1. Concepts of the Standard Model: renormalisation FK8022, Lecture 2 Core text: Quarks and leptons: an introductory course in modern physics, Halzen and Martin Further reading: Basics of QCD perturbation theory, D.E. Soper (hep-ph/9702203)

2. Lecture outline Infinities arising from loops – Regularisation – Reparameterisation of charge – Renormalisation  finite charge, amplitude  Renormalisation scale  Running of couplings

3. A simple QED process e e   p1p1 p2p2 p4p4 p3p3 q

4. Adding a higher order process + e e    ee ee

6. Renormalised charge

7. Bare and renormalised charges

8. Defining the coupling at a scale

10. Renormalisation scale [] -- + higher orders (A)(B)(C)

11. Higher orders }

12. Scale dependence of amplitude r(Q2)r(Q2) Order of calculation e  scattering Single photon exchange

13. Running coupling

15. Consistency check

16. Divergence in amplitude from higher order loop Make finite by redefining coupling/charge and remove infinity For a complete expansion the amplitude is independent of  r Renormalisation Finite expression for running of coupling/charge with  r Coupling/charge depends on an arbitrary kinematic scale  r for a given interaction

17. Renormalisation works: electromagnetic coupling running Interaction distance (momentum transfer) 2 /GeV 2 Tiny shift

18. Physical interpretation of the running coupling + e-e- e-e- e-e- e-e- e-e- e-e- e-e- e +,e - Distance scale Coupling

19.  Renormalisation of other quantities Different loop diagrams cause other physical quantities to be renormalised. Eg mass Masses also run:

20. What about the strong force ? qq qq qq qq qq qq qq qq q,q + other higher order diagrams

21. Asymptotic freedom ss Momentum transfer / GeV Interaction distance

22. Renormalisation scale setting

24. Scale variation

25. Summary and discussion (1)

26. Summary and discussion (2)

27. Summary and discussion (3)