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MARKUS KLUTE: Welcome
back to 8.701.

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So in this lecture, we look
at the interaction of W bosons

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with quarks, or the charged
weak interaction of quarks.

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Let's just make a
number of observations.

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Now, we observe that the weak
interaction respects the lepton

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generation, meaning
that a W couples

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to an electron and
an electron neutrino,

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but not an electron
muon neutrino.

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But in the case of the quarks,
there is violation of this.

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So there is a disrespect
of the quark generation

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when it comes to the
interaction with Ws.

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So when you investigate
these two diagrams here,

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you find that the W couples to
the V quark and the U quark,

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but it also couples
to the S quark.

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S quark and the U
quark, all right?

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In order to encapsulate this,
we have to make a correction.

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And the corrections
are typically

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called cosine theta C
and sine theta C. Theta

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C is the Cabibbo angle,
so theta Cabibbo.

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Turns out this is rather
small, so it's a decrease.

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So it is a correction,
a small correction.

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We studied the
partial decay width

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of the kaon in leptonic
decays over the partial decays

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to a pion of the pion
in leptonic decays.

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We found that there
is a set of forms

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that are the form factor
for the pion decay

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and a form factor
for the kaon decay.

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It turns out that
the form factors are

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due to this
additional correction,

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so you find the tangent-square
of the Cabibbo angle

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as part of this correction.

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Good.

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So far, so good.

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Now we have made an observation.

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We haven't explained
anything yet.

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We can make one more
observation, or discuss one,

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and that's the decay of neutral
kaons to a pair of muons.

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It turns out that those
are not very likely.

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Even so, you would expect that
the amplitude has a factor here

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of sine and cosine of
Cabibbo, so the amplitude

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should be on the order of sine
theta Cabibbo times cosine

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theta Cabibbo.

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So when this was
studied, the charm quark

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hadn't been discovered.

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And the explanation to why
this decay is suppressed

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comes from the fact that there
is a second diagram here,

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where we just replace the U
quark in this rule with a C

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quark.

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This diagram contributes there's
a minus sign to the amplitude.

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And therefore, those two
diagrams, they cancel.

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Right?

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They are the same magnitude,
about the same magnitude,

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and they have an opposite sign.

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So this was the first
indication that there

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must be a force
quark contributing

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to this kind of process.

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That's the charm quark.

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Let me now try to understand
what's going on here.

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Why is the W coupling modified?

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Or why is not the full
down quark or charm

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and strange quark participating
in the weak interaction?

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We can do the following here.

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We can rewrite-- we note
that the weak interaction

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eigenstate, the eigenstate which
participate in the [INAUDIBLE],,

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is not the eigenstate
of the particle

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itself, the so-called
mass eigenstate.

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So we have to write
the weak eigenstate

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as the linear combination of the
mass eigenstate or [INAUDIBLE]..

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This can be done
in this matrix form

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here, where we simply
multiply the weak eigenstates

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with a matrix, and just
basically rotate it

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into the mass eigenstate.

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So this was proposed by
Cabibbo, and rather successful.

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But it didn't incorporate the
third-generation particle.

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And this was done by
Kobayashi and Maskawa,

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who generalized the scheme
and proposed the so-called CKM

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matrix, the easier
form Cabibbo--

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Cabibbo, Kobayashi, and Maskawa.

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Because of constraints
we'll discuss

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in one of the
recitations, this matrix

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can be parameterized as only
three independent angles

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and one complex phase as
independent parameters.

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So you can choose
different parameterization

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to capture that there's only
four parameters in this matrix,

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which has nine components.

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One is by thinking
about this matrix

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as three independent rotations
and this complex phase here.

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In terms of
numerical values, you

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see that the diagonal
elements of this matrix

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are very close to 1,
meaning that this mixing is

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on the block sector of
the [INAUDIBLE] effect.

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You find that those next-nearest
off-diagonal elements are

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on the order of 20%, and the
next-to-next off-elements

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are even smaller.

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OK?

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This leads us, then,
to the discussion

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that we can use different
parameterization in order

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to capture [INAUDIBLE].

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We already discussed the
standard parameterization,

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which you can really think
about three different rotation.

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And the values of those
angles are give here, together

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with the value of
this additional phase.

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Another way to look at this
is the so-called Wolfenstein

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parameterization.

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And this captures
the fact that it

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seems like that there's
a correction being

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applied to the actual particle.

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So you find elements
of the order of lambda.

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Lambda is about 22%.

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And you find elements
which are in the order of 1

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minus the lambda-square
correction.

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And then there is elements which
are of lambda-square and lambda

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3rd power.

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So this captures the
matrix, and then there's

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higher-order corrections to
that which are of order lambda

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to the 4th power.

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OK?

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Because there's constraints on
this matrix-- and specifically,

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unitarity constraints, meaning
that we have three generations

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that will make a mixing of
those three mass eigenstates

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to weak eigenstate-- then
unitarity, the total number

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of particles in this
discussion, is conserved.

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This will change if there
would be, for example, a force

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generation particle.

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So the study of weak-charged
interaction with quarks

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helps us to understand
whether or not there

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might be a force generation.

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We'll not go into
too much detail here,

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but also, the complex
phase explains

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part of our understanding
of CP violation.

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And we might discuss this in a
little bit of a later lecture.

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But nevertheless, what
we can achieve from here

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is those unitarity
constraints, just simply

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summing over the
matrix elements,

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the scalar product
of matrix elements.

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And those where the
contribution vanishes,

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so those where j
and k are not equal,

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those can be represented
as a triangle.

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That's kind of interesting.

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You can just rewrite this.

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You just say that those
three elements of the sum

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are equal to 0.

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Then you normalize
by one element.

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In this case here,
normalize by Vcd, Vcb.

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And so then this makes
this point being 0,

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and so we have
this nice triangle

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here, which has three
angles, alpha, beta,

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and gamma, and this
point here, rho and eta.

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And so this is a nice
way to illustrate

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actual measurements of
the elements of the CKM

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matrix [INAUDIBLE].

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And without actually explaining
how we do this experiment,

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you can assume that
all measurements have--

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well, you can understand
that all measurements

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have to do with the weak
interaction with quarks.

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That's how we have access
to the CKM matrix elements.

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Sometimes this results in
the modification of masses

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or splitting of mass
states, and sometimes

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the direct measurement
cause a recoupling.

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When you put all of those
measurements back together,

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you can look at this.

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So we see our triangle here.

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We see this point, eta and
rho, which is given here

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in this right-angular plane.

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And you see various number of
measurements which correspond

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to elements of this CKM matrix.