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

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

3. 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. 4 Euler Lagrange Equation

5. 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. 6 In Quantum Field Theory

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

8. 8 Local Gauge Invariance and QED

9. 9 Massless vector field(spin 1) Example

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

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

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

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

14. 14 Lattice gauge theory for gluons

15. 15 Fermion doubling problem of quarks on the lattice

16. 16 Solutions of Fermion doubling problem

17. 17 Action with quarks

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

19. 19 1- choose the lattice spacing - close to the continuum - computation costs 2- Choose a quark formulation and number of quark flavors 3- generating an ensemble of gluon configurations - Try to go near small masses - computation costs 4- calculation of quark propagators on each gluon configuration 5- combination of quark propagators to form hadron correlators 6- Determination of lattice spacing in Gev(lattice calibration) 7- extrapolation of hadron masses as a function of bare quark masses 8- repeat the calculation using several lattice spacing to compare with physical results at the limit of a 0 9- compare with experiment or give a prediction for experiment Steps of typical lattice calculation

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

21. 21 The ratio of inverse lattice spacing

22. 22

23. 23 Charmonium spectrum in quenched approximation  c   J PC

24. 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.