Determination of the θ23 octant in long baseline neutrino experiments within and beyond the standard model

This is an electronic reprint of the original article. This reprint may differ from the original in pagination and typographic detail. Author(s): Title: Year: Version: Please cite the original version: All material supplied via JYX is protected by copyright and other intellectual property rights, and duplication or sale of all or part of any of the repository collections is not permitted, except that material may be duplicated by you for your research use or educational purposes in electronic or print form. You must obtain permission for any other use. Electronic or print copies may not be offered, whether for sale or otherwise to anyone who is not an authorised user. Determination of the θ23 octant in long baseline neutrino experiments within and beyond the standard model Das, C. R.; Pulido, João; Maalampi, Jukka; Vihonen, Sampsa Das, C. R., Pulido, J., Maalampi, J., & Vihonen, S. (2018). Determination of the θ23 octant in long baseline neutrino experiments within and beyond the standard model. Physical Review D, 97(3), Article 035023. https://doi.org/10.1103/PhysRevD.97.035023 2018 Determination of the θ23 octant in long baseline neutrino experiments within and beyond the standard model C. R. Das* Bogoliubov Laboratory of Theoretical Physics, Joint Institute of Nuclear Research, Joliot-Curie 6, 141980 Dubna, Moscow region, Russia João Pulido† Centro de Física Teórica das Partículas, Instituto Superior T´ecnico (CFTP-IST), Avenida Rovisco Pais, P-1049-001 Lisboa, Portugal Jukka Maalampi‡and Sampsa Vihonen§ University of Jyvaskyla, Department of Physics, P.O. Box 35, FI-40014 University of Jyvaskyla, Finland (Received 23 October 2017; published 28 February 2018) The recent data indicate that the neutrino mixing angle θ23 deviates from the maximal-mixing value of 45°, showing two nearly degenerate solutions, one in the lower octant (LO) (θ23 <45°) and one in the higher octant (HO) (θ23 >45°). We investigate, using numerical simulations, the prospects for determining the octant of θ23 in the future long baseline oscillation experiments. We present our results as contour plots on the (θ23 −45°, δ)–plane, where δis the CP phase, showing the true values of θ23 for which the octant can be experimentally determined at 3σ,2σand 1σconfidence level. In particular, we study the impact of the possible nonunitarity of neutrino mixing on the experimental determination of θ23 in those experiments. DOI: 10.1103/PhysRevD.97.035023 I. INTRODUCTION Many solar, atmospheric, reactor, and accelerator neutrino experiments have firmly established the existence of neutrino oscillations. Neutrino oscillations can be parametrized in terms of six physical variables, namely by three mixing angles θ12,θ23, and θ13, a phase δCP, and two squared-mass differences Δm2 21 ¼m2 2−m2 1and Δm2 31 ¼ m2 3−m2 1. These parameters are by now experimentally quite precisely determined, with the exception of the CP phase δCP. As to the mixing angle θ23, one still do not know in which octant it lies (θ23 <45°orθ23 >45°). Also the order of the masses of three light neutrinos (ν1,ν2,ν3) remains unknown, namely whether it is m3≥m1;m 2(normal hierarchy, NH) or m3≤m1;m 2(inverted hierarchy, IH). As to the mixing angle θ23, also known as the atmospheric angle, the fits on global data indicate that θ23 deviates from the maximal-mixing value 45° showing two degenerate solutions, a low-octant (LO) solution with θ2 23 <45° and a high-octant (HO) solution with θ23 >45° [1–4]. This octant degeneracy is one of many parameter degeneracies that hamper the interpretation of neutrino oscillation data [5]. The NOνA experiment has recently excluded the maximal-mixing value θ23 ¼45° at the 2.6σ confidence level [6]. Two statistically degenerate values for sin2θ23 were found, 0.404þ0.030 −0.022 and 0.624þ0.022 −0.030 , which both explain the data on muon neutrino disappearance at the 68% confidence level. Earlier experimental results are compatible with θ23 ¼45°[7–10]. The prospects of resolving the θ23 octant in next generation experiments have been studied in, e.g., [11–17]. Identifying the true value of θ23 is an important goal for future experiments, given its importance for understanding the mechanism behind neutrino masses and mixing. For example, one symmetry of the neutrino sector under the interchange of νμand ντwould predict θ23 to have the