Energy density and path-length dependence of the fractional momentum loss in heavy-ion collisions at $\sqrt{s_{\mathrm{N}\mathrm{N}}}$ from 62.4 to 5020 GeV

In this work a study of the fractional momentum loss ($S_{\mathrm{l}\mathrm{o}\mathrm{s}\mathrm{s}}$) as a function of the characteristic path-length ($L$) and the Bjorken energy density times the equilibration time (${ϵ}_{\mathrm{B}\mathrm{j}}{\tau }_0$) for heavy-ion collisions at different $\sqrt{s_{\mathrm{N}\mathrm{N}}}$ is presented. The study has been conducted using inclusive charged particles from intermediate to large transverse momentum ($5<p_{\mathrm{T}}<20$ GeV/$c$). Within uncertainties and for all the transverse momentum values which were explored, the fractional momentum loss linearly increases with $({ϵ}_{\mathrm{B}\mathrm{j}}{\tau }_0{)}^{3\mathrm{/}8}$$L$. The functional form of $S_{\mathrm{l}\mathrm{o}\mathrm{s}\mathrm{s}}$ vs. $({ϵ}_{\mathrm{B}\mathrm{j}}{\tau }_0{)}^{3\mathrm{/}8}$$L$ seems to be universal. Moreover, for identified charged hadrons a linear relationship between $S_{\mathrm{l}\mathrm{o}\mathrm{s}\mathrm{s}}$ and $L$ is also observed. The behaviour of data could provide important information aimed to understand the parton energy loss mechanism in heavy-ion collisions and some insight into the expected effect for small systems.