Effects of isovector scalar \delta-meson on hypernuclei

# Effects of isovector scalar δ-meson on hypernuclei

## Abstract

We analyze the effects of meson on hypernuclei within the frame-work of relativistic mean field theory. The meson is included into the Lagrangian for hypernuclei. The extra nucleon-meson coupling () affects the every piece of physical observables, like binding energy, radii and single particle energy of hypernuclei. The lambda mean field potential is investigated which is consistent with other predictions. Flipping of single particle energy levels are observed with the strength of in the considered hypernuclei as well as normal nuclei. The spin-orbit potentials are observed for considered hypernuclei and the effect of on spin-orbit potentials is also analyzed. The calculated single- binding energies () are quite agreeable with the experimental data.

###### pacs:
21.10.-k, 21.10.Dr, 21.80.+a

## I Introduction

Normal nuclei are quite informative for showing the distinctive features of nucleon-nucleon (NN) interaction. The knowledge on NN interaction may be extended to hyperon-nucleon (YN) or hyperon-hyperon (YY) interaction by injecting one or more strange baryon to bound nuclear system (1); (2); (3); (4); (14); (6); (7); (8); (9); (10); (11); (12); (13); (14); (15); (16). The injected hyperon originates a new quanta of strangeness and makes a more interesting nuclear system with increasing density (2). Unlike to nucleons, a hyperon is not Pauli blocked owing to strangeness quantum number and resides at the centre of the nucleus. Hyperons are used as an impurity in nuclear systems to reveal many of the nuclear properties in the dimension of strangeness  (17); (18); (19). For this, a slightly unbound normal nucleus can be bound by addition of particle (20); (21).

To understand the structure of strange system, it is necessary to evaluate the contribution of YN interaction. But, due to short life-time of hyperon, only limited information on YN scattering data is available which is a major consequence of the experimental difficulties (22). For this purpose, more theoretical data are needed to explore the strangeness physics. However, extensive efforts on theoretical basis have been made to enrich the knowledge about YN interaction using relativistic and non-relativistic mean field approaches. For example, Skyrme Hartree-Fock (SHF) (23); (24); (1); (25); (7), deformed Hartree-Fock (DHF) (26); (21), Skyrme Hartree-Fock with BCS approach (27) and relativistic mean field (RMF) formalism (31); (32); (28); (29); (20); (3); (4); (30); (33); (34).

From last three decades, the relativistic mean field theory reproduces the experimental data on binding energy, root mean square (rms) radius, and quadrupole deformation parameter for finite nuclei throughout the periodic chart (41); (42); (35); (36); (37); (38); (39); (40). Here the degrees of freedom are nucleons and mesons. To deal with hypernuclei, one has to incorporate the meson-hyperon interaction to the relativistic Lagrangian. The most successful RMF model of Boguta and Bodmer, included the , , and mesons along with the nonlinear coupling of meson, which simulates the three-body interaction (43). The meson takes care the neutron-proton asymmetry, while the Coulomb interaction is taken care by the electromagnetic field produced by the protons. Although, conventional RMF model is quite successful, but it is recently realized that the isovector-scalar meson, which arises from the mass and isospin asymmetry of proton and neutron is very important for nuclear system with much difference in neutron N and proton Z number (44); (45); (46). The main objective of the present study is to see the effects of meson for some selected hypernuclear systems. For this purpose, we evaluate the contribution of meson in hypernuclear system and make a comparison with normal nuclei.

The paper is organized as follows: Section II gives a brief description of relativistic mean field formalism for hypernuclei with inclusion of meson. The results are presented and discussed in Section III. Selection of and coupling constant is also discussed in this section. The paper is summarized in Section IV.

## Ii Formalism

The RMF Lagrangian for hyperon-nucleon-meson many-body system including the meson is written as (3); (4); (28); (29); (47); (20); (48); (49):

 L = LN+LΛ, (1)
 LN = ¯ψi{iγμ∂μ−M}ψi+12(∂μσ∂μσ−m2σσ2) (2) − 13g2σ3−14g3σ4+12(∂μδ∂μδ−m2δδ2) − gs¯ψiψiσ−gδ¯ψi→τψi→δ−14ΩμνΩμν+12m2wVμVμ − gw¯ψiγμψiVμ−14BμνBμν+12m2ρ→Rμ→Rμ − 14FμνFμν−gρ¯ψiγμ→τψi→Rμ − e¯ψiγμ(1−τ3i)2ψiAμ, LΛ = ¯ψΛ{iγμ∂μ−mΛ}ψΛ−gωΛ¯ψΛγμψΛVμ (3) − gσΛ¯ψΛψΛσ,

where and denote the Dirac spinors for nucleon and particle, whose masses are M and respectively, and , are meson coupling constants. Because of zero isospin, the hyperon does not couple to and mesons. The quantities , , and are the masses for , , and mesons. The field for the -meson is denoted by , meson by , meson by and -meson by . The quantities , , , and =1/137 are the coupling constants for the , , , mesons and photon, respectively. We have and self-interaction coupling constants for mesons. The field tensors of the vector, isovector mesons and of the electromagnetic field are given by

