 
We've seen in the standard
model that the massless vector
bosons-- the W3 and
the B-- mix together
to form a massive vector,
the Z boson, and also
the massless photon.
Now, let's see how this
fits in with what we already
know about how light
interacts with matter.
Let's begin with the up quark.
The up quark has charge
2/3, so it interacts
with the photon with
strength 2/3 times
the electric charge e.
Now, breaking that up into
the left and the right handed
components of the
up quark, there
are two diagrams,
which are just added.
Note the left-handed up quark
interacts with the photon
and the right-handed up quark
interacts with the photon.
In both cases, the
strength of the interaction
is 2/3 times the
electric charge e.
However, the photon
is a combination
of the B and the W3.
Let's see how that works in
the case of the right-handed up
quark, which is
a little simpler.
Now, we know that
the W3 just doesn't
interact with
right-handed particles.
So the only part of the photon
which actually interacts
with the right-handed
up is the part
involving the B. The mixing
angle, cos theta weak,
multiplies that interaction.
Now we can move that
mixing angle from the B,
from the field the B, to be part
of the interaction strength.
That's because all
of these quantities
are just multiplied together.
So the total interaction
strength of the right-handed
up with the B is given
by g prime, the strength
of the interaction, the cosine
of the weak mixing angle,
and then what we call the weak
charge of the right-handed up.
That's denoted by Y of uR.
This leads to an equation.
Since both sides must
be equal, the strength
of these instructions
must be equal.
We learn that Y(uR)
times g prime times
the cosine of the
weak mixing angle
must equal 2/3 times
the electric charge.
 
Now, filling in the definition
of the weak mixing angle,
this equation takes
the following form.
On the right hand
side, we still have
2/3 times the electric charge.
But now, on the left
hand side, we just
have the interaction
strength g and g prime
and the weak charge of
the right-handed up.
 
The solution of these equations
that we're going to use
is that the weak charge of
the right-handed up is 2/3,
and the electric charge
is given by g prime g
over the square root of g
squared plus g prime squared.
This is one way in which we
see that the electromagnetic
interaction is unified
with the weak forces.
The strength of the
electromagnetic interaction
is just given by a
combination of the strength
of the weak forces.
 
Now let's move on
to understand what
happens in the case of
the left-handed up quark.
In this case, things are a
little bit more complicated
because both the B
and the W3 interact
with the left-handed up.
So we can expand
out the interaction
of the left-handed up with the
photon in terms of two terms.
One involves the field
B times the cosine
of the weak mixing angle.
The strength is given
by the weak charge
of the left-handed up y (uL)
times the strength of the B
coupling g prime.
The other term, which
is simply summed,
involves W3 times the sine
of the weak mixing angle.
As we said before, the
strength of that coupling
is just a half times g.
Now, as before, we can move the
cosine of the weak mixing angle
from the particle
B, and think of it
as just part of the
strength of the interaction.
We could do just the
same for the sine
of the weak mixing angle.
Now, filling in the
definitions of the cosine
of the weak mixing angle and the
sine of the weak mixing angle,
we see that in
both cases, we just
get the electric charge out.
This leads to a simple
equation that 2/3 times
the electric charge
equals the weak charge
of the left-handed up times the
electric charge plus 1/2 times
the electric charge.
The solution of this
is simple, and it
is that the weak charge of
the left-handed up is 1/6.
So we find that the weak charge
of the left-handed up is 1/6,
but the weak charge of the
right-handed up is 2/3.
The weak hypercharge
of the left-handed up
and the right-handed up
are completely different.
So before symmetry
breaking, we should
think of the left-handed
up and the right-handed up
as being simply
different particles.
This is true of all the
fermions in the standard model.
Before the electroweak
symmetry is broken,
their left-handed components and
their right-handed components
have nothing to do
with one another.
They're simply
different particles.
It's only after
electroweak symmetry
breaking that the charges
of these particles,
their electric
charges, are equal.
Then they combine and
form a massive fermion.
This mass also comes about
through their interaction
with the Higgs boson.
So let's look at how matter
interacts with the Higgs boson.
Here we see the interaction
of up quarks and down quarks
with the Higgs.
On the left is the interaction
of the up and the Higgs.
The strength of that coupling
is proportional to the mass
of the up quark.
It's proportional to the
mass of the up quark divided
by the vacuum expectation
value of the Higgs.
Notice that in this
interaction, the left-handed
up changes into a
right-handed up.
 
