Combined Gemini-South and HST photometric analysis of the globular cluster NGC 6558. The age of the metal-poor population of the Galactic Bulge

NGC~6558 is a low-galactic latitude globular cluster projected in the direction of the Galactic bulge. Due to high reddening, this region presents challenges in deriving accurate parameters, which require meticulous photometric analysis. We present a combined analysis of near-infrared and optical photometry from multi-epoch high-resolution images collected with Gemini-South/GSAOI+GeMS (in the $J$ and $K_S$ filters) and HST/ACS (in the F606W and F814W filters). We aim to refine the fundamental parameters of NGC~6558, utilizing high-quality Gemini-South/GSAOI and HST/ACS photometries. Additionally, we intend to investigate its role in the formation of the Galactic bulge. We studied the impact of two differential reddening corrections on the age derivation. When removing as much as possible the Galactic bulge field star contamination, the isochrone fitting combined with synthetic colour-magnitude diagrams gives a distance of $8.41^{+0.11}_{-0.10}$ kpc, an age of $13.0\pm 0.9$ Gyr, a reddening of E($B-V$)$\,\,=0.34\pm0.02$, and a total-to-selective coefficient R$_V = 3.2\pm0.2$ thanks to the simultaneous near-infrared$-$Optical synthetic colour-magnitude diagram fitting. The orbital parameters showed that NGC~6558 is confined whitin the inner Galaxy and it is not compatible with a bar-shape orbit, indicating that it is a bulge member. The old age of NGC~6558, combined with similar metallicity and a blue horizontal branch in the Galactic bulge, indicates that it is part of the moderately metal-poor globular clusters. Assembling the old and moderately metal-poor ([Fe/H]$\,\,\sim-1.1$) clusters in the Galactic bulge, we derived their age-metallicity relation with star formation stars at $13.6\pm0.2$ Gyr and effective yields of $\rho=0.007\pm0.009\,\, Z_\odot$ showing a chemical enrichment ten times faster than the ex-situ globular clusters branch.