content_image/410997.jpg content_image/410998.jpg content_image/411000.jpg content_image/411001.jpg content_image/410999.jpg content_image/411002.jpg
<img src='content_image/410997.jpg'>

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, 

<img src='content_image/410998.jpg'>

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. 

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 

## A. Oscillation Probabilities 

<img src='content_image/411000.jpg'>

<img src='content_image/411001.jpg'>

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, 

<img src='content_image/410999.jpg'>

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, 

<img src='content_image/411002.jpg'>

where α = e, τ, or s and the orthogonality constraint ∑ $_{i}$U$_{ai}$ U ∗ bi = 0 has been used to eliminate the matrix ele- 

ments corresponding to the first and second mass states. 

