# The Ground-State Calculations for Some Nuclei by Mesonic Potential of Nucleon-Nucleon Interaction

Authors: K. M. Hanna; S. H. M. Sewailem; R. Hussien; L. I. Abou-Salem; Asmaa G. Shalaby
Journal: Advances in High Energy Physics
 (2020)

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## Abstract

The interaction of nucleon-nucleon (NN) has certain physical characteristics, indicated by nucleon, and meson degrees of freedom. The main purpose of this work is calculating the ground-state energies of12H and 24He through the two-body system with the exchange of mesons (π, σ, and ω) that mediated between two nucleons. This paper investigates the NN interaction based on the quasirelativistic decoupled Dirac equation and self-consistent Hartree-Fock formulation. We construct a one-boson exchange potential (OBEP) model, where each nucleon is treated as a Dirac particle and acts as a source of pseudoscalar, scalar, and vector fields. The potential in the present work is analytically derived with two static functions of meson, the single-particle energy-dependent (SPED) and generalized Yukawa (GY) functions; the parameters used in meson functions are just published ones (mass, coupling constant, and cutoff parameters). The theoretical results are compared to other theoretical models and their corresponding experimental data; one can see that the SPED function gives more satisfied agreement than the GY function in the case of the considered nuclei.


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## Body

## 1. Introduction

One of the aims of nuclear structure theory is to derive the ground-state properties. Such properties are related to the constituents of matter, which are represented in the physics of elementary particles with their characteristics (electric charge, mass, spin, etc.) and how each particle interacts with others [1]. Yukawa (1934) introduced an assumption of some sort of field to be the reason of attraction between proton and neutron. This field is quantized, characterized by the force of the short range, and its mass equals 300 times of electron called Yukawa particle (meson). Meson is a Greek word, which means intermediate, and this is the right description for meson which transmits the nuclear force between hadrons; meson can participate in weak, strong, and electromagnetic interactions with a net electric charge. All mesons are unstable and their lifetimes reach to hundredths of microseconds. Each meson is characterized by quantum numbers, principal number (n=0,1,⋯), orbital angular momentum l=n−1 (indicating the orbiting of quarks around each other), magnetic number (m=−l,⋯,l), and spin s=0 for singlet state (1 for triplet state). The description of quantum numbers can be illustrated by using the nuclear shell model [2].The interaction between each nucleon with all other nucleons generates an average potential field where each nucleon moves. The rules of Pauli exclusion principle govern the occupation of orbital quantum states in the shell model and postulate that under the meson exchange between two nucleons, the wave function is a totally antisymmetrical product wave function. The interacted nucleons have a potential (the nuclear mean field) characterized by its dependence on the position coordinates. There is an ability to calculate the mean-field potential for electrons or in nuclei; the calculation methods are very similar, but the interactions are different. The NN interaction is a fundamental problem in nuclear physics. It had a variant success in describing the nuclear properties; this determination ranged from the empirical picture to fit the experimental data and to derive it microscopically from the bare NN potential. Thus, there is no unique NN potential to be the start point [3–5].A microscopic description was provided in nuclear models to include the elementary interaction between nucleons. The original attempts to find the fundamental theory of nuclear forces [6–10] were not so successful. The reason for their failure was the pion dynamics which has been restricted by the chiral symmetry [11, 12]. The quantum description and field methods were included in making the potential structure such as the Partovi-lomon model [13], Stony Brook-group [14], Paris-group [15], Nijmegen-group [16], and Bonn-group potentials [12]. The successful theoretical models were based on OBEP, including one-meson exchange and multimeson exchange, plus short-range phenomenology. Inside a nucleus, there is a fact that nucleons move quasi-independently from one to another which achieves the concept of nuclear mean field (NMF); this fact relies on Hartree-Fock interaction.The microscopic description of degrees of freedom related to nucleon and meson has to depend on a relativistic quantum field to include the full structure of the medium (spin structure) which is associated with the fermion field of Dirac equation [17] and the bound-state energies. This can be found by solving the Dirac equation which leads to investigating the ground-state energies of nuclei which is ensured by the calculation of the NMF potential with Dirac-Hartree-Fock [18].It is known that the NN interaction can be distinguished into three parts: the first part is the long range atr≥2 fm originated from pseudoscalar mesons; the second part is the medium range at 1fm≤r≤2fm, which mainly comes from the exchange of scalar meson (σ which is a fictitious scalar meson responsible for attraction); and the third part is the short range at r≤1fm, from the vector meson (ρ,ω,⋯) exchange. In order to have a potential of NN interaction, there were many models that serve this point [19–25]. After little development in nuclear properties, the dominant part of the interaction is central, having a strong repulsion at a short range (r≤0.7 fm) and attraction force at an intermediate range (>1 fm). There is a cancelation of major static effects between vector and scalar mesons to maintain the stability of nucleus [26].Now, a variant number of pseudoscalar, scalar, and vector mesons are found; the vast advance of OBEP models related to NN interaction not only for free parameter reduction but also for the accuracy and fitting them with experimental data [12, 27]. The development of quantum field theory and boson field Lagrangian by Heisenberg, Pauli, Dirac, and Rosenfeld in 1930 allows the meson field coordinates to depend on themselves by Yukawa in 1935. Firstly, Yukawa suggested the conjunction of a scalar field coordinate of mesons and then extended to include vector fields by Proca (1936), and Kemmer (1938) embraced the pseudoscalar, axial vector, and antisymmetric tensor. Till now, there are a large number of modifications in the vector-scalar combinations as well as pseudoscalar and pseudovector mesons.The present work represents a motivated model of determining the ground state for deuteron (2H) and helium (4He) based on the NN interaction, the potential related to OBEP with the exchange of pseudoscalar meson (π), scalar meson (σ), and vector meson (ω). This potential is derived analytically with two static functions of mesons; it relies on the Dirac-Hartree-Fock equation. Then, we compare the obtained theoretical results with others and their corresponding experimental data.This paper is arranged as follows. Section2 is devoted to explain the theoretical analysis in details, with three subsections, 2.1, 2.2, and 2.3. Subsection 2.1 refers to how this model uses the Hartree-Fock equation with the Dirac Hamiltonian and how it deals with the wave functions. Subsection 2.2 is related to the mathematical treatment of each term in the Hamiltonian equation. Subsection 2.3 represents the potential of our model. Section 3 represents the results of the potential and ground-state energy for the selected nuclei. Finally, Section 4 is the conclusion.


