Study of the ground state energies of some nuclei using hybrid model

The quark-quark QQ interaction as a perturbed term to the nucleon-nucleon interaction NN without any coupling between them is studied in a hybrid model. This model is used to calculate the ground-state energies of 2H1 and 4He2 nuclei. In a semi-relativistic framework, this model is encouraged for light nuclei and the instanton induced interaction by using the QQ potential and the NN interaction for a small scale around the hadron boundaries. This hybrid model depends on two theories, the one-boson exchange potential OBEP and the Cornell-dressed potential CDP for QQ. A small effect of quark-quark interaction is obtained on the values of the ground state energies, around 6.7 for 2H1 and 1.2 for 4He2 by using the considered hybrid model.

mesons outside it can be founded based on the variational concept of physics. In the present work, The ground state energies for some light nuclei can be calculated successfully by using the considered hybrid model.
In section II, we introduce the theory of N N interactionthrough the one-boson exchange potential with investigations and motivations of the formula. In section III, we have a brief look at the QQ interaction, and the reason for choosing Cornell dressed potential is mentioned to make the idea of the hybrid model possible. In section IV, where the theoretical analysis for the construction of one-boson exchange potential through the exchange of two, three, and four mesons is clarified. Section V shows the theoretical analysis of QQ interaction and the final form of CDP . Finally, in sections VI and VII, the obtained results and conclusion are given.

II. THEORY OF N N THROUGH THE OBEP
The start of using the fact that there is no unique potential for determination of the effective N N potential, leads to exist different forms with different methods. So, this work concerned to show the effect of our potential which is published in previous work [1]. Our potential is constructed with the idea of one-boson exchange and also depends on the motion of nucleons in the nucleus. This motion produced a field called nuclear mean field, the interaction between nucleons is controlled with Pauli principle and nuclear shell-model. We considered the spatial exchange between two nucleons so, the non-local field is determined with Hartree-Fock approximation. Since the Fock effect is demonstrated in the non-vanishing spatial components for the vector part of the potential. Our potential implies Dirac-Hartree-Fock method to determine the wave function and energy of a quantum N-body system in a stationary state. We classified our potential as a semi-relativistic model because of neglecting the fourth power of momentum to simplify the formula and it will be included in following work. Our potential is associated with Bonn group to have the meson's function and its parameters. To calculate the ground state energies of Hartree-Fock approximation, we need to minimize the total energy of single particle potential by the Steepest descent methods directly to have the lowest energy. It is demanded a modification for nucleons wave functions and energy. The modification of wave function is demonstrated by Clebsch-Gordon coefficients, and Talmi-Moshinsky harmonic oscillator bracket, affecting on the radial, spin and isotopic wave functions. We use the formalism of second quantization just as a convenient way of handling antisymmetric wave function. This formalism referred as a representation of the occupation number, hence it leads to be represented in the Fock-state basis which can be constructed by filling up each single-particle state with a certain number of identical particles. As a real space basis, we write the antisymmetric wave functions in a Slater-determinant. Second quantization give us the ability to displace the wave function as a Dirac-state and do the same for Slater-determinant. So, we use operators to specify the occupied orbitals and the field operators to define the coordinates for the real space representation. It is noticed that, the atom in a quantum state of energy E depends only on that energy through the Boltzmann factor and not on any other property of the state when we represent this atom in complete thermal equilibrium to determine the ground state energy for the considered nuclei. We use this fact to neglect the tensor force for the Deuteron nucleus and calculate its wave function in S-state only. The fact of being the vector mesons and QCD affected on the nuclear properties at short range, hence the studying of nucleon-nucleon interaction through OBEP should not be enough. The exchange of bosons with OBE potential models comes about more than size of nucleon or equal to the inter-nucleon distances. We have the effect of QCD dynamics at distance less than or almost around the boundaries of hadron and that is necessary for the description of nucleon-nucleon interaction. The quantitative theoretical models can analyze the experimental data based on the degrees of hadrons over the last three decades [2][3][4][5][6][7] and also the quark degrees of freedom in QCD models are a successful models for the description of the nuclear properties [8][9][10][11]. These models analyze the static properties of baryon successfully. We are concerned to add the quark degrees of freedom as a perturbed term to the meson degrees of freedom and have a hamiltonian equation of two parts as following.