maximal mixing value of 45° (see, e.g., [18]). In that case, the νμand ντflavors would have an equal weight in the ν3 mass state. In some models the octant of θ23 is directly related to the neutrino mass hierarchy, e.g., in the model considered in [19], the mass hierarchy for the higher octant is normal and for the lower octant the mass hierarchy is inverted. *[email protected] †[email protected] ‡[email protected] §[email protected] Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI. Funded by SCOAP3. PHYSICAL REVIEW D 97, 035023 (2018) 2470-0010=2018=97(3)=035023(8) 035023-1 Published by the American Physical Society In interpreting the data, one should take into account the possibility of beyond the standard model effects, which may appear as nonunitarity of the mixing of light neutrinos. In particular, the possible existence of sterile neutrinos and nonstandard neutrino interactions would influence the experimental determination of θ23 and the octant where the angle lies. The effects of some new physics in octant determination have been studied in future long baseline neutrino experiments in, e.g., [20–24]. In this paper, we will consider the determination of the mixing angle θ23 in long baseline neutrino experiments taking possible nonunitarity effects into account. First, we will update our previous studies [25,26], done for the standard model case with three conventional neutrinos. We apply the results of the most recent fits for the values of mixing parameters and use as our benchmark the setup of theproposedDeepUndergroundNeutrinoExperiment (DUNE). Our main goal is to find out how the possible nonunitarity of the mixing matrix of the light neutrinos affect the sensitivity of the experiments in identifying the octant of θ23. Such nonunitarity may arise, e.g., if one adds to the particle content of the standard model sterile neutrinos that mix with the conventional neutrinos. We also study to which extent the still unknown value of the CP phase interferes with the determination of the octant of θ23, and we present our results for both the normal and inverted mass hierarchy. This work is organized as follows. In Sec. II we will present a brief review of the formalism we will use. In Sec. III the simulation method is described. The results of simulations are presented and discussed in Sec. IV. Section Vcontains a summary and conclusions. II. BRIEF REVIEW OF THE FORMALISM We start our analysis by investigating the prospects for θ23 octant determination in long baseline experiments in the case of standard neutrinos. For these, the matter evolution is determined by the Hamiltonian which in the mass basis reads H¼1 2E 0 B B @ 00 0 0Δm2 21 0 00Δm2 31 1 C C A þU†0 B B @ VCC þVNC 00 0VNC 0 00VNC 1 C C A U: ð1Þ Here Uisthe conventionalneutrino mixing matrix (PMNS matrix), VCC ¼ffiffiffi 2 pGFNeand VNC ¼−GFNn/ffiffiffi 2 pare respectively the charged current and the neutral current matter potentials. Owing to their smallness, the most common understanding of the origin of neutrino masses lies in the assumption of a new physics scale associated with the general seesaw mechanism, whereby heavy right-handed neutrinos are added to the particle content of the Standard Model. These SUð3ÞC⊗SUð2ÞL⊗Uð1ÞYsinglet neutrinos mix with the standard neutrino flavors νe,νμ,ντand, although in the likely case they are too heavy to be kinematically produced, they should leave traces at the energies within experimental reach. These traces appear in the oscillation probabilities through the n×nunitary matrix Uconnecting the neutrino mass and flavor eigenstates which generalizes the conventional 3×3Umatrix of the standard case. The U matrix can be written in the form [27] U¼NS TV ð2Þ where Nand Sare (3×3) submatrices that contain respectively the mixing in the light (active) neutrino sector and the active-sterile mixing. Submatrices Tand Vdefine the mixing of the sterile states with the active and sterile states respectively. In this way, Eq. (1) is modified to H¼1 2E 0 B B @ 00 0 0Δm2 21 0 00Δm2 31 1 C C A þN†0 B B @ VCC þVNC 00 0VNC 0 00VNC 1 C C A N: ð3Þ The unitarity of Uimplies that the matrix describing the mixing in the light sector, namely N, is no longer unitary. In [28], the matrix Nwas presented in the form N¼NNPU, where Uis the conventional unitary 3×3matrix and NNP the triangular matrix NNP ¼0 B @ α11 00 α21 α22 0 α31 α32 α33 1 C A;ð4Þ parametrizing the deviations from unitarity. The description of