12 U = R$_{34}$ ( θ$_{34}$ ) R$_{24}$ ( θ$_{24}$, δ$_{2}$ ) R$_{14}$ ( θ$_{14}$ ) R$_{23}$ ( θ$_{23}$ ) R$_{13}$ ( θ$_{13}$, δ$_{1}$ ) R$_{12}$ ( θ$_{12}$, δ$_{3}$ ) = R$_{34}$ ( θ$_{34}$ ) R$_{24}$ ( θ$_{24}$, δ$_{2}$ ) R$_{14}$ ( θ$_{14}$ ) R$_{23}$ ( θ$_{23}$ ) R$_{13}$ ( θ$_{13}$, δ$_{1}$ ) , (6) U =   $^{U$_{e}$$_{1}$}$ U$_{e}$$_{2}$ c$_{14}$s$_{13}$e − iδ 1 s$_{14}$ U$_{µ}$$_{1}$ U$_{µ}$$_{2}$ − s$_{14}$s$_{13}$e − iδ 1 s$_{24}$e − iδ 2 + c$_{13}$s$_{23}$c$_{24}$ c$_{14}$s$_{24}$e − iδ 2 U$_{τ}$$_{1}$ U$_{τ}$$_{2}$ − s$_{14}$c$_{24}$s$_{34}$s$_{13}$e − iδ 1 − c$_{13}$s$_{23}$s$_{34}$s$_{24}$e iδ 2 + c$_{13}$c$_{23}$c$_{34}$ c$_{14}$c$_{24}$s$_{34}$ U$_{s}$$_{1}$ U$_{s}$$_{2}$ − s$_{14}$c$_{24}$c$_{34}$s$_{13}$e − iδ 1 − c$_{13}$s$_{23}$c$_{34}$s$_{24}$e iδ 2 − c$_{13}$c$_{23}$s$_{34}$ c$_{14}$c$_{24}$c$_{34}$   $^{.}$ (7) P ( ν$_{α}$ → ν$_{β}$ ) = δ$_{αβ}$ − 4 ∑ j>i R ( U ∗ $_{αj}$U$_{βj}$ U$_{αi}$U ∗ $_{βi}$) sin $^{2}$Δ$_{ji}$ +2 ∑ j>i I ( U ∗ $_{αj}$U$_{βj}$ U$_{αi}$U ∗ $_{βi}$) sin 2Δ$_{ji}$ , (10) | ν$_{α}$ 〉 = ∑ j U ∗ $_{αj}$e − im 2 $_{j}$L/ (2 E ) | ν$_{j}$, 0 〉 , (8) A = 4 ∑ j =1 U ∗ $_{µj}$U$_{xj}$ e − im 2 $_{j}$L/ (2 E $^{)}$. (9) 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}$ | $^{2}$| U$_{µ}$$_{3}$ | $^{2}$sin$^{2}$Δ$_{43}$ + | U$_{µ}$$_{4}$ | $^{2}$(1 − | U$_{µ}$$_{3}$ | 2 − | U$_{µ}$$_{4}$ | $^{2}$) sin$^{2}$Δ$_{41}$ } , P$_{ν}$$_{µ}$$_{→}$$_{ν}$$_{α}$ = 4 R { | U$_{µ}$$_{3}$ | $^{2}$| U$_{α}$$_{3}$ | $^{2}$sin$^{2}$Δ$_{31}$ + | U$_{µ}$$_{4}$ | $^{2}$| U$_{α}$$_{4}$ | $^{2}$sin$^{2}$Δ$_{41}$ + U ∗ µ $_{4}$U$_{α}$$_{4}$ U$_{µ}$$_{3}$U ∗ α $_{3}$(sin$^{2}$Δ$_{31}$ − sin $^{2}$Δ$_{43}$ + sin $^{2}$Δ$_{41}$ ) } +2 I { U ∗ µ $_{4}$U$_{α}$$_{4}$ U$_{µ}$$_{3}$U ∗ α $_{3}$(sin 2Δ$_{31}$ − sin 2Δ$_{41}$ + sin 2Δ$_{43}$) } , (11) A. Oscillation Probabilities ments corresponding to the first and second mass states. where α = e, τ, or s and the orthogonality constraint ∑ $_{i}$U$_{ai}$ U ∗ bi = 0 has been used to eliminate the matrix ele- where the δ$_{k}$ are the three CP-violating Dirac phases and the last equality reflects the assumption of degeneracy in mass states 1 and 2. Thus, the mixing matrix can be written as, Here c$_{ij}$ = cos θ$_{ij}$ and s$_{ij}$ = sin θ$_{ij}$ , and only the elements of the matrix that appear in the oscillation probabilities given below have been expressed in terms of the mixing angles and phases. In this equation U$_{µj}$ is the element of the mixing matrix describing the coupling between the muon flavor state and mass state j . By squaring the amplitudes and using the unitarity of U , one obtains the oscillation probabili- ties for the different channels, The oscillation probabilities are derived following the example in Ref. [34]. The evolution of a neutrino with initial flavor state α is given by where the sum is over the mass states, E is the neutrino energy, L is the distance traveled by the neutrino and m$_{j}$ is the mass of state j . As the NuMI beam is almost en- tirely ν$_{µ}$ or ν$_{µ}$ , only the amplitudes and probabilities for ν$_{µ}$ → ν$_{x}$ , where x represents e, µ, τ, or s , are described. The amplitude for ν$_{µ}$ → ν$_{x}$ is given by where Δ$_{ji}$ ≡ ( m 2 j − m 2 $_{i}$) L/ (4 E ) and R (), I () designate the real and imaginary parts of the amplitude products. Because Δ$_{21}$ ≪ Δ$_{31}$ the first and second mass states are treated as degenerate and the factors of sin Δ$_{21}$ can be set to 0. This degeneracy also implies that Δ$_{42}$ = Δ$_{41}$ and Δ$_{32}$ = Δ$_{31}$. Equation (10) can be expanded for the different oscillation scenarios,