Where the hamiltonian of the nucleon-nucleon interaction is H N N and the hamiltonian of the quark-quark interaction is H QQ . So, we study the OBEP with the exchange of three and four mesons as it represents the nucleon-nucleon interaction, and the Cornell dressed potential as the quark-quark interaction for constructing more realistic model of the nuclear properties.

III. THEORY OF QQ THROUGH THE CDP
The simulation of quark-quark interaction phenomenon in a semi-relativistic frame work shows that the long range part of this interaction increasing linearly with the distance and called it confinement, and the short range of interaction is a result of Coulomb-like interaction (one-gluon exchange). The idea of considering the contribution of the constant potential is dominant than the other contributions to the quark-quark potential worth good thinking of it as in [12,13], and a good spectra of mesons and baryons are obtained. At first, very good results for the charmonium spectrum obtained from a simple form of a potential called Funnel potential or Cornell potential [14,15]. The Hamiltonian of hadrons containing light quark should simultaneously define a number of relativistic corrections. The momentum-dependent corrections as well as a non-local kinetic energy (there is no commutation with faster than light and it is compatible with special relativity) are required to be included in the effects of relativistic kinematics of the potential of the potential energy operator [16]. These relativistic kinematics are included in the Bethe-Salpter equation, neglecting the spin effect which introduce non-local modifications of the relative coordinate. The spinless Salpter equation has the form, This is suitable for baryon where H is the total energy of the system, V is the central potential between two particles (i, j) and p is their relative momentum. In case of baryon m i is the constituent masses of quarks with the same mass of u and d quarks (isospin symmetry is maintained). The central potential is the Cornell-potential, The factor half is related to the half rule, k is the Coulomb parameter, a is the string constant and C is additive constant equals zero in the heavy quark sector. To solve the spinless-Salpter equation, [16] expanded the wave function in terms of a complete set of basis functions according to Rayleigh, Ritz and Galerkin method as the previous radial wave function in OBEP . Many symmetries are slightly broken in nature as it can give rise the classical solutions to a particular symmetry-breaking amplitude. This amplitude is similar to the tunneling effect, indeed the classical solutions of the equation of motion can sometimes describe the tunneling through a barrier. The classical solution of equations of motion was introduced in Yang-Mills theory, known as 'instanton' term. The computation of the quantum effects of instantons was introduced by 't Hooft [17] firstly. In [12,13], the authors went with instanton induced interaction (it is a solution to the equations of motion of the classical field theory, it is supposed to be critical points of the action for such quantum theories), In non-relativistic quark model, it is assumed that this model is based on the confinement potential and a residual interaction. The residual interaction is related to the reduction of the one-gluon exchange OGE. One able to compute the residual interaction by 't Hooft force from instanton effects [17,18]. Ref [14] proposed a model of quarks interaction with the replacement of the traditional OGE potential by a non-relativistic limit of 't Hooft's interaction. The residual interaction is observed by 't Hooft as an expansion of the Euclidean action around the single instanton solutions under the assumption of zero mode in the fermion sector. This interaction has an effective lagrangian with effective potential between two quarks. The instanton calculus can be summarized by four steps [19].
• The gluon fields can not deform the instantons into classical solutions continuously.
• The perturbative gluon diagrams can not cover the effective interaction between quarks which caused by instantons.
• The instanton calculus denotes as a non-perturbative method for the calculation of path integrals, which are represented in the fluctuations around the instanton and change the action. All of this is normally done in the Gaussian approximation.
• The instanton effects in QCD realized that instanton is similar to be described as 4-dimensional gas of pseudo particles, then use the summation over the instanton gas.
