution normalization of the PN cross section. with the MC prediction for the best-fit θ13 value in the Oscillation analysis.—The neutrino oscillation parameters alternative analysis. are evaluated using a binned extended maximum-like- The significance for a nonzero θ13 is calculated to be lihood fit. The likelihood consists of four components: a 7.3σ, using the difference of log likelihood values between normalization term (L ), a term for the spectrum shape the best-fit θ value and θ 0. An alternative method of norm 13 13 ¼ (Lshape), a systematics term (Lsyst), and a constraint term calculating the significance, by generating a large number (Lconst) from other measurements of toy MC experiments assuming θ 0, also returns a Neutrino Oscillations with the MINOS, MINOS+, T2K,13 ¼ and NOvA Experiments. 15 ⃗ ⃗ ⃗ L Nobs; x;⃗ o;⃗ f Lnorm Nobs; o;⃗ f × Lshape x⃗ ; o;⃗ f Data ð Þ¼ ð Þ ð Þ 10 Best fit × L f⃗ × L o⃗ ; (3) Background component systð Þ constð Þ 5 where Nobs is the number of observed events, x⃗ is a set of 180 kinematic variables, o⃗ represents oscillation parameters, 1 and f⃗ describes systematic uncertainties. In the fit, the 150 Data 0.8 likelihood is integrated over the nuisance parameters to 120 Best fit obtain a marginalized likelihood for the parameters of 0.6 interest. 90 0.4 Lnorm is calculated from a Poisson distribution using the 60 mean value from the predicted number of MC events. Angle (degrees) 30 0.2 Lsyst f⃗ constrains the 27 systematic parameters from the ND280ð Þ fit, the SK-only cross section parameters, and the 0 0 5 0 500 1000 1500 SK selection efficiencies. Table II shows the uncertainties Momentum (MeV/c) on the predicted number of signal νe events. The Lshape term Figure 11. The T2K electron momentum versus angle distribution for 28 single- uses x pe; θe to distinguish the νe signal from back- FIG. 3 (color online). The (pe, θe) distribution for νe candidate ¼ð Þ rec ring electron events, together with the MC expectation in [7]. The best fit value of 2 grounds. An alternative analysis uses x Eν , the recon- eventssin with2θ13 the= 0MC.140 inprediction the normal usinghierarchy the case primary is used for method the expectation. best-fit structed neutrino energy. In order to combine¼ the results value of sin22θ 0.140 (normal hierarchy). 13 ¼ from [8]. In order to be sensitive to δ , T2K uses the value of θ , 0.098 0.013, from CP 13 ± 061802-6reactor experiments in PDG2012 [9]. The 2∆ ln L in the fit as a function of δ is − CP extracted, and is shown in Figure 13. The T2K measurement, together with the reactor θ value prefers δ = π/2 with an exclusion of 0.19π < δ < 0.80π( π < δ < 13 CP − CP − CP 0.97π and 0.04π < δ < π) with normal (inverted) hierarchy at 90 % C.L. This − − CP may be a hint of CP violation in neutrinos.

4. Precise measurements of oscillation parameters

4.1. T2K

The most precise measurement of θ23 has been carried out [4] by T2K based on the data set of 6.57 1020 POT. First, a single-ring muon sample is selected by requiring one × k muon-type Cherenkov ring with momentum greater than 200 MeV/c in Super-K. The details of the event selection are found in [4]. One hundred twenty events are selected while the expectation without neutrino oscillation is 446.0 22.5 (syst.). The neutrino ± energy for each event is calculated under the quasi-elastic (QE) assumption using the

The number of rings corresponds to the number of observed particles in Super-K k PHYSICAL REVIEW LETTERS week ending PRL 112, 061802 (2014) Neutrino Oscillations with the MINOS, MINOS+, T2K,14 and FEBRUARY NOvA Experiments. 2014 16 1 Data 8 Best fit Background component 0.5 )

6 π 2 Fit region < 1250 MeV ( 0 ∆m32>0 CP

δ 68% C.L. 90% C.L. candidate events e 4 -0.5 Best fit ν PDG2012 1σ range -1 2 1

Number of 0.5 0

0 500 1000 1500 >2000 ) π

2 value of 7.3σ. These significances were calculated using a FIG.Figure 5 (color 12. The online). allowed regions The for 68% sin 2θ13 andas a function90% C.L. of δCP for allowed the normal mass regionshierarchy for sin case22 (top)θ , and as the a invertedfunction one of (bottom)δ fromassuming [7]. The normal value of θ13 from test statistic having fixed values for θ23 and δCP. For any reactor experiments13 in PDG2012 is shown as theCP shaded region values for these parameters, consistent with their present hierarchy (top) and inverted hierarchy (bottom). The solid line 2 uncertainties, the significance remains above 7σ. represents the best fit sin 2θ13 value for given δCP values. The 2 2 As the precision of this measurement increases, theexpressionvalues of sin θ23 and Δm32 are varied in the fit with the constraint from [30]. The2 shaded region2 shows2 the average θ13 value from uncertainty from other oscillation parameters becomes m (mn Eb) m + 2(mn Eb)El the PDG2012Erec = p[9]−. − − l − , (7) ν 2(m E El + p cos θ ) increasingly important. The uncertainties on θ23 and n − b − l l 2 where m is the proton mass, m the neutron mass, m the lepton mass, E the lepton Δm32 are taken into account in the fit by adding a Lconst p n l l 16 2energy, and2Eb = 27 MeV the binding2 energy of a nucleon3 inside2 a O nucleus. Figure 14 term and marginalizing the likelihood over θ23 and Δm32. sin θ23 0.5, and Δm32 2.4 × 10− eV . The MC aver- 2 shows the neutrino¼ energy of the observed¼ 120 events with the MC expectations for The Lconst term is the likelihood as a function of sin θ23 and aged value of 2Δ ln L at δCP π=2 is 2.20 (4.10) for the m2 , obtained from the T2K ν disappearance measure-neutrinonormal oscillations. (inverted)− hierarchy case,¼ and the probability of Δ 32 μ Using the number of events and the neutrino energy spectrum, the oscillation ment [30]. The value of δCP and the hierarchy are held obtaining2 a value2 greater or equal to the observed value is parameters (sin θ23, ∆m32(13)) are estimated with an un-binned maximum likelihood fixed in the fit. Performing the fit for all values of δCPfit, for the34.1% normal (33.4%). (inverted) With mass thehierarchies. same MC Details settings, of the method the expected are found in [4]. 2 the allowed 68% and 90% C.L. regions for sin 2θ13 areThe result90% is shown C.L. in exclusion Figure 15. region is evaluated to be between 2 +0.055 obtained as shown in Fig. 5. For δCP 0 and normal The0. best35π fitand value0.63 withπ the(0.09 1Dπ 68and % confidence0.90π) radians intervals for are the sin θ normal23 = 0.514 0.056 2 2 3 2 − ¼ (0.511 0.055) and ∆m = 2.51 0.10 (∆m = 2.48 0.10) 10− eV for the (inverted) hierarchy case, the best-fit value with a 68% C.L. ±(inverted) hierarchy32 case.± 13 ± × 2 0.044 0.051 normal (inverted) hierarchy case. The result is consistent with the maximal possible is sin 2θ13 0.136þ0.033 (0.166þ0.042). With the current statistics, the¼ correlation− between− the ν disappearance anddisappearance probability and is more precise than previous measurements, especially μ for sin2 θ . ν appearance measurements in T2K is negligibly small. 23 e 6 m2 >0 Constraints on δCP are obtained by combining our results ∆ 32 2 with the θ13 value measured by reactor experiments. The 5 ∆m32<0 additional likelihood constraint term on sin22θ is defined 90% CL (∆m2 >0) 13 4 32 2 2 2 2 as exp sin 2θ13 0.098 = 2 0.013 , where 0.098 90% CL (∆m32<0) f−ð − Þ ½ ð Þg 2 lnL and 0.013 are the averaged value and the error of sin 2θ13 ∆ 3 -2 from PDG2012 [9]. The 2Δ ln L curve as a function of − 2 δCP is shown in Fig. 6, where the likelihood is marginalized 2 2 2 over sin 2θ13, sin θ23, and Δm32. The combined T2K and 1 reactor measurements prefer δCP π=2. The 90% C.L. 0 limits shown in Fig. 6 are evaluated¼ by− using the Feldman- -1 -0.5 0 0.5 1 δ (π) Cousins method [31] in order to extract the excluded CP region. The data exclude δCP between 0.19π and 0.80π FIG. 6 (color online). The 2Δ ln L value as a function of δCP ( π and 0.97π, and 0.04π and π) with normal (inverted) − − − − for normal hierarchy (solid line) and inverted hierarchy (dotted hierarchy at 90% C.L. 2 2 line). The likelihood is marginalized over sin 2θ13, sin θ23, and The maximum value of 2Δ ln L is 3.38 (5.76) at m2 . The solid (dotted) line with markers corresponds to the − Δ 32 δCP π=2 for the normal (inverted) hierarchy case. This 90% C.L. limits for normal (inverted) hierarchy, evaluated by ¼ value is compared with a large number of toy MC experi- using the Feldman-Cousins method. The δCP regions with values 2 ments, generated assuming δCP π=2, sin 2θ13 0.1, above the lines are excluded at 90% C.L. ¼ − ¼

061802-7 PHYSICAL REVIEW LETTERS week ending PRL 112, 061802 (2014) 14 FEBRUARY 2014 1 Data 8 Best fit Background component 0.5 )

6 π 2 Fit region < 1250 MeV ( 0 ∆m32>0 CP

δ 68% C.L. 90% C.L. candidate events e 4 -0.5 Best fit ν PDG2012 1σ range -1 2 1

Number of 0.5 0

0 500 1000 1500 >2000 ) π

Reconstructed neutrino energy (MeV) ( 0 CP δ rec 2 FIG. 4 (color online). The Eν distribution for νe candidate -0.5 ∆m32<0 2 events with the MC prediction at the best fit of sin 2θ13 0.144 rec ¼ (normal hierarchy) by the alternative binned Eν analysis. -1 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 2 sin 2θ13 value of 7.3σ. These significances were calculated using a FIG. 5 (color online). The 68% and 90% C.L. allowed 2 test statistic having fixed values for θ23 and δCP. For any regions for sin 2θ13, as a function of δCP assuming normal values for these parameters, consistent with their present hierarchy (top) and inverted hierarchy (bottom). The solid line 2 uncertainties, the significance remains above 7σ. represents the best fit sin 2θ13 value for given δCP values. The 2 2 As the precision of this measurement increases, the values of sin θ23 and Δm32 are varied in the fit with the constraint uncertainty from other oscillation parameters becomes from [30]. The shaded region shows the average θ13 value from the PDG2012 [9]. increasingly important. The uncertainties on θ23 and 2 Δm32 are taken into account in the fit by adding a Lconst 2 2 2 3 2 term and marginalizing the likelihood over θ23 and Δm32. sin θ23 0.5, and Δm32 2.4 × 10− eV . The MC aver- 2 ¼ ¼ The Lconst term is the likelihood as a function of sin θ23 and aged value of 2Δ ln L at δCP π=2 is 2.20 (4.10) for the 2 − ¼ Δm32, obtained from the T2K νμ disappearance measure- normal (inverted) hierarchy case, and the probability of ment [30]. The value of δCP and the hierarchy are held obtaining a value greater or equal to the observed value is fixed in the fit. Performing the fit for all values of δCP, 34.1% (33.4%). With the same MC settings, the expected 2 the allowed 68% and 90% C.L. regions for sin 2θ13 are 90% C.L. exclusion region is evaluated to be between obtained as shown in Fig. 5. For δCP 0 and normal 0.35π and 0.63π (0.09π and 0.90π) radians for the normal (inverted) hierarchy case, the best-fit value¼ with a 68% C.L. (inverted) hierarchy case. is sin22θ 0.136 0.044 (0.166 0.051). With the current 13 þ0.033 þ0.042 Neutrino Oscillations with the MINOS, MINOS+, T2K, and NOvA Experiments. 17 ¼ − − statistics, the correlation between the νμ disappearance and ν appearance measurements in T2K is negligibly small. e 6 m2 >0 Constraints on δCP are obtained by combining our results ∆ 32 2 with the θ13 value measured by reactor experiments. The 5 ∆m32<0 additional likelihood constraint term on sin22θ is defined 90% CL (∆m2 >0) 13 4 32 2 2 2 2 as exp sin 2θ13 0.098 = 2 0.013 , where 0.098 90% CL (∆m32<0) f−ð − Þ ½ ð Þg 2 lnL and 0.013 are the averaged value and the error of sin 2θ13 ∆ 3 -2 from PDG2012 [9]. The 2Δ ln L curve as a function of − 2 δCP is shown in Fig. 6, where the likelihood is marginalized 2 2 2 over sin 2θ13, sin θ23, and Δm32. The combined T2K and 1 reactor measurements prefer δCP π=2. The 90% C.L. 0 limits shown in Fig. 6 are evaluated¼ by− using the Feldman- -1 -0.5 0 0.5 1 δ (π) Cousins method [31] in order to extract the excluded CP region. The data exclude δCP between 0.19π and 0.80π FIG. 6Figure (color 13. online).The 2∆ Theln L value2 asln a functionL value of δ asCP awith function the reactor of θδ13 constraints ( π and 0.97π, and 0.04π and π) with normal (inverted) − Δ CP for normalfor normal hierarchy and inverted (solid hierarchies− line) and from inverted[7]. The likelihood hierarchy is marginalized (dotted over θ13, − − − θ and ∆m2 . The 90 % C.L. is evaluated by using the Feldman-Cousins method. hierarchy at 90% C.L. line). The23 likelihood32 is marginalized over sin22θ , sin2θ , and The δCP regions with values above the 90 % C.L. lines are13 excluded.23 The maximum value of 2Δ ln L is 3.38 (5.76) at m2 . The solid (dotted) line with markers corresponds to the − Δ 32 δCP π=2 for the normal (inverted) hierarchy case. This 90% C.L. limits for normal (inverted) hierarchy, evaluated by ¼ value is compared with a large number of toy MC experi- using the Feldman-Cousins method. The δCP regions with values 2 ments, generated assuming δCP π=2, sin 2θ13 0.1, above the lines are excluded at 90% C.L. week ending PRL 112, 181801 (2014)¼ − PHYSICAL¼ REVIEW LETTERS 9 MAY 2014

