arXiv:1403.0867v2 [hep-ex] 10 May 2014 FERMILAB-PUB-14-028-E-PPD Combined analysis of νµ disappearance and νµ → νe appearance in MINOS using accelerator and atmospheric neutrinos P. Adamson,8 I. Anghel,15, 1 A. Aurisano,7 G. Barr,21 M. Bishai,3 A. Blake,5 G. J. Bock,8 D. Bogert,8 S. V. Cao,29 C. M. Castromonte,9 D. Cherdack,30 S. Childress,8 J. A. B. Coelho,30, 6 L. Corwin,14 D. Cronin-Hennessy,18 J. K. de Jong,21 A. V. Devan,32 N. E. Devenish,27 M. V. Diwan,3 C. O. Escobar,6 J. J. Evans,17 E. Falk,27 G. J. Feldman,10 M. V. Frohne,11 H. R. Gallagher,30 R. A. Gomes,9 M. C. Goodman,1 P. Gouffon,24 N. Graf,22, 13 R. Gran,19 K. Grzelak,31 A. Habig,19 S. R. Hahn,8 J. Hartnell,27 R. Hatcher,8 A. Himmel,4 A. Holin,16 J. Huang,29 J. Hylen,8 G. M. Irwin,26 Z. Isvan,3, 22 C. James,8 D. Jensen,8 T. Kafka,30 S. M. S. Kasahara,18 G. Koizumi,8 M. Kordosky,32 A. Kreymer,8 K. Lang,29 J. Ling,3 P. J. Litchfield,18, 23 P. Lucas,8 W. A. Mann,30 M. L. Marshak,18 N. Mayer,30, 14 C. McGivern,22 M. M. Medeiros,9 R. Mehdiyev,29 J. R. Meier,18 M. D. Messier,14 D. G. Michael,4, ∗ W. H. Miller,18 S. R. Mishra,25 S. Moed Sher,8 C. D. Moore,8 L. Mualem,4 J. Musser,14 D. Naples,22 J. K. Nelson,32 H. B. Newman,4 R. J. Nichol,16 J. A. Nowak,18 J. O’Connor,16 M. Orchanian,4 R. B. Pahlka,8 J. Paley,1 R. B. Patterson,4 G. Pawloski,18, 26 A. Perch,16 S. Phan-Budd,1 R. K. Plunkett,8 N. Poonthottathil,8 X. Qiu,26 A. Radovic,32, 16 B. Rebel,8 C. Rosenfeld,25 H. A. Rubin,13 M. C. Sanchez,15, 1 J. Schneps,30 A. Schreckenberger,29, 18 P. Schreiner,1 R. Sharma,8 A. Sousa,7, 10 N. Tagg,20 R. L. Talaga,1 J. Thomas,16 M. A. Thomson,5 X. Tian,25 A. Timmons,17 S. C. Tognini,9 R. Toner,10, 5 D. Torretta,8 G. Tzanakos,2, ∗ J. Urheim,14 P. Vahle,32 B. Viren,3 A. Weber,21, 23 R. C. Webb,28 C. White,13 L. Whitehead,12, 3 L. H. Whitehead,16 S. G. Wojcicki,26 and R. Zwaska8 (The MINOS Collaboration) 1Argonne National Laboratory, Argonne, Illinois 60439, USA 2Department of Physics, University of Athens, GR-15771 Athens, Greece 3Brookhaven National Laboratory, Upton, New York 11973, USA 4Lauritsen Laboratory, California Institute of Technology, Pasadena, California 91125, USA 5Cavendish Laboratory, University of Cambridge, Madingley Road, Cambridge CB3 0HE, United Kingdom 6Universidade Estadual de Campinas, IFGW-UNICAMP, CP 6165, 13083-970, Campinas, SP, Brazil 7Department of Physics, University of Cincinnati, Cincinnati, Ohio 45221, USA 8Fermi National Accelerator Laboratory, Batavia, Illinois 60510, USA 9Instituto de F´ısica, Universidade Federal de Goia´s, CP 131, 74001-970, Goiaˆnia, GO, Brazil 10Department of Physics, Harvard University, Cambridge, Massachusetts 02138, USA 11Holy Cross College, Notre Dame, Indiana 46556, USA 12Department of Physics, University of Houston, Houston, Texas 77204, USA 13Department of Physics, Illinois Institute of Technology, Chicago, Illinois 60616, USA 14Indiana University, Bloomington, Indiana 47405, USA 15Department of Physics and Astronomy, Iowa State University, Ames, Iowa 50011 USA 16Department of Physics and Astronomy, University College London, Gower Street, London WC1E 6BT, United Kingdom 17School of Physics and Astronomy, University of Manchester, Oxford Road, Manchester M13 9PL, United Kingdom 18University of Minnesota, Minneapolis, Minnesota 55455, USA 19Department of Physics, University of Minnesota Duluth, Duluth, Minnesota 55812, USA 20Otterbein College, Westerville, Ohio 43081, USA 21Subdepartment of Particle Physics, University of Oxford, Oxford OX1 3RH, United Kingdom 22Department of Physics and Astronomy, University of Pittsburgh, Pittsburgh, Pennsylvania 