The normalization condition is also the same: ∫ ( dz/z ) ψ 2 a 1 = 1. Similarly, the decay constant F$_{a}$$_{1}$ is given by (5.16), but with ρ replaced by a$_{1}$ : The pion can be found by solving (5.19) and (5.20) simultaneously, subject to the boundary conditions φ ′ ( z$_{m}$ ) = 0 , φ (0) = 0 and π (0) = 0. The pion decay constant can be found by considering the axial current correlator of (1.36): According to (5.23), Π$_{A}$( − q $^{2}$) will have a pole at q 2 = m 2 $_{π}$. In the limit m$_{π}$ = 0, Π$_{A}$( − q $^{2}$) ∼ − f 2 $_{π}$/q $^{2}$. Hence, on comparison with (5.16), we conclude From this simple model we can predict six quantities ∗ : the mass and decay constants of the ρ mesons, the pions, and the a$_{1}$ mesons. We still have three inputs: the quark condensate, the quark mass, and the position of the infrared cutoff. Performing a best-fit analysis gives a rms error of 9 %.(Note ϵ$_{rms}$ = ∑ $_{O}$(( δO/O ) $^{2}$/n ) 1 / 2 with O the observable and n equal to the number of predictions minus the number of inputs, so here n = 3).The results are displayed in table 5.2. $^{∗}$in [2] they claim to predict seven quantities; the seventh being the π − ρ coupling, g$_{ρππ}$ . However this term would presumably receive an important contribution from F 3 terms in the holographic action, which haven’t been included. It could be argued, though, that a good agreement between the prediction for g$_{ρππ}$ and its experimental value in the absence of F 3 terms indicates that these terms have little influence 79 5.4 Results f 2 π = − 1 g 2 5 ∂$_{z}$A (0 , ϵ ) ϵ (5.24) F 2 a 1 = 1 g 2 5 ( $_{ψ}$′ a $_{1}$( ϵ ) /ϵ ) 2 (5.22) 〈 0 | A$_{µ}$ (0) | π ( q ) 〉 = if$_{π}$q$_{µ}$ (5.23)