 
My name is Donal O'Connell.
Today we're going to build
on what we've already
learned about the physics
of the Higgs mechanism
and see it working in
the context of our best
current description of
particle physics, which
we call the standard model.
So let's recap with a simple
version of the Higgs mechanism,
which involves
vector fields, which
I call A and draw like this,
and a scalar field, phi,
which I draw with dashed lines.
Phi is the Higgs fields
in this simple model.
Let's think about the number
of different kinds of particle
states in this simple model.
The vector field,
A, can fluctuate
in two different directions.
These are its two
different polarizations,
so it can fluctuate up
and down, for example,
or it can fluctuate from
the left to the right.
So there are two kinds
of particle state
for the massless vector field.
The scalar field in the simplest
model we've considered also
had two states.
They were phi 1 and phi 2.
So the total number of
states in this model is 4.
After symmetry breaking,
the scalar field, phi,
takes on an expectation value,
which breaks the symmetry.
Consequently, the vector
field, A becomes massive.
Now, a massive vector
fields has 3 polarizations.
The total number of
polarizations in the model
has to be the same after
symmetry breaking, just
the same as it was
before symmetry breaking.
Before symmetry breaking,
we had 4 polarizations,
so after symmetry
breaking, there
must be 1 polarisation
left in the scalar field.
This we interpret as the
physical Higgs boson.
Let's look at the interactions
in this model in a little bit
more detail.
There are two kinds of
interaction between the scalar
field and the vector field
that we'll be interested in.
The first of these
interactions involves
two scalar fields, phi,
interacting with one vector
field, A. The second
interaction on the right
involves two scalar
fields, phi, interacting
with two vector fields, A.
The second of these
interactions is
more important for
understanding why
it is that the vector field
gets a mass when the scalar
field breaks the symmetry.
The second interaction
is associated
with a term in the potential
energy of this form.
It has two powers of
the scalar field, phi,
and two powers of the vector
field, A. In addition,
in the potential
energy of the model,
there is the scalar
potential, which
has the property that
it breaks the symmetry.
 
The scalar field's potential
energy is minimised when
the field takes on a nonzero
value at all points in space.
We call this value the field's
vacuum expectation value,
and we write it as v. When
the scalar field, phi, takes
on it's vacuum expectation value
v, the structure of potential
gets modified.
The scalar potential is now
minimised, and meanwhile,
the interaction term between
two scalar fields and two gauge
fields becomes a harmonic
oscillator potential
for the gauge field.
You see, the gauge field now is
just multiplied by constants.
So we can't shift
the vector field
without changing the physics.
Because of this harmonic
oscillator potential
for the vector field,
the vector field
takes on a mass, which
is given by g times v.
As we learned before, we need
two kinds of vector particle
to explain beta decay.
These are the massive
W minus and W plus.
Let's now see how we
get these from the Higgs
mechanism in the standard model.
In the standard model, there
are three kinds of symmetry.
Only two are relevant
for beta decay.
The first of these
symmetries is SU(2).
Since the group is SU(2),
there are three massless vector
particles associated
with that symmetry.
We call these three
massless vectors
the W's: W plus,
W minus, and W3.
There is also a U of 1 symmetry.
Because it's U(1),
there is simply
one vector field associated
with the symmetry.
This is the massless
vector field, B.
Now, in the standard model,
to break the symmetry,
there are 4 scalar fields.
In honour of Peter Higgs, we'll
call these scalar fields H,
that is H1 up to H4.
 
Let's look at the
SU(2) interactions
in a little bit more detail.
SU(2) is associated with
three massless vector bosons,
which we call the W's.
These cause interactions
between pairs of particles.
For example, they
cause interactions
between up quarks, u,
and down quarks, d.
One such interaction
allows an up quark
to split into a down
quark and a W plus.
That interaction has
strength which we call g.
Similarly, a down
quark can become
an up quark and a W minus.
This occurs with the same
strength, which we call g.
 
The W3 boson interacts
such that the type of quark
remains the same,
but it distinguishes
between up and down
quarks by having a minus
sign between the strengths
of the interaction.
 
In more detail, the
strength of the interaction
of a W3 with an up quark
is g/2, while the strength
of the W interaction with
down quarks is minus g/2.
 
One very peculiar
thing about nature
is that the SU(2)
interaction only
operates on the left-handed
component of a fermion.
 
So in these diagrams,
rather than the up quark
and the down quark
interacting with the W3,
in fact, it's only the
left-handed component
of the up and down quark,
which interacts with W3.
 
