Prediction of a $Z_c(4000)$ $D^* \bar D^*$ state and relationship to the claimed $Z_c(4025)$

After discussing the OZI suppression of one light meson exchange in the interaction of $D^* \bar D^*$ with isospin I=1, we study the contribution of two pion exchange to the interaction and the exchange of a heavy vector $J/\psi$. We find this latter mechanism weak but enough to barely bind the system in J=2 with a mass around 4000 MeV, while the effect of the two pion exchange is a net attraction but weaker than that from $J/\psi$ exchange. We discuss this state and try to relate it to the $Z_c(4025)$ state, above the $D^* \bar D^*$ threshold, claimed in an experiment at BES from and enhancement of the $D^* \bar D^*$ distribution close to threshold. Together with the results from a recent reanalysis of the BES experiment showing that it is compatible with a J=2 state below threshold around 3990 MeV, we conclude that the BES experiment could be showing the existence of the state that we find in our approach.