## 2. Theoretical Analysis

There are several models which determine the structure and properties of the nucleus through the strong force between nucleons in the nucleus [28, 29]. The states of the nucleus are bounded due to this strong force. The nature of nucleon interactions can be described by studying it as a two-body problem. The general wave equation used in such models has the form
(1)H^Ψ=EΨ,where H^ represents the general Hamiltonian operator and E is the eigen energy. We studied the interaction through two bodies via OBEP between two fermions (nucleons), so the convenient representation of the energy is the relativistic form of the Dirac equation.Thus, the accurate interaction of the nuclear system can be described by the Dirac Hamiltonian which includes all fermion interactions and is given by [30–32]
(2)H^=∑iAcα→i.p→i+βi−Imic2−T^+12∑i≠jAVij.SinceI is the unit matrix, α→ and β are (4×4) Dirac matrices, mi is the nucleon mass, p→i is the momentum of the system, T^ is kinetic energy operator, and Vij is the potential energy between fermions’ pairs, we ignore three and many body interactions in the present work. The total kinetic energy of the nucleon equals the total energy subtracted from the rest of the mass energy [33].
(3)T^=E−Mc2,where M=Ami, in which A is the number of nucleons and E is the total relativistic energy which has the form
(4)E=Mc21+p2M2c2.The kinetic energy can be decomposed into two contributions: the first one is the relative space contributionT^r, and the other is the center of mass contribution T^cm [34, 35].
(5)T^=∑iT^r−T^cm=∑iApi22mA−∑iApi22mA.The second part of Equation (5) can be neglected. This neglects the center of mass term ∑iApi2/2mA according to [36]. Applying the binomial theorem for E and substituting into Equation (3), the relativistic kinetic energy T^ takes the form
(6)T^=∑iApi22mA=12m∑i=1Api2−2mA∑i<jApij2,where pij=1/2pi−pj is the relative momentum of the two nucleon systems. By substituting Equation (6) into Equation (2), this leads to the effective nuclear Hamiltonian operator.
(7)H^=∑iAcα→i.p→i+βi−Imic2−12mp2+∑i<jAVij+2mA∑i<jApij2.In Hartree-Fock theory, we seek the best single state given by the lowest energy expectation value of this Hamiltonian.


### 2.1. Variational and Modified Hartree-Fock Wave Function

One is able to ensure the antisymmetry of the fermions’ wave functions with the aid of a Slater determinant and Hartree product to have the convenient form in calculating the ground-state energy as the following wave function which is suitable for fermions [33]. So, the wave function of nucleus Ψr becomes
(8)Ψr=1A!detψir→i,where ψi is the nucleon wave function which can be expanded as
(9)ψir→i=∑αCiαFαr→iwhere Ciα is the oscillator constant and Fαr→i is the oscillator wave function which has two components, radial component Φα and spin component χα.
(10)Fa=ΦaXa.The two components have the following relation between them [17, 37], as
(11)χ=1−ε−v2Mc2σ→·p→2mcϕ.The principle of antisymmetry of the wave function was not clarified by the Hartree method only, but the accurate picture of the ground-state energy calculations should have the Hartree-Fock approximation besides the Slater determinant as in Equation (8) for the wave function. To facilitate the calculation of the wave function, we will use Equation (11) which enables replacing between the two parts of the oscillator wave function with the quantities of 1, 2, and 3. Here, we are dealing with the ground state, so ε−v/c2 makes the value of the second term very small and can be neglected.
(12)χ≅σ→·p→2mcϕThe wave functions for two nucleonsi and j have the formula for bra part Φariϕγrj and ket part ϕβriϕδrj as the bracket needs two wave functions in each side of the bracket:
(13)ϕαriϕγrj=∑mlαmsα∑mlγmsγlαsαmlαmsαjαMαlγsγmlγmsγjγMγϕnαlαmlαriϕnγlγmlγrjχmsα1/2χmsγ1/2P^TαP^Tγ,where lαsαmlαmsαjαMα is the Clebsch-Gordon coefficient, χmsα1/2 is the spin function, and P^Tα is the function of isotopic spin. The two wave functions depend on ri and rj which can be merged to one wave by changing the special coordinates for it that converts to the relative and center of mass coordinates (see Appendix A for more details). Then, we have the formula
(14)ϕαriϕγrj=∑mlαmsα∑mlγmsγ∑lS∑λμ∑nlNL∑mM∑sms∑TlαsαmlαmsαjαMαlγsγmlγmsγjγMγlαlγmlαmlγλμnαlαnγlγNLnllSmlmSJMLlMmλμsαsγmsαmsγSMssαsγTαTγTMTϕnlmrϕNLMRχmsSi,jP^Ti,j,where nαlαnγlγ∣NLnl is the Talmi-Moshinsky bracket and ϕnlmr=RnlmYnlm, with radial function Rnlm and Ynlm the spherical harmonics; the same treatment happens to the ket part. The bracket of the spin function is χmsSi,j∣χmsSi,j=1, and the isotopic function is P^Ti,j∣P^Ti,j=1. This formula is convenient for two-body interaction as in Deuteron, and the number of nucleons of the Helium nucleus should be emerged in an equation through adding Σi<j=14. The bracket for spherical functions equals one as ϑ and φ are not affected here, but the distance r does. We have the solution of radial wave function as an oscillator with the Laguerre function (Leigh, Ritz, and Galerkin) method [38] where the wave function can be expanded in terms of a complete set with basis set:
(15)Rnlm=2n!Γn+l+3/21/21b3/2rblexp−12rb2Lnl+1/2rb2.l represents the angular momentum, Lnl+1/2 is the associated Laguerre polynomial [39], and the length parameter b=ℏc2/mc2ℏω, where m is the mass of the considered particles (nucleons) and ω is the oscillator frequency. The simplest shell model should have the overall size of the nucleus through the scale of this parameter, and it is related to the number density of nucleons or equivalent to ℏω=45A−1/3−25A−2/3 according to the equilibrium density A of the even-even nucleus [40].