the unitarity violation therefore requires three real parameters αii which are close to unity and three complex ones αij (i≠j), which are close to zero. A slightly different notation for the prefactor matrix NNP was given in [29], NNP ¼0 B @ 1−αee 00 αμe1−αμμ 0 ατeατμ 1−αττ 1 C A:ð5Þ Here αll0directly parametrize deviations from the unitarity and since these deviations are known to be small one DAS, PULIDO, MAALAMPI, and VIHONEN PHYS. REV. D 97, 035023 (2018) 035023-2 has αll ≪1and jαll0j≪1. The difference between the parametrizations (4) and (5) is purely aesthetic, but as both of them are used parallel in the literature, we give them both here for convenience. Since current experiments involve mainly electron and muon neutrinos, only α11,α22 and α21 (or equivalently αee, αμμ and αμe) need to be considered, hence four new parameters are effectively required in the analysis. Constraints for αij and αll0are given in Refs. [29,28]. No constraint exists, however, for the off-diagonal phases. One should note that the nonunitary of the mixing of the neutrino flavors νe,νμ, and ντwould, in general, affect the determination of the mixing angles θ12,θ23, and θ31 from the existing neutrino oscillation data. However, in the triangular parametrizations of Eqs. (4) and (5) the nonunitarity effects disappear in the leading order and are hence negligible in comparison with the uncertainties of the experimental results used in the fits done [29]. Hence, the matrix Uhas in good approximation the same numerical form as is obtained in the standard analysis of the data where the unitarity assumed to hold. Our analysis, which follows the lines presented in [25],is based on numerical simulations where we utilize the GL O BES software [30,31]. Let us note that whereas analytical expressions for survival and conversion probabilities both in vacuum and matter have been given in the literature [32,33], for the nonunitarity effect in neutrino oscillations only the vacuum expressions exist [28]. Brief discussions of the matter potential in the nonunitary case are given in Refs. [34,35]. III. NUMERICAL METHODS We evaluate the effect of nonunitary mixing on the determination of θ23 octant by simulating a long baseline oscillation experiment with the DUNE specifications with the GL O BES program. Since the nonunitary mixing matrix is not available in the standard GL O BES package, we modify the program by introducing our own add-on, which replaces the standard PMNS mixing matrix with its nonunitary version given by Eq. (4), or alternatively Eq. (5), and replaces the standard Hamiltonian shown in Eq. (1) with its nonunitary version (3). The octant discovery potential is evaluated for a given θ23 value as Δχ2ðθ23Þ¼χ2ð90°−θ23Þ−χ2ðθ23Þ;ð6Þ where χ2ðθ23Þevaluates the chi-square distribution at the given true value θ23, whilst in χ2ð90°−θ23Þit is evaluated at the wrong octant solution 90°−θ23. This leaves the subtraction of the two, Δχ2, an approximate chi-square distribution with one degree of freedom, and hence the sensitivity for ruling out the wrong octant at 1σ,2σ, and 3σ confidence levels is reached at Δχ2¼1, 4, and 9, respectively. The simulation of the DUNE setup is performed using the same experimental configuration that was used in the DUNE conceptual design report [36] and was published in Ref. [37]. The octant discovery potential is evaluated using Eq. (6) as described above. In this work, we assess the sensitivity to the θ23 octant in DUNE in four different scenarios. On the one hand, we update the octant sensitivity plots for the standard model case, where no sterile neutrinos exist and the oscillations would follow the standard three-neutrino paradigm. On the other hand, we also evaluate the octant sensitivity in scenarios, where sterile neutrinos do exist, manifesting themselves as nonunitarity of the mixing matrix of the three active neutrinos. The effects of sterile neutrinos to the oscillations would depend on the scale of the lightest sterile mass. For the standard oscillation parameters, we employ the best-fit values and their associated errors, which have been obtained from the experimental data collected from the past and ongoing neutrino experiments (see Ref. [38]). We take the central values and standard deviations of these parameter best-fits and take them into account as Gaussian distributions. For the mixing angles the Gaussian distributions are set for sin2θ12, sin2θ13, and sin2θ23. These values are