A New Model to Describe Quarkonium Systems under Modified Cornell Potential at Finite Temperature in pNRQCD

. In the present work, the three-dimensional modified radial Schrödinger equation is analytically solved. The nonrelativistic interactions under new modified Cornell potential (NMCP, in short ) at ﬁnite temperature, are extended to the symmetries of nonrelativistic noncommutative space phase (NRNSP, in short), using the generalized Bopp’s shift method in the case of perturbed nonrelativistic quantum chromodynamics (pNRQCD). W generalize this process by adding multi-variable coupling potentials


Introduction
It is well known that the Cornell potential model was one of the most popular models for studying the interactions in such systems as quarkonium (heavy quarkonia) consisting of heavy quark and antiquark (charmonium , bottomonium and mesons in the nonrelativistic quantum chromodynamics NRQCD). It is consisting of two terms. One of the terms is responsible for the Coulomb interaction of quarks and the other corresponds to the strong interaction, which provides confinement, it is the first potentials proposed to describe the interaction between heavy quarks [1][2][3][4][5][6][7][8][9][10]. Furthermore, the Cornell model has been extraordinarily successful in describing hadronic phenomenology, in addition to its success in describing a huge amount of experimental data including masses, widths, radiative and strong transitions [11][12]. The problem of calculating the energy spectra for the Schrodinger equation with various types of potentials such as Cornell potential and the Killingbeck potential at finite temperature has been attracting interest for recent years [9][10][11]. The main objective is to develop the research article for Abu-Shady [11] and expand it to a large symmetry known by the noncommutative space phase (NCSP, in short) to achieve a more accurate physical vision so that this study becomes valid in the field of nanotechnology. And on the other hand, to explore the possibility of creating new applications and more profound interpretations in the sub-atomics and nano scales using the new version of the modified Cornell potential, which has the following form: We refer to this term � �⃗��⃗ in the materials and methods section. The new structure of NCSP based on new canonical commutations relations in both Schrödinger SP and Heisenberg HP, respectively, as follows (Throughout this paper, the natural units = ℏ = will be used) [13][14][15][16][17][18]: where the indices , ≡ 1,3 and ℏ is the effective Planck constant. This means that the principle of uncertainty for Heisenberg generalized to include another two new uncertainties related to the positions � � , � � and the momenta's �̂,̂�, in addition to the ordinary uncertainty of � � ,̂�.
The very small two parameters and (compared to the energy) are elements of two antisymmetric real matrixes, parameters of noncommutativity and ( * ) denote to the Weyl Moyal star product, which is generalized between two arbitrary functions ( , ) ( , ) to the new form ̂( �,) �( �,) ≡ ( * )( , ) in 3D-NCSP symmetries [18][19][20][21][22][23][24][25]: The second and the third terms in the above equation are present the effects of (space-space) and (phase-phase) noncommutativity properties. It should be noted that noncommutativity was introduced firstly by Heisenberg in 1930 [26] and then by Syndre in 1947 [27]. However, the new operators ̂( ) = � � ∨̂�( ) in HP are depending on the corresponding new operators ̂= � ∨̂ in SP from the following projections relations: here = ∨ and ( ) = � ∨ �( ), while the dynamics of new systems ( ) dt are described from the following motion equations in NCSP: The operators � and � are present the quantum Hamiltonian operators for MCP and NMCP in the QM and its extension NCSP, respectively. This paper consists of five sections and the organization scheme is given as follows: In the next section, the theory part, we briefly review the SE with modified Cornell potential at finite temperature on based to ref. [11][12]. Section 3 is devoted to studying the modified Schrödinger equation MSE by applying the generalized Bopp's shift method and obtaining the NMCP and the modified spin-orbit operator at finite temperature. Then, we applied the standard perturbation theory to find the quantum spectrum of the ground state, the first excited state and the ℎ excited-state produced by the effects of modified spin-orbit and modified Zeeman interactions. After that, in the fourth section, a discussion of the main results is presented in addition to determining the new formula of mass spectra of the quarkonium system in 3D-NCSP symmetries. Finally, in the last section, a summary and conclusions are presented.

