BOTTOMONIA UNDER EFFECT THREE INSPIRED QCD POTENTIALS IN THE FRAMEWORK OF NON-RELATIVISTIC QUARK MODEL

In this paper, we have studied the spectrum of bottomonium mesons behavior under the effect of three types of potentials inspired by Quantum Chromodynamics. In addition, other properties like Hyperfine splitting behavior, and Fine splitting behavior have been studied. We used these potential models within the non-relativistic quark model to present this study. We found that our expectations are consistent with experimental data and other theoretical works as well we presented new conclusions regarding the spectrum of unseen bottomonium states for S, P, and D-wave bottomonia. And we have expected other their characteristics.


INTRODUCTION
Investigation of heavy quarkonia systems like charmonium, bottomonium, and toponium, offers a clear understanding of particle physics, the standard model, and the quantification characterization of the QCD quantum chromodynamics theory [1,2,3,4].But because of the tiny lifetime for the top quark approximately equals 0.5 × 10 −24 s, it is very hard to appear in nature [5].So we can say the bottomonia are the heaviest mesons that the experiments discovered them over a long time.From that, the bottomonium family holds a substantial place in the hadronic particles and participates effectively in the inspecting of strong interactions.
So clear, it is hard to assign the whole spectrum of bottomonia.As a part of fact, we are still at the beginning of this road.Until now many lower bottomonium states that are under the threshold of creation of the B B pair have not appeared.But the running of the Belle II [18] and upcoming colliders will open the hope door towards this challenge, and we foresee the appearance of new bottomonium states.To achieve that, we need to probe the bottomonia with many theoretical techniques that provide us with accurate expectations and inspired QCD.There are many techniques have utilized for example; the QCD sum rule [19,20], Bethe-Salpeter [21], the Regge phenomenology [22,23,24,25], the method of perturbative QCD [26],the lattice QCD [27,28,29], and the coupled-channel model [30,31,32,33] in addition to utilizing the versions of relativistic [34,35], relativized [36,37,38], semi-relativistic [39,40], and finally the non-relativistic [41].
Our purpose in this paper is to obtain the accurate mass spectrum of bottomonia in addition to the characteristics of hyperfine splitting behavior and fine splitting behavior.We suggest applying three QCDinspired potentials, the first one is the simplest.For more accurate results, we use the second one, which takes into account the spin-spin interactions between quarks.Here appears the hyperfine splitting behavior.To obtain the best potential, which is more sophisticated and more accurate, we apply the third one, which EEJP. 2(2023) includes spin-orbit interactions and tensor interactions, and here the fine-splitting behavior appears, and the results become more accurate.The theoretical frame for this work is in the next section, while we offer our findings and discussion in section 3. Our conclusions appear in the last section.

THE THEORETICAL FRAME
We adopt the non-relativistic approximation, which includes the Hamiltonian that rules the dynamics of mesons, consisting of the kinetic energy part T and the potential energy part V which considers the phenomenological interactions of the constituent quarks.We can write the Hamiltonian operator Ĥ as: Ĥnr = Tnr + V. ( We can express the wave functions of mesons as the eigenfunctions in the Schrödinger equation as follows: Ĥnr ψ = Eψ. (2) 2.1.The Kinetic Energy of Bottomonia We can treat the bottomonium mass center of motion as the non-relativistic kinetic energy.The expression of the non-relativistic kinetic energy operator is as the next [42]: The m b and mb are the masses of the bottom quark and the bottom antiquark.The µ and P are the reduced mass and the relative momentum of bottomonium meson, respectively.The constituent quark mass is used to provide us with suitable calculations of bottomonium properties that could be performed.We can compare the findings to the experimental data [43] to appear how our models work.

