}{\omega } \,.$$

We show in figure [**2**]{} the spectral functions for the bottomonium vector states at four illustrative temperatures. In these plots one can clearly see the following situation:

-   $\bullet$ at $T = 200$ MeV three peaks corresponding to $ 1S$, $2S$ and $3S$ states;

-   $\bullet$ at $T = 260$ MeV two peaks corresponding to the melting of the $3S$ state;

-   $\bullet$ one peak at $T = 340 $ MeV where only the $1S$ states survives and

-   $\bullet$ at $T = 660 $ MeV the complete melting of the states.

We present in the appendix [**A**]{} a more detailed picture of the melting process by showing more plots that illustrate the temperature evolution of the spectral function. From this analysis one can infer that the states $1S, 2S $ and $3S$ melt at different temperatures, as expected. In particular, the $1S$ states survives at temperatures much larger than the critical temperature. The complete disappearance of the $1 S$ states happens at $ T \sim 600 $ MeV, corresponding to $ T/T_c \sim 3.2  $. For the $ 2S$ state there will be no trace of the peak for temperatures above $ T \sim 360 $ 