J/$\Psi$-J/$\Psi$ scattering cross sections of Quadratic and Cornell Potentials

We study the scattering of J/$\Psi$-J/$\Psi$ mesons using Quadratic and Cornell potentials in our tetraquark ($c$$\bar{c}$$c$$\bar{c}$) system. The system's wavefunction in the restricted gluonic basis is written by utilizing adiabatic approximation and Hamiltonian is used via quark potential model. Resonating group technique is used to get the integral equations which are solved to get the unknown inter-cluster dependence of the total wavefunction of our tetraquark system. T-Matrix elements are calculated from the solutions and eventually the scattering cross sections are obtained using the two potentials respectively. We compare these cross sections and find that the magnitude of scattering cross sections of Quadratic potential are higher than Cornell potential.


Introduction
The J/Ψ meson, a bound state of a charm quark and its anti-particle was discovered in 1974 and since then, it has been studied extensively from a theoretical and experimental point of view. In heavy ion collisions at RHIC [1] and recently at LHC [2], its production mechanisms have been explored. Perhaps, the most significant phenomenon as a result of heavy ion collisions is a formation of the particular state of matter called the quark-gluon plasma (QGP). A fundamental effect associated with the QGP medium is known as the color screening i.e. the interaction range of heavy quarks decreases with the increase in surrounding temperature [3]. As a result, the potential between heavier quarks cc or bb gets screened due to the deconfinement of other quarks and gluons. The consequent separation of heavy quarks leads to suppressed quarkonia yields. J/ Ψ suppression, an idea first put forward by theorists T. Matsui and H. Satz [4] in 1986, is considered one of the indicators of the formation of QGP. Since charm quark is more abundantly produced in heavy-ion experiments as compared to bottom quark, researchers initially thought that J/ Ψ could also be used for measuring the temperature of QGP. However, due to technical issues associated with J/Ψ production it is an unsuitable QGP temperature probe. Among J/ Ψ's dominant decay

Wave function and the Hamiltonian
We write the state function of our tetraquark system by utilizing adiabatic approximation [40] as follows where, Q c : position of COM (center of mass) of our complete system, Q 1 : vector joining COM of (1,3) and (2,4) clusters, u 1 : vector representing position of quark 1 w.r.t. quark3 in the cluster (1,3), v 1 : vector representing position of quark 2 w.r.t. quark4 in the cluster (2,4). Identical expressions are given for the clusters (1,4) and (2,3) via Q 2 , u 2 and v 2 . Likewise, (1,2) and (3,4) are defined by Q 3 , u 3 and v 3 . Hence, The following diagrams show the possible topologies of quark anti-quark clusters: As we are considering the system in centre of mass reference frame ψ c (Q c ) has no role in the dynamics of the four quark system. χ k (Q k ) is an unknown in the radial part of total wavefunction and ξ k (u k ),ζ k (v k ) are the predefined Gaussian wave functions. |k f , |k s and |k c are flavor, spin and color parts of the wave function respectively. The Hamiltonian of our (cccc) system is written where m andP denote quark mass and linear momentum respectively. Also, F i = λ i /2 for quark and F i = −λ * i /2 for an anti-quark, where λ's are the well-known Gell-Mann matrices. The potentials v(r ij ) used for qq pair wise interaction are Quadratic Potential [27] v ij = Cr 2 ij +C (7) where i, j = 1, 2,3,4 and Cornell Potential (Coulombic plus linear potential) [28] is written as with i, j = 1, 2,3,4 where α s is the strong coupling constant and b s is the string tension (Flux tube model) The mesonic size d appearing in ξ k (u k ) and ζ k (v k ) can be adjusted in a manner such that the Gaussian ground state wave function of quadratic potential approximates the ground state wave function of Cornell potential. Hence, their overlaps become unity for a fitted value of parameter d [32].

Integral equations and their solutions
We employ the RGM (resonating group method) technique [39] and take variations only in the χ k (Q k ) factor of the total wave function Ψ T , and by making the use of linear independence of these variations both w.r.t. the two values of k, i.e. k=1,2 and w.r.t. all possible continuous values of Q k in δΨ T |H − E c |Ψ T = 0 we get, from eq.1, the following two integral equations The operator (Ĥ -E c ) is identity in the flavour basis whereas the overlap factors are also the (Ĥ -E c ) operator's flavour matrix elements. So, our potential energy matrix in spin and colour basis is The spin overlaps are given as where V qq is a vector meson. Now, we discuss the Kinetic energy operator. In spin spaceK where k=1,2. Opening the summation over l, we have the following two equations respectively In the first integral equation's diagonal part, Q 1 ,u 1 and v 1 are linearly independent so we take χ 1 (Q 1 ) outside the integral. Since χ 1 (Q 1 ) is the only unknown so we can integrate the remaining integrands. In the off-diagonal part of (12), we replace u 1 ,v 1 with Q 2 and Q 3 whereas u 2 ,v 2 is replaced by a linear combination of Q 1 ,Q 2 and Q 3 . Q 1 ,Q 2 and Q 3 form a set of linearly independent vectors. The same steps apply for the integral equation (13). After performing some differentiations and integrations, we obtain the following two equations from (12) and (13).
where, written up to accuracy 4, Similarly, we also have We Fourier transform (14) w.r.t. Q 1 and applying the Born approximation, we use inside the integral to get Here, χ 1 (P 1 ) is the Fourier transform of χ 1 (Q 1 ). We write the formal solution of (18) as with, If x-axis is chosen along P 1 and z-axis in such a way that the xz-plane becomes the plane containing P 1 and P 2 , the aforementioned equation takes the following form where, in the rectangular coordinates, Here ϕ is the angle between P 1 and P 2 . As we are considering elastic scattering, so (20), we can write the 1,2 element of the T-matrix [40] as follows and, M 1 = M 2 = 3ω 2 For the total spin averaged cross sections, we make use of the following relation [42] where J and s 1 , s 2 denote the total spin of the two outgoing mesons and spins of the two incoming mesons respectively. In our situation J = 0 and s 1 = s 2 = 1. Thus, for i = 1, j = 2 we obtain Similarly, for i = 2, j = 1