Now that we've seen how
the particles interact
in the standard model, let's
look in a little bit more
detail at how symmetry
is broken in the theory.
The process is much the
same as in the simple model
we previously discussed.
Diagrammatically, the process
involves two scalar fields,
in this case, the Higgs scalar
H, and two vector fields.
There are a couple of vector
fields involved in this case.
The vectors are
the B and the W's.
The diagram here is
associated with a term
in the potential
energy of the model.
Now, the symmetry is broken
when the Higgs field takes
on a vacuum expectation
value, v. When that happens,
the term in the potential
energy takes on a form here.
It involves the vector field,
B, the vector field W3,
and the vacuum of
the expectation
value, v of the Higgs field.
The B interacts with
strength g prime,
while the W3 interacts
with strength minus g.
The minus comes about
because our convention
is that the part of the
Higgs field that breaks
electroweak symmetry
interacts with the W3 just
like a down quark, so
it gets this minus sign.
All of this stuff gets squared.
Now, it's easy to manipulate
these terms into this form
here.
So we can pull out two
parts of v, like so,
and we can also
choose to pull out
a combination of the couplings
g and g prime in this form here.
Everything else that
remains we call Z.
So here's the
definition of Z. It's
a combination of
the W3 vector field
and the B vector field
with some coefficients.
So it's a linear combination
of vector fields.
Z itself is a vector field.
We see here that Z is
a massive vector field.
This equation is a simple
harmonic oscillator term
for this Z. The
combination here,
v squared times g squared
plus g prime squared
must therefore be the
square of the mass of Z.
So we learn that the Z boson
mass is given by this formula
here.
The Z boson is a combination of
the W3 and the B vector bosons
that originally
appeared in the model.
We can understand this
combination quite simply.
To do that, we fit these
combinations of the couplings,
g and g prime in a triangle.
 
The hypotenuse of the
triangle is the square root
of g squared plus
g prime squared.
The other two sides
of the triangle
are just given by the couplings
themselves, g and g prime.
Now, let's call this
angle theta W. Theta
W is known as the
weak mixing angle.
The reason for that, is
that the cosine and the sine
of this mixing angle
are the quantities here
that appear in the
combinations of W3 and B.
The cosine of theta weak is
simply g over the hypotenuse
here.
 
Similarly, the
sine of theta weak
is g prime over the hypotenuse.
 
So this combination
here for the Z boson
is just cosine theta weak times
W3 minus sine theta weak times
B. This is a
combination of W3 and B,
which gets mass in
Higgs mechanism.
However, there's still the
combination of W3 and B
that doesn't get mass
in the Higgs mechanism.
This is just the orthogonal
combination of W3 and B,
and it's given by sine theta
weak times W3 plus cosine theta
weak times B.
This vector field, A, is
what we call the photon.
It didn't get mass in
the Higgs mechanism,
so it is still a
massless particle,
and it's the particle that you
are using to see me right now.
Now, let's look at the total
number of states in the model,
both before symmetry breaking
and after symmetry breaking.
Before symmetry breaking, the
particles are easy to list.
We have 3 kinds of massless
W. We had 1 massless B,
and we had 4 kinds of
Higgs field, H1 up to H4.
The number of states is
easy to compute as well.
Because there are 3
massless W's, each
massless W having 2
polarisation-- remember,
these are massless
vector bosons--
they contribute a
total of 6 states.
The massless B vector
boson contributes 2 states.
The 4 Higgs components
contribute a total of 4 states.
So adding it up, there's a
total of 12 states in the model.
 
After symmetry breaking,
we interpret these states
a little bit differently.
We already described
that there's
a massive Z and a
massless photon,
the A. These contribute 3
states for the massive Z
and 2 states for the
massless photons.
Now similarly, the
W plus and minus
get mass through
the Higgs mechanism,
so they contribute
a total of 6 states.
Now, we need to get
12 states in total,
but so far we've only found 11.
So we need one more state.
This state is the
Higgs scalar state, H,
which was recently
discovered at CERN.
One interesting thing
about this model
is that we only wanted to
get a W plus and a W minus
to explain beta decay.
But along the way, we found
an extra massive boson,
the Z boson.
This was a prediction
of the model,
and it was discovered
at CERN in 1983.
In the 1964 paper, Peter Higgs
made an important prediction.
We have it highlighted here.
Higgs says, "It is worth noting
that an essential feature
of the type of theory which
has been described in this note
is the prediction of incomplete
multiplets of scalar and vector
bosons."
It's just such a scalar boson
that has been discovered.
It forms an incomplete
multiplet in the sense
that we've only seen one
of these scalar bosons.
But the symmetries would lead
us to think that there is more.
For example, before
symmetry breaking,
our theory consisted
of 4 scalar bosons.
But after symmetry breaking,
only one of them is left.
This incomplete
multiplet scalar bosons
is what we interpret as
the physical Higgs scalar.
 