### 2.2. The Handling of the Kinetic Energy Term

Using Equations (9), (10), and (7), we obtain the relativistic modified Hartree-Fock equations, and we apply the Lagrange multiplier method for seeking the minimum point of the expression:
(16)∑iαβhiCiα∗CiβFα∣Fβ~=∑iαβCiα∗CiβFαr∣cα→i.p→i+βi−Imic2−12mpi2∣Fβ+∑i<j∑αγβδCiα∗CiβCjγ∗CjδFαFγ∣2Ampij2+Vij∣FβFδ~.Differentiating Equation (16) with respect to Ciα∗ which is the conjugate of the oscillator constant, one has
(17)∑iαβCiβFα∣cαi.pi+βi−Imic2−12mpi2∣Fβ+∑i<j∑αγβδCiβCjγ∗CjδFαFγ∣2Ampij2+Vij∣FβFδ~−∑iαβhiCiβFα∣Fβ=0.Treating the first bracket asH1,
(18)∑iαβCiβFα∣H^1∣Fβ=∑iαβCiβFα∣cα→i.p→i+βi−Imic2−12mpi2∣Fβ.Taking into account Dirac matrices [26, 41], α=0σσ0, β=I00−I, the unit matrix I=1001 and pi=p:
(19)Fα∣H1∣Fβ=ϕα∣σ→.p→σ→.p→2m∣ϕβ+ϕα∣σ→.p→σ→.p→2m∣ϕβ−ϕα∣σ→.p→σ→.p→2m∣ϕβ−ϕα∣p22m∣ϕβ−ϕα∣σ→.p→σ→.p→p22m4m2c2∣ϕβ=0.We ignore the last term for both simplicity and avoiding the fourth power of momentum and speed of lightp4/8m3c4; hence, we have the kinetic term being vanished.


