Delineating Effects of Nuclear Symmetry Energy on the Radii and Tidal Polarizabilities of Neutron Stars

What can we learn about the density dependence of nuclear symmetry energy $E_{\mathrm{s}\mathrm{y}\mathrm{m}}(\rho )$ from precise measurements of the radius ($R_{1.4}$) and/or tidal polarizability (${\mathrm{\Lambda }}_{1.4}$) of canonical neutron stars (NSs) with a mass of 1.4 M${}_{\odot }$? With the $E_{\mathrm{s}\mathrm{y}\mathrm{m}}(\rho )$ parameterized using three parameters $L$, $K_{\mathrm{s}\mathrm{y}\mathrm{m}}$, and $J_{\mathrm{s}\mathrm{y}\mathrm{m}}$ which have the asymptotic meaning of being respectively the slope, curvature, and skewness of symmetry energy at saturation density, we found that, while both the $R_{1.4}$ and ${\mathrm{\Lambda }}_{1.4}$ depend strongly on the slope $L$, the $K_{\mathrm{s}\mathrm{y}\mathrm{m}}$ and $J_{\mathrm{s}\mathrm{y}\mathrm{m}}$ parameters characterizing the high-density behavior of $E_{\mathrm{s}\mathrm{y}\mathrm{m}}(\rho )$ also play appreciable roles. Thus, there is not a simple relation between the ${\mathrm{\Lambda }}_{1.4}$/$R_{1.4}$ and $L$ alone. Precise measurements of just the ${\mathrm{\Lambda }}_{1.4}$ and $R_{1.4}$ can not completely determine the $E_{\mathrm{s}\mathrm{y}\mathrm{m}}(\rho )$ but limit combinations of its parameters. In particular, stringent constraints approximately independent of the $J_{\mathrm{s}\mathrm{y}\mathrm{m}}$ on the $L$-$K_{\mathrm{s}\mathrm{y}\mathrm{m}}$ correlations can be obtained. However, infinite combinations of the larger (smaller) $L$ and smaller (larger) $K_{\mathrm{s}\mathrm{y}\mathrm{m}}$ can lead to the same ${\mathrm{\Lambda }}_{1.4}$ and $R_{1.4}$. Additional observables including those from terrestrial nuclear experiments are thus necessary to break this degeneracy in order to completely determine the density dependence of nuclear symmetry energy $E_{\rm{sym}}(\rho )$.