 Ωμν = ∂μVν−∂νVμ, Bμν = ∂μRν−∂νRμ, Fμν = ∂μAν−∂νAμ. (4)

The classical variational principle is used to solve the field equations for bosons and Fermions. The Dirac equation for the nucleon is written as:

 [−iα.∇+V(r)+β(M+S(r))]ψi=ϵiψi, (5)

where V(r) and S(r) represent the vector and scalar potential, defined as

 V(r)=gωV0(r)+gρτ3ρ0(r)+e(1−τ3)2A0(r), (6)

and

 S(r)=gσσ(r)+τ3gδδ0(r), (7)

where subscript n, p for neutron and proton, respectively. The Dirac equation for particle is

 [−iα.∇+β(mΛ+gσΛσ(r))+gωΛV0(r)]ψΛ=ϵΛψΛ. (8)

The field equations for bosons are

 {−△+m2σ}σ(r) = −gσρs(r)−g2σ2(r)−g3σ3(r) −gσΛρΛs(r), {−△+m2ω}V0(r) = gωρv(r)+gωΛρΛv(r), {−△+m2δ}δ3(r) = −gδρs3(r), {−△+m2ρ}R03(r) = gρρ3(r), −△A0(r) = eρc(r). (9)

Here , , and are the scalar and vector density for and field in nuclear and hypernuclear system which are expressed as

 ρs(r) = ∑i=n,p¯ψi(r)ψi(r), ρΛs(r) = ∑¯ψΛ(r)ψΛ(r), ρv(r) = ∑i=n,pψ†i(r)ψi(r), ρΛv(r) = ∑ψ†Λ(r)ψΛ(r). (10)

The scalar density for field is

 ρs3(r) = ∑i=n,p¯ψi(r)τ3iψi(r). (11)

The vector density for field and charge density are expressed by

 ρ3(r) = ∑i=n,pψ†i(r)γ0τ3iψi(r), ρc(r) = ∑i=n,pψ†i(r)γ0(1−τ3i)2ψi(r). (12)

The various rms radii are defined as

 ⟨r2p⟩ = 1Z∫r2pd3rρp, ⟨r2n⟩ = 1N∫r2nd3rρn, ⟨r2m⟩ = 1A∫r2md3rρ, ⟨r2Λ⟩ = 1Λ∫r2Λd3rρΛ, (13)

for proton, neutron, matter and lambda rms radii respectively and , , and are their corresponding densities. The charge rms radius can be found from the proton rms radius using the relation taking into consideration the finite size of the proton. The total energy of the system is given by

 Etotal = Epart(N,Λ)+Eσ+Eω+Eδ+Eρ (14) +Ec+Epair+Ec.m.,

where is the sum of the single particle energies of the nucleons (N) and hyperon (). , , , , , and are the contributions of meson fields, Coulomb field, pairing energy and the center-of-mass energy, respectively. We use NL3* parameter set through out the calculations (50).

We adopt the relative and coupling to find the numerical values of meson coupling constants. The relative coupling constants for and field are defined as and . We use the value of the relative coupling as from the naive quark model (51); (52). For used NL3* parameter set, we take the relative coupling value as  (30). In present calculations, to take care of pairing interaction the constant gap BCS approximation is used and the centre of mass correction is included by the formula .

## Iii Results and Discussions

The calculated results are shown in Table 1 and Figs. (111) for both normal nuclei and hypernuclei. We study the effect of meson on some selected hypernuclei, like Ca, Zr and Pb. To demonstrate the effect of on hypernuclei, we make a comparison with their normal nuclear (Ca, Zr and Pb) counter parts.

### iii.1 Strategy to fit gρ and gδ:

The NL3* parametrization used in RMF is fitted phenomenologically. All the masses and their coupling constants are adjusted to reproduce some specific experimental data. Therefore, it is not just to add one more parameter like , to study it’s effect keeping all other parameters of NL3* as fixed. It might be possible that the physics described by may already be inbuilt in the sub parameters of NL3* and the inclusion of meson coupling may lead towards a double counting.

In this regard, we might expect a connection between and since both the coupling constants are isospin dependent. In such a situation, there are two possible ways for this problem to avoid the double counting: (i) to consider a dependency on both and couplings. In this case, modify the parameter to fit an experimental data which is linked to both and for each new given value of , such as binding energy or (ii) to get a completely new parameter set including this interaction to consider as a new degree of freedom from the beginning, i.e., start from an ab initio calculations as done in Ref. (54).