15260, USA 23Rutherford Appleton Laboratory, Science and Technologies Facilities Council, Didcot, OX11 0QX, United Kingdom 24Instituto de F´ısica, Universidade de S˜ao Paulo, CP 66318, 05315-970, S˜ao Paulo, SP, Brazil 25Department of Physics and Astronomy, University of South Carolina, Columbia, South Carolina 29208, USA 26Department of Physics, Stanford University, Stanford, California 94305, USA 27Department of Physics and Astronomy, University of Sussex, Falmer, Brighton BN1 9QH, United Kingdom 28Physics Department, Texas A&M University, College Station, Texas 77843, USA 29Department of Physics, University of Texas at Austin, 1 University Station C1600, Austin, Texas 78712, USA 30Physics Department, Tufts University, Medford, Massachusetts 02155, USA 31Department of Physics, University of Warsaw, Hoz˙a 69, PL-00-681 Warsaw, Poland 32Department of Physics, College of William & Mary, Williamsburg, Virginia 23187, USA (Dated: May 13, 2014) We report on a new analysis of neutrino oscillations in MINOS using the complete set of accelerator and atmospheric data. The analysis combines the νµ disappearance and νe appearance data using the three-flavor formalism. We measure |∆m232| = [2.28 − 2.46] × 10−3 eV2 (68% C.L.) and sin2 θ23 = 0.35 − 0.65 (90% C.L.) in the normal hierarchy, and |∆m232| = [2.32 − 2.53] × 10−3 eV2 (68% C.L.) and sin2 θ23 = 0.34 − 0.67 (90% C.L.) in the inverted hierarchy. The data also constrain δCP , the θ23 octant degeneracy and the mass hierarchy; we disfavor 36% (11%) of this three-parameter space at 68% (90%) C.L. PACS numbers: 14.60.Pq The study of neutrino oscillations has entered a preci- sion era in which the experimental data can be used to probe the three-flavor framework of mixing between the neutrino flavor eigenstates (νe, νµ, ντ ) and mass eigenstates (ν1, ν2, ν3). In the standard theory, neutrino mixing is described by the unitary PMNS matrix [1], pa- rameterized by three angles θ12, θ23, θ13 and a phase δCP . The oscillation probabilities additionally depend on the two mass-squared differences ∆m232 and ∆m221, where ∆m2ij ≡ m2i − m2j . The current generation of experiments has measured all three mixing angles and the mass-squared differences using accelerator, atmospheric, reactor and solar neutrinos [2]. Most recently, the smallest mixing angle, θ13, has been measured precisely by reactor neutrino experiments [3–5]. However, the picture is not yet complete. The value of δCP , which determines the level of CP violation in the lepton sector, has not yet been measured. It is also not known whether the neutrino mass hierarchy is normal (∆m232 > 0) or inverted (∆m232 < 0), whether sin2 2θ23 is maximal, or if not, whether the mixing angle θ23 lies in the lower (θ23 < π/4) or higher (θ23 > π/4) octant. These unknowns, which are essential to a complete understanding of neutrino mass and mixing, can be probed by long-baseline neutrino experiments. The MINOS long-baseline experiment [6] has published measurements of oscillations using accelerator and atmospheric neutrinos and antineutrinos. The oscillations ob- served by MINOS are driven by the larger mass-squared difference ∆m232; hence, many features of the data can be described by an effective two-flavor model with a single mass-squared difference ∆m2 and mixing angle θ. In this approximation, the νµ and νµ survival probabilities are: P (νµ → νµ) ≈ 1 − sin2 2θ sin2 ∆m2Lν 4Eν , (1) where Lν is the neutrino propagation distance and Eν is the neutrino energy. A previous two-flavor analysis of νµ and νµ disappearance using the combined accelerator and atmospheric data from MINOS yielded |∆m2| = 2.41+−00..0190 × 10−3 eV2 and sin2 2θ = 0.950+−00..003356 [7]. The statistical weight of the data now enables MINOS to con- strain the full three-flavor model of νµ and νµ disappearance. The uncertainty on ∆m2 is approaching the size of the smaller mass-squared difference, ∆m221, which is neglected in the two-flavor model. Moreover, the precise knowledge of θ13 enables an analysis of the data based on the full set of mixing parameters. In this paper we present the three-flavor analysis of the combined MINOS data. In the three-flavor framework, the oscillations are driven by two mass-squared differences ∆m232 and ∆m231, where ∆m231 = ∆m232 + ∆m221. The interference between the resulting two oscillation frequencies leads to terms in the oscillation probabilities that depend on all the mixing parameters. The leading-order νµ and νµ survival probabilities in vacuum take the same form as the two-flavor approximation in Eq. (1), with the effective parameters given by [8]: sin2 2θ =4 sin2 θ23 cos2 θ13(1 − sin2 θ23 cos2 θ13), ∆m2 =∆m232 + ∆m221 sin2 θ12 + ∆m221 cos δCP sin θ13 tan θ23 sin 2θ12. (2) The exact symmetries of the two-flavor model under θ → π/2 − θ and ∆m2 → −∆m2 lead to approximate degeneracies in the octant of θ23 and mass hierarchy in the three-flavor formalism. For neutrinos traveling through matter, the propagation eigenstates are modified by the MSW effect [9]. In this case, the mixing angle θ13 is replaced by a modified version, θM , given by [10]: sin2 2θM = sin2 2θ13 sin2 2θ13 . + (A − cos 2θ13)2 (3) The size of th√e matter effect is determined by the parameter A ≡ ±2 2GF neEν /∆m231, where GF is the Fermi weak coupling constant, ne is the density of electrons and the sign of A is positive (negative) for neutrinos (antineutrinos). Equation (3) shows that sin2 2θM is maximal at A = cos 2θ13. This condition leads to the resonant enhancement of νµ ↔ νe oscillations, which can significantly alter the magnitude of νµ disappearance. The effect is present for neutrinos in the normal hierarchy and for antineutrinos in the inverted hierarchy. An MSW res- onance is predicted to occur in multi-GeV, upward-going atmospheric neutrinos, which travel through the earth’s mantle [11]. MINOS is the first experiment to probe this resonance by measuring νµ and νµ interactions separately with atmospheric neutrinos, yielding sensitivity to the mass hierarchy and θ23 octant. MINOS [12] has previously reported measurements of νe and νe appearance in accelerator νµ and νµ beams. Measurements of νµ → νe appearance in accelerator neutrinos have also been published by T2K [13]. Both results are based on three-flavor analyses. For accelerator neu- trinos, the νµ → νe appearance probability in matter, expanded to second order in α ≡ ∆m221/∆m231 (≈ 0.03), 2 is given by [14]: P (νµ → νe) ≈ sin2 θ23 sin2 2θ13 sin2 ∆(1 − A) (1 − A)2 + αJ˜ cos(∆ ± δCP ) sin ∆A A sin ∆(1 (1 − − A) A) + α2 cos2 θ23 sin2 2θ12 sin2 ∆A A2 . (4) In this expression, J˜ ≡ cos θ13 sin 2θ13 sin 2θ12 sin 2θ23, ∆ ≡ ∆m231Lν/4Eν and the plus (minus) sign applies to neutrinos (antineutrinos). The first term in Eq. (4) is proportional to sin2 θ23 and breaks the θ23 octant degeneracy. In addition, the dependence on A is sensitive to the mass hierarchy and the second term in the expansion is sensitive to CP violation. In this paper, we strengthen the constraints on δCP , the θ23 octant and the mass hierarchy obtained from the MINOS appearance data [12] by combining the complete MINOS disappearance and appearance data and by exploiting the improved precision on θ13 from reactor experiments. In the MINOS