Continuing our discussion on weak interactions,
we discussed the fact that weak interactions
violate parity. They also violate charge conjugation
symmetry and the combined symmetry of charge
conjugation and parity is conserved in most
of the; in most of the weak interactions.
Now let us look at it little; in a little
more details. Let us look at the parity of
quantum number how we can assign the parity
to the wave functions.
Quantum mechanically, wave function let us
say r theta phi; let us consider the case
where this can be separated into the radial
part and the angular parts; the angular part
is called the spherical harmonics like in
the case of hydrogen atom. What does this;
what does parity do to this? r theta phi the
polar coordinates under parity goes to r pi
minus theta and pi plus phi. How do we see
that?
Let us look at the theta r is just the magnitude
of the position vector which remains the same
whether it is in one direction or opposite
to that. So, parity will not do anything on
that that is scalar quantity. Now theta let
us take the y z plane and consider the position
vector lying in that. In that case this is
your theta the polar angle and under parity
it will go to minus r and this angle is now
theta and what is the polar angle of minus
r.
It is the angle measured from z axis which
is pi minus theta. It is clear. So, under
parity theta goes to pi minus theta. For the
case of azimuthal angle, let us go to the
x y plane. So, it is the x y plane, in the
x y plane projection of the position vector
let me call that rho which is equal to r sin
theta cos phi x plus sin phi y as you know
let us say it makes an angle in this particular
situation a phi with the x axis, under parity
it will go to the opposite direction with
phi here and what is the angle that minus
rho makes with x axis it is pi plus phi.
So, that is what you have here. And what about
Y l m theta phi? How does it go? To what does
it go into under parity?
Just note that this is proportional to Legendre
polynomial; associated Legendre polynomial
as a function of cos theta and exponential
i m phi. These are the phi and theta dependence
of the spherical polar coordinates; spherical
harmonics Y l m. And the azimuthal angle under
parity goes to pi plus phi and e power i m
phi will go to e power i m phi plus pi under
parity of course, which is equal to e power
i m; let me write it as i pi m e power i m
phi, which is equal to minus 1; e power i
pi is minus 1; power m e power i m phi. There
is a factor of minus 1 power m that the e
power i m factor i m phi factor picks up for
the azimuthal part of the wave function picks
up.
What about the polar angle part under parity,
it will go to P l m with cos theta; cosine
of pi minus theta now and without going into
analyzing it we will just give you the result
it is equal to minus 1 over l plus m. You
have to just look at how the Legendre are
associated Legendre polynomials behave under
theta are going to pi minus theta. P l m cos
theta. And together, Y l m theta phi under
parity goes to Y l m pi minus theta pi plus
phi is equal to minus 1 power l plus m plus
m l plus 2 m, minus 1 power 2m is always positive.
So, we do not have to worry about it, so it
is minus 1 power l Y l m theta phi.
So, this factor decides whether the total
wave function is even under parity or odd
under parity. If it is even under parity;
if l is odd then it is an odd function of
parity immersion under parity if it is even
if l is even then it is even under parity.
Now, assumption to start with that strong
interactions and electromagnetic interactions
conserve parity. This need to be checked experimentally
and experimentally we have not seen any violation,
Parity violation in strong interaction or
electromagnetic interaction. But for this
we need to assign intrinsic parities 
to various different particles. When you talk
about intrinsic parity, we do not have any
wave function like this that we can and orbital
angular momentum that we can think about.
The particle is a particle which has maximally
it can have a spin for this thing.
So, without going into how exactly that comes
about there is the question that will ask
is looking at various different reactions,
is it possible to assign intrinsic parities
to particles, different particles consistently,
so that the strong interaction and electromagnetic
interactions conserve parity. Answer is yes
this can be done consistently. So, what we
had to do is, to say psi the particle wave
function 
how does it undergo, how does it change under
parity, it can either pick up a plus sign
or a minus sign. So, this is called even parity
and this is the odd parity.
So, even and odd parity for these particles
can be assigned consistently. So, how do we
do that convention.
To start with spin half particles like proton,
neutron, electron, muon, etcetera neutrinos
etcetera all have positive parity. Anti particles
spin half, like anti proton, anti neutron,
positron, mu plus etcetera have negative parity
or I should I can write as even parity and
odd parity. Other particles so, consistently
considering reactions.
For example, pions, pi plus pi minus pi 0
all have odd parity. We will not go into how
one determines this or how one actually assigned
this there is a consistent way of doing it.
This has been done.
