Dyonic Einstein–Maxwell-scalar black holes: the cold, the hot and the plunge

Abstract We investigate dyonic nonlinearly scalarized black holes in Einstein–Maxwell-scalar theory. The domain of existence of scalarized dyonic black holes consists of three branches. The cold branch and the hot branch bifurcate at a minimal value of the charge, analogous to the purely electrically charged scalarized black holes. However, the presence of both charges allows for regular extremal black holes, leading to a third branch featuring a sudden plunge in Hawking temperature. This demonstrates that the dyonic sector exhibits a qualitatively different domain of existence from the purely electric nonlinear scalarization case. This behavior can be traced to the fact that the presence of both electromagnetic charges introduces a factor $$\Delta (\phi )$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>Δ</mml:mi> <mml:mo>(</mml:mo> <mml:mi>ϕ</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> in the source term of scalar field equations that vanishes when the coupling function $$f(\phi )$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>f</mml:mi> <mml:mo>(</mml:mo> <mml:mi>ϕ</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> equals the ratio of the charges for some value of the scalar field $$\phi _c$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>ϕ</mml:mi> <mml:mi>c</mml:mi> </mml:msub> </mml:math> . The scalar field of extremal black holes assumes precisely this value at the horizon, $$\phi _H=\phi _c$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>ϕ</mml:mi> <mml:mi>H</mml:mi> </mml:msub> <mml:mo>=</mml:mo> <mml:msub> <mml:mi>ϕ</mml:mi> <mml:mi>c</mml:mi> </mml:msub> </mml:mrow> </mml:math> . We demonstrate the plunge for the coupling function $$f(\phi )=\exp (\alpha \phi ^3)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>f</mml:mi> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>ϕ</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo>=</mml:mo> <mml:mo>exp</mml:mo> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>α</mml:mi> <mml:msup> <mml:mi>ϕ</mml:mi> <mml:mn>3</mml:mn> </mml:msup> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> which provides a minimal representative of nonlinear scalarization models.