Research Article | Open Access
R. Hussien, Sh. M. Sewailem, L. I. Abou-Salem, "Study of the Ground-State Energies of Some Nuclei Using Hybrid Model", Advances in High Energy Physics, vol. 2021, Article ID 9915801, 14 pages, 2021. https://doi.org/10.1155/2021/9915801
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 semirelativistic 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 and 1.2 percentage for 2H1 and 4He2, respectively nuclei.
One of the fundamental problems of the nuclear structure is the derivation of the ground-state energies through different methods, such as properties related to the constituents of matter, which are represented in the physics of elementary particles with their characteristics and how each particle interacts with others. The interaction between each nucleon with all other nucleons generates an average potential field where each nucleon moves. The rules of the 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 the antisymmetrical product wave function. The calculation of the nuclear mean-field potential with Dirac-Hartree-Fock qualifies the description of nucleon-nucleon interaction to be successful microscopically. The interaction between two nucleons has three regions with three ranges. The first region originated from pseudoscalar meson, the second region was related to the scalar meson, and the third region was caused by the exchange of vector meson besides the effects of quantum chromodynamics (QCD). The nucleon-nucleon potential has no definite method to determine it. The Bonn group potential known as one-boson exchange potential is supposed to be the suitable model for this interaction because of the reduction of free parameters and fitting them accurately with the experimental data.
On the other hand, the quark degrees of freedom are under the dynamics of QCD. The interaction between quarks has various forms of potentials, and these forms have to regard the quark properties (confinement and asymptotic properties). The mechanism of the one-gluon exchange approach is dominant at the short range with two parts. The linear confinement at a long distance and a part of the asymptotic property represented in the pairing force acting only on the quark-antiquark states. The constituents of baryons composed of quarks can use a semirelativistic potential model that refers to their interaction, including the instanton-induced forces. The instanton-induced model is used to describe baryons composed of light quarks that are demanded in the considered baryons. This interaction resembles the tunneling phenomena as it can be affected outside the hadron for a short scale comparing with the confinement scale. In the used model, we have two contributions in the potential form, the one-gluon exchange part and the exchange of pseudo particles between quark-antiquark pair. The possibility of proposing a hybrid model with no coupling between quarks inside the baryon and 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 2, we introduce the theory of NN interaction through the one-boson exchange potential with investigations and motivations of the formula. In Section 3, 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 4, where the theoretical analysis for the construction of one-boson exchange potential through the exchange of two, three, and four mesons is clarified. Section 5 shows the theoretical analysis of QQ interaction and the final form of CDP. Finally, in Sections 6 and 7, the obtained results and conclusion are given.
2. Theory of NN through the OBEP
The start of using the fact that there is no unique potential for determination of the effective NN 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 . 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, and the interaction between nucleons is controlled with Pauli principle and nuclear shell model. We considered the spatial exchange between two nucleons, so the nonlocal field is determined with the Hartree-Fock approximation, since the Fock effect is demonstrated in the nonvanishing spatial components for the vector part of the potential. Our potential implies the 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 semirelativistic 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 the Bonn group to have the meson’s function and its parameters. To calculate the ground-state energies of the 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 for a modification of nucleon wave functions and energy.
The modification of wave function is demonstrated by Clebsch-Gordan 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 gives 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 internucleon distances. We have the effect of QCD 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 [2–8], and also, the quark degrees of freedom in QCD models are successful models for the description of the nuclear properties [9–12]. 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 the following:
where the Hamiltonian of the nucleon-nucleon interaction is and the hamiltonian of the quark-quark interaction is . 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.