experiment, the accelerator neutrinos are produced by the NuMI facility [15], located at the Fermi National Accelerator Laboratory. The complete MINOS accelerator neutrino data set comprises exposures of 10.71 × 1020 protons-on-target (POT) using a νµ-dominated beam and 3.36 × 1020 POT using a νµ-enhanced beam [7]. These data were acquired in the “low energy” NuMI beam configuration [15], where the neutrino event energy peaks at 3 GeV. The spectrum and composition of the beam are measured using two steel-scintillator tracking detectors with toroidal magnetic fields. The Near and Far detectors are located 1.04 km and 735 km downstream of the production target, respectively. The 5.4 kton Far Detector is installed 705 m (2070 m water-equivalent) underground in the Soudan Underground Laboratory and is equipped with a scintillator veto shield for rejection of cosmic-ray muons. These features have enabled MINOS to collect 37.88 kton-years of atmospheric neutrino data [16]. The oscillation analysis uses charged-current (CC) interactions of both muon and electron neutrinos. These events are distinguished from neutral-current (NC) backgrounds by the presence of a muon track or electromagnetic shower, respectively. The events also typically contain shower activity from the hadronic recoil system. The selection of accelerator νµ CC and νµ CC events is based on a multivariate k-Nearest-Neighbor classification algorithm using a set of input variables characterizing the topology and energy deposition of muon tracks [17]. The selected events are separated into contained-vertex neutrinos, with reconstructed interaction positions inside the fiducial volume of the detectors, and non-fiducial muons, in which the neutrino interactions occur outside the fidu- cial volume or in the surrounding rock. The containedvertex events are further divided into candidate νµ and νµ interactions based on the curvature of their muon tracks. In the oscillation fit, the events are binned as a function of reconstructed neutrino energy. For containedvertex events, this is taken as the sum of the muon and hadronic shower energy measurements; for non-fiducial muons, the muon energy alone is used as the neutrino energy estimator. To improve the sensitivity to oscillations, the contained-vertex νµ events from the νµ-dominated beam are also binned according to their calculated energy resolution [18–20]. The predicted energy spectra in the Far Detector are derived from the observed data in the Near Detector using a beam transfer matrix [21]. The selection of accelerator νe CC and νe CC events is based on a library-event-matching (LEM) algorithm that performs hit-by-hit comparisons of contained-vertex shower-like events with a large library of simulated neutrino interactions [22–24]. The events are required to have reconstructed energies in the range 1−8 GeV, where most of the νe and νe appearance is predicted to occur. The 50 best-matching events from the library are used to calculate a set of classification variables that are combined into a single discriminant using an artificial neural network. The selection does not discriminate between νe and νe interactions. The selected events are binned as a function of the reconstructed energy and LEM discriminant. The background contributions from NC, νµ CC and νµ CC interactions, and intrinsic νe CC and νe CC interactions from the beam, are determined using samples of Near Detector data collected in different beam configurations. The backgrounds in the Far Detector are calculated from these Near Detector components [25]. The rates of appearance in the Far