Here, we are not interested to make a new parameter by inclusion of this interaction but our motive is just to extract the contribution of meson in hypernuclei and corresponding normal nuclei. For this, we adopt the first approach to analyze the effect of on hypernuclei. The combination of and are chose in such a way that for a given value of , the combined contribution of and (by adjusting ) reproduces the physical observable which exactly match with the original predictions when was not included. We implement this scheme on binding energy which is the best physical observables to see every effects in the nuclear system. So, we choose the binding energies of Ca, Zr, Pb hypernuclei and corresponding their normal nuclei to consider as an experimental data. By inclusion of , the binding energies change from their original predictions. To bring back the NL3* binding energies for considered nuclei and hypernuclei, we modifiy the coupling. In this way, we get various combinations of (, ) for different given values of . As we have already mentioned, the combinations of and are possible because of both of the coupling constants are linked with isospin.

### iii.2 Binding energy, radii and single particle energy

Before going to task on meson, it is necessary to check the reliability of the parameter which is going to be used. For this purpose, we calculate the total binding energy (BE), single lambda binding energy () and radii for some selected hypernuclei whose experimental data are available. After analyzing Table 1, we found that the lambda binding energy (for s- and p-state) are quite comparable with the experimental data. For example, the of N is 13.8 MeV in our calculation, and the experimental value is (13.760.16) MeV. It is obvious that the hyperon exhibits its strange behaviour and enhance the binding of nucleons in hypernucleus. The other thing is, with increasing the mass number the lambda density becomes smaller in respect to nucleon density and as a result lambda radius () grows up. This observation is reflected in Table 1, where increases with increasing the nuclear number.

In this section, we analyze the effects of meson on considered hypernuclei and make their comparison with normal nuclei to demonstrate the affects, which is the central theme of the paper. For the same, we calculate the binding energy (BE), root mean square neutron (), proton (), charge () and matter radius (), and energy of first and last filled orbitals of Ca, Zr, Pb and Ca, Zr, Pb with various combinations of and .

In Fig. 1 (a) and (c), we have shown the binding energy difference of Ca and Ca between the two solutions obtained with (=0) and (), i.e.

 ΔBE=BE(gρ,gδ=0)−BE(gρ,gδ), (15)

here is the binding energy at in the adjusted combination of () and is the binding energy with non-zero value of in the adjusted combination which reproduce the same binding as pure NL3*. Here, the value of used in adjusted combination with is different from the actual value given in original NL3* parameter set. In other words, we can say that, this strategy evolve a new parameter set with extra coupling constant , which also reproduces exactly same physical observables as NL3* set. Using this procedure, the contribution of meson in binding energy is obtained from . Similarly, the effect of meson in radius for both nuclei and hypernuclei can be seen from:

 Δr=r(gρ,gδ=0)−r(gρ,gδ), (16)

where is the radius at in the adjusted combination of () and is the radius in adjusted combination of and with non zero value of , produces exactly same experimental value as pure NL3*. The magnitude of with respect to for Ca, Zr, Pb and their hypernucleus Ca, Zr, Pb are shown in Figs. 13. The same procedure has adopted to estimate the contribution of meson on single particle energy for considered hypernuclei and their non-strange counter parts, which are shown in Figures. 46. The difference in single particle energy () for a particular level is expressed as

 Δϵ=ϵ(gρ,gδ=0)−ϵ(gρ,gδ), (17)

where is the single-particle energy for adjusted combination (, ) with , and is energy of the occupied level with non zero value of .

From Figs.16, it is evident that the binding energies, radii, single particle energies and spin-orbit splitting of nuclei and hypernuclei are affected with . Because of the presence of hyperon, the contribution of meson in binding energies, radii and single particle energies are less in hypernuclei compared to normal nuclei. In other words, we can say that meson affects the physical observables less in strange nuclei relative to nonstrange nuclei. In contrary to this, the proton and charge radii are affected more in hypernuclei compared to normal nuclei. From the overview of on radii, we find that and are in opposite trend with , , and that’s why the magnitude of differences of and increases with decreasing the asymmetry of the system by addition of hyperon. A very small reduction on lambda radius is observed with increasing strength of as shown in Figs. 1 3, while the lambda potential is completely unaffected by . It may happen because of the rearrangement of the levels due to presence of lambda particle. It is to be noticed that there are no convergence solutions beyond .

In Fig. 4, we have shown the change in single particle energy of the first () and last ( and ) occupied orbitals for Ca, and Ca. In the same way, the change in first () and last occupied levels ( and ) for Zr and Zr with the strength of is shown in Fig. 5. We also get the same trend in the magnitude of single particle energy difference for first () and last occupied levels ( and ) in Pb and corresponding their normal nucleus (Pb) which are displayed in Fig. 6. The magnitude of the difference of single particle energy for both neutron and proton (first and last occupied) orbitals of considered hypernuclei is small comparable to normal nuclei. Owing to zero isospin of hyperon, the lambda orbit () is unaffected with the strength of .

