Extension to a theory with scalar-coupled Euler–Heisenberg terms

Determine whether the same scalarized-extremal-black-hole picture holds for the theory with both scalar coupling functions set to the quartic polynomial form g(φ)=f(φ)=1−αφ²+βφ⁴, including the behavior and connection of its non-extremal and extremal scalarized solutions.

Background

The paper analyzes the generic Einstein–Euler–Heisenberg–scalar theory with a scalar coupling to the Maxwell term while taking the Euler–Heisenberg coupling function f(φ)=1. It studies exponential and quartic-polynomial choices for the Maxwell coupling, constructs scalarized non-extremal and extremal black holes, and characterizes their accumulation near extremality, constant and nonconstant scalar hair, and the relation between the solution families.

The authors leave unresolved whether the same qualitative picture persists when the scalar also couples to the Euler–Heisenberg nonlinear-electrodynamics terms through the same quartic-polynomial function, namely when g(φ)=f(φ)=1−αφ²+βφ⁴. This would require determining whether the conclusions about extremal scalarization, secondary hair, branch structure, and the meeting of scalarized solution domains remain valid in that extended theory.