Review of Feynman rules for QED - Department of Physicsdermisek/QFT_08/qft-II-11-1p.pdf · Review...
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Transcript of Review of Feynman rules for QED - Department of Physicsdermisek/QFT_08/qft-II-11-1p.pdf · Review...
Review of Feynman rules for QED
external lines:
incoming electron
outgoing electron
vertex and the rest of the diagram
incoming positron
outgoing positron
incoming photon
outgoing photon
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vertex
draw all topologically inequivalent diagrams
for internal lines assign momenta so that momentum is conserved in each vertex (the four-momentum is flowing along the arrows)
propagators
for each internal photon
one arrow in and one out
the arrow for the photon can point both ways
for each internal fermion
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sum over all the diagrams and get
spinor indices are contracted by starting at the end of the fermion line that has the arrow pointing away from the vertex, write or ; follow the fermion line, write factors associated with vertices and propagators and end up with spinors or .
assign proper relative signs to different diagrams
follow arrows backwards!
draw all fermion lines horizontally with arrows from left to right; with left end points labeled in the same way for all diagrams; if the ordering of the labels on the right endpoints is an even (odd) permutation of an arbitrarily chosen ordering then the sign of that diagram is positive (negative).
additional rules for counterterms and loops
The vector index on each vertex is contracted with the vector index on either the photon propagator or the photon polarization vector.
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and use covariant derivatives where:
Scalar electrodynamicsbased on S-61
Consider a theory describing interactions of a scalar field with photons:
is invariant under the global U(1) symmetry:
we promote this symmetry to a local symmetry:
so that
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A gauge invariant lagrangian for scalar electrodynamics is:
The Noether current is given by:
depends explicitly on the gauge field multiplied by e = electromagnetic current
New vertices:
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external lines:
incoming selectron
outgoing selectron
vertex and the rest of the diagram
incoming spositron
outgoing spositron
Additional Feynman rules:
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Let’s use our rules to calculate the amplitude for :
and we use to calculate the amplitude-squared, ...
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Loop corrections in QEDbased on S-62
Let’s calculate the loop corrections to QED:
adding interactions results in counterterms
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The exact photon propagator:
the sum of 1PI diagrams with two external photon lines (and the external propagators removed)
we saw that we can add or ignore terms containing
the free photon propagator in a generalized Feynman gauge or gauge:
Feynman gauge
Lorentz (Landau) gauge
The observable amplitudes^2 cannot depend on which suggests:
(we will prove that later)
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In the OS scheme we choose:
and so we can write it as:
is the projection matrixwe can write the propagator as:
summing 1PI diagrams we get:
has a pole at with residue
to have properly normalized states in the LSZ160
the integral diverges in 4 spacetime dimensions and so we analytically continue it to ; we also make the replacement to keep the coupling dimensionless:
see your homework
is transverse :)
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