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We investigate the relationship between the hard X-ray photon index $\Gamma$and the Eddington ratio ($\xi=L_{X}(0.5-25 \rm keV)/L_{Edd}$) in six X-raybinaries (XRBs) with well constrained black hole masses and distances. We findthat different XRBs follow different anti-correlations between $\Gamma$ and$\xi$ when $\xi$ is less than a critical value, while $\Gamma$ and $\xi$generally follow the same positive correlation when $\xi$ is larger than thecritical value. The anti-correlation and the positive correlation may suggestthat they are in different accretion modes (e.g., radiatively inefficientaccretion flow (RIAF) and standard disk). We fit both correlations with thelinear least-square method for individual sources, from which the crosspoint oftwo fitted lines is obtained. Although the anti-correlation varies from sourceto source, the crosspoints of all sources roughly converge to the same pointwith small scatter($\log \xi=-2.1\pm0.2, \Gamma=1.5\pm 0.1$), which maycorrespond to the transition point between RIAF and standard accretion disk.Motivated by the observational evidence for the similarity of the X-rayspectral evolution of ultraluminous X-ray sources (ULXs) to that of XRBs, wethen constrain the black hole masses for seven ULXs assuming that their X-rayspectral evolution is similar to that of XRBs. We find that the BH masses ofthese seven luminous ULXs are around $10^{4}\msun$, which are typicalintermediate-mass BHs (IMBHs). Our results are generally consistent with the BHmasses constrained from the timing properties (e.g., break frequency) or themodel fitting with a multi-color disk.

The hard X-ray spectral evolution in X-ray binaries and its application to constrain the black hole mass of ultraluminous X-ray sources