In the first analogy of super conductivity with the Bardee-Cooper-Schrieffer theory [20]. When the interaction between fermions(nucleons) and light quarks are attracted strongly at the short range, this interaction can rearrange the vacuum and the ground state affected by it which resembles the effect of super conductivity. Then, the short range interaction can bind these constituent light quarks into hadrons without confinement in order to make quantitative predictions for hadronic observable. It is clarified that instanton is represented as a tunneling event between vacua [21].

IV. THEORETICAL ANALYSIS OF OBEP
The general form which describe the ground state energy of the considered system is the following.
Where H is the Hamiltonian and E is the total energy of the system.
Defining T is kinetic energy and V is the potential energy. Hence the Hamiltonian of fermions interacting via the potential V ij . Thus the accurate Hamiltonian interaction of the nuclear system can be described by Dirac to represent the number of fermions s' interaction where this Hamiltonian is [22][23][24][25], Since α and β are 4 × 4 Dirac matrices, c is the speed of light, m i is the nucleon mass, T ij is the relative kinetic energy and p is the momentum operator. See appendix A for the details of relative kinetic energy calculations to have the following equation [26][27][28][29].
Substituting the last equation in Eq (5), thus the relativistic Hamiltonian operator for bound nucleons which interact strongly through the potential can be expressed as following.
In Hartree-Fock theory, we seek for the best state giving the lowest energy expectation value of this hamiltonian to determine the ground state energy of the considered nuclei. One able to ensure the antisymmetry of the fermions' wave functions with the aid of Slater Determinant introduced in(1929) and Hartree product to have the convenient form in calculating the ground state energy as the following wave function which is suitable for fermions [23], Where the wave function of all nucleons Ψ(r), and the wave function for i-nucleon Ψ i (r). The wave function for nucleon i depends on the oscillator parameter as, Where C iα is the oscillator constant and F α is the wave function of two components.
With the wave function for radial component Φ α , and the spin component χ α . The principle of antisymmetry of the wave function was not completely explained by the Hartree method according to Slater and Fock independently. So the accurate picture in calculating the ground state energy is the Hartree-Fock approximation.
Where C iβ is the occupation number or the oscillator number( For a system consisting of fermions, or particles with half-integral spin, the occupation numbers may take only two values; 0 for empty states or 1 for filled states). The two components of the wave functions have the following relation between them [30,31].
Using the relation Eq (12), where ε is external energy which equals zero, here we are deal with ground state and c 3 makes the value so small and can be neglected.
Differentiate Eq (11) with respect to C iα , the F β F δ has a sign defines the exchange that happening between the two nucleons, hence substituting Treating with the 1st part of Eq (15) give us the coming formula Taking into account Dirac matrices [32] with defining the wave functions in bracket as Substituting α, β in H 1 and have the result The 1st term of kinetic energy tends to zero and this result has agreement with another calculations in [33] After the treatment of the kinetic terms are done, the residual Hamiltonian of the expectation value becomes, The popular form of the force between two nucleons is cleared according to meson exchanges. The potential form of one boson exchange V ij between two nucleons (i, j) based on the degrees of freedom associated with three mesons, pseudoscalar, scalar, and vector mesons.
The Dirac representation for functions of mesons will be used and Dirac matrices corresponding to pauli spin matrices [22,34], according to this representation we use V ω (r) = V ρ (r). Where Substituting Eq. (20) and Eq. (24) into Eq. (19) to get the expectation value by three potentials V π , V σ and V ω .
According to the relation between φ , χ in Eq. (13), one obtain Defining the momentum for each nucleon (i, j) We will apply some important relations [36] 1.