Detector are derived from the νµ CC and νµ CC spectra measured in the Near Detector [12]. Atmospheric neutrinos are separated from the cosmicray muon background using selection criteria that identify either a contained-vertex interaction or an upwardgoing or horizontal muon track [26, 27]. For containedvertex events, the background is further reduced by checking for associated energy deposits in the veto shield. The event selection yields samples of contained-vertex and non-fiducial muons, which are each separated into candidate νµ CC and νµ CC interactions. These samples of muons are binned as a function of log10(E) and cos θz, where E is the reconstructed energy of the event in GeV and θz is the zenith angle of the muon track. This twodimensional binning scheme enhances the sensitivity to the MSW resonance. The results remain in close agreement with the two-flavor analysis of νµ and νµ disappearance, in which these data were binned as a function of log10(L/E) [7]. A sample of contained-vertex showers is also selected from the data, composed mainly of NC, νe CC and νe CC interactions. These events are grouped into a single bin, since they have negligible sensitivity to 3 ∆m322 (10-3eV2) 2.8 MINOS νµ disappearance + νe appearance 10.71 ×1020 3.36 ×1020 POT POT νµ-dominated beam νµ-enhanced beam 2.6 37.88 kt-yr atmospheric neutrinos 2.4 2.2 Normal hierarchy Inverted hierarchy -2.2 -2.4 -2.6 -2.8 0.3 Best fit 68% C.L. 90% C.L. 0.4 0.5 0.6 sin2θ23 0.7 -2∆log(L) -2∆log(L) 6 4 2 0 2.2 6 4 2 0 0.3 Profile of likelihood surface Normal hierarchy Inverted hierarchy 90% C.L. 68% C.L. 2.3 2.4 2.5 |∆m232| (10-3 eV2) Profile of likelihood surface Normal hierarchy Inverted hierarchy 90% C.L. 68% C.L. 0.4 0.5 sin2θ23 0.6 2.6 0.7 FIG. 1. The left panels show the 68% and 90% confidence limits (C.L.) on ∆m232 and sin2 θ23 for the normal hierarchy (top) and inverted hierarchy (bottom). At each point in this parameter space, the likelihood function is maximized with respect to sin2 θ13, δCP and all of the systematic parameters. The −2∆ log(L) surface is calculated relative to the overall best fit, which is indicated by the star. The right panels show the 1D likelihood profiles as a function of ∆m232 and sin2 θ23 for each hierarchy. The horizontal dotted lines indicate the 68% and 90% C.L. oscillations but constrain the overall flux normalization. The predicted event rates in each selected sample are calculated from a Monte Carlo simulation of atmospheric neutrino interactions in the Far Detector [16, 28]. The cosmic-ray muon backgrounds are obtained from the observed data by reweighting the events tagged by the veto shield according to the measured shield inefficiency [26]. For all the data samples, the predicted event spectra in the Far Detector are reweighted to account for oscillations, and the backgrounds from ντ and ντ appearance are included. The oscillation probabilities are calculated directly from the PMNS matrix using algorithms optimized for computational efficiency [29]. The probabilities account for the propagation of neutrinos through the earth. For accelerator neutrinos, a constant electron density of 1.36 mol cm−3 is assumed. For atmospheric neutrinos, the earth is modeled by four layers of constant electron density: an inner core (r < 1220 km, ne = 6.05 mol cm−3); an outer core (1220 < r < 3470 km, ne = 5.15 mol cm−3); the mantle (3470 < r < 6336 km, ne = 2.25 mol cm−3); and the crust (r > 6336 km, ne = 1.45 mol cm−3). This four-layer approximation reflects the principal structures of the PREM model [30]. Comparisons with a more de- tailed 42-layer model yield