Interferometric signatures of the temperature dependence of the specific shear viscosity in heavy-ion collisions

Recent work has shown that a temperature dependence of the shear-viscosity-to-entropy ratio $\eta /s$ influences the collective flow pattern in heavy-ion collisions in characteristic ways that can be measured by studying hadron transverse momentum spectra and their anisotropies. Here we point out that it also affects the pair momentum dependence of the Hanbury-Brown–Twiss (HBT) radii (the source size parameters extracted from two-particle intensity interferometry) and the variance of their event-by-event fluctuations. This observation establishes interferometric signatures as useful observables to complement the constraining power of single-particle spectra on the temperature dependence of $\eta /s$.

Figure 1

The different parametrizations of $(\eta /s)\left(T\right)$ used in this work, taken from Ref. [1].

Figure 2

${\overline{R}}_{ij}^2$ vs $K_T$ from single-shot hydrodynamics, for ideal and viscous fluids with the temperature dependent shear viscosities $(\eta /s)\left(T\right)$ shown in Fig. 1. Note the almost perfect overlap of the green (dashed) and blue (dotted) curves in (b), (c).

Figure 3

Same as Fig. 2, but leaving out in Eq. (4) the viscous correction $\delta f$ at freeze out. Comparison of Figs. 2, 2 and 3, 2 shows that the sensitivity of ${\overline{R}}_s^2\left(K_T\right)$ and ${\overline{R}}_{os,2}^2\left(K_T\right)$ to the temperature dependence of $(\eta /s)\left(T\right)$ below $T_{\mathrm{tr}}$ is almost entirely due to the theoretically poorly controlled viscous correction $\delta f$ at freeze out, and that any sensitivity to the temperature dependence of $(\eta /s)\left(T\right)$ during the dynamical evolution, i.e., before freeze out, is strongly weighted at high temperatures, i.e., early times. The sensitivity to $\delta f$ at freeze out of the HBT radii probing the outward and longitudinal dimensions of the emission function (b), (c) is negligible.

Figure 4

The ensemble averaged $\langle R_{ij}^2\rangle$ as a function of pair momentum $K_T$, for the same choices of $(\eta /s)\left(T\right)$ as in Figs. 2 and 3. This figure should be compared with Fig. 2.

Figure 5

Same as Fig. 4, but without the $\delta f$ correction. This figure should be compared with Fig. 3.

Figure 6

The fractional change $(\langle R_{ij}^2\rangle -{\overline{R}}_{ij}^2)/{\overline{R}}_{ij}^2$ of the squared HBT radii due to event-by-event fluctuations, as a function of pair momentum $K_T$. See text for discussion.

Figure 7

(a) The emission function $S(x,K)$, for three values of $K_T$ as indicated, $Y=0$ and integrated over ${\Phi }_K$ and ${\eta }_s$, for a single fluctuating event from the 0–10% centrality range, evolved with ${(\eta /s)}_{\mathrm{HH}}\text{−}\mathrm{LQ}$. Colors code the emission intensity, normalized to its maximum value on the freeze-out surface. (b) The emission function $S(x,K)$ for the smooth average single-shot hydrodynamic event corresponding to 0–10% centrality, for the same three values of $K_T,Y=0$ and integrated over ${\Phi }_K$ and ${\eta }_s$, using the same color code as in (a). The width of the bands arises from plotting the $(\tau ,r)$ freeze-out contours for all spatial angles ${\varphi }_s$ on top of each other. The left and right columns of panels show results for evolution with ${(\eta /s)}_{\mathrm{HH}}\text{−}\mathrm{LQ}$ and ${(\eta /s)}_{\mathrm{HH}}\text{−}\mathrm{HQ}$, respectively. See text for discussion.

Figure 8

For each cell on the freeze-out surface the radial velocity is plotted against its radial position, color coded as in Fig. 7 for its emission intensity normalized by its maximal value on the surface. (a), (d), (g) in the left column are for the single fluctuating event, evolved with ${(\eta /s)}_{\mathrm{HH}}\text{−}\mathrm{LQ}$, shown in Fig. 7, for the same three $K_T$ values as indicated. (b), (e), (h) in the middle column and (c), (f), (i) in the right column are for the ensemble-averaged event shown in Fig. 7, evolved with single-shot hydrodynamics using ${(\eta /s)}_{\mathrm{HH}}\text{−}\mathrm{LQ}$ (middle column) and ${(\eta /s)}_{\mathrm{HH}}-\mathrm{HQ}$ (right column), respectively. See text for discussion.

Figure 9

The normalized widths of the event-by-event distributions of HBT radii as functions of pair momentum $K_T$, for ideal fluid evolution and viscous dynamics with the four parametrizations shown in Fig. 1.

Figure 10

Same as Fig. 9, but without the viscous correction $\delta f$ to the freeze-out distribution function. One sees that for the normalized width of the distribution of sideward radii $R_s^2$, deleting $\delta f$ removes all sensitivity to the temperature dependence of $\eta /s$ in the hadronic phase, while all other normalized widths are only weakly affected by $\delta f$.