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MARKUS KLUTE: Welcome
back to 8.701.

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So we continue our discussion
on the Feynman Calculus,

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and continue now talking
about Fermi's Golden Rule.

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The heart of calculating
decay rates and cross sections

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is Fermi's Golden Rule.

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And it simply tells
you how we can

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use the calculation
of amplitudes

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and the available phase space to
make assessment of decay rates

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and cross sections.

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So the amplitude M holds all
the dynamical information.

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And we can see that we can
calculate the amplitude

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by evaluating Feynman diagrams
directly using Feynman rules.

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The available phase space
holds is a kinematic factor,

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and it depends on
masses and the energies

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and the momentum of
the particles involved.

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And then again, Fermi's
Golden Rule simply

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says that the transition rate or
decay rates and cross sections

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are given by the product
of the phase space

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and the square of the amplitude.

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How does this now look like?

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If you look at the
Golden Rule for Decays,

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here we suppose
having one particle

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decaying into a second,
third, fourth, n-th particle.

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This is that the
decay rate is given

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by matrix element, the amplitude
squared, and a term which

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is C, the phase space factor.

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There's also a
factor here in front.

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This S has to count-- we have
to account for the fact that we

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might have the same
particle in the final state,

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or the same particle
occurring multiple times.

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And we have to make sure that
we don't have double counting.

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In this double counting
has to be correct.

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If all particles are different,
this extra factor is 1.

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We'll look at this
some more later.

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Now, at first glance, this
looks like rather complicated.

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But if you try to
assess now what

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those individual
terms mean, you will

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see that it's very accessible.

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So when we try to calculate
a use from this golden rule

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for the case, we to integrate
over all outgoing particle

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four-momenta.

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But we have three
kinematical constraints.

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The first one is that
the outgoing particles

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have to be on mass.

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So they have to
be on mass shell.

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We talked about this issue of
virtual particle [? inertia ?]

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before.

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But simply, they have to--

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the energy of the particle
has to follow this condition.

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This is a delta
function, which simply

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means that this gives
us, if the argument is

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0, the delta
function, which was 1.

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The argument is non-zero.

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The function returns 0.

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So this [INAUDIBLE]
simplifies this term here.

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So this first part here in
our Fermi's Golden Rule simply

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accounts for the fact
that outgoing particles

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have to be on mass shell.

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Outgoing particles also have
to have positive energies.

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And this explains our second
factor here, this factor.

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And so this factor is
the heavy side function,

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and this is simply 0
for negative values

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and 1 for positive values.

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And the last one means
that energy and momentum

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of the particle
have to conserve.

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So the first particle
minus the second,

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third, and so on for each
of the component for energy

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and for those [? three ?]
components of the momenta

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have to be 0 for
this to return 1.

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So again, another
delta function.

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That's also factors of pi.

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And the simple rule here
is for each delta function,

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you have to account for a
vector of 2 pi in your function.

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So this basically
explains already

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everything we see
here on this slide.

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So now we can calculate.

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And I recommend to have a
look at Griffiths chapter

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6 for this.

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If you look at two
particle decays.

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So one particle
into two particles.

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This simplifies
quite tremendously

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because of all the
delta functions here

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and the heavy side functions.

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This equation simplifies
directly to a factor.

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You have the momentum
of the particle

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here, and a matrix element.

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Again you have this
statistical factor here

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to account for the fact that
there might be duplicates

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and you want to keep track
of the statistical factor.

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For scattering, the equation
looks almost the same.

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So you have the same
phase space factor.

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Almost the same
phase space factor.

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Matrix element.

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Again, this is
the transition way

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it is given by, as Fermi's
Golden Rule tells us,

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by the matrix elements squared.

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And the phase space factor.

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This overall effect,
as we look at later,

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they are slightly different.

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But here, as a rule,
we want to make sure

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that we have a way to
assess cross section.

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Did this make sense?

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We'll see later in more detail.

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For two body decays in
the center of mass frame.

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You know that the
initial momenta

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in the center of mass frames
of particle 1 and particle 2

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have to be the same.

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The outgoing momenta
also have to be the same.

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They don't necessarily have to
be the same as the [INAUDIBLE]

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But if any uses of the
differential cross-section

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can be calculated quite
straightforwardly.

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Again, there's a
matrix element squared.

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You have the final state
momenta, the initial state

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momenta.

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And divide this by the sum of
the energies squared together

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with an extra factor here.

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So what we have seen here is
just we have looked at it.

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I didn't explain
how we got to this.

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But we have seen
from this Golden

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Rule, which helps us to assess.

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And we have seen how we can
calculate the phase space

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factor.

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The next lecture, we now
see how we can calculate

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the matrix element itself.

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And we'll start doing this by
using a toy experiment or toy

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model, such that the discussion
simplifies, algebra simplifies

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quite a bit.