Similarly, the down quark
interaction strength
with the Higgs boson is
proportional to the mass
of the down quark.
And also, the
left-handed down quark
changes into a
right-handed down quark
during the interaction
with the Higgs boson.
Now, after electroweak
symmetry breaking,
the Higgs takes on its
vacuum expectation value.
There is still one
Higgs scalar left over.
That Higgs scalar interacts
with up and down quarks
in just the same way
as the full Higgs field
does prior to electroweak
symmetry breaking.
But there's one additional
effect after symmetry breaking.
That is that the Higgs take on
its vacuum expectation value.
So if we plug that in, we
get interactions like this.
This is the kind of attraction
that ties the up left
and the up right together
into just one up quark.
The interaction is proportional
to the mass of the up quark.
Similarly, the left-handed
down and the right-handed down
are tied together
into one particle
through this interaction.
The interaction strength in
the case of the down quarks
is proportional to the
mass of the down quark.
Let's look at how the
fermions in the standard model
behave before and after
symmetry breaking.
Before symmetry breaking,
these are the particles.
There are left-handed
ups and left-handed downs
that form a pair and
interact with SU(2).
In addition, there's
a right-handed up
and a right-handed down.
These don't interact with SU(2).
In addition to the quarks,
there is the leptonic sector.
This is comprised of a
right-handed electron
and another pair, the
left-handed neutrino
and the left-handed electron.
This left-handed pair
interacts with SU(2)
while the right-handed electron
does not interact with SU(2).
 
After symmetry breaking,
the left-handed up
and the right-handed up pair
up and become the up quark, u.
In just the same way,
the left-handed down
and the right-handed down pair
up and form the down quark, d.
Again, in exactly the same
way, the left-handed electron
and the right-handed
electron pair
up and for the
usual electron, e.
Finally, as far
as we know, there
are only left-handed
neutrinos in nature.
So after symmetry breaking,
there is a neutrino.
But it's just a
left-handed particle.
 
Now, there's one further
surprise in the standard model,
and that's that this structure
is replicated three times.
So this family is
just one generation
of three generations
in the standard model.
Here are what the
generations looks
like after symmetry breaking.
The first generation is just
what we talked about before.
There's an up quark
and a down quark.
In addition, there's a
electron and a neutrino.
The second generation
has these other particles
whose properties are exactly the
same as the first generation,
but they have different masses.
In this generation,
there are two kinds
of quark-- the charm quark
and the strange quark.
There's also a particle
of the same properties
of the electron.
It's just heavier.
We call it the muon.
In addition, there's
another neutrino
called a muon neutrino.
The third generation consists of
the heaviest set of particles.
The quarks are called the top
quark, t, and the bottom quark
b.
Meanwhile, there's
yet another heavy copy
of the electron called the tauon
and another neutrino, the tau
neutrino.
These particles again
other than their mass have
exactly the same properties
as the familiar up and down
quarks, the electron, and the
first generation neutrino.
We've now learned all of
the details of physics
at the deepest level we
currently understand it.
So let's use this knowledge
to understand how the Higgs
boson was discovered at Cern.
To do that, there are two things
that we have to understand.
The first is, how is the
Higgs boson produced?
And the second is,
how did we know
that a Higgs boson was produced?
The key thing to
understanding all of this
is the statement that the
Higgs boson couples to mass.
We've already seen this.
Before symmetry
breaking, the Higgs field
couples to particles
in a way which
is proportional to their mass.
After the symmetry
breaking, this coupling
leads to two phenomena.
The first is the
particles get a mass.
And the second is that
the Higgs scalar interacts
with particles,
again, in a way which
is proportional to their mass.
Since the Higgs
couples to mass, we
need to know a little bit
about what particles are heavy
and what particles are light.
So here's a table
of various particles
in the standard model,
including their mass,
and measuring the mass in
units of gigaelectron-volts
over c squared.
Gigaelectron-volts
is a unit of energy.
So since E equals mc
squared, gigaelectron-volts
over c squared is
a unit of mass.
The heaviest particle in this
example is the top quark.
Its mass is 170
GeV over c squared.
The next heaviest
particle is the Z,
which has a mass of
91 GeV over c squared.
After that, there's the
W boson with a mass of 80
and the bottom quark,
which has a mass of four.
The heaviest partner of
the electron is the tauon.
It has a mass of 1.7
GeV over c squared,
which is just a little heavier
than the charm quark, which
has a mass of 1.2.
The next heaviest
particle is the muon,
followed by the strange quark,
the down quark, the up quark.
Then we arrive at the electron.
In these units,
the electron mass
is 0.00051 GeV over c squared.
After the electron, the
next lightest particles
are the neutrinos.
We haven't measured the mass of
neutrinos, but it's very light.
So for our purposes,
it's nearly zero.
Finally, the gluon
and the photon
are exactly massless particles.
One thing to notice
about this is
that there's quite
a large spread
in the masses of the particles.
The top quark mass is
about 10 to the five times
the up quark mass.
That's a factor of
100,000 in the masses.
So we've seen that the
top quark is the heaviest
particle in the standard model.
Let's use that knowledge
to see how the Higgs
boson was produced at the LHC.
Now, the LHC
collides two protons.
So some protons are accelerated
and brought together
until they collide.
Now, these protons aren't
elementary particles.
They're composed of a
bunch of other particles.
 
So a proton consists of
two ups and a down all held
by a sea of gluons.
 