Overview of the eigenfunctions and the energy eigenvalues for the modified Cornell potential at finite temperature in QM
We shall recall briefly in this section, the time-independent Schrödinger equation for the MCP at finite temperature [11]: The relative spatial coordinate between the two quarks is , and are purely phenomenological constants of the model. In a thermal medium of a positive temperature , the potential is modified by color screening which can be parameterized in the form [7,11]: Here ( ) is the Debye screening mass. The expand with Taylor series around = 0 gives [11]: where = ( ), C = a-1/2 2 ( ) and D = 1/2 ( ). If we insert this potential into the Schrödinger equation, the radial part function ( ) = ( ) is given as [11,12]: here = + the reduced mass for the quarkonium particle for example ( , , , , ) and . The complete wave function ( , , ) = ( ) ( , ) is given by [11,12]: In addition, the energy of the potential in Eq. (8) [11]: and is a normalization constant while 0 ≡ 1/ is the characteristic radius of meson and is a natural number accounting for the radial excitation while is a non-negative integer number that represents the orbital angular momentum.

Solution of MSE for new modified Cornell potential at finite temperature in pNRQCD
In this section, we shall give an overview or a brief preliminary for NMCP in (3D-NCSP) symmetries. To perform this task the physical form of MSE, it is necessary to replace ordinary threedimensional Hamiltonian operators � � , �, complex wave function � respectively. In addition to replacing the ordinary product with the Weyl Moyal star product, which allows us to construct the MSE in 3D-NCSP symmetries as [28][29][30][31]: The generalized Bopp's shift method has been successfully applied to relativistic and nonrelativistic noncommutative quantum mechanical problems using modified Dirac equation ( The new generalized positions and momentum coordinates � � ,̂� in 3D-NCSP are defined in terms of the commutative counterparts � , � in ordinary quantum mechanics via, respectively [29][30][31][32]: The above equation allows us to obtain the two operators (̂2,̂2) in 3D-NCSP symmetries [28][29][30][31]: The two couplings �⃗ �⃗ and �⃗ → are � 12 + 23 + 13 � and� 12 + 23 + 13 �, respectively and ( , and ) are the three components of the angular momentum operator �⃗ while = /2. Thus, the reduced Schrödinger equation (without star product) can be written as: The new operator of Hamiltonian � � ,̂� can be expressed as: Where () denote to the NMCP in 3D-NCSP symmetries: Again, applying Eq. (15) to find the three terms (, �−� and(−̂2)), which will be used to determine the NMCP (), as follows: Substituting, Eq. (19) into Eq. (18), gives the new modified Cornell potential in 3D-NCSP symmetries as follows: By making the substitution above equation into Eq. (17), we find the global our working new modified Hamiltonian operator () satisfies the equation in 3D-NCSP symmetries: where the operator � , � is just the ordinary Hamiltonian operator with modified Cornell potential in commutative quantum mechanics: while the rest part ( , ,̄) (The perturbative Hamiltonian operator) is proportional with two infinitesimals parameters ( and ̄) : Thus, we can consider per-( ) as a perturbation term compared with the principal Hamiltonian operator � , � in 3D-NCSP symmetries.

The exact modified spin-orbit operator for heavy quarkonium systems under NMCP in pNRQCD:
In this subsection, we will apply the same strategy, which we have seen exclusively in some of our published scientific works [29][30][31]. Under such a particular choice, one can easily reproduce both couplings (   { ( + 1) − ( + 1) − 2} corresponding = + 1 (spin great), = (spin middle) and = − 1 (spin little), respectively, then, one can form a diagonal (3 × 3)matrix for NMCP in 3D-NCSP symmetries, with diagonal elements� � 11 , � � 22 and � � 33 are given by: We will see them in detail in the next subsection. After profound calculation, one can show that the new radial function ( )satisfying the following differential equation for modified Cornell potential: Through our observation of the expression of ( ), which appear in the equation (23), we see it as proportionate to two infinitesimals parameters ( and̄), thus, in what follows, we proceed to solve the modified radial part of the MSE that is, equation (28) by applying standard perturbation theory to find acceptable solutions at the first order of two parameters and . The proposed solutions for MSE under NMCP include energy corrections, which are produced automatically from two principal physical phoneme's, the first one is the effect of modified spin-orbit interaction and the second is the modified Zeeman effect while the stark effect can appear in the linear part of new modified Cornell model.

The exact modified spin-orbit spectrum for heavy quarkonium system under NMCP in pNRQCD
The purpose here is to give a complete prescription to determine the energy level of the ground state, the first excited state and the ℎ excited state, of heavy quarkonium systems. We first find the corrections ( ( 1 ( ), , , , , 2 , 1 ), ( 2 ( ), , , , , 2 , 1 ) and ( 3 ( ), , , , , 2 , 1 )) for heavy quarkonium system such as (charmonium and bottomonium) mesons that have the quark and antiquark flavor under new modified Cornell potential at finite temperature, which have three polarities up and down = + 1 (spin great), = (spin middle) and = − 1 (spin little), respectively, at the first order of two parameters ( and ). Moreover, by applying the perturbative theory, in the case of perturbed nonrelativistic quantum chromodynamics pNRQCD, we obtained the following results: We have used the orthogonality property of the spherical harmonics∫ ( , ) ′ ′ ( , ) ( ) = ′ ′ . Now, we can rewrite the above equations to the simplified new form: For the ground state, we have, the expressions of the 4-factors � = 1,4� will be simplified to the following form: . It is convenient to apply the following special integral [34]: With conditions ( ⟩0, ⟩0 and ⟩0) while � � denoting to the ordinary Gamma function.

