Review of Feynman rules for QED
vertex and the rest of the diagram
external lines:
incoming electron
outgoing electron
incoming positron
outgoing positron
incoming photon
outgoing photon
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the arrow for the photon can point both ways
vertex
one arrow in and one out
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
for each internal fermion
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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
.
follow arrows backwards!
The vector index on each vertex is contracted with the vector index on either
the photon propagator or the photon polarization vector.
assign proper relative signs to different diagrams
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).
sum over all the diagrams and get
additional rules for counterterms and loops
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Scalar electrodynamics
based 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:
and use covariant derivatives
where:
so that
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A gauge invariant lagrangian for scalar electrodynamics is:
The Noether current is given by:
multiplied by e = electromagnetic current
depends explicitly on the gauge field
New vertices:
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Additional Feynman rules:
vertex and the rest of the diagram
external lines:
incoming selectron
outgoing selectron
incoming spositron
outgoing spositron
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vertices:
incoming selectron
outgoing selectron
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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 QED
based 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 free photon propagator in a
generalized Feynman gauge or
gauge:
the sum of 1PI diagrams with two
external photon lines (and the
external propagators removed)
Feynman gauge
Lorentz (Landau) gauge
The observable amplitudes^2 cannot depend on
we saw that we can add or
ignore terms containing
which suggests:
(we will prove that later)
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and so we can write it as:
is the projection matrix
we can write the propagator as:
summing 1PI diagrams we get:
has a pole at
with residue
In the OS scheme we choose:
to have properly normalized states in the LSZ
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Let’s now calculate the
at one loop:
extra -1 for fermion loop; and the trace
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we ignore terms linear in q
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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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the integral over q is straightforward:
imposing
fixes
and
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