### 2.3. The Construction of the Potential through One-Boson Exchange

After the treatment of the kinetic energy, the two-body Hamiltonian becomes(20)H=∑i<j2AmPij2+Vij.The first term in the right hand side is the remainder part results from the treatment of kinetic term in the Dirac equation as Equation (6), and Vijr is the potential according to the two-body interaction. The relativistic form of one-meson exchange potential between two nucleons i,j based on the degrees of freedom was associated with three, pseudoscalar, scalar, and vector mesons:
(21)Vijr=Vπr+Vσr+Vωr,(22)Vpsr=γioγi5γjoγ5jJps,Vσ=−γioγjoJσ,Vωr=γioγjoγi←μγ→jμJω,where γ→iμγ→jμ=γioγjo−γ→iγ→j.
(23)γio=100−1γ5=iγoγ1γ2γ3=0II0γ→i=I00−I0σi−σi0.The Dirac representation for mesons’ functions will be used to have Dirac matrices corresponding to Pauli spin matrices [30, 42]. Substitute Equations (23) and (22) into Eq. (21) to get Vπ, Vσ, and Vω. We add π-meson as a pseudoscalar one to the previous two mesons because it ties the mesons with the nucleus as it is the fare one. We seek for the stability of the nucleus, and the exchange of pion meson increases the stability of the nucleus. The attractive behavior is represented in scalar σ meson, and the repulsive behavior is represented in vector ω meson. So, the physics of nucleon potential is maintained. The Fock exchange between the two wave functions spatially is introduced after interaction in the potential, briefly in the (~) symbol on the right part (ket part).
(24)FαFγVijFβFδ~=FαFγVπrFβFδ~+FαFγVσrFβFδ~+FαFγVωrFβFδ~=ϕαϕγJπχβχδ~+ϕαχγJπ−ϕβχδ~+χαϕγJπχβ−ϕδ~+χαχγJπ−ϕβ−ϕδ~−ϕaϕγJσϕβϕδ~+ϕαχγJσχβϕδ~+χαϕγJσϕβχδ~−χαχyJσχβχδ~+ϕαϕγJωϕβϕδ~+ϕαχγJωχβϕδ~+χaϕγJωϕβχδ~+χaχγJωχβχδ−ϕαϕγJωσ→i⋅σ→jχβχδ~−ϕαχγJωσ→i⋅σ→jϕβχδ~−χαϕγJωσ→i⋅σ→jχβϕδ~−χαχγJωσ→i⋅σ→jϕβϕδ~.According to the relation betweenϕ and χ in Equation (12) and defining the momentum for each nucleon (i,j) [33, 43], pi=pr+1/2pR, pj=−pr+1/2pR, and pi=p′i, pj=p′j, and pr=p.Substituting those relations into Equation (B.1), We will apply some important relations [44] on this equation. We have used two static functions for the meson degree of freedom in the NN interaction, GY and SPED, with (k=π,σ,ω). These forms were used to carry out our calculations for a Hartree-Fock problem (HF). The first function [30] is represented by
(25)JkGY=gkℏcexp−μkrr−exp−λkrr1+λk2−μk22λkr.We have the following:gk2 is the meson-nucleon coupling constant, λk is a parameter related to the structure function of the form factor, and μk=mc/h is the range of i-meson associated with the meson mass. The second function has the form [33]
(26)JkSPED=gkℏcλk2λk2−μk2exp−μkrr−exp−λkrr.The details to obtain the following equation is explained in AppendicesB and C:
(27)FαFγ∣Vij∣FβFδ~=ϕαϕγ−Jσ+Jω+14m2c22Jσp2−2ℏ2dJσdrddr+2rdJσdrS→.L→+2Jωp2−2ℏ2dJωdrddr+2rdJωdrS·L−6Jωp2+6ℏ2dJωdrddr−6rdJωdrS→·L→+Jωσ→i·σ→j2ℏ2S·p2−Jωσ→i·σ→jp2+2ℏ2S·p2Jωσ→i·σ→j−p2Jωσ→i·σ→j+14m2c2−Jωσ→i·σ→j2ℏ2S·PR2+Jωσ→i·σ→jpR2+1/2pR2Jσ+1/2pR2Jω+−ℏ2m2c22SS+1−3dJπdrddr−1mc22ℏ2S·n^2−1Jπℏω2n+l+32−12mω2r2ϕβϕδ~.With total spin operatorS and the meson function Jr, to simplify the solution and get the result, we suppose the nucleons of equal masses, so the relative mass μ=m1m2/m1+m2 and center mass M=m1+m2.
(28)FαFγ∣Vij∣FβFδ=ϕαϕγ−Jσ+Jω+18μ2c2−ℏ2dJσdrddr+1rdJσdrℏ22JJ+1−LL+1−SS+1+2ℏ2dJωdrddr−2rdJωdrℏ22JJ+1−LL+1−SS+1+14μc2Jσrℏω2n+l+3/2−1/2μω2r2−2Jωrℏω2n+l+3/2−1/2μω2r2+42SS+1−32ℏ2S.n^2−1Jωℏω2n+l+3/2−1/2μω2r2+1Mc2−22SS+1−32ℏ2S.n^2−1Jωℏω2N+L+3/2−1/2Mω2R2+Jσ+Jωℏω2N+L+3/2−1/2Mω2R2Jω+−ℏ2m2c22SS+1−3dJπdrddr−1mc22ℏ2S.n^2−1Jπℏω2n+l+32−12mω2r2ϕβϕδ~.We substituteS·n^ from [45]. The determination of the energy eigen values requires the diagonalization of the Hamiltonian matrix whose elements are calculated with the functions of Equation (7). We have Equation (28) to show that our model can determine a satisfied result for S state with the meson functions Jσ,Jω, and Jomega; the value of mesons’ wave functions depends on distance r which is determined as follows:
(i)
In the case of repulsive meson exchangeω, the lower value of (r) is taken as the hadron radius≃0.5 fm and the upper value is calculated using the following equation, where R is the range of meson and μ is the mass of meson:
(29)R=1μ
(ii)
In the case of the attractive meson exchangeπ and σ, the upper limit of the previous case is taken as the lower limit in this case and the upper limit is determined using Equation (29)In the present work, we have applied our model to calculate the ground states of2H and 4He nuclei (A=2, A=4), respectively. We have determined for two nucleons and four nucleons in 1S1/2-state according to nXj where n=1, j=l+s, and X represents the state. Table 1 shows the group of parameters used for π, σ, and ω mesons. The set of parameters are I and II that include mass μ, coupling constant (g), and the cutoff λ parameters.Table 1
The meson parameters for OBEP for different sets.
Ref
Meson
Mass (MeV)
Coupling constantgi
Cutoff parameterλ (MeV)
Set I
[46]
π
138.03
14.9
2000
σ
700
16.07
2000
ω
782.6
28
1300
Set II
[46]
π
138.03
14.40
1700
σ
710
18.37
2000
ω
782.6
24.50
1850We have determined the ratioR, to ensure the accuracy between the calculated results and the experimental data [47].
(30)R=Etheor.Eexp.,where Etheor is the calculated ground state and Eexp the experimental one. We can also determine the binding energy per nucleon E/A for the studied nuclei as [48]
(31)EA=−Eg.s.A,with the mass number A and the total ground state energy Eg.s..


