MAGOO 2-Point Correlation function derivation
I am studying the famous MAGOO review and I am trying to understand section 3.3 where we calculate the correlation functions from the AdS side. In subsection 3.3.1 the authors go through the 2-point functions.
The given action is $$ S=\frac{\eta}{2}\int{d^5x\sqrt{g}\left[(\partial\phi)^2+m^2\phi^2\right]} $$ where we used the Euclidean action and the metric is given in Poincare coordinates as $$ ds^2=\frac{R^2}{z^2}(dx^2+dz^2) $$ again, where $dx^2$ is the Euclidean part of the metric and $dz^2$ gives the radial part of AdS space. Then the fields are expressed in Bessel functions as $$ \phi(x,z)=\frac{z^2K_{\Delta-2}(pz)}{\epsilon^2K_{\Delta-2}(p\epsilon)}e^{ipx} $$ Where $K_{\Delta-2}(pz)$ is the modified Bessel function of the second kind. My problem is in deriving equation (3.40) where we substitute the field in the action and then compute the integral. 402
Do we first take the limit of the Bessel function with $z\rightarrow0$ and then do the integration?
My attempt: $$ S= \frac{\eta}{2}\int{d^4xdz\left[\frac{R^3}{z^3}(\nabla\phi)^2+\frac{R^3}{z^3}(\partial_z\phi)^2+\frac{R^5}{z^5}m^2\phi^2\right]} $$ then if I substitute $\phi$ everything is a mess. I have also took the limit of the field for $z\rightarrow0$ giving $$ \phi(z\rightarrow 0)\rightarrow\frac{\Gamma(\Delta-2)e^{ipx}}{2\epsilon^2K_{\Delta-2}(p\epsilon)}\left(\frac{p}{2}\right)^{2-\Delta}z^{4-\Delta} $$
My progress
So far I have used the fact that the derivative of the specific bessel function has the following closed form $$ \partial_z K_\nu(pz) = \frac{\nu}{z}K_{\nu}(pz)-pK_{\nu+1}(pz) $$ then carefully taking the derivatives and collecting the terms I computed the following integral $$ S = \frac{\eta}{2\epsilon^4 K^2_{\Delta-2}(p\epsilon)}\int{d^4x dz e^{2ipx}\left[-p^2R^3 z K^2_{\Delta-2}(pz)+R^3\left(\Delta^2 K^2_{\Delta-2}(pz)+p^2 z K^2_{\Delta-1}(pz)-2\Delta p K_{\Delta-2}(pz)K_{\Delta-1}(pz)\right)+R^5 m^2 z^{-1}K^2_{\Delta-2}(pz)\right]} $$ Now each integral in $z$ indeed has the form $\int{dz z^n K^2_{\nu}(pz)}$. Then we note that the $z$ integral is in the range $(\epsilon,\infty)$. I understand that we should take this integral and then expand in $\epsilon$ but I am not sure how to compute the integral. 402