presented in Table I [39]. In addition to the standard oscillation parameters, we also need to consider the new physics parameters αij,i, j¼1,2,3,andαll0,l;l0¼e,μ,τ, as indicated in Eqs. (4) and (5), respectively. Since no significant signs of physics TABLE I. The experimental best-fit values and standard deviations for the standard neutrino oscillation parameters. These values are shown for both mass hierarchies and are taken from a recent global analysis [38]. Note that Δm2 3lstands for Δm2 31 in normal hierarchy (NH) and Δm2 32 in inverted hierarchy (IH). Parameter Central value (NH) Error (NH) Central value (IH) Error (IH) sin2θ12 0.306 0.012 0.306 0.012 sin2θ13 0.02166 0.00075 0.02179 0.00076 sin2θ23 0.441 0.027 0.587 0.024 δCP (°) 261 59 277 46 Δm2 21 (10−5eV2)7.50 0.19 7.50 0.19 Δm2 3l(10−3eV2)2.524 0.040 −2.514 0.041 DETERMINATION OF THE θ23 OCTANT IN LONG …PHYS. REV. D 97, 035023 (2018) 035023-3 beyond the standard model has been observed in oscillation experiments, there exist only upper bounds on the different αparameters. In this work we consider the possibility of sterile neutrino induced new physics by allowing the αij and αll0parameters to have Gaussian distributions, where central values are set at zero and standard deviations to match the appropriate upper bounds. The standard threeneutrino paradigm is restored by setting α11,α22,α33 ¼1 and α21,α31,α32 ¼0, or alternatively αll0¼0for all l;l0¼e,μ,τcombinations. We start the investigation on the new physics scenarios by assuming that all three sterile neutrinos are too massive to be produced in the experiment. In this scenario, the sterile neutrinos do not contribute to the oscillations kinematically, but they affect the neutrino oscillation probabilities through the nonunitarity of the 3×3mixing matrix that controls the mixing of active neutrinos. These bounds have been evaluated for nonunitary mixing in two independent references; for the αij basis they are provided in Ref. [35] and for αll0in Ref. [29], and they are both shown in Table II. We also consider the scenario where at least one of the sterile neutrinos is sufficiently light to be produced kinematically in the experiment. In such case, sterile neutrinos could contribute to the oscillations in two different ways depending on their mass range. If the lightest sterile neutrino has its mass m4in the range such that Δm2 41≡ m2 4−m2 1∼0.1–1eV2, the active neutrinos could oscillate to the sterile neutrino state between the near and far detectors. It is also possible, however, that the oscillations to the sterile neutrino are too rapid to be observed in the far detector, i.e., they average out, but sufficiently light to occur before the near detector. In either case, not all constraints used in deriving the upper bounds in Table II are applicable, and therefore new bounds must be derived. This topic has been thoroughly reviewed in Ref. [29], where appropriate bounds were provided for αll0, l;l0¼e,μ,τin mass ranges Δm2 41 ∼0.1–1eV2and Δm2 41 ≥100 eV2. These bounds are shown in Table III. IV. RESULTS We restrict our study to four different scenarios that could take place in the presence (or absence) of physics beyond the standard model. These scenarios include the standard three-neutrino paradigm, nonunitary neutrino mixing, mixing with a light sterile neutrino, and finally, a scenario where no constraints are set for the new physics parameters. A. The standard model case In the standard model there exist three active neutrinos (νe,νμ, and ντ) and no sterile neutrinos. Using the best-fit values and errors presented in Table Iwe plotted the 1σ,2σ, and 3σconfidence level contours for various possible true values of θ23 and δCP in both NH and IH. The results are presented in Fig. 1. We obtained this figure by keeping θ23 and δCP fixed to their assigned values, whilst the other four oscillation parameters (θ12,θ13,Δm2 21,Δm2 31) and the matter density were included in the χ2minimization. Fig. 1is to be read as follows. The white regions correspond to the θ23 and δCP values where the octant of θ23 can be determined by DUNE at a 3σconfidence level or better. In the colored region the sensitivity falls below 3σ, 2σ,or1σ, where the latter two are indicated by the dashed and solid lines, respectively. In other words, if the values of θ23 and δCP fall in the region outside the band bordered by the dashed (solid) lines the octant of θ23 can be determined at a 2σ(3σ) confidence level