Abstract

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, utilising high-quality Gemini-South/GSAOI and HST /ACS photometries. Additionally, we intend to investigate its role in the formation of the Galactic bulge. We performed a meticulous differential reddening correction to investigate the effect of contamination from Galactic bulge field stars. To derive the fundamental parameters — age, distance, reddening, and the total-to-selective coefficient — we employed a Bayesian isochrone fitting. The results from high-resolution spectroscopy and RR Lyrae stars were implemented as priors. For the orbital parameters, we employed a barred Galactic mass model. Furthermore, we analysed the age-metallicity relation to contextualise NGC 6558 within the Galactic bulge's history. We studied the impact of two differential reddening corrections on the age derivation. When removing as much as possible of the Galactic bulge field star contamination, the isochrone fitting combined with synthetic colour-magnitude diagrams gives a distance of $8.41^ $ kpc, an age of $13.0 0.9$ Gyr, and a reddening of E($B-V$)$\,\,=0.34 We derived a total-to-selective coefficient of R$_V = 3.2 thanks to the simultaneous near-infrared$-$optical synthetic colour-magnitude diagram fitting, which, aside from errors, agrees with the commonly used value. The orbital parameters showed that NGC 6558 is confined within the inner Galaxy and it is not compatible with a bar-shape orbit, indicating that it is a bulge member. Assembling the old and moderately metal-poor Fe/H clusters in the Galactic bulge, we derived their age-metallicity relation with star formation starting at $13.6 Gyr and effective yields of $ Z_ odot$. The old age derived for NGC 6558 is compatible with other clusters with similar metallicity and a blue horizontal branch in the Galactic bulge, which constitute the moderately metal-poor globular clusters. The age-metallicity relation shows that the starting age of star formation is compatible with the age of NGC 6558, and the chemical enrichment is ten times faster than the ex situ globular cluster branch.