10 Since the proof of the diagonalization is somewhat technical, we will proceed in reverse order, first defining the three sectors of the model, together with the conditions that the corresponding wave-functions have to satisfy, then show how the three sectors are derived by the request of diagonalizing the KK expanded Lagrangian. The three sectors are: • A left charged sector coming from the expansion of the ( W 1 , 2 L )$_{M}$ fields and the brane pseudo-scalars π 1 , $^{2}$. The explicit form of the expansion is The wave-functions of the vector fields satisfy the equation of motion: where we defined the differential operator: and the set of BCs: The pseudo-scalar profiles are fixed by the conditions: Note that in this sector no massless solution is allowed; in fact, eq. (15) together with the Neumann BC at y = 0 (17) imply that a massless mode must have a constant profile, and a constant, massless solution cannot satisfy the BC at y = πR (18). Also note that eq. (15) and the BCs (17) and (18) are diagonal in the isospin index, so we have f 1 L n = f 2 $_{L n}$. W 1 , 2 $_{L µ}$( x, y ) = ∞ ∑ n =0 f 1 , 2 $_{L n}$( y ) W 1 , 2( n ) Lµ ( x ) , W 1 , 2 L $_{5}$( x, y ) = ∞ ∑ n =0 g 1 , 2 $_{L n}$( y ) G 1 , 2 ( n ) L ( x ) , π 1 , $^{2}$( x ) = ∞ ∑ n =0 c 1 , 2 n G ( n $^{)}$( x ) . (14) ˆ Df 1 , 2 L n = − m 2 $_{L n}$f 1 , 2 $_{L n}$, (15) ˆ D ≡ ∂$_{y}$ ( b ( y ) ∂$_{y}$ ( · )) , (16) ∂$_{y}$f 1 , 2 L n = 0 at y = 0 , (17) ( ˜ g 2 g 2 $_{5}$∂$_{y}$ − b ( πR ) m 2 L n + ˜ g $^{2}$˜ v 2 4 ) f 1 , 2 L n = 0 at y = πR. (18) g 1 , 2 L n = 1 m$_{L n}$ ∂$_{y}$f 1 , 2 $_{L n}$, c 1 , 2 n = ˜ v 2 m$_{L n}$ f 1 , 2 L n ∣ $_{πR}$. (19)