From Special Relativity to Feynman Diagrams.pdf

Coming back to the our second order amplitude we see

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Coming back to the our second order amplitude, we see that the first order ampli- tude ( 12.325 ) is changed as follows: ψ out | S ( 1 ) + S ( 3 ) | ψ in = i e c mc 2 E p E p V e ¯ u ( p , s ) γ μ + f μ ( p , p ) u ( p , r ) ˜ A ext μ ( k ), (12.329) where f μ ( p , p ) , the finite remainder of the second-order vertex part, is the radia- tive correction to the first-order electron scattering. This is not the only correction to the first-order scattering. There is a further correction arising from the vacuum polar- ization graph of Fig. 12.23 . One can show that the external field will get replaced by A ( ext ) μ ( k ) A ( ext ) μ ( k ) 1 α 15 π k 2 m 2 c 2 ,
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534 12 Fields in Interaction which amounts just to adding 1 5 to 3 8 in the last term of ( 12.326 ). We now show that the first term of ( 12.326 ), depending on the function F 2 , computed at zero momentum transfer ( k 2 = 0 ) , represents the effect of an anomalous electron magnetic moment to the amplitude. To this end let us rewrite the current u ( p μ u ( p ) in the three-level part ( 12.197 ) of ( 12.329 ) using ( 12.303 ). As shown in Sect.12.5.6 by evaluating the non-relativistic limit of the tree amplitude, the term contributing to the magnetic coupling is the one proportional to γ μν k μ ˜ A ext ν which has the following form: i 1 cV e e 2 mc ¯ u ( p μν k μ ˜ A ext ν ( k ) u ( p ), where we have used the non-relativistic approximation E p E p mc 2 . The factor e /( mc ) = ge /( 2 mc ) represents the gyromagnetic ratio that we have computed earlier. If we add the second order correction represented by the first term in ( 12.326 ) we end up with i 1 cV e e 2 mc ( 1 2 mcF 2 ) ¯ u ( p μν k μ ˜ A ext ν ( k ) u ( p ). We see that the gyromagnetic ratio has acquired a correction of the form: e mc e mc ( 1 2 mcF 2 ) = 2 e 2 mc 1 + α 2 π = ge 2 mc , (12.330) corresponding to a corrected g -factor: g = 2 1 + α 2 π . This result was first obtained by Schwinger in 1948. The quantum deviation μ = ( g 2 ) e 2 mc s , of the electron magnetic moment from its classical value, due to perturbative correc- tions, is usually referred to as the electron anomalous magnetic moment . Nowadays the very high precision measurements [10] of g 2 provide the most stringent tests of QED (the agreement between theory and experiment is to within ten parts in a billion). 41 There is another experimental result which is successfully predicted by quantum electrodynamics, which is worth mentioning without entering into heavy technical details. It is the splitting of the 2 s 1 / 2 and 2 p 1 / 2 levels in hydrogen atom, which was 41 Since in order to test QED predictions for higher order corrections to a given quantity (like the g - factor ), a high-precision determination of the coupling constant α is needed, one uses the QED formulas to experimentally determine α. QED is then tested by comparing the values of α determined from different experiments.
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12.8 A Pedagogical Introduction to Renormalization 535 first measured by Lamb and Retherford in 1947 and is known as the Lamb shift .
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