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Beyond-mean-field effects on the symmetry energy and its slope from the low-lying dipole response of $^{68}$Ni

We study low-energy dipole excitations in the unstable nucleus $^{68}$Ni with the beyond-mean-field (BMF) subtracted second random-phase-approximation (SSRPA) model based on Skyrme interactions. First, strength distributions are compared with available experimental data and transition densities of some selected peaks are analyzed. The so-called isospin splitting is also discussed by studying the isoscalar/isovector character of such excitations. We estimate then in an indirect way BMF effects on the symmetry energy of infinite matter and on its slope starting from the BMF SSRPA low-lying strength distribution. For this, several linear correlations are used, the first one being a correlation existing between the contribution (associated with the low-energy strength) to the total energy-weighted sum rule (EWSR) and the slope of the symmetry energy. BMF estimates for the slope of the symmetry energy can be extracted in this way. Correlations between such a slope and the neutron-skin thickness of $^{68}$Ni and correlations between the neutron-skin thickness of $^{68}$Ni and the electric dipole polarizability times the symmetry energy are then used to deduce BMF effects on the symmetry energy.