Including these relations in potential equation. we substitute every term by using the relation of angular momentum L = r × p, σ = 2 S (where S is the total spin operator), p = −ı ∇, and ∇J σ = 1 r ( dJσ dr )r. According to the previous relations, and where σ j 2 = σ x 2 + σ y 2 + σ z 2 = 1 as triplet case for two nucleons With total spin operator S and the meson function J(r), using [37] ( Quantum mechanics have a magnificent tool, this tool is the harmonic oscillator which is capable of being solved in closed form, it has generally useful approximations and exact solutions of different problems [38]. It solves the differential equations in quantum mechanics. We have the energy of Harmonic Oscillator( ω(2n+ l + 3/2)) which equals the kinetic energy( p 2 2m ) added to the potential energy((1/2)mω 2 x 2 ) to simplify the solution and get the result. It is slitted in relative harmonic oscillator energy ω(2n + l + 3 2 ) = p 2 2µ + 1 2 µω 2 r 2 [29,39], with ω that is the angular frequency, and center of mass contribution in harmonic oscillator energy ω(2N + L + 3 2 ) = p 2 2M + 1 2 M ω 2 R 2 .We suppose the nucleons have average masses mn+mp 2 , so the relative mass µ = m1m2 m1+m2 = m 2 , and center mass M = m 1 + m 2 = 2m.
The wave functions of the two nucleons Eq (26)should be treated as following .
See appendix B to have the final formula.
The bracket n α l α n γ l γ |N Lnl represents the Talmi Moshinsky bracket. The same treatment for the ket part |φ β (r i )φ δ (r j ) to have we have the wave function φ N LM (R) = R N LM (R)Y N LM (ϑ, ϕ) as radial part (R) and angular part (Y ) for center of mass coordinates, the wave function φ nlm (r) = R nlm (r)Y nlm (ϑ, ϕ) as radial part (R), and angular part (Y ) for relative coordinates. The two-nucleons interaction formula through the exchange of four mesons where p ij = p and A is the mass number of the required nuclei. We define the bracket χ s ms (i, j)|χ s ms (i, j) = 1, P T (i, j)|P T (i, j) = 1 and Y N LM Y nlm |Y N LM Y nlm = 1 as the terms of equation depend on (r).
We have the formula of radial wave function which involves the length parameter b = mω with angular frequency ω and the associated Lageurre polynomial L The differentiation of Radial function equals Defining the operator S.n [37]as The meson degrees of freedom have some static functions J k for description, but here we choose GY and SP ED for meson k and (k = π, σ, ω, ρ).
Where the meson-nucleon coupling constant g 2 k , the cut off λ k and the mass of the meson is associated with µ k = mc . The second function has the form [23],

V. THEORETICAL ANALYSIS OF CDP
There is an explicit spin dependence for the instanton interaction unlike one-gluon exchange, it contains a projector on spin S = 0 states. The distribution of this interaction represented with δ( r) replaced by a Gaussian function with range Λ.
with Λ is the range of the pairing force(QCD scale parameter).
Where g,ǵ are two dimensioned coupling constants according to quark flavors.This equation is under condition of (l = s = 0, I = 0), where l, s, I denotes angular momentum, spin and isotopic spin quantum numbers respectively for nn pair, and the form of instanton contributions represents as [40]. The pairing force depends on the value of the parameters g andǵ, if we set g for strange flavor with symbol s andg for non-strange flavor with symbol n. The Hamiltonian contains diagonal parts in the isoscalar space (|nn ,|ss ).
Having the coupling constant as,g The parameter g ef f denotes the strength and is defined as [15], Where d o (ρ) is a function instanton density of the instanton size ρ, For three colors and three flavors this quantity is given in [15], m o i is the current mass of flavor i and the quark condensate for this flavor is c i = ( 2 3 )π 2 q i q i , q i q i (non-vanishing expectation values). The integration over ρ c which is the maximum size of the instanton.
The constituent masses are the re-normalization of quarks' masses which demonstrate the contribution of the constituent masses [15].
m o n is the current mass of non-strange quark, ∆m n is the contribution of constituent mass [12] with free parameter δ n added to the running masses. The contribution of the constituent masses has the following formula.
It is important to replace the dimensional instanton size [13] as x = Λρ with a dimensionless quantity with using the definition of d o (ρ), This dimensionless integration should still have small value of ln ln-term. It is involved in the parameterś g, ∆m n .g The functions α 9 (x c ), α 11 (x c ), α 13 (x c ) given in [13], the m o s is the constituent mass of strange flavor and also the c s is the quark condensate related to the strange flavor. It is supposed that the quark as an effective degrees of freedom is dressed by the gluon and quark-anti quark pair clouds(constituent masses) and it is natural to express the probability density of quark configuration as a Gaussian function around its average position.
