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/ )

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

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

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

Renormalised charge

Bare and renormalised charges

Defining the coupling at a scale

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

Higher orders }

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

Running coupling

Consistency check

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

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

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

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

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

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

Renormalisation scale setting

Scale variation

Summary and discussion (1)

Summary and discussion (2)

Summary and discussion (3)