The Potential Models
From a time the first detection of the charmonium states in 1974, the first system of quark and antiquark, it became prototypical of the exotic positronium atom e + e − for meson spectroscopy [44,45,46].There is a similarity between them roughly [1].So the initial attempt started to improve positronium-like potential, to obtain a fine static potential of quarkonium, which is known as a vector color coulomb-like potential.The improved color coulomb-like potential depends on QCD spirit.In this potential, the effectiveness of quark interaction is in the short distances [47,48].According to the rules of quantum chromodynamics (QCD) theory, the behavior of short distance has dominated via one-gluon exchange (OGE) interaction [49,1].Hence the color coulomb-like potential takes the following formula: where α s and −4 3 are the running strong coupling constant and the color factor, respectively.From the phenomenological side, we must consider the quark confinement at the long ranges [36,42], where one of the significant properties of strong interactions is confinement, which is generally acknowledged.From QCD rules, the confining potential increases with increasing the inter-quark distance [50,1].So we can write the confinement potential as: From the previous, we can obtain the conventional potential, which collects the vector color Coulomb-like potential as in Eq.4 and the scalar linear potential as in Eq.5.This is the first suggested potential that we can write as: We will investigate b b system spectra under the effect of this potential as a minimal potential model, as shown in Table 2; column 4.
Here, we go ahead a step toward the front to consider the spin-spin interactions between b b systems, which takes the following relation: where When we add Eq.7 to Eq.6, we acquire the second potential that takes the following form: where as S is the total spin of the meson [51].The second potential considers the hyperfine behavior that arises due to spin-spin interaction between quarks, so it differentiates between the triplet spin states and singlet spin states.Now we take into account two significant parts spin-orbit interactions and tensor interactions.They have the next formulas [52]: and where the spin-orbit matrix elements of the L.S operator is determined by where L and J are orbital angular momentum and total angular momentum quantum numbers.This operator has a diagonal base |J, L, S⟩, and T represents the tensor operator [53]: From the equations (9,11,12), we can write the third potential as follows: This potential considers fine splitting behavior due to taking into account the spin-orbit and the tensor interactions in addition to the hyperfine splitting behavior due to spin-spin interaction.
We have determined the utilized parameters in these potentials by fitting the corresponding spectrum of bottomonium states for each different potential, as in Table 1.We can have high expectations concerning the bottomonia spectrum (b b), as shown in Table 2 due to, these modified parameters tabulated in Table 1.
Table 1.The modified parameters were utilized in three potentials to obtain the masses of b b-bottomonium states

Model
The potentials in the (NRQM)

RESULTS AND DISCUSSION
In this paper, we remedied Schrodinger's equation using the matrix method to obtain its numerical solution.Using the previous three potentials, the bottomonium mesons' characteristics are probed in the frame of the non-relativistic kinetic energy.We have obtained Hamiltonian values for the nS(n ≤ 4), nP (n ≤ 3), and nD(n ≤ 3) bottomonium states with the spectroscopic notation n 2S+1 L J .The mass and other characteristics are introduced in Tables (2 : 4).EEJP.2(2023)

The Mass Spectrum of Bottomonium Mesons
We investigated the spectrum of S, P, and D-Wave bottomonium mesons under the influence of three potentials.This study includes thirty-six seen and unseen meson states.Fitting with seventeen mesonic states in the last update of data, due to it yields new parameters in each potential to get the best new theoretical bottomonium spectra in three QCD-inspired potentials types, as shown in Table 1.These newly yielded theoretical spectra were compared with the current experimental data [43] and compared to other works for calculated spectra [54,55,56].
The χ 2 values for the expectations of the three potential types are (0.0010), (0.0005), and (0.0002), respectively.So, one observes that the first potential type provides us an agreement with the experiment, although it is a blind model concerning the multiplets of bottomonium states that own the same orbital angular momentum L quantum number with different spin quantum number S and different total angular momentum J as in column 4 of Table 2.When we go ahead a step toward more accuracy, we find the second potential type that supplies us with more accuracy than the first one, but one notices it is a blind relative to the multiplets that own the same orbital angular momentum L quantum number, with a different total angular momentum quantum number J, as seen in column 5 of Table 2. Now we continue ahead another step toward more accuracy, and we have found that the third type of potential has achieved the most accuracy concerning these potentials, in addition, this potential provides us with the ability to distinguish between the multiplets of bottomonia that own the same quantum number L and with different quantum numbers S and J, as shown in Table 2 column 6.
The masses of bottomonia are plotted in Fig. 1 and Fig. 2 with the principle quantum number (n).Where the masses are extracted within the third potential type; based on its parameters which are in Table 1.The figures illustrate the masses of S, P, and D-wave states with triplet spin (in Fig. 1) and singlet spin (in Fig. 2) with various values of J.The graphs demonstrate that the mass spectra increase along with increasing the principle quantum number (n).Here, we find (n) has a strong influence on the S, P, and D-wave bottomonia masses when particles have the same S and J quantum numbers.
Figures 6 and 7 illustrate the theoretical masses of S, P-wave bottomonium states (Υ(nS) and η(nS)) and (χ (nS)and h(nS)), respectively.They provide us with a comparison between the practical data and the theoretical expectations using the third potential, tabulated in Table 2. we find a good consistency between them and observe a tiny distinction between them.
The other studies for the bottomonium spectra employed a variety of models, including the variational method with a single Gaussian trial wavefunction in a Cornell potential model in the relativistic Quark Model framework used by (Virendrasinh Kher et al.) [54].While (B.Chen et al.) utilized the RFT model [55], meanwhile the Non-Relativistic with Screened Potential Model is applied by (W.J. Deng et al.) [56].And (M.Wurtz et al.) use the LATTICE Field Theory estimations [29].We find from Table .2 that the three potentials to which we applied them agree in general in the most estimations with these models, but the third one owns the most agreement with the most states.

