New scalarization of the Einstein–Euler–Heisenberg black hole

Abstract The spontaneous scalarization of the Einstein–Euler–Heisenberg (EEH) black hole is performed in the EEH-scalar theory by introducing an exponential scalar coupling (with $$\alpha $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>α</mml:mi> </mml:math> coupling constant) to the Maxwell term. Here, the EEH black hole as a blad black hole is described by mass M and magnetic charge q with an action parameter $$\mu $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>μ</mml:mi> </mml:math> . A choice of $$\mu =0.3$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>μ</mml:mi> <mml:mo>=</mml:mo> <mml:mn>0.3</mml:mn> </mml:mrow> </mml:math> guarantees a single horizon with unrestricted magnetic charge q . The onset scalarization of this black hole appears for a positive coupling $$\alpha $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>α</mml:mi> </mml:math> with an unlimited magnetic charge q . However, there exists a difference between $$q\le 1$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>q</mml:mi> <mml:mo>≤</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:math> and $$q&gt;1$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>q</mml:mi> <mml:mo>&gt;</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:math> onset scalarizations. We notify the presence of infinite branches labeled by the number of $$n=0,1,2,\ldots $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>n</mml:mi> <mml:mo>=</mml:mo> <mml:mn>0</mml:mn> <mml:mo>,</mml:mo> <mml:mn>1</mml:mn> <mml:mo>,</mml:mo> <mml:mn>2</mml:mn> <mml:mo>,</mml:mo> <mml:mo>…</mml:mo> </mml:mrow> </mml:math> of scalarized charged black holes by taking into account the scalar seeds around the EEH black hole. We find that the $$n=0$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>n</mml:mi> <mml:mo>=</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> </mml:math> fundamental branch of all scalarized black holes is stable against the radial perturbations, while the $$n=1$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>n</mml:mi> <mml:mo>=</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:math> excited branch is unstable.