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 $E_{\rm{sym}}(\rho )$ from precise measurements of the radius ($R_{\rm{1.4}}$) and/or tidal polarizability ($\Lambda_{1.4}$) of canonical neutron stars (NSs) with a mass of 1.4 M$_\odot$? With the $E_{\rm{sym}}(\rho )$ parameterized using three parameters $L$, $K_{\rm{sym}}$, and $J_{\rm{sym}}$ 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 $R_{\rm{1.4}}$ and $\Lambda_{1.4}$ depend strongly on the slope $L$, the $K_{\rm{sym}}$ and $J_{\rm{sym}}$ parameters characterizing the high-density behavior of $E_{\rm{sym}}(\rho )$ also play appreciable roles. Thus, there is not a simple relation between the $\Lambda_{\rm{1.4}}$/$R_{\rm{1.4}}$ and $L$ alone. Precise measurements of just the $\Lambda_{\rm{1.4}}$ and $R_{\rm{1.4}}$ can not completely determine the $E_{\rm{sym}}(\rho )$ but limit combinations of its parameters. In particular, stringent constraints approximately independent of the $J_{\rm{sym}}$ on the $L$-$K_{\rm{sym}}$ correlations can be obtained. However, infinite combinations of the larger (smaller) $L$ and smaller (larger) $K_{\rm{sym}}$ can lead to the same $\Lambda_{\rm{1.4}}$ and $R_{\rm{1.4}}$. 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 $E_{\rm{sym}}(\rho )$.