Hayward boson stars exist only when sqrt(beta)/Q > 1.49661, exactly where the vacuum Hayward spacetime has no horizon.
Analytical Study of Charged Boson Stars with Large Scalar Self-couplings
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We give good approximate analytic solutions for spherical charged boson stars in the large scalar-self-coupling limit in general relativity. We show that if the charge $e$ and mass $m$ of the scalar field nearly satisfy the critical relation $e^2\approx Gm^2$ (where $G$ is the Newton constant), our analytic expressions for stable solutions agree well with the numerical solutions.
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Hayward Boson Stars
Hayward boson stars exist only when sqrt(beta)/Q > 1.49661, exactly where the vacuum Hayward spacetime has no horizon.