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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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 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

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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