Finite nuclei in an extended Nambu-Jona-Lasinio model

By treating nucleons as the basic building blocks, it was shown that the properties of finite nuclei and nuclear matter below and around the nuclear saturation density $n_{0}\approx 0.16$ fm^-3 can be well understood, e.g., using relativistic density functional theories Meng (2016), nonrelativistic density functional theories Dutra et al. (2012), or microscopic many-body theories Hebeler (2021). Being the fundamental constituents of nucleons, the quark degrees of freedom seems irrelevant and nucleons can be treated as point particles since exciting a nucleon takes $\sim$300 MeV while exciting a nucleus takes $\sim$10 MeV. Nevertheless, in relativistic density functional theories Meng (2016), there exist strong attractive scalar potential ($\sim$440 MeV) and repulsive vector potential ($\sim$360 MeV) to accommodate the spin-orbit splitting of single particle energies in finite nuclei. In microscopic many-body theories Hebeler (2021), the many-body interactions were shown to be important to achieve the saturation properties of nuclear matter, which effectively account for the internal excitations of nucleons. In fact, the structures of nucleons in nuclear medium deviate from those of free nucleons, e.g., the European Muon Collaboration (EMC) effect European Muon Collaboration (1983). It is thus favorable to consider directly the quark degrees of freedom in nuclear matter.

To account for the contributions of quarks in nucleons, the quark-meson-coupling (QMC) model was then proposed considering the effects of the three valence quarks confined in bags so that the EMC effect can be understood Guichon et al. (2018). Nevertheless, at larger densities, the boundaries between baryons become less distinctive as the number of exchanged quarks increases Oka and Yazaki (1980); Chen et al. (2007); Fukushima and Kojo (2016); McLerran (2009); Park et al. (2021); Park and Lee (2022). It is thus insufficient to treat nucleons as independent particles and a unified description based on quark degrees of freedom is favorable. Extensive efforts were then devoted to this field in the past decades, e.g., the string-flip model Horowitz and Piekarewicz (1991), color molecular dynamics Maruyama and Hatsuda (2000); Yasutake and Maruyama (2024), quark-diquark approach in the framework of Nambu-Jona-Lasinio (NJL) model Bentz and Thomas (2001), and the hadronic SU(3) nonlinear $\sigma$ model including quarks Dexheimer and Schramm (2010). A direct modeling of quarkyonic matter was recently proposed McLerran and Reddy (2019); Margueron et al. (2021), while a complete field model was later developed by synthesizing the Walecka model and quark-meson model Cao and Liao (2020); Cao (2022). By combining relativistic mean field (RMF) models and equivparticle models with density-dependent quark masses, the quarkyonic matter and quarkyonic transition were investigated as well Xia et al. ; Xia et al. (2023).

In our previous study, by considering baryons as cluster of quarks we have extended the NJL (eNJL) model to describe baryonic matter, quark matter, and their transitions in a unified manner, where various first-order phase transitions and the microscopic structures of thier mixed phases were investigated Xia (2024); Xia et al. (2024). The Dirac sea, the spontaneous chiral symmetry breaking, and the quark degrees of freedom can then be treated in a unified manner in the framework of the eNJL model. In this work, based on the eNJL model, we further investigate the properties of finite nuclei in mean field approximation (MFA) and solve the Dirac equations of nucleons assuming nuclei are spherically symmetric. It is found that our model generally reproduces the binding energies of the 2495 nuclei ($A>2$) from the 2016 Atomic Mass Evaluation (AME2016) Audi et al. (2017); Huang et al. (2017); Wang et al. (2017), while the obtained charge radii of 906 nuclei are systematically smaller than the experimental values Angeli and Marinova (2013).

The paper is organized as follows. In Sec. II we present our theoretical framework, including the Lagrangian density of the eNJL model and the formalism of fixing the properties of spherically symmetric finite nuclei in MFA. The obtained properties of finite nuclei are presented in Sec. III. We draw our conclusion in Sec. IV.

For fixed neutron $(N)$ and proton $(Z)$ numbers, we then solve the Dirac Eq. (13), nucleon mass Eq. (7), quark mass Eq. (8), mean field vector potentials Eq. (14), boson field Eqs. (16-19), and densities Eq. (20) in the eNJL model iteratively in coordinate space with a box size of $12.8~{}{\rm fm}$ and a grid distance of $0.1~{}{\rm fm}$. Once convergency is reached, the energy and binding energy can be fixed by Eqs. (27) and (28).

As an example, in Fig. 1 we present the obtained density profiles and mean field potentials of ²⁰⁸Pb, where the adopted parameters are indicated in Table. 1. It is found that the potential depths and density profiles are generally consistent with those obtained with RMF models, which would lead to correct description of finite nuclei. Note that we have used a rather large $\omega$ meson mass $m_{\omega}=2500$ MeV to prevent unrealistic density fluctuations caused by the six-point (‘t Hooft) interaction, where $m_{\omega}$ should be viewed as an effective $\omega$ meson mass including the contributions of the ‘t Hooft term. Meanwhile, since the quantum fluctuations of nucleons have been considered, here $m_{\omega}$ can be smaller than those in Ref. Xia et al. (2024) adopting TFA for nucleons.