## 2.1. Variational and Modified Hartree-Fock Wave Function

One is able to ensure the antisymmetry of the fermions’ wave functions with the aid of a Slater determinant and Hartree product to have the convenient form in calculating the ground-state energy as the following wave function which is suitable for fermions [33]. So, the wave function of nucleus Ψr becomes
(8)Ψr=1A!detψir→i,where ψi is the nucleon wave function which can be expanded as
(9)ψir→i=∑αCiαFαr→iwhere Ciα is the oscillator constant and Fαr→i is the oscillator wave function which has two components, radial component Φα and spin component χα.
(10)Fa=ΦaXa.The two components have the following relation between them [17, 37], as
(11)χ=1−ε−v2Mc2σ→·p→2mcϕ.The principle of antisymmetry of the wave function was not clarified by the Hartree method only, but the accurate picture of the ground-state energy calculations should have the Hartree-Fock approximation besides the Slater determinant as in Equation (8) for the wave function. To facilitate the calculation of the wave function, we will use Equation (11) which enables replacing between the two parts of the oscillator wave function with the quantities of 1, 2, and 3. Here, we are dealing with the ground state, so ε−v/c2 makes the value of the second term very small and can be neglected.
(12)χ≅σ→·p→2mcϕThe wave functions for two nucleonsi and j have the formula for bra part Φariϕγrj and ket part ϕβriϕδrj as the bracket needs two wave functions in each side of the bracket:
(13)ϕαriϕγrj=∑mlαmsα∑mlγmsγlαsαmlαmsαjαMαlγsγmlγmsγjγMγϕnαlαmlαriϕnγlγmlγrjχmsα1/2χmsγ1/2P^TαP^Tγ,where lαsαmlαmsαjαMα is the Clebsch-Gordon coefficient, χmsα1/2 is the spin function, and P^Tα is the function of isotopic spin. The two wave functions depend on ri and rj which can be merged to one wave by changing the special coordinates for it that converts to the relative and center of mass coordinates (see Appendix A for more details). Then, we have the formula
(14)ϕαriϕγrj=∑mlαmsα∑mlγmsγ∑lS∑λμ∑nlNL∑mM∑sms∑TlαsαmlαmsαjαMαlγsγmlγmsγjγMγlαlγmlαmlγλμnαlαnγlγNLnllSmlmSJMLlMmλμsαsγmsαmsγSMssαsγTαTγTMTϕnlmrϕNLMRχmsSi,jP^Ti,j,where nαlαnγlγ∣NLnl is the Talmi-Moshinsky bracket and ϕnlmr=RnlmYnlm, with radial function Rnlm and Ynlm the spherical harmonics; the same treatment happens to the ket part. The bracket of the spin function is χmsSi,j∣χmsSi,j=1, and the isotopic function is P^Ti,j∣P^Ti,j=1. This formula is convenient for two-body interaction as in Deuteron, and the number of nucleons of the Helium nucleus should be emerged in an equation through adding Σi<j=14. The bracket for spherical functions equals one as ϑ and φ are not affected here, but the distance r does. We have the solution of radial wave function as an oscillator with the Laguerre function (Leigh, Ritz, and Galerkin) method [38] where the wave function can be expanded in terms of a complete set with basis set:
(15)Rnlm=2n!Γn+l+3/21/21b3/2rblexp−12rb2Lnl+1/2rb2.l represents the angular momentum, Lnl+1/2 is the associated Laguerre polynomial [39], and the length parameter b=ℏc2/mc2ℏω, where m is the mass of the considered particles (nucleons) and ω is the oscillator frequency. The simplest shell model should have the overall size of the nucleus through the scale of this parameter, and it is related to the number density of nucleons or equivalent to ℏω=45A−1/3−25A−2/3 according to the equilibrium density A of the even-even nucleus [40].


## 2.2. The Handling of the Kinetic Energy Term

Using Equations (9), (10), and (7), we obtain the relativistic modified Hartree-Fock equations, and we apply the Lagrange multiplier method for seeking the minimum point of the expression:
(16)∑iαβhiCiα∗CiβFα∣Fβ~=∑iαβCiα∗CiβFαr∣cα→i.p→i+βi−Imic2−12mpi2∣Fβ+∑i<j∑αγβδCiα∗CiβCjγ∗CjδFαFγ∣2Ampij2+Vij∣FβFδ~.Differentiating Equation (16) with respect to Ciα∗ which is the conjugate of the oscillator constant, one has
(17)∑iαβCiβFα∣cαi.pi+βi−Imic2−12mpi2∣Fβ+∑i<j∑αγβδCiβCjγ∗CjδFαFγ∣2Ampij2+Vij∣FβFδ~−∑iαβhiCiβFα∣Fβ=0.Treating the first bracket asH1,
(18)∑iαβCiβFα∣H^1∣Fβ=∑iαβCiβFα∣cα→i.p→i+βi−Imic2−12mpi2∣Fβ.Taking into account Dirac matrices [26, 41], α=0σσ0, β=I00−I, the unit matrix I=1001 and pi=p:
(19)Fα∣H1∣Fβ=ϕα∣σ→.p→σ→.p→2m∣ϕβ+ϕα∣σ→.p→σ→.p→2m∣ϕβ−ϕα∣σ→.p→σ→.p→2m∣ϕβ−ϕα∣p22m∣ϕβ−ϕα∣σ→.p→σ→.p→p22m4m2c2∣ϕβ=0.We ignore the last term for both simplicity and avoiding the fourth power of momentum and speed of lightp4/8m3c4; hence, we have the kinetic term being vanished.


