From Special Relativity to Feynman Diagrams.pdf

Is that the arrow on an external fermionic leg

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is that the arrow on an external fermionic leg indicates the flow of negative charge (electron charge), which is clearly opposite to the flow of the positron charge. One of them contributes to the integrand in ( 12.171 ) a term of the form: −¯ v( p + , r + μ v( q + , s + ) ¯ u ( q , s ν u ( p , r ) e i ( p + q + ) · x e i ( p q ) · y , to be contracted with the photon propagator, the other a similar term with x y . The minus sign in the above expression originates from the definition of normal ordering for fermions: : dd c c := − d c cd . Consider now the two terms in which d and d originate from field operators com- puted in different vertices. They are also related by an exchange of the two vertices and thus give equal contributions to the integral ( 12.171 ). Each of them describes a process in which the incoming electron and positron lines converge on a same vertex, where the two particles are both destroyed (by c and d , respectively). They annihilate,producingavirtualphotonwhichpropagatesuptothesecondvertexwhere it originates the couple of outgoing electron and positron (created by c and d , respectively), see Fig. 12.5 b. This is thus an annihilation process rather than a diffu- sion one. Its contribution to the integrand in ( 12.171 ) is a term of the form: ¯ u ( q , s μ v( q + , s + ) ¯ v( p + , r + ν u ( p , r ) e i ( p + p + ) · y e i ( q + + q ) · x ,
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484 12 Fields in Interaction Fig.12.5 Two second-order contributions to the e e + scattering amplitude: The diffusion , ( a ), and the annihilation, ( b ) diagrams to be contracted with the photon propagator. The integration over x and y yields the conservation of 4-momentum at each vertex. We have thus found two distinct contributions to this integral, one describing a diffusion and an other an annihilation process. In the former case the momentum of the photon is p = p q = q + p + , while in the latter p = p + p + = q + + q (the sign of p is irrelevant since the integral is invariant upon changing p → − p and x y ). Upon integration over x and y and the photon momentum p we end up with a single delta function implementing the conservation of the total momentum p + p + = q + + q . By factoring this delta function out, just as we did in the case of the electron-electron scattering we derive the expression for the matrix element of T ( 2 ) : i ψ out | T ( 2 ) | ψ in = ( ie ) 2 4 m 2 × −¯ v( p + , r + μ v( q + , s + ) i ( p q ) 2 ¯ u ( q , s μ u ( p , r ) + ¯ u ( q , s μ v( q + , s + ) i ( p + p + ) 2 ¯ v( p + , r + μ u ( p , r ) , (12.172) where we have used the properties ¯ v( p + , r + )( p + q + )v( q + , s + ) = 0 and ¯ v( p + , r + ) ( p + + p ) u ( p , r ) = 0 which descend from ( 12.161 ) and ( 12.160 ). Consider now an electron–muon scattering: e + μ −→ e + μ . (12.173) The interaction Hamiltonian is obtained by writing the electric current as the sum of the electron and the muon ones, as in ( 12.141 ), in which “particle q ” (which however now is no longer a “spectator”) is the muon ( q = e = −| e | < 0): H I ( x ) ≡ − e : [ ψ( x μ ψ( x ) + ψ (μ) ( x μ ψ (μ) ( x ) ] A μ ( x ) : . (12.174)
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12.5 Amplitudes in the Momentum Representation
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