e^{-k^ 2 z^ 2}}{z^ 5}  \,.
 \label{densities2}
 \end{aligned}$$

The critical temperature, where the two actions densities are equal, depends on the infrared parameter $k$ of the soft wall background. This parameter is flavor dependent. The formation of the plasma occurs when the lightest hadrons dissociate. So, we consider the confinement/deconfinement transition to be determined by the soft wall background that fits the masses of the $\rho$ vector mesons. In the present model $\rho$ vector mesons can be described taking as in ref. [@Braga:2015jca] $ 1/z_0 = 12.5 $ GeV and reproducing the calculation of the mass reviewed in section 2. One finds, using the parameter $ k $ = 0.388 GeV as in [@Herzog:2006ra], that the model with UV cut off leads to a mass of 777.6 MeV for the $1 S $ state.

The corresponding critical temperature is $ T_c = 191 $ MeV, the same result of ref. [@Herzog:2006ra]. In figure [**1** ]{} we show the difference $ \Delta V = V_{BH \, AdS}  \,-\, V_{th\, AdS}\, $ between the action densities of eqs. (\[densities1\]) and (\[densities2\]) as a function of the temperature. The critical temperature corresponds to the point where the curve crosses the temperature axis.

 Vector 