Probing the Symmetry Energy with the Spectral Pion Ratio

Many neutron star properties, such as the proton fraction, reflect the symmetry energy contributions to the equation of state that dominate when neutron and proton densities differ strongly. To constrain these contributions at suprasaturation densities, we measure the spectra of charged pions produced by colliding rare isotope tin (Sn) beams with isotopically enriched Sn targets. Using ratios of the charged pion spectra measured at high transverse momenta, we deduce the slope of the symmetry energy to be $42<L<117\mathrm{MeV}$. This value is slightly lower but consistent with the $L$ values deduced from a recent measurement of the neutron skin thickness of ${}^{208}\mathrm{Pb}$.

Figure 1

Measured and calculated pion spectra. The red lines are the calculated pion spectra after adjusting the $\mathrm{\Delta }$ potential to reproduce the pion multiplicities. The blue lines differ from the red lines in that the pion optical potential has been removed. The nucleon potentials in these simulations correspond to $L=80\mathrm{MeV}$ and $\mathrm{\Delta }{m}_{np}^{*}=0$.

Figure 2

Single pion spectral ratios for ${}^{132}\mathrm{Sn}+{}^{124}\mathrm{Sn}$ (top) and ${}^{108}\mathrm{Sn}+{}^{112}4\mathrm{Sn}$ (bottom) reactions. The curves are dcQMD predictions from different $L$ and $\mathrm{\Delta }{m}_{np}^{*}$ values listed in the bottom panel.

Figure 3

Correlation contours between $L$ and $\mathrm{\Delta }{m}_{np}^{*}/\delta$ extracted from the single pion spectral ratio of the neutron rich ${}^{132}\mathrm{Sn}+{}^{124}\mathrm{Sn}$ and near symmetric ${}^{108}\mathrm{Sn}+{}^{112}\mathrm{Sn}$ reactions. The green shaded region lies within the 68% confidence level for data with $p_T>200\mathrm{MeV}/c$. The dotted blue lines denote contours corresponding to the 95% confidence level.

Figure 4

Transverse momentum spectra of the double pion ratio. The shaded region covers dcQMD predictions within $1\sigma$ of the most probable values of $L$ and $\mathrm{\Delta }{m}_{np}^{*}$ values.