Charged qOS-extremal black hole and its scalarization by entropy function approach

We investigate scalarization of charged quantum Oppenheimer–Snyder extremal (cqOSe)-black hole described by mass M , quantum parameter α α and magnetic charge P in the Einstein–Gauss–Bonnet-scalar theory with a nonlinear electrodynamics term. This scalarization may be equivalent to scalarization of quantum Oppenheimer–Snyder (qOS)-extremal black hole whose action is still unknown when replacing (3α P²)1/4 ( 3 α P 2 ) 1 / 4 by 3 M /2. Focusing on the onset of scalarization, we find the single branch of scalarized cqOS extremal (scqOSe)-black holes. To obtain a scalar cloud (seed) for the single branch, however, we have to consider its near-horizon geometry of the Bertotti–Bobinson (BR) spacetime. In this case, two scalar clouds for positive and negative coupling constant λ λ are found to represent two branches. Applying Sen’s entropy function approach to this theory, we obtain the entropy which is the only physical quantity to describe the scqOSe-black holes. We find that the positive branch is preferred than the negative branch.