Convergence properties of $T'$-Expansion Scheme: Hadron Resonance Gas and Cluster Expansion Model

In this study, we assess the effectiveness and robustness of the recently proposed $T'$-expansion scheme for expanding the equation of state of strongly interacting matter to finite density, by comparing its performance relative to the conventional Taylor expansion method in various effective QCD models. We use baryon number density and its susceptibilities to calculate the expansion coefficients in the $T'$-expansion scheme with and without the Stefan-Boltzmann limit correction. Our methodology involves comparing truncation orders to exact solutions to assess the scheme's accuracy. We utilize Ideal, Excluded Volume, and van der Waals formulations of the Hadron Resonance Gas (HRG) model at low temperatures, and the Cluster Expansion Model at higher temperatures. Our findings indicate that the $T'$-expansion scheme offers superior convergence properties near and above the chiral crossover temperature, where the chiral-criticality-inspired scaling $(\partial/ \partial T)_{\mu_B} \sim (\partial^2/\partial \mu_B^2)_T$ holds. However, it shows limited improvement in the HRG models, indicating that it may not be the most suitable choice for describing the hadronic phase.