MARKUS KLUTE: Welcome back to 8.701.
So in this section, we look at electroweak unification.
So the aim is to combine the weak
and the electromagnetic interactions.
The issues we can see here are first,
the strength of the interactions are very different.
This can be mitigated by the fact
that we have heavy gauge bosons involved,
and we have seen our heavy particles
as being used as mediators to kind of change
the strength of the interaction.
So this might not be a big issue.
The second problem is that the structure of the coupling
is very different.
We have seen for QED that they are the vector coupling
and for weak interaction that they
are the vector-axial coupling.
This 1 minus gamma 5 is vector minus axial coupling.
So one way to mitigate this problem
is to simply absorb this 1 minus gamma 5 term in the definition
of the particle spinors.
And I have to warn you, this is a little misleading.
And I think that led also to some of the confusion we
had in the class before.
So what we are doing here simply is, we take our spinor,
and we project out with the 1 minus gamma 5 term
what the left-handed component of this spinor is.
This is just a projection and the definition of this.
And we can do this for antiparticles as well as
for the right-handed components as well.
Good.
So now we can look at the current again.
And we look at this weak current that you have seen you can
write as our antineutrino here, gamma mu times 1 minus gamma 5
1/2 times e.
Now, if we now define our particles which gamma matrices,
we find that this simplifies quite a bit,
because we now find the current, which can be simply written
as a vector current.
So we mitigated this quite nicely.
So what now happens to our electromagnetic interaction
here?
So we have an electron coupling to a photon.
We can project out a right-handed and left-handed
component, and then we have to add them together again.
When we do this, you find that this component
of the left-handed component, or the current
corresponding to the left-handed particle
and the current corresponding to the right-handed particle.
There's no mixed term here because
of the way gamma matrices or gamma 5 matrices multiply.
So that's nice.
This also explains why the helicity is not changed in QED.
You basically see this from the algebra
involved in those equations.
Good.
So far, we haven't done anything.
We have just changed the notation.
So we can go one step further.
And we use the concepts we introduced
when we talked about QCD or the strong isospin.
And since we can nicely describe those currents here
of those particles, we can maybe see
if we can write a neutrino and an electron
as part of a duplet.
And when we do this, we rewrite the currents,
the positively charged and the negatively charged current
as simply the left-handed components of those duplets.
We introduced a new matrix here, tau plus and tau minus.
And they're simply combinations of tau 1 and tau 2,
which are, in fact, the Pauli matrix.
This is just a relabelling as well.
So there's a lot of relabelling going on, not to confuse you.
But we have simply written this current
as positively charged weak current
and negatively charged current, where
we rotate a neutrino into an electron or an electron
into a neutrino using the weak interaction.
Great.
So now we can write this current here
as the third component of this current.
And we see when we write down the third component
of this current using tau 3 here,
we find that something looks--
something which looks like a neutral current.
So this is something like a neutral current, where
we have a neutrino coupling to a neutrino
and a left-handed electron coupling
to a left-handed electron, on a vertex vector
[?] where there's an interaction that was going on.
This is not quite the full story yet.
Let me remind you about the definition
we used also in isospin, which is
the Gell-Mann-Nishijima equation, which connected
the electric charge to the isospin and the strangeness
of the particle.
And we do the very same thing.
We have an isospin component and a so-called hypercharge
component.
It's similar to the strangeness we had before.
And Q is the electric charge of the particle involved.
And so with this, we can now define an iso--
a hyperspin current, which is given as 2 times
the electromagnetic current minus 2 times
the third component of the--
third component of the weak current here.
And so now we find interesting effects here.
There's a new component which also couples
to right-handed particles.
So the missing part here--
some of you might have seen this already--
is this neutral current in the upper equation
didn't connect or didn't have a contribution
from right-handed particle.
And since there's right-handed electron which
coupled to Z boson, there needed to be
this kind of additional term.
So now we have a current which includes
the right-handed particles as well.
So that's great.
We can generalize this by writing those duplets
for all particles we know.
There's an additional kind of caveat here.
We haven't talked about this too much.
We have to consider the fact that mass eigenstates are not
really the same eigenstates which participate
in the weak interaction.
Right now we can ignore this.
We'll later come back to this question.
And then we can write the three components
of our isospin current and our hypercharge current as well.
Note that this EM current here is our electromagnetic current.
Good.
So now we rewrote this, and we find somehow very close
a consistent picture.
We find that there is a charged current,
and then there is a neutral current
in the weak interaction.
But what we actually wanted to do
is combine the weak interaction and
electromagnetic interaction.
Now, let's look at this again and start over again.
So we have an isospin current here,
which copies to the three components--
the triplet, the isospin triplet.
So this is a W1, W2, W3 triplet.
And then we have the singlet here
which couples to the hypercharge.
Very good.
So now, if I try to identify now components which we already
know the first thing we can do, we
have to make sure that we find our W plus and W minus
bosons again.
And they're simply linear combinations
of the W1's and the W2's.
And then the next thing I have to do
is I have to find my electromagnetic interaction.
And we can do that by binding this A-- this is the photon--
as a linear combination of the third component
of our triplet, isospin triplet, and our singlet.
And what you see here is that there's actually
mixing going on.
So we basically rotate those with this mixing angle, which
we already introduced, sine omega weak
mixing angle theta omega--
sorry, theta W.
So we find that the photon can be made out
of a mixing of the third component of the isospin
triplet and the singlet component B mu.
And similarly, we can find the Z boson
as the other component in this mixing, the other state we
find in this mixing.
The way we find those mixings here
is through the couplings we already know,
that we find this omega g times sine
theta W is equal to g prime cosine omega W.
And that's equal to the electromagnetic coupling.
And then similarly, we find a solution for gz.
So what we have seen now is that apparently we
are able to combine the weak and the electromagnetic interaction
by mixing--
by introducing the weak isospin and by mixing isospin triplet
components with a singlet component.
And so we find a picture which is
consistent with the W plus, a W minus photon, and the Z boson.
So that's very nice.