## 2.3. The Construction of the Potential through One-Boson Exchange

After the treatment of the kinetic energy, the two-body Hamiltonian becomes(20)H=∑i<j2AmPij2+Vij.The first term in the right hand side is the remainder part results from the treatment of kinetic term in the Dirac equation as Equation (6), and Vijr is the potential according to the two-body interaction. The relativistic form of one-meson exchange potential between two nucleons i,j based on the degrees of freedom was associated with three, pseudoscalar, scalar, and vector mesons:
(21)Vijr=Vπr+Vσr+Vωr,(22)Vpsr=γioγi5γjoγ5jJps,Vσ=−γioγjoJσ,Vωr=γioγjoγi←μγ→jμJω,where γ→iμγ→jμ=γioγjo−γ→iγ→j.
(23)γio=100−1γ5=iγoγ1γ2γ3=0II0γ→i=I00−I0σi−σi0.The Dirac representation for mesons’ functions will be used to have Dirac matrices corresponding to Pauli spin matrices [30, 42]. Substitute Equations (23) and (22) into Eq. (21) to get Vπ, Vσ, and Vω. We add π-meson as a pseudoscalar one to the previous two mesons because it ties the mesons with the nucleus as it is the fare one. We seek for the stability of the nucleus, and the exchange of pion meson increases the stability of the nucleus. The attractive behavior is represented in scalar σ meson, and the repulsive behavior is represented in vector ω meson. So, the physics of nucleon potential is maintained. The Fock exchange between the two wave functions spatially is introduced after interaction in the potential, briefly in the (~) symbol on the right part (ket part).
(24)FαFγVijFβFδ~=FαFγVπrFβFδ~+FαFγVσrFβFδ~+FαFγVωrFβFδ~=ϕαϕγJπχβχδ~+ϕαχγJπ−ϕβχδ~+χαϕγJπχβ−ϕδ~+χαχγJπ−ϕβ−ϕδ~−ϕaϕγJσϕβϕδ~+ϕαχγJσχβϕδ~+χαϕγJσϕβχδ~−χαχyJσχβχδ~+ϕαϕγJωϕβϕδ~+ϕαχγJωχβϕδ~+χaϕγJωϕβχδ~+χaχγJωχβχδ−ϕαϕγJωσ→i⋅σ→jχβχδ~−ϕαχγJωσ→i⋅σ→jϕβχδ~−χαϕγJωσ→i⋅σ→jχβϕδ~−χαχγJωσ→i⋅σ→jϕβϕδ~.According to the relation betweenϕ and χ in Equation (12) and defining the momentum for each nucleon (i,j) [33, 43], pi=pr+1/2pR, pj=−pr+1/2pR, and pi=p′i, pj=p′j, and pr=p.Substituting those relations into Equation (B.1), We will apply some important relations [44] on this equation. We have used two static functions for the meson degree of freedom in the NN interaction, GY and SPED, with (k=π,σ,ω). These forms were used to carry out our calculations for a Hartree-Fock problem (HF). The first function [30] is represented by
(25)JkGY=gkℏcexp−μkrr−exp−λkrr1+λk2−μk22λkr.We have the following:gk2 is the meson-nucleon coupling constant, λk is a parameter related to the structure function of the form factor, and μk=mc/h is the range of i-meson associated with the meson mass. The second function has the form [33]
(26)JkSPED=gkℏcλk2λk2−μk2exp−μkrr−exp−λkrr.The details to obtain the following equation is explained in AppendicesB and C:
(27)FαFγ∣Vij∣FβFδ~=ϕαϕγ−Jσ+Jω+14m2c22Jσp2−2ℏ2dJσdrddr+2rdJσdrS→.L→+2Jωp2−2ℏ2dJωdrddr+2rdJωdrS·L−6Jωp2+6ℏ2dJωdrddr−6rdJωdrS→·L→+Jωσ→i·σ→j2ℏ2S·p2−Jωσ→i·σ→jp2+2ℏ2S·p2Jωσ→i·σ→j−p2Jωσ→i·σ→j+14m2c2−Jωσ→i·σ→j2ℏ2S·PR2+Jωσ→i·σ→jpR2+1/2pR2Jσ+1/2pR2Jω+−ℏ2m2c22SS+1−3dJπdrddr−1mc22ℏ2S·n^2−1Jπℏω2n+l+32−12mω2r2ϕβϕδ~.With total spin operatorS and the meson function Jr, to simplify the solution and get the result, we suppose the nucleons of equal masses, so the relative mass μ=m1m2/m1+m2 and center mass M=m1+m2.
(28)FαFγ∣Vij∣FβFδ=ϕαϕγ−Jσ+Jω+18μ2c2−ℏ2dJσdrddr+1rdJσdrℏ22JJ+1−LL+1−SS+1+2ℏ2dJωdrddr−2rdJωdrℏ22JJ+1−LL+1−SS+1+14μc2Jσrℏω2n+l+3/2−1/2μω2r2−2Jωrℏω2n+l+3/2−1/2μω2r2+42SS+1−32ℏ2S.n^2−1Jωℏω2n+l+3/2−1/2μω2r2+1Mc2−22SS+1−32ℏ2S.n^2−1Jωℏω2N+L+3/2−1/2Mω2R2+Jσ+Jωℏω2N+L+3/2−1/2Mω2R2Jω+−ℏ2m2c22SS+1−3dJπdrddr−1mc22ℏ2S.n^2−1Jπℏω2n+l+32−12mω2r2ϕβϕδ~.We substituteS·n^ from [45]. The determination of the energy eigen values requires the diagonalization of the Hamiltonian matrix whose elements are calculated with the functions of Equation (7). We have Equation (28) to show that our model can determine a satisfied result for S state with the meson functions Jσ,Jω, and Jomega; the value of mesons’ wave functions depends on distance r which is determined as follows:
(i)
In the case of repulsive meson exchangeω, the lower value of (r) is taken as the hadron radius≃0.5 fm and the upper value is calculated using the following equation, where R is the range of meson and μ is the mass of meson:
(29)R=1μ
(ii)
In the case of the attractive meson exchangeπ and σ, the upper limit of the previous case is taken as the lower limit in this case and the upper limit is determined using Equation (29)In the present work, we have applied our model to calculate the ground states of2H and 4He nuclei (A=2, A=4), respectively. We have determined for two nucleons and four nucleons in 1S1/2-state according to nXj where n=1, j=l+s, and X represents the state. Table 1 shows the group of parameters used for π, σ, and ω mesons. The set of parameters are I and II that include mass μ, coupling constant (g), and the cutoff λ parameters.Table 1
The meson parameters for OBEP for different sets.
Ref
Meson
Mass (MeV)
Coupling constantgi
Cutoff parameterλ (MeV)
Set I
[46]
π
138.03
14.9
2000
σ
700
16.07
2000
ω
782.6
28
1300
Set II
[46]
π
138.03
14.40
1700
σ
710
18.37
2000
ω
782.6
24.50
1850We have determined the ratioR, to ensure the accuracy between the calculated results and the experimental data [47].
(30)R=Etheor.Eexp.,where Etheor is the calculated ground state and Eexp the experimental one. We can also determine the binding energy per nucleon E/A for the studied nuclei as [48]
(31)EA=−Eg.s.A,with the mass number A and the total ground state energy Eg.s..


