Abstract
The ultimate goal of determining the density dependence of nuclear symmetry energy is to understand the poorly known isovector nuclear interaction in neutron-rich matter at various densities. At the mean-field level, the potential part of the symmetry energy is determined by the density and momentum dependence of the symmetry potential while the kinetic part depends on the isoscalar nucleon effective mass. Going beyond the mean-field, the short-range correlation due to both the hard core and the tensor force acting predominantly between deuteron-like neutron-proton pairs affects the nucleon momentum distribution. The latter, in turn, affects both the kinetic and potential parts of the symmetry energy. Different isovector observables from various experiments, such as n-skin, radii of neutron stars, $n/p$ ratio of low- and high-momentum nucleons, depend differently on different parts of the symmetry energy. It is thus important to perform detailed diagnosis of all isovector observables to completely determine all parts of the symmetry energy. As examples, we will discuss in this talk, how covariance analysis can help us find out which parts of and at what densities the symmetry potential determines the n-skin of $^{208}$Pb and the radii of neutron stars, how the short-range correlation may reduce the kinetic symmetry energy and lead to an isospin quartic term in the Equation of State of neutron-rich matter.