Safe essential scalar-tensor theories

We discuss the renormalisation group flow of all essential couplings of quantum gravity coupled to a shift-symmetric scalar field at fourth order in the derivative expansion. We derive the global structure of the phase diagram, and identify a bounded region in theory space which is both asymptotically safe in the ultraviolet, and connects to standard effective field theory in the infrared. Our system thus satisfies the weak-gravity bound. The allowed infrared behaviour of the essential four-scalar coupling is restricted by requiring an ultraviolet completion. This bound can be saturated by a theory without free parameters, which gives a concrete example for a fully predictive scalar-tensor theory.