2500 16 Prediction before ND νµ constraint 061802-7 14 2000 Prediction after ND νµ constraint DATA

1500 Data 12 νµ+νµ CCQE

Events 1000 10 νµ+νµ CC non-QE + CC 500 8 νe νe NC 0 6 0 0.5 1 1.5 2 2.5> 3 4 Muon Momentum (GeV/c) Events/0.10 GeV 2 2500 0 2000 1.5 MC Best-fit 1500

Events 1 1000 0.5 Ratio to no 500 oscillations 0 < 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1 01234>5 Muon cos(θ) Reconstructed ν Energy (GeV)

FIG. 1 (color online). The momentum and angular distributions FIG.Figure 2 (color). 14. The Reconstructed neutrino energy energy spectrum spectrum for single-ring for single-ring muon events in [4]. (Top) for muons in ND280’s CC-0π selection. The predicted distribu- μ-likeThe SKobserved events. and Top: expected The spectra observed with spectrum the event and categories expected in the simulation. tions before and after the ND280 fit are overlaid on both figures. spectrum(Bottom) with The interaction ratio of the modes observed for spectrum the T2K to thebest no-oscillation fit. Bottom: hypothesis with Thethe ratio best-ﬁt of the case. observed spectrum (points) to the no-oscillation hypothesis, and the best oscillation fit (solid). muon momentum and angle relative to the beam direction for the CC-0π sample and the improvement in data and MC agreement when using the best-fit parameters. The fit uses decay electrons. Correlated selection efficiency parameters the neutrino interaction model to extrapolate ND280 are assigned for six event categories: ν ν¯ CCQE in μ þ μ measurements, primarily covering cos θμ > 0.5, over three energy bins, ν ν¯ CC non-QE, ν ν¯ CC, and ð Þ μ μ e e SK’s 4π angular acceptance. NC events. An energyþ scale uncertainty ofþ 2.4% comes The fit gives estimates for 16 beam flux parameters at from comparing reconstructed momenta in data and MC for SK, the seven common cross section parameters, and their cosmic-ray stopping muons and associated decay electrons, covariance. Using the ND280 data reduces the uncertainty and from comparing reconstructed invariant mass in data on the expected number of events at SK due to these and MC simulations for π0’s produced by atmospheric parameters from 21.6% to 2.7%. neutrinos. Systematic uncertainties in pion interactions in SK measurements.—An enriched sample of νμ CCQE the target nucleus (FSI) and SK detector (SI) are evaluated events, occurring within 2 μs to 10 μs of the expected by varying pion interaction probabilities in the NEUT − þ neutrino arrival time, is selected in SK. We require low cascade model. These SK detector and FSI SI uncer- activity in the outer detector to reduce entering backgrounds. tainties produce a 5.6% fractional error in theþ expected We also require: visible energy > 30 MeV, exactly one number of SK events (see Table I). reconstructed Cherenkov ring, μ-like particle identification, Oscillation fits.—We estimate oscillation parameters reconstructed muon momentum > 200 MeV, and ≤ 1 using an unbinned maximum likelihood fit to the SK 2 2 reconstructed decay electron. The reconstructed vertex must spectrum for the parameters sin θ23 and either Δm32 2 ð Þ be in the fiducial volume (at least 2 m away from the ID or Δm13 for the normal and inverted mass hierarchies, walls) and we reject “flasher” events (produced by inter- respectively, and all 45 systematic parameters. The fit uses mittent light emission inside phototubes). More details 73 unequal-width energy bins, and interpolates the spec- about the SK event selection and reconstruction are found trum between bins. Oscillation probabilities are calculated in [22]. using the full three-flavor oscillation framework. Matter The neutrino energy for each event is calculated under effects are included with an Earth density of ρ 2.6 g=cm3 ¼ the quasielastic assumption as in [2] using an average [45], δCP is unconstrained in the range −π; π , and other 16 ½ 2 binding energy of 27 MeV for nucleons in O. The Ereco oscillation parameters are fit with constraints sin θ13 2 ð 2Þ¼ distribution of the 120 selected events is shown in Fig. 2. 0.0251 0.0035, sin θ12 0.312 0.016, and Δm21 The MC expectation without oscillations is 446.0 22.5 7.50 Æ0.20 × 10−5 ðeV2=cÞ¼4 [46].Æ Figure 2 shows the¼ Æ ð Æ Þ (syst.) events, of which 81.0% are νμ ν¯μ CCQE, 17.5% best-fit neutrino energy spectrum. The point estimates of are ν ν¯ CC non-QE, 1.5% are NC,þ and 0.02% are ν the 45 nuisance parameters are all within 0.25 standard μ þ μ e þ ν¯e CC. The expected resolution in reconstructed energy for deviations of their prior values. ν ν¯ CCQE events near the oscillation maximum Two-dimensional confidence regions in the oscillation μ þ μ is ∼0.1 GeV. parameters are constructed using the Feldman-Cousins Systematic uncertainties in the analysis are evaluated method [47], with systematics incorporated using the with atmospheric neutrinos, cosmic-ray muons, and their Cousins-Highland method [48]. Figure 3 shows 68% and

181801-5 week ending Neutrino OscillationsPRL 112, with181801 the MINOS, (2014) MINOS+, T2K,PHYSICAL and NOvA Experiments. REVIEW18 LETTERS 9 MAY 2014

4 CCQE 3 90% CL lnL 2 ∆ -2∆lnLcrit. Nieves multinucleon (×5)

-2 1 68% CL 00 pionless ∆-decay (×5) 4.2

) 68% (dashed) and 90% (solid) CL Contours 4 4 /c

2 T2K [NH] T2K [IH] 3.8

eV SK I-IV [NH] MINOS 3-flavor+atm [NH] -3 3.6 Arbitrary Units 3.4 3.2 [IH] (10 2 13 3 -1 -0.5 0 0.5 m

∆ QE 2.8 Ereco - Etrue (GeV) 2.6 [NH]/ 32 2 2.4 FIG. 4 (color online). The difference between the reconstructed m ∆ 2.2 energy assuming QE kinematics and the true neutrino energy. 2 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7 01234 True QE events with energies below 1.5 GeV show little bias 2 -2∆lnL sin (θ23) while multinucleon events based on [43] and NEUT pionless Δ decay (shown scaled up by a factor of 5) are biased towards lower 2 2 FIG.Figure 3 (color). 15. T2K TheContours 68% of and oscillation 90% C.L. parameters confidence sin θ23 regionsversus ∆ form energies.for 68 % 2 2 2 | 32(13)| sinandθ 9023 %and C.L.Δ region.m (NH) The or one-dimensionalΔm (IH). The profile SK likelihoods[49] and MINOS are also shown for each ð Þ 32 13 [7]oscillation90% C.L. parameter regions in for the NH top and are rightshown windows. for comparison. The 1D 2∆ T2K ln L’criticals values for − 1Dthe profile normal likelihoods mass hierarchy for are each shown oscillation in the windows. parameter The Super-Kamiokande separately and [14] mightand introduce a bias on the oscillation parameters as areMINOS also shown [54] results at the are top also and shown right for overlaid comparison. with light blue lines large as a few percent [28–43]. We are the first oscillation and points representing the 1D −2Δ ln Lcritical values for NH at 68% and 90% C.L. experiment to consider the potential bias introduced by multinucleon interactions including potential cancellation 4.2. Joint analysis of νµ disappearance and νe appearance samples in MINOS from measurements at the near detector. At T2K beam Beginning in90% 2005 confidence (2003 for collection regions for of atmospheric the oscillation data), parameters the MINOS for experimentenergies, most interactions produce final-state nucleons has providedboth precision normal measurements and inverted of the hierarchies. oscillation The parameters 68% and ∆m 90%2 and sin2below(2θ) for SK’s Cherenkov threshold, making multinucleon 2 effective definitionsexpected of these sensitivity parameters curves in are a two-neutrino each 0.04 wider approximation. in sin θ23 Most recently,interactions indistinguishable from quasielastic (QE) inter- 3 2 3 2 ð Þ MINOS hasthan quoted these 2.28 contours. x 10− < ∆ Anm32 alternative< 2.46 x 10 analysis− eV (68% employing confidence)actions. and a Even if the additional nucleon does not leave the | 2 | 90% C.L. rangea binned for θ23 likelihoodof 0.37 < sin ratioθ13 gave< 0.64 consistent (both normal results. mass Also hierarchy),nucleus, using the multinucleon mechanism alters the kinematics a complete 3-neutrinoshown are description 90% confidence of the data regions [54]. Atmospheric from other data recent and appearanceof the out-going lepton, distorting the reconstructed neu- data are includedexperimental in the combined results. fits.Statistical In particular, uncertainties the νe dominateappearance datatrino and energy which assumes QE kinematics (see Fig. 4) in atmosphericT2K data’s provide, error budget. in principle, sensitivity to additional information concerningaddition to increasing the overall QE-like event rate. mass hierarchyWe and CPcalculate phase. one-dimensional As an example, the (1D) atmospheric limits using data sample, a new dividedThe into T2K neutrino interaction generator, NEUT, includes neutrino andmethod antineutrino inspired samples by for Feldman-Cousins up-going multi-GeV[47] events,and showsCousins- differentan matter effective model (pionless Δ decay) that models some, effects for normalHighland and[48] invertedthat marginalizes hierarchies.In over the the current second sample, oscillation these additionalbut not all, of the expected multinucleon cross section. In sensitivitiesparameter. are limited, asToy shown experiments by the presentation are used in Figureto calculate 16, and theorder fitted to evaluate the possible effect on the oscillation 2 2 values of ∆−m232Δ andln Lcritical sin θ23values,are consistent above which [54]. Thea parameter fit results value are obtained is analysis, using we perform a Monte Carlo study where the | | 2 2 constraints fromexcluded, external for data. each valueIn particular, of sin aθ23 value. These of sin toyθ13 experi-= 0.0242 existing0.0025 effective model is replaced with a multinucleon 2 ð 2 Þ ± has been takenments from draw a weighted values for averageΔm32 ofor reactorΔm13 experimentin proportion values to the [55], andprediction solar based on the work of Nieves [43] going up to 2 oscillation parameterslikelihood are for taken fixed from sin [56].θ23 , marginalized over systematic 1.5 GeV in energy. We used this modified simulation to ð Þ parameters. The toy experiments draw values of the 45 make ND280 and SK fake data sets with randomly chosen systematic parameters from either Gaussian or uniform systematic uncertainties but without statistical fluctuations, 2 2 distributions. We generate Δm32 or Δm13 limits with the and performed oscillation analyses as described above on same procedure. Figure 3 shows the 1D profile likelihoods each of them, allowing ND280 fake data to renormalize the for both mass hierarchies, with the −2Δ ln Lcritical MC SK prediction. The mean biases in the determined oscil- estimates for NH. lation parameters are < 1% for the ensemble, though the 2 2 The 1D 68% confidence intervals are sin θ23 sin θ23 biases showed a 3.5% rms spread. 0.055 2 ð Þ¼ ð Þ 2 0.514−þ0.056 (0.511 0.055) and Δm32 2.51 0.10 Conclusions.—The measurement of sin θ23 2 Æ −3 2 4 ¼ Æ 0.055 ð Þ¼ Δm 2.48 0.10 × 10 eV =c for the NH (IH). 0.514þ0.056 (0.511 0.055) for NH (IH) is consistent with 13 − Æ Theð best¼ fit correspondsÆ Þ to the maximal possible disap- maximal mixing and is more precise than previous mea- 2 pearance probability for the three-flavor formula. surements. The best-fit mass-squared splitting is Δm32 2 −3 2 ¼4 Effects of multinucleon interactions.—Inspired by more 2.51 0.10 IH∶ Δm13 2.48 0.10 × 10 eV =c . precise measurements of neutrino-nucleus scattering PossibleÆ multinucleonð knockout¼ inÆ neutrino-nucleusÞ inter- [50–53], recent theoretical work suggests that neutrino actions produces a small bias in the fitted oscillation interactions involving multinucleon mechanisms may be a parameters and is not a significant uncertainty source at significant part of the cross section in T2K’s energy range present precision.

