  It has recently been shown that nontrivial couplings between a scalar and the
Gauss-Bonnet invariant can give rise to black hole spontaneous scalarization.
Theories that exhibit this phenomenon are among the leading candidates for
testing gravity with upcoming black hole observations. All models considered so
far have focused on specific forms for the coupling, neglecting scalar
self-interactions. In this work, we take the first steps towards placing this
phenomenon on a more robust theoretical footing by considering the
leading-order scalar self-interactions as well as the scalar-Gauss-Bonnet
coupling. Our approach is consistent with the principles of effective field
theory and yields the simplest and most natural model. We find that a mass term
for the scalar alters the threshold for the onset of scalarization, and we
study the mass range over which scalarized black hole solutions exist. We also
demonstrate that the quartic self-coupling is sufficient to produce scalarized
solutions that are stable against radial perturbations, without the need to
resort to higher-order terms in the Gauss-Bonnet coupling function. Our model
therefore represents a canonical model that can be studied further, with the
ultimate aim of developing falsifiable tests of black hole scalarization.