3. Theory of QQ through the CDP
The simulation of quark-quark interaction phenomenon in a semirelativistic framework shows that the long range part of this interaction is increasing linearly with the distance and is called 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 [13, 14], 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 are called Funnel potential or Cornell potential [15, 16]. The Hamiltonian of hadrons containing light quark should simultaneously define a number of relativistic corrections. The momentum-dependent corrections as well as a nonlocal kinetic energy (there is no commutation with faster than light, and it is compatible with special relativity) is required to be included in the effects of relativistic kinematics of the potential of the potential energy operator . These relativistic kinematics are included in the Bethe-Salpeter equation, neglecting the spin effect which introduces nonlocal modifications of the relative coordinate. The spinless Salpeter equation has the form
This is suitable for baryon where is the total energy of the system, is the central potential between two particles , and is their relative momentum. In the case of baryon, is the constituent mass of quarks with the same mass of and quarks (isospin symmetry is maintained). The central potential is the Cornell potential,
The factor half is related to the half rule, is the Coulomb parameter, is the string constant, and is additive constant equals zero in the heavy quark sector. To solve the spinless Salpeter equation,  expanded the wave function in terms of a complete set of basis functions according to Rayleigh, Ritz, and Galerkin methods as the previous radial wave function in OBEP. Many symmetries are slightly broken in nature as it can give rise to 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  firstly. In [13, 14], the authors went with instanton-induced interaction (it is a solution to the equations of motion of the classical field theory, and it is supposed to be critical points of the action for such quantum theories). In nonrelativistic 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 is able to compute the residual interaction by ‘t Hooft force from instanton effects [18, 19]. Ref  proposed a model of quark interaction with the replacement of the traditional OGE potential by a nonrelativistic 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 . (i)The gluon fields cannot deform the instantons into classical solutions continuously(ii)The perturbative gluon diagrams cannot cover the effective interaction between quarks which caused by instantons(iii)The instanton calculus denotes as a nonperturbative 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(iv)The instanton effects in QCD realized that instanton is similar to be described as -dimensional gas of pseudo particles; then, use the summation over the instanton gas
In the first analogy of super conductivity with the Bardeen-Cooper-Schrieffer theory , when the interaction between fermions (nucleons) and light quarks is 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 .
4. Theoretical Analysis of OBEP
The general form which describes the ground-state energy of the considered system is the following: where is the Hamiltonian and is the total energy of the system.
defining as kinetic energy and as the potential energy. Hence, the Hamiltonian fermions are interacting via the potential . Thus, the accurate Hamiltonian interaction of the nuclear system can be described by Dirac to represent the number of fermions’ interaction where this Hamiltonian is [23–26] as follows: where and are Dirac matrices, is the speed of light, is the nucleon mass, is the relative kinetic energy, and is the momentum operator.
Substituting the last equation in Equation (6), thus the relativistic Hamiltonian operator for bound nucleons which interact strongly through the potential can be expressed as follows:
In the 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 is 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 : where the wave function of all nucleons is and the wave function for -nucleon is . The wave function for nucleon depends on the oscillator parameter as where is the oscillator constant and is the wave function of two components. where the wave function for radial component is and the spin component is . 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 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: for empty states or for filled states). The two components of the wave functions have the following relation between them [31, 32].
Using the relation Equation (13), where is external energy which equals zero, here we deal with ground state and makes the value so small and can be neglected.
Differentiate Equation (12) with respect to , and the has a sign defines the exchange that happening between the two nucleons, hence substituting and .
Treating with the 1st part of Equation (16) gives us the coming formula.
Taking into account Dirac matrices  with defining the wave functions in bracket as and , where unit matrix
Substituting in will have the following result:
The 1st term of kinetic energy tends to zero, and this result has an agreement with another calculations in  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 between two nucleons is based on the degrees of freedom associated with three mesons, pseudoscalar, scalar, and vector mesons.
According to the relation between and in Equation (14), one obtains
We will apply some important relations 
Include these relations in potential equation. We substitute every term by using the relation of angular momentum , (where is the total spin operator), , and . According to the previous relations, where as triplet case for two nucleons,
With total spin operator and the meson function , using  , and . Quantum mechanics have a magnificent tool; this tool is the harmonic oscillator which is capable of being solved in closed form, and it has generally useful approximations and exact solutions of different problems . It solves the differential equations in quantum mechanics. We have the energy of harmonic oscillator which equals the kinetic energy added to the potential energy to simplify the solution and get the result. It is slitted in relative harmonic oscillator energy [30, 40], with that is the angular frequency and center of mass contribution in harmonic oscillator energy . We suppose the nucleons have average masses , so the relative mass , and center mass .
