Auxiliary Function Approach for Determining Symmetry Energy at Supra-saturation Densities

Abstract

Nuclear symmetry energy $E_{\rm{sym}}(ρ)$ at density $ρ$ is normally expanded or simply parameterized as a function of $χ=(ρ-ρ_0)/3ρ_0$ in the form of $E_{\rm{sym}}(ρ)\approx S+Lχ+2^{-1}K_{\rm{sym}}χ^2+6^{-1}J_{\rm{sym}}χ^3+\cdots$ using its magnitude $S$, slope $L $, curvature $K_{\rm{sym}}$ and skewness $J_{\rm{sym}}$ at the saturation density $ρ_0$ of nuclear matter. Much progress has been made in recent years in constraining especially the $S$ and $L$ parameters using various terrestrial experiments and astrophysical observations. However, such kind of expansions/parameterizations do not converge at supra-saturation densities where $χ$ is not small enough, hindering an accurate determination of high-density $E_{\rm{sym}}(ρ)$ even if its characteristic parameters at $ρ_0$ are all well determined by experiments/observations. By expanding the $E_{\rm{sym}}(ρ)$ in terms of a properly chosen auxiliary function $Π_{\rm{sym}}(χ,Θ_{\rm{sym}})$ with a parameter $Θ_{\rm{sym}}$ fixed accurately by an experimental $E_{\rm{sym}}(ρ_{\rm{r}})$ value at a reference density $ρ_{\rm{r}}$, we show that the shortcomings of the $χ$-expansion can be completely removed or significantly reduced in determining the high-density behavior of $E_{\rm{sym}}(ρ)$. In particular, using two significantly different auxiliary functions, we show that the new approach effectively incorporates higher $χ$-order contributions and converges to the same $E_{\rm{sym}}(ρ)$ much faster than the conventional $χ$-expansion at densities $\lesssim3ρ_0$. Several quantitative demonstrations using Monte Carlo simulations are given.