Start our discussion on the Weak Interactions
and some of the discrete symmetries associated
with it. So, we had been discussing the electromagnetic
interactions and then we saw that there are
convenient ways to actually represent this
by using diagrams, to describe the scattering
of charged particles.
Now, we know there is this weak interaction,
as already introduced somewhere in the beginning
of the discussion that we had. We said it
was observed that certain nuclei undergo beta
decay. An example of weak interaction is beta
decay. Say for example, Cobalt 60 with mass
number 60 undergoes beta decay by converting
itself into Nickel and emitting electron and
a neutral particle.
When you consider the nucleon level of interaction,
this is essentially conversion of one of the
neutrons to proton by emitting an electron
and the neutral particles. Electric charge
is conserved on both sides. So, is what is
called the Lepton number and barian number.
At the quark level we know we can think about
this as a u quark and a d quark and the d
quark converting into u quark u quark and
d quark plus e minus plus nu e bar. Or you
can say that one of the d quarks converts
itself into a u quark and an electron plus
nu e bar. And we had been describing the transition
amplitudes in terms of current interactions.
Current-current interaction.
So, Fermi thought that it is possible to understand
weak interactions also in a similar footing.
He said the invariant amplitude can be written
in terms of two currents, one is an n p current
and the other is basically electron and neutrino.
So, these currents can be written in corresponding
in terms of the corresponding spinners, u
p bar gamma mu u n, u e bar gamma mu. It should
be the anti neutrino that is coming with it,
because it is the lepton and anti lepton together
that will be emitted not the lepton and the
lepton. So, we will have a v nu bar associated
with.
So, this is basically the kind of picture
that Fermi thought is possible, with of course
some kind of constant or coupling which connects
these two currents. He did not introduce any
exchange particle between; any propagation
any propagator or any particle propagating
between these two. So that was one thing.
Other thing was the structure of the current
was very similar to that of the that used
in the case of electromagnetic interactions,
this is what Fermi proposed.
But then it turned out that it is not able
to describe all aspects of beta decay, and
then investigation of other possibilities
leads to the following. With the same kind
of the spinners and the conjugate spinner
the bar u bar and u and v and v bar, what
are the other objects that we can construct
similar to these currents j mu j mu? So, possibilities
are, we can make scalar quantity psi bar psi,
which is a scalar under the Lorentz transformation.
We can actually have a pseudo scalar; by pseudo
scalar we mean something which changes sign
under parity.
So, it is gamma 5 psi where we define gamma
5 as i, gamma 0, gamma 1, gamma 2, gamma 3
product of all the 4 gamma matrices multiplied
by an i. We will not go into the detailed
properties of this gamma 5 etcetera; we will
not also go into how we say that and what
is the transformation property of this object
under Lorentz transformation. We will just
take it for granted that gamma 5, when put
between psi bar and psi the whole object transforms
like a pseudo scalar. Then we have the familiar
vector which is psi bar gamma mu psi and axial
vector which is psi bar gamma 5 gamma mu psi.
So, vector and axial vector. How do they differ?
A vector when you take the spatial components
that is gamma 1, gamma 2, gamma 3 part of
psi bar gamma mu psi, they transform under
parity like any ordinary vector; position
vector x or momentum vector p or any other
ordinary vector, which means they change sign
under parity. So, but axial vector is something
which will not change sign under parity. The
spatial components of the axial vector will
not change sign under parity similar to orbital
angular momentum or spin orbital spin angular
momentum. We will come to that in a little
while.
But then there is another object a tensor
of second rank, that can be constructed out
of the gammas and psi bar and psi, which is
for short written in this fashion, psi bar
sigma mu nu psi where sigma mu nu is a second
rank tensor operator. Gamma nu tensor gamma
mu gamma nu minus gamma nu gamma mu the anti-symmetric
tensor obtained i over 2.
These are the different type of what are called
the by lineage set of by lineage with proper
Lorentz transformation structure, that can
be constructed from using gamma mu and psi
and psi bar. Again, we will not go into the
details of why this particular set and how
we confine, say that this is; we cannot add
anything more to it again let us take it for
granted that that is the way it is.
So, this is; with this now we can also think
about other possibilities for the invariant
amplitude m. In particular we can think about
the pseudo scalars for example, sorry pseudo
the axial vectors for example. How do we decide
all these things? It was sort of own by because
of various experimental reasons one of which
is what is called the Parity Violation Observed
in beta decay.
This was suggested actually by Lee and; T.
D. Lee and C.N. Yang. And they later on got
Nobel Prize for theoretical support that they
gave for that arguments that, they put forward
for that which was experimentally proven.
So, the experiment that actually proved or
showed first that the parity is violated in
beta decay was conducted by W u and others.
