Constraining the high-density nuclear symmetry energy with the transverse-momentum-dependent elliptic flow

Within the newly updated version of the ultrarelativistic quantum molecular dynamics (UrQMD) model, the transverse-velocity dependence of the elliptic flow of free nucleons from ${}^{197}\mathrm{Au}$+${}^{197}\mathrm{Au}$ collisions at the incident energy 400 MeV$/$nucleon is studied within different windows of the normalized c.m. rapidity $y_0$. It is found that the elliptic flow difference $v_2^n-v_2^p$ and ratio $v_2^n$/$v_2^p$ of neutrons versus protons are sensitive to the density dependence of the symmetry energy, especially the ratio $v_2^n$/$v_2^p$ at small transverse velocity in the intermediate rapidity intervals $0.4<|y_0|<0.6$. By comparing either transverse-momentum-dependent or integrated FOPI/LAND elliptic flow data of nucleons and hydrogen isotopes with calculations using various Skyrme interactions, all exhibiting similar values of isoscalar incompressibility but very different density dependences of the symmetry energy, a moderately soft to linear symmetry energy is extracted, in good agreement with previous UrQMD or Tübingen QMD model calculations but contrast with results obtained with ${\pi }^{-}/{\pi }^{+}$ yield ratios in the literature.

Figure 1

Density dependence of the symmetry energy for Skyrme interactions Skz2, SV-mas08, MSL0, SV-sym34, Ska35s25, and SkI1 (lines). Symmetry energies used in previous UrQMD [23] (lines with solid symbols), in LQMD [22] (line with open squares), and in IBUU04 [21] (line with open triangles) calculations are also shown for comparison.

Figure 2

Elliptic flow $v_2$ of protons and neutrons in semicentral ($0.25<b_0<0.45$) ${}^{197}\mathrm{Au}$+${}^{197}\mathrm{Au}$ collisions at $E_{\mathrm{lab}}=400$ MeV/nucleon as a function of the normalized transverse component $u_{t0}$ of the four-velocity. Calculations with Skz4 and SkI1 (lines) for the two rapidity windows $|y_0|<0.4$ (a) and $0.4<|y_0|<0.6$ (b) are shown and, in the top panel, compared with the FOPI data for protons (stars) reported in Ref. [36].

Figure 3

Dependence of the elliptic flow difference [panels (a) and (b)] and ratio [ panels (c) and (d)] of free neutrons vs free protons produced in semicentral ${}^{197}\mathrm{Au}$+${}^{197}\mathrm{Au}$ collisions at $E_{\mathrm{lab}}=400$ MeV$/$nucleon as a function of the normalized transverse velocity $u_{t0}$ for two rapidity windows $|y_0|<0.4$ and $0.4<|y_0|<0.6$.

Figure 4

(a) Elliptic flow ratio of neutrons vs hydrogen isotopes ($Z=1$) as a function of the transverse momentum $p_t/A$, calculated with the indicated nine Skyrme forces for central ($b<7.5$ fm) ${}^{197}\mathrm{Au}$+${}^{197}\mathrm{Au}$ collisions at $E_{\mathrm{lab}}=400$ MeV/nucleon in the midrapidity interval $|y_0|\le 0.5$ in comparison with the FOPI/LAND data (shaded area) reported in Ref. [23]; (b) the same quantity calculated with the indicated seven Skyrme forces for the intermediate rapidity interval $0.25\le |y_0|\le 0.75$.

Figure 5

The total ${\chi }^2$ characterizing the fit results obtained with the 21 studied Skyrme forces as a function of the slope parameter $L$. The smooth curve is a quadratic fit to the total ${\chi }^2$, and the horizontal dashed line is used to determine the error of $L$ within a 2-$\sigma$ uncertainty.

Figure 6

The elliptic flow differences $v_2^n-v_2^p$ (a) and $v_2^n-v_2^H$ (b) and the elliptic flow ratios $v_2^n/v_2^p$ (c) and $v_2^n/v_2^H$ (d) produced in moderately central ($b<7.5$ fm) ${}^{197}\mathrm{Au}$+${}^{197}\mathrm{Au}$ collisions at $E_{\mathrm{lab}}=400$ MeV/nucleon are shown as a function of the slope parameter $L$. In each plot, the gray-shaded region indicates the $p_t/A$-integrated experimental data [23], full squares denote UrQMD calculations with the studied set of Skyrme forces, while the lines represent linear fits to the calculations.