181801-6 Neutrino Oscillations with the MINOS, MINOS+, T2K, and NOvA Experiments. 19

2.8 6 MINOS µ disappearance + e appearance Profile of likelihood surface 20 20 10.71 ×10 POT µ-mode, 3.36 ×10 POT µ-mode Normal hierarchy 37.88 kt-yr atmospheric neutrinos Inverted hierarchy 2.6 4

log(L) 90% C.L.

2.4 -2 2 )

2 68% C.L.

eV 2.2 -3 0 Normal hierarchy 2.2 2.3 2.4 2.5 2.6 2 -3 2

(10 Inverted hierarchy |m32| (10 eV ) 2 32 -2.2

m 6

Profile of likelihood surface Normal hierarchy -2.4 Inverted hierarchy 4

log(L) 90% C.L. -2.6 -2 2 68% C.L. Best fit 68% C.L. -2.8 90% C.L. 0 0.3 0.4 0.5 0.6 0.7 0.3 0.4 0.5 0.6 0.7 2 2 sin 23 sin 23

Figure 16. MINOS 3-flavor oscillation parameters using both disappearance and appearance data. Left: Confidence level contours for assumed normal and inverted hierarchies, computed using 2∆ln(L) w.r.t. the overall best fit point (star). Right: − One-dimensional likelihood profiles for the parameters. All results from [54].

4.3. Joint analysis of νµ disappearance and νe appearance samples in T2K The oscillation probability of ν ν depends on many oscillation parameters: sin2 θ , µ → e 13 sin2 θ , ∆m2 and δ ; that of ν ν depends mainly on sin2 θ and ∆m2 . Therefore, 23 32 CP µ → µ 23 32 all oscillation parameters can be efficiently extracted by fitting two data samples of νe and νµ simultaneously. For this purpose, the T2K collaboration developed analysis techniques to fit both νe and νµ samples. One method is based on the ∆ log L method, and the other is based on the Markov Chain Monte Carlo (MCMC) method. The θ13 constraint from PDG2012 [9] is applied in the analysis. 2 2 2 With the ∆ log L method, T2K measures sin 2θ13, sin θ23, ∆m32(13) and δCP as shown in Figure 17. The results are consistent with those shown in Section 3.2 and 4.1, and the correlations between parameters are properly treated. With the MCMC method, the quantities 2∆ ln L( ln L(δ ) ln L(best fit values)) − ≡ CP − in the fit, as a function of δCP, are evaluated as shown in Figure 18. The best fit value and the preferred regions at 90 % C.L. are consistent with the result of the νe only sample shown in Figure 13. In Figure 19, the credible intervals calculated in the MCMC method are shown in the sin2 θ versus ∆m2 plane, for both normal and inverted mass hierarchy cases. 23 | 32| The results are compared with other measurements by Super-Kamiokande [14] and MINOS [54]. The T2K best fit point is found to lie in the normal mass hierarchy as shown in Figure 19. Neutrino Oscillations with the MINOS, MINOS+, T2K, and NOvA Experiments. 20 ) )

4 3 4 3 /c /c

2 Normal Hierarchy 2 Normal Hierarchy Inverted Hierarchy Inverted Hierarchy eV eV

-3 2.8 -3 2.8 (10 (10 2 2 m m

Δ 2.6 Δ 2.6

2.4 2.4

2.2 2.2

2 2 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.015 0.02 0.025 0.03 0.035 2 2 sin θ23 sin θ13

CP 3 δ 0.035 Normal Hierarchy Inverted Hierarchy 13

2 θ

Normal Hierarchy 2 Inverted Hierarchy sin 0.03 1

0 0.025

-1

0.02 -2

-3 0.015 0.015 0.02 0.025 0.03 0.035 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7 2 2 sin θ13 sin θ23

FIG. 34: 68%Figure (dashed) 17. andT2K 90% Contours (solid) CLof oscillation regions from parameters the analysis calculated that includes with the results ∆ log L 2 from reactormethod experiments for both normal with di and↵erent inverted mass hierarchy hierarchy assumptions cases [34]: (top-left) using sin2 withθ23 versus 2 2 2 2 ∆m32(13) , (top-right) sin θ13 versus ∆m32(13) , (bottom-left) sin θ13 versus δCP, respect to| the best-fit| point, the2 one from the fit2 | with normal| hierarchy. The parameter and (bottom-right) sin θ23 versus sin θ13. The 90 % (68 %) C.L. are shown in the m2 representssolid (dashed)m2 linesor withm2 thefor normalbest fit pointand inverted shown by mass the hierarchy mark. assumptions | | 32 13 respectively.

5. Systematic Uncertainties C. Alternate frequentist analysis 5.1. T2K Beam In acceleratorIn order to neutrino thoroughly beam cross-check experiments, the analysis understanding described above, of an the alternate properties frequentist of the neutrinojoint ﬁt beam analysis is very was performed important. which An di↵ experimenters in the treatment is usually of the designed systematic to errors. cancel This ﬁrst- order uncertainties of the neutrino beam by adopting the ”two detectors” technique, in which one detector, located near the beam production90 point, is used to monitor the beam and the other, far detector, studies neutrino oscillations. By normalizing the neutrino events with the near detector measurement, the systematic uncertainties of the beam and the neutrino cross sections are largely canceled. Even with the cancellation, for a Neutrino Oscillations with the MINOS, MINOS+, T2K, and NOvA Experiments. 21

2 7

χ excluded at 90% CL Δ 6 0.5 Normal Hierarchy Inverted Hierarchy 5 FC 90 % critical χΔ 2 (NH) FC 90 % critical χΔ 2 (IH)

⇥ 4 / 0 CP

⇤ 3

2 -0.5

1 excluded at 90% CL

-1 0 0.02 0.04 0.06 0.08 0.1 -1 -0.5 0 0.5 1 2 δCP(π) sin (13)

T2K+Reactor 68% Credible Region T2K Only 68% Credible Region FIG. 37: Profiled 2 as a function of with the results of the critical 2 values for T2K+Reactor 90% Credible Region T2K Only 90% Credible Region CP 2 T2K+Reactor BestFigure Fit Point 18. T2K(Left) Only Best Fit The Line T2Kthe credible normal and invertedintervals hierarchies in the for the sin jointθ fit13 withversus reactorδ constraint,CP plane with the 2 FIG. 38: Credible regions for sincalculated✓13 and CP for in T2K-only the MCMC and T2K+reactor method combined [34]. The 68 % intervalsexcluded regions are found show overlaid. by the dashed lines analyses. These are constructedand by marginalizing the 90 % over intervals both massare hierarchies. by the For the solid lines. These are constructed by marginalizing T2K-only analysis, the best fitover line is both shown insteadmass of hierarchies. the best fit point becauseThe red the lines are extracted only from the T2K data set, analysisand has the little blacksensitivity line to CP and. the point are extracted from both the T2K data set and the reactor θ13 constraint. (Right) The 2∆ ln L value as a function of δCP in the MCMC over the mass hierarchy; in particular, the most probable value line appears to be o↵−set method with both νe and νµ samples and reactor θ13 constraint [34]. The δCP regions from the center of the credible region. This is because the most probable value line is for with values above the 90 % C.L. lines are excluded. the preferred inverted hierarchy, and the credible intervals are marginalized over hierarchy.

Fig. 40 shows the posterior probability for CP with 68% and 90% credible intervals for the

T2K+reactor combined analysis. Figure 41 shows comparisons of SK ⌫µ CC and ⌫e CC candidateprecision events with the measurement, best-fit spectra produced understanding from the T2K-only and T2K+reactorof the beam itself is essential. combined analyses. Each best-fit spectrum is formed by calculating the most probable value for the predictedIn number the of eventsT2K in each experiment, energy bin, using all the of theneutrino MCMC points from beam is simulated by incorporating real the correspondingmeasurements analysis. The fit of spectrum hadron for ⌫µ CC production, events does not change among appreciably which large contributions come from the when the reactor prior is included, but the ⌫e CC fit spectrum shows a noticeable reduction in theCERN number of events. NA61 experiment [10, 11]. The uncertainties of hadron production decaying

into neutrinos are directly97 related to the systematic uncertainty in the neutrino beam. The uncertainty of the neutrino beam flux at the far detector in the T2K experiment is shown in Figure 20. The fractional uncertainty of the beam is at the 10 15 % 93 ∼ level, of which the largest component is still the uncertainty in hadronic interactions. The second largest component is due to the combined uncertainties of proton beam parameters, alignment of the beam line components and off-axis angle. In T2K, the beam stability and the off-axis angle are directly monitored using the neutrino beam monitor: INGRID [12]. As shown in Section 5.2, the relative beam flux uncertainty between the near and far detectors can be reduced to the 3 % level including the uncertainty in cross sections constrained by the near detector.

5.2. Constraints by the T2K near detector measurements The uncertainty of neutrino cross sections is not small, especially in the GeV energy ∼ region. Although the uncertainty is typically at the 20 % level, the ﬁrst order ∼ uncertainties of the beam and cross sections can be canceled by adopting the two detector technique, as described in Section 5.1. For this purpose, MINOS, MINOS+, NOvA and T2K have sophisticated near detectors which collect large amounts of neutrino data to measure the neutrino beam ﬂux and neutrino cross sections with high precision. Neutrino Oscillations with the MINOS, MINOS+, T2K, and NOvA Experiments. 22 ) 4 /c 2 3.2 eV -3 SK joint OA

(10 3 2 32 m Δ 2.8 T2K joint OA

2.6

2.4

2.2 MINOS joint OA

0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7 ) 4 -2 /c 2 eV -3 -2.2 MINOS joint OA (10 2 32

m -2.4 Δ

-2.6

-2.8 T2K joint OA

-3 SK joint OA

0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7 2 sin θ23

FIG. 36: 68%Figure (dashed) 19. T2K and 90% 68 % (solid) (dashed) CL regions and 90 for % (solid)normal CL (top) regions and inverted contours (bottom) of oscillation parameters sin2 θ versus ∆m2 for normal (top) and inverted (bottom) mass mass hierarchy combined with23 the results from32 reactor experiments in the (sin2✓ , m2 ) hierarchy [34]. The best ﬁt point is shown as the black mark in the23 normal32 mass spacehierarchy. compared to The the Super-Kamiokande results from the Super-Kamiokande [14] and MINOS [128] [54] and results MINOS are also [129] shown for comparison. experiments.

