content_image/324536.jpg content_image/324537.jpg content_image/324538.jpg content_image/324539.jpg content_image/324540.jpg
## 5.2 Fixing the g$_{5}$ coupling 

The idea in this section is to calculate the vector current two-point function in the holographic scheme, and then by equating the result to that obtained from perturbative QCD we can hence fix g$_{5}$ . 

We introduce the vector field as V = ( A$_{L}$ + A$_{R}$ ) / 2, choose the gauge V$_{r}$ ( x, r ) = 0, and by using the Euler-Lagrange equation on the holographic action (5.2), we obtain an equation of motion: 

<img src='content_image/324536.jpg'>

We then substitute (5.5) back into the action (5.2), and we are left with the boundary term 

<img src='content_image/324537.jpg'>

So, by the claim made above (5.4), equation (5.6) is the generating func- tional for QCD. To obtain the vector current two-point function, we simply functionally differentiate (5.6) twice with respect to the source V$_{0}$ , with V$_{0}$ defined in V $^{µ}$( q, z ) = V ( q, z ) V µ 0 ( q ). ( V µa 0 ( q ) is the Fourier transform of the source of the vector current J a µ = ¯ qγ$_{µ}$t $^{a}$q ).Doing the necessary maths, we get 

<img src='content_image/324538.jpg'>

<img src='content_image/324539.jpg'>

For large Euclidean Q $^{2}$( ≡ − q $^{2}$) we only need to know V ( q, r ) near the boundary, 

<img src='content_image/324540.jpg'>

