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

Abstract

Nuclear symmetry energy $E_{\mathrm{sym}}(\rho )$ at density $\rho$ is normally expanded or simply parameterized as a function of $\chi =(\rho -{\rho }_0)/3{\rho }_0$ in the form of $E_{\mathrm{sym}}(\rho )\approx S+L\chi +2^{-1}{K}_{\mathrm{sym}}{\chi }^2+6^{-1}{J}_{\mathrm{sym}}{\chi }^3+\cdots$ using its magnitude $S$, slope $L$, curvature $K_{\mathrm{sym}}$ and skewness $J_{\mathrm{sym}}$ at the saturation density ${\rho }_0$ 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 $\chi$ is not small enough, hindering an accurate determination of high-density $E_{\mathrm{sym}}(\rho )$ even if its characteristic parameters at ${\rho }_0$ are all well determined by experiments/observations. By expanding the $E_{\mathrm{sym}}(\rho )$ in terms of a properly chosen auxiliary function ${\Pi }_{\mathrm{sym}}(\chi ,{\Theta }_{\mathrm{sym}})$ with a parameter ${\Theta }_{\mathrm{sym}}$ fixed accurately by an experimental $E_{\mathrm{sym}}({\rho }_{\mathrm{r}})$ value at a reference density ${\rho }_{\mathrm{r}}$, we show that the shortcomings of the $\chi$-expansion can be completely removed or significantly reduced in determining the high-density behavior of $E_{\mathrm{sym}}(\rho )$. In particular, using two significantly different auxiliary functions, we show that the new approach effectively incorporates higher $\chi$-order contributions and converges to the same $E_{\mathrm{sym}}(\rho )$ much faster than the conventional $\chi$-expansion at densities $\lesssim 3{\rho }_0$. Several quantitative demonstrations using Monte Carlo simulations are given.

arXiv commentDiscussions and references added. Phys. Rev. C in press

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