Scalarization of Einstein–Euler–Heisenberg black hole with multiple horizons

Abstract Scalarizations of the Einstein–Euler–Heisenberg (EEH) black hole (EEHBH) with multiple horizons are investigated in the EEH-scalar theory by introducing a quadratic scalar coupling to the Maxwell term. For mass $$M=1$$ M = 1 and Euler–Heisenberg parameter $$\\mu =0.03$$ μ = 0.03 , the magnetically charged EEHBH admits four horizon families (low, cold, negative, and hot), with triple horizons appearing in the narrow band of magnetic charge $$q\\in [0.95,1.0065]$$ q ∈ [ 0.95 , 1.0065 ] . The onset scalarization around the low, cold, and high horizons is then analyzed for the magnetic charge $$q=0.5,\\,1,\\,2$$ q = 0.5 , 1 , 2 , implying infinite branches of scalarized black holes for each case. We construct the three fundamental branches of scalarized black holes. From the positivity condition of their mass, we find the upper bounds on primary scalar charges $$q_{s}$$ q s for scalarized low and cold horizons. These bounds determine the allowable regions for the Hawking temperature and entropy. Furthermore, we perform a time-domain stability analysis and find that the instabilities arise only at small scalar charge regime. Therefore, stable and physically viable scalarized black holes exist in an intermediate window of the primary scalar charge for low and cold horizon solutions and a lower bound for hot horizon solution.