similar oscillation results. The oscillation parameters are determined by applying a maximum likelihood fit to the data. The parameters ∆m232, sin2 θ23, sin2 θ13 and δCP are varied in the fit. The mixing angle θ13 is subject to an external constraint of sin2 θ13 = 0.0242 ± 0.0025, based on a weighted average of the published results from the Daya Bay [31], RENO [4] and Double Chooz [5] reactor experiments. This constraint is incorporated into the fit by adding a Gaussian penalty term to the likelihood function. The fit uses fixed values of ∆m221 = 7.54 × 10−5 eV2 and sin2 θ12 = 0.307 [32]. The impact of these two parameters is evaluated by shifting them in the fit according to their uncertainties; the resulting shifts in the fitted values of ∆m232 and sin2 θ23 are found to be negligibly small. The likelihood function contains 32 nuisance parame- ters, with accompanying penalty terms, that account for the major systematic uncertainties in the simulation of the data [16, 23, 33]. The fit proceeds by summing the separate likelihood contributions from the νµ disappearance [7] and νe appearance [12] data sets, taking their systematic parameters to be uncorrelated. Figure 1 shows the 2D confidence limits on ∆m232 and sin2 θ23, obtained by maximizing the likelihood function 4 Mass hierarchy ∆m232 < 0 ∆m232 < 0 ∆m232 > 0 ∆m232 > 0 θ23 octant θ23 < π/4 θ23 > π/4 θ23 < π/4 θ23 > π/4 ∆m232 / 10−3eV2 −2.41 −2.41 +2.37 +2.35 sin2 θ23 0.41 0.61 0.41 0.61 sin2 θ13 0.0243 0.0241 0.0242 0.0238 δCP /π 0.62 0.37 0.44 0.62 −2∆ log(L) 0 0.09 0.23 1.74 TABLE I. The best-fit oscillation parameters obtained from this analysis for each combination of mass hierarchy and θ23 octant. Also listed are the −2∆ log(L) values for each of the four combinations, calculated relative to the overall best-fit point. at each point in this parameter space with respect to sin2 θ13, δCP and all of the systematic parameters. Also shown are the corresponding 1D likelihood profiles as a function of ∆m232 and sin2 θ23. The 68% (90%) confidence limits (C.L.) on these parameters are calculated by taking the range of negative log-likelihood values with −2∆ log(L) < 1.00 (2.71) relative to the overall best fit. This yields |∆m232| = [2.28 − 2.46] × 10−3 eV2 (68% C.L.) and sin2 θ23 = 0.35 − 0.65 (90% C.L.) in the normal hierarchy; and |∆m232| = [2.32 − 2.53] × 10−3 eV2 (68% C.L.) and sin2 θ23 = 0.34 − 0.67 (90% C.L.) in the inverted hierarchy. The data disfavor maximal mixing (θ23 = π/4) by −2∆ log(L) = 1.54. The measurements of ∆m232 are the most precise that have been reported to date. The data also constrain δCP , the θ23 octant degeneracy and the mass hierarchy. Table I lists the best-fit oscillation parameters for each combination of octant and mass hierarchy, and the differences in negative log-likelihood relative to the overall best fit. Assuming θ23 > π/4 (θ23 < π/4), the data prefer the inverted hierarchy by −2∆ log(L) = 1.65 (0.23). The combination of normal hierarchy and higher octant is disfavored by 1.74 units of −2∆ log(L), strengthening the previous constraints from νe and νe appearance [12]. Figure 2 shows the 1D likelihood profile as a function of δCP for each of the four possible combinations. The data disfavor 36% (11%) of the parameter space defined by δCP , the θ23 octant, and the mass hierarchy at 68% (90%) C.L. In summary, we have presented the first combined analysis of νµ disappearance and νe appearance data by a long-baseline neutrino experiment. The results are based on the complete set of MINOS accelerator and atmospheric neutrino data. A combined analysis of these data sets yields