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MARKUS KLUTE: Welcome
back to 8.701.

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So in this lecture,
we are going to start

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looking at an example
of the QED process,

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for which we can now, with
all the tools we have in hand,

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calculate the matrix elements'
transition amplitude.

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All right.

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In more general terms, we
can look at all the examples.

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And they are listed here--

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second-order processes and
one third-order process.

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We are going to discuss them
in more detail as we go along.

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This is really just to
give you some feedback

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for the different
kinds of processes

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we're going to look at.

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So the first one is
elastic scattering.

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And muon-electron
scattering, that's

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the one process we're going
to look in more detail.

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Why?

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Because this is
the simplest case.

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For this process, there is
only one leading-order diagram,

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which is exactly
the one shown here.

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For other processes where have
the same particle interacting,

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we find that we do have to
consider multiple diagrams--

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for example, this one here,
where we have electron

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and electron scattering.

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And so we have to calculate
not just this leading diagram,

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which looks exactly like the
one for the muon scattering,

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but we also have to
include the [INAUDIBLE]

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where we change the
outgoing electron leg.

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And so on.

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And other processes
are including

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electron-positron
scattering, which

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is caused Bhabha scattering,
Compton scattering, which

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we discussed the
kinematics for already,

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but also inelastic processes
like pair annihilation or pair

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production.

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There's a very
interesting diagram here,

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which is the third-order
diagram, which

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is responsible for the
anomalous magnetic moment.

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And we'll talk more
about that when

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we talk about
higher-order interactions.

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So let's have a look at this
electron-muon scattering

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process.

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So only one diagram contributes
at the second order.

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And so you have an electron
and a muon scattering

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via the exchange of a photon.

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This is after all
of QED diagram.

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So now, how do we calculate
the matrix element?

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We simply just follow
the Feynman rules--

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Feynman rules as we
discussed them before.

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And if you want to do this now,
you draw your Feynman diagram.

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It's always very good and useful
to draw a Feynman diagram first

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and label accordingly.

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That's super-useful if you
want to systematically evaluate

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this process.

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And then you start going
backwards from an outgoing leg

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back to the initial leg.

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And you see this part here.

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You have the u3, the
third particle here,

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the vertex vector, and
the first particle.

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Then you have a propagator
here for your photon.

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It's given by minus i g
mu,nu divided by q square.

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And then you analyze
the second part here.

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Here you find the first
particle, vertex vector,

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and the second particle.

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For each of those
lines, you have

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to make sure that
energy and momentum is

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converted into those
[INAUDIBLE] delta functions.

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And then the last
part you have to do,

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integrate over your momentum.

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All right.

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That's already the end.

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The next step in
your list of rules

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is carry out the integration.

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Integrate over q.

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That drops your delta
function, but you

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are left with one delta
function which you are also

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supposed to drop, which then
gives you your matrix element.

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Now, here we're already done.

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If you now further want
to evaluate this diagram,

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you actually have to be more
explicit about the spinors

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involved.

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What needs to be done
now is have a discussion

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on how to handle the
spin of the particles,

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meaning being explicit
about the spinors.

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And in order to do
that, we'll discuss

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how we treat spin, how
we have to treat spin,

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either in an experiment
where the spin

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of the initial particle is
known or an experiment where

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we have to average over
all possible spin states.

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So that's part of the
next lecture discussion.