14  (GeV) reco E 0 5 10 15 20 Events/GeV 20 40 60 80 100 Far Detector Data Monte Carlo Expectation NC Background  2  eV -3 10  | = 2.43 2 32 m |   = 1 23 2 2 sin 2. Active-sterile mixing when m 4 ≫ m 3 B. Fitting active-sterile oscillations to the data TABLE II: Best-fit points and uncertainty ranges obtained for the active-sterile oscillation models. Results are shown with and without ν$_{e}$ appearance at the CHOOZ limit. All angles are given in degrees. The quantity f$_{s}$ is defined as the fraction of disappearing ν$_{µ}$ that could transition to ν$_{s}$ and is given at the 90% C.L. in this table. The values of f$_{s}$ in the m 4 ≫ m 3 model are evaluated for E$_{ν}$ = 1 . 4 GeV. FIG. 11: The reconstructed energy spectrum of charged- current selected events at the far-detector (points with sta- tistical uncertainties). The Monte Carlo prediction assuming standard three-flavor oscillations (solid line) is shown with the 1 standard deviation systematic uncertainty on the pre- diction indicated by the shaded region. The prediction for the small background of misidentified neutral-current events in this sample (hatched region) is also shown. In the m$_{4}$ ≫ m$_{3}$ case, the fourth mass eigenstate is as- sumed to be much larger than the third; consequently the values of sin $^{2}$Δ$_{41}$ and sin $^{2}$Δ$_{43}$ average to 1 $_{2}$. Addition- ally, sin 2Δ$_{41}$ and sin 2Δ$_{43}$ average to 0. In this model Δ m 2 43 is assumed to be O (eV $^{2}$) such that the regime of rapid oscillations and thus the averages mentioned are valid at the far-detector site, while ensuring no detectable depletion of ν$_{µ}$ occurs at the near-detector. Such models have recently received attention in the literature [12, 13]. Using the above simplifications reduces the number of parameters in this model by two, and allows Eq. (11) to be written as where the second term of Eq. (11) does not appear be- cause of the assumptions that θ$_{14}$ = 0 ◦ and δ$_{2}$ = 0. P$_{ν}$$_{µ}$$_{→}$$_{ν}$$_{µ}$ = 1 − 4 { | U$_{µ}$$_{3}$ | 2 ( 1 − | U$_{µ}$$_{3}$ | 2 − | U$_{µ}$$_{4}$ | 2 ) sin $^{2}$Δ$_{31}$ + | U$_{µ}$$_{4}$ | 2 2 (1 − | U$_{µ}$$_{4}$ | $^{2}$) } , P$_{ν}$$_{µ}$$_{→}$$_{ν}$$_{α}$ = 4 R {( | U$_{µ}$$_{3}$ | $^{2}$| U$_{α}$$_{3}$ | 2 + U ∗ µ $_{4}$U$_{α}$$_{4}$ U$_{µ}$$_{3}$U ∗ α 3 ) sin $^{2}$Δ$_{31}$ + | U$_{µ}$$_{4}$ | $^{2}$| U$_{α}$$_{4}$ | 2 2 } , (13) χ 2 = 2 N ∑ i $_{=1}$[ e$_{i}$ − o$_{i}$ + o$_{i}$ ln o$_{i}$ e$_{i}$ ] + 5 ∑ j =1 ϵ 2 j σ 2 j . (14) nuisance parameters in the far-detector fits. By fitting for the systematic parameters simultaneously using both near and far-detector data, the effect of the uncertainties is substantially reduced due to significant cancellations of uncertainties between the two detectors. The data are compared to Monte Carlo predictions based on the probabilities in Eqs. (12) and (13) using the χ 2 statistic appropriate to small sample sizes, The best-fit values for the mixing angles in the two models as well as the χ 2 for each are shown in Table II. A list of the systematic effects for the fit parameters is presented in Table III. The latter table shows that for each mixing angle evaluated in the four-neutrino models considered, the uncertainties introduced by the five most significant sources of systematic error are relatively small compared to the total uncertainty ranges obtained from the fits, as summarized in Table II. The best-fit values obtained for each model for the mass splitting, | Δ m 2 $_{31}$| , agree well with the result found in Ref. [10]. The one-dimensional projections of the Δ χ 2 Here e$_{i}$ is the expected number of events, assuming os- cillations among four flavors, in bin i of the energy spec- trum, and o$_{i}$ is the observed number of events in that bin. The second sum is the contribution to χ 2 from the parameters describing the systematic uncertainties; the nuisance parameter ϵ$_{j}$ is the shift from the nominal fit value for the j -th source of systematic uncertainty and σ$_{j}$ is the uncertainty associated with that source. Both the neutral-current spectrum shown in Fig. 9 and the charged-current spectrum shown in Fig. 11 are used to obtain the oscillation parameters that best fit the data. The neutral-current spectrum provides information on the mixing angles for mixing between active and sterile neutrinos while the charged-current spectrum provides constraints on the mixing angle θ$_{23}$ and the mass splitting | Δ m 2 $_{31}$| . The charged-current-like spectrum of Fig. 11 is statistically consistent with that presented in Ref. [10] given the different fiducial volumes and event separation procedures used in the two analyses. The five system- atic uncertainties described in Sec. VI are included as Model θ 13 χ $^{2}$/D.O.F. θ 23 θ 24 θ 34 f$_{s}$ m 4 = m 1 0 47.5/39 45 . 0 +9$_{.}$0 − 8$_{.}$9 0 . 1 +28$_{.}$7 − 0$_{.}$1 0.51 12 46.2/39 47 . 1 +8$_{.}$8 − 11$_{.}$0 23 . 0 +22$_{.}$6 − 24$_{.}$1 0.55 m 4 ≫ m 3 0 47.5/38 45 . 0 +9$_{.}$0 − 8$_{.}$9 0 . 0 +7$_{.}$2 − 0$_{.}$0 0 . 1 +28$_{.}$7 − 0$_{.}$1 0.52 12 46.2/38 47 . 1 +8$_{.}$8 − 11$_{.}$0 0 . 0 +7$_{.}$2 − 0$_{.}$0 23 . 0 +22$_{.}$6 − 24$_{.}$1 0.55