SHOTGLAS II. MUSE spectroscopy of blue horizontal branch stars in the core of $\omega$ Centauri and NGC 6752

In this work, we characterize the properties of HB stars in the GCs $\omega$ Cen and NGC 6752. We use dedicated model atmospheres and synthetic spectra grids computed using a hybrid LTE/NLTE modeling approach to fit the MUSE spectra of HB stars hotter than 8000 K in both clusters. The spectral fits provide estimates of the effective temperature, surface gravity, and helium abundance of the stars. The model grids are further used to fit the HST magnitudes, meaning the spectral energy distributions (SED), of the stars. From the SED fits, we derive the average reddening, radius, luminosity, and mass of the stars in our sample. The atmospheric and stellar properties that we derive for the stars in our sample are in good agreement with the theoretical expectations. In particular, the stars cooler than $\sim$15 000 K follow neatly the theoretical predictions on the radius, log $g$, and luminosity for helium-normal models. In $\omega$ Cen, we show that the majority of these cooler HB stars cannot originate from a helium-enriched population with $Y>$0.35. The properties of the hotter stars (radii and luminosities) are still in reasonable agreement with theoretical expectations, but the individual measurements have a large scatter. We use three different diagnostics, namely the position of the G-jump and changes in metallicity and helium abundances to place the onset of diffusion in the stellar atmospheres at Teff between 11 and 11.5 kK. Our sample includes two stars known as photometric variables, we confirm one to be a bona fide extreme HB object but the other is a blue straggler star. Finally, unlike what has been reported in the literature, we do not find significant differences between the properties of the stars in both clusters. We showed that our analysis method combining MUSE spectra and HST photometry of HB stars in GC is a powerful tool to characterize their stellar properties.