precision measurements of ∆m232 and sin2 θ23, along with new constraints on the three-parameter space defined by δCP , the θ23 octant, and the mass hierarchy. This work was supported by the U.S. DOE; the United Kingdom STFC; the U.S. NSF; the State and University of Minnesota; Brazil’s FAPESP, CNPq and CAPES. We are grateful to the Minnesota Department of Natural Resources and the personnel of the Soudan Laboratory and Fermilab for their contributions to the experiment. We thank the Texas Advanced Computing Center at The University of Texas at Austin for the provision of computing resources. -2∆log(L) 5 MINOS νµ disappearance + νe appearance 10.71 3.36 ×1020 ×1020 POT POT νµ-dominated beam νµ-enhanced beam 37.88 kt-yr atmospheric neutrinos 4 ∆m232<0, θ23<π/4 ∆m232<0, θ23>π/4 ∆m232>0, θ23<π/4 3 ∆m232>0, θ23>π/4 90% C.L. 2 68% C.L. 1 0 0.0 0.5 1.0 1.5 2.0 δCP / π FIG. 2. The 1D likelihood profile as a function of δCP for each combination of mass hierarchy and θ23 octant. For each value of δCP , the likelihood function is maximized with respect to sin2 θ13, sin2 θ23, ∆m232 and all of the systematic parameters. The horizontal dashed lines indicate the 68% and 90% confidence limits. ∗ Deceased. [1] Z. Maki, M. Nakagawa, and S. Sakata, Prog. Theor. Phys. 28, 870 (1962); B. Pontecorvo, Sov. Phys. JETP 26, 984 (1968); V. N. Gribov and B. Pontecorvo, Phys. Lett. B 28, 493 (1969). [2] J. Beringer et al. (Particle Data Group), Phys. Rev. D 86, 010001 (2012). [3] F. An et al. (Daya Bay), Phys. Rev. Lett. 112, 061801 (2014). [4] J. K. Ahn et al. (RENO), Phys. Rev. Lett. 108, 191802 (2012). [5] Y. Abe et al. (Double Chooz), Phys. Rev. D 86, 052008 (2012). [6] D. G. Michael et al. (MINOS), Nucl. Instrum. Meth. A 596, 190 (2008). [7] P. Adamson et al. (MINOS), 5 Phys. Rev. Lett. 110, 251801 (2013). [8] H. Nunokawa, S. Parke, and R. Zukanovich Funchal, Phys. Rev. D 72, 013009 (2005). [9] L. Wolfenstein, Phys. Rev. D 17, 2369 (1978); S. P. Mikheev and A. Y. Smirnov, Sov. J. Nucl. Phys. 42, 913 (1985). [10] C. Giunti, C. W. Kim, and M. Monteno, Nucl. Phys. B 521, 3 (1998). [11] S. Palomares-Ruiz and S. T. Petcov, Nucl. Phys. B 712, 392 (2005). [12] P. Adamson et al. (MINOS), Phys. Rev. Lett. 110, 171801 (2013). [13] K. Abe et al. (T2K), Phys. Rev. Lett. 112, 061802 (2014). [14] A. Cervera et al., Nucl. Phys. B 579, 17 (2000). [15] K. Anderson et al., FERMILAB-DESIGN-1998-01 (1998). [16] P. Adamson et al. (MINOS), Phys. Rev. D 86, 052007 (2012). [17] R. Ospanov, Ph.D. thesis, University of Texas at Austin (2008). [18] J. Mitchell, Ph.D. thesis, University of Cambridge (2011). [19] S. J. Coleman, Ph.D. thesis, College of William & Mary (2011). [20] C. J. Backhouse, D.Phil. thesis, University of Oxford (2011). [21] P. Adamson et al. (MINOS), Phys. Rev. D 77, 072002 (2008). [22] J. P. Ochoa Ricoux, Ph.D. thesis, California Institute of Technology (2009). [23] R. B. Toner, Ph.D. thesis, University of Cambridge (2011). [24] A. P. Schreckenberger, Ph.D. thesis, University of Min- nesota (2013). [25] J. A. B. Coelho, Ph.D. thesis, Universidade Estadual de Campinas (2012). [26] J. D. Chapman, Ph.D. thesis, University of Cambridge (2007). [27] B. P. Speakman, Ph.D. thesis, University of Minnesota (2007). [28] G. D. Barr et al., Phys. Rev. D 70, 023006 (2004). [29] J. Kopp, Int. J. Mod. Phys. C 19, 523 (2008). [30] A. M. Dziewonski and D. L. Anderson, Phys. Earth Planet. Interiors 25, 297 (1981). [31] F. P. An et al. (Daya Bay), Chin. Phys. C 37, 011001 (2013). [32] G. L. Fogli et al., Phys. Rev. D 86, 013012 (2012). [33] P. Adamson et al. (MINOS), Phys. Rev. Lett. 106, 181801 (2011). 6