Results with Quadratic Potential
To fit the parameters for quadratic potential, we first took the spin averaging over E i to obtain the value of ω and c for the set of mesons η c (1S), η c (2S), J/ψ(1S) and J/ψ(2S). Here E i = (ω/2)(4n + 2l + 3) + c [46]. By using the fitted value of ω = 0.303 GeV and c = 2.61 GeV the constant C = −(3/16)(2µω 2 ), mesons sizes d = 1/2µω andC = −(3/4)(c − 2m) are obtained. The constituent quark mass is taken from ref. [47] and the meson mass is obtained from [42].  The results clearly indicate that with an increase in T c the total cross section gradually decreases.

Results with Cornell potential
For Cornel potential the parameters α s and C are adjusted by minimizing the χ 2 between the masses taken from [42] and a spectrum generated by using Cornell in the quark potential model for the mesons η c , J/ψ, h c , χ c0 , χ c1 and χ c2 . The other parameters i.e string tension, σ and quark mass are taken from [45].

Results with Coulombic plus Quadratic Potential
After a discussion of Cornell and Quadratic potentials respectively, we also incorporate Coulombic plus Quadratic potential in our study. It is defined as where i, j = 1, 2,3,4 For Coulombic plus Quadratic potential the parameters α s ,C andC are adjusted by minimizing the χ 2 between the masses taken from [42] and a spectrum generated by using the Coulombic plus Quadratic potential in the quark potential model for the mesons η c , J/Ψ ,h c ,χ c0 ,χ c1 ,and χ c2 . The quark mass is taken from [45].   It is observed that if we replace linear confinement with the quadratic in the Coulombic plus linear potential, then the magnitude of the cross sections lie in between the two i.e. the quadratic and the Cornell potentials. As noted in our earlier work [30] , the meson sizes obtained via two potential models show a remarkable difference in magnitude i.e. the meson size 'd' calculated for Quadratic potential is about 1.5 times greater than its value for the Cornell potential. A possible explanation could take into account the respective shapes of the two potentials for any given value of energy E. Thus, for a specific energy E, the value of classical turning point r 0 (E = V (r 0 )) is greater for Quadratic potential than the Cornell potential and beyond classical turning point (where E < V (r)) the state-function dampens rapidly. This means that a greater value of classical turning point for quadratic potential gives us a larger rms radius compared to the Cornell potential. The rms radii for the Cornell and Quadratic potentials are d = 0.995 GeV −1 and d = 1.49 GeV −1 (where d = 1 2µω [27]) respectively. Lastly, we can also study the effect of scattering angles on the aforementioned cross sections for quadratic potential. The results for different angles (such as ϕ = 0 • ,30 • ,60 • ,90 • ) between the two incoming waves can be plotted. However, it can be shown that varying the scattering angle has no effect whatsoever on the respective cross sections i.e. the graphs for different values of scattering angle ϕ overlap if plotted simultaneously. Hence, the scattering cross sections are independent of the angles between the two incoming waves.

Conclusions
We have considered J/ Ψ -J/Ψ scattering using Quadratic and Cornell potentials. The qualitative behavior of both graphs is identical i.e. with an increase in total COM energy the scattering cross sections gradually decrease. However, by using quadratic potential we obtain higher magnitudes of scattering cross sections. It is pertinent to mention that throughout our discussion, we have reported our model-dependent results upto three significant figures.
At low energies only S-wave scattering is significant. The contribution to scattering cross sections from other partial waves, I > 0, is negligible. As we are studying the case of S-wave, so it can be seen that as the energy increases, scattering cross sections decrease which is in accordance with the experimental results (lattice QCD simulations) for other scatterings. The quadratic confinement is a good approximation of more realistic Cornell potential as far as the properties of conventional hadrons are concerned. However, this comparison shows that as far as the tetraquark systems are concerned the quadratic confinement does not appear to be a good approximation of Cornell confinement. So, in order to find out the properties of four quark systems like the spectra of four quark states, it is recommended not to use the quadratic confinement as a replacement of Cornell potential for future studies.
Since heavy quarkonia decay strongly, it is very difficult to directly measure the dissociation cross sections in hadron scattering experiments. As we mentioned in our introduction, the cross sections are calculated using theoretical approaches. One such theoretical approach called the quark-interchange model [31] has been successfully supported with light hadron scattering data (I = 2 ππ [31], I = 3/2 Kπ [43], I = 0, 1 K, N [44]). While we have retained basic features of quark-interchange model, our improvement to certain treatments in it suggest that our calculations may be reasonably sound as far as the heavy hadron-hadron scattering is concerned. However, a direct comparison with experimental data on heavy hadron interaction is not possible. In future, it may be useful to check out our predicted cross sections using detailed Monte-Carlo simulations. If our work turns out to be reasonably accurate, then it will be clearly beneficial to include our insights in simulations involving hadron processes in heavy-ion collisions and other studies concerning tetraquark states.