Scalar-vector Lagrangian without nonlinear self-interactions of bosonic fields in the relativistic mean-field theory

A new Lagrangian model without nonlinear scalar self-interactions in the relativistic mean-field (RMF) theory is proposed. Introducing terms for scalar-vector interactions (SVI), we have developed a RMF Lagrangian model for finite nuclei and nuclear matter. It is shown that by inclusion of SVI in the basic RMF Lagrangian, the nonlinear sigma^3 and sigma^4 terms can be dispensed with. The SVI Lagrangian thus obtained provides a good description of ground-state properties of nuclei along the stability line as well as far away from it. This Lagrangian model is also able to describe experimental data on the breathing-mode giant monopole resonance energies well.

have been introduced [15,16] to describe finite nuclei. Attempts have been made to broaden the basis of the RMF Lagrangian by including terms of higher orders in the scalar and vector fields with inclusion of interaction terms amongst various mesonic degrees of freedom [17,18,19]. Density-dependent meson couplings [20,21,22,23,24] have been introduced with a view to modify density dependence of the nuclear interaction in an explicit form with a good degree of success. This requires inclusion of additional parameters to model density-dependence of meson couplings. Notwithstanding the above, the RMF theory serves as an ideal platform for an effective field theoretical approach for many-body problems of nuclei with sufficient space for innovation.
An upsurge in experimental data especially in the domain of extreme regions of the periodic table provides an incentive to devise new and improved approaches and models to be able to describe the same. Savushkin et al. [17] have incorporated various mesonmeson interactions in their approach especially those between σ and ω meson in addition to nonlinear couplings of both these mesonic fields. This problem has been approached [18] from a more general point of view by taking expansion in and interactions amongst various mesonic fields. This approach has led to an improvement for finite nuclei and nuclear matter with a larger number of parameters required.
In this work, we have sought to explore the possibility of dispensing with the nonlinear scalar self-couplings which have so far remained essential for finite nuclei. We ask ourselves: whether it is possible to mock the scalar self-couplings and their inherent density dependence in nuclei by employing meson-meson interactions especially between σ and ω mesons instead?
Keeping the issue of renormalizability in abeyance, we have added couplings between σ and ω mesons of the form σω 2 + σ 2 ω 2 to the basic (linear) RMF Lagrangian based upon exchange of σ, ω and ρ mesons. Properties of nuclear matter for the scalar-vector interaction (SVI) of this form were explored in ref. [25]. Recently, the Lagrangian model SIG-OM with the inclusion of the coupling of the form σ 2 ω 2 whilst retaining the scalar self-couplings σ 3 + σ 4 has been developed [26]. In the present work, we have narrowed down the space by excluding the self-couplings at the expense of the scalar-vector meson-meson couplings.
The basic RMF Lagrangian density that describes nucleons as Dirac spinors interacting with the meson fields is given by [1] where M N is the bare nucleon mass and ψ is its Dirac spinor. Nucleons interact with σ, ω, and ρ mesons, with coupling constants being g σ , g ω and g ρ , respectively. The photonic field is represented by the electromagnetic vector A µ . The effective Lagrangian for finite nuclei that is used commonly is given by The nonlinear σ-meson self-couplings which have so far been an integral part of the RMF Lagrangian are given by The parameters g 2 and g 3 are the nonlinear couplings of σ-meson in the conventional σ 3 + σ 4 model [10]. Here, we put g 2 = g 3 = 0, thus eliminating the self-couplings U NL of σ meson.
Instead, we introduce the meson-meson interaction terms of the form where g 4 and g 5 represent the respective coupling constants for meson-meson interactions between σ and ω mesons. The effective Lagrangian in our case is then The corresponding Klein-Gordon equations can be written as where the effective meson masses m * σ and m * ω can be obtained as These equations represent an implicit density dependence of σ and ω meson masses and effectively that of the nuclear interaction therein.
The parameters of the new Lagrangian model SVI are obtained by a multi-dimensional search in the parameter space by fitting experimental binding energies and charge radii of a set of a few nuclei (cf. [5] for a detailed procedure). The nuclei included are 16  The ω and ρ meson masses have been fixed at their empirical values.
