Scalar electrodynamics - Department of Physics | Indiana...
Transcript of Scalar electrodynamics - Department of Physics | Indiana...
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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 gaugeLorentz (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 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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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)
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There is only one diagram contributing at one loop level:
fictitious photon mass
the photon propagator in the Feynman gauge:
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following the usual procedure:
we get:
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we can impose by writing:
we set Z’s to cancel divergent parts
fixed by imposing:
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Finally, let’s evaluate the diagram contributing to the vertex:
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combining denominators...
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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.
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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
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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:
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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:
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we can set since these terms come from the finite piece
infrared regulator is needed
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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
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antisymmetric under , and so it doesn’t contribute when we integrate over Feynman’s parameterssymmetric under
Gordon identity
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putting everything together
we get:
where the form factors are:
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where the form factors are:
can be further simplified .... but we will be mostly interested in the values for :
the fine-structure constant