Constraining the density dependence of the symmetry energy using the multiplicity and average $p_T$ ratios of charged pions

The charged pion multiplicity ratio in intermediate-energy heavy-ion collisions, a probe of the density dependence of symmetry energy above the saturation point, has been proven in a previous study to be extremely sensitive to the strength of the isovector $\mathrm{\Delta }\left(1232\right)$ potential in nuclear matter. As there is no knowledge, either from theory or experiment, about the magnitude of this quantity, the extraction of constraints on the slope of the symmetry energy at saturation by using exclusively the mentioned observable is hindered at present. It is shown that, by including the ratio of average $p_T$ of charged pions $\langle p_T^{\left({\pi }^{+}\right)}\rangle /\langle p_T^{\left({\pi }^{-}\right)}\rangle$ in the list of fitted observables, the noted problem can be circumvented. A realistic description of this observable requires accounting for the interaction of pions with the dense nuclear matter environment by the incorporation of the so-called $S$-wave and $P$-wave pion optical potentials. This is performed within the framework of a quantum molecular dynamics transport model that enforces the conservation of the total energy of the system. It is shown that constraints on the slope of the symmetry energy at saturation density and the strength of the $\mathrm{\Delta }$(1232) potential can be simultaneously extracted. A symmetry energy with a value of the slope parameter $L>50$ MeV is favored, at $1\sigma$ confidence level, from a comparison with published FOPI experimental data. A precise constraint will require experimental data more accurate than presently available, particularly for the charged pion multiplicity ratio, and better knowledge of the density and momentum dependence of the pion potential for the whole range of these two variables probed in intermediate-energy heavy-ion collisions.

Figure 1

Density dependence of the pionic $S$-wave potential at fixed momentum, $p=0.125$ $\mathrm{GeV}/c$ (left-hand panels), and its momentum dependence at fixed density, $\rho /{\rho }_0=1.0$ (right-hand panels). The total $S$-wave potential has been split into its isoscalar (bottom panels, labeled “s”) and isovector (top panels, labeled “v”) components. In addition to the potentials extracted from pionic atom data, the behavior of the theoretical model of Nieves et al. [43] and of the chiral perturbation theory inspired effective model discussed in the text are also presented. The value of the isospin asymmetry parameter has been set to $\beta =0.20$, close to that of the ${}^{197}\mathrm{Au}$ nuclei for which the heavy-ion simulations have been performed.

Figure 2

The same as in Fig. 1, but for the $P$-wave potential. In this case the results of two theoretical models are shown, namely the one of Nieves et al. [43] and that of Garcia-Recio et al. [56]. As discussed in the text the momentum dependence of the $P$-wave potentials extracted from pionic atom data has been extrapolated using the momentum dependence of the latter theoretical model. The presented results correspond to the case of uniform nuclear matter of given density and isospin asymmetry $\beta =0.2$. Consequently, the density gradient term that appears in Eq. (7) does not contribute.

Figure 3

Ratio $R$ of the density gradient component of the $P$-wave pion potential $\left(V_{\mathrm{grad}}\right)$ vs the component proportional to $p^2$ $\left(V_{\text{non-loc}}\right)$ for the ${\pi }^{-}$ (bottom) and ${\pi }^{+}$ (top) mesons. Results for various choices of the $P$-wave potential parameter set (see Table 2) are presented. The calculation has been performed for a ${}^{197}\mathrm{Au}$ nucleus whose density profile can be parametrized by the simple expression $\rho \left(r\right)={\rho }_0/\{1+exp[(r-R)/a]\}$, with ${\rho }_0=0.165{\mathrm{fm}}^{-3},R=6.40$ fm, and $a=0.60$ fm. The value of the coordinate $r$ is chosen such as to maximize the magnitude of the density gradient, which for the chosen parametrization occurs at the location at which $\rho ={\rho }_0/2$ and hence $r=R$. The spread of the results for the chosen parameters sets of the $P$-wave potential is essentially given by the variation of the magnitude of $\mathrm{Im}{C}_0$ between the different potentials.

Figure 4

Total ${\pi }^{-}$ (lower panel) and ${\pi }^{+}$ (upper panel) potentials in uniform nuclear matter of isospin asymmetry $\beta =0.2$ for several values of the pion momentum (expressed in $\mathrm{GeV}/c)$. The chiral perturbation theory inspired effective model was chosen for the $S$-wave part, while for the $P$-wave part the potential labeled Batty-1 in Table 2 has been selected. The other possible combinations yield qualitatively the same behavior. The repulsion generated by the density dependence of the $S$-wave isovector strength ${\overline{b}}_1$ overcomes eventually the attraction in the $P$-wave channel leading to a transition from a net attractive to a net repulsive potential for ever-increasing, with the pion momentum, values of the density.

