From Special Relativity to Feynman Diagrams.pdf

After this general discussion let us come back to the

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After this general discussion, let us come back to the case of the QED. At one- loop order besides the three divergent diagrams a,b,c of Fig. 12.13 , corresponding to the electron and photon self-energy parts and vertex part discussed before, which are of second-order in the coupling constant, there is also at one loop a divergent fourth-order diagram, represented in Fig. 12.13 d, which is referred to as the photon- photon system and an order-three three-photon vertex, see Fig. 12.13 e. Applying ( 12.257 ) to diagrams a,b, and c we immediately conclude by power counting that the electron self-energy part is linearly divergent ( D G = 1 ) , the photon self-energy part is quadratically divergent ( D G = 2 ) and the vertex part is logarithmically divergent ( D G = 0 ). As far as the photon–photon system is concerned, it is logarithmically divergent, while the three-photon vertex is linearly divergent D G = 1 . However, an explicit evaluation of the former diagram, shows that the coefficient of its divergent part is exactly zero. Therefore we shall disregard this diagram in the following. As far as the three-photon vertex is concerned, it is zero being odd under charge conjugation and thus would violate the charge conjugation symmetry (recall from Chap. 11 that the photon is odd under charge conjugation: η C = − 1 ) . Actually, by the same argument, one can show that all diagrams with an odd number of external photons is zero (Furry’s theorem). Renormalizability of QED means that all the divergences appearing in the pertur- bative expansions can be eliminated. As previously anticipated, a complete account
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514 12 Fields in Interaction of the full renormalization program to all orders is rather heavy and complicated and would be outside the scope of this pedagogical introduction. In the following we shall therefore limit ourselves to apply the renormalization program to the self- energy and vertex parts given in Fig. 12.13 which correspond to 1-loop insertions and are therefore of second order in the coupling constant . We believe that even in this restricted framework the main ideas used in the full renormalization program, to all orders, can be understood. Thus far we have been dealing with divergences as if they were well defined quantities. Actually, in order to make sense of divergent integrals, and their manipu- lations, it is important, as a first step, to make such divergent integrals finite by some regularization procedure. The general procedure is the following. One first separates the divergent integral into two parts, 28 where the first part is still divergent, but the divergence is entirely contained in a set of divergent constants , that is in a set of integrals which do not depend on the external momenta, the second part instead is completely finite and, in general, will depend on the external momenta. To show how this separation can be made we quote the following simple example.
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