13 1 1 1 2 2 2 3 3 3 4 4 4 1  = m 4 m 3  >> m 4 m 3  = m 4 m mass between the active and sterile components. This conse- quence comes from the SNO results [6] which indicate that any coupling between ν$_{s}$ and the active neutrinos occurs in the third and fourth mass eigenstates. There- fore, in this case the mass difference in which oscillations between active and sterile neutrinos could arise is too small to be observed in terrestrial baselines and it is not explored further in this analysis. FIG. 10: Schematic representation of the possible mass spec- tra for models including four mass eigenstates. In the left panel, the first and fourth eigenstates are degenerate, the cen- ter panel has the fourth mass eigenstate much heavier than the third, and the right panel has the third and fourth eigen- states as degenerate. Only the scenarios illustrated in the left and center panels are considered in this analysis. An alteration of the oscillation probability of Eq. (11) is predicted for neutrinos which traverse dense matter. By virtue of their neutral-current coherent forward scat- tering with nucleons, active neutrinos acquire an effective matter potential whereas sterile neutrinos do not. The effective potentials are identical for the ν$_{µ}$ and ν$_{τ}$ fla- vors so that matter effects vanish for ν$_{µ}$ → ν$_{τ}$ mixing. For ν$_{µ}$ → ν$_{s}$ oscillations, however, the overall nonzero matter potential yields MSW-type modifications to the mixing angle and oscillation length such that, in normal (inverted) hierarchy scenarios, the oscillation probability will be suppressed (enhanced) relative to ν$_{µ}$ → ν$_{τ}$ [35]. Observations of neutrinos with energies above 12 GeV that have traveled through several thousand kilometers of dense material, such as those described in Ref. [8], would be sensitive to this matter effect. By contrast, the beam-induced neutrinos monitored at the MINOS far site originate from a E$_{ν}$ spectrum highly peaked between 2 and 6 GeV and travel 735 km through the Earth’s crust. For the effective mixing angles and oscillation lengths of neutrinos in the NuMI beam, the ν$_{µ}$ → ν$_{s}$ matter effect only gives rise to subpercent distortions of the neutrino oscillation probability that are mostly confined to neu- trino energies below 2 GeV. Distortions of this magnitude are of no consequence for the studies reported here, and so matter effects involving sterile neutrinos have been neglected. As can be seen from Eqs. (7) and (11), mixing between three active and one sterile neutrino in the most general case involves ten parameters which are mostly unknown: five mixing angles, three mass-squared splittings, and two CP -violating phases. The following simplifying assump- tions are made to reduce the number of possible parame- ters. First, θ$_{13}$ is eliminated as a free parameter by only considering two values, θ$_{13}$ = 0 ◦ or θ$_{13}$ = 12 $^{◦}$, where the latter is the CHOOZ limit at the MINOS measured value of | Δ m 2 $_{32}$| . When θ$_{13}$ is set to the CHOOZ limit, δ$_{1}$ is taken to be 3 π/ 2 as that is the value of the CP -violating phase that maximizes the ν$_{e}$ appearance probability. Sec- ond, the additional CP -violating phase is removed by set- ting δ$_{2}$ = 0 as MINOS has no sensitivity to this phase. Finally, nonzero values of θ$_{14}$ do not measurably change the oscillation probabilities, so that angle is set to 0 ◦ as well. In addition to the parameters represented in the mix- ing matrix and oscillation probability equations, there is one more free parameter, namely the mass of ν$_{4}$ rela- tive to the other mass states. To limit the number of free parameters in the models, this analysis only exam- ines possibilities for the relative mass of ν$_{4}$ that allow the oscillation probabilities to depend only on | Δ m 2 $_{31}$| . The possible hierarchies are illustrated in Fig. 10 and only the normal mass hierarchy is shown. An inverted mass hierarchy would change the relative locations of the pair ν$_{1}$ and ν$_{2}$ to ν$_{3}$ . The fourth mass state could either be degenerate with the first mass state, as seen in the left panel of the figure, much more massive than the third mass state, as seen in the middle panel, or degenerate with the third mass state, as shown in the right panel. In the scenario with the third and fourth eigenstates being nearly degenerate there would be no discernible mixing This model is equivalent to the phenomenological model presented by MINOS in Ref. [17]. In the m$_{4}$ = m$_{1}$ case, the first and fourth mass eigen- states are assumed to be degenerate. Because the first and second eigenstates are also treated as degenerate, the second and fourth states are degenerate as well. These degeneracies allow one to set θ$_{14}$ = θ$_{24}$ = 0 ◦ in Eq. (7), which reduces the number of parameters in the model to four. There are no ν$_{e}$ or ν$_{µ}$ components in the fourth mass eigenstate, however there is a ν$_{s}$ component in the third mass eigenstate. Using these simplifications, the oscillation probabilities become 1. Active-sterile mixing when m 4 = m 1 P$_{ν}$$_{µ}$$_{→}$$_{ν}$$_{µ}$ = 1 − 4 | U$_{µ}$$_{3}$ | 2 ( 1 − | U$_{µ}$$_{3}$ | 2 ) sin $^{2}$Δ$_{31}$ , P$_{ν}$$_{µ}$$_{→}$$_{ν}$$_{α}$ = 4 | U$_{µ}$$_{3}$ | $^{2}$| U$_{α}$$_{3}$ | $^{2}$sin$^{2}$Δ$_{31}$ (12)