By Forshaw J.
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Additional resources for An introduction to QED and QCD
2 Lowest order Feynman diagrams for electron–electron scattering. Other calculations of cross sections or decay rates will follow the same steps we have used above. You draw the diagrams, write down the amplitude, square it and evaluate the traces (if you are using spin sum/averages). There are one or two more wrinkles to be aware of, which we will meet below. 5 Electron–Electron Scattering Since the two scattered particles are now identical, you can’t just replace m µ by me in the calculation we did above.
In fact, there is a mapping, called a covering, from SU(2) to SO(3) which preserves the group property: that is if U ∈ SU(2) is mapped to f (U) ∈ SO(3), then f (UV ) = f (U)f (V ). In the SU(2) → SO(3) case, two elements of SU(2) are mapped on to every element of SO(3). Whenever a group G has the same Lie algebra as a simply connected group S there must be such a covering S → G. The double covering of SO(3) by SU(2) underlies the behaviour of spin-1/2 and other half-odd-integer spin particles under rotations: they really transform under SU(2), and rotating them by 2π only gets you half way around SU(2), so you pick up a minus sign.
The divergences contained in the counterterms cancel the infinities produced by the loop integrations, leaving a finite answer. The old A and ψ are called the bare fields, and e and m are the bare coupling and mass. Note that to maintain the original form of L, you want Z 1 = Z2 , so that the ∂/ and eˆA / terms combine into a covariant derivative term. This relation does hold, and is a consequence of the electromagnetic gauge symmetry: it is known as the Ward identity. Let me stress again that renormalisation is not about sweeping infinities under the carpet.
An introduction to QED and QCD by Forshaw J.