1. Lattice Quantum Chromodynamics By Leila Joulaeizadeh 19 Oct. 2005 1- Literature : Lattice QCD , C. Davis Hep-ph/0205181 2- Burcham and Jobes

2. Outline • - Introduction • Hamilton principle • Local gauge invariance and QED • Local gauge invariance and QCD • Lattice QCD calculations • Some results • Conclusion

3. What is Quantum Chromodynamics and why LQCD? • Strong interaction between coloured quarks by exchange of coloured gluon • Gluons carry colour so they have self interaction • Self interaction of gluons , nonabelian group SU(3) • - QCD is a nonlinear theory so there is no analytical solution and we should use numerical methods

4. Euler Lagrange Equation

5. For motion of a point like particle with mass m in a central potential: Physical systems will evolve in such a way to minimize the action Hamilton Principle

6. In Quantum Field Theory

7. Examples Scalar field (spin 0 particle) Spinor field(spin 1/2 particle)

8. Local Gauge Invariance and QED

9. Example Massless vector field(spin 1)

10. Local Gauge Invariance and QCD Non-Abelian nature of SU(3) Gluon self interaction term

11. Diagrams representing propagation of free quark and gluon and their interaction

12. Lattice QCD O : operator whose expectation value we want to calculate

13. Lattice gauge theory for gluons x X+1 x X+1 x

14. Lattice gauge theory for gluons

15. Fermion doubling problem of quarks on the lattice

16. Solutions of Fermion doubling problem

17. Action with quarks

18. Relating lattice results to physics Make the correlators of quarks by using  matrices r

19. Steps of typical lattice calculation 1- choose the lattice spacing - close to the continuum - computation costs2- Choose a quark formulation and number of quark flavors3- generating an ensemble of gluon configurations - Try to go near small masses - computation costs4- calculation of quark propagators on each gluon configuration5- combination of quark propagators to form hadron correlators6- Determination of lattice spacing in Gev(lattice calibration) 7- extrapolation of hadron masses as a function of bare quark masses8- repeat the calculation using several lattice spacing to compare with physical results at the limit of a 09- compare with experiment or give a prediction for experiment

20. Some results of lattice QCD calculations The spectrum of light mesons and baryons in the quenched approximation

21. The ratio of inverse lattice spacing

23. c   JPC Charmonium spectrum in quenched approximation

24. Summary • Photons don’t carry any colour charge, so QED is analytically solvable. • Gluons do carry colour charge,so to solve the QCD theory, approximations are proposed • (e.g. Lattice calculation method ). • - There is a fermion doubling problem in lattice which can be solved by various methods. • In order to obtain light quark properties, we need bigger computers and the • calculation costs will be increased. • Quenched approximation is reasonable in order to decrease the computation costs.