92 K. ABE et al. PHYSICAL REVIEW D 88, 032002 (2013) 0.3 the anomalous field show deviations from the nominal flux Total (a) of up to 4%, but only for energies greater than 1 GeV. Pion Production Kaon Production The absolute horn current measurement uncertainty is 2% 0.2 Secondary Nucleon Production and arises from the uncertainty in the horn current monitoring. Production Cross Section We simulate the flux with 5kAvariations of the horn current, and the effect on theÆ flux is 2% near he peak.

0.1

Fractional Error F. Off-axis angle constraint from INGRID The muon monitor indirectly measures the neutrino beam direction by detecting the muons from meson decays, while 0 the INGRID on-axis neutrino detector directly measures the 10-1 1 10 Neutrino Oscillations with the MINOS, MINOS+, T2K, and NOvAE (GeV) Experiments. 23 neutrino beam direction. The dominant source of uncertainty K. ABE et al. PHYSICALν REVIEW D 88, 032002on (2013) the beam direction constraint is the systematic uncertainty on the INGRID beam profile measurement, corresponding to 0.3 0.3the anomalous field show deviations from the nominal flux Total a 0.35 mrad uncertainty. We evaluate the effect on the flux (a) of up toTotal 4%, but only for energies(b) greater than 1 GeV. Pion Production Pion Production when the SK or ND280 off-axis detectors are shifted in the The absolute horn current measurement uncertainty is 2% Kaon Production Kaon Production simulation by 0.35 mrad. 0.2 Secondary Nucleon Production 0.2and arisesSecondary from the Nucleon uncertainty Production in the horn current monitoring. Production Cross Section We simulateProduction the Cross flux Section with 5kAvariations of the horn current, and the effect on theÆ flux is 2% near he peak. G. Summary of flux model and uncertainties The T2K flux predictions at the ND280 and SK detectors 0.1 0.1 have been described and are shown in Fig. 5. We use the Fractional Error

Fractional Error F. Off-axis angle constraint from INGRID flux predictions as inputs to calculate event rates at both the The muon monitor indirectly measures the neutrino beam ND280 and SK detectors. To evaluate the flux related direction by detecting the muons from meson decays, while uncertainties on the event rate predictions, we evaluate 0 0the INGRID on-axis neutrino detector directly measures the -1 -1 the fractional uncertainties on the flux prediction in bins 10 1 10 neutrino10 beam direction.1 The dominant10 source of uncertainty Eν (GeV) Eν (GeV) of energy for each neutrino flavor. The bin edges are on the beam direction constraint is the systematic uncertainty (i) # : 0.0, 0.4, 0.5, 0.6, 0.7, 1.0, 1.5, 2.5, 3.5, 5.0, 7.0, 0.3 on the INGRID beam profile measurement, corresponding to " Figure 20. Fractional uncertaintyFIG. 7 (color of the online). T2K neutrino The fractional beam hadron flux: interaction (left) muon errors 30.0 GeV Total a 0.35 mrad uncertainty. We evaluate the effect on the flux (b) on #" (a), #e (b) flux predictions at SK. Pionneutrino Production component and (right) electron neutrinowhen the component SK or ND280 [6]. off-axis detectors are shifted in(ii) the#": 0.0, 1.5, 30.0 GeV Kaon Production simulation by 0.35 mrad. (iii) #e: 0.0, 0.5, 0.7, 0.8, 1.5, 2.5, 4.0, 30.0 GeV Secondary Nucleon Production 0.2 D. Horn and target alignment and uncertainties (iv) #e: 0.0, 2.5, 30.0 GeV CC0π - DataProduction Cross Section CC1π - Data CC Other - Data We choose coarse binning for the antineutrino fluxes since µ µ 1 70 1 18 µ

θ 1 θ The horns areG. aligned Summary relative of to ﬂux the model primary and beam uncertainties line 16 θ 14 they make a negligible contribution for the event samples cos cos

60 with uncertainties ofcos 0.3 mm in the transverse x direction 0.9 0.9 The14 T2K0.9 flux predictions at the ND28012 and SK detectorsdescribed in this paper. The neutrino flux has finer bins 0.1 50 and 1.0 mmhave in the been transverse describedy direction and are shown and beam in Fig. direc-5. We use the Fractional Error 12 10 around the oscillation maximum and coarser bins where 0.8 0.8 40 tion. The precision10 0.8 of the horn angular alignment is flux predictions as inputs to calculate event8 rates at boththe fluxthe prediction uncertainties are strongly correlated. 30 0.2 mrad. After installation8 in the first horn, both ends of 0.7 0.7 ND280 and0.7 SK detectors. To evaluate6 the flux related the target wereuncertainties surveyed,6 on and the the event target rate was predictions, found to be we evaluate 0 20 4 Events/(100 MeV/c * 0.01)

Events/(100 MeV/c * 0.01) 4 tilted from its intended orientation by 1.3 mrad. We haveEvents/(100 MeV/c * 0.01) 0.6 10-1 0.6 the fractional0.6 uncertainties on the flux prediction in bins 1 10 10 2 ND280 Flux not included this misalignment2 in the nominal flux calcu- νµ Eν (GeV) of energy for each neutrino flavor. The bin edges are 0.5 0 0.5 0 0.5 0 SK Flux 0 1000 2000 3000 4000 5000 0 1000 2000lation,3000 4000 but5000 the(i) effect# : 0.0,0 is 0.4,1000 simulated 0.5,2000 0.6,3000 and 0.7,4000 included 1.0,5000 1.5, as 2.5, an 3.5, 5.0, 7.0, 0.2 νµ p (MeV/c) p (MeV/c) " p (MeV/c) µ uncertainty.µ We also simulate linear and angularµ displace- FIG. 7 (color online). The fractional hadron interaction errors 30.0 GeV SK νe Flux on #" (a), #e (b) flux predictions at SK. ments of the horns within their alignment uncertainties and (ii) #": 0.0, 1.5, 30.0 GeV Figure 21. The muon momentumevaluate versus the the effect scattering on the flux. angle The for total CC0π alignment(left), CC1 uncer-π (iii) #e: 0.0, 0.5, 0.7, 0.8, 1.5, 2.5, 4.0, 30.0 GeV (middle) and CC Other (right)tainty samples on in the T2K flux data is less than 3% near the flux peak. 0.1

D. Horn and target alignment and uncertainties (iv) #e: 0.0, 2.5, 30.0 GeV Fractional Error We choose coarse binning for the antineutrino fluxes since The horns are aligned relative to the primary beam line E. Horn current, field and uncertainties with uncertainties of 0.3 mm in the transverse x direction they make a negligible contribution for the event samples We assumedescribed a 1=r dependence in this paper. of the The magnetic neutrino field flux in thehas finer bins and 1.0In mm the in the T2K transverse experiment,y direction the and near beam detector direc- called ND280 [13] is located at 280 m 0 flux simulation.around The the validity oscillation of this maximum assumption and coarser is con- bins where -1 tion. The precision of the horn angular alignment is 10 1 10 from the beam production target, in the samefirmed direction by measuringthe flux as prediction the the farhorn detector, field uncertainties using at a Hall 2.5 are degreesprobe. strongly The correlated. 0.2 mrad. After installation in the first horn, both ends of E (GeV) maximum deviation from the calculated values is 2% for ν theoff-axis. target were ND280 surveyed, consists and the of target two Fine was found Grained to be Detectors (FGD), three Time Projection the first horn and less than 1% for the second and third tiltedChambers from its intended (TPC), orientation an Electromagnetic by 1.3 mrad. WeCalorimeter have system (ECAL), Side Muon Range FIG. 8. The fractional uncertainties on the ND280 #", SK #" horns. Inside the inner conductorND280 of a spare Flux first horn, we and SK # flux evaluated for the binning used in this analysis. not included this misalignment in the0 nominal flux calcu- νµ e Detectors (SMRD) and a π detector (P0D).observe Except an anomalous for the field SMRD, transverse the to detectors the horn axis are with This binning is coarser than the binning shown in Fig. 7 and lation, but the effect is simulated and included as an 0.2 SK νµ Flux located inside a dipole magnet of 0.2 Ta magnetic maximum strength field. For of 0.065 neutrino T. Flux simulationsenergies around including includes the correlations between merged bins. uncertainty. We also simulate linear and angular displace- SK νe Flux ments1 GeV, of the the horns dominant within their neutrino alignment interaction uncertainties and is charged-current (CC) quasi-elastic (QE) evaluatescattering, the effect and on the the flux. second The dominant total alignment one uncer- is CC 1 π production. In the higher energy tainty on the flux is less than 3% near the flux peak. 0.1 032002-10 region, deep-inelastic scattering (DIS) becomes dominant.Fractional Error In ND280, the following three eventE. categories Horn current, have field been and uncertaintiesmeasured: CC0π, CC1π and CC Other where the CC0π Wesample assume is a for1=r CCdependence QE events, of the CC1 magneticπ for field CC in 1 theπ production and CC Other for DIS. The flux simulation. The validity of this assumption is con- 0 distributions of muon momentum and scattering angle relative10-1 to the neutrino1 beam 10 firmed by measuring the horn field using a Hall probe. The E (GeV) maximumare shown deviation in Figure from the 21 calculated for data. values The is 2%neutrino for interaction models and theν neutrino thebeam first horn flux and are less tuned than to 1% match for the the second observed and third distributions in Figure 21. After tuning, the FIG. 8. The fractional uncertainties on the ND280 #", SK #" horns.uncertainties Inside the inner of the conductor neutrino of a event spare first rates horn, are we summarizedand SK #e influx Table evaluated 2. The for the uncertainties binning used in this analysis. observe an anomalous field transverse to the horn axis with This binning is coarser than the binning shown in Fig. 7 and a maximum strength of 0.065 T. Flux simulations including includes the correlations between merged bins.

032002-10 Neutrino Oscillations with the MINOS, MINOS+, T2K, and NOvA Experiments. 24

Figure 22. The uncertainty of the neutrino events in the T2K far detector as a function of neutrino energy [34]. (Left) νe candidate events and (right) νµ candidate events. of π hadronic interactions in the far detector and the detector systematic error in the far detector are also shown.

Table 2. Fractional uncertainties (%) of the number of neutrino events in the T2K far detector [34]. The uncertainties of cross sections are categorized into two parts: One is constrained by the ND280 measurement, and the other is independent of ND280.

Sources νe candidates νµ candidates ﬂux + cross sections (ND280 constrained) 3.2 2.7 cross sections (ND280 independent) 4.7 5.0 π interactions in the far detector 2.5 3.0 Far detector systematic 2.7 4.0 Total 6.8 7.7

In T2K, the number of observed νe (νµ) events is 28 (120) with a systematic uncertainty of 6.8 (7.7) %. In Figure 22, the uncertainties on the expected energy

distributions of both νe and νµ events are shown before and after constraint by the ND280 measurement. Today, the T2K sensitivity is not very limited by systematic errors which will also rapidly improve for the future CP violation measurement. In the near future, T2K expects that the total systematic uncertainty can be reduced down to 5 % or less.

5.3. MINOS Systematics MINOS measurements have signiﬁcant statistical error. For example, in the combined beam and atmospheric analysis a total of 3117 beam-generated contained-vertex charged Neutrino Oscillations with the MINOS, MINOS+, T2K, and NOvA Experiments. 25

current events, distributed across the entire neutrino energy spectrum, are used [46]. The appearance analysis finds 172 events. Notwithstanding these small samples, MI- NOS has performed complete and detailed analysis of the systematics of the parameter measurements. The sensitivity of the results to systematics also benefits from the great similarity between the MINOS near and far detectors. Here we summarize the most important beam-related systematics and their effects.

The MINOS 3-neutrino combined νµ disappearance and νe appearance paper [54] describes the use of 32 systematic effects as nuisance parameters in the final fits. Of these 13 concern the atmospheric neutrinos; these are not discussed further in this section.

