Minimum mass, maximum charge and hyperbolicity in scalar Gauss-Bonnet gravity

We study the loss of hyperbolicity of perturbation equations for black hole solutions of scalar Gauss-Bonnet gravity. We consider a class of coupling functions allowing for static black hole solutions with arbitrary small masses. For masses below a minimum value, such solutions become unphysical, because the perturbation equations become elliptic; this arguably corresponds to the loss of validity of the effective field theory. We analyse the dependence of this minimum mass on the parameters of the theory, finding that with an appropriate choice of the coupling function, such mass can be chosen arbitrarily small. However, this does not correspond to larger deviations from general relativity, since observable quantities like the black hole scalar charge are bounded by above.