1 / ( D 1 − D 2 ) 1 / ( D 2 − D 3 ) 1 / ( D 3 − D 1 ) dx 1 /dA x 1 x 2 − 2 x 1 y 2 + y 1 y 2 − x 1 x 2 + x 1 x 3 + y 1 y 2 − y 1 y 3 − x 1 x 3 + 2 x 1 y 3 − y 1 y 3 dx 2 /dA − x 1 x 2 + 2 x 2 y 1 − y 1 y 2 x 2 x 3 − 2 x 2 y 3 + y 2 y 3 x 1 x 2 − x 2 x 3 − y 1 y 2 + y 2 y 3 dx 3 /dA − x 1 x 3 + x 2 x 3 + y 1 y 3 − y 2 y 3 − x 2 x 3 + 2 x 3 y 2 − y 2 y 3 x 1 x 3 − 2 x 3 y 1 + y 1 y 3 dy 1 /dA − x 1 x 2 + 2 x 2 y 1 − y 1 y 2 − x 1 x 2 + x 1 x 3 + y 1 y 2 − y 1 y 3 x 1 x 3 − 2 x 3 y 1 + y 1 y 3 dy 2 /dA x 1 x 2 − 2 x 1 y 2 + y 1 y 2 − x 2 x 3 + 2 x 3 y 2 − y 2 y 3 x 1 x 2 − x 2 x 3 − y 1 y 2 + y 2 y 3 dy 3 /dA − x 1 x 3 + x 2 x 3 + y 1 y 3 − y 2 y 3 x 2 x 3 − 2 x 2 y 3 + y 2 y 3 − x 1 x 3 + 2 x 1 y 3 − y 1 y 3 d (ln J ) /dA − x 1 + x 2 + y 1 − y 2 − x 2 + x 3 + y 2 − y 3 x 1 − x 3 − y 1 + y 3 TABLE I: dx i /dA , dy i /dA , and d (ln J ) /dA are expressed as sums of terms in 1 / ( D 1 − D 2 ), 1 / ( D 2 − D 3 ), and 1 / ( D 3 − D 1 ). Experimentally, it is known that δ$_{0}$ = m 2 − m 2 1 ∼ = 7 × 10 − $^{5}$eV $^{2}$, Δ$_{0}$ = m 2 3 − m 2 ∼ = 3 × 10 − $^{3}$eV $^{2}$, so that δ$_{0}$/ Δ$_{0}$ ≪ 1 (we assume the “normal” ordering of neutrino masses. The “inverted” case can be similarly treated). Note that these values are relevant to long baseline experiments since A = √$_{2}$$_{G}$$_{F}$$_{n}$$_{e}$$_{E}$ ∼ (7 . 6 × 10 − $^{5}$eV $^{2}$)( E/GeV )( ρ/gcm − $^{3}$). region, the explicit, approximate, solutions are Since δ$_{0}$ ≪ Δ$_{0}$, we expect that the three-flavor prob- lem can be approximated by a pair of well separated two-flavor problems [10]. Indeed, the structure of the differential equations in Table I shows that the vari- ables ( x$_{i}$, y$_{i}$ ) evolve slowly as a function of A except for two regions, where D$_{1}$ ≈ D$_{2}$ and D$_{2}$ ≈ D$_{3}$ , cor- responding to the two resonance regions. More pre- cisely, let us denote by ( A$_{0}$, A$_{l}$, A$_{i}$, A$_{h}$, A$_{d}$ ) the values of A in vacuum ( A$_{0}$ = 0), at the lower resonance ( A$_{l}$, [ d ( D$_{1}$ − D$_{2}$ ) /dA ]$_{A}$$_{l}$ = 0), intermediate range ( A$_{i}$ ), higher resonance ( A$_{h}$, [ d ( D$_{2}$ − D$_{3}$ ) /dA ]$_{A}$$_{h}$ = 0), and for dense medium ( A$_{d}$ ). Rapid evolution for ( x$_{i}$, y$_{i}$ ) only occurs for A ≈ A$_{l}$ and A ≈ A$_{h}$ . For 0 ≤ A < A$_{i}$ , we need only to keep the terms ∝ 1 / ( D$_{1}$ − D$_{2}$ ) in Table I. It is seen that Given W$_{0}$ (Eq. (10)), with x$_{i}$$_{0}$ + y$_{i}$$_{0}$ ∼ = 0, Eq. (20) yields x$_{i}$ + y$_{i}$ ∼ = 0. Thus, we expect that for 0 ≤ A < A$_{i}$ , while the individual variables ( x$_{1}$, x$_{2}$, y$_{1}$, y$_{2}$ ) are rapidly