From Special Relativity to Feynman Diagrams.pdf

Two second order contributions to the compton

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Two second-order contributions to the Compton scattering amplitude (4), which, however, give an equal contribution upon integration over x and y . We can thus focus of (3) and write: ψ out | S ( 2 ) | ψ in = ( ie ) 2 d 4 xd 4 y × q 1 , i | q 2 , s | : ψ( x μ S F ( x y ν ψ( y ) A μ ( x ) A ν ( y ) : | p 1 , i | p 2 , r . (12.182) Expanding the free field in creation an annihilation operators, we can convince our- selves that the only terms contributing to the matrix element have the form: c a ca , which destroys the initial photon and electron (operators a, c , respectively) and cre- ates the outgoing ones (operators a , c ). Their non-vanishing contributions have the general form: 0 | c ( q 2 , s ) a ( q 1 , i ) c a caa ( p 1 , i ) c ( p 2 , r ) | 0 = [ a ( q 1 , i ), a ][ a , a ( p 1 , i ) ]{ c ( q 2 , s ), c }{ c , c ( p 2 , r ) } . (12.183) Let us note, however, that there are two terms of the form c a ca : One in which a comes from A μ ( x ) (and thus a from A μ ( y ) ), the other in which a comes from A μ ( y ) (and thus a from A μ ( x ) ). The former describes a process in which the incoming electron emits the outgoing photon ( q 1 , i ) in y and absorbs the incoming one in x , see Fig. 12.8 a, while in the latter the incoming photon ( p 1 , i ) is absorbed in y and the outgoing one emitted in x , see Fig. 12.8 b. Using, for each term, ( 12.183 ), and eliminating, by integration, the delta functions arising from the commutators and anticommutators, we end up with: ψ out | S ( 2 ) | ψ in = ( ie ) 2 d 4 xd 4 y d 4 p ( 2 π) 4 2 m × ¯ u ( q 2 , s μ i p m γ ν u ( p 2 , r μ ( p 1 , i ν ( q 1 , i ) e i ( p 2 q 1 p ) · y e i ( q 2 p 1 p ) · x + ¯ u ( q 2 , s μ i p m γ ν u ( p 2 , r ν ( p 1 , i μ ( q 1 , i ) e i ( p 1 + p 2 p ) · y e i ( q 2 + q 1 p ) · x , (12.184)
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488 12 Fields in Interaction We note that under the exchange p 1 q 1 ; μ ( p 1 , i ) μ ( q 1 , i ) the total matrix element remains invariant. This invariance is known as crossing symmetry , the graph (a) being referred to as the crossed term of graph (b). The integrations over x and y implement the conservation of momentum at each vertex, while the integration over the momentum p of the virtual electron yields the global delta function ( 2 π) 4 δ 4 ( p 1 + p 2 q 1 q 2 ). The matrix element of T ( 2 ) reads: i ψ out | T ( 2 ) | ψ in = ( ie ) 2 2 m × ¯ u ( q 2 , s ) γ μ i p 2 q 1 m γ ν + γ ν i p 1 + p 2 m γ μ u ( p 2 , r ) × ε μ ( p 1 , i ν ( q 1 , i ) , (12.185) Let us now verify that the above result does not depend on our gauge choice for the electromagnetic potential, namely that it is not affected by a gauge transformation A μ A μ + μ . In momentum space a gauge transformation amounts to adding an unphysical component to ε μ : ε μ ( p , i ) −→ ε μ ( p , i ) + χ( p ) p μ . (12.186) To show that such component gives no contribution to ( 12.185 ), let us replace any of the photon polarization vectors (e.g. ε μ ( p 1 , i ) ) by the corresponding 4-momentum p 1 μ and prove that the resulting expression is zero. It suffices to prove that the following quantity vanishes: ¯ u ( q 2 , s ) γ μ i p 2 q 1 m γ ν + γ ν i p 1 + p 2 m γ μ u ( p 2 , r ) p 1 μ = ¯ u ( q 2 , s ) p 1 i p 2 q 1 m γ ν + γ ν
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