Acoustic scaling of linear and mode-coupled anisotropic flow; implications for precision extraction of the specific shear viscosity

The $n\mathrm{th}$-order linear flow coefficients $v_n^L(n=2,3,4,5)$, and the corresponding nonlinear mode-coupled ($\mathrm{mc}$) coefficients $v_{4,(2,2)}^{\mathrm{mc}}$, $v_{5,(2,3)}^{\mathrm{mc}}$, $v_{6,(3,3)}^{\mathrm{mc}}$, and $v_{6,(2,2,2)}^{\mathrm{mc}}$, are studied for Pb+Pb collisions at $\sqrt{s_{NN}}=2.76$ TeV. Both sets of coefficients indicate a common acoustic scaling pattern of exponential viscous modulation, with a rate proportional to the square of the harmonic numbers and the mean transverse momenta (respectively), and inversely proportional to the cube root of the charged particle multiplicity $\left[{\left(N_{\mathrm{ch}}\right)}^{1/3}\right]$, which characterizes the dimensionless size of the systems produced in the collisions. These patterns and their associated scaling parameters provide new stringent constraints for eccentricity-independent estimates of the specific shear viscosity $\frac{\eta }{s}\left(T\right)$ and the viscous correction to the thermal distribution function for the matter produced in the collisions. They also give crucial constraints for extraction of the initial-state eccentricity spectrum.

Figure 1

Comparison of $(v_n^L/{\varepsilon }_n^{\prime })$ vs ${\left(N_{\mathrm{ch}}\right)}^{-1/3}$ for the linear harmonics (left panel), and $v_{n,(i,j)}^{\mathrm{mc}}/{\varepsilon }_{n,(i,j)}^{\mathrm{mc}}$ and $v_{n,(i,i,j)}^{\mathrm{mc}}/{\varepsilon }_{n,(i,i,j)}^{\mathrm{mc}}$ vs ${\left(N_{\mathrm{ch}}\right)}^{-1/3}$ (respectively) for the nonlinear mode-coupled harmonics for Pb+Pb collisions at at $\sqrt{s_{NN}}=2.76\mathrm{TeV}$. The lines represent a simultaneous exponential fit to the data, following Eqs. (11) and (12). The ALICE data are taken from Refs. [37, 38].

Figure 2

Same as Fig. 1 but for ATLAS data [22]; the $\beta$ prefactors $n^2$ and $(i^2+j^2)$ are indicated in the figure.

Figure 3

(a) $v_3$ vs $N_{\mathrm{ch}}$ for several $〈p_T〉$ selections as indicated; (b) $v_3/{\varepsilon }_3$ vs ${\left(N_{\mathrm{ch}}\right)}^{-1/3}$ for the data shown in (a). The dashed lines represent an exponential fit to the data for the selections $〈p_T〉=$ 0.7 and 3.5 GeV/c respectively. (c) $n({\beta }_n-{\beta }^0)$ vs ${〈p_T〉}^2$ (see text); the slopes $n{\beta }_n$, are obtained from fits to the eccentricity-scaled data, similar to that shown in (b). The ATLAS data used in the plots are taken from Ref. [22].

Figure 4

(a) ${\beta }^0$ vs ${〈p_T〉}^2$; the ${\beta }^0$ values are extracted from plots of $ln[(v_3/{\varepsilon }_3)/(v_2/{\varepsilon }_2)]$ vs ${\left(N_{\mathrm{ch}}\right)}^{-1/3}$, for several $〈p_T〉$ selections (see text). (b) ${(v_2/{\varepsilon }_2)}^{1/2}/{(v_3/{\varepsilon }_3)}^{1/3}$ vs ${〈p_T〉}^2$ for 15–20% central Pb+Pb collisions. The dashed lines in both panels are drawn to guide the eye. The ATLAS data used in the plots are taken from Ref. [22].