Now, let us go to the other interaction which
is basically the weak interaction. We already
listed the elementary particles known to us.
Let us write down again that here; elementary
particles. We have electron, let us explicitly
write down the left handed and right handed
ones; and we have mu e L corresponding to
an electron, we have left handed muon, nu
mu L corresponding neutrino left handed tau
lepton and mu tau L and we have a right handed
electron, there is no right handed neutrino
corresponding to that. We have right handed
muon, we have right handed tau particle.
And we do not have any no right handed neutrinos.
This is because they are not found. In experiments,
we are not found right handed neutrinos we
have only found left handed neutrinos and
right handed and the neutrinos in beta decays
and other weak interactions.
What about the quarks? We have u L, d L, c
L the charm quark strange quark left handed,
top quark left handed, bottom quark left handed.
We also have right handed quarks all of them;
c R, s R top and bottom. All of them. And
again, what is observed is that only left
handed particles take part in weak interactions.
This is an observation experimental observation.
Now, to conform to that, we consider the weak
interaction and the gauge transformation 
corresponding to the gauge weak interaction,
the gauge group is identified after some studies
as SU 2 gauge group. SU 2, since only the
left handed particles transform under SU 2
weak interactions I consider we will write
the gauge group as SU 2 L; L just be noting
that we made to remind us that only left handed
particles take part in this. And we can club
this particles together.
Psi corresponding to the left handed neutrino,
psi corresponding to the electron can be clubbed
together and considered as a doublet under
SU 2 L. Whereas, the right handed field has
no such partner and the wave function corresponding
to around the field corresponding to that
is a singlet under SU 2 L. When I say singlet
which means that it is invariant or do not
take part in the interaction corresponding
to a SU 2 L which is weak interaction.
This is also similar to the isospin that we
had discussed earlier in the case of the flavor
isospin. For example, we considered the proton
and neutron has 2 different projections of
same object, which we could call as nucleon.
We could consider something called isospin
and the isospin of nucleon is half and plus
half projection of that is proton, minus of
projection of that is neutron.
Now, in exactly the same way we can consider
here the SU 2 L isospin or the weak isospin.
This is completely different from the isospin
that we had considered in the case of strong
interactions, but let us call it isospin,
because that is basically a standard terminology
now. So, under isospin we can categorize these
fields and we just now mentioned that the
wave or the field corresponding to nu e L
and e L transform as a doublet, which means
isospin of this is equal to; let me denote
this by I; equal to 1 by 2 and isospin 3,
the third component of nu e L is equal to
plus half, and that of electron is minus half.
Whereas when we consider psi e r we have a
singlet, and the isospin of e R is equal to
0. And isospin third component is also equal
to 0. That is why we call this singlet; singlet
under su 2 L. Similarly, other fields also
can be clubbed. So, let us write that down.
I will forget about the psi for the time being,
because the notation is becoming too much
and I mean more and more complicated or clumsy
if I write all the labels.
So, I will forget about psi here write only
the notation nu e L e L just to denote the
corresponding fields. Similarly, we have nu
mu L and nu tau L the left handed ones. And
all others e R mu R and tau R singlets. And
in the case of quarks you will we will club
u L and d L, c L and s L top left handed and
bottom left handed.
Right handed fields are all singlets, t R;
let me start from the beginning u R, d R,
c R, s R, t R, b R. All these are singlets.
So, this is how we club this; write this,
explicitly as the, in terms of the multiplets
under SU 2 L. That is one thing. Now study
is; lot of investigation had gone into understanding
the weak interactions using the gauge symmetry
similar to the quantum chromodynamics for
strong interactions and quantum electrodynamics
for the electromagnetic interactions.
We will not go into the history or the details
of all those studies, but discuss only the
end product. At the end, we see that we have
to consider the electric electromagnetic interaction
and the weak interaction together, let us
call it electro weak interaction. Some kind
of a unification between the electromagnetic
and the weak interaction has to be considered
for consistency and the gauge group that we
should consider is not just SU2, but a direct
product of SU 2 and U 1.
Let us call this interaction or the charge
corresponding to that as hyper charge. SU
2 is the same as isospin weak isospin, whereas,
while, but the weak itself is now to be considered
as to be understood as used under electroweak
interactions there is a mix up between s u
2 isospin and hyper charge in both electromagnetic
interaction as well as weak interaction. Such
unification is not very unfamiliar to us.
If you remember your school days you will
see that magnetism and electrostatics and
electricity are taught different as different
topics.
Especially the magneto statics and electrostatics.
We have electric fields giving rise to all
electrostatics. Force and dynamics under that.
Similarly, we have magnetic field B giving
rise to magneto static. But at high energies
or for moving charged particles when the motion
is or the velocity of the particle is large
enough, so that we cannot neglect the effects
due to this motion then we cannot separate
the electricity and magnetism from each other
we have to consider electromagnetic theory
or electromagnetic interactions between charged
particles and Maxwell’s equation follows.
You see that induced electric; changing electric
field induces magnetic field the changing
magnetic field induces electric field etcetera
and then clubbing all these together you realize
that you have to discuss electromagnetic interactions
rather than electric and magnetic interactions
separately. Unification of weak and electromagnetic
interactions is another step in this direction.
