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=mc 2 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 ( n 2 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 Master Slave Index, Entry Matrix Construction Diagonalization (Master Processor) Output Eigenvalue Next n Index Index, Entry Entry Subroutine Entry Subroutine Index Entry

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. 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. Results: Positronium Eigenvalues converge to -0.500638 ± 3.741×10 -5. For simplicity of calculation, α and ħ have been set equal to one. The result must then be multiplied by the factor Fit = -0.500638 + 2.4395 x -0.81524

20. Results: TDA Eigenvalues converge to 3.31843 ± 0.000785. The results of this calculation are in units of gluon mass, m g, 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, m g, 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