Universal geometrical scaling of the elliptic flow

The presence of scaling variables in experimental observables provides very valuable indications of the dynamics underlying a given physical process. In the last years, the search for geometric scaling, that is the presence of a scaling variable which encodes all geometrical information of the collision as well as other external quantities as the total energy, has been very active. This is motivated, in part, for being one of the genuine predictions of the color glass condensate formalism for saturation of partonic densities. Here we extend these previous findings to the case of experimental data on elliptic flow. We find an excellent scaling for all centralities and energies, from the BNL Relativistic Heavy Ion Collider to the CERN Large Hadron Collider, with a simple generalization of the scaling previously found for other observables and systems. Interestingly the case of the photons, difficult to reconcile in most formalisms, nicely fits the scaling curve. We discuss the possible interpretations of this finding in terms of initial or final state effects.

Figure 1

(a) $v_2$ divided by the product ${\epsilon }_1{Q}_s^{A}L$ for 10–20%, 20–30%, 30–40%, and 40–50% Au-Au collisions at 200 GeV [30], for 10–20%, 20–30%, 30–40%, and 40–50% Pb-Pb collisions at 2.76 GeV [3] in terms of $\tau$. The solid black line is a fit to data according to Eq. (9). (b) Ratio of Pb-Pb 10–20%, Pb-Pb 40–50% at 2.76 TeV [3], Au-Au 20–30% and Au-Au 30–40% at 200 GeV [30] over Pb-Pb 30–40% at 2.76 TeV [3] versus $\tau$.

Figure 2

(a) $v_2$ of $\pi$, $k$, and $p$ divided by the product ${\epsilon }_1{Q}_s^{A}L$ for 10–20%, 20–30%, 30–40%, and 40–50% Au-Au collisions at 200 GeV [41], for 10–20%, 20–30%, 30–40%, and 40–50% Pb-Pb collisions at 2.76 GeV [42] versus $\tau$, for $\tau <1$. The solid line is a fit to Eq. (10). (b) Ratio of the experimental points over the fitting function Eq. (10) versus $\tau$, for $0.1<\tau <1$.

Figure 3

$v_2$ prediction for $pp$ collisions at $\sqrt{s}=14$ TeV for impact parameters values of $b=0.5$ fm (solid black curve) and $b=0.7$ fm (dashed red curve) as a function of $p_T$.

Figure 4

$v_2$ divided by the product ${\epsilon }_1{Q}_s^{A}L$ for direct photons at 0–20% and 20–40% Au-Au collisions at 200 GeV [47] and direct photons at 0–40% Pb-Pb collisions at 2.76 GeV [48] plotted as a function of $\tau$. The solid black line is the scaling curve of Fig. 1. We have included all the data with $p_T<Q_s$.