or better. B. The nonunitary mixing case In the case of nonunitary mixing, the sensitivity to the θ23 octant is evaluated by using Eq. (4) or Eq. (5) to calculate the mixing matrix and Eq. (3) to construct the corresponding Hamiltonian. We take parameters αij,i,j¼1, 2, 3, and αll0,l;l0¼e, μ,τ, into account as Gaussian priors, where central values are set to match the standard three-neutrino case and the standard deviations are taken from the 1σbounds corresponding to the 90% CL ones presented in Table II. In Fig. 2, we present the 1σ,2σ, and 3σcontours for the octant determination under nonunitary mixing in the αll0 TABLE II. Bounds on nonunitary parameters in both αij and αll0representations, taken from [35,29], respectively. The bounds are given in 90% and 2σconfidence levels. Parameter Upper bound (90% CL) Parameter Upper bound (2σCL) α11 0.9974 αee 1.3×10−3 α22 0.9994 αμμ 2.2×10−4 α33 0.9988 αττ 2.8×10−3 jα21j2.6×10−2jαμej6.8×10−4 jα31j2.0×10−3jατej2.7×10−3 jα32j1.5×10−2jατμj1.2×10−3 TABLE III. Bounds on nonunitary parameters in αll0repre- sentation, taken from [29]. In this scenario the constraints would correspond to mixing with a light sterile neutrino in two mass scales: Δm2 41 ∼0.1–1eV2(left column) and Δm2 41 ≥100 eV2 (right column). The constraints are presented at a 95% confidence level. Parameter Δm2 41 ∼0.1–1eV2Δm2 41 ≥100 eV2 αee 1.0×10−22.4×10−2 αμμ 1.4×10−22.2×10−2 αττ 1.0×10−11.0×10−1 jαμej1.7×10−22.5×10−2 jατej4.5×10−26.9×10−2 jατμj5.3×10−21.2×10−2 DAS, PULIDO, MAALAMPI, and VIHONEN PHYS. REV. D 97, 035023 (2018) 035023-4 basis. The sensitivity plots are shown both in the NH and IH cases. We also studied the sensitivities in the αll0basis, and found the results to be nearly identical to the ones obtained in the αij basis. C. The light sterile neutrino case Using the bounds derived for the mixing with a light sterile neutrino of Δm2 41 ≥100 eV2mass range, shown in the centre column of Table III, we obtained the sensitivity contours shown in Fig. 3. We also studied the case of 0< Δm2 41 <1eV2by using the bounds presented in the right column of Table III, but we did not find any significant difference to the Δm2 41 ≥100 eV2case. D. The unconstrained new physics case Since it is unknown what may lie beyond the standard model, it is also necessary to discuss the scenario where FIG. 1. Octant determination in DUNE under the standard three-neutrino paradigm. The white regions show the values of θ23 and δCP at which the octant of θ23 could be determined at a 3σCL or better. In the colored regions, conversely, the significance falls under 3σ. The 1σand 2σCL contours are shown with dashed and solid lines, and the sensitivities are presented for both NH (blue, left panel) and IH (red, right panel) mass orderings. FIG. 2. Octant determination in DUNE under nonunitary mixing. The white regions show the values of θ23 and δCP at which the octant of θ23 could be determined at a 3σCL or better. In the colored regions, conversely, the significance falls under 3σ. The 1σand 2σCL contours are shown with dashed and solid lines, and the sensitivities are presented for both NH (blue, left panel) and IH (red, right panel) mass orderings. DETERMINATION OF THE θ23 OCTANT IN LONG …PHYS. REV. D 97, 035023 (2018) 035023-5 none of the bounds that have been derived for nonunitary mixing or light sterile neutrinos may apply. An example of this situation could be a scenario where the sterile neutrinos are accompanied by nonstandard interactions (see e.g. [35]) mediated by new Higgs particles, as is the case in left-right symmetric models. Due to its unknown nature, we consider an arbitrary form of new physics by calculating the χ2values with no constraints on the α parameters. We calculated the sensitivity of DUNE to the θ23 octant in DUNE after removing all priors that concern αij where, i, j¼1, 2, 3. The results are presented in Fig. 4. One notices that maximizing the effects of nonunitary and light sterile neutrinos roughly worsens the sensitivity of DUNE to the octant in terms of the angle. In order to get an understanding on how the magnitude of the αparameters affects the worsening of the sensitivity to the θ23 octant in the event where only α21 is taken FIG. 3. Octant determination in DUNE in the presence of light sterile mixing with Δm41 ≥100 eV2. The white regions show the values of θ23 and δCP at which the octant of θ23 could be determined at a 3σCL or better. In the colored regions, conversely, the