W u and others conducted an experiment in
1957, result of which is published in physical
review volume 105 page 1413. The experiment
was basically conducted with a radioactive
sample of cobalt, which as we said in the
earlier the discussion, could decay into nickel
and electron and anti neutrino. Parity says
that; say how things behave under spatial
inversion for that, when we say spatial inversion,
we need to have some reference direction about
which you invert this thing.
So, that such a direction was needed here,
then for us the cobalt 60 nucleus has a large
nuclear spin. So, the nuclear spin of cobalt
60 is let me denote it by I Co 5. And the
nickel it is four. Very large nuclear spin
associated with both the nuclei, and this
could be a reference direction. First thing
was to actually align all this in a particular
direction. For that they use large magnetic
field and cool the sample to a low temperature,
then observed their decay properties.
So, if you look at the spin configuration
of cobalt, let us say this is spin of the
cobalt that we are considering. That is what
we have the initial state and in the final
state we have the spin of the nickel and then,
you also have electrons and neutrinos coming
out. We can consider cobalt and nickel to
be at rest and electron and neutrino to come
out. So, to get the specific spin configuration;
to agree with the spin angular momentum conservation,
we need the spins of the outgoing particle
to be aligned along with the spin of the cobalt,
which is also the direction of the spin of
the nickel which is coming out.
So, now one possibility is that electron comes
out opposite to the spin direction of the
cobalt and neutrino or anti neutrino goes
with the momentum goes along the direction
of the cobalt 60 spin. So, let me actually
call this reference direction as Z direction
and this is basically can be; you need some
heavy magnetic high magnetic field in that
direction for the spins to be aligned in this
fashion. And here, this thin line represents
momentum direction linear momentum and thick
line is denotes the spin direction for either
of the particle. So, let me call this configuration
1. Possible also to have a second configuration
in which case, I cobalt is in the same direction
as the Z cap again and I nickel is also in
the same direction.
Now, particles which are coming out will again
have the same spin directions that along the
Z direction, but it could be the electron
going in the positive direction now z direction
now and anti neutrino coming out in the negative
Z direction. This is another possibility.
If you apply parity to this; what is parity?
We already mentioned that, but parity let
me denote the operator by operation by P.
It changes r to minus r position vector and
also; r goes to minus r under parity and p
linear momentum goes to minus p, whereas,
if you look at the angular momentum, that
goes to what? Remember L is orbital angular
momentum is r cross p; both r and p changes
sign. So, it goes to minus r cross minus p
which is equal to plus r cross p. So, it is
equal to L.
So, this is the orbital angular momentum.
Spin is similar to the orbital angular momentum.
So, if I denote it by sigma I will say that
this also goes unchanged under parity, because
it is similar in all ways to orbital angular
momentum. So, both sigma and L are axial vectors,
whereas the position vector and linear momentum
are ordinary polar vectors as they are called.
Consider particle with spin aligned along
the linear momentum.
Under parity what you expect is, that the
spin is; spin direction is unchanged while
the momentum direction is opposite. So, as
we mentioned in a in an earlier occation,
this particle with momentum aligned or the
spin aligned along the momentum are called
a right handed particle and the other ones
aligned opposite to the linear momentum are
called left handed particles. So, the parity
operating on a right handed system; particle
takes it to a right left handed particle.
In an experiment what it means is that, if
parity is a good quantum number; is a good
symmetry of the interaction; of the system
as a whole including the interactions, then
if configuration 1 happens, the other possibility,
configuration 2, which is obtained by applying
parity on configuration 1. When you apply
configuration parity and configuration 1 you
get the spin of the cobalt unchanged, spin
of nickel unchanged, spin of electron unchanged,
a spin of neutrino anti neutrino unchanged,
the directions of it. Whereas, the direction
of electron momentum linear momentum is changed
goes to a opposite direction. In configuration
1 it was along the negative z direction and
in configuration 2 it is along the z axis,
similarly for the anti-neutrino.
So, you can say you can get configuration
2 from configuration 1 by applying a parity
operation. Experimentally the statement is
that, if configuration 1 is observed that
is if you see electrons coming out opposite
to the direction of the spin of cobalt, then
you should see electrons coming out along
the spin of the cobalt also. You remember
that what we are working with is a quantum
mechanical system and therefore, all the statements
are statistical in nature.
In the sense that you observe a large number
of such interactions such decays, then say
that out of many such this things; say 1000
decays, you expect approximately 50 percent
of the time electron comes out along the spin
direction of the cobalt along the z axis and
50 percent of the time opposite to this thing.
Or if you take the positive z hemisphere and
negative z hemisphere, you will see that with
all the uncertainties and experimental details
taken into account. You will you should see
approximately the same number of electrons
coming out in the negative z hemisphere and
the positive z hemisphere.
Wu’s experiment: If parity is conserved.
