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

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:

external lines:

incoming selectron

outgoing selectron

vertex and the rest of the diagram

incoming spositron

outgoing spositron

Additional Feynman rules:

vertices:

incoming selectron outgoing selectron

Let’s use our rules to calculate the amplitude for :

and we use to calculate the amplitude-squared, ...

Loop corrections in QEDbased on S-62

Let’s calculate the loop corrections to QED:

adding interactions results in counterterms

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 gaugeLorentz (Landau) gauge

The observable amplitudes^2 cannot depend on which suggests:

(we will prove that later)

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 LSZ

Let’s now calculate the at one loop:

extra -1 for fermion loop; and the trace

we ignore terms linear in q

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 :)

the integral over q is straightforward:

imposing fixes

and

Let’s now calculate the fermion propagator at one loop:

the exact propagator in the Lehmann-Källén form:

no isolated pole with well defined residue

it is a signal of an infrared divergence associated with the massless photon; a simple way out is to introduce a fictitious photon mass. After adding contributions to the cross section from processes that are indistinguishable due to detector inefficiencies it is safe to take ; it turns out that in QED we do not have to abandon the OS scheme.

using this procedure we can write the exact propagator as:

a simple pole at with residue one implies: , we use these conditions to fix and .

sum of 1PI diagrams with 2 external lines (and ext. propagators removed)

There is only one diagram contributing at one loop level:

fictitious photon mass

the photon propagator in the Feynman gauge:

following the usual procedure:

we get:

we can impose by writing:

we set Z’s to cancel divergent parts

fixed by imposing:

Finally, let’s evaluate the diagram contributing to the vertex:

combining denominators...

continuing to d dimensions

evaluating the loop integral we get:

the infinite part can be absorbed by Z

the finite part of the vertex function is fixed by a suitable condition.

The vertex function in QEDbased on S-63

For the vertex function we can impose a physically meaningful condition:

momentum conservation allows all three particles to be on shell:

and so we can define the electron charge via:

consistent with the definition given by Coulomb’s law

exact propagator and exact vertex approach their tree level values as

Consider electron-electron scattering:

finite when

physically, means that the electron’s momentum changes very little during the scattering; measuring the slight deflection in the trajectory of the electron is how we can measure the coefficient in the Coulomb’s law.

Our on-shell condition enforces and so the condition imposed on the vertex function can be written as:

Now we use our condition to completely determine the vertex function:

we can use the freedom to choose the finite part of

fixed by imposing:

we can set since these terms come from the finite piece

infrared regulator is needed

To calculate e-e scattering we need the vertex function for arbitrary ;

we need to calculate:

using

we can rewrite it in terms of and

antisymmetric under , and so it doesn’t contribute when we integrate over Feynman’s parameterssymmetric under

Gordon identity

putting everything together

we get:

where the form factors are:

where the form factors are:

can be further simplified .... but we will be mostly interested in the values for :

the fine-structure constant