The wave functions of the two nucleons of Equation (28) should be treated as follows:
See Appendix B to have the final formula.
The bracket represents the Talmi-Moshinsky bracket. The same treatment for the ket part to have We have the wave function as radial part and angular part for center of mass coordinates, the wave function as radial part , and angular part for relative coordinates. For the two-nucleon interaction formula through the exchange of four mesons where and is the mass number of the required nuclei, we define the bracket , and as the terms of equation depend on .
We have the formula of radial wave function which involves the length parameter with angular frequency and the associated Laguerre polynomial .
The differentiation of Radial function equals
Define the operator  as
The meson degrees of freedom have some static functions for description, but here we choose GY and SPED for meson and . where the meson-nucleon coupling constant , the cut-off , and the mass of the meson are associated with . The second function has the form 
5. Theoretical Analysis of CDP
There is an explicit spin dependence for the instanton interaction unlike one-gluon exchange, and it contains a projector on spin states. The distribution of this interaction represented with replaced by a Gaussian function with range . where is the range of the pairing force (QCD scale parameter). where and are two dimensioned coupling constants according to quark flavors. This equation is under condition of , where , , and denote angular momentum, spin, and isotopic spin quantum numbers, respectively, for pair, and the form of instanton contributions represents as . The pairing force depends on the value of the parameters and , if we set for strange flavor with symbol and for nonstrange flavor with symbol . The Hamiltonian contains diagonal parts in the isoscalar space .
having the coupling constant as
The parameter denotes the strength and is defined as  where is a function instanton density of the instanton size . For three colors and three flavors, this quantity is given in , is the current mass of flavor , and the quark condensate for this flavor is , (nonvanishing expectation values). The integration over which is the maximum size of the instanton is where
The constituent masses are the renormalization of quarks’ masses which demonstrate the contribution of the constituent masses .
is the current mass of nonstrange quark, and is the contribution of constituent mass  with free parameter added to the running masses. The contribution of the constituent masses has the following formula.
It is important to replace the dimensional instanton size  as with a dimensionless quantity with using the definition of .
This dimensionless integration should still have small value of -term. It is involved in the parameters and .
In the functions given in , the is the constituent mass of strange flavor and also the 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-antiquark 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 is the probability density not the instanton size as previous with the size parameter, and it is dependent on the quark mass flavor ( for nonstrange flavor and for strange flavor). The operator for the quark in positions and is replaced by effective one after double convolution of the bare operator with the density functions and . This can be performed by using the dressed expression of the bare operator which depends only on the relative distance between quarks .
The convolution procedure supposed to remain the center of mass fixed during it and that the tends to a delta function at the limit of an infinitely large .
This formula resembles the previous form of the probability density of Gaussian form, but with parameter . 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 and is also a Gaussian function. After convolution with the quark density, the Cornell-dressed potential has the following form: with the error function and , where and .
6. Results and Discussion
Table 1 represents the group of parameters used for mesons. The sets of parameters are I and II that include mass of meson, the coupling constant , and the cut-off parameter .
The parameters listed in Table 2 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 2H1 and helium 4H2. So, we include the interaction between two baryons which is bounded in a hadron, and each baryon contains 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 in attractive attitude at the medium range. The attractive behavior has 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 and QCD effects. The potential is elaborated to calculate the ground-state energies for the 2H and 4He nuclei. We have examined the OBEP to calculate the ground-state energy of 2H and 4He 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, mesons, mesons, and .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 is simple and the 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 quark-quark 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 antiquarks 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 MeV in the case of 2H and MeV in the case of 4He around the boundaries of the hadron. Our results are shown in Tables 3 and 4 with the effect of CDP with parameters of Table 2 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 SPED.