Hyperfine Splitting Behavior ∆M Hyp.Sp.
The hyperfine splitting force plays an influential role in the calculations of the mass of bottomonia.So, we have to take it into account.This force started with the appearance through the second potential, so the second potential provides us with more accurate calculations for masses than the first potential, as evident in column 5 of Table 2.However, its accuracy is further improved in the third potential to supply us with more accurate calculations, as extremely evident in column 6 of Table 2.We notice that the difference between the multiplets decreases by increasing (n) for S-wave bottomonia; the same thing concerning P-wave, but in the S-wave bottomonia the triplet spin particles have greater mass than the singlet spin particles.While the opposite happens for the P-wave bottomonia, the spectrum mass of singlet spin particles is heaviest than triplet spin particles.While in the D-wave bottomonia, the mass difference is constant between triplet, and singlet spin particles, and compared to the singlet spin particles, the triplet spin particles are heavier.Fig. 3 shows the relation between the hyperfine splitting mass and the quantum number (n) in comparison with experimental data for the S-wave in a graph (a) and also for the P-wave in graph (b) according to the third potential.

Fine Splitting Behavior ∆M F.S.
The role of fine splitting force, which appears in the third potential is very significant for estimations of the masses of bottomonium mesons.That matter makes us enter this force in our calculations of mass spectra, not only in the bottomonium spectrum but in all hadronic spectra.Consequently, we find the mass estimations are the most accurate.Also, it can distinguish between the multiplets that have the same L and S, but they have various J because of spin-orbit force and tensor force, as is very clear in Table 2; column 6.Now we find the 1 3 P 1 -1 3 P 0 is the dominant partial splitting where it is of ( ∼ = 59%) relative to total splitting concerning the 1P-level.Also, the 2 3 P 1 -2 3 P 0 takes ( ∼ = 55%) to be the dominant partial splitting relative to the 2P-level.Table 2. Theoretical spectra of b b states in GeV under the influence of three QCD-inspired potentials compared with experimental data [43] and with theoretical results of Refs.[ [54], [55], [56], [29]].In this work, we propose applying the three potentials in the framework of the non-relativistic quark model to probe the bottomonia spectra and obtain accurate yields.Our perspective depends on the nonrelativistic framework that is suitable generally for the heavy mesons sector and particularly for bottomonium mesons because they are the heaviest mesons.The expectations of all these potentials present overall agree with practical data and with the theoretical expectations of other groups.The first one involves the one-gluon exchange interaction (like color coulomb potential) in addition to the scalar linear confinement potential, which provides us sensible findings where its uncertainty almost equals 0.0010, but it blinds concerning splitting between the multiples of bottomonium states which have the different S, and the J, but the same L quantum numbers.When we add the spin-spin interactions of quarks, we obtain the second potential, which is more accurate than the first one.Its χ value almost equals 0.005 and also exhibits hyperfine splitting behavior.
The third potential achieves our computational strategy that aims towards more accurate outputs using sophisticated treatments concerning the bottomonia spectrum, and we can extend and apply it to other heavy mesons.This potential is the most complex, but it is the most accurate one, with a χ value is almost 0.002.Additionally, it indicates the fine splitting behavior for bottomonium spectra.Our expected masses of unseen   So we use the yieldings from the third potential to study the hyperfine s plitting b ehavior a nd t he fine splitting behavior of the S, P, and D-wave bottomonia spectrum.This study of the bottomonia multiples behaviors agrees with the experimental data and presents significant p redictions; w e c an u se t hem t o observe unseen bottomonium states in the future.

( a )
The masses of S, P, and D states for J=1.(b) The masses of P and D states for J=2.(c) The masses of D states for J=3.(d)The masses of P states for J=0.

Figure 1 .
Figure 1.The Masses of S, P, and D states for QCD-inspired V III , which have spin one and various total angular momentum J versus n quantum number

Figure 2 .
Figure 2. The Masses of S, P, and D states for QCD-inspired V III , which have spin zero and various total angular momentum states J versus n quantum number a) The theo.ratio of 1P fine splitting mesons (b) The exp. ratio of 1P fine splitting mesons.

Figure 4 .
Figure 4.The theoretical and experimental fine splitting behavior of one spin 1P bottomonium states (a) The theo.ratio of 2P fine splitting mesons (b) The exp. ratio of 2P fine splitting mesons.

Figure 5 .Figure 6 .Figure 7 .
Figure 5.The theoretical and experimental fine splitting behavior of one spin 2P bottomonium states

Table 4 .
Theoretical fine splitting (Theo.∆M F.Sp. ) and Experimental fine splitting (Exp.∆M F.Sp. ) of [b b] states in MeV for P, D-Wave studied states.
bottomonia states, via the third model, can provide good benefits t o d iscovering t hese s tates i n t he incoming experiments.