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Theoretical study reveals constant scalar hair solutions in extremal Einstein–Maxwell-scalar systems, suggesting difficulty in finding primary scalar hair.
We study scalarized extremal black holes in the Einstein–Maxwell-scalar (EMS) theory with two different scalar couplings to two U(1) fields. This theory is inspired by the dyonic EMS theory. Two scalarized extremal black holes are found with constant scalar hair. We confirm that these are precisely obtained from the extremal scalarization and entropy function approaches. This may imply that it is not easy to find extremal black holes with primary scalar hair.
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Chew et al. (2026) studied this question.
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