Probing the path-length dependence of parton energy loss via scaling properties in heavy ion collisions

The scaling property of large-$p_\perp$ hadron suppression, $R_{\rm{AA}}(p_\perp)$, measured in heavy ion collisions at RHIC and LHC leads to the determination of the average parton energy loss $\langle \epsilon \rangle$ in quark-gluon plasma produced in a variety of collision systems and centrality classes. Relating $\langle \epsilon \rangle$ to the particle multiplicity and collision geometry allows for probing the dependence of parton energy loss on the path-length $L$. We find that $\langle \epsilon \rangle \propto L^\beta$ with $\beta=1.02^{+0.09}_{-0.06}$, consistent with the pQCD expectation of parton energy loss in a longitudinally expanding quark-gluon plasma. We then demonstrate that the azimuthal anisotropy coefficient divided by the collision eccentricity, $v_2/\textrm{e}$, follows the same scaling property as $R_{\rm{AA}}$. This scaling is observed in data, which are reproduced by the model at large $p_\perp$. Finally, a linear relationship between $v_2/\textrm{e}$ and the logarithmic derivative of $R_{\rm{AA}}$ is found and confirmed in data, offering an additional way to probe the $L$ dependence of parton energy loss using coming measurements from LHC Run 3.