Where ρ i (r) is the probability density not the instanton size as previous with γ i the size parameter and it is dependent on the quark mass flavor(n for non-strange flavor and s for strange flavor). The operator for the quark in positions r 1 and r 2 is replaced by effective one after double convolution of the bare operator with the density functions ρ i and ρ j . This can be performed by using the dressed expressionÕ ij (r) of the bare operator O ij (r) which depends only on the relative distance r ij = r i − r j between quarks [12].
The convolution procedure supposed to remain the center of mass fixed during it and that the ρ ij tends to a delta function at the limit of an infinitely large γ ij [41].
This formula resembles the previous form of the probability density of Gaussian form, but with parameter γ ij . The convolution (a function derived from two given functions by integration that expresses how the shape of one is modified by the other) of two Gaussian functions with size parameter γ i and γ j is also a Gaussian function. After convolution with the quark density, the Cornell dressed potential has the following form, With the error function erf and γ ij = γ 2 ).

VI. RESULTS AND DISCUSSION
In table I represents the group of parameters used for (π, σ, ω, ρ) mesons. The set of parameters are I,II that include mass of meson, the coupling constant (g) and the cut-off parameter (λ).
Tab. I: The meson parameters for OBEP for different sets [42]. The parameters listed in table II is related to the quark-quark potential through the CDP which is added to the nucleon-nucleon potential in the hybrid model.
The two-body force is a simple model to reveal the hidden physics of the atomic systems and also the nuclear systems. Our work boils down to simple fact of constructing more realistic model that contains all possible degrees of freedom in some light nuclei such as Deuteron 2 H 1 and Helium 4 H 2 . So, we include the interaction between two baryons which is bounded in a hadron and each baryon contain three bounded quarks. The nucleon-nucleon interaction is well introduced by the exchange of mesons with the OBE model. At long range of this interaction, it is supposed to be due to the exchange of pion-meson(pseudoscalar meson) followed by the effect of scalar meson (σ) in attractive attitude at the medium range. The attractive behavior have to face an opposite behavior to maintain the stability of nuclei, so, the short range of this interaction is affected by a repulsive behavior due to the exchange of vector mesons such as (ω, ρ) and QCD effects. The potential is elaborated to calculate the ground state energies for the 2 H, 4 He nuclei. We have examined the OBEP to calculate the ground state energy of H 2 and He 4 nuclei using two static meson functions (GY, SP ED) with two sets of parameters listed in table I which shows the different sets of the used parameters and for different exchange mesons, (σ, ω) mesons, (π, σ, ω) mesons and (π, σ, ω, ρ).We have mentioned that there is an effect of QCD at the short range of nucleon-nucleon interaction via three bounded quarks interacted between each other. The so-called Funnel potential or Cornell potential which is the simple and best model for the description of Charmonium system, but in our hybrid model without coupling between mesons and quarks, it gives too high values and the effect of our model is a destructive one. When we tried to apply the idea of hybrid model with the aid of the instanton induced interaction, it really gives us a transition probability for the interaction of quarkquark interaction in small scale comparing with the confinement scale. It is indeed similar to the tunneling effect with possibility of treating the instanton interaction as a field configuration between quarks and anti-quarks in the ground states. This interaction is also applied on the light quarks not only the quark-antiquark. So, it is useful for us in our model as the proton or neutron is a hadron of three light quarks. The instanton interaction is included in the CDP , giving us a small value ranged between −0.15M eV in case of 2 H and −0.25M eV in case of 4 He around the boundaries of the hadron. Our results shown in table V and table VI with the effect of CDP with parameters of table II besides the exchange of mesons through OBEP . Generally, the effect is encouraged and it improves the ratios for the ground state energies of the Deuteron and Helium nuclei in all cases with different parameters of meson degrees of freedom and different functions GY and SP ED. We have determined the ratio Rat, to ensure the accuracy between the calculated results and the experimental data.
