Rapidity decorrelation of anisotropic flow caused by hydrodynamic fluctuations

We investigate the effect of hydrodynamic fluctuations on the rapidity decorrelations of anisotropic flow in high-energy nuclear collisions using a $(3+1)$-dimensional integrated dynamical model. The integrated dynamical model consists of twisted initial conditions, fluctuating hydrodynamics, and hadronic cascades on an event-by-event basis. We consider several choices of the cutoff which is the free parameter of fluctuating hydrodynamics within the current modeling. To understand the rapidity decorrelation, we analyze the factorization ratio in the longitudinal direction. Comparing the factorization ratios between fluctuating hydrodynamics and ordinary viscous hydrodynamics, we find a sizable effect of hydrodynamic fluctuations on rapidity decorrelations. We qualitatively describe the experimental data of the factorization ratios. We also propose to calculate the Legendre coefficients of the flow magnitude and the event-plane angle to understand the decorrelation of anisotropic flow in the longitudinal direction.

Figure 1

Factorization ratio of the initial eccentricities, (a) $r_2^{\varepsilon }({\eta }_s^a,{\eta }_s^b)$ and (b) $r_3^{\varepsilon }({\eta }_s^a,{\eta }_s^b)$ in $\mathrm{Pb}+\mathrm{Pb}$ collisions at $\sqrt{s_{NN}}=2.76$ TeV. The parameters of initial conditions are $C/{\tau }_0=41{\mathrm{fm}}^{-1}$, $\alpha =0.16$, $\mathrm{\Delta }\eta =3.2$, and ${\sigma }_{\eta }=1.9$. The space-time rapidity for reference is ${\eta }_s^b=4.5$. The results are shown for centrality 0–5% (filled square), 5–10% (filled circle), 10–20% (open square), 20–30% (filled triangle), 30–40% (open circle), and 50–60% (open diamond).

Figure 2

Charged-hadron multiplicity normalized by the number of the participant pair, $(d{N}_{\mathrm{ch}}/d{\eta }_p)/(N_{\mathrm{part}}/2)$, as a function of the number of participants. The results from ideal hydrodynamics (open square), viscous hydrodynamics (open circle), fluctuating hydrodynamics–$\lambda 1.0$ (filled square), fluctuating hydrodynamics–$\lambda 1.5$ (filled circle), and fluctuating hydrodynamics–$\lambda 2.0$ (filled triangle) are compared with experimental data (open diamond) obtained by the ALICE Collaboration [81].

Figure 3

Pseudorapidity distributions of charged hadrons in $\mathrm{Pb}+\mathrm{Pb}$ collisions at $\sqrt{s_{NN}}=2.76$ TeV. Centralities are (a) 0–5%, (b) 5–10%, (c) 10–20%, (d) 20–30%, (e) 30–40%, (f) 40–50%, (g) 50–60%, and (h) 60–70%. The symbols are the same as in Fig. 2. The experimental data are taken from Refs. [82, 83].

Figure 4

$v_2\left\{2\right\}\left(p_T\right)$ in $\mathrm{Pb}+\mathrm{Pb}$ collisions at $\sqrt{s_{NN}}=2.76\text{TeV}$ for centrality 0–5%, 30–40%, and 40–50%. The symbols are the same as in Fig. 2 The experimental data are taken from Refs. [20, 21].

Figure 5

Factorization ratio in the longitudinal direction, $r_2({\eta }_p^a,{\eta }_p^b)$, in $\mathrm{Pb}+\mathrm{Pb}$ collisions at $\sqrt{s_{NN}}=2.76$ TeV for (a) 0–5% and (b) 20–30% centralities. The rapidity region for reference is $3.0<{\eta }_p^b<4.0$. The results from ideal hydrodynamics (open square), viscous hydrodynamics (open circle), fluctuating hydrodynamics–$\lambda 1.0$ (filled square) and fluctuating hydrodynamics–$\lambda 1.5$ (filled circle) are compared with experimental data (open diamond) obtained by the CMS Collaboration [50].

Figure 6

Centrality dependence of factorization ratio $r_2({\eta }_p^a,{\eta }_p^b)$ in $\mathrm{Pb}+\mathrm{Pb}$ collisions at $\sqrt{s_{NN}}=2.76$ TeV. The rapidity regions of two-particle correlation functions are taken as $2.0<{\eta }_p^a<2.5$ and $3.0<{\eta }_p^b<4.0$. The symbols are the same as in Fig. 5. The experimental data are taken from Ref. [50].

Figure 7

Factorization ratio $r_3({\eta }_p^a,{\eta }_p^b)$ in $\mathrm{Pb}+\mathrm{Pb}$ collisions at $\sqrt{s_{NN}}=2.76\text{TeV}$ for (a) 0–5% and (b) 20–30% centralities. The symbols are the same as in Fig. 5. The experimental data are taken from Ref. [50].

Figure 8

Centrality dependence of factorization ratio $r_3({\eta }_p^a,{\eta }_p^b)$ in $\mathrm{Pb}+\mathrm{Pb}$ collisions at $\sqrt{s_{NN}}=2.76$ TeV. The rapidity regions of two-particle correlation functions are taken as $2.0<{\eta }_p^a<2.5$ and $3.0<{\eta }_p^b<4.0$. The meaningful result for 50–60% centrality from viscous hydrodynamic model was not obtained due the insufficient statistics. The symbols are the same as in Fig. 5. The experimental data are taken from Ref. [50].

Figure 9

Centrality dependence of the first-order Legendre coefficients (a) $A_2^1$ and (b) $B_2^1$ for elliptic flow ($n=2$). The results from the viscous hydrodynamic model (open circle) and the fluctuating hydrodynamic model (filled circle) are shown for comparison.

Figure 10

Centrality dependence of the second-order Legendre coefficients (a) $A_2^2$ and (b) $B_2^2$ for elliptic flow ($n=2$). The symbols are the same as in Fig. 9.

Figure 11

Cutoff parameter dependence of the Legendre coefficients. The first and the second-order Legendre coefficients for the magnitude and the event-plane angle of elliptic flow parameters, (a) $A_2^1$, (b) $A_2^2$, (c) $B_2^1$, and (d) $B_2^2$, from viscous hydrodynamics and fluctuating hydrodynamics–$\lambda 1.0$, $\lambda 1.5$, $\lambda 2.0$, and $\lambda 2.5$ are shown. The results from viscous hydrodynamics are plotted at $1/\lambda =0$ (or $\lambda =\infty$). The results are shown for centrality 0–10% (open square), 20–30% (filled square), and 50–60% (open diamond).