The exact modified magnetic spectrum for heavy quarkonium systems under NMCP in pNRQCD
Further to the important previously obtained results, now, we consider another important physically meaningful phenomenon produced by the effect of new modified Cornell potential at finite temperature in perturbative NRQCD related to the influence of an external uniform magnetic field �⃗ . To avoid the repetition in the theoretical calculations, it is sufficient to apply the following replacements: �⃗ → �⃗ and ⃗ → �⃗ Allow us to make the changes �− � 2 3 + 2 − � �⃗ +� We have (− ≤ ≤ + ), which allows us to fix (2 + 1) values for discreet numbers . It should be noted that the results obtained in Eq. (42) could find it by direct calculation = ⟨ ( , , )� ( , , )� ( , , )⟩that takes the following explicit relation: (43) Then we find the corrections produced by the operator ( , , ) for the ground state and other excited states repeating the same calculations in the previous subsection.

Main Results
In the previous sub-sections, we obtained the solution of the modified Schrödinger equation for new modified Cornell potential, which is given in Eq.
Thus, the ordinary kinetic term for Cornell potential (− 2 ) and ordinary interaction ( , ) − ( , ) are replaced by a new modified form of the kinetic term 2 and new modified interactions modified to the new form � , and � in 3D-NCSP symmetries. On the other hand, it is evident to consider the quantum number takes (2 + 1) values and we have also two values for( = ± 1, ), thus every state in usually three-dimensional space of energy for heavy quarkonium system under NMCP will be 3�2(2 + 1)�sub-states. To obtain the total complete degeneracy of energy level of the modified Cornell potential in 3D-NCSP symmetries, we need to sum for all allowed values of . Total degeneracy is thus, Note that the obtained new energy eigenvalues � − , − , − �� , � = − , + �, , , , 2 , 1 � now depend on new discrete atomic quantum numbers ( , , , ) and in addition to the parameters ( , , )of the modified Cornell potential. It is pertinent to note that when the atoms have spin − 0, the total operator can be determined from the interval| − | ≤ ≤ | + |, which allows us to obtain the eigenvalues of the operator ( 2 − − ) as ( , , ) ≡ 0 and then the nonrelativistic energy spectrum � − , − , − �� , � = − , + �, , , , 2 , 1 � reads: One of the most important applications, in the extended model of pNRQCD, is to calculate the modified mass spectra of the heavy quarkonium systems (the mass of the quarkonium bound state), such as charmonium and bottomonium mesons, that have the quark and antiquark flavor in the symmetries of NCSP under NMCP at finite temperature. To achieve this goal, we generalize the traditional formula = 2 + to the new form: here is the bare mass of quarkonium or twice the reduced mass of the system. Moreover, Here is the heavy quarkonium system at a finite temperature under modified Cornell potential in commutative quantum mechanics, which is defined in [11]. If we consider ( , ) → (0,0), we recover the results of the commutative space of the ref. [11] obtained for the modified Cornell potential, which means that our calculations are correct. Our obtained results are in good agreement with the already existing literature in NRNCSP symmetries [35]. The novelty in this work is that the generalized Bopp's shift method successfully applies to find the solution of the 3-radial modified Schrödinger Equation at finite temperature in the symmetries of NRNCSP.

Conclusion
In the present work, the 3-dimensional modified Schrodinger equation is analytically solved using the generalized Bopp's shift method and standard perturbation theory. The modified Cornell potential at finite temperature is extended to include the effect of the noncommutativity space phase based on Refs. [11][12]; we resume the main obtained results: � − , − , − �� , � = − , + �, , , , 2 , 1 � for the ground state, the first excited state and the ℎ excited state with (spin−1 and spin − 0) for heavy quarkonium system under influence of new modified Cornell model at finite temperature is obtained.

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We have obtained at a finite temperature ( ≠ 0) the modified mass of the quarkonium system M nc cp which equal the sum of corresponding value in CQM and two perturbative terms proportional with two parameters ( and ).
Through the high-value results, which we have achieved in the present work, we hope to extend our recently work physics for further investigations of elementary particles physics and other characteristics of quarkonium at a finite temperature [11] among others.