Calculation of Dissociation Temperature of Nucleon Using Gaussian Expansion Method

The first study of the dissociation temperature of nucleon in hot QCD medium in the framework of constituent quark model is presented. The temperature-dependent potential energy of the three quark system, taking as the internal energy of the system are obtained from the free energy of the system, and the temperature-dependent free energy is derived based on Debye-H\"uckel theory. The lattice QCD results of the free energy for heavy three-quark system are employed and extended to the light three-quark system. The Schr\"{o}dinger equation for nucleon is solved with the help of Gaussian expansion method and the dissociation temperature of the nucleon is determined according to the temperature dependence of binding energy and radii. Comparing with the dissociation temperature of $J/\psi$, the dissociation temperature of nucleon is higher. So, nucleon is more difficult to melt than Charmonium.


I. INTRODUCTION
The relativistic heavy-ion collider experiments show that human beings may have produced quark-gluon plasma (QGP) in the laboratory [1]. It is generally believed that some quark bound states can survive in QGP. In thermal Quantum chromodynamics (QCD), the thermal properties of QGP can be determined by studying the behavior of these quark bound states in hot medium. In 1986, Satz pointed out that the suppression of J/ψ could be a signature of QGP formation in the relativistic heavy ion collisions [2]. Since then, many people have systematically studied the melting of Charmonium and Bottomonium. But the dissociation temperature of nucleon, the lightest baryon, has not been studied systematically. This is due to the difficulties of solving threebody system and obtaining the temperature dependent quark-quark interaction potential within the nucleon.
QCD is the fundamental theory of strong interaction. It works well in the perturbative region but it does not work in the non-perturbative region. It is difficult for us to use QCD to study the thermal properties of quark bound states directly. In this case, people have to use the model [3][4][5][6] to study the dissociation of quarkonium states. In high temperatures and density, the interaction between quarks is screened [7] and the binding energy will be decreased. As a result, the nucleon will start to melt when the binding energy become low enough. The melting of nucleon can be solved by Schrödinger equation of three body. For the calculation, we need the interacting potentials among quarks of the nucleon in the hot medium, which is temperature-dependent. Unfortunately, the potentials are not yet well understood up to now. The free energy of a static heavy three-quark system F qqq (r, T ) can be calculated in lattice QCD and the internal energy can be obtained by using thermodynamic relation. In the present approach, the needed potential is assumed to be the internal energy, i.e. V = F + sT with s being the entropy density s = −∂F/∂T . The temperature-dependent form of F qq (r, T ) can be constructed based on Debye-Hückel theory [8], as having been done in Ref. [7]. Then, we can determine the T-dependent parameter in the free energy by fitting it with the lattice data. According to the relation between F qq (r, T ) and F qqq (r, T ), we can obtain the free energy of heavy three-quark system. We assume the conclusions are applicable to light quark system due to the flavor independence of the strong interaction. After constructing the potential of nucleon system at finite temperature, we can obtain the temperature dependence of binding energies and radii by solving the corresponding Schrödinger equation. The dissociation temperature is the point where the binding energy decreases to zero. The Gaussian expansion method (GEM), which is an efficient and powerful method in few-body system [9], is employed to calculate dissociation temperature of nucleon in this paper. This paper is organized as follows. In Sec.II, we show the rationality of GEM on studying the melting of Charmonium and Bottomonium by comparing our results with others. In Sec.III, we construct the potential of nucleon and apply GEM to solve corresponding Schrödinger equation. In Sec.IV, we show the results at quenched and 2-flavor QCD, respectively. Sec.V contains summary and conclusion. Before studying the dissociation of nucleon, we test the rationality of GEM on studying dissociation temperatures of quarkonium by comparing our results with others. To compare with Satz's results, the potential of quarkonium at finite temperature we use is the same as Satz's work [4]. The results on dissociation temperature of Charmonium, Bottomonium in Ref. [4] and our calculation results are listed in TableI, TableII, which show our results are consistent with that in Ref. [4]. So GEM can give accurate results on the dissociation temperature of quarkonium. Giving accurate binding energy and wave function [9] makes GEM very suitable for studying dissociation temperature of quark bound states (more detail can be found in Appendix A). In the following, we will use this method to calculate the dissociation temperature of nucleon.

A. Constituent Quark Model
The constituent quark model is a non-relativistic quark model [10]. In the constituent quark model, baryons are formed by three constituent quarks, which are confined by a confining potential and interact with each other [11]. The potential of baryon can be described by a sum of the potential of corresponding two-quark system. In Kaczmarek's work [12], it has been calculated in lattice QCD that the potential of diquark system is about half of that of corresponding quark-antiquark system, i.e. V qq = 1 2 V qq . The simplest and most frequently used po-tential for a qq system is the Cornell potential [7], where α is the coupling constant, and σ is the string tension. In the present work, we neglect the spin-dependent part of potential here. Thus our Hamiltonian is written as where m i is the constituent quark mass of the i-th quark, and T cm is the kinetic energy of center-of-mass frame (cm). r ij = r i − r j is the relative coordinate between i-th quark and j-th quark. In this model, the mass of light quark (u and d quark) we use is 300 MeV. The parameters of Cornell potential we use are: α = 1.4, √ σ = 0.131 GeV. Solving the corresponding Schrödinger equation, HΨ total = E m Ψ total , with GEM, we can get the mass E m and corresponding wave function of nucleon Ψ total . We define the radii of nucleon as with where r i is the distance between the center of nucleon and i-th quark. Using the calculated wave function, we can calculate the radii of nucleon. The calculating mass and radii of nucleon are 939 MeV and 0.83933 fm, respectively. While the corresponding experimental data are about 939 MeV and 0.841 fm. We can see this model gives a good estimation of the properties of nucleon even if the spin-dependent part is neglected. So it is reasonable for us to use this potential model to study the dissociation of nucleon. Of course, we need notice that the spindependent part plays an important role in the baryon spectrum.

