Physics 535 lecture notes: - 8 Sep 27th, 2007
Homework: Griffiths: 4.8, 4.10, 4.11, 4.29
1) Review spin, orbital and total angular momentum
Many symmetries that we observe in nature are associated with conservation laws.
Often these kinds of symmetries are categorized by group theory.
Spin conservation comes from the symmetry between spin up and spin down particles.
An interaction with one spin up and one spin down particle would be no different if the
spins are switched. In group theory it can be represented by the SU(2) group.
Spin is integral in particle physics. All particles that we deal with will have spin quantum
numbers and conservation of spin will effect what sorts of interactions are possible and
with what probability possible interactions occur. Therefore, spin will be included in the
wave functions used in particle physics calculations.
Orbital angular momentum and spin angular momentum are also connected in that they
stem from very similar groups and can be combined into total angular momentum just as
in the classical systems. Composite particles will be in states of spin and angular
momentum. Excited states of angular momentum have enough additional energy to be
considered as separate particles.
A given particle will be in states of spin, orbital angular and total angular momentum.
These states are described by quantum numbers for the angular momentum s, l and j and
the projection of the angular momentum on the z axis ms, ml and mj respectively. The
projection quantum number runs from negative to positive the associated angular
momentum quantum number in integer steps.
These quantum numbers can be extracted via operators. For instance Sz and S2 which
yield quantum numbers of s(s+1) 2 and sl respectively.
To add angular momentums:
1) The z components
 simplyadd m=m1+m2
2) The j(or l,s) components may be aligned j=j1+j2, opposite, j=|j1-j2| or anywhere
in between, j=|j1-j2|,|j1-j2|+1…j1+j2.
Angular momentums and angular momentum projections will be conserved. However,
it’s usually the total that is conserved and there can be probabilities involved since you
there might be various possible combinations in the sum each with it’s own probabilities.
The coefficients that determine the probability of each state are called the ClebschGordan coefficients and can be looked up in the particle databook. They can also be
derived by starting with the top state, which is trivial, using step down operators.
In the case of combining quarks.
|s,m>=|s1,m1>|s2,m2>
|1,1>=|1/2,1/2>|1/2,1/2>
|1,0>=(1/2)(|1/2,-1/2>|1/2,1/2>+|1/2,-1/2>|1/2,1/2>)
|1,-1>=|1/2,-1/2>|1/2,-1/2>
vector mesons – symmetric wave function – example s=1 0 m = 776 MeV/c2
|0,0>=(1/2)(|1/2,-1/2>|1/2,1/2>-|1/2,-1/2>|1/2,1/2>)
pseudoscalar mesons – antisymmetric wavefunction ,example s=0 0 m = 136 MeV/c2
Can also add orbital angular momentum. Example s=0 l=1 B m = 1229 MeV/c2
For the pion since the two quarks are in a ms=0 state they can decays to spin 1 photons,
but the spins have to be oppositely aligned.
Similarly the rho can decay into two pions if in the ms=0 state. This is a strong decay so
it dominates. If not in a ms=0(spins not aligned) the decay can still happen. However,
there is going to be an angular distribution involved. Decay is favored along the axis
where the ms=0. You end up with pions with relative l=1 angular momentum to
conserve total angular momentum.
2) Isospin
Another such spin like conservation principle comes from the similarity between down
and up quarks. They both have approximately the same mass and both interact with the
strong force the same way! Therefore there should be a conserved quantity, isospin, that
will govern what sort of interactions are allowed and at what probabilities. Isospin
conservation was noticed before it was even understood that the proton and neutron are
made up of quarks from just noting that the proton and neutron were very similar and
treating them as two of the same type of particle with different isospins. This can even be
extended to strange particles since the strange mass is not that different. However, the
predictions start to be slightly less accurate.
Isospin:
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
For the proton and neutron |I,I3>
P=|1/2,1/2>
N=|1/2,-1/2>
and for the pions
Pi+=|1,1>
Pi0=|1,0>
Pi-=|1,-1>
Example: Consider the pion and nucleon colliding via the strong force. At first glance
these all such processes happen at the same rate, especially for the elastic processes. To
see the isospin effects we need a process that takes place through a eigenstate of the
combined isospin quantum number.
