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PROFESSOR: Welcome
back to 8.701.

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So in this lecture, we talk
about the Higgs mechanism.

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As you might know,
the Higgs boson

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was discovered in 2012
by the LHC Experiment,

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but the theoretical
discovery of the Higgs boson

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happened much, much
earlier than that did.

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In the mid 1960s, Peter
Higgs and a few others

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proposed a mechanism
which gives rise

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to masses of the gauge
boson, the w and the z boson.

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And the Higgs boson,
or the Higgs field

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then can also be used to
give masses to the fermions.

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So let's have a look
at this and start

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with a simple observation.

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When we have written down our
Lagrangian for a simple spin

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one field, gauge
field, like a photon,

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we find that we want
to have local gauge

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invariant for this
equation, which

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means that we can
do a local gauge

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transformation of our
fields, and the physics

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should be unchanged of this.

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So the physics, meaning that the
description by the Lagrangian,

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should be invariant under
this transformation.

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All right?

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The problem, however,
is that, if you

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want to have a spin one
gauge field which is massive,

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you have to have terms in
your Lagrangian, like this one

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here, where you have a
mass term for your fields.

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So in general, this is not
possible without breaking gauge

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invariants, and this
is a guiding principle

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of our Lagrangian theory.

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So this is a real bummer.

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So if you want a
stopping point, so you

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have a beautiful theory which
describes all the interactions,

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but one important
characteristic, the masses

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of the particle is missing.

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But you are able to
actually do this,

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not by adding specific mass term
but by breaking the symmetry,

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by breaking the
local gauge symmetry.

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There's various ways to do
this, and one of the ways

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is to use spontaneous
symmetry breaking.

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So what is spontaneous
symmetry breaking?

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Imagine you have a
symmetry of rotation,

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a symmetry like this pen here,
and by applying some force

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on top, this pen would bend.

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And by bending
this pen, it bends

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in one specific direction, that
breaks the rotational symmetry.

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Another way to look at this
is to just let this pen drop.

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Let it go to its ground state,
lowest possible energy state,

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and it will land
somewhere on the table

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and by doing this
breaking the symmetry,

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and it does this spontaneously.

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Let's look at spontaneous
symmetry breaking

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in a toy model first.

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So what we're going
to do here just

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add a complex scalar field and
the corresponding potential

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for this field.

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Potential is shown here,
and this general potential

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can have multiple forms.

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The first form would just
be this parabola here.

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This is a solution where mu
square, this term, mu square,

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is greater than 0.

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In this term, there's
this unique minimum.

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The minimum is here at
0, and because of that,

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the mass of this field
would be equal to 0,

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and the mas of our gauge field
would also be equal to 0.

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But what happens now if we
have through this potential

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a breaking of the symmetry?

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So in this case here, the
vacuum itself, the lowest energy

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state, breaks the symmetry.

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You go away from the
0 point, and you're

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breaking the symmetry.

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So this minimum is at
v over square root 2.

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v is the vacuum expectation
value of this field,

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and you can simply
rewrite then this field

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itself by evolving it
around its minimum.

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And so you find two fields
here, this chi and this h.

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The h is already kind of
pointing towards the Higgs

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boson, and the vacuum
expectation value.

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Now, if you add this back
into your Lagrangian--

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and I'll do this again.

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This is shown here, but
also on the next slide--

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you can start to
identify terms which

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look like mass terms
for your particle.

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And the first one
is here which can

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be identified as a mass
term for our gauge field.

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The mass is e time v.

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e times v is the strength of
the coupling of this gauge field

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e times the value of the
vacuum expectation value.

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All right?

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So this is interesting.

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So we used this new scalar
field to break spontaneously

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the symmetry, and
then the mass term

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appears which is
proportional to the strength

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of the coupling and the
vacuum expectation value.

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So the mass is generated
through the spontaneous symmetry

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breaking and the
coupling to the field.

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You also find a mass
term for the six field

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here, for this h field.

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This is not the Higgs boson.

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It's just a field
which looks like it,

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and so this mass term is here.

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But remember that mu
squared is less than 0,

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and then this chi, or the
so-called Goldstone boson,

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its mass is 0.

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But then we have those
terms left over here,

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which we cannot really
interpret it very well.

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And it's possible to remove them
by choosing a specific field.

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So we do a gauge transformation
by just relabelling things,

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and then the new Lagrangian
is independent of this field.

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Just as a reminder, Goldstone,
the Goldstone boson,

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you find those Goldstone bosons
in many places in physics.

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And Jeffrey Goldstone is
a retired faculty at MIT,

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so you might, in the spring
or next summer, you'll him

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walking across the corridors.

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So we find our new Lagrangian
which has our mass terms here,

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which has a term
for the Higgs field,

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and has our potential
for the Higgs field.

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So this specific gauge
we just decided to use,

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this so-called unitarity
gauge, and it's

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important to note that the
Lagrangian itself contains

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all physical particles.

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But this chi, the
Goldstone boson, is gone,

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and the lingo we
sometimes use here

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is that the Goldstone
boson has been

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eaten by the physical bosons.

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And the way it has
been eaten is through

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the longitudinal
polarization of this boson.

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It's the equivalent of saying
that it has acquired mass.

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So the pocket guys here for
spontaneous symmetry breaking

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is such that spontaneous
symmetry breaking of a u1 gauge

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symmetry by a non-zero
vacuum expectation

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value of a complex scalar
results in a massive gauge

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boson and one real
massive scalar field.

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So we created mass,
but as a side product,

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we also have an
additional field.

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And that field itself has a
mass term, so it's massive.

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The second scalar we
had just disappeared.