There are 4 dominant systematic effects in νµ disappearance in MINOS. These are, with representative values (from [42]): (i) Hadronic shower energy (7% in the oscillation maximum region), (ii) µ energy (2 -3%, depending on technique), (iii) Relative normailzation (1.6%), and (iv) Residual neutral current (NC) contamination (20%) The systematic knowledge of the muon energy includes measurements by range (in the MINOS steel, 2%), and by momentum extracted from curvature in the MINOS magnetic field (3%). The relative normalization error of 1.6% is derived from knowledge of fiducial masses and relative reconstruction efficiencies [42].

Additional systematics are taken into account for νe appearance, affecting both signal and background predictions which are compared with the observed data. Many of the large number of systematic checks have small or negligible effects on the measured parameters; nevertheless they are incorporated in the final fitting procedures as discussed above. Illustrative values of the systematic effects are quoted in [45], where the effect of the relevant uncertainties on the far detector background prediction for νe appearance in the νµ beam are given as: (i) Energy scale: This includes both relative energy scale differences between the

near and far detectors and the absolute energy scale. The former affects the νe background prediction by directly impacting the extrapolation from data, and is the most important single systematic (2%). The latter enters via its effects on the event selection process, and is less important. The combined effect on the background is 2.7% (ii) Normalization: This term refers to effects relating to the relative fiducial masses and exposures of the two detectors. It is quoted as 1.9%

(iii) ντ cross section: A poorly known pseudo-scalar form factor [57] causes a background uncertainty of 1.7%. (iv) All others: Small eﬀects due to, for example, neutrino ﬂuxes and cross-sections (which largely cancel due to the functional identity of the near and far detectors) Neutrino Oscillations with the MINOS, MINOS+, T2K, and NOvA Experiments. 26

and hadronic shower modeling in neutrino interactions. [58]. The sum of these small eﬀects is < 1%, showing once more the eﬀectiveness of extrapolating from near detector data.

The ﬁnal uncertainty on the νe background is 4%, to be compared with its statistical uncertainty of 8.8%. The numbers cited here apply to the νµ beam mode only, with similar, but slightly higher values for theν ¯µ beam mode. In addition, there is a systematic error of approximately 5% on the appearance signal selection eﬃciency, studied with CC events in which the muon data has been replaced with a simulated electron.

6. Additional Measurements

6.1. Additional Measurements in MINOS In addition to the primary mission of MINOS and MINOS+, which is the understanding of the three-neutrino oscillation sector, the MINOS detectors and NuMI beam line are capable of a wide variety of additional measurements which enrich our understanding of the physics of neutrinos, and other areas. Measurements published by the MINOS collaboration include: (i) Searches for additional sterile neutrinos using charged and neutral current interactions [47] [48], (ii) Measurement of neutrino cross-sections [50], (iii) Tests of fundamental symmetries and searches for non-standard interactions. [51], (iv) Studies of cosmic rays at both near and far detectors [52]. In this section we discuss briefly the first item, searches for sterile neutrinos, and present an example of a fundamental symmetry test. Anomalies seen in short baseline experiments [53] and others have generated great interest in the possibility of a fourth neutrino which would not have Standard Model interactions. Neutral currents in the MINOS detectors are visible as hadronic showers without an accompanying lepton. All active flavors of neutrinos produce neutral currents equivalently, so that the three-neutrino oscillation phenomenon should not cause any depletion with respect to expectations in the observed far detector spectrum. This is indeed seen to be the case, as documented in [48] which measures a limit on the fraction

of neutrinos which can have oscillated to sterile neutrinos, fs, of fs < 22%. More recently, preliminary analysis of further data has generated limits on the sterile mixing

angle θ24 which extend the range of previous experiments [49]. A fundamental test of CPT symmetry is the equivalence of oscillation parameters

obtained from νµ andν ¯µ. The magnetized MINOS detector can perform an event-by- event comparison of these parameters enabling an accurate test of this prediction. The resulting allowed regions (from [46]) are displayed in figure 23. The difference in ∆m2 2 2 +0.24 3 2 | | obtained, in a two-flavor model, is ∆m ¯ ∆m = (0.12 0.26) 10− eV . | | − | | − × Neutrino Oscillations with the MINOS, MINOS+, T2K, and NOvA Experiments. 27

Figure 23. MINOS comparison of νµ andν ¯µ oscillations from [46]. The 90% confidence level allowed regions and best fit values are shown for νµ andν ¯µ oscillations, and for a fit in which the parameters are assumed to be identical.

6.2. Additional Measurements in T2K In addition to neutrino oscillation studies, T2K conducts various measurements on neutrino-nucleus cross sections. As described in Section 5.2, the understanding of neutrino cross sections is important to reduce systematic uncertainties of neutrino oscillation measurements, which could improve the sensitivity to neutrino oscillations. The cross sections measured in T2K are summarized in Table 3. As the first step, T2K measures the muon neutrino charged current (CC) inclusive cross sections with the T2K off-axis near detector (ND280) [15] and the on-axis near detector (INGRID) [17]. In future, these analyses will be more sophisticated to measure exclusive channels, such as CC-QE, CC 1π production, and CC-coherent π, as energy dependent differential cross sections. With INGRID, there are two types of neutrino detector with different target materials. One has an iron target, and the other has a plastic (CH) target. By using two targets, the CC inclusive cross sections on iron and plastic, and the ratio of cross sections are measured. The CC-QE cross sections are also measured with the CH target [18]. With ND280, the electron neutrino CC inclusive cross sections can also be measured [16, 20], using the powerful particle identification performance of TPC and ECAL. In the analysis [20], the electron neutrino contamination in the beam is measured relative to the prediction in the simulation. T2K also has divided the measurement into two contributions: One is the electron neutrino from kaon decay and the other is from muon decay. The neutral current (NC) gamma production cross section in neutrino-oxygen interaction has also been measured [20] by using the far detector, Neutrino Oscillations with the MINOS, MINOS+, T2K, and NOvA Experiments. 28

Super-Kamiokande. All the results are consistent with the predictions in the neutrino interaction generator libraries NEUT [21] and GENIE [22]. Finally, with ND280, the electron neutrino disappearance sample was searched to investigate neutrino oscillations to sterile neutrinos [23].

Table 3. Neutrino Cross Sections measurements in T2K for charged current (CC) and neutral current (NC) inclusive processes. The measurements of cross sections are given per nucleon. The ratio of cross sections is also shown in some measurements. cm2 Mode Results ( nucleon or the ratio) < Eν > (GeV) Reference −39 νµ CC inclusive (6.91 0.13(stat) 0.84(syst)) 10 0.85 [15] ± ± × −38 νe CC inclusive (1.11 0.09(stat) 0.18(syst)) 10 1.3 [16] ± ±+0.189 × −38 ∼ νµ CC inclusive on F e (1.444 0.002(stat)−0.157(syst)) 10 1.51 [17] ± +0.178 × −38 νµ CC inclusive on CH (1.379 0.009(stat) (syst)) 10 1.51 [17] ± −0.147 × νµ CC ratio of F e/CH 1.047 0.007(stat) 0.035(syst) 1.51 [17] ± +2.03± −39 νµ CC-QE on CH (10.64 0.37(stat)−1.65(syst)) 10 0.93 [18] ± +1.82 × −39 νµ CC-QE on CH (11.95 0.19(stat) (syst)) 10 1.94 [18] ± −1.47 × NC γ (1.55+0.65) 10−38 0.63 [19] −0.33 × νe CC ratio σν CC /σprediction = 1.01 0.10 1.3 [20] e ± ∼ νe (K decay) CC ratio σν CC /σprediction = 0.68 0.30 — [20] e ± νe (µ decay) CC ratio σν CC /σprediction = 1.10 0.14 — [20] e ±

7. The Frontier and future sensitivity

7.1. NOvA The NOvA experiment is the principal appearance-mode long-baseline experiment at Fermilab. It makes use of the off-axis NuMI beam as discussed in previous sections. The far and near detectors minimize passive mass, moving toward the ideal of a totally active fiducial volume which will be realized with future detectors, such as e.g. DUNE liquid argon TPC’s. A totally active detector typically has better performance because all charged particles in neutrino interactions are reconstructed properly with good efficiency. In the case of NOvA the active medium is liquid scintillator mixed into oil. The basic segmentation of the detector is planes of 3.6 cm x 5.6 cm PVC tubes, separated by thin walls. Light is collected in each tube by a looped wavelength shifting fiber that is directed onto a single pixel of an avalanche photodiode (APD). The APD’s are cooled

by a hybrid thermoelectric/water system to a temperature of 15◦ C. − There are a total of 344,064 tubes arranged in an interchanging pattern of horizontal and vertical planes, giving an active structure 15.6 m x 15.6 m, and 60 m long. The active scintillator mass is 8.7 ktonnes with 5.3 ktonnes devoted to the PVC support structure, ﬁber readout, and other structural components, giving a total far detector mass of 14.0 ktonnes. Figure 7.1 shows a typical NOvA far detector event. In addition to the far detector, a functionally equivalent near detector is located at Fermilab. This detector consists of horizontal planes 4.0 m x 4.0 m, together with a Neutrino Oscillations with the MINOS, MINOS+, T2K, and NOvA Experiments. 29

Figure 24. Charged-current neutrino interaction observed in the NOvA Far Detector (Courtesy NOvA Collaboration).

steel muon catcher to assist in measurement of the spectrum of CC events. The expected average occupancy of the near detector is 6 events per spill at full intensity, which ≈ will be separated by their time of occurrence, as is already successfully done in MINOS and MINERvA.

The relatively large value of θ13 provides the opportunity for a rich harvest of physics results for NOvA via both disappearance and appearance, with both neutrino and antineutrino beams. Here we focus on the appearance channel, with its rich information about θ13, mass hierarchy, and δCP . Disappearance measurements are also highly sensitive and give excellent information on the octant of θ23. During early running, combination with MINOS+ will be exploited, as discussed separately below.

As discussed in earlier sections, the measurements of νe appearance of NOvA are significantly affected by matter effects, potential CP violation, mass hierarchy, and the distinctions between neutrinos and anti-neutrinos. A useful tool for understanding these dependencies is the biprobability plot, in which one axis displays the appearance probability for νe and the other the probability forν ¯e. Figure 25 shows this situation graphically, for the case of maximal mixing. The results are clearly separated for many values of δCP , with areas of overlap for the regions around δCP = π/2 (δCP = 3π/2) for normal (inverted) mass hierarchy. Moreover, the subleading effects cause a separation in the appearance probabilities depending on the octant of θ23. The sensitivity of NOvA to the octant depends on the value of θ23, and somewhat on both the hierarchy and the 2 value of δCP . For a value of sin 2θ23 of 0.95, there is considerable sensitivity for all values of those parameters, exceeding 95% CL for significant portions of the parameter space with the nominal exposure, shown in Figure 26. Neutrino Oscillations with the MINOS, MINOS+, T2K, and NOvA Experiments. 30

NOΝA1Σ and 2Σ Contours for Starred Point NOΝA1Σ and 2Σ Contours for Starred Point NOΝA NOΝA Nominal Run NOΝA NOΝA Nominal Run 810km Baseline 3.6E21 PoT Ν&Ν 810km Baseline 3.6E21 PoT Ν&Ν 2 2 sin 2Θ13#0.09 sin 2Θ13#0.09 0.08 2 0.08 2 sin 2Θ23#1.00 sin 2Θ23#0.95 " # " #

" Inverted 0.06 0.06 # " !

# Inverted

# # e e $" )Ν

)Ν " Μ Μ Ν ! Ν " " ! 2Σ P P 0.04 " 0.04 # $ ! # 1Σ 2Σ ! " 1Σ %$ # ! $ %$ # 0.02 Normal 0.02 $ ! ∆#0 Normal ! ∆#0 " ∆#Π 2 " ∆#Π 2 # ∆#Π # ∆#Π $ ∆#3Π 2 $ ∆#3Π 2 0.00 0.00 0.00 0.02 0.04 0.06 0.08 ! 0.00 0.02 0.04 0.06 0.08!

P ΝΜ)Νe ! P ΝΜ)Νe !

