The symmetry energy contribution to the nuclear equation of state impacts various phenomena in nuclear astrophysics, nuclear structure, and nuclear reactions. Its determination is a key objective of contemporary nuclear physics, with consequences for the understanding of dense matter within neutron stars. We examine the results of laboratory experiments that have provided initial constraints on the nuclear symmetry energy and on its density dependence at and somewhat below normal nuclear matter density. Even though some of these constraints have been derived from properties of nuclei while others have been derived from the nuclear response to electroweak and hadronic probes, within experimental uncertainties-they are consistent with each other. We also examine the most frequently used theoretical models that predict the symmetry energy and its slope parameter. By comparing existing constraints on the symmetry pressure to theories, we demonstrate how contributions of three-body forces, which are essential ingredients in neutron matter models, can be determined.

We describe a relation between the symmetry energy coefficients $c_{\mathrm{sym}}(\rho )$ of nuclear matter and $a_{\mathrm{sym}}(A)$ of finite nuclei that accommodates other correlations of nuclear properties with the low-density behavior of $c_{\mathrm{sym}}(\rho )$. Here, we take advantage of this relation to explore the prospects for constraining $c_{\mathrm{sym}}(\rho )$ of systematic measurements of neutron skin sizes across the mass table, using as example present data from antiprotonic atoms. The found constraints from neutron skins are in harmony with the recent determinations from reactions and giant resonances.

A precise determination of the neutron skin $\Delta${}rnp of a heavy nucleus sets a basic constraint on the nuclear symmetry energy ($\Delta${}rnp is the difference of the neutron and proton rms radii of the nucleus). The parity radius experiment (PREX) may achieve it by electroweak parity-violating electron scattering (PVES) on Pb208. We investigate PVES in nuclear mean field approach to allow the accurate extraction of $\Delta${}rnp of Pb208 from the parity-violating asymmetry APV probed in the experiment. We demonstrate a high linear correlation between APV and $\Delta${}rnp in successful mean field forces as the best means to constrain the neutron skin of Pb208 from PREX, without assumptions on the neutron density shape. Continuation of the experiment with higher precision in APV is motivated since the present method can support it to constrain the density slope of the nuclear symmetry energy to new accuracy.

The information on the symmetry energy and its density dependence is deduced by comparing the available data on the electric dipole polarizability ${\alpha }_D$ of ${}^{68}\text{Ni}$, ${}^{120}\text{Sn}$, and ${}^{208}\text{Pb}$ with the predictions of the random-phase approximation, using a representative set of nuclear energy density functionals. The calculated values of ${\alpha }_D$ are used to validate different correlations involving ${\alpha }_D$, the symmetry energy at the saturation density $J$, the corresponding slope parameter $L$, and the neutron skin thickness $\mathrm{\Delta }{r}_{np}$, as suggested by the droplet model. A subset of models that reproduce simultaneously the measured polarizabilities in ${}^{68}\text{Ni}$, ${}^{120}\text{Sn}$, and ${}^{208}\text{Pb}$ are employed to predict the values of the symmetry energy parameters at saturation density and $\mathrm{\Delta }{r}_{np}$. The resulting intervals are $J=30–35$ MeV, $L=20–66$ MeV; and the values for $\mathrm{\Delta }{r}_{np}$ in ${}^{68}\text{Ni}$, ${}^{120}\text{Sn}$, and ${}^{208}\text{Pb}$ are in the ranges 0.15–0.19, 0.12–0.16, and 0.13–0.19 fm, respectively. The strong correlation between the electric dipole polarizabilities of two nuclei is instrumental to predict the values of electric dipole polarizabilities in other nuclei.

Although ab initio calculations of relativistic Brueckner theory lead to large scalar isovector fields in nuclear matter, at present, successful versions of covariant density functional theory neglect the interactions in this channel. A new high-precision density functional DD-ME$\delta${} is presented which includes four mesons, $\sigma${}, $\omega${}, $\delta${}, and $\rho${}, with density-dependent meson-nucleon couplings. It is based to a large extent on microscopic ab initiocalculations in nuclear matter. Only four of its parameters are determined by adjusting to binding energies and charge radii of finite nuclei. The other parameters, in particular the density dependence of the meson-nucleon vertices, are adjusted to nonrelativistic and relativistic Brueckner calculations of symmetric and asymmetric nuclear matter. The isovector effective mass mp*$-${}mn* derived from relativistic Brueckner theory is used to determine the coupling strength of the $\delta${} meson and its density dependence.

We study the electric dipole polarizability ${\alpha }_D$ in ${}^{208}$Pb based on the predictions of a large and representative set of relativistic and nonrelativistic nuclear mean-field models. We adopt the droplet model as a guide to better understand the correlations between ${\alpha }_D$ and other isovector observables. Insights from the droplet model suggest that the product of ${\alpha }_D$ and the nuclear symmetry energy at saturation density $J$ is much better correlated with the neutron skin thickness $\Delta r_{np}$ of ${}^{208}$Pb than the polarizability alone. Correlations of ${\alpha }_{D}J$ with $\Delta r_{np}$ and with the symmetry energy slope parameter $L$ suggest that ${\alpha }_{D}J$ is a strong isovector indicator. Hence, we explore the possibility of constraining the isovector sector of the nuclear energy density functional by comparing our theoretical predictions against measurements of both ${\alpha }_D$ and the parity-violating asymmetry in ${}^{208}$Pb. We find that the recent experimental determination of ${\alpha }_D$ in ${}^{208}$Pb in combination with the range for the symmetry energy at saturation density $J=[31\pm {\left(2\right)}_{\mathrm{est}}]$ MeV suggests $\Delta r_{np}\left({}^{208}\mathrm{Pb}\right)=0.165\pm {\left(0.009\right)}_{\mathrm{expt}}\pm {\left(0.013\right)}_{\mathrm{theor}}\pm {\left(0.021\right)}_{\mathrm{est}}\mathrm{fm}$ and $L=43\pm {\left(6\right)}_{\mathrm{expt}}\pm {\left(8\right)}_{\mathrm{theor}}\pm {\left(12\right)}_{\mathrm{est}}$ MeV.