The space is a Poincaré AdS chart: $$ds^2 \,\,= \,\, \frac{R^2}{z^2}  \,  \Big(  - dt^2 + dz^2  + d\vec{x}\cdot d\vec{x}  \Big)   \,.
 \label{metric1}$$

The second input parameter of the model, that is not present in the usual formulation of the soft wall model, is introduced by calculating the correlators at a finite position $ z = z_0$ instead of taking the boundary to be at $z=0$. The parameter $1/z_0$ is interpreted as an UV energy scale. A similar approach appeared in ref. [@Evans:2006ea] but for light vector mesons.

One considers the action of eq. (\[vectorfieldactionzeroptemp\]) to be defined in the region $ z_0 \le z <  \infty $, then the on shell action takes the form $$I_{on \, shell }\,=\, - \frac{1}{2 {\tilde g}_5^2}  \, \int d^4x \,\,\Big[ \, \frac{e^{- k^2 z^ 2  }}{z} \eta^{\mu \nu} V_\mu \partial_z V_ \mu \,\Big] 
{\Big \vert}_{_{ \! z \to z_0 }} 
 \,,
\label{onshellactionzerotemp}$$ where $ {\tilde g}_5^2 =  g_5^2 /R $ is the relevant dimensionless coupling of the vector field and $\eta^{\mu \nu} $ is the Minkowski metric.

The gauge $V_z = 0 $ is used, so that the boundary values of the other remaining components of the vector field: $ V^0_{\mu}(x) =
