investigations of qcd glueballs
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Investigations of QCD Glueballs. Denver Whittington Anderson University Advisor: Dr. Adam Szczepaniak Indiana University Summer 2003. Introduction: QCD. Quantum Electrodynamics (QED) Electromagnetic Interaction Electric Charge Positive/Negative Quantum Chromodynamics (QCD) - PowerPoint PPT PresentationTRANSCRIPT
Investigations of QCD Glueballs
Denver WhittingtonAnderson University
Advisor: Dr. Adam SzczepaniakIndiana University
Summer 2003
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
Introduction: QCD
Electromagnetic interactions are mediated by photons.
Strong Interactions are mediated by gluons.
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.
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.
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)
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
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.
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.
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
Positronium: Schrödinger Equation in Momentum Space
yyy
ppp
yy
yy
yy
dyyE
pp
ppdp
pE
dE
pE
2
22/
0
2
2
2
0
2
3
2
2
tantan
tantanln
coscos22
tan
ln22
2
4
2
2
ˆ
ppp
p
x
ppp
xxx
Positronium: Solution by Matrix Diagonalization
nn
iy
ny
yy
yy
yy
yyaE
i
ji
ji
jiij
jji
42
1 and
2
tantan
tantanln
coscos22
tan where, 2
22
Aχχ
Thus, diagonalization of the matrix A yields eigenvalues which are the energies of the symmetric states of the system.
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.
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
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.
Numerical Computation: Parallel Processing Framework
Master
SlaveSlave
Index, Entry
Matrix Construction
Diagonalization (Master Processor)
Output Eigenvalue
Nex
t n
Index
Index
Index, Entry
EntrySubroutine
EntrySubroutine
Index
Index
EntryEntry
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.
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
.2 22
2
n
Fit = -0.500638 + 2.4395 x -0.81524
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.
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.
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.
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.