Figure 5

Impact of the modifications of the model described in Sec. 2a on the multiplicity ratio of ${\pi }^{-}$ and ${\pi }^{+}$ mesons as compared with the result of Ref. [15]. The ordinate $x$ parametrizes the stiffness of the symmetry energy, negative and positive values corresponding to a stiff and a soft asy-EOS, respectively. The horizontal band depicts the experimental result of the FOPI Collaboration [21].

Figure 6

Average $p_T$ ratio of charged pions (left panel) and average $p_T$ of all charge states of the $\pi$ meson (right panel) as a function of the stiffness parameter $x$ in central ${}^{197}\mathrm{Au}+{}^{197}\mathrm{Au}$ collision at an impact energy of 400 MeV/nucleon. Results for the VEC (dashed-dotted curves), LEC (dashed curves), and GEC scenarios are presented. For the case of the GEC scenario results for two strengths of the Coulomb interaction are shown: the one of Ref. [15] (dashed-double-dotted curves) and the one used in this study (full curves). The FOPI experimental result for the PAPTR [20] is depicted by a horizontal band.

Figure 7

Impact of the $S$-wave pion potential on pion multiplicities (left panel) and pion average transverse momenta (right panel) in mid-central collisions (3.35 fm $<$ $b$ $<6.00$ fm) of ${}^{197}\mathrm{Au}+{}^{197}\mathrm{Au}$ at an incident energy of 400 MeV/nucleon for various choices of the $S$-wave potential, as presented in Sec. 2b. The following kinematical cuts have been applied: $p_T<0.33$ $\mathrm{GeV}/c$ and $\left|y\right|<1.75$. The curves labeled “No $S$-wave potential” were obtained by omitting any contributions due to the pion optical potential. Experimental data [22] are represented by horizontal bands, with their widths representing only the statistical uncertainties (systematic uncertainties were not available).

Figure 8

Impact of the $P$-wave pion potential on pion multiplicities (left panel) and pion average transverse momenta (right panel) in mid-central collisions (3.35 fm $<$ $b$ $<6.00$ fm) of ${}^{197}\mathrm{Au}+{}^{197}\mathrm{Au}$ at an incident energy of 400 MeV/nucleon for various choices of the $P$-wave potential, as presented in Sec. 2b. The following kinematical cuts have been applied: $p_T<0.33$ $\mathrm{GeV}/c$ and $\left|y\right|<1.75$. The calculations labeled “No $P$-wave potential” include the impact of the effective model $S$-wave pion potential alone. The same remark for the experimental values (horizontal bands), as the one made in the caption of Fig. 7, holds true.

Figure 9

Impact of the various $S$ (left panel) and $P$ (right panel) wave pion potentials introduced in Sec. 2b on pion multiplicity ratio (PMR) and pion average $p_T$ ratio (PAPTR) in mid-central collisions (3.35 fm $<$ $b$ $<6.00$ fm) of ${}^{197}\mathrm{Au}+{}^{197}\mathrm{Au}$ at an incident energy of 400 MeV/nucleon. The meaning of the curves labeled “No $S$-wave potential” and “No $P$-wave potential” is the same as in Figs. 7 and 8, respectively.

Figure 10

Time dependence of the average transverse momentum of ${\pi }^{-}$ (top), ${\pi }^0$ (middle), and ${\pi }^{+}$ (bottom) mesons in mid-central heavy-ion collisions of ${}^{197}\mathrm{Au}+{}^{197}\mathrm{Au}$ at an incident energy of 400 MeV/nucleon for several choices for the total potential experienced by pions besides Coulomb. The value of the stiffness parameter has been set to $x=0$ and the Batty-1 parametrization of the $P$-wave pion potential has been selected where indicated. The labels bear the following meaning: “no pion pot”, both components of the pion potential have been omitted; “$S$-wave”, only the effective model $S$-wave pion potential has been included; “$S$+$P$ (non-grad)”, the $S$-wave and nongradient terms of the $P$-wave potential have been included; and finally “$S$+$P$ (full)”, the $S$ wave and the full $P$ wave (both the gradient and nongradient terms) are taken into account. The vast majority of the pions that escape into detectors are emitted at moments ulterior to $t=30$ $\mathrm{fm}/c$.