After analyzing the single particle spectra for both nuclei and hypernuclei, we notice that the orbitals make a shift with the strength of . In case of Ca, the levels are flipped with in hypernucleus and normal nucleus also. It is shown in Fig. 8, the Zr spectra pretend the flipping between and levels for both strange and nonstrange nuclei, however the strength is low, while the same orbitals ( and ) for neutron goes apart from each other with increasing the strength of . The same trend as Ca is observed for Pb and its normal nucleus as shown in Fig 9. The proton level close to flip with , and the neutron levels ( and ) also show the flipping with a very little change in the value of single-particle energy . In the analysis of neutron and proton single particle energy levels, we find that the trend of proton and neutron orbits are opposite to each other. This nature gives rise to effect of change in neutron and proton radius in opposite trend.

It is worthy to mention that the radius of Ca is slightly more than that of Ca (i.e. =3.4776 of Ca and =3.4771 of Ca (55); (56)), and this is difficult to explain by most of the nuclear models. We expact that similar anomaly may be occured in hyper-calcium (Ca and Ca) also and can be solved by the additional meson degree of freedom to the model. This mechanism can be used to solve the well known radius anomaly of Ca and Ca.

The neutron and lambda mean field potential for considered hypernuclei are plotted in Fig 10. The lambda central potential depth is found to be 32.87, 30.41 and 31.95 MeV for Ca, Zr and Pb, respectively. It is to be noticed that the amount of lambda potential is 3840% of nucleon potential. There are many of the calculations (51); (57); (58) in prediction of lambda potential depth and our results are consistent with these predictions. It is shown in Fig. 10 that both the potentials have similar shape but different depth. It is also found that the lambda potential is completely unaffected with the strength of meson coupling.

### iii.3 Spin-orbit splitting

The spin-orbit interaction plays a crucial role in order to investigate the structural properties of normal as well as hypernuclei developed by the exchange of scalar and vector mesons (59); (33). It is well known that the spin-orbit force in hypernuclei is weaker than normal nuclear system (51); (33); (60). Here, we study the spin-orbit potential for nucleon () and hyperon () in hypernuclei and also analyze the effect of on spin-orbit interaction. The spin-orbit potentials are displayed in Fig. 11 for considered hypernuclei. To see the effect of , we make a plot with =0.0 and for =8.0, which is the largest allowed strength of delta-meson coupling. Figure 11 reveals that the spin-orbit potential for hyperons is weaker than their normal counter parts and these results are consistent with existing predictions  (51); (33); (60). It is clearly seen from the Fig. 11 that the delta-meson coupling does not have any valuable impact on spin-orbit interaction. Actually, no change in spin-orbit potential is observed for Ca and Zr hypernuclei. Rather than this, the spin-orbit potentials in Pb hypernucleus is affected by a very little amount. This trend reflects that the measurable effect of on spin-orbit interaction can be observed from a system with large isospin asymmetry for example, heavy or superheavy nuclei and hypernuclei.

## Iv Summary and Conclusions

In summary, we study the contribution as well as importance of meson coupling in non-linear RMF model for hypernuclei. The lambda potential depth is found to be consistent with other predictions (51); (57); (58). The calculated for considered nuclei are quite agreeable with the experimental data. In the present calculation, we have included it to reveal the effects of coupling strength on hypernuclei which are found to be significant. It is clear to say that affects every piece of physical observables of hypernuclei, like binding energy, radii, single particle energy and spin-orbit splitting for nuclear system with NZ, but the magnitude of affects is less comparable to normal nuclei. Contrary to this, the proton and charge radii are affected relatively more than normal nuclear case. A very small reduction in lambda radius is also observed with increasing strength of meson coupling. However, the lambda potential is completely unaffected by meson coupling strength due to zero isospin nature of particle. The variation of spin-orbit interaction is discussed in respect of -meson coupling. This coupling does not have any significant impact on spin-orbit potential for considered hypernuclei but reflects that its impact would be measurable for a system with large isospin asymmetry. It is clearly seen that the contribution of meson is more effective with the magnitude of asymmetry of the system. From the given results, it is concluded that meson has indispensable contribution not only in asymmetric nuclei but also for hypernuclei.

The meson coupling may prove to be a significant degree of freedom for resolving the charge radius anomaly which is appeared in Ca and Ca and also if happened in corresponding hypernuclei. The production of Ca hypernucleus is possible in future due to advanced experimental facilities across the world.

## Acknowledgments

One of the author (MI) would like to acknowledge the hospitality provided by Institute of Physics, Bhubaneswar during the work.

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