Lagrangian

1. fermionも加える Lagrangian

2. Feynman rules gluon propagator vertices FP ghost vertex propagator quark vertex propagator

3. Feynman rules internal lines propagators gluon quark FP ghost gluon propagator vertices FP ghost vertex propagator quark vertex propagator

4. Feynman rules internal lines propagators gluon quark FP ghost gluon propagator vertices FP ghost vertex propagator quark vertex propagator

5. Feynman rules internal lines propagators gluon quark FP ghost gluon = FP ghost quark

6. Feynman rules gluon vertices 3 gluon vertex 4 gluon vertex gluon propagator vertices FP ghost vertex propagator quark vertex propagator

7. Feynman rules gluon vertices 3 gluon vertex 4 gluon vertex 3 gluon vertex 4 gluon vertex

8. Feynman rules ghost-gluon vertex fermion-gluon vertex gluon propagator vertices FP ghost vertex propagator quark vertex propagator

9. Feynman rules ghost-gluon vertex fermion-gluon vertex ghost-gluon vertex fermion-gluon vertex

10. external lines gluon 1 FP ghost incoming outgoing quark anti-quark loop fermion loop -1 T matrix

11. loop diagrams dimension

12. convenient formulae

13. + finite terms + finite terms + finite terms

15. higher loops

16. 4 ghost gluon vertex part 収束 ghost ghost 2 gluon vertex part 収束 収束

17. relation among numbers = # of internal fermion lines, = # of internal boson lines = # of external boson lines = # of external fermion lines, = # of loops, = # of differentiations at vertices, divergent if depends only on the numbers of external lines

18. primitive divergences gluon self energy part quark self energy part ghost self energy part quark quark gluon vertex part ghost ghost gluon vertex part 4 gluon vertex part 3 gluon vertex part

19. renormalization r: renormalized quantity ZX: renormalization constant :renormalized Lagrangian, :counter term

20. counter term

21. addition to the Feynman rules

23. renormalization There always exist counter terms for divergences. e.g. The divergences are polynomials in momentum. BPHZ renormalization (Bogoliubov Parasiuk Hep Zimmermann) The divergences can be absorbed by The gauge symmetry (or BRS symmetry) ensures the Slavnov Taylor identities and The divergences can be absorved by

24. at one loop level + finite terms

25. + finite terms + + + finite terms

26. + + + finite terms + finite terms

27. + + + + + finite terms

28. + finite terms

29. Renormalization Condition In absorbing divergences by there remain ambiguities of finite constants. The condition to fix the ambiguities is called renormalization condition. The scheme with the renormalization condition is called renormalization scheme minimal subtraction scheme modified minimal subtraction scheme momentum subtraction scheme on-shell renormalization scheme Observable relations do not depend on the renormalization scheme