System-size dependence of the viscous attenuation of anisotropic flow in $p$ + Pb and Pb + Pb collisions at energies available at the CERN Large Hadron Collider

The elliptic and triangular flow coefficients ($v_n,n=2,3$) measured in Pb + Pb ($\sqrt{s_{{}_{\mathrm{NN}}}}=2.76\mathrm{TeV}$) and $p+\mathrm{Pb}(\sqrt{s_{{}_{\mathrm{NN}}}}=5.02\mathrm{TeV}$) collisions are studied as a function of initial-state eccentricity (${\varepsilon }_n$) and dimensionless size characterized by the cube root of the midrapidity charged hadron multiplicity density ${\langle N_{\mathrm{ch}}\rangle }^{1/3}$. The results indicate that the influence of eccentricity ($v_n\propto {\varepsilon }_n$) observed for large $\langle N_{\mathrm{ch}}\rangle$ is superseded by the effects of viscous attenuation for small $\langle N_{\mathrm{ch}}\rangle$, irrespective of the colliding species. Strikingly similar acoustic scaling patterns of exponential viscous modulation with a damping rate proportional to $n^2$ and inversely proportional to the dimensionless size are observed for the eccentricity-scaled coefficients for the two sets of colliding species. The resulting scaling parameters suggest that, contrary to current predilections, the patterns of viscous attenuation as well as the specific shear viscosity $〈\frac{\eta }{s}\left(T\right)〉$ for the matter created in $p+\mathrm{Pb}$ and Pb + Pb collisions are comparable.

Figure 1

(a) $v_n$ and ${\varepsilon }_n$ vs $\langle N_{\mathrm{ch}}\rangle$ for Pb + Pb collisions at $\sqrt{s_{{}_{\mathrm{NN}}}}=2.76$. The filled and open symbols indicate centrality selected (I) and $N_{\mathrm{ch}}$-selected (II) measurements, respectively. The dotted and dashed curves show ${\varepsilon }_n$ values which are normalized to $v_n$ at $\langle N_{\mathrm{ch}}\rangle \sim 1600$; (b) $v_n/{\varepsilon }_n$ vs ${\langle N_{\mathrm{ch}}\rangle }^{-1/3}$ for the data shown in panel (a). The dashed lines represent fits to the eccentricity-scaled data following Eq. (3); (c) $[(v_3/{\varepsilon }_3)/(v_2/{\varepsilon }_2)]$ vs ${\langle N_{\mathrm{ch}}\rangle }^{-1/3}$ for the data shown in panel (b). The dashed line represents a fit to the data following Eq. (3).

Figure 2

(a) $v_n$ and ${\varepsilon }_n$ vs $\langle N_{\mathrm{ch}}\rangle$ for $N_{\mathrm{ch}}$-selected (II) measurements of $v_n$ in $p+\mathrm{Pb}$ collisions at $\sqrt{s_{{}_{\mathrm{NN}}}}=5.02\mathrm{TeV}$. The dotted and dashed curves show ${\varepsilon }_n$ values which are normalized to $v_n$ at $\langle N_{\mathrm{ch}}\rangle \sim 100$; (b) $v_n/{\varepsilon }_n$ vs ${\langle N_{\mathrm{ch}}\rangle }^{-1/3}$ for the data shown in panel (a). The dashed lines represent fits to the eccentricity-scaled data following Eq. (3); (c) $ln[(v_3/{\varepsilon }_3)/(v_2/{\varepsilon }_2)]$ vs ${\langle N_{\mathrm{ch}}\rangle }^{-1/3}$ for the data shown in panel (b). The dashed line represents a fit to the data following Eq. (3).

Figure 3

Comparison of $ln[(v_3/{\varepsilon }_3)/(v_2/{\varepsilon }_2)]$ vs ${\langle N_{\mathrm{ch}}\rangle }^{-1/3}$ for centrality selected (I) and $N_{\mathrm{ch}}$-selected (II) Pb + Pb measurements and $N_{\mathrm{ch}}$-selected (II) $p+\mathrm{Pb}$ measurements. The dashed line represents a fit to the combined data sets, following Eq. (3).