18 general, without needing to specify b ( y ). In fact, using eq. (54), it is not difficult to show that the functions ˜ C ( y, ω ) and ˜ S ( y, ω ) obey the integral equations: from which we can derive a low-energy expansion (small ω ): We will substitute expansions (60), (61) in eq. (57), keeping terms up to O ( ω $^{4}$), which - as we will soon show - will reproduce the Standard model plus corrections of order m 2 $_{Z}$/ ¯ M $^{2}$, where ¯ M is given by 1 : and, as we will show later, is of the order of the mass of the lightest resonance that we have integrated out. After the substitution, the bulk Lagrangian becomes 1 Notice that the parameter ¯ M can be related to the integrals introduced in [74], 1 ¯ M 2 = 1 πR ∫ πR 0 dy ∫ πR y $_{dz z b}$− $^{1}$( z ) (62) L (2) bulk = − πR 2 g 2 5 ( ˜ W a $_{µ}$( p $^{2}$η µν − p $^{µ}$p$^{ν}$)(1 − p 2 ¯ M $_{2}$) ˜ W a ν + ˜ B$_{µ}$ ( p $^{2}$η µν − p $^{µ}$p$^{ν}$)(1 − p 2 ¯ M $_{2}$) ˜ B$_{ν}$ ) ; (63) 1 ¯ M 2 = I 2 ( πR ) − I 1 ( πR ) I 1 ( y ) = 1 πR ∫ y 0 ∫ z 0 dz $^{′}$z $^{′}$b− $^{1}$( z $^{′}$) , I 2 ( y ) = ∫ y 0 dz $^{′}$z $^{′}$b− $^{1}$( z $^{′}$) where ˜ C ( y, ω ) = 1 − ω 2 ∫ πR y $_{dy}$′$_{b}$ − $^{1}$( $_{y}$′$_{)}$ ∫ πR y ′ $_{dy}$′′ ˜ C ( $_{y}$′′$_{, ω}$$_{)}$ (58) ˜ S ( y, ω ) = ω ∫ πR y $_{dy}$′$_{b}$ − $^{1}$( $_{y}$′$_{)}$ − ω 2 ∫ πR y $_{dy}$′$_{b}$ − $^{1}$( $_{y}$′$_{)}$ ∫ πR y ′ $_{dy}$′′ ˜ S ( $_{y}$′′$_{, ω}$$_{)}$$_{,}$ (59) ˜ C ( y, ω ) = 1 − ω 2 ∫ πR y $_{dy}$′$_{y}$ ′$_{b}$− $^{1}$( $_{y}$′$_{)}$ + ω 4 ∫ πR y $_{dy}$′$_{b}$ − $^{1}$( $_{y}$′$_{)}$ ∫ πR y ′ $_{dy}$′′ ∫ πR y ′′ $_{dy}$′′′$_{y}$ ′′′$_{b}$− $^{1}$( $_{y}$′′′$_{) +}$$_{. . .}$ (60) ˜ S ( y, ω ) = ω πR y $_{dy}$′$_{b}$ − $^{1}$( $_{y}$′$_{)}$ − ω 3 ∫ πR y $_{dy}$′$_{b}$ − $^{1}$( $_{y}$′$_{)}$ ∫ πR y ′ $_{dy}$′′ ∫ πR y ′′ $_{dy}$′′′$_{b}$ − $^{1}$( $_{y}$′′′$_{) +}$$_{. . .}$ (61)