Lecture 11: Feb 28th 2012
Reading: Griffiths chapter 4,6 (skipping 5)
Homework: 4.29, 4.37, 6.2, 6.3 Thursday Mar 1st
1) Isospin
Isospin is a symmetry related to flavor and the strong interaction. The proton and the
neutron, or the up, down, and to some degree the strange quarks interact the in the same
way with the strong interaction. They have the same possible colors and the same
probability of interaction with the gluon. Therefore, they should have identical
interactions but conserving weak flavor. This symmetry is approximate because the u
and d quark and certainly the strange quark have different masses. Thus for instance pair
production interactions are different.
Define I and I3
I is going to lead to 2I+1 states delineated by I3=Q-1/2(A+S) which goes from –I to I in
integer steps
Q = charge, A = baryon number and S=strangeness
Note I3 formula only works for mesons and baryons.
This formula essentially conserves the total flavor projection in a similar way to to how
spin projection keeps track of the total spin projection. Quark are consdired two aspects
of the same particle to the strong force just as spin up and spin down particles are
considered two aspects of the same particle to the electromagnetic force.
Multiplets of isospin particles. Found by adding together the isospin of the individual
quarks as angular momentums or spins are combined.
Pions |I,I3> I = ½ + ½ = 1, I3 = -1, 0, 1
Pi+: u d :
|1,1> = |½ ,½>|½ ,½>
Pi0: u u or d d :|1,0> = (1/2)(|½ ,½>|½ ,-½> + |½ ,½>|½ ,-½>)
Pi-= d u :
|1,-1> = |½ ,-½>|½ ,-½>
There is also singlet state.
eta : u u or d d :|0,0> = (1/2)(|½ ,½>|½ ,-½> - |½ ,½>|½ ,-½>)
Protons and neutrons
|I,I3>: I = ½ + ½ - ½, I3 = -½,½.
P=|1/2,1/2>
N=|1/2,-1/2>
Delta particles
I = ½ + ½ + ½ = 3/2, I3 = -3/2, -1/2, 1/2, 3/2.
++: uuu: |3/2,3/2>
+: uud |3/2,1/2>
0: udd, |3/2,-1/2>
-: ddd |3/2,-3/2>
Note that the two middle particles can be thought of as total isospin excited states of the
proton and neutron. In this case, as with spin and angular momentum, when we do
scattering experiments to determine the isospin we can only measure the isospin
projection. However we can observe a consequence of being in the I=3/2 state in that the
mass of the delta baryons is larger.
Isospin helps us to figure out what combinations of quarks are allowed, which will have
similar properties relative to the strong force, and whether they are expected to be more
or less massive. These particles can also be in ground or excited spin and angular
momentum states. To classify a particle and understand it’s interactions all these
quantum numbers need to be determined.
Since isospin is conserved in strong interaction it will have dynamical implication on
strong scattering interactions. Isospin projection will have to be conserved which can
decrease the probability of certain interactions occurring.
Example: Consider the pion and nucleon colliding via the strong force. At first glance
these all such processes happen at the same rate. To see the isospin effects we need a
process that takes place through a definite eigenstate of the isospin quantum number, An
example is pion nucleon at the CM energy of the .
Consider
pi+p: pi+p -> ++ -> pi+p
pi-p: pi-p -> 0 -> pi-p
To determine the probabilities we need to express the pi p eigenstates as a combination of
the  and p/n eigenstates
pi+p=|1,1>|1/2,1/2>= |3/2,3/2> = ++
pi+p can only go the the ++. This expresses conservation of isospin projection.
However, isospin projection is dependent on the quark content so it more fundamentally
expresses conservation of charge(and strangeness – quark flavor).
pi-p=|1,-1>|1/2>,1/2>=(1/3)|3/2,-1/2> - (2/3)|1/2,-1/2>
0=|3/2,-1/2>
p=>|1/2>,1/2>
However, at any energy the transition through a proton is forbidden by energy and
momentum conservation.
The probability to transition through to a 0 is 1/3 (reduced because the proton transition
is not allowed).
