content_image/311682.jpg content_image/311683.jpg content_image/311681.jpg
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$_{1}$ : 

<img src='content_image/311682.jpg'>

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): 

<img src='content_image/311683.jpg'>

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 

## 5.4 Results 

<img src='content_image/311681.jpg'>

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. 