For pi+p: pi+p -> ++ -> pi+p
For pi-p: pi-p -> 0 -> pi-p
or : pi-p -> 0 -> pi0n
pi+p=|1,1>|1/2,1/2>= ++ =|3/2,3/2>
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>
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. This is representative of the quarks involved in that in the later case there are two
udd combinations that could be made and the probability has to be divided up between
them. However there is a charge exchange process pi-p to pi0n that happens twice as
often so total rate is only 3 times as large.
pi0n=|1,0>|1/2>|-1/2>=(2/3)|3/2,-1/2> + (1/3)|1/2,-1/2>
You get a factor of 1 from the pi-p to delta and 2 from the delta to pi0n.
3) Parity
Parity symmetry, P, is the inversion of all coordinates through the axis. It can be
simplified for conceptual purposes to the refection in a mirror by removing one of the
coordinates. 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 Partity operator and identity operator. This is unlike the SU(2) and
SO(3) groups which contain an infinite number of rotations. The eigenstates of the parity
have just two eigenvalues, +1 and -1.interactions that respect parity must conserve parity,
where parity is accessed in a multiplicative way. 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 must conserve parity, where total parity is the product of
the parity of the individual particles involved.
Spins are altered by parity operation. The direction of the spin will be reversed by the
parity operator.
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. There is an additional factor
for particles with orbital angular momentum of (-1)l, since the wave functions are being
inverted.
Mesons in the l=0, j=0 state with antiparralles spins have parity -1 and are called
pseudoscalars
This differentiates them from a true scalar particle such as the Higgs.
Mesons in the l=0, j=1 state with parallel spins have parity -1 are vectors – they have the
same spin and parity as the photon.
Mesons in the l=1 states are pseudovectors,
Baryons have parity 1 while the anti-baryons have negative parity.
Example: pi0 -> 
Spin conservation spin 0 pion decays to two spin 1 photon.
The spins of the photons must be anti parallel to conserve spin.
Partiy 1: If both photon had spin oriented along their direction of motion (helicity 1) 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. This is also
true of the spin 1 parity -1 rho, which can decay to two spin 0 parity -1 pions. The rho
can be in various states of spin alignment, or polarization, and the decay will have
specific angular distributions depending on the initial polarization state. We learned
about this in the spin discussion as well.
A clearer example is the spin 0 parity -1 eta, which decays to three three spin 0 parity -1
pions, but not to two, with no intrinsic spin to monitor. -1 = -1*-1*-1
3) Parity Violation
Parity is maximally violated in weak decays. For instance, in cobalt 60 the electron from
beta decay is emitted preferentially in direction of the nuclear spin but not in the opposite
direction as in its mirror image process.
Furthermore in pion decays to muons and anti-neutrinos the spin on of the muons is
oriented with its direction of flight so therefore the antineutrino must also have it’s spin
oriented along it’s direction of flight. The antineutrino is said to be right-handed. This
means that all neutrinos must be left-handed or have spin oriented opposite they’re
direction of flight. There are no right-handed neutrinos and thus for this interaction to
even happen there must be 100% parity violation.
Note that the decay to elections is suppressed. The antineutrino is in the right-handed
state. The right-handed antineutrino and the right-handed positron couples to the weak
force and the similarly the left-handed neutrino and electron couple to the weak force.
However, the muon being heavier and moving more slowly has more of a chance of
being force into the “wrong” state, actually being in the correct state in the frame where
the weak interaction took place. This is called helicity suppression.
4) 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 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.
Example: pi0 -> 
+1 = -1*-1
Note C ignores the l angular momentum between the photons.
Example eta
Eta -> pi0pi0pi0
+1 = +1*+1*+1
Example rho
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.
In addition you can form charge conjugation interaction 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.
5) Charge conjugation violation
The weak force clearly violates charge conjugation. Charge conjugating an interaction
with a left-handed neutrino will give a left-handed anti-neutrion which does not exist.
Though in most cases the weak force does obey the combination of charge conjugation
and party, CP.
Next Time G parity CP