Researchers have studied static, spherically symmetric axion-dilaton solutions within the tree-level four-dimensional string effective action. Utilizing the sigma-model structure of the scalar sector, they showed that the full family of static, spherically symmetric, and asymptotically flat solutions is obtained as the $SL(2,\mathbb R)$ orbit of the pure-dilaton FJNW solution. This analysis allowed for the characterization of the associated axion-dilaton charge space.

The study also addressed the perturbative regime of these solutions by comparing the onset of $α'$ curvature corrections with that of string-loop corrections. A key aspect of the research was the application of the string-black-hole correspondence criterion to the FJNW family, evaluating the local $α'$ curvature diagnostic at the physical size $R_{\rm typ}$ of a highly excited string state. With the normalization fixed on the Schwarzschild branch, it was found that Schwarzschild saturates the nominal $α'$ threshold at the correspondence surface, whereas every scalar-haired FJNW branch lies above it.

More generally, and independently of the common overall normalization within the matching prescription, every nonzero-hair branch was observed to have a larger curvature diagnostic at the correspondence surface than Schwarzschild. Away from the correspondence point, scalar-haired exteriors may remain under $α'$ control at sufficiently weak local self-gravity and may provide perturbatively controlled long-distance fields of highly excited string states whenever the local string coupling also remains weak.