Where E theor is the calculated ground-state and E exp the experimental one. We can also determine the binding energy per nucleon B.E A for the studied nuclei as [43], with the mass number A, and the total ground state energy E g.s. . The results of the OBEP are listed in tables (III−IV) in comparison with other theoretical and 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 SP ED functions. At first, the OBEP depended on the cancelation of σ-meson and ω-meson, the results is satisfied for the ground state energies of 2 H nucleus as in table III with two different sets of meson s' parameters, but here also tried to include the OBEP through the exchange of three mesons (π, σ, ω) and four mesons (π, σ, ω, ρ), results are listed in table III with two static functions for the meson; GY , SP ED functions. The preferable value of deuteron ground state is in case of using three mesons by using SP ED function for parameters I. It is noticed that the case of exchange three mesons gives closer value than the case of exchange of four mesons and demonstrating that the effect of π meson as an attractive one to be clear than the effect of ρ meson. This behavior is reasonable for light nuclei.  4 He nucleus, we have the value of SP ED function with the exchange of two, three mesons in the hybrid model with the parameters I to be the preferred one. It is obvious from table V and table VI the ground energy is close to the data in case of SPED function for set I and set II in comparison with the experimental data. The 4 He nucleus has little different manner, the theoretical values of the hybrid model are more cleared than in 2 H nucleus. It is noticed that the quark-quark interaction improves the values with GY function. We concluded that the used model is well-defined and compatible with the data and even than other models see [50,51].

VII. CONCLUSION
In the framework of quasi relativistic formulation, the meson exchange potential helps in obtaining a potential with few number of parameters to calculate the ground state for the light nuclei Deuteron and Helium using two (σ, ω), three (π, σ, ω) and four (π, σ, ω, ρ) mesons exchange. In addition, it was shown that a self-consistent treatment of the semi-relativistic 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 semi-relativistic 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). The work with OBEP in Dirac-Hartree-Fock equation gives a close relationship to other recent approaches, based upon different formalisms which tended to support this direction. The nuclear properties are being clear in our trail to include more two mesons to describe the NN interaction through our potential. SP ED function has an 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 NN interaction with different ranges of energies in following search. The ground state energies for 2 H and 4 He nuclei are successfully determined through this work, and gives us a hope to continue with more massive nuclei. QCD model is based on one-gluon exchange process besides the interaction of instanton that supplemented the Confinement. The Cornell dressed potential represents the interaction between quarks through the exchange of pseudoscalar mesons (instantons) under controlling of one-gluon exchange process. Our semi-relativistic hybrid model is encouraged for light nuclei, and the instanton induced interaction caused to construct a link of quark-quark interaction to the nucleon-nucleon interaction for small scale around the hadron boundaries. The effect of adding the QQ interaction on the ground state energies is ranged from 6.7 for 2 H nucleus to 1.2 for 4 He nucleus, this is small effect and it is expected to be vanished for massive nuclei.

VIII. CONFLICT OF INTEREST
quantum number , M γ = m lγ + m sγ andP Tα is the function of isotopic spin. The two wave functions are not connected and depend on r i ,r j so, the two wave functions need to be connected φ α (r i )φ γ (r j )| = m lα ms α m lγ ms γ λµ (l α s α m lα m sα |j α M α )(l γ s γ m lγ m sγ |j γ M γ ) (l α l γ m lα m lγ |λµ) φ nα lαm lα (r i )φ nγ lγ m lγ (r j )| χ 1/2 ms α χ 1/2 ms γ | P TαPTγ | (B.2) With λ = l α + l γ and µ = m lα + m lγ , we can change the special coordinates for each wave functions to become one wave, that depends on relative mass and center of mass. As L gives the total orbital quantum number in center of mass , l gives the total orbital quantum number in relative coordinates and S = s i + s j is the total spin. Relative to the spin functions and isospin functions to be connected ,we have to use them as following.