Based on the parameter set indicated in Table. 1, we then carry out systematic investigations on the properties of finite nuclei in the framework of eNJL model. In Fig. 2 we present the binding energies per nucleon ($B/A$) of $\beta$-stable nuclei, which are compared with the experimental values from AME2016 Audi et al. (2017); Huang et al. (2017); Wang et al. (2017). The obtained binding energies for $\beta$-stable nuclei are generally consistent with the experimental values with slight deviations at certain mass numbers, e.g., at $A=16$, 20, 32, 50, 62, 80, 88, 140, etc.

To show the the deviations more clearly, in Fig. 3 we present the differences ($\Delta B=B-B_{\mathrm{exp}}$) between the theoretical binding energies and the experimental values of the 2495 nuclei in AME2016 Audi et al. (2017); Huang et al. (2017); Wang et al. (2017). It is found that our model generally reproduces the binding energies of the 2495 nuclei ($A>2$) from AME2016 with the root-mean-square deviations $\sqrt{\sum_{i}\Delta B_{i}^{2}/2495}=5.38$ MeV. Evidently, as indicated in Fig. 3, the nuclei around the magic numbers $N(Z)=28$, 50, and 82 are over bound, where the binding energy may exceed the experimental values by up to $\sim$10 MeV for double magic nuclei. Additionally, in our model we have observed spurious shell closures at $N(Z)=34$, 58, and 92, leading to over bound nuclei in these regions as well. Beside the over bound nuclei at magic numbers and spurious shell closures, the binding energy for nuclei at $Z\approx 4$, 12, 44, 65, 87 and $N\approx 4$, 12, 25, 44, 72, 101, 146 are too small compared with the experimental values. As will be illustrated later, those deviations are caused by too large shell closures with the nuclei being too compact.

In Fig. 4 we present the single neutron separation energies $S_{n}=B(Z,A)-B(Z,A-1)$ in O, Ca, Ni, Sn, and Pb isotopes. It is found that our eNJL model generally reproduces the experimental neutron separation energies as well as the trends with respect to the variations of $N$. The odd-even staggering is also obtained and generally consistent with the experimental values, where the simple formula in Eq. (26) well describes the pairing correlations in finite nuclei. The shell closures at $N=8$, 20, 28, 82, and 126 can be easily identified with the sudden decrease of $S_{n}$ as we increase $N$. Nevertheless, the extent of decline in $S_{n}$ is slightly larger than the experimental values, which is mainly attributed to the too large shell gaps at $N=28$, 82, and 126. The spurious shell closures at $N=34$, 58, 92, and 114 are also identified with unrealistic decline in $S_{n}$, leading to slight deviations of the obtained neutron separation energies from the experimental values.

To show the (spurious) shell closures predicted by our eNJL model more clearly, in Fig. 5 we present the single particle energies of protons and neutrons in ¹³⁵Sn and ²⁰⁸Pb. It is found that our eNJL model generally reproduces the magic numbers in nuclei at $N(Z)=2$, 8, 20, 28, 50, 82, and 126, while spurious shell closures is identified as well at $N(Z)=34$, 58, 92, and 114. Combined with the single neutron separation energies $S_{n}$ in O isotopes, it can be easily identified that the spin-orbit splitting of the level 1$d$ is too large, leading to a slightly too large decline in $S_{n}$ at $N=14$. Similar situation is observed for other orbitals, e.g., the spacings in the pairs of levels 1$f_{7/2}$-1$f_{5/2}$, 1$f_{5/2}$-2$p_{3/2}$, 1$g_{7/2}$-2$d_{5/2}$, 3$s_{1/2}$-1$h_{9/2}$, 1$i_{13/2}$-2$f_{5/2}$, 2$f_{5/2}$-3$p_{3/2}$, and 3$p_{1/2}$-1$i_{11/2}$ are too large. This problem may be solved in our future work by carefully calibrating the density dependence of coupling constants so that the too large level spacings can be reduced with the spurious shell closures eliminated Wei et al. (2020).

Finally, in Fig. 6 we present the charge radii $R$ of 906 nuclei predicted by the eNJL model, the differences $\Delta R=R-R_{\mathrm{exp}}$ and relative differences $\Delta R/R$ with respect to the experimental values $R_{\mathrm{exp}}$ Angeli and Marinova (2013) are indicated in the upper panels as well. In general, the charge radii of 906 nuclei predicted by our eNJL model are systematically smaller than the experimental values with the root-mean-square deviations $\sqrt{\sum_{i}\Delta R_{i}^{2}/906}=0.127$ fm. This could be the reason of too large level spacings as indicated in Fig. 5, which should be improved in our future investigations.

Assuming spherical symmetry and neglecting the center-of-mass corrections, we investigate systematically the properties of finite nuclei based on an extended NJL (eNJL) model that treats baryons as clusters of quarks Xia (2024); Xia et al. (2024). The Dirac sea, the spontaneous chiral symmetry breaking, and the quark degrees of freedom are considered self-consistently in the eNJL model. The binding energies of the 2495 nuclei ($A>2$) from AME2016 Audi et al. (2017); Huang et al. (2017); Wang et al. (2017) are reproduced using the eNJL model with the root-mean-square deviations 5.38 MeV. It is found that the deviations are mainly attributed to the too large shell gaps at $N(Z)$ = 28, 50, 82, 126 and 34, 58, 92. The charge radii of 906 nuclei are obtained and compared with the experimental values, which are systematically smaller with the root-mean-square deviations 0.127 fm. In our future works, by carefully calibrating the density dependence of coupling constants and considering deformations with microscopic collective corrections from the nucleons in the Fermi sea and quarks in the Dirac sea, we expect the deviations can be further reduced.