Anisotropy scaling functions in heavy-ion collisions: Insights into the ultracentral flow puzzle and constraints on transport coefficients and nuclear deformation

Anisotropy scaling functions derived from comprehensive measurements of transverse momentum- and centrality-dependent anisotropy coefficients $v_2(p_T,\text{cent})$ and $v_3(p_T,\text{cent})$ in $\mathrm{Pb}+\mathrm{Pb}$ collisions at 5.02 and 2.76 TeV, $\mathrm{Xe}+\mathrm{Xe}$ collisions at 5.44 TeV, and $\mathrm{Au}+\mathrm{Au}$ collisions at 0.2 TeV offer new insights into the ultracentral flow puzzle. These functions integrate diverse measurements into a single curve, clarifying anisotropy attenuation throughout the entire $p_T$ and centrality range. They reveal the influence of initial-state eccentricities (${\varepsilon }_n$), dimensionless size ($\mathbb{R}$), radial flow, viscous correction to the thermal distribution function (${\delta }_f$), the medium's stopping power ($\widehat{q}$), and specific shear viscosity ($\eta /s$) on the observed anisotropies. This analysis not only enhances understanding of transport coefficients but also provides crucial constraints on nuclear deformation.