1. Isospin
Idea originally introduced in nuclear physics to explain observed symmetry between protons and neutrons (e.g. mirror nuclei have similar strong interaction properties)
Now understood as a consequence of very similar masses of u and d quarks, and conservation of flavour in strong interactions.
(electric and weak charges of u and d are different so do not respect isospin)
Still useful concept in particle physics for analysing relationships amongst different strong interaction processes
Isospin follows same algebra as normal spin:
Isospin doublet:
I = 1/2 , I3 = +1/2, -1/2
Isospin triplet:
I=1, I3=-1,0,+1

2. Isospin Two nucleon system: I = 1, I3=-1,0,+1 triplet
I=0, I3=0 singlet
Now analyse pion-nucleon scattering by considering different combinations of I=1 (pion) and I=1/2 (nucleon).
Same rules as combination of angular momentum (CG coefficients)

3. Isospin Relative cross section of : (1) (2)
no cross term since Isospin is conserved:
ratio of cross sections (1) and (2) depends whether reaction goes through I=3/2 or I=1/2 state - depends on energy.
At energy of 1232 MeV, we have the I =3/2 resonances
and
(1)
(2)

4. Isospin Can extend idea of isospin to strange, charm, bottom mesons,
considering heavy quarks as spectators