Now, these ups and downs
are not the only quarks
that are existing in the proton.
That's because
gluons are constantly
splitting into other quarks.
The gluons can come
along and interact
with a pair of quarks which
then reform the gluon.
So in addition to the
valence up and down quarks,
there's continuously a
selection of other quarks
being created inside the proton.
Now, if you want to create
a Higgs boson by smashing
together two protons,
you might think
that you've got to get the
Higgs boson for its interactions
with the ups and the downs.
But that's not the
most important process.
Because the Higgs interacts
so much more strongly
with the top quark,
it's much more
probable to get the Higgs
boson from an interaction
where the gluon has split
into pairs of top quarks
and to get a Higgs boson from
the interactions of the Higgs
with the top quark.
This interaction
is much stronger.
There's two reasons for that.
The first is that the top
quark has a large mass.
Its 10 to the five times
the mass of the up quark.
But these diagrams
represent amplitudes.
The probability, the quantum
mechanical probability
of a process occurring, is
the square of the amplitude.
So this factor of 10
to the 5 is squared.
That's a factor of 10 to the 10.
Now we've learned about how the
Higgs is produced at the LHC.
The process of Higgs production
is usually called gluon fusion
because two gluons are required,
making a top quark which
then radiates a Higgs boson.
But before we really understand
the behaviour of the Higgs
at the LHC, we also need to
know how the Higgs decays.
The Higgs is not
a stable particle.
So at the LHC,
experimentalists face the task
of understanding whether
a Higgs was produced based
on the evidence contained
in the stable particles
that they are able to detect.
So now let's talk
a little bit about
the various different kinds
of Higgs decay processes.
We've seen that the most
important interaction
between the Higgs and
matter in the standard model
involves the top quark.
So you might think that the most
important decay of the Higgs
involves the Higgs falling
apart into two top quarks,
as shown in this picture.
In fact, this process
is just not allowed.
The reason is that there is
not enough energy in the Higgs
to make two tops.
So the reason for that is
just that, on the left hand
of this picture, we
have a Higgs boson.
In its rest frame, the
only energy available
is in the mass of
the Higgs boson.
So since the mass
of the Higgs is
125 GeV over c squared
or so, the energy
available on the left
hand side is 125 GeV.
On the right hand
side, the top quark
has a mass of about
170 GeV over c squared.
So on the right, I need a
mass-energy of, in total,
more than 340 GeV.
The "more than" is because
these two tops are in motion
relative to one another.
So 340 is the minimum
amount of energy
I would need to
make these two tops.
Since I just don't
have enough energy,
the process is forbidden.
 
The next heaviest quark
in the standard model
is the bottom quark b.
It has a mass of 4
GeV over c squared.
So in this picture on the
right hand side, I need a mass
- I need an energy
of more than 8 GeV.
Since the Higgs has an energy
of 125 GeV in its rest frame,
this process is
perfectly allowed
and is, in fact, the most
important decay process
of the Higgs in
the standard model.
Are there any other
decays of the Higgs boson
other than to matter
in the standard model?
Well, the answer is yes.
For example, we've seen that the
production process of the Higgs
that's most relevant to the LHC
involves two gluons interacting
to form a top which
radiates a Higgs.
Now, we can just turn
this picture around.
Now, we have a
diagram where a Higgs
is decaying to two gluons.
This is a perfectly
allowed process
because gluons are exactly
massless particles.
 
In a similar way, we can have
an interaction of the Higgs
where it decays
into two photons.
Again, photons are massless,
so this is perfectly allowed.
 
Now we've seen
that the Higgs can
decay into the massless vector
bosons in the standard model--
the gluon and the photon.
What about the
massive vector bosons?
Well, let's consider a Higgs
going to W plus W minus.
On the left, we have available
energy of 125 GeV as usual.
On the right, well,
since the mass of the W
is 80 GeV over c squared, I need
an energy of at least 160 GeV.
So you might think that
this process is forbidden.
However, since 160 GeV is
quite close to 125 GeV,
this process is allowed.
But there's a little caveat.
That is that one of the
W's shouldn't really
be thought of as a particle.
Since there isn't
quite enough energy
to make the particle of the
W, what really happens here
is that we just create some
disturbance in the W field.
One of the W's
becomes a particle.
The other one, this
disturbance in the field,
falls apart very quickly.
But since it decays like a W,
to our experimental colleagues,
it looks just like a W.
Similarly, we consider the decay
of the Higgs to two Z bosons.
Since the mass of the
Z is a little larger
than the mass of the w, this
process is a little bit more
disfavored.
In addition, since there
are two W's but only one Z,
this process is always at
least half as important
as the process of the
Higgs decaying to two W's.
So let's put it
together and see which
decays are the most important.
Here's a table of
the various Higgs
decays in their
order of importance.
As we said, the most
important decay,
in the sense it's the most
probable decay of the Higgs
boson, is to two b quarks.
The next most probable
decay is to two W's.
After that, the Higgs can
decay to gluons, to tauons,
to two Z bosons, to two charm
quarks, and to two photons.
Of course, there are
more allowed decays,
but these ones are
the ones that are
most important for understanding
the Higgs at the LHC.
 
