From Special Relativity to Feynman Diagrams.pdf

Gauge invariance with respect to the incoming photon

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gauge invariance with respect to the incoming photon of momentum k = p p requires the current V μ to be conserved (i.e. divergenceless), namely: k μ V μ ( p , p ) = ( p p ) μ V μ ( p , p ) = 0 . (12.299) When summing all the contributions to a given amplitude coming from S -matrix terms of orders differing by two units, we will have to sum contributions from two diagrams differing just in the substitution of a tree vertex by a one loop vertex. Adding up the two terms amounts to effectively replacing in the lowest order one: γ μ μ ( p , p ), μ ( p , p ) γ μ + μ ( p , p ). (12.300) The quantity μ ( p , p ) then represents a second order correction to a vertex, whose integral expression in ( 12.297 ) has a logarithmic divergence for large values of the integration variable q , representing the momentum of a virtual electron. The matrix μ ( p , p ) is referred to as the second order corrected vertex . There are other cor- rections to the vertex, obtained by inserting self-energy parts in the legs of the three diagram. These are in principle accounted for by using the exact propagators for the electrons and the photon. Expanding μ ( p , p ) in p , p μ ( p , p ) = L γ μ + f μ ( p , p ), (12.301) where L γ μ = μ ( 0 , 0 ) and one can isolate the divergent part L , which is a constant, from the finite remainder f μ ( p , p ). The second order corrected vertex μ ( p , p ) consequently splits as follows:
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528 12 Fields in Interaction μ ( p , p ) = ( 1 + L μ + f μ ( p , p ). (12.302) The ambiguity in the definition of L is fixed as follows. Let us first show that, on the general grounds of Lorentz covariance, if p = p , the current V μ ( p , p ) = ¯ u ( p ) μ ( p , p ) u ( p ) is proportional, through a constant, to ¯ u ( p μ u ( p ). By Lorentz covariance we can indeed convince ourselves that μ ( p , p ), which is a spinorial matrix depending on p , can only be combination of the matrices p μ 1 and γ μ . Using then the property 38 ¯ u ( p μ u ( p ) = p μ m ¯ u ( p ) u ( p ), (12.304) we conclude that V μ ( p , p ) = ¯ u ( p ) μ ( p , p ) u ( p ) = f 0 ¯ u ( p μ u ( p ), (12.305) f 0 being a constant. Actually, using Lorentz covariance and the gauge invariance con- dition ( 12.299 ), one can show that the current V μ ( p , p ) can only have the following general form V μ ( p , p ) = ¯ u ( p ) F 1 ( k 2 μ + F 2 ( k 2 μν k ν u ( p ), (12.306) where k = p p . 39 It follows that V μ ( p , p ) = F 1 ( 0 ) ¯ u ( p μ u ( p ) , so that F 1 ( 0 ) = f 0 . We now fix the ambiguity in L by requiring L = f 0 , which implies u ( p , s ) f μ ( p , p ) u ( p , s ) = 0 . (12.307) Let us observe that the vertex μ ( p , p ) contains in general the coupling constant e 0 and a factor Z 2 Z 1 2 3 originating from the wave function renormalization of the electron and photon fields L I 0 = e 0 ¯ ψ 0 γ μ ψ 0 A 0 μ = e 0 Z 2 Z 1 2 3 ¯ ψγ μ ψ A μ . (12.308) We conclude that the logarithmic divergence in the vertex part correction can be absorbed in a charge (or coupling constant) renormalization as follows: 38 To show this use the general identity ¯ u ( p μ u ( p ) = 1 2 m u ( p μ pu ( p ) + u ( p ) p γ μ u ( p ) = u ( p ) p μ + p μ 2 m γ μν ( p ν p ν ) 2 m u ( p ), ( 12 . 303 ) where we have written γ μ γ ν = η μν + γ μν , and γ μν being defined as [ γ μ , γ ν ] / 2 .
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