If this reaction happens at the energy of the Delta particles then the pi+p process has an
amplitude three times as larger than the pi-p process at the delta particle energy. The
cross section actually goes as the square of the amplitude so this process happens 9 times
as often - you go from pi+p to the delta particle and then back again. Note when you
observe unusual dynamical features in interactions it is often a consequence of some type
of structure or symmetry, in this case the structure and symmetries of the quarks with
regards to the strong force.
2) Parity
Parity symmetry, P, is the inversion of all coordinates through the axis. The
electromagnetic and strong interactions conserve parity. An interaction transformed
using the parity operation happens at the same rate as the initial interaction.
This is what we call a discrete transformation and can be represented by a finite group.
In this case the Parity operator and identity operator. This is unlike the SU(2) and SO(3)
groups which contain an infinite number of rotations. As with the other cases when there
is a transformation of the system that leaves the probabilities of interaction unchanged
there is a conserved quantity and eigenstates with quantum numbers of that property.
The eigenstates of the parity have just two eigenvalues, +1 and -1. This could be deduced
form the fact that clearly P2 = I = 1. For this to work +1 and -1 are the only possible
eigenvalues. Interactions that respect parity conserve parity, where parity is accessed in a
multiplicative way. i.e. interactions that respect parity conserve total parity, where total
parity is the product of the parity of the individual particles involved.
Note, any operation P such that P2 = I is a valid representation of the parity group.
Therefore, in addition to inversion of all three coordinates, inversion or one or the
equivalent inversion of two coordinates to is also a parity inversion and a parity
respecting interaction will also be invariant under all of these transformations. We will
often use the idea of a parity mirror to investigate parity ideas.
The parity of the various particles can be determined as follows. The parity is the
composite of its constituents. Quarks have positive parity and antiquarks negative parity.
Photon and other vector particles also have negative parity.
Angular momentum will be altered by the parity operation.
There is an additional parity factor for particles with orbital angular momentum of (-1)l,
since the wave functions are being inverted.
Mesons in the s= 0, l=0, j=0 state with anti-parallel spins have parity -1 and are called
pseudoscalars. The parity is simply set by the by the quark content. This differentiates
them from a true scalar particle such as the Higgs, which has parity 1.
Mesons in the s=1, l=0, j=1 state with parallel spins also have parity -1 are vectors. Again
the parity is determined by the quark content. They have the same intrinsic spin as the
photon and the same parity as the photon.
Mesons with spins anti-parallel in the s=0, l=1, j=1 are pseudo vectors or axial vectors
and have positive parity. There is an extra factor of (-1)l and thus mesons in general have
parity: -1*(-1)l
Baryons have parity 1 while the anti-baryons have negative parity set by their quark
content.
Mesons have parity -1*(-1)l
Baryons have parity 1*(-1)l
Anti-baryons have parity -1*(-1)l
The photon, a vector particle, has parity -1
The scalar Higgs had parity 1
Consequences:
Example: pi0 -> 
Spin conservation spin 0 pion decays to two spin 1 photons.
The spins of the photons must be anti parallel to conserve spin.
Parity 1: If both photons had spin oriented along their direction of motion (helicity) then
the parity operation will invert them to be oriented opposite to their direction of motion.
These two interactions have to occur at the same rate.
Parity 2: The pion is parity -1 and the photons are parity -1. Hey! This doesn’t conserve
parity. You also have to consider the correlation of angular momentum between the
photons. If they are oppositely polarized there is an additional factor of -1. An l=1 but
ml = 0 state. In other cases the relative angular momentum between the particles will
result in a specific angular distribution for the decay.
Parity 3: A clear example is the spin 0 parity -1 eta, which decays to three spin 0 parity -1
pions, but not to two. With no intrinsic spin or angular momentum to contribute to a final
state angular momentum then only intrinsic parity conservation needs to be checked -1 =
-1*-1*-1
3) Charge conjugation.
Charge conjugation, C, converts each particle into its antiparticle. It does not affect spin
or angular momentum. This symmetry tells us that charge conjugated interactions should
take place at the same rate as the initial interaction.
