Effects of $\delta$ mesons in relativistic mean field theory

The effect of $\delta$- and $\omega \text{−}\rho$-meson cross couplings on asymmetry nuclear systems are analyzed in the framework of an effective field theory motivated relativistic mean field formalism. The calculations are done on top of the G2 parameter set, where these contributions are absent. To show the effect of $\delta$ meson on the nuclear system, we split the isospin coupling into two parts: (i) $g_{\rho }$ due to $\rho$ meson and (ii) $g_{\delta }$ for $\delta$ meson. Thus, our investigation is based on varying the coupling strengths of the $\delta$ and $\rho$ mesons to reproduce the binding energies of the nuclei ${}^{48}\mathrm{Ca}$ and ${}^{208}\mathrm{Pb}$. We calculate the root mean square radius, binding energy, single particle energy, density, and spin-orbit interaction potential for some selected nuclei and evaluate the $L_{\mathrm{sym}}$ and $E_{\mathrm{sym}}$ coefficients for nuclear matter as function of $\delta$- and $\omega \text{−}\rho$-meson coupling strengths. As expected, the influence of these effects are negligible for the symmetric nuclear system, but substantial for the contribution with large isospin asymmetry.

Figure 1

Binding energy (BE), root mean square radius, and first ($1s^{n,p}$) and last ($1f^n$, $2s^p$) occupied orbits for ${}^{48}\mathrm{Ca}$ using various ($g_{\rho }$, $g_{\delta }$, ${\Lambda }_v$) combinations of Table 1.

Figure 2

Same as Fig. 1 for ${}^{208}\mathrm{Pb}$.

Figure 3

The neutron and proton density with radial coordinate $r\left(\mathrm{fm}\right)$ at different combinations of ($g_{\rho }$, $g_{\delta }$) in (a) and with ${\Lambda }_v$ in (c). The variation of spin-orbit potential for proton and neutron are shown in (b) and (d) by keeping the same $g_{\delta }$ and ${\Lambda }_v$ as (a) and (c), respectively.

Figure 4

Same as Fig. 3 for ${}^{208}\mathrm{Pb}$.

Figure 5

Variation of nucleonic effective masses, binding energy per particle (BE$/A$) and pressure density as a function of $g_{\delta }$ on the saturation density of the G2 parameter set for nuclear matter.

Figure 6

Energy per particle and pressure density with respect to baryon density at various combinations of $g_{\delta }$ from Table 1.

Figure 7

Symmetry energy $E_{\mathrm{sym}}$ (MeV) of symmetric nuclear matter with respect to density by taking different value of $g_{\delta }$ sets. The heavy ion collision (HIC) experimental data [63] (shaded region) and nonrelativistic Skyrme GSkII [64] and Skxs20 [65] predictions are also given. ${\Lambda }_v=0.0$ is taken.

Figure 8

Constraints on $E_{\mathrm{sym}}$ with its first derivative, $L_{\mathrm{sym}}$, at saturation density for symmetric nuclear matter. The experimental results of HIC [63], PDR [70, 71], and IAS [72] are given. The theoretical prediction of the finite range droplet model (FRDM) and Skyrme parametrization are also given [73], SHF [59].

Figure 9

Symmetry energy $E_{\mathrm{sym}}$ (MeV), slope coefficients $L_{\mathrm{sym}}$ (MeV), and $K_{\mathrm{sym}}$ (MeV) at different sets of $g_{\rho }$ and $g_{\delta }$ as given in Table 1 with ${\Lambda }_v=0.0$.

Figure 10

The mass and radius of a neutron star at different values of $g_{\delta }$. (a) $M/M_{\odot }$ with neutron star density (gm/cm${}^3$), (b) $M/M_{\odot }$ with neutron star radius (km).