No realistic wormholes from ghost-free scalar-tensor phantom dark energy

It is proved that no wormholes can be formed in viable scalar-tensor models of dark energy admitting its phantom-like ($w<-1$) behaviour in cosmology, even in the presence of electric or magnetic fields, if the non-minimal coupling function $f(\Phi)$ is everywhere positive and the scalar field $\Phi$ itself is not a ghost. Some special static, spherically symmetric wormhole solutions may exist if $f(\Phi)$ is allowed to reach zero or to become negative, so that the effective gravitational constant becomes negative in some region making the graviton a ghost. If $f$ remains non-negative, such solutions require severe fine tuning and a very peculiar kind of model. If $f<0$ is allowed, it is argued (and confirmed by previous investigations) that such solutions are generically unstable under non-static perturbations, the instability appearing right near transition surfaces to negative $f$.