33 Appendix A: Derivation of the conditions for the KK expansion We will now show how eqs. from (14) to (32) can be derived from the request that the effective 4D Lagrangian is diagonal. Throughout the following calculation, we will only need to work with the bilinear gauge part of the action (8). Expanding the gauge fields as in eq. (13) without assuming anything a priori on the form of the functions f a j and g a j and the constants c a j and carrying out the integration with respect to the extra dimension, we get where we defined the matrices: 2 k k ∣ In the expanded Lagrangian (A1), it is possible to recognize vector and scalar kinetic-like terms, vector mass-like terms and vector / would-be goldstone mixings. However, all those terms are in general not diagonal with respect to the KK number. This is of course a direct consequence of the general nature of the expansion (13). However, if the expanded theory is to be consistent, it must be possible to obtain the actual physical degrees of freedom - with explicitly diagonal mass and kinetic terms - by defining appropriate linear combinations of the modes V ( j ) µ and G ( j $^{)}$. We then introduce a still general basis change in field space: and require the Lagrangian (A1) to be diagonal in terms of the new degrees of freedom ˜ V ( j ) µ and ˜ G ( j $^{)}$. This means that the matrices R $^{T}$AR , R $^{T}$BR , S $^{T}$CS and R $^{T}$DS (all the fields are V ( j ) µ = R$_{jk}$ ˜ V ( k ) µ ; G ( j ) = S$_{jk}$ ˜ G ( k $^{)}$, (A3) L (2) = − 1 4 V ( j ) µν V ( k ) $^{µν}$A$_{jk}$ − 1 2 V ( j ) µ V ( k ) $^{µ}$B$_{jk}$ − 1 2 ∂$_{µ}$G ( j $^{)}$∂ $^{µ}$G ( k $^{)}$C$_{jk}$ + V ( j ) µ ∂ $^{µ}$G ( k $^{)}$D$_{jk}$ , (A1) A$_{jk}$ = 1 g 2 5 ∫ πR 0 dy ( f a $_{L j}$f a L k + f a $_{R j}$f a R k ) + 1 ˜ g $_{2}$f a $_{L j}$f a L k ∣ πR + 1 ˜ g ′ $_{2}$f 3 $_{R j}$f 3 R k ∣ $_{πR}$; B$_{jk}$ = 1 g 2 5 ∫ πR 0 dy b ( y ) ( ∂$_{y}$f a $_{L j}$∂$_{y}$ f a L k + ∂$_{y}$f a $_{R j}$∂$_{y}$ f a R k ) + ˜ v 2 4 b ( πR ) ( f a $_{L j}$f a L k + f 3 $_{R j}$f 3 R k − 2 f 3 $_{L j}$f 3 R k ) ∣ ∣ $_{πR}$; C$_{jk}$ = 1 g 2 5 ∫ πR 0 dy b y ) ( g a $_{L j}$g a L k + g a $_{R j}$g a R k ) + c a $_{j}$c a $_{k}$; D$_{jk}$ = 1 g 2 5 πR 0 dy b ( y ) ∂$_{y}$ ( f a $_{L j}$g a L k + f a $_{R j}$g a R k ) − ˜ v 2 ( f a $_{L j}$c a k − f 3 $_{R j}$c 3 k )∣ $_{πR}$. (A2)