11 E reco (GeV) N Data S NC B ν$_{µ}$ CC B $^{ν}$τ CC B ν$_{e}$ CC 0 − 3 141 125.1 13.3 1.4 2.3 (12.4) 3 − 120 247 130.4 84.0 4.9 16.0 (32.8) 0 − 3 R = 0 . 99 ± 0 . 09 ± 0 . 07 − 0 . 08 3 − 120 R = 1 . 09 ± 0 . 12 ± 0 . 10 − 0 . 13 0 − 120 R = 1 . 04 ± 0 . 08 ± 0 . 07 − 0 . 10 rate assuming null (maximally-allowed) ν$_{e}$ appearance of less than 8% (18%) at 90% confidence level. The follow- ing sections explore the extent to which the data allow  (GeV) reco E 0 5 10 15 20 Events/GeV 20 40 60 Far Detector Data  = 0 13 /2  = 3  = 12 13  CC Background 2  eV -3 10  | = 2.43 2 32 m |   = 1 23 2 2 sin TABLE I: Values of R, N Data , S NC , and the contributions to B CC for various reconstructed energy ranges. The numbers in parentheses are calculated including ν$_{e}$ appearance at the CHOOZ limit, as discussed in the text. The first uncertainty in the value of R shown is the statistical uncertainty, the sec- ond is systematic uncertainty, and the third is due to possible ν$_{e}$ appearance. FIG. 9: The reconstructed energy spectrum of neutral-current selected events at the far-detector (points with statistical un- certainties). The Monte Carlo prediction assuming standard three-flavor oscillations is also shown, both with (dashed line) and without (solid line) ν$_{e}$ appearance at the CHOOZ limit. The shaded region indicates the 1 standard deviation system- atic uncertainty on the prediction. The hatched region shows the Monte Carlo prediction for the background of misidenti- fied charged-current events in this sample. oscillations between the active flavors and one sterile neu- trino or oscillations in combination with neutrino decay. VIII. OSCILLATIONS INCLUDING A STERILE NEUTRINO FLAVOR Mixing of the three active neutrino flavors with one sterile neutrino requires the addition of one mass eigen- state. The probability for mixing between any two flavors is described by the neutrino mixing matrix [32], U , which defines the rotation from the mass basis into the flavor basis. Thus, the mixing matrix needs to be expanded by one row and one column to accommodate the additional mass and flavor states. The expanded 4 × 4 mixing ma- trix contains six mixing angles and six phases, with three of the phases being Majorana phases that are not rele- vant to an oscillation experiment [33]. The matrix can be written as a product of six independent rotation matri- ces about the Euler axes R$_{ij}$ , where ij refers to the plane in which a particular rotation takes place. The ordering of the rotation matrices is arbitrary, reflecting that the mixing matrix can be parameterized in many ways. The ordering described below was chosen to make the analysis more straightforward. background from all flavors, and S$_{NC}$ is the extrapolated number of neutral-current interactions [17]. The values of S$_{NC}$ and contributions to B$_{CC}$ are shown in Table I. The disappearance of ν$_{µ}$ occurs mainly for true neu- trino energies < 6 GeV [10]. While the true energy of a neutrino interacting through the neutral-current pro- cess cannot be measured, the data can be separated into two samples whose reconstructed energy roughly discrim- inates between neutrinos with true energies greater than and less than 6 GeV. According to the Monte Carlo simulation, events with E$_{reco}$ < 3 GeV have a me- dian true neutrino energy of 3.1 GeV while events with 3 GeV < E$_{reco}$ < 120 GeV have a median true neutrino energy of 7.9 GeV. The values of R calculated for these ranges in E$_{reco}$ are shown in Table I. For all data samples, R differs from unity by less than 1 standard deviation. The effect of ν$_{e}$ appearance on the value of R is treated as an uncertainty in this analysis and is indicated by the last uncertainty in Table I. Because ν$_{e}$ -charged-current events are almost always identified as neutral-current, the effect of ν$_{e}$ appearance at the far-detector is to decrease the value of R , since the predicted background will be larger than for the case of null ν$_{e}$ appearance. The measured values of R for each energy range in- dicate that neutrino oscillations among the active fla- vors describes the observed data quite well. Over the full energy range, 0 − 120 GeV, a value of R = 1 . 04 ± 0 . 08(stat.) ± 0 . 07(syst.) − 0 . 10( ν$_{e}$ ) is measured, corre- sponding to a depletion of the total neutral-current event MINOS is designed to precisely measure | Δ m 2 $_{32}$| but has no sensitivity to Δ m 2 $_{21}$. Consequently, the mass states 1 and 2 are treated as degenerate in this analy- sis. When two mass states are degenerate the rotation in the ij -plane is unphysical and the corresponding mixing angle, θ$_{ij}$ , vanishes from the oscillation probabilities [12]. Additionally, the matrix should be of the form such that the U$_{e}$$_{3}$ component of the mixing matrix goes to zero when θ$_{13}$ = 0 ◦ to distinguish an electron component in the third mass eigenstate from effects of sterile neutrinos. For those reasons, the general form of the mixing matrix used by the current analysis is written as