## 3. Results and Discussion

Table1 represents the group of parameters used for π, σ, and ω mesons. The set of parameters are I and II that include the mass of meson, the coupling constant (g), and the cutoff parameter (λ). The potential is elaborated to calculate the ground-state energies for the H2, He4 nuclei. The results are listed in Tables 2 and 3 in comparison with the experimental data. The ratio between the present work and experimental one is estimated for both cases, in other words, by using the potential extracted from GY and SPED functions.Table 2
The ground-state energy of2H with the aid of Table 1.
Parameter sets [46]
Meson exchange
Present work (GY)
Present work (SPED)
Others
exp. [49–51]
Ratio GY
Ratio SPED
E/A GY
E/A SPED
E/Aexp. [52]
I
σ, ω
−2.916
−2.041
-2.215[53]
1.311
0.918
1.458
1.0205
II
−3.486
−1.973
−2.224
1.567
0.887
1.743
0.9865
1.112
I
π, σ, ω
−2.199
−2.248
-2.220
0.989
1.011
1.0995
1.124
II
−2.168
−2.204
±0.179[54]
0.975
0.991
1.084
1.102Table 3
The ground-state energy of4He with the aid of Table 1.
Parameter sets [46]
Meson exchange
Present work (GY)
Present work (SPED)
[55]
exp. [47, 56]
Ratio GY
Ratio SPED
E/A GY
E/A SPED
E/A exp.
I
σ, ω
−22.372
−20.238
1.0966
0.992
5.593
5.0595
II
−22.751
−21.556
-21.385
-20.4±0.3
1.115
1.057
5.6877
5.389
5.1
I
π, σ, ω
−22.637
−20.375
1.109
0.999
5.659
5.0937
II
−21.871
−20.337
1.072
0.997
5.4677
5.08425We have examined potential Equation (28) to calculate the ground-state energy of H2 and He4 nuclei using two static meson functions (GY and SPED) with two sets of parameters listed in Table 1 which shows the different sets of the used parameters and for different exchange mesons, σ and ω mesons and π, σ, and ω mesons. The potential for different cases is plotted in Figures 1−8.Figure 1
The potential with GY and SPED functions, respectively, of2H nuclei as OBEP through the exchange of σ and ω mesons, with parameter I.
(a)
(b)Figure 2
The potential with GY and SPED functions, respectively, of4He nuclei as OBEP through the exchange of σ and ω mesons, with parameter I.
(a)
(b)Figure 3
The potential with GY and SPED functions, respectively, of2H nuclei as OBEP through the exchange of σ and ω mesons, with parameter II.
(a)
(b)Figure 4
The potential with GY and SPED functions, respectively, of4He nuclei as OBEP through the exchange of σ and ω mesons, with parameter II.
(a)
(b)Figure 5
The potential with GY and SPED functions, respectively, of2H nuclei as OBEP through the exchange of π, σ, and ω mesons, with parameter I.
(a)
(b)Figure 6
The potential with GY and SPED functions, respectively, of4He nuclei as OBEP through the exchange of π, σ, and ω mesons, with parameter I.
(a)
(b)Figure 7
The potential with GY and SPED functions, respectively, of2H nuclei as OBEP through the exchange of π, σ, and ω mesons, with parameter II.
(a)
(b)Figure 8
The potential with GY and SPED functions, respectively, of4He nucleus as OBEP through the exchange of π, σ, and ω mesons, with parameter II.
(a)
(b)Figures1−8 illustrate the potential Vr in MeV versus r in fm for H2 and He4 nuclei for both cases (GY and SPED) and for different sets of parameters (I and II) (Table 1).Potential energy Equation (28) is illustrated in Figures 1–8 by two sets of parameters. We have checked them with different meson exchange function GY and SPED. So, we categorize our results into two groups: for the set I parameter with GY and SPED calculated within σ and ω and for the set II parameter the same above for both nuclei (H2 and He4).All cases are calculated again withinσ, ω, and π, in other words, by adding the third exchange meson “π” which works attractively at large r. Figures 1–4 represent category “I, II” for two-meson exchange, respectively, and Figures 5–8 represent category “I, II” for three-meson exchange. Figure 1(a) shows the potential by using the GY meson function in which the effect of repulsive potential due to ω meson appears at quite large distance, while the attractive part does not appear clearly as the depth of the potential is very small, the attractive part began with r~2.0fm near to the diameter of H2 nuclei and finished at r~0.7fm.Figure1(b) is calculated by using the SPED meson function; in this case, a significant attractive potential began with r~1.1fm and ended at r~0.25fm. (Figure 2) represents the same manner of H4 nuclei with different values of the potential; the transferred point between the attractive and repulsive parts is similar to the one in (Figure 1), and the beginning point of the attractive part is controlled by the diameter of nuclei.It