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MARKUS KLUTE: Welcome
back to 8.701.

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So again, we have now
all tools in place

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to do a next round of
cross-section calculations.

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We have seen how to set
up a matrix element.

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We have seen how to build spin
average or to treat the spin,

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and then specifically
to calculate

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spin average amplitudes
using [INAUDIBLE]..

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All right, I'm not saying
that this is all easy now,

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but you have seen
all necessarily

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elements to calculate a
cross-section for QED process.

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So let's summarize.

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So we have seen that we can set
up some matrix element using

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Feynman rules for QED.

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We have seen how to set up the
spin average matrix element

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squared using the traces.

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Now we would have to
evaluate the traces in order

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to derive this formula here.

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So I'll spare you a precise
discussion of this step here,

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but you can actually follow
this quite straightforwardly.

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Let me just step back a
little bit before we proceed.

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My goal for the class is
not to have you calculate

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all kinds of
cross-section processes,

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but to understand
how you would do it,

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for the purpose of
really understanding

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where dependencies come from and
where this kind of calculation

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has its limitations.

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The first part is
you want to see

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what is the dependencies
on the couplings involved.

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You see this g
squared, for example.

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That's a rather
important effect.

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You also want to see, so
Fermi's this golden rule,

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how we get actually into the
cross-section from the matrix

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element calculation.

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So if you ever had to
calculate a matrix element,

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am I going to ask you to
do this once, maybe twice,

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as part of the homework set.

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I encourage you to open
the book, follow the rules,

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look up tricks, how
to work with traces.

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And then you should get
to a reasonable solution

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in a reasonable amount of time.

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But here for the purpose
of this discussion,

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we want to just have a look
at a few specific cases

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where we make assumptions
and simplifications

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to the discussion.

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So the first one is
called Mott scattering.

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So here, again, we are at this
example of a spin-half particle

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scattering with a
spin-half particle--

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a different spin-half particle,
so an exchange of a photon.

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So we used the example of
an electron-muon scattering,

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but this muon here could also
be a proton or any other nuclei

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we spin off.

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The assumption for Mott
scattering we are using

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is that the mass of
this particle, the muon,

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is much heavier than the
mass of the electron.

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And that's true the muon 200
times heavier than an electron.

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A proton is even heavier.

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Any heavier nuclei of this
feels even heavier than this.

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In Mott scattering, we
also make the assumption

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that the momenta
involved are lower

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than the mass of
the heavy particle

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and that the recoil of the
heavy nuclei, or the muon,

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can be neglected.

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If we do that, we can then write
the differential cross-section

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using Fermi's golden rule
as a spin average matrix

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element squared divided
by 2 pi M squared.

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OK.

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If you then use this
kinematic information,

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you basically start from
this matrix element here.

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And then you use those
vectors, those four vectors,

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for your momentum of the
first, second, third,

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and fourth particle.

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You find that many of the
vectors are simplifying to ME.

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So p2 times p3 is ME.

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And so are many of the others.

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And there is a few
important factors.

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For example, p1 minus p3
squared is minus 4p squared sine

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squared theta half.

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And similarly, p1 times p3.

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So you put this all in--

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again, starting from
this very formula we

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just had discussed before--

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and you put all
the simplifications

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and you get this matrix
element, which already that

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looks much more manageable.

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There's an M squared,
there's a p squared,

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there's a cosine
squared theta half term,

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and some factor which depends
on the moment times the mass.

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And if you then add this
to Fermi's golden rule,

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you find this equation
for your Mott scattering.

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Again, this is the
scattering of two

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different spin-half particles
where one is much heavier.

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The outgoing momenta are small.

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And the recoil of the heavier
particles can be neglected.

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So this Mott's
formula describes,

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for example, the Coulomb
scattering, so the scattering

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this photon on the electric
charge of a nuclei.

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And the scattering particle
is not too heavy and not too

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energetic, like an electron.

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We also assume that everything
involved here is point-like.

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We haven't had any discussion
on the charge distribution

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of the nuclei or anything.

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We assume that this is
a point-like particle.

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OK, we can further
discuss now the case

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where the initial state
particles are non-relativistic.

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So here our momentum
formulas simplify.

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This is simply M squared,
p amplitudes is 2ME.

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And alpha is q1 times q2.

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Those are the electric charges.

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And so then our
differential cross-section

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further simplifies to
something you've already seen.

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The Lorentzian cross-section
is equal to q1 times

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q2 divided by 4
times the energy sine

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squared theta half squared.

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And we have seen that as already
the Rutherford scattering

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cross-section when we discussed
cross-section measurements

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in a geometrical kind of thing.

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So this closes a loop here in
our cross-section discussion

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how we can think
about those things.

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The Rutherford cross-section
is nothing else

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but a big billiard ball being
hit by a small billiard ball

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and looking at how the
cross-section differentially

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kind of evolves out this setup.

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All right, in this sequence
we have a little bit more

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of a discussion.

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What happens now if we
induce higher-order terms

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and how can we think
about those solutions?

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And then have two extra
lectures and where

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we go back and
discuss spin, and also

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how we can actually understand
this in a Lagrangian setup.