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

In this work a study of the fractional momentum loss ($S_{rm loss}$) as a function of the characteristic path-length ($L$) and the Bjorken energy density times the equilibration time ($epsilon_{rm Bj}tau_{0}$) for heavy-ion collisions at different $sqrt{s_{rm NN}}$ is presented. The study has been conducted using inclusive charged particles from intermediate to large transverse momentum ($5<p_{rm T}<20$ GeV/$c$). Within uncertainties and for all the transverse momentum values which were explored, the fractional momentum loss linearly increases with $({epsilon_{rm Bj}tau_{0}})^{3/8}$$L$. The functional form of $S_{rm loss}$ vs. $({epsilon_{rm Bj}tau_{0}})^{3/8}$$L$ seems to be universal. Moreover, for identified charged hadrons a linear relationship between $S_{rm loss}$ 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.