Now, let us look at one particular reaction
observation of K 0 decay. It was observed
that it could decay K into 2 pi ons; pi zeroes
or pi plus plus. It actually mostly decays
into this. But there was a small part of this
decaying into pi plus pi minus pi 0. Something
like 1 in 10 power minus 7 decays to 3 pions.
Now if pions has odd parity, parity minus
1 the final state of the first reaction decay
gives together parity plus minus into minus
the second reaction it is minus into minus
into minus 3 times and that for together it
is minus 1.
So, the K 0 decays is to final stage with
positive parity mostly, but also there is
a small decay probability for it to decay
into negative parity final state; which in
other ways says that K 0 is a particle which
cannot be assigned a proper parity to the
one or the other way to interpret it is that,
parity is not conserved in such decays. It
can be gained do positive or negative verities
there was also observation of similar; the
particle K plus going into pi plus pi 0 or
going into pi plus pi minus pi plus. 3 pions
and 2 pions.
Again, one final state is, even parity the
other one is odd parity. In fact, historically
speaking initially these particles were thought
of to be 2 different particles, because parity
was violated in the final state. Or, they
decay into 2 different final states with 2
different final parities. But then it was
suggested by T D Lee and C N Yang that parity
is actually parity could be thought of violated
in weak decays like this. And then further
on it was experimentally found as we saw earlier
that parity is indeed violated in C P variation
and this fetched Nobel prize for Lee and Yang
in 1957. Combined C P variation is also observed
in the weakly case weak interactions.
So, for example, if you consider again K meson
system, and more recently B meson system.
Both of these so, C P violation.
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There are dedicated experiments; there had
been dedicated experiments running for a long
time; many decades; investigating the C P
violation in the K meson system. Experiments
accelerators were tuned to electron positron
colliders for example, could be tuned to energies
which will abundantly produce K mesons. Different
excited states of K mesons also. And similarly
one could tune this collider energies to B
meson masses and you can produce these masons
abundantly.
They will decay in different channels looking
at their decay properties. Decay channels
one can study the C P properties and then
it has been found that C P violation is a
feature of K meson system, and B meson system.
C P violation, unlike the parity variation
is somewhat smaller. When I say smaller, what
I mean is the following. Consider the case
of beta decay; in beta decay electrons coming
out are always observed to be left handed
there is no right handed electron which is
coming, which is observed in the beta decay.
Whereas, parity conservation or parity transformation
would take left handed electron to right handed
electron. So, if parity was a good symmetry
or if things were symmetric under parity then
equal number of these 2 left handed and right
handed electrons would have been still in
beta decay.
So, non-observation of right handed electrons
or left handed positrons in beta decay says
that parity is violated and maximally. That
is there is no right handed electron at all
observed. But in the case of C P variation
when we consider 2 processes which; one of
which can be the C P transformed state of;
case of the other one; experimentally we see
both of these. But the probability for both
of these to occur are not the same. That is
the kind of situation that we see in experiments.
For example, if something happens at hundred
percent some decay particular decay happens
at the rate of say hundred in a particular
time interval for a particular sample. A sample
which can be obtained by C P transformation
decays to the C P conjugate of that final
stage not 100 times, but something like say
90.
So, there is an asymmetry between the C P
conjugated state and the original state. Or
the interaction; a particular interaction
and the C P conjugated interaction. So, this
asymmetry if it is not equal to 0 then we
can say that C P is violated and it is seen
that it is not it is never like 50 percent
or 100 percent it is always something like
a few percent or less than a few percent that
way.
We will see now the theoretical developments
and then eventually we will conclude that
whatever we have observed the C P violation
as well as parity violation as well as C P
violation, whatever we have observed so, far
can be accommodated in our theoretical framework.
So, let us look at the theory in a theoretical
development again.
We did talk about the theory of beta decay
which was proposed by Fermi. So, Fermi’s
theory was that you consider the invariant
amplitude like two currents interacting with
each other. u e bar gamma mu v nu bar and.
So, this is one current which is the electron
and the neutrino current and the other one
is neutron, proton, neutron going into proton
so, that is u p bar gamma mu u n.
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This cannot have parity violation. It is because,
it is a vector current vector current interaction.
To introduce parity violation one can look
at the interaction in a similar fashion, with
a slight change. The current is now u e bar
gamma mu 1 minus gamma 5 by 2 v nu bar u p
bar gamma mu 1 minus gamma 5 by 2 u n. So,
Fermi’s theory is not; I mean it is modified,
but it is not modified drastically. So, instead
of a vector current vector current interaction
now we have a mixture of a vector and an axial
vector current.