This is a second discrete symmetry. Only particles that are their own antiparticles are
eigenstates of charge conjugation. Similarly since C has two elements, C and I, and C2=I,
C has eigenvalues +1 and -1. C is conserved as a product as is parity.
The photon has C -1. The mesons that are also their own antiparticles have C (-1)l+s.
Pseudoscalars mesons with s=0 and l=0 have C +1.
Vector mesons with s=1 and l=0 have C -1.
Mesons that are not their own antiparticles are not in eigenstates of the C operator so you
don’t need to check for charge conjugation eigenvalue conservation. Though the charge
conjugated interaction should still take place at the same rate.
Example: pi0 s=0, l=0
pi0 -> 
+1 = -1*-1
Note C ignores the l angular momentum between the photons. The relative angular
momentum is not changed by conjugating the charges.
Example eta s=0, l=0
Eta -> pi0pi0pi0
+1 = +1*+1*+1
Example rho s=1, l=0
Rho -> pi0 pi0
-1  +1*+1
Not allowed by charge conjugation
However rho -> pi+pi- is allowed since pi+ and pi- are not C eigenstates so you don’t
have to conserve C. Note the rho -> pi+pi- decay must have relative angular momentum
l=1 between the pions to conserve total angular momentum and parity.
In addition you can form charge conjugation interactions from combinations of particles
and anti-particles.
ppbar -> pi+pi-pi0
In this interaction the charged pions must have similar energies or it’s not charge
conjugation symmetric.
4) C, P and CP Violation
The weak force violates both parity and charge conjugation.
The historical example is if you measure the electron decay direction in the weak decay
of the 60Co atom where the spin has been aligned upwards the electron is emitted
downward. The mirror image interaction where two coordinates are inverted does not
exists. i.e. if the electron is emitted along z then inverting x and y reverses the spin
vector but not the electron direction so that the electron is emitted along the direction of
the spin.
The electron has spin oriented oppositely to its momentum (helicity) and is classified as a
left-handed particle. The antineutrino has spin oriented along its direction of travel and is
classified as right handed. Similarly, positrons are generally right handed and all
neutrinos are left-handed. Note, electrons and positrons do not have a purely right or left
handed helicity since you can apply a velocity frame transformation that will reverse the
direction of the velocity and thus the helicity. If the neutrinos were mass-less they would
have an absolute helicity since any velocity transformation would still leave them moving
at the speed of light in the same direction. The fact that neutrinos are mass-less actually
breaks the standard model in which only (right) left handed (anti) neutrinos would exist
and interact via the weak force.
Interaction via weak force clearly violate charge conjugation. Charge conjugating an
interaction with a left-handed neutrino will give a left-handed anti-neutrino, which does
not exist.
The weak force carriers have a V-A, vector-axial(or pseudo)vector form. The axial part
of the force carrier has the opposite parity compared typical force carriers and thus
intrinsically flips the parity of the interaction leading to parity violation.
Though in most cases the weak force does obey the combination of charge conjugation
and party, CP. We expected the product of C and P to always be conserved for the weak
force. Charge conjugated weak force diagrams did not exist but the charge conjugated
parity flipped version did exist.
+  + 
happens as often as the charge conjugated and parity inverted
-  - u
Note also that this interaction does not preserve helicity for the muon but helicity is not
necessarily perfectly conserved for massive particle. However, this does lead to a
reduction in the probability of the interaction referred to as helicity suppression. This
suppression is much greater for the decay to electrons, where the electron is moving
closer to the speed of light and is less often in the correct helicity state.
However CP is violated in some common decays.
K  - e+ 
happens 3.3x10-3 more often than
K  + e- u
CP violation was surprising but welcome. Charge conjugation violation would not
explain the matter-anti-matter asymmetry of the universe because though charge
conjugated weak interactions didn’t exist the additionally parity flipped interactions did.
If there were interactions where matter particles decayed more often than antimatter
particles then it might explain the baryon asymmetry of the universe.
Research concentrates in two directions. Direct CP violation matter decays that happen.
more often than an anti-matter decays.
Bs(b s )  K+pi- 39% more than Bs( b s) -> K-pi+
Oscillation processes that convert matter to anti-matter.