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The Goldstone boson
has been eaten

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by the longitudinal component
of the gauge field itself.

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All right.

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That was a simplified toy model.

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Let's look at the
standard model.

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So now, here, we have
to generalize from u1

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to su2 or su-n gauge groups.

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The scalar field is now an
n-dimensional fundamental

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representation of that group
for the standard model.

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That will be su2.

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The gauge fields are n squared
minus one-dimensional joint

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representations,
like our photon,

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for example, or our
unmixed w boson field.

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And the Lagrangian looks
very similar to the one we

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just before with our potential.

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Again, we have this
mu squared term.

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We also have a lambda term here,
and then we require local gauge

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invariants again.

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OK.

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So now, for the standard
model, again, su2

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cross u1 gauge groups.

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We introduce a complex field,
complex six field in su2.

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It's a duplex, meaning that
it has four components.

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It's complex and
has two components,

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which there's a total of
four components to it.

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So we already know that we
want to use m square less

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than 0 for our potential to
allow for spontaneous symmetry

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breaking to occur.

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The minimum is then at
1 over square root 2.

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0 for the upper component,
and v, the vacuum expectation,

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for the lower component.

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That's a choice already.

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Again, why mu
square less than 0?

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Because if we would have chosen
to use a positive value for mu

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square, we wouldn't have
spontaneously broken

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the symmetry.

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OK?

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So we need to have
potential which looks

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like this Mexican hat here.

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All right.

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So now, what happens now
to our w and z bosons.

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We have discussed
electroweak mixing already.

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Good.

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So that was the first step.

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Now, we will understand
where the mass terms actually

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come from.

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So now, we just did the
spontaneous symmetry breaking,

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and now we are looking at what
happens now if we also couple

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the Higgs field to the bosons.

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So again, we write
this this way,

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and then we just
try to find terms.

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It's really like a
mechanical writing

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of the individual terms,
and you find again

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terms which have a vacuum
expectation value here

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and the coupling here.

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And you find the coupling to
the u1 term and the coupling

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to the su2 and the coupling
to the u1 term representative

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to the coupling to the
original photon field

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and our gauge field for su2.

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All right.

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The rest is rewriting
and identifying terms.

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If you do this-- and this
is like a couple of pages

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of writing, fine--

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but if you do this, you
find, again like before,

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that you find the first
and second component

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of our su2 gauge field.

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It gives us the charge at the
boson, the w plus and the w

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minus.

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00:11:06,420 --> 00:11:08,640
And then the z
boson and the photon

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mixtures of the third
component and the field b.

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00:11:15,030 --> 00:11:16,080
All right?

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00:11:16,080 --> 00:11:18,930
So these are all
physical fields,

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00:11:18,930 --> 00:11:22,860
and then we try to identify the
mass terms you find for the w.

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00:11:22,860 --> 00:11:25,830
That the w mass is
proportional or equal

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00:11:25,830 --> 00:11:33,630
to the coupling strands of
the su2 group times the vacuum

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00:11:33,630 --> 00:11:35,610
expectation value over 2.

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And the mass of the
z boson is given

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by both couplings times
the square root of the sums

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00:11:45,600 --> 00:11:49,350
of the square times v over 2.

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00:11:49,350 --> 00:11:50,160
OK.

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00:11:50,160 --> 00:11:52,560
If you're trying to look for
a mass term for the photon,

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00:11:52,560 --> 00:11:56,310
you find none, meaning that
the photon is massless.

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00:11:56,310 --> 00:11:59,110
And then we can look again
at weak mixing angles,

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00:11:59,110 --> 00:12:01,160
and they are now
defined directly

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00:12:01,160 --> 00:12:04,560
through the couplings in
those two gauge groups.

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00:12:04,560 --> 00:12:06,900
The masses of the
w and the z bosons

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00:12:06,900 --> 00:12:12,312
are related via this weak
mixing angle, cosine theta w.

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00:12:12,312 --> 00:12:13,400
All right.

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00:12:13,400 --> 00:12:16,020
Those elements we
already saw before.

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00:12:16,020 --> 00:12:18,850
Now, we find that the
masses of the gauge bosons

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00:12:18,850 --> 00:12:21,180
are given by
spontaneous symmetry

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00:12:21,180 --> 00:12:24,260
breaking via the vacuum
expectation value

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and the strength of the
coupling of the gauge field

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to the Higgs field.

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00:12:31,510 --> 00:12:32,080
All right.

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00:12:32,080 --> 00:12:36,550
So in summary, we started
with a complex scalar field.

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00:12:36,550 --> 00:12:41,130
A representation of su2 is
four degrees of freedom.

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00:12:41,130 --> 00:12:44,550
The Higgs vacuum expectation
value breaks the symmetry

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00:12:44,550 --> 00:12:46,260
spontaneously.

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00:12:46,260 --> 00:12:50,760
The w plus and w minus and
the z boson require mass,

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00:12:50,760 --> 00:12:54,480
and the three Goldstone
bosons are each absorbed

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00:12:54,480 --> 00:12:57,540
into the w's and the z bosons.

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00:12:57,540 --> 00:13:00,270
We also find an
additional scalar Higgs

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00:13:00,270 --> 00:13:02,770
boson that remains.

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00:13:02,770 --> 00:13:07,110
And so that was
the understanding

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00:13:07,110 --> 00:13:09,300
in the '60s and '70s,
and the standard model

247
00:13:09,300 --> 00:13:10,770
was further developed.

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00:13:10,770 --> 00:13:12,930
And then it took us
all the way to 2012

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00:13:12,930 --> 00:13:15,650
to actually find this
new scalar particle,

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00:13:15,650 --> 00:13:18,110
the Higgs boson itself.