So, let me write down one of these currents
say the J e nu electron neutrino current as
u e bar gamma mu 1 minus gamma 5 by 2 v nu
bar. This can be written as 1 over 2 u e bar
gamma mu v nu bar, minus u e bar or 1 over
2 into u e bar gamma mu gamma 5 v nu bar.
First one is a vector current, second one
is an axial vector current. So, there is a
vector minus axial vector we call it V mu
A mu.
So, this basically the V minus A structure,
the V minus A structure of the current accommodates
parity violation. How? Because, under parity
we will transform in a slightly different
way compared to A mu. So, there would not
be any particular way to particular parity
transformation property for V minus A. So,
that indirectly that this theoretical structure
will when you compute the cross section or
the decay probabilities you will see that
this will lead to violation of parity. That
this particular structure and the parity conjugated
structure will not give the same decay probability
or transition amplitude. C P variation on
the other hand requires slightly more involved
analysis, we will come to that somewhat later.
Now, let us look at the weak interaction further.
n going to p, but we know n has one d quark
another d quark and another u quark and proton
has one u quark another d quark a d quark
and another u quark. So, one can add the quark
level, think about a d quark converting into
a u quark, and then electron and antineutrino
are created. We want to understand it in a
way similar to the case of electromagnetic
interaction that we had considered. Meaning
we want to actually connect these 2 coins
by some propagator something taking this information
from this conversion of the d quark to u quark
that information goes to that is connecting
the electron, neutrino current.
This is again thought of as if happening through
the exchange of something; propagation of
something; similar to the photon propagation
in the case of electromagnetic interaction.
We denote this by W minus. W for some weak
particle or weak boson. Why boson? Because
it turns out that this is a vector particle
which has spin one. So, spin one integer;
spin boson, it is a similar to the interpretation
of photon as the quantum or the representation
of the electromagnetic field here we consider
w particle as the representation as the as
the quantum picture of the weak field.
So, w is a particle when you get; I mean which
is the quantized form of the weak field. And
we call this w. This need to have a charge
because electric charge is changed when you
convert a d quark to a u quark. So, to start
with d quark has minus 1 by 3 electric charge
this converted into, at this first vertex
here, converted into a u quark with 2 by 3
electric charge plus w goes on with minus
1 electric charge. Sum it up, we will see
that right hand side has minus 1 plus 2 by
3. So, this is one bit which is equal to minus
1 by 3 charge.
Similarly, when w goes to electron and antineutrino,
anti neutrino does not have any charge w has
minus charge bringing into that vertex which
is taken away by the electron. ok. So, that
is w minus going into e minus plus antineutrino.
So, this is roughly the kind of split of this
process. There are 2 currents associated with
these, one associated with the conversion
of d to u and production of w and this w is
then decays to e minus and nu bar. So, w minus
is the weak gauge boson.
We can look at other processes. It is possible
to have a Beta plus decay also,
which is essentially p going to n plus positron
plus neutrino and this can be thought of as
conversion of a u quark into a d quark plus
electron plus a positron plus anti neutrino.
So, this is picturized as u quark into d quark
with the w plus now, because u has a charged
2 by 3 go into d with charge minus 1 by 3
plus w plus 1. So, that charges are added
up properly, correctly. This will go into
positron, when a positron goes out we put
the arrow in the opposite direction. So, it
is e plus and nu e.
So, positron takes away the charge which is
the w plus brings to that second vertex. So,
w plus goes to e plus plus nu e. So, we have
another gauge boson w plus. So, weak; there
are 2 weak gauge bosons now. W plus and w
minus it turns out that one is the antiparticle
of the other.
Then you will see that; for the theory to
be consistent when we discuss the theoretical
model it will be clear we need another neutral
weak gauge boson. This was in fact a prediction
of the theoretical framework; the standard
model we will come to that in a little while,
when we actually discuss the formal framework.
So, we have here w plus, w minus and a z 0,
3 weak gauge bosons, right. One is a neutral
gauge boson the other 2 are charged.
Let us consider some other examples of Weak
processes.
We have been talking about the muon, which
has which is like electron, but massive, more
massive. It can decay into electron and electron
type of anti neutrino and muon type of neutrino.
In all these we will consider the lepton number
being conserved lepton; whenever when we say
lepton number, muon type particles will have
the lepton number called muon lepton number,
which is different from electron type lepton
number. Individually they are conserved.