So, we can think about weak and electromagnetic
interaction as unified; considered to be a
unified theory at sufficiently large energies.
Thinking in terms of moving charged particles
has larger energy compared to charged particle
at rest.
So, we will not go into any further motivations
in this direction, but tell that this is what
we are going to consider. The symmetry group
consider it is s u 2 cross u 1, L to denote
that only the left handed particles are going
to be affected by transformation under s u
2 and y to denote that the charge corresponding
to u 1 is what called hyper charged denoted
by y. How do we assign this hyper charge?
There is a consistent way of doing it this
is due to the fact that we need to get the
electrodynamics from this. We will see that
later that we have to actually describe the
photons from this one right, because this
includes electrodynamics what are photons.
As I said photon now is a mixture of s u 2
L and u 1 by gauge fields. It will be seen
later clearly. But that tells us that the
corresponding charges whatever charges are
also related. This relation is given by the
charge of electromagnetic interaction Q is
equal to I 3, the third projection of isospin
which is the charge corresponding to s u 2
plus y by 2, where y is the hyper charge.
Now, let us look at left handed electron and
neutrino doublet. Q of nu e L is equal to
0, we know that it is a neutral particle.
I 3 of this is plus half and that is sufficient
to fix y to be equal to minus 1. If I take
y equal to minus 1, I have minus 1 by 2 in
the second term of the relation above and
I 3 is plus 1 by 2, total is 0. Similarly,
for electron Q is known to be equal to minus
1, I 3 is equal to minus half which again
gives you y equal to minus 1.
This indeed should be so, this is because
the all the multiplieds s u 2 multiplied will
have the same hyper charged while we will
see this is so. So, when we write down the
transformations this is also the reason that
we have written the neutral component as the
first component I 3 equal to half and the
negatively charged e as the I 3 equal to minus
half. To start with we would have wondered
if you were keenly observing it why I wrote
nu first hand and then e l second hand what
would happen.
If the other way around if it was the other
way around, if it is the other way around
this relation would change; that is the only
thing. but if we stick to this conventional
relation then basically we have this basically
this way of representing the doublet there
is no other way of representing the doublet
here.
You will also see that because I 3 changes
by one unit when you go from one component
to another component nu e to e in the same
multiplet; the charge should also reduce or
change by one unit when you go from one component
to other component. So, that is always the
case. This is because the multiplet has the
same hyper charged and whenever you change
I 3 by 1 unit Q will be changing by one unit.
What about the right hand electron? Charge
is the same as minus 1. I 3 is equal to 0.
Therefore, y of e L is equal to minus 2.
So, that is it. There is a difference in the
hyper charged assignment. Sorry this is the
right handed electron that we are talking
about. There is a difference in the hyper
charge of electron, which is right handed
and electron which is left handed. So, there
is a difference between the left handed and
the right handed electron, the charges in
terms of the hyper charge there is a quantum
number which makes left handed and right handed
electron distinguishable, which is the hyper
charge and also the isospin of course, the
isospin is another one. These 2 charges; these
2 quantum numbers allow us to distinguish
or help us distinguish the left handed and
the right handed particles.
That indeed is the basic reason why they interact
differently under weak interactions, which
is a combination of s u 2 L and u 1 y now.
The electromagnetic charge or electric charge
of electron is minus 1 irrespective of whether
it is right handed or left handed. This is
the reason why we cannot distinguish the left
handed and the right handed electrons under
electromagnetic interaction. But they are
distinguishable under weak interaction. The
same story goes through for other particles
as well. Like, you can; so for the other leptons,
muons and the tau lepton and the corresponding
neutrinos the story is exactly the same; replication
of this. For quarks because their electric
charge is now different.
Let us consider you well up type lefts charge
left up type quark has a charge of plus 2
by 3 and isospin third component is equal
to plus half. And that tells us that y is
equal to; so, relation let me write here this
Q equal to I 3 plus y by 2 or y is equal to
2 Q minus I 3 which is equal to 2 times 2
by 3 minus 1 by 2, which is equal to 1 by
6, 2 into 1 by 6, is 1 by 3.
Similarly, for down type quacks I 3 is equal
to minus half, charge is equal to minus 1
by 3, y equal to; again, it should be equal
to 1 by 3 let us see Q minus I 3 which is
equal to minus 1 by 2 into minus 1 by 3 plus
1 by 2, which is again equal to 1 by 3, which
should indeed be.
Right handed Q of right handed charged quark;
u right handed is 2 by 3, I 3 equal to 0 that
gives y equal to twice Q, which is equal to
4 by 3. And similarly for down type right
handed it is 1 by 3, I 3 equal to 0 that will
give you y equal to minus 2 by 3. So, these
are the hyper charged assignments of the quark
doublet and the quark singlets.
The up type quark right handed will have all
the quark doublet us left handed will have
1 by 3 as their hyper charge; and right handed
up quark c R and t R all have hyper charge
4 by 3; and down type quark s quark and b
quark all right handed have hyper charge minus
2 by 3. All right. So, this is done we know
what is the what are the hyper charges, what
are the isospins of all these particles, now
we are in a position to construct the Lagrangian
which is invariant and a the SU 2 cross U
1 transformation. We will follow a similar
when approach similar to what we had done
in the case of is QED and strong interaction
QCD. We will do that in the next class.
