2025MNRAS.539.2185M

This study examines the flux and photon index distributions of 11 Very High Energy (VHE) Flat Spectrum Radio Quasars (FSRQs) using over 16 yr of Fermi-LAT $\gamma$-ray data. The distributions reveal double lognormal profiles in both flux and index, primarily in the 3-d and 7-d binnings, supporting the 'two-flux-state hypothesis' for blazars. These profiles, which become insignificant at 30-d binning, suggest that shorter time-scales are better at capturing distinct states, while longer time-scales smooth out shorter variations. Most VHE FSRQs exhibit a 'harder-when-brighter' trend, where the photon index decreases during high-flux states, suggesting efficient particle acceleration and possibly reduced radiative cooling. In contrast, two sources display a 'softer-when-brighter' behaviour, likely due to enhanced radiative cooling in high photon density environments. Additionally, we observe that the Spearman rank correlation between flux and photon index strengthens with increasing time bin sizes, indicating more pronounced correlations over longer time-scales. This possibly indicates that, on shorter time-scales, flux variations are driven by a combination of photon index changes and normalization effects. Averaging flux over longer durations minimizes the effect of normalization variation, thereby enhancing the observed correlation. We also compare the flux and index distributions of VHE and non-VHE FSRQs, emphasizing the differences in their variability and emission patterns.