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MARKUS KLUTE: All right,
so welcome back to 8.801.

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We'll continue our discussion
on Feynman calculus.

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And here we dive
into toy theory.

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So this theory is a toy.

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And it's just an example to
illustrate Feynman rules.

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What we do, the
simplification we employ here

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is leaving out the spin
of the particle involved.

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We consider the spin, we add
another algebraic complication

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which is quite now confusing.

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So we leave this out for now.

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We come back later to this.

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So we're supposed to have three
kinds of particles involved

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here-- particle A,
B, and C. And so we

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can have a primary vertex where
these all three particle are

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interacting like shown here.

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So particle A, might decay
into particle B and C.

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You can assume that
particle A is heavier

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than the sum of B and C, so
this is schematically allowed.

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We might also have corrections
involved here as shown here.

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And what we'd be interested
in is now, for example,

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calculating the lifetime
of this particle A.

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We might do this just for
this primitive vertex.

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Or we might do this for this
complicated set of corrections.

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We might also be interested
in calculating scattering

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processes where particle A
is scattered with particle A

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and produces particle
B and particle B.

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Or we scatter particle A
with particle B and so on.

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So in this theory, at
the end of the lecture,

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we have all tools in
hand to calculate this.

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No worries-- I will not
leave you alone with this.

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This lecture, we go
through the recipe.

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And then later on we'll see
how we actually apply this.

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So let's look at this recipe.

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So the recipe has
a number of steps.

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And the key is to just
follow those steps in order

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to get to the desired result.

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The first step is to label
incoming and outgoing

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four-momenta of particles.

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We label them with
p1, p2, and up to pn.

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We also want to label
all internal momenta.

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So we have an
internal line, then we

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want to label this internal
momenta with q1, q2, and so on.

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We want to add
arrows to each line

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to keep track of what is a
positive direction, as we

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discussed before, particles
might travel backwards in time.

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Those are typically
antiparticles.

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And for those we, have to
make sure that we correctly

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account for the momenta.

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For each vertex,
we have a factor.

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We write this factor
minus ig, where

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g is the coupling constant.

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So this is a measure
of the strength

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of the interaction involved.

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Then we have a propagator.

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So for each internal
line, the internal lines

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are also called propagator.

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We write down a factor, i
over qj squared minus mj

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squared t squared.

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Note that qj
squared doesn't have

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to be mj squared c squared,
meaning that the parity can

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be off shell, off mass shell.

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You also see that there is a
complication in the integral

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when you actually have those
vectors being the same.

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You want to make sure that
energy and momentum is

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conserved.

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So for each vertex, you
write down a delta function

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with the conditions.

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This is for this three vertex
where momentum of the first one

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plus the second plus
the third is equal to 0.

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Only then the value of
the delta function is 1.

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Remember, there's a
minus sign somewhere,

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most likely here
for this k1 value.

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Then you want to integrate
overall internal momenta.

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So for each internal
line, we write a factor--

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1 over 2 pi to the fourth power.

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d4 is on your momenta.

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And then, this all will
result in a delta function,

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which you just eliminate.

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And you do that
by multiplying it.

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You erase this delta
function and you replace it

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by a factor i.

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So this seems like
very confusing.

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Why do you add delta
functions first, and then

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you erase them later?

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Note in Fermi's
golden rule, we use

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the squared of the amplitude.

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And you also saw that the
[? phase-based ?] factors

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already have this kind of
delta functions included.

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So we get out of the
complications that we

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don't really know what to
square of the delta function is

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by erasing, by adding the
i and then keeping track

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of the momentum conservation,
this conservation here when we

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apply the
[? phase-based ?] factor.

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And then, voila, you just
calculated a matrix element.

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All right, so those
are the rules.

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Now the key is to practice
how to apply them.

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So what we do next is to
practice using this toy

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experiment in how to
calculate the matrix element,

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the [? phase ?] base, and the z
decay rates and cross sections.

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And then as a following step, we
will see how this all unfolds.

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Then we have a real
series, like QED,

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like the weak interaction
and the strong interaction.