Dynamical couplings, dynamical vacuum energy and confinement/deconfinement from R²-gravity

We study within Palatini formalism an f(R)-gravity with f(R)=R+αR² interacting with a dilaton and a special kind of nonlinear gauge field system containing a square-root of the standard Maxwell term, which is known to produce confinement in flat space-time. Reformulating the model in the physical Einstein frame we find scalar field effective potential with a flat region where the confinement dynamics disappears, while in other regions it remains intact. The effective gauge couplings as well as the induced cosmological constant become dynamical. In particular, a conventional Maxwell kinetic term for the gauge field is dynamically generated even if absent in the original theory. We find few interesting classes of explicit solutions: (i) asymptotically (anti-)de Sitter black holes of non-standard type with additional confining vacuum electric potential even for the electrically neutral ones; (ii) non-standard Reissner-Nordström black holes with additional constant vacuum electric field and having non-flat-space-time "hedgehog" asymptotics; (iii) generalized Levi-Civita-Bertotti-Robinson "tube-like" space-times.