1. Study of Charmonium States in Vacuum and High Density Medium
Juan Alberto Garcia,
The University of Texas at El Paso
Cyclotron REU 2009
Advisor: Dr. Ralf Rapp

2. Overview Some words about Quarks The Quark Gluon Plasma
The Charmonium System
The Schrödinger Equation
The Hydrogen Atom
Cornell Potential
Color Screening Potentials
Lattice QCD Potential

3. Quarks Proton Quarks Gluons
Nucleons and other hadrons are made up of quarks.
Gluons
Quarks interact with one another via the Strong Force; described by Quantum Chromodynamics (QCD).
The Force is mediated by Gluons.
Each quark and gluon carry a color charge, which is conserved in Strong interactions.

4. Quarks are confined to one another, by a phenomenon called “quark confinement”.
This happens because the force between becomes constant with distance.
Quarks also undergo the Annihilation-Creation process in which a quark-antiquark pair is either converted or created from or into a gluon.

5. The Quark Gluon Plasma
QCD predicts a new phase of quark matter above temperatures of about 170 MeV (2*1012 K), the Quark Gluon Plasma (QGP).
In this state quarks are deconfined from the hadrons that contain them and form a hot and dense plasma.
The picture shows a Au-Au collision at RHIC in which a QGP is believed to have been created, but for this to happed a probe its needed!!

6. Charmonium
Hadrons containing heavy quarks(charm and bottom) have been identified as possible probes for the QGP.
This is believed because if the QGP is created, charmonium should be suppressed which is one of the signals of formation of the QGP.
Charmonium consists of a charm quark and an anticharm antiquark pair in a bound state.
Since the charm has a large mass a non relativistic approach is valid for its study.

7. The Schrödinger Equation
Describes the time evolution of the physical state of a quantum system.
In its time independent form, it becomes an eigen value equation.
If a central potential is used then the PDE is separable and can be set in the form Ψ(r, θ, φ)=R(r)*Θ(θ)*Φ(φ), where R(r) is the equation for the radial part of the solution.

8. The Hydrogen Atom (Test)- +
A proton and electron pair form a bound state called Hydrogen-1 (Hydrogen atom).
Proton
Electron
Photon
Not to scale.
Schrödinger’s equation can be solved for the hydrogen system analytically.
The electromagnetic potential is used in this scenario to solve the Schrödinger radial equation.

9. Methods
In order to solve Schrödinger’s equation, for the given potential; it was rewritten in finite difference form (FDF).
FDF consists in approximating the derivatives using finite difference relations.
With these approximations, one is able to evolve the system from a given initial condition.

10. Shooting
A solution to the problem must satisfy boundary conditions which are: R(0)=C and R(∞)=0.
The solution can not be obtained by a simple evolution of our initial condition because the boundary conditions can only be satisfied with a discrete number of Energies.
The algorithm is based on the Shooting method:
R(0)=C E1 E2 E3 E4
Choose initial condition and initial energy.
Solve Equation for given energy
If condition for a solution is met then keep eigen energy and quit.
If the function has the different sign as the previous one, decrease the energy the previous amount and decrease energy steps by a factor of 2.
Increase energy by a small amount.
Go back to step 2.

11. Back to Hydogen
Hydrogen was solved to test the numerical accuracy of our program because it had an analytic solution it can be compared to.
n=1, l=0
As can be seen on the left solution does diverge but only after large distances where effects can be neglected
n=1, l=0

12. Plots for n=1, l=0; n=2, l=0; n=2, l=1.

13. The Cornell Potential
A potential to simulate non relativistic quark interactions in Vacuum
Two terms: “coulomb” which accounts for one-gluon exchange; and linear which accounts for quark confinement.
The potential does not contain “String Breaking”.
Revisa esta …
As the quarks are separated, the system stops being energetically favored and the quarks go to a lower state of energy by forming bound states with lighter quarks. This is referred as “String Breaking”.

14. Vacuum Potential J/ψ ----> 3.0969 ψ’ ----> 3.659
A charm bare mass of GeV was used in order to match the total mass of the J/ψ to which is about its real mass.
The total mass is given by 2 times the bare mass plus the Eigen Energy of the State.
α=.212
σ=.422 GeV2
J/ψ ---->
ψ’ ---->
χc---->
n=3, l=2---->

15. fm fm fm fm

16. Color Screening
Interacting matter of sufficient temperature and pressure is predicted to undergo a transition to a state of deconfined quarks and gluons (the QGP).
We say deconfinement occurs when color charge screening becomes strong enough that it shields the quark binding potential with any other quark or anti quark.

17. Cornell potential with screening mass
The Screening mass is defined as μ(T)= 1/rD; where rD is the color screening length.
The screening length is the distance at which the color force becomes suppresed
r radius
μ(T) temperature dependent “Screening mass”
σ GeV2 α

18. This is expected because μ(T) increases as the temperature increases
From the graph we can see that as expected the radius increases with the screening mass, and data tells us that the system is completely dissolved with a screening mass higher than about 600MeV.
This is expected because μ(T) increases as the temperature increases fm

19. Lattice QCD Potential
Computed by numerical simulations of QCD at finite temperatures
Vacuum limit potential is used to match bare mass.
Bare mass is set so that J/ψ mass matches GeV
r1=.317 fm
mass = >
E 1 = l= 0

20. Medium Potential
V(r, T)

21. As can be seen in the table both BE and KE decrease as T increases.
By comparing the energies with the screening threshold, we can calculate the Kinetic and Binding Energies for Each state.
As can be seen in the table both BE and KE decrease as T increases.
All other states have been dissolved. -
T (Tc)
Mass (GeV)
BE (GeV)
KE (GeV)
1.2
3.3506
.2031
.4263
1.8
3.0999
.0576
.3009
2.4
3.0073
.0466
.2546

22. Acknowledgements Advisor: Dr Ralf Rapp Dr. Riek Felix Xingbo Zhao
Dr. Sherry Yennello

23. References
R. Rapp and H. van Hees Heavy Quark Diffusion as a probe for the Quark-Gluon Plasma. arXiv: v2.
Stephen Gasiorowicz “Quantum Physics”.
E. Eichten, K. Gottfried, T. Kinoshita, K. D. Lane, T. M. Yan Charmonium: Comparison with Experiment. Physical Review D.
F. Karsch, M.T. Mehr, H. Sartz Color Screening and deconfinement for bound States of Heavy quarks. Zeitschnft fur Physik C.