Investigations of QCD Glueballs

1. Investigations of QCD Glueballs Denver Whittington Anderson University Advisor: Dr. Adam Szczepaniak Indiana University Summer 2003

2. Introduction: QCD • Quantum Electrodynamics (QED) • Electromagnetic Interaction • Electric Charge • Positive/Negative • Quantum Chromodynamics (QCD) • Strong Interaction • Color Charge • Red/Anti-Red, Blue/Anti-Blue, Green/Anti-Green

3. Introduction: QCD • Electromagnetic interactions are mediated by photons. • Strong Interactions are mediated by gluons.

4. Introduction: Glueballs • As a consequence of QCD, gluons themselves interact strongly. • This allows them to form hybrid mesons and particles of pure radiation called glueballs. • The simplest glueball consists of two gluons.

5. Approximation: Ground-State • The lowest energy at which a glueball may exist is the ground-state energy of a two-constituent-gluon glueball. • Approximation of this energy involves more than the interaction of the two gluons.

6. Approximation: Virtual Medium • E=mc2 and ΔEΔt=ħ/2 • Virtual Particles • Many-Body Problem

7. Approximation: Methods • Single gluon, no vacuum interactions • Two gluons, no vacuum interactions • Single gluon plus virtual gluon interactions • “in-virtual-medium” gluon • Two gluons plus virtual gluon and virtual glueball interactions • Two “in-virtual-medium” gluons, no vacuum interactions • Tamm-Dancoff Approximation (TDA) • Two “in-virtual-medium” gluons plus interactions with virtual glueballs • Random Phase Approximation (RPA)

8. Approximation: TDA • Properties of constituent gluons adjusted for individual vacuum interactions • Two-body problem • Schrödinger equation • Solution involves diagonalization of a symmetric matrix based on the Hamiltonian

9. Approximation: RPA • Extension of Tamm-Dancoff Approximation • Addition of glueball interactions with virtual particles. • Many-body problem • Solution involves diagonalization of a non-symmetric matrix.

10. Approximation: Goal • As the complexity of the approximation increases, the contributions from the extra effects become negligible. • If the TDA and RPA methods yield similar results, the effects of the vacuum on glueballs beyond interactions with the constituent gluons can be ruled negligible. • Goal: To investigate the role of these many-body effects on the ground-state energy of a two-constituent-gluon glueball.

11. Positronium: An Example Calculation • Electron • Positron (Anti-electron) • Bound Electromagnetically • Instructive Example to Understand Computation • Similar System to Two-Gluon Glueball • Numerical Solution Is Similar

12. Positronium: Schrödinger Equation in Momentum Space

13. Positronium: Solution by Matrix Diagonalization • Thus, diagonalization of the matrix A yields eigenvalues which are the energies of the symmetric states of the system.

14. Positronium: Extrapolation of Ground-State Energy • As the interval is more finely partitioned, the matrix becomes larger and the summation approaches the integral. • As the matrix size increases, the eigenvalues will converge to the true ground-state energy. • Plotting eigenvalues vs. matrix size and fitting a curve allows extrapolation of the energy.

15. Numerical Computation: Parallel Processing with MPI • Large matrix sizes (n = 100 to 2000) • Long construction time (n2 elements) • Parallel processing • Evaluate multiple elements simultaneously • Message-Passing Interface (MPI) • Subroutine library for creating a parallel processing environment on a network of computers

16. Numerical Computation: Parallel Processing Framework • Master Processor sends indices to Slave Processors. • Slave Processors compute and return entry, then acquire a new pair of indices. • Master Processor diagonalizes matrix and outputs lowest eigenvalue. • Program loops for a new matrix size.

17. Numerical Computation: Parallel Processing Framework Matrix Construction Master Index, Entry Index, Entry Slave Index Slave Index Index Entry Index Entry Entry Subroutine Entry Subroutine Next n Diagonalization (Master Processor) Output Eigenvalue

18. Numerical Computation: Parallel Processing Framework • Positronium Approximation and TDA produce symmetric matrices. • Evaluate upper half of entries plus diagonal. • Use diagonalization subroutine for symmetric matrix. (faster) • RPA produces non-symmetric matrix. • Evaluate all entries. • Use diagonalization subroutine for general matrix.

19. Results: Positronium Fit = -0.500638 + 2.4395 x -0.81524 • Eigenvalues converge to -0.500638 ± 3.741×10-5. • For simplicity of calculation, αand ħ have been set equal to one. The result must then be multiplied by the factor • The resulting approximation for the ground-state energy of positronium is -6.811 ± 5.05×10-4 eV, which agrees favorably with the accepted value of -6.805 eV.

20. Results: TDA • Eigenvalues converge to 3.31843 ± 0.000785. • The results of this calculation are in units of gluon mass, mg, which is between 0.5 and 0.6 GeV. Fit = 3.31843 + -0.499722 x -0.352296 • The resulting Tamm-Dancoff approximation for the ground-state energy of a two-constituent-gluon glueball is between 1.659 and 1.992 GeV.

21. Results: RPA • Eigenvalues converge to 3.31728 ± 0.000785. • The results of this calculation are in units of gluon mass, mg, which is between 0.5 and 0.6 GeV. Fit = 3.31728 + -0.498628 x -0.351692 • The resulting random phase approximation for the ground-state energy of a two-constituent-gluon glueball is between 1.658 and 1.991 GeV.

22. Results: Comparison • The positronium example produces the correct ground-state energy. • The program frameworks produce correct results and can be used for the TDA and RPA methods. • The TDA and RPA methods both calculate the ground-state energy of a two-constituent-gluon glueball as approximately 3.32 gluon masses.

23. Conclusions • Agreement between Tamm-Dancoff and random phase approximations. • Two-constituent-gluon glueball mass is approximately 3.32 gluon masses (1.658 to 1.992 GeV). • Vacuum effects beyond interactions with the individual constituent gluons seem to be negligible.

24. Acknowledgements • Dr. Adam Szczepaniak, Indiana University • Dr. Andrew Bacher, Indiana University Cyclotron Facility • Dr. Mark Pickar, Minnesota State University, Mankato • Fellow REU Students Work supported in part by the National Science Foundation and the U.S. Department of Energy