So, let us see lepton number electron type
it is equal to 0 here, 1 here, minus 1 here,
0 here. So, total in the initial and final
it is 0. L mu similarly is 1 here, 0 here,
0 here, 1 here. So, theoretically that is
why we do not have muon going to e minus nu
e bar and nu mu bar. And similarly, in the
beta decay it is the anti neutrino which is
coming out not the neutrino.
Experimentally one can actually determine
whether one particular thing is an antineutrino
or an neutrino, by for example, considering
it is reaction with other particles. For example,
the antineutrino which is coming out from
beta decay will not give a reaction like say
for example so, this is one week process;
diagrammatically we can represent this as
a muon going into nu mu and emitting a w mu
which will go into electron and electron type
anti neutrino.
So, as we said in the case of electromagnetic
(read as weak) interactions we do not mix
the type of the particles. So, with muon only
muon type neutrino connects. Similarly, with
electron only the electron type neutrino or
antineutrino connected. And there are no;
the fermion lines are continuous from one
end to the other. So, this is one way to understand
this. And once you know the rules with this;
associated with this like in the case of the
Q E D we had written down the Feynman rules
for Q E D.
Similarly, in the case of weak interactions
also we will write down the Feynman rules.
Once you get the Feynman rules, then from
the diagram one will be able to compute the;
write down their invariant amplitude for this
process and subsequently compute the probability
for it to happen or the decay width.
We can think about other processes which also
involve weak interaction we had seen this
process we had seen this in electromagnetic
interaction. There we had written down or
drawn the diagram e plus e minus goes to mu
plus mu minus through the exchange of photon
represented by gamma. It can also happen through
the exchange of weak gauge bosons. Same final
states mu minus and mu plus in the final shape
same initial state, but the intermediate propagator
is now different. It is the z 0 that can carry
this information. This is the weak neutral
current interaction. The neutral; the propagator
is neutral here and there is no current and
the charge change along the current. Charge
remains the same all along the current.
Now, so, when we consider e plus e minus go
into mu plus mu minus we should be considering
both of these. Indeed the electromagnetic
interaction here dominates compared to the
weak interactions, but this statement is;
statement depends on at what energies the
electron and positron are colliding each other.
So, many times this becomes very important.
For example, the neutral gauge bosons z 0
has a mass of 91, around 91 GeV.
So, if you tune the energy of the electron
and positron center of mass energy to 91 GeV,
it turns out that most of this reaction will
go through the exchange of z 0 that there
will be a production of z 0 which subsequently
decays is to mu plus and mu minus. We will
try to look at some of these processes later.
But here we will just try to indicate that
along with the electromagnetic interaction
in many of these reactions we will have to
also consider the weak interactions.
In fact, the neutral gauge boson and the neutral
currents like what we have just now written
down, currents, were discovered by the framework
of standard model which we will come to later
and it was subsequently discovered and that
was a big triumph of the model and gave us
a lot of confidence in our analysis. There
are also other processes like e plus e minus
going to 2 neutrinos. A neutrino and an antineutrino
of this type.
There is no way that this can go through using
electromagnetic interaction, in electromagnetic
interactions. But in weak interaction it is
possible to have production of z boson or
exchange of z boson between nu e and n u e
bar. Well rather than exchange here one can
think about production of z 0 and decaying
into; which decays to nu nu. Similarly, e
plus e minus go into nu plus nu minus mu plus
mu minus through photon can also be thought
of as annihilation of electron and positron
and creation of a photon which then pair creates
a mu and; mu minus and mu plus.
So, this is possible, but as I said it is
not possible to how e minus e plus annihilating
to a photon pair creating a neutrino and antineutrino.
This is because photons do not interact with
neutral fermions; neutral particles photons
only; photons onl; interact only with charged
particles at least directly like this. We
will see that when we consider the higher
order perturbation there are corrections to
this statement. Photons can interact with
neutral particles through higher order processes
higher order perturbation process.
This is not allowed. Incidentally this can
also go to the same initial state electron
and positron can also go to mu type neutrino
and mu type antineutrino without being in
conflict with any of those Lepton number conservation.
So, because it is f 0 which is coupling to
any type of neutrinos, but again you cannot
have nu e and nu mu bar together it is either
nu e nu nu e bar or nu mu nu mu bar or even
nu tau nu tau bar.
We will stop our discussion here. And then,
further go on with our discussion on the weak
interactions in the next class.