can be noticed that the depth of the attractive potential extracted by the SPED function is greater than the one from the GY function for both nuclei (Figures1 and 2). But it seems that, using set II, the behavior of two functions is close to each other from the transferred point and the depth; the difference between two nuclei is still the values of the attractive potential, and ω meson is the master here in which the repulsive potential has more higher values and the effect of σ has been damped (see Figures 3 and 4). The effect of the third meson exchange π meson is added as shown in Figures 5 and 6 for set I and Figures 7 and 8 for set II. Firstly, again for set I, Figures 5 and 6 represent the potential that behaves as the same before for two-meson exchange, in which the attractive part increased very slowly. The depth of the attractive potential increased significantly, π meson flies at more than r~1.5fm, and the transferred point has different values from the one in the two-meson exchange by using the SPED function. On the other hand, for set II as shown in Figures 7 and 8, GY and SPED have an improvement in their transferred point and depth than in the two-meson exchange; the SPED function is still better than the GY function.We observe from Tables2 and 3 two sets of parameters and how many mesons to be exchanged between two nucleons and meson exchanges with two functions, and we listed our results for each one for the selected nuclei. Meanwhile, if the ratio tends to unite, the ground energies would be close to the experimental data. The preferable theoretical value of the 2H nucleus in case of using two mesons is the SPED function for parameter I, and by using three mesons, we have the value of SPED in parameter II to be more accurate than others. It is obvious from Tables 2 and 3 that the ground energy is close to the data in the case of SPED function for set I and set II in comparison with the experimental data. The 4He nucleus has a little different manner; the theoretical value of the SPED function for two mesons in parameter I is better than the value of the GY function. In case of handling three mesons, SPED function in parameter II is the best as shown in Table 3. Using the ratio relation is useful to ensure that our results for the SPED function is better than the GY function, and our attempt to include more two mesons in OBEP analytically is successful in result improvement. The ratio is getting a better result for going on more massive nuclei and encouraged for our potential. We concluded that the used model is well-defined and compatible with the data and even with other models (see [57, 58]). The deuteron ground-state energy is quiet a little different from the numerical data in [54, 59] as a good sign for our constructing potential analytically. The calculation of binding energy per nucleon serves our idea of being the OBEP with three and four mesons in the case of SPED function and gives satisfied values for Deuteron and Helium nuclei comparing with the experimental one.


## 4. Conclusion

In the framework of quasirelativistic formulation, the meson exchange potential helps in obtaining a potential with few numbers of parameters to calculate the ground state for the light nuclei Deuteron and Helium using two- (σ and ω) and three- (π, σ, and ω) meson exchange. In addition, it was shown that a self-consistent treatment of the semirelativistic nucleon wave function in nuclear state has a great importance in calculations. The difference in masses of σ and ω mesons would not seriously change the main aspect of the concept of relativistic or semirelativistic interaction, providing an average potential of cancelation of the repulsive meson (ω) and the attractive meson (σ) in conjunction with a weak long-range effect (π). This work with OBEP in the Dirac-Hartree-Fock equation gives a close relationship to other recent approaches, based upon different formalisms which tended to support this direction. The ground-state energies for 2H and 4He nuclei are successfully determined through this work and give us a hope to continue with more massive nuclei. The nuclear properties are being clear in our trail to include two more mesons to describe the NN interaction through our potential. The SPED function has a good ability to give us the better shapes of our potential and also better values for energies. We hope that our potential represents a base for the NN interaction with different ranges of energies in the following search.



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