" # Figure 25." Biprobability# plot showing the eﬀect of mass hierarchy, δCP , and νµ vs ν¯µ exposure. The left panel shows the expected appearance probabilities for νe (¯νe) when the mixing angle θ23 = π/4 (maximal mixing.) The right panel shows the same for assumed non-maximal mixing with θ23 < π/4 and θ23 > π/4. (Courtesy NOvA Collaboration).

NOνA octant determination NOνA octant determination

) 2 θ 2 θ θ π 20 ) 2 θ 2 θ θ π 20 σ sin 2 13=0.095, sin 2 23=0.95, 23< /4 36×10 POT σ sin 2 13=0.095, sin 2 23=0.95, 23> /4 36×10 POT 5 5 4.5 Inverted 4.5 Inverted 4 Normal 4 Normal 3.5 3.5 3 3 2.5 2.5 2 2 1.5 1.5 1 1 0.5 0.5 0 0 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 significance of octant determination ( δ π significance of octant determination ( δ π CP / CP /

Figure 26. NOvA sensitivities for resolution of the octant of non-maximal θ23, 2 assumed to be at sin 2θ23 = 0.95. Left (right) panel shows the sensitivity for θ23 < π/4 (θ23 > π/4). (Courtesy NOvA Collaboration).

As suggested by Figure 25, a principle goal of NOvA is to gain information about the mass hierarchy of the neutrino eigenstates. We can see the sensitivity of a ”standard” expected exposure in the NuMI beam in Figure 27. Hierachy and CP violation information are coupled, leading to the right panel of the ﬁgure, in which the fractional coverage of ”CP-space” 0 2π at which the hierarchies can be separated is shown as a − function of the signiﬁcance of the separation.

Because of parameter ambiguities, the study of CP violation in neutrino oscillations is particularly challenging. Full resolution of the problem may require the very large Neutrino Oscillations with the MINOS, MINOS+, T2K, and NOvA Experiments. 31

NOνA hierarchy resolution NOνA hierarchy resolution

) sin22θ =0.095, sin22θ =1.00 × 20 sin22θ =0.095, sin22θ =1.00 × 20 σ 13 23 36 10 POT 13 23 36 10 POT 4 1 0.9 3.5 Inverted Normal 0.8 3 Inverted 0.7 CP

δ Normal 2.5 0.6 2 0.5 0.4 1.5

Fraction of 0.3 1 0.2 0.5 0.1 0 0 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 0 0.5 1 1.5 2 2.5 3 3.5 4 significance of hierarchy resolution ( δ π σ CP / Significance of hierarchy resolution ( )

Figure 27. Sensitivity of NOvA to the neutrino mass hierarchy. The left panel shows the sensitivity as a function of δCP /π. The right panel shows the fraction of δCP space covered for various degrees of statistical signiﬁcance σ. Both plots are for maximal θ23. Approximate one third of the δCP range gives a hierarchy determination at 95% C.L. (Courtesy NOvA Collaboration).

Example NOνA contours 36×1020 POT Example NOνA contours 36×1020 POT 2 θ 2 θ θ π δ π 2 θ 2 θ θ π δ π sin 2 13=0.095, sin 2 23=1.00, Normal, 23> /4, =3 /2 sin 2 13=0.095, sin 2 23=1.00, Normal, 23> /4, = /2 0.7 CP 0.7 CP

0.65 0.65

0.6 0.6

0.55 0.55 23 23 θ θ

2 0.5 2 0.5 sin sin 0.45 0.45

0.4 Inverted, 1σ CL Normal, 1σ CL 0.4 Inverted, 1σ CL Normal, 1σ CL Inverted, 2σ CL Normal, 2σ CL Inverted, 2σ CL Normal, 2σ CL 0.35 0.35 True parameters True parameters 0.3 0.3 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 δ π δ π CP / CP /

2 Figure 28. Examples of joint NOvA sensitivity contours for sin θ23 and δCP , for a nominal run of 36 1020 protons on target in the 700 kW NuMI beam. In the × left panel θ23 is maximal, and δCP is chosen for maximum separation between the hierarchies. The inverted hierarchy and some values of δCP are disfavored. A less- favorable case is shown in the right panel, where δCP is chosen to illustrate the diﬃculty of distinguishing the hierarchies. In both-cases θ13 is taken as an external input (Courtesy NOvA Collaboration).

detectors of Hyper-K and DUNE, currently under discussion. However, particularly in favorable cases, important information can be obtained. Figure 28 illustrates the situ- 2 ation by plotting the simultaneous signiﬁcance of two quantities, sin θ23 and δCP , for both hierarchies. In confused cases information is obtained about the likely correlations of δCP and the hierarchy, and less ambiguous cases will favor speciﬁc regions of δCP .

7.2. MINOS+ The MINOS experiment, as described in this paper, has studied the region in L/E near the oscillation minimum in detail, using primarily low-energy settings of the NuMI Neutrino Oscillations with the MINOS, MINOS+, T2K, and NOvA Experiments. 32

Figure 29. Ratio of oscillated to unoscillated predictions for νµ charged-current events in the MINOS+ experiment, as a function of exposure in terms of protons on target (POT). The statistical precision in the neutrino energy region of 5-7 GeV is much improved by the high rates available from the NuMI complex (Courtesy MINOS+ Collaboration).

beam. As discussed in section 2.2 the requirements of oﬀ-axis kinematics for the NOvA experiment lead to a need for a higher energy on-axis setting of the on-axis beam. This beam, with its associated larger event rates in both MINOS near and far detectors, is exploited by the MINOS+ experiment. Fig 29 shows the expected structure of the data spectrum which will be obtained by MINOS+. The experiment, will collect on the order of 3000 CC and 1200 NC events for each exposure of 6.0 1020 protons on the × NuMI target (roughly annually). These large event rates allow a varied physics program.

Representative physics goals of the MINOS+ experiment include:

(i) Precise veriﬁcation of the expected spectral shape of the oscillation phenomenon, using the disappearance technique, especially in its transition region in energy between 5 and 7 GeV. (ii) Utilization of the precise spectrum together with MINOS and NOvA data to continue improving understanding of the oscillation parameters. (iii) Further study of the possibilities of sterile neutrinos, using both CC and NC disappearance. (iv) Improved precision on searches for exotic phenomena. Figure 30 shows an example of (ii), considering all the NuMI program data expected to be obtained in the 2015 time frame. Comparison with Figure 16 shows improved determination of the parameters, especially in the case of the normal mass hierarchy. Neutrino Oscillations with the MINOS, MINOS+, T2K, and NOvA Experiments. 33

MINOS DATA MINOS DATA

) ) & MINOS+, NOvA SIMULATION 2 2 2.8 & MINOS+ SIMULATION 2.8 MINOS: All atmospheric and beam data MINOS: All atmospheric and beam data MINOS+: 5.3 ×1020 POT ν -mode, 17.0 kt-yr atmospheric ν MINOS+: 5.3 ×1020 POT ν -mode, 17.0 kt-yr atmospheric ν eV µ µ eV × 20 ν -3 -3 2.6 2.6 NOvA: 2.5 10 POT µ-mode

(10 (10 2.4 2.4 2 32 2 32 m m ) ) ∆ ∆

2 2.2 2 2.2 Normal hierarchy Normal hierarchy eV eV -3 -3 0.3Inverted hierarchy0.4 0.5 0.6 0.7 0.3Inverted hierarchy0.4 0.5 0.6 0.7 sin2θ sin2θ (10 23 (10 23 ) )

2 2 -2.2 -2.2 2 32 2 32 m m eV eV ∆ ∆ -3 -3 -2.4 -2.4

(10 (10 -2.6 -2.6 2 32 2 32 m m

∆ ∆ -2.8 68% C.L. 90% C.L. -2.8 68% C.L. 90% C.L.

0.3 0.4 0.5 0.6 0.7 0.3 0.4 0.5 0.6 0.7 2θ 2θ sin 23 sin 23

Figure 30. Expected near-term precision of determination of the oscillation 2 2 parameters sin θ23 and ∆m32 for the MINOS+ experiment. The left panel shows the results when MINOS and MINOS+ data are combined, and the right panel includes the expected data from the NOvA experiment. The contours are generated using as inputs the best-ﬁt MINOS values from [54] (Courtesy MINOS+ Collaboration).

7.3. Future prospects of T2K The approved beam for the T2K experiment is 7.8 1021 protons on target (POT). × The results of T2K reported in this paper are based on 6.6 1020 which is only 8 % of × the original goal . In the near future, the J-PARC accelerator plans to increase the ¶ repetition rate of the acceleration cycle by updating the power supply system. With the upgrade, the beam power of J-PARC will reach 750 kW, and T2K will accumulate the design beam within several years. In this section, we show the physics sensitivity of T2K with 7.8 1021 POT. In T2K, there are two beam operation modes: one is neutrino × beam and the other is anti-neutrino beam. Since the fraction of the anti-neutrino beam to the neutrino is not ﬁxed yet, we will show both possibilities. The current goals of T2K with 7.8 1021 POT are × Initial measurement of CP violation in neutrinos up to a 2.5σ level of signiﬁcance. • T2K collected 1.0 1021 POT on March 26th, 2015. ¶ × Neutrino Oscillations with the MINOS, MINOS+, T2K, and NOvA Experiments. 34

Precision measurement of oscillation parameters in the νµ disappearance with • 2 4 2 2 precision of δ∆m 10− eV and δ sin 2θ 0.01 ; also determination of θ 32 ' 23 ' 23 octant at 90 % C.L. if θ 45◦ > 4◦. | 23 − | Contribution to the determination of the mass hierarchy. •

7.3.1. Neutrino events with 7.8 1021 protons on target (POT) ×

+ Based on the analysis method in [6, 8], the expected number of νe and νµ events are 2 shown in Table 4 and 5 using the following neutrino oscillation parameters: sin 2θ13 = 2 2 2 5 2 2 3 2 0.1, sin θ23 = 0.5, sin 2θ12 = 0.8704, ∆m12 = 7.6 10− eV , ∆m32 = 2.4 10− eV , 2 × | | × δCP = 0, and ∆m32 > 0.

21 Table 4. The expected number of νe events in T2K with 7.8 10 POT for the × neutrino beam mode and 7.8 1021 POT for the ant-neutrino [35]. × Signal Signal Beam CC Beam CC NC Beam Mode Total ν ν ν ν ν + ν ν + ν NC µ → e µ → e e e µ µ neutrino beam 291.5 211.9 2.4 41.3 1.4 34.5 anti-neutrino beam 94.9 11.2 48.8 17.2 0.4 17.3

21 Table 5. The expected number of νµ events in T2K with 7.8 10 POT for each × beam operation mode [35].

CCQE CC non-QE CC νe + νe Beam Mode Total ν (ν ) ν (ν ) ν (ν ) ν (ν ) NC µ µ µ µ µ µ → e e neutrino beam 1,493 782 (48) 544 (40) 4 75 anti-neutrino beam 715 130 (263) 151 (138) 0.5 33

7.3.2. CP sensitivity

Since the electron neutrino appearance is sensitive to CP violation, the variation

of the number of electron neutrino events with δCP parameters is shown in Figure 31. In maximum, we expect a 27 % change, compared to no CP violation with δCP = 0. Hereafter, we assume the beam exposure to be 50 % for the neutrino beam and 50 % for the anti-neutrino. We also assume sin2 2θ = 0.10 0.005 as the ultimate θ value 13 ± 13 from reactor experiments. In the case of the maximum CP violation (δ = 90◦), the CP − T2K sensitivity for δ = 90◦ is shown in Figure 32 with 90 % C.L. CP − + This study was conducted before T2K developed the special π0 rejection algorithm. So, the number of NC background is higher, compared to the results in Section 3.2 35

δCP = 0° 30 δCP = 90°

δCP = 180° 25 δCP = -90°

10 150 5 100 0 Events / 50 MeV 7.8e21 POT 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 50

) Reconstructed Energy (GeV) ° Neutrino Oscillations with ( the0 MINOS, MINOS+, T2K, and(a) ⌫ NOvA-mode running Experiments. 35 CP δ -50 35 35NH, no Sys. Err.