changing, x$_{3}$ ≃ y$_{3}$ ≃ O ( ϵ ) stay small, as do the combi- nations x$_{1}$ + y$_{1}$ ∼ = x$_{2}$ + y$_{2}$ ∼ = 0, and x$_{1}$ + x$_{2}$ ≃ constant, y$_{1}$ + y$_{2}$ ≃ constant. The differential equations can then be approximated, with δ ≡ D$_{2}$ − D$_{1}$ , by It follows that where b$_{l}$ ∼ = 2( x$_{10}$ + x$_{20}$ ) $^{2}$. So, in the lower resonance Thus, as A goes from A$_{i}$ through A$_{h}$ to A$_{d}$ , the changes for ( x$_{i}$, y$_{j}$ ) are: x$_{2}$ ≃ 1 / 2 → 0; y$_{2}$ ≃ − 1 / 2 → 0; x$_{3}$ ≃ 0 → 1 / 2; and y$_{3}$ ≃ 0 → − 1 / 2. Our results can be summarized by the matrices W at with b$_{h}$ = 2( x$_{20}$ + x$_{30}$ ) 2 and c$_{h}$ = Δ$_{0}$( x$_{20}$ − x$_{30}$ ). The solutions are together with x$_{2}$ + y$_{2}$ ≃ x$_{3}$ + y$_{3}$ ≃ 0, while x$_{1}$ and y$_{1}$ are slowly varying so that x$_{1}$ ≃ y$_{1}$ 0 throughout. With Δ ≡ D$_{3}$ − D$_{2}$ , the approximate equations near A ≈ A$_{h}$ are then with c$_{l}$ = δ$_{0}$ ( x$_{10}$ − x$_{20}$ ). Also, x$_{1}$ + y$_{1}$ ∼ = x$_{2}$ + y$_{2}$ ∼ = 0, x$_{3}$ ∼ = y$_{3}$ ∼ = 0. From W$_{0}$ , we have b$_{l}$ = 2( x$_{10}$ + x$_{20}$ ) 2 ∼ = 1 / 2. We see thus, as A sweeps through the lower resonance region, δ goes through a minimum, x$_{1}$ decreases and x$_{2}$ rises while keeping x$_{1}$ + x$_{2}$ ≃ 1 / 2. After the resonance, for large A ( A ≫ δ$_{0}$ ), δ ≃ A , x$_{1}$ → 0, and x$_{2}$ → 1 / 2. A similar analysis can be done for the region A$_{i}$ < A < A$_{d}$ . Here, the starting values ( A$_{l}$ ≪ A < A$_{h}$ ) are x$_{1}$ ≃ y$_{1}$ ≃ 0, x$_{3}$ ≃ y$_{3}$ ≃ 0, x$_{2}$ → 1 / 2, y$_{2}$ → − 1 / 2. The differential equations are dominated by terms pro- portional to 1 / ( D$_{2}$ − D$_{3}$ ), and they satisfy δ = [2 b$_{l}$A 2 − 4 c$_{l}$A + δ 2 $_{0}$] 1 / $^{2}$, x$_{1}$ = 1 2 [( x$_{10}$ + x$_{20}$ ) − ( b$_{l}$A − c$_{l}$ ) /δ ] , x$_{2}$ = 1 2 [( x$_{10}$ + x$_{20}$ ) + ( b$_{l}$A − c$_{l}$ ) /δ ] , (24) d ( x$_{1}$ − y$_{2}$ ) dA = d ( x$_{2}$ − y$_{1}$ ) dA = d ( x$_{3}$ − y$_{3}$ ) dA = 0 . (21) d ( x$_{1}$ − y$_{1}$ ) dA = d ( x$_{2}$ − y$_{3}$ ) dA = d ( x$_{3}$ − y$_{2}$ ) dA = 0 . (25) dx$_{2}$ dA ∼ = − 4 x$_{2}$x$_{3}$ Δ ∼ = − dx$_{3}$ dA , d Δ dA ∼ = 2( x$_{3}$ − x$_{2}$ ) , (26) dx$_{1}$ dA ∼ = − 4 x$_{1}$x$_{2}$ δ ∼ = − dx$_{2}$ dA , dδ dA ∼ = 2( x$_{2}$ − x$_{1}$ ) . (22) d dA [ x$_{1}$x$_{2}$δ $^{2}$] = 0 , d dA [( x$_{1}$ − x$_{2}$ ) δ ] = − 2( x$_{1}$ + x$_{2}$ ) 2 ≡ − b$_{l}$, (23) Δ = [2 b$_{h}$A 2 − 4 c$_{h}$A + Δ 2 $_{0}$] 1 / $^{2}$, x$_{2}$ = 1 2 [( x$_{20}$ + x$_{30}$ ) − ( b$_{h}$A − c$_{h}$ ) / Δ] , x$_{3}$ = 1 2 [( x$_{20}$ + x$_{30}$ ) + ( b$_{h}$A − c$_{h}$ ) / Δ] , (27) 3