Long-term evolution of BH-ULX candidates: An `unusual' $L_{\mathrm{d}\mathrm{i}\mathrm{s}\mathrm{c}}$-$T_{\mathrm{c}\mathrm{o}\mathrm{l}}$ correlation associated with spectral states

We present the long-term spectral evolution of eight black hole ultra-luminous X-ray sources (BH-ULXs), namely NGC1313 X-1, NGC5408 X-1, NGC6946 X-1, IC342 X-1, NGC55 ULX1, NGC4395 ULX1, NGC5204 X-1 and NGC4190 ULX1 using {\it XMM-Newton} monitoring data spanning over a decade or more. An in-depth spectral modeling with thermal Comptonization ({\it nthComp}) and standard disc ({\it diskbb}) components reveals NGC5204 X-1, IC342 X-1, NGC4190 ULX1 and NGC1313 X-1 exhibiting harder spectral characteristics with dominant effect of Comptonization ($F_{nth}>F_{disc}$, ${\mathrm{\Gamma }}_{nth}\lesssim 2$). However, NGC6946 X-1 and NGC55 ULX1 remain in a disc-dominated state ($F_{disc}\sim 2F_{nth}$, ${\mathrm{\Gamma }}_{nth}\gtrsim 2$), while NGC5408 X-1 shows intermediate spectral characteristics. The spectral analyses indicate an anti-correlation between disc luminosity ($L_{disc}$) and temperature ($T_{col}$) for all sources except NGC5204 X-1. These anti-correlations follow a relation $L_{disc}\propto T_{col}^{\alpha }$ with steeper exponents of $\alpha =-6.01\pm 0.25$ (NGC55 ULX1), $-8.93\pm 0.11$ (NGC6946 X-1), and $-10.31\pm 0.10$ (NGC5408 X-1) for sources with softer or intermediate spectral characteristics. For harder sources, NGC1313 X-1 and IC342 X-1, the combined results provide $\alpha =-3.58\pm 0.04$. However, for NGC5204 X-1, a positive correlation is found, yielding $\alpha =1.4\pm 0.1$, suggesting that the emission mechanism is associated with the transition from the standard disc' to the slim disc’ scenario. These findings suggest that the observed $L_{disc}-T_{col}$ correlations, along with the overall spectro-temporal properties of BH-ULXs, seems to be governed by disc-corona-wind driven accretion processes at various inclinations. Finally, we report a QPO-like feature ($\sim 20$ mHz) with $rms\mathrm{\%}\sim 6.6$, Q-factor $\sim 6.7$ and significant $2.8\sigma$ in NGC55 ULX1.