δ = 0° NH, w/ Sys. Err. δCP = 0° -100 CP 30 30IH, w/o Sys. Err. = 90 δCP = 90° δCP ° = 180 IH, w/ Sys. Err. δ = 180° -150 δCP ° CP 25 25 = -90 δCP = -90° δCP ° 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 20 20 2 sin 2θ13 15 15 (a) 50% ⌫-mode only. 10 10

5 5 150 0 0 Events / 50 MeV 7.8e21 POT Events / 50 MeV 7.8e21 POT 0.0 0.2 0.4 0.6 0.8100 1.0 1.2 1.4 1.6 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 Reconstructed Energy (GeV) Reconstructed Energy (GeV) 50

(a) ⌫-mode) running (b)⌫ ¯-mode running °

Figure 31. ( The0 reconstruction energy of expected T2K electron neutrino appearance

CP 21 35 events withFig. various 2: ⌫eδCPappearanceparameters reconstructed [35]. (Left) energy Neutrino spectra beam in Super-K operation for with 7.8 10 POT in either 21 δ 21 ⇥ 7.8 10 POT and (Right)δ = 0° Anti-neutrino beam operation with 7.8 10 POT. 2 × -50⌫-mode orCP⌫ ¯-mode at variousNH, values no Sys. of Err. assumed true×CP with sin ✓23 =0.5. 30 δ = 90° CP NH, w/ Sys. Err. -100 δ = 180° CP IH, w/o Sys. Err. 25 δ = -90° 280 4.3. SensitivitiesCP for CP-violating term, non-maximal ✓23, and ✓23 octant -150 IH, w/ Sys. Err. 20 281 The sensitivities for CP violation, non-maximal ✓23, and the octant of ✓23 (i.e., whether the 282 mixing0.00 0.05 angle 0.10✓23 0.15is less 0.20 than 0.25 or 0.30 greater 0.35 0.40 than 45) depend on the true oscillation parameter 15 2 2 283 values. Fig. 7 shows the expected for the sin = 0 hypothesis, for various true values sin 2θ13 CP 2 2 10 284 of CP and sin ✓23. To see the dependence more clearly, is plotted as a function of CP 2 285 for various(b) values 50%⌫ of¯-mode sin ✓23 only.in Fig. 8 (normal MH case) and Fig. 9 (inverted MH case). For 5 286 favorable sets of the oscillation parameters and mass hierarchy, T2K will have greater than 287 90% C.L. sensitivity to non-zero sin CP. 0 Events / 50 MeV 7.8e21 POT 0.0 0.2 0.4 0.6 0.8150 1.0 1.2 1.4 1.6 14 Reconstructed Energy (GeV) 100 (b)⌫ ¯-mode running 50 )

° 21 Fig. 2: ⌫e appearance reconstructed energy spectra in Super-K for 7.8 10 POT in either ( ⇥ ⌫-mode or⌫ ¯-mode at various values of assumed true0 with sin2 ✓ =0.5.

CP CP 23 δ -50 NH, no Sys. Err. NH, w/ Sys. Err. 280 4.3. Sensitivities for CP-violating term, non-maximal-100 ✓23, and ✓23 octant IH, w/o Sys. Err. 281 The sensitivities for CP violation, non-maximal ✓23, and the octant of ✓23 (i.e., whether the IH, w/ Sys. Err. 282 mixing angle ✓23 is less than or greater than 45-150) depend on the true oscillation parameter 2 283 values. Fig. 7 shows the expected for the sin = 0 hypothesis, for various true values CP0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 2 2 284 of CP and sin ✓23. To see the dependence more clearly, is plotted2 as a function of CP 2 sin 2θ13 285 for various values of sin ✓23 in Fig. 8 (normal MH case) and Fig. 9 (inverted MH case). For 286 favorable sets of the oscillation parameters and mass(c) hierarchy, 50% ⌫ T2K-, 50% will have⌫ ¯-mode. greater than Figure 32. The 90 % contour region of the δ versus sin2 2θ plane in the expected 287 90% C.L. sensitivity to non-zero sin CP. CP 13 T2K sensitivity with 7.8 1021 POT [35]. POT are assumed to be equally distributed 14 × to the neutrino2 beam mode (3.9 1021 POT) and the anti-neutrino (3.9 1021 POT). Fig. 3: Expected CP vs. sin 2✓13 90% C.L.× intervals, where (a) and (b)× are each given for 50% of the full T2K POT, and (c) demonstrates the sensitivity of the total T2K POT with 50% ⌫-mode plus 50%⌫ ¯-mode running. Contours are plotted for the case of true = 90 CP and NH. The blue curves are ﬁt assuming the correct MH(NH) , while the red are ﬁt assuming the incorrect MH(IH), and contours are plotted from the minimum 2 value for both MH assumptions. The solid contours are with statistical error only, while the dashed contours include the systematic errors used in the 2012 oscillation analysis assuming full correlation between15 ⌫- and⌫ ¯-mode running errors. 0.65 0.65 5 0.60 0.60 5 4

23 0.55 23 0.55 4 θ θ 2 2

3 2 2 χ χ

0.50 0.50 3 Δ Δ 2 0.45 0.45 2 True sin True sin

0.40 1 0.40 1

0.35 0.35 -150 -100 -50 0 50 100 150 -150 -100 -50 0 50 100 150

True δCP (°) True δCP (°) (a) Normal mass hierarchy. (b) Normal mass hierarchy. Neutrino Oscillations100% ⌫-mode. with the MINOS, MINOS+, T2K,50% and⌫-, NOvA 50%⌫ ¯-mode. Experiments. 36

0.65 0.650.65 10 0.65 5 6 0.60 5 0.60 0.60 0.60 8 4 5 23 23 23 0.55 0.5523 0.55 4 0.55 θ θ θ θ 2 2 2 2 6 4 2 2 3 2 2 χ χ χ χ

0.50 0.500.50 3 0.50 Δ Δ

Δ Δ 3 4 2 0.45 0.45

0.45 True sin 0.45 2 True sin True sin True sin 2 2 0.40 1 0.400.40 1 0.40 1

0.35 0.350.35 0.35 -150 -100 -50 0 50 100 150 -150-150 -100 -100 -50 -50 0 0 50 50 100 100 150 150 -150 -100 -50 0 50 100 150 True δ (°) True δ (°) True δCP (°) TrueCP δCP (°) CP (a) Normal mass hierarchy. (c)(b) Inverted Normal mass mass hierarchy. hierarchy. (d) Inverted mass hierarchy. 2 2 100% ⌫-mode. 50%Figure100%⌫-,⌫ 50%-mode. 33. ⌫ ¯T2K-mode. expected χ diﬀerence50% ⌫ between-, 50%⌫ ¯ the-mode. true point (δCP, sin θ23) and the 2 point δCP = 0 [35]. The map of χ diﬀerence shown in color is calculated assuming no 2 2 0.65 10 Fig. 7: The0.65 expectedsystematic errors.for the The sin solidCP contours= 0 hypothesis, show the 90 in % the C.L.CP sensitivitysin ✓ with23 plane. statistical The error only, while the dashed contours include the 2012 T2K systematic error. (Left) 2 map shown in color is calculated assuming6 no systematic errors. The solid contours 0.60 0.60 the contour in the case of Normal mass hierarchy, and (Right) in the case of Inverted 8 show the 90% C.L. sensitivity with statistical error only, while the dashed contours include mass hierarchy. POT are assumed5 to be equally distributed to the neutrino beam the 2012 T2K systematic error.21 The dashed contour does not appear21 in (a) because T2K 23 0.55 23 0.55 mode (3.9 10 POT) and the anti-neutrino (3.9 10 POT).

θ θ × × 2 6 does not2 have 90% C.L. sensitivity in this4 case. 2 2 χ χ

0.50 0.50

Δ 3 Δ 4 In reality, the sensitivity to CP violation also depends on θ23. In Figure 33, as the 0.45 0.45 True sin 308 Generally,True sin the e↵ect of the systematic2 errors2 is reduced by running with combined ⌫-mode2 T2K sensititivy, we show the χ difference between the true point with (δCP, sin θ23) 3092 and⌫ ¯-mode. When running 50% in ⌫-mode and 50% in⌫ ¯-mode, the statistical 1 uncertainty 0.40 and the0.40 hypothesis test point with δ = 0. 2 2 CP1 3 2 310 of sin ✓23 and m32 is 0.045 and 0.04 10 eV , respectively, at the T2K full statistics. ⇥ 2 0.35 311 It should0.35 be noted that the sensitivity to sin ✓23 shown here for the current exposure -150 -100 -50 0 50 100 150 7.3.3. Precision20 -150 -100 measurements -50 0 50 100 of 150 neutrino oscillation parameters True δ (°) 312 (6.57 10 POT) isTrue significantly δ (°) worse than the most recent T2K result [14], and in fact the CP ⇥ CP 21 313 recent result is quite close to the final sensitivity (at 7.8 10 POT) shown. This apparent (c) Inverted mass hierarchy. (d)Since Inverted most mass of hierarchy. the T2K measurements are limited⇥ by statistics, more data improve 314 discrepancy comes from three factors. About half of the di↵erence between the2 expected 100% ⌫-mode. the50% precision⌫-, 50%⌫ ¯-mode. of the measurements. Among the oscillation parameters, sin θ23 and 315 sensitivity2 and observed result is due to an apparent statistical fluctuation, where fewer T2K ∆m32 are interesting because of their relatively larger uncertainty compared to other 2 | | 2 Fig. 7: The expected for the sin CP = 0 hypothesis, in the CP sin ✓23 plane.19 The 2 2 parameters. Figure 34 shows the expected precision of sin θ23 and ∆m32 as a function 2 | | map shown in color is calculated assumingof POT no systematic for the normal errors. mass The hierarchy solid contours case. Hereafter, the total exposure in T2K is show the 90% C.L. sensitivity with statisticalassumed error only, to bewhile 7.8 the10 dashed21 POT contours which are include equally distributed to the neutrino beam mode × the 2012 T2K systematic error. The dashed(3 contour.9 10 does21 POT) not and appear the in anti-neutrino (a) because (3 T2K.9 1021 POT). The statistical uncertainty of does not have 90% C.L. sensitivity in this case. 2 × 2 3 × 2 sin θ23 and ∆m32 is 0.045 and 0.04 10− eV , respectively, at the T2K full statistics. | | 2 × The precision of sin θ13 is influenced by the precision of other oscillation parameters 2 including sin θ23 and δCP. 308 Generally, the e↵ect of the systematic errors is reduced by running with combined ⌫-mode An interesting question about θ23 is which θ23 is exactly 45◦ or not. In the case of 309 and⌫ ¯-mode. When running 50% in ⌫-mode and 50% in⌫ ¯-mode, the statistical 1 uncertainty 2 2 3θ23 =2 45◦, which octant does the value of θ23 fit in, (θ23 > 45◦ or θ23 < 45◦)? Figure 35 310 of sin ✓23 and m32 is 0.045 and 0.04 10 eV6 , respectively, at the T2K full statistics. ⇥ shows2 the region where T2K can reject the maximum mixing θ23 = 45◦ and the regions 311 It should be noted that the sensitivity to sin ✓23 shown here for the current exposure 20 where T2K can reject one of the octants of θ23. The octant of θ23 is determined at 90 % 312 (6.57 10 POT) is significantly worse than the most recent T2K result [14], and in fact the ⇥ C.L. if θ23 4521◦ > 4◦. 313 recent result is quite close to the final sensitivity (at 7| .8 −10 |POT) shown. This apparent ⇥ 314 discrepancy comes from three factors. About half of the di↵erence between the expected

315 sensitivity and observed result is due to an apparent statistical ﬂuctuation, where fewer T2K 19 0.10 0.10 0.09 Stat. Err. Only 0.09 Stat. Err. Only 2012 Sys. Errs. 0.08 Projected Sys. Errs. 0.08 Projected Sys. Errs. 0.07 0.07 Full T2K POT Full T2K POT 0.06 0.06

Precision 0.05 Precision 0.05 σ σ

1 0.04 1 0.04 23 23

θ 0.03 θ 0.03 2 2 0.02 0.02 sin sin 0.01 0.01 21 21 0.00 ×10 0.00 ×10 0 1 2 3 4 5 6 7 8 9 10 0 1 2 3 4 5 6 7 8 9 10 POT POT Neutrino Oscillations(a) 100% ⌫ with-mode. the MINOS, MINOS+, T2K,(b) and 50% NOvA⌫,50%¯⌫ Experiments.-mode. 37