The parameters of the Lagrangian obtained as a result of a free variation in the multidimensional space are shown in Table I. We have obtained two parameter sets SVI-1 and SVI-2 which are deemed as appropriate for ground-state binding energies and charge radii of nuclei. The parameter of the forces NL-SH and NL3 with the nonlinear scalar couplings are also shown for comparison.
The nuclear matter properties of SVI-1 and SVI-2 are shown in the lower section of Table   I. The sets SVI-1 and SVI-2 are close to each other in the nuclear matter properties with a slight difference in the incompressibility with K = 264 MeV for SVI-1 and K = 272 MeV for SVI-2. There is only a minor difference in the effective mass m * . The m * values for SVI interactions are clearly higher than those of the Lagrangian sets NL-SH and NL3 with the scalar self-interactions.
The saturation density for both SVI-1 and SVI-2 is slightly higher than that of NL-SH and NL3. One notable difference between the two SVI sets is the difference in the asymmetry energy J (or a 4 ). One can note that even in an interaction that is different from the scalar self-interactions, it is not possible to bring down the asymmetry energy in the acceptable range of 30-33 MeV.
The binding energy of key spherical nuclei along with a few representative ones as obtained with SVI-1 and SVI-2 is shown in Table II. For a comparison, we also show the results due  Table   II. A marked improvement with SVI interactions is in the binding energy of doubly magic We have calculated the ground-state properties of the isotopic chains of Sn and Pb.
Especially, the chain of Sn isotopes offers experimental binding energies over the whole   Fig. 1. For the BCS pairing, neutron pairing gaps have been obtained from the experimental masses of neighbouring nuclei.
The problem of the arches and a predominance of shell energy at the magic numbers is well-known. It pervades both the microscopic theories as well as macroscopic-microscopic mass formulae. Viewing the results for Sn isotopes in Fig. 1(a) This pattern is also visible for the isotopic chain of Pb in Fig. 1(b). Both SVI-1 and SVI-2 exhibit a significant improvement in the binding energies over NL3. The rms deviation of the theoretical values with SVI-1 and SVI-2 is 0.88 MeV and 0.69 MeV, respectively. This is much smaller than the corresponding value of 1.82 MeV with NL3 for the Pb isotopes. Thus, SVI interactions provide a better description of the binding energies of the Pb isotopes. In comparison, NL3 values overestimate the data near the magic number and gives a wellformed arch about the magic number. With NL3 divergences of the binding energies near the magic number are displayed strongly as has been observed also for the Sn isotopes above.
It is a matter of further investigation as to what ingredients in the RMF theory would lead to divergences or a lack thereof at shell closures.
The charge radii of Pb isotopes and the anomalous kink in charge radii represent a characteristic feature related to shell structure of nuclei. It was shown that the RMF theory with NL-SH reproduced the anomalous kink successfully [7]. This feature has since been demonstrated by all the Lagrangian models in the RMF theory. In order to discern the behaviour of SVI in this respect, we show in Fig. 2  In order to test the applicability of the new model for nuclei away from the line of βstability, we have performed axially deformed RMF calculations for a number of nuclei.
The nuclei encompass representative cases from 36 Si to 196 Pt, which includes nuclei from medium masses through the rare-earth region to higher masses. The results of calculations with SVI-1 and SVI-2 are shown in Table IV. The experimental data on the binding energies are taken from the recent high-precision mass measurements on Si [28], Sr [29] and Mo [30].
The binding energy of other nuclei has been taken from the 2003 compilation of atomic masses [31]. The original Walecka model (linear) [1] with the nucleon-meson couplings of σ and ω mesons has been instructive for achieving saturation of nuclear matter. With the inclusion of nonlinear scalar self-couplings, the saturation is achieved at nuclear matter properties viz., the compressibility within the acceptable range (cf. nuclear matter properties in Table   I  For a comparative analysis of the Lagrangian models involved, we have carried out con-