By Forshaw J.

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**Extra resources for An introduction to QED and QCD**

**Example text**

2 Renormalisation of Quantum Chromodynamics Qcd is a theory of interactions between spin-1/2 quarks and spin-1 gluons. It is a non-Abelian gauge theory based on the group SU(3), with Lagrangian 1 L = − Gaµν Ga µν + 4 gauge fixing and f ψ f (iD / − mf )ψf + ghost terms . 6) Here, a is a colour label, taking values from 1 to 8 for SU(3), and f runs over the quark flavours. 7) where the f abc are the structure constants of SU(3) and the T a are a set of eight independent Hermitian traceless 3 × 3 matrix generators in the fundamental or defining representation (see the pre school problems and the quantum field theory course).

Normal ordering means we are to put all creation operators to the left of all the annihilation operators. If this means moving an anticommuting (fermion) operator through another such operator then we need to remember to pick up a minus sign. After a bit of algebra we can get H= d3 k 1 [b† (k)bα (k) + d†α (k)dα (k)]. 20) If we had tried to impose commutation relations, the dd † term would have entered with a minus sign in front, which would signal that something has gone wrong. In particular, it would mean that d † creates particles of negative energy.

2. These both contain a loop with one photon propagator, behaving like 1/k 2 at large momentum k, and two electron propagators, each behaving like 1/k. 2). 3) arises because there is a divergence associated with the coupling of each electron in the scattering process). 2). 2? Now we have two photon and two electron propagators, leading to d 4k I∼ . 3 Primitive divergences of qed. 4 Diagram containing a primitive divergence. This time the integral is convergent. Detailed study like this reveals that, for qed, ultraviolet divergences always cancel in relations between physically measurable quantities.