Expected result; number of electrons in the
or along z direction, which is the direction
of the cobalt 60 spin approximately statistically;
number of electrons opposite to or along the
negative z axis opposite to positive z axis.
Observation: Mostly electrons are emitted
along negative z axis z direction opposite
to the cobalt 60 spin. Conclusion: parity
is violated.
And you can see that later on other weak interactions,
confirm that they all violate the parity.
So, parity violation is a common feature of
weak interaction ok. In fact, this was a very
important experimental confirmation of theoretical
expectation and it was very important to understand
the interaction weak interactions ok.
Let us look at how to accommodate how one
accommodates that in a theoretical framework.
So, we will not go into the details, but let
us give some flavor of this as this kind of
right handed 
particles say electrons are denoted by e R
and can be represented by psi R, which can
be written as psi operated by an operator
1 minus gamma 5 which we had defined earlier
by 2 or in terms of the spinners it is u a
spinner u it is u R equal to 1 plus of course
1 plus gamma 5 by 2 u.
Similarly, you can represent the 
left handed electron by e L; and that can
be represented by wave function psi L 1 minus
gamma 5 by 2 psi, for u L is 1 minus gamma
5 by 2 u. This half factor is for un-polarized
without any specific left handed or right
handed or a mixture of these two can be written
as psi L plus psi R equal amount of these
two. You can you see that when you add this
to the gamma 5 factor drops out. This is how
you can represent them and then this is the
kind of thing that will come in the current
for example, when you consider only the left
handed electrons in interactions.
Similarly, for antiparticles, it is again
the same notation v R for the right handed
ones and v L for the left handed ones and
v L is 1 plus gamma 5 by 2. We know it is
the other way around v R is equal to 1 minus
gamma 5 by 2 v. How do we conclude this etcetera;
take us slightly far from our main focus in
this course? So, we will not go into that,
you have you can take any standard introduction
to introductory book on high energy physics
or particle physics and that will all be describing
these aspects. So, we will not do that in
these discussions. Now similar to parity there
is another important symmetry called Charge
Conjugation which is 
again important in weak interactions.
Essentially it is a symmetry that takes a
particle to an anti-particle. For example,
electron goes to positron under C or positron
goes to electron under c or muon goes to mu
plus under c ok.
Now, let us look at one particular process
weak process pi on with charge plus 1 decays
to mu on plus the neutrino corresponding to
the muon. muon further decays to electron
and electron type neutrino and muon type neutrino
muon type anti neutrino and then the other
neutrino already exist. Let me call this process
1. Experimentally it is observed that we can
measure the let us say like the earlier beta
decay experiment say we can find out what
is the spin orientation of the electron or
the positron coming out of this and it is
found that the positron coming out is right
handed one always.
Now, consider another process which is obtained
by applying charge conjugation to process
1. So, pi plus goes to its antiparticle pi
minus we already discussed this earlier, that
pi minus is an antiparticle of pi plus mu
minus nu mu bar goes to e minus nu e bar plus
nu mu plus nu mu bar; let me call this process
2 p 2. Expected that electron coming out of
this is also a right handed particle. It is
e R which is coming on. Why? Because charge
conjugation operation will not do anything
on the spin or the momentum. It only changes
the type of the particle from particle to
antiparticle and antiparticle to particle.
So, if it is the positron with right handed
or the spin aligned along the momentum that
is coming out in process 1. If charge conjugation
is a good symmetry that nothing changes under
charge conjugation, then we expect this electron
coming out in process 2 to be also right handed.
What is observed experimentally is that, electron
is always, but left handed. This is that charge
conjugation is violated c violation.
%%%%
So, there are other examples also, enough
example lot of examples in weak interactions,
where one can see that the charge conjugation
is not a good symmetry which means that the
different processes which are related by charge
conjugation do not happen at the same rate
at least or some of them do not happen at
all like in this case.
But now, let us look at this same set of positron
and electron coming out. So, e plus under
charge conjugation is expected to go to, e
minus with the same momentum and spin, but
now let me apply parity to it and that should
give me opposite momentum, but the spin orientation
the same particle remains the same what is
that? It is an electron with spin oriented
opposite to the momentum.
So, start with its a right handed positron,
under charge conjugation goes to a right handed
electron, under parity goes to a left handed
electron and that is fine that is observed.
So, if in the combined charge conjugation
and parity transformation, it seems to be
alright. So, combined C P is a good, when
I say good symmetry, it means that processes
interaction which conserves the C P. This
is what we mean by C P is conserved or C P
is a good symmetry C P is conserved. So, most
of the weak interactions, conserve charge
conjugation and parity together.
But then later on it is found that C P is
also violated. When I say C P, it is a combined
charge conjugation and parity symmetry is
also violated in weak interactions. To understand
this, we will need to discuss some more aspects
of parity and charge conjugation and we will
do that in the next discussion.