B. Wave Function
Here, we solve the Schrödinger equation with GEM. In this method, three sets of Jacobi coordinates (Fig.1) are introduced to express three-quark wave function. The Jacobi coordinates in each channel c(c = 1, 2, 3) are defined as FIG. 1. Three sets of Jacobi coordinates for a three-body system [9].
where x i is the coordinate of the i-th quark and (i, j, k) are given by Table III. The total wave function is given as a sum of three rearrangement channels (c = 1 − 3) where the index α represents (s, S, l, L, I, n, N ). Here s is the spin of the (i, j) quark pair, S is the total spin, l and L are the orbital angular momentum for the coordinate r and R, respectively, and I is the total orbital angular momentum. The wave function for channel c is given by as given in Ref. [11]. The orbital wave function Φ where the range parameters, ν n and λ N , are given by ν n = 1/r 2 n , r n = r 1 a n−1 (n = 1, ..., n max ), λ N = 1/R 2 N , R N = R 1 A N −1 (N = 1, ..., N max ); (13) In Eqs. (11) and (12), N nl (N N L ) [9] denotes the normalization constant of Gaussian basis. The coefficients C c,α of the variational wave function, Eq. (8), are determined by Rayleight-Ritz variational principle.

C. Potential model for nucleon
The potential of nucleon at zero temperature has been discussed above and its parameters have been determined by fitting the properties of nucleon. To determine the dissociation temperature of nucleon, we need the potential in hot medium, i.e. V qqq (r, T ) (the index q represents u or d quark). Here, we assume that the potential is just the internal energy where s is the entropy density s = −∂F qqq /∂T . In Refs. [13][14][15], Kaczmarek's works show that the color singlet free energies of the heavy three-quark system (F 1 qqq ) can be described by the sum of antitriplet free energies of the corresponding diquark system (F 3 qq ) plus self energy contributions when the temperature is above T c . It can be expressed as where P = i<j R ij and the self energy F q (T ) = 1 2 F 3 qq (∞, T ). In Ref. [12], O.
Kaczmarek's work suggests a simple relation between free energies of anti-triplet qq states and color singlet qq The form of F 1 qq can be obtained based on studies of screening in Debye-Hückel theory. It can be written as [7] (17) where screening mass µ and the parameter κ are temperature-dependent, and K 1/4 [x] is the modified Bessel function. We can determine the T-dependent µ and κ by fitting F 1 qq (r, T ) to the lattice result obtained in quenched [16] and 2-flavor [17] QCD. At r = ∞, the free energy F 1 qq (T ) is wirtten as Thus, the form of µ(T ) is given as function of F (T ) Once we obtain the temperature dependence of µ(T ), we fit Eq. (17) to the lattice data to obtain κ(T ). The results for µ(T ) and κ(T ) are shown in Fig. 2 and Fig.  3, respectively. In Fig. 4, we show our fit curves (solid lines) together with the lattice results. We can see that the resulting F 1 qq (r, T ) fits the lattice results quite well for all r and in a broad range of temperatures from T c to 4T c in the two cases. For higher temperature, the resulting F 1 qq (r, T ) cannot be fitted quite well to the lattice results in quenched QCD. There can be higher order corrections to Poisson equation [7]. To obtain the binding energies of nucleon, we define a effective potential as Combining Eqs. (14)(15)(16)20), we can get a relation between effective potential and free energies of qq Replacing the potential term, 3 1=i<j 1 2 V (r ij ), in Eq.
(2) with this effective potentialṼ qqq (R, T ), we can get a new Hamiltonian for nucleon at finite temperature written as Solving corresponding Schrödinger euqation, with GEM, we can get the binding energies ∆E(T )(= − (T )) and corresponding wave function at finite temperature. Using the wave function, we can calculate the T-dependent radii according to Eq. (4).

IV. NUMERICAL RESULTS
In Fig. 5, we show the resulting binding energies behaviour for nucleon in quenched and 2-flavor QCD. We can see there is little difference between the two lines. When they vanish, the nucleon no longer exists. So ∆E(T ) = 0 determines the dissociation temperature. From Fig. 5, we get the dissociation temperature in quenched and 2-flavor QCD are about 3.0T c and 3.3T c , respectively; in Fig. 6, we show the corresponding nucleonic radii. The dissociation temperature determined from Fig. 6 is consistent with that determined from Fig.  5. It is seen that the divergence of the radii defines quite well the different dissociation points in the two cases. The resulting dissociation temperatures have a little difference between the two cases.

V. SUMMARY AND CONCLUSION
The free energies of quark-antiquark system we construct based on Debye-Hückel theory at finite temperature fit the lattice results quite well from T c to 4T c , but not well for higher temperature. According to Kaczmarek's works, we can get a relation between color singlet free energy of heavy qqq system F 1 qqq and color singlet free energy of heavy qq system F 1 qq , written as about 3.0T c and 3.3T c , respectively. There are a little difference between the two results. Comparing with J/ψ, the dissociation temperature of nucleon is higher. So, nucleon is more difficult to melt than charmonium. For the potential, we neglect the spin-dependent part in this work which may has some effects to the resulting dissociation temperature. The effects arising from spindependent part deserve further studies.