Figure 11

The ratio of charged pion multiplicities (left panel) and average transverse momenta (right panel) as a function of the stiffness parameter $x$ for six different choices of the strength of the isovector component of the $\mathrm{\Delta }$(1232) potential in nuclear matter [see Eq. (6)]. The results correspond to central $(b<2.0$ fm) ${}^{197}\mathrm{Au}+{}^{197}\mathrm{Au}$ collisions. The full experimental FOPI results (including both systematical and statistical errors) of Refs. [20, 21] are depicted by horizontal bands.

Figure 12

Contour plots for the ${\chi }^2/\mathrm{dof}$ value of the comparison theory vs experiment for the case of the PMR (left panel) and PMR+PAPTR (right panel). In the latter case, contributions due to the two observables are added with the same weights. Curves are labeled according to the corresponding value of the ${\chi }^2/\mathrm{dof}$ quantity.

Figure 13

Sensitivity of the extracted constraints for the stiffness parameter $x$ and strength of the isovector $\mathrm{\Delta }$(1232) potential $V_v$ to different choices for the $S$-wave (left panel) and $P$-wave (right panel) pion potentials. The calculations help quantify the impact of uncertainties in the energy and density dependence of these potentials on the quantities of interest. For that purpose the $1\sigma$ confidence level (CL) contour plots of the quantity ${\chi }^2/\mathrm{dof}$ determined by comparing theoretical and experimental results for the observables PMR and PAPTR for different choices for the pion $S$-wave (left panel) or $P$-wave (right panel) potentials are plotted. The case for which pion potential contributions are completely omitted (“no pion pot”) or only the $S$-wave component is included (“only $S$ Eff Mod”) are also shown (left panel). Result for a version of the $P$-wave potential with the gradient terms omitted is also shown (“$P$ Batty-1 no grad” in the right panel). Additionally, for the Batty-1 parametrization of the $P$-wave potential the $1\sigma$ CL contour curve determined by artificially decreasing the experimental uncertainties for PMR and PAPTR to $3\%$ and $1.5\%$, respectively, is also plotted and labeled “$P$ Batty-1 low exp err” (right panel).

Figure 14

Rapidity (left-hand panel) and transverse momentum (right-hand panel) spectra of charged pions in mid-central ${}^{197}\mathrm{Au}+{}^{197}\mathrm{Au}$ collisions at impact energy of 400 MeV/nucleon. Theoretical results for ${\pi }^{-}$ (full curves) and ${\pi }^{+}$ (dashed curves) mesons are compared with their experimental counterpart [22] depicted by diamond and triangle shaped symbols, respectively. In addition to the impact parameter selection $(3.35\mathrm{fm}<b<6.0$ fm) the indicated kinematical cuts were imposed in each case. The value of the stiffness parameter has been set to $x=-1$. Systematic uncertainties were not accounted for in the plotted experimental error bars.

Figure 15

Theoretical transverse momentum PMR spectra in mid-central ${}^{197}\mathrm{Au}+{}^{197}\mathrm{Au}$ collisions at an impact energy of 400 MeV/nucleon compared to its FOPI Collaboration experimental value [22]. The effective model $S$-wave and the Batty-1 $P$-wave pion potentials have been accounted for in the simulations. Results for five values of the stiffness parameter $x$ are plotted. Remarks made in the caption of Fig. 14 hold also for the case presented here.

Figure 16

The impact of the $S$-wave and $P$-wave components of the optical pion potentials on the transverse momentum PMR in mid-central ${}^{197}\mathrm{Au}+{}^{197}\mathrm{Au}$ collisions for two different values of the impact energy: $T_{\mathrm{lab}}=250$ MeV/nucleon (left panel) and $T_{\mathrm{lab}}=400$ MeV/nucleon (right panel). For the latter case the experimental FOPI result [22] is shown as a band. The value of the asy-EOS stiffness parameters has been set to $x=0$ $(L=61$ MeV). The same kinematical cuts as in Fig. 14 have been applied.

Figure 17

Theoretical predictions for pion multiplicity (lower panels) and average transverse momentum (upper panels) ratios in ${}^{108}\mathrm{Sn}+{}^{112}\mathrm{Sn}$ (left panels) and ${}^{124}\mathrm{Sn}+{}^{132}\mathrm{Sn}$ (right panels) central collisions at an impact energy of 270 MeV/nucleon as a function of the stiffness parameter $x$ and for the various choices of the isovector $\mathrm{\Delta }\left(1232\right)$ isobar discussed in the text. No kinematical cuts have been applied on spectra.