-3 -3 0.10 0.10×10 ×10

Stat. Err. Only ) 0.14 Stat. Err. Only ) 0.14

0.09 2 0.09 Stat. Err. Only 2 Stat. Err. Only 2012 Sys. Errs. 2012 Sys. Errs. 0.08 Projected Sys. Errs. 0.120.08 ProjectedProjected Sys. Sys. Errs. Errs. 0.12 Projected Sys. Errs. 0.07 0.07 Full T2K POT 0.10 Full T2K POT 0.10 0.06 0.06 0.08 0.08 Precision 0.05 Precision 0.05 σ σ Full T2K POT Full T2K POT

1 0.04 0.06 1 0.04 0.06 Precision (eV Precision (eV 23 23 σ σ

θ 0.03 θ 0.03 2 2

1 0.04 1 0.04

0.02 2 32 0.02 2 32 sin sin

m 0.02 m 0.02 0.01 0.01 Δ Δ 21 21 21 21 0.00 ×10 0.000.00 ×10×10 0.00 ×10 0 1 2 3 4 5 6 7 8 9 10 00 1 1 2 2 3 3 4 4 5 5 6 6 7 7 8 8 9 9 10 10 0 1 2 3 4 5 6 7 8 9 10 POT POTPOT POT (a) 100% ⌫-mode. (b)(c) 50% 100%⌫,50%¯⌫-mode.⌫-mode. (d) 50% ⌫,50%¯⌫-mode. 2 2 Figure 34. T2K expected sensitivity to sin θ23 and ∆m32 as a function of POT for -3 -3 | | ×10 ×10 the normal mass hierarchy2 [35]. The2 solid lines are without systematic error and the Fig. 12: The uncertaintydashed one on with sin the✓ systematic23 and m error32 plotted from the as T2K a function2012 analysis. of T2K POT POT. are assumed Plots ) 0.14 Stat. Err. Only ) 0.14

2 2 Stat. Err. Only assume the trueto oscillation be 3.9 1021 parametersPOT for the given neutrino in Table beam mode 3. The and solid 3.9 10 curves21 for includethe anti-neutrino. statisti- 2012 Sys. Errs. × × 0.12 Projected Sys. Errs. cal errors0.12 only, while theProjected dashed Sys. curves Errs. assume the 2012 systematic errors (black) or the 0.10 projected0.10 systematic errors (red). A constraint based on the ultimate reactor precision is included. 0.08 0.08 Full T2K POT Full T2K POT 0.06 0.06 Precision (eV Precision (eV

σ 348 case theσ systematic error on the predicted number of events in Super-K is about 7% for the

1 0.04 1 0.04 349 ⌫µ and ⌫e samples and about 14% for the⌫ ¯µ and⌫ ¯e samples. These errors were calculated 2 32 0.652 32 0.65

0.65m m 0.65 0.02 350 by reducing0.02 the 2012 oscillation analysis errors by removing certain interaction model and Δ Δ 21 21 0.00 351×10 cross0.60 section0.00 uncertainties from both the ⌫×e10- and ⌫µ-mode0.60 errors, and by additionally scaling 0.60 0 1 2 3 4 5 6 7 8 9 10 0.600 1 2 3 4 5 6 7 8 9 10 352 all ⌫µ-mode errors down by a factor of two. Errors for the⌫ ¯µ- and⌫ ¯e-modes were estimated

POT 353 23 0.55 POT 23 0.55 23 to be twice23 those of the ⌫µ- and ⌫e-modes, respectively. These reduced ⌫-mode errors are

0.55 θ 0.55 θ θ θ 2 2 2 (c) 100% ⌫-mode. 354 in fact2 very close(d) 50% to the⌫,50%¯ errors⌫-mode. used for the oscillation results reported by T2K in 2014, 0.50 0.50 0.50 355 where the0.50 T2K oscillation analysis errors have similarly been reduced by improvements in 2 2 Fig. 12: The uncertainty on sin ✓23 and356 understandingm32 plotted as the a functionrelevant interactions of T2K POT. and Plots cross sections. 0.45 0.45

0.45 True sin 0.45 True sin assumeTrue sin the true oscillation parameters given in TableTrue sin 3. The solid curves include statisti- 24 cal errors only, while the dashed curves assume the 2012 systematic errors (black) or the 0.40 0.400.40 0.40 projected systematic errors (red). A constraint based on the ultimate reactor precision is included.0.35 0.350.35 0.35 -150 -100 -50 0 50 100 150 -150-150 -100 -100 -50 -50 0 0 50 50 100 100 150 150 -150 -100 -50 0 50 100 150 True δ (°) True δ (°) True δCP (°) TrueCP δCP (°) CP 348 case the systematic error on the predicted number of events in Super-K is about 7% for the (a) Normal mass hierarchy. (a)(b) Normal Normal mass mass hierarchy. hierarchy. (b) Normal mass hierarchy. 349 ⌫µ and ⌫e samples100% and⌫-mode about 14% for the⌫ ¯µ and⌫ ¯e samples.50%100%Figure These⌫-,⌫-mode. 50% 35. errors⌫ ¯-mode.(Left)The were calculated shaded region is where50% T2K⌫-, has 50% more⌫ ¯-mode. than a 90 % C.L. 350 by reducing the 2012 oscillation analysis errors by removingsensitivity certain tointeraction reject maximal model mixing and [35]. (Right) The shaded region is where T2K has more than a 90 % C.L. sensitivity to reject one of the octants of θ23 [35]. The mass 351 cross0.65 section uncertainties from both the ⌫e- and0.65⌫µ0.65-mode errors, and by additionally scaling 0.65 hierarchy is considered unknown in the normal mass hierarchy case. The shaded region 352 all ⌫ -mode errors down by a factor of two. Errors for the⌫ ¯ - and⌫ ¯ -modes were estimated µ µis calculatede assuming no systematic errors (statistical error only), and the dashed 0.60 0.600.60 0.60 353 to be twice those of the ⌫µ- and ⌫e-modes, respectively. Thesecontours reduced show the⌫-mode sensitivity errors including are the systematic errors. POT are assumed to be 21 21 354 in fact very close to the errors used for the oscillation results3.9 10 reportedPOT for by the T2K neutrino in 2014, beam mode and 3.9 10 for the anti-neutrino. 23 23 23 0.55 0.5523 0.55 × 0.55 × θ θ θ θ 2 2 355 where2 the T2K oscillation analysis errors have similarly2 been reduced by improvements in

356 understanding0.50 the relevant interactions and cross0.50 sections.0.50 0.50 24 0.45 0.450.45 0.45 True sin True sin True sin True sin

0.40 0.400.40 0.40

0.35 0.350.35 0.35 -150 -100 -50 0 50 100 150 -150-150 -100 -100 -50 -50 0 0 50 50 100 100 150 150 -150 -100 -50 0 50 100 150

True δCP (°) TrueTrue δCP δ CP(°) (°) True δCP (°) (c) Inverted mass hierarchy. (d) Inverted(c) Inverted mass hierarchy. mass hierarchy. (d) Inverted mass hierarchy. 100% ⌫-mode 50% ⌫-, 50%⌫ ¯100%-mode.⌫-mode. 50% ⌫-, 50%⌫ ¯-mode.

Fig. 10: The region, shown as a shaded area,Fig. where 11: TheT2K region, has more shown than as a a90 shaded % C.L. area, sensitivity where T2K has more than a 90% C.L. sensitivity to reject maximal mixing. The shaded regionto reject is calculated one of the assuming octants of no✓23 systematic. The shaded errors region is calculated assuming no systematic (the solid contours show the 90% C.L. sensitivityerrors (the with solid statistical contours error show only), the and 90% the C.L. dashed sensitivity with statistical error only), and the contours show the sensitivity including thedashed 2012 systematic contours show errors. the sensitivity including the 2012 T2K systematic errors.

340 4.5. E↵ect of reduction of the systematic error size

341 An extensive study of the e↵ect of the systematic error size was performed. Although the

342 actual e↵ect depends on the details of the errors, here we summarize the results of the

335 indicate the earliest case for T2K to343 observestudy. CP As violation. given in Table If the 2, systematic the systematic error error size onis the predicted number of events in Super- 336 negligibly small, T2K may reach a higher344 K sensitivity in the 2012 at an oscillation earlier stage analysis by running is 9.7% in for 100% the ⌫e appearance sample and 13% for the ⌫µ 337 ⌫-mode, since higher statistics are expected345 disappearance in this case. However, sample. with projected systematic

338 errors, 100% ⌫-mode and 50% ⌫-mode346 + 50%In Sec.⌫ ¯-mode 4.4 we running showed give the essentially T2K sensitivity equivalent with projected systematic errors which are

339 sensitivities. 347 estimated based on a conservative expectation of T2K systematic error reduction. In this 22 23 Neutrino Oscillations with the MINOS, MINOS+, T2K, and NOvA Experiments. 38

Figure 36. The regions where the wrong mass hierarchy is expected to be rejected at 90 % C.L. by the NOvA measurement (blue line) and the NOvA + T2K measurements (black shaded regions) [35]. (Left) the true hierarchy is normal and (right) it is inverted. The T2K POT are assumed to be 3.9 1021 POT for the neutrino beam mode and × 3.9 1021 for the anti-neutrino. The NOvA POT (3.6 1021) are also assumed to be × × distributed to the neutrino beam mode and the anti-neutrino equally.

7.3.4. Mass Hierarchy

Because of the relatively short baseline ( 300 km) of T2K, the experiment is less ∼ sensitive to the mass hierarchy (more sensitive to CP). However, the measurement of T2K (sensitive to CP) can contribute to improving the mass hierarchy sensitivity of NOvA by helping untangle the two eﬀects of CP and mass hierarchy in NOvA. Figure 36 shows the 90 % sensitivity region for mass hierarchy with the T2K and NOvA measurements. The sensitivity is really expanded by adding the T2K measurements, especially for δ 90◦( 90◦) case in the normal (inverted) hierarchy case. CP ∼ −

8. Conclusions; Building the future

We have presented the results and prospects of neutrino oscillation measurements by the present generation of experiments: MINOS/MINOS+, T2K and NOvA. The phenomenology of neutrinos is being rapidly revealed by these experiments with a dramatical improvement of the precision of neutrino oscillation parameters in 10 years. In the standard neutrino oscillation scenario, all three mixing angles have been measured and found to be large enough to explore CP violation. Surprisingly, the new data from the accelerator experiments is beginning to become sensitive to CP violation, when coupled with the precise knowledge of the mixing angles. Upcoming results from on-going experiments, especially T2K and NOvA, will have large impact for the following reasons. The measurements of T2K and NOvA individually are the most sensitive to CP • Neutrino Oscillations with the MINOS, MINOS+, T2K, and NOvA Experiments. 39

violation. By combining both, the sensitivity will be further improved. In order to explore CP violation, the precision of mixing angles is essential. In • particular, the value of θ23 plays a key role in specifying the complicated parameter space of delta-CP and the mass hierarchy. These experiments are the most sensitive

to θ23. The NOvA experiment has some sensitivity to the mass hierarchy. By combining • with T2K, the sensitive region will be expanded. The future neutrino experiment, Hyper-Kamiokande and DUNE, will have greatly • expanded sensitivity to CP violation. The experiences of T2K and NOvA, together with the improvement of systematic uncertainties are key inputs to the future experiments. Going beyond the standard neutrino oscillation scenario, unexpected phenomena may appear in the most sensitive experiments. Thus, there is discovery potential for MINOS/MINOS+, T2K and NOvA at any time. Success of on-going experiments is building the future.

Acknowledgements

Fermilab is operated by Fermi Research Alliance, LLC under Contract No. DE-AC02- 07CH11359 with the United States Department of Energy. J-PARC is operated by KEK and JAEA with MEXT in Japan. An author (TN) acknowledges the support of MEXT and JSPS in Japan with the Grant-in-Aid for Scientific Research A 24244030 and for Scientific Research on Innovative Areas 25105002 titled “Unification and Development of the Neutrino Science Frontier”. Fruitful discussions with members of the T2K, MINOS+, and NOvA collaborations are gratefully acknowledged.

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