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MARKUS KLUTE: Let's
come back to 8.701.

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So in the previous
video, we talked

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about higher-order
diagrams and we

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looked at how we can
classify those contributions

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to the total matrix element.

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We didn't do any of
the calculations,

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or we didn't calculate the
Feynman diagram itself.

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And I'm not actually planning
to do this in the lecture.

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What we want to do
here is investigate

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one of the features, a very
important features of having

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those higher-order corrections.

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So if we look at this
higher-order diagram here,

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one of the specific
ones where we

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have a self correction,
self energy correction

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to the propagator C here.

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We find this loop
here in the middle.

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And if you were to-- and we
have all the tools at hand

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to actually do the
calculation-- if you

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were to calculate the amplitude,
we find this term here.

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So very good.

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So let's investigate.

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The first part is this
finite element here,

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which we can rewrite
as q cubed times

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a finite element dq times
all the angles which

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we have to integrate around.

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So we find this q cubed.

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If you look at under
in this fraction here,

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we find a q squared
times q squared.

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And if you go to just
very large values of q,

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that's the only
thing which remains.

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So we have an integral, 1
over q to the fourth power,

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times q to the third power.

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And then we have to integrate
this from 0 to infinity.

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So if you do this,
you know that this

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results in a logarithmic term.

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And if you have this
evaluated at infinite,

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we find that it diverges.

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So the result of the
integral is infinity.

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That is a real problem.

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If you calculate the
scattering process,

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the result is
better not infinite.

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The cross-section
shouldn't be infinite.

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The lifetime shouldn't be 0.

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So that is a real problem.

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And that actually caused
this entire theory

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to not really make much
progress for quite some time,

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because you were not actually
able to calculate anything.

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The solution is to
introduce a cutoff.

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So what happens now if
we don't just [? jump ?]

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to the integration
[INAUDIBLE],, but to some scale.

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And so you introduce this
additional factor here

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in the integral.

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And you just calculate
the integral up

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to a cutoff scale m.

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And then you have
an additional term

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you have to in principle
evaluate from m to infinite.

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And you find that that
additional element is still

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infinite.

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You can evaluate
all the other parts.

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And it turns out if you
are smart and introduce

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the cutoff, the theory,
the calculation still

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remain sensible, meaning
that they perform fine

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under Lorentz transformation.

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All the physics intuition
we have is fine.

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You just have the
issue that still there

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is a contribution to this
integral, which is infinite.

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It turns out now by miracle
that you can redefine, re-scale,

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or re-normalize the physical
objects in your calculation

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such that it appears that there
is a correction to your masses

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or a correction
to your couplings.

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So what you find
then is that there

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is a component which is
your physical value, which

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is a bare mass and the bare
coupling, your coupling

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constant, plus some correction.

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There's still a problem
that those corrections

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at infinite scale are infinite.

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However, when we
do experiments, we

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are performing them
at a specific scale.

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And so this problem of if
you go to really high scales,

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things get out of hand, is
actually not a real problem

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when you compare the theoretical
prediction with the experiment.

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There's an interesting
feature here.

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When you actually look in
the running or the evolution

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of your coupling,
which is shown here,

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the function of
energy that shows this

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as a logarithm of
the energy, you

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can do this for the
electromagnetic, for the weak,

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and for strong interactions.

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And note here that this is
an inverse of the coupling.

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They all run.

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They all are dependent
and have to be

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evaluated at a specific scale.

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But unfortunately at
very high mass scale,

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they don't all appear to
converge in the same spot.

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It is interesting to
know that if I introduce

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new particles along
the scale here,

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note that this is 10 to 10
GeV, this new particle will

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change the behavior of the
running of the couplings.

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The energy behavior
of the coupling

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changes if I introduce
new couplings.

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And you can already
understand this

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because I would introduce
new diagrams which

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contribute in this way.

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And then those
result in a change

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in the running behavior
of the [? plot. ?]

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So one of the ideas
for new physics,

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which we might discuss in
the very last lecture of this

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class, is that by introducing
new particles along the way,

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you are actually able to combine
all of the couplings involved--

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here, electromagnetic,
weak, and strong--

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at a specific and
specific scale,

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and then have a combined,
unified theory describing

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all of the physics we discuss
in nuclear and particle physics.

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So that would be great.

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That is new physics, and we
don't know if this is realized.

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However, what is realized
in our calculations

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is that the physical masses
and couplings we observe,

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they are evaluated
at a specific scale.

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And they do run as function of
a scale as shown in this plot

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here.

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We will look at this
very specifically

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at the running of those
three interactions--

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the electromagnetic, the
weak, and the strong.

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And we can also
observe this when

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we study the masses involved.

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They're have to be evaluated
or are evaluated in experiments

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at a specific scale.