Scalarizations of qOS-extremal black hole and Aretakis instability

We study scalarization of quantum Oppenheimer-Snyder (qOS)-extremal black hole in the Einstein-Gauss-Bonnet-scalar theory. This black hole is described by mass ($M=4\sqrt{\alpha }\mathrm{/}3\sqrt{3}$) with $\alpha$ quantum parameter. From studying onset of scalarization, we find the appearance of the single branch of scalarized qOS-extremal black holes. To obtain the tachyonic scalar cloud being a seed to generate the single branch of scalarized qOS-extremal black holes, we consider the near-horizon geometry of the Bertotti-Bobinson (BR) spacetime. In this case, we find that the appearance of a large scalar cloud at the horizon ($\rho =0$) is a new feature to represent onset scalarization of extremal black holes for tachyon with negative mass, but it is not related to the Aretakis instability of a propagating scalar with standard mass around the BR spacetime, showing polynomial instability of the ingoing time $v$. The Aretakis instability is connected to the scalar cloud with standard mass, indicating the blow-up at the horizon.