significance falls under 3σ. The 1σand 2σCL contours are shown with dashed and solid lines, and the sensitivities are presented for both NH (blue, left panel) and IH (red, right panel) mass orderings. FIG. 4. Octant determination in DUNE with unconstrained αll0parameters. The white regions show the values of θ23 and δCP at which the octant of θ23 could be determined at a 3σCL or better. In the colored regions, conversely, the significance falls under 3σ. The 1σand 2σCL contours are shown with dashed and solid lines, and the sensitivities are presented for both NH (blue, left panel) and IH (red, right panel) mass orderings. DAS, PULIDO, MAALAMPI, and VIHONEN PHYS. REV. D 97, 035023 (2018) 035023-6 into account, whereas the other alpha parameters are excluded from the χ2calculation. In Fig. 5we show the octant sensitivity as a function of the 1σupper bound of jα21j, and allow the phase of α21 vary freely in the range ½0;2π. Clearly, the contours in Fig. 5show that the constraint on α21 affects the octant determination when α21 ≳10−2. V. CONCLUSIONS We have presented the sensitivity to the determination of the θ23 octant (θ23 ≤π/4 or θ23 ≥π/4) in DUNE in four different scenarios. On the one hand, we have updated the 1σ,2σ, and 3σconfidence level contours for the standard model, where oscillations are constituted between three active neutrinos. On the other hand, we have also given these contours for three different scenarios where the octant sensitivity is interfered by sterile neutrinos and other potential sources for physics beyond the standard model. We analyzed these scenarios by parametrizing the new physics with the methods that were originally introduced in Refs. [28,29] to describe nonunitarity of the light neutrino mixing matrix. We found that the nonunitarity of the mixing matrix caused the sensitivity θ23 octant to decrease from the standard model case. Nevertheless, due to the strictness of the existing bounds for the nonunitarity parameters αij,i,j¼1,2,3derivedinRef.[35] and for αll0, l;l0¼e,μ,τderived in Ref. [29] the observed drop in the octant sensitivity was found to be very small. The worsening of the octant sensitivity due to sterile neutrino was found larger than this. The sensitivity was calculated in this case using the bounds on αll0given in Ref. [29]. The worsening of the sensitivity was found to be less than 1° in each octant. We found the decrease in sensitivity due to the light sterile neutrino to be substantially less significant than that reported in Ref. [21] where the impact of a sterile neutrino with mixing angles θ14 ¼θ24 ¼9° and θ34 ¼0°was considered in the determination of the θ23 octant in DUNE. Evidence of this sensitivity decrease can be seen from the comparison between our Fig. 1with Fig. 3 of Ref. [21]. When converted to the nonunitarity formalism (see the Appendix of Ref. [28]), this kind of sterile neutrino would imply nonunitarity whose parameter values lie close to the existing bounds we presented for 0<Δm2 41 <1eV2 in Table III. On the other hand, our investigation takes into account all possibilities for light sterile neutrinos, whereas the authors of Ref. [21] consider a specific model. Thus our results are in this respect more general, therefore statistically favored by comparison and hence the difference between the two sets. If the model of Ref. [21] is realized in nature, then the ability of DUNE to tell the θ23 octancy is deteriorated. We also tested how the octant sensitivity changed when the new physics parameters αij were left unconstrained. This type of simulation corresponds to a new physics scenario, where sterile neutrinos are associated with other new physics effects, not taken into account in Refs. [29,35] when deriving the bounds for the nonunitary and light sterile mixing effects. An example of this could be nonstandard interactions involved in the neutrino propagation. Our simulations showed that in the worst case the octant could be determined at 3σCL or better for θ23 ≲41.0°andθ23 ≳48.5° for the normal hierarchy to be compared with the bounds θ23 ≲43.5°andθ23 ≳46.5°of the standard case. FIG. 5. Octant determination in DUNE as function of the 1σupper bound on jα21j. The phase of α21 is allowed to vary freely in the range ½0;2πand the other alpha parameters are set to correspond to the standard three-neutrino paradigm. DETERMINATION OF THE θ23 OCTANT IN LONG …PHYS. REV. 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