content_image/315565.jpg content_image/315571.jpg
<img src='content_image/315565.jpg'>

we performed for the vector sector, the a$_{1}$ decay constants are given by 

<img src='content_image/315571.jpg'>

Looking at equations (6.17) and (6.18) we see that the pion and longitudinal gauge fields mix. This makes finding numerical solutions to (6.17) and (6.18) very hard. In principle we’d have to search for solutions with two unknowns: the pion mass, − q $^{2}$, and the ratio of the φ and π fields at r = 0. So instead, we use the Gell-Mann-Oakes-Renner relation ( m 2 $_{π}$f 2 π = 2 m$_{q}$c ) [145], to fix m$_{π}$ . Here we introduce this relation by hand, but it is possible to show that in the geometry studied in chapter 5, the equation can be derived explicitly [2]. This relation eliminates one of our free parameters, and makes the numerical search much easier. 

This model has two free parameters: m and b . m represents the quark mass (this is a model with only two flavours of quark, which we have as- sumed to be of equal mass), and b , which in the original supergravity solu- tion represented the size of the conformal symmetry breaking. Here then it roughly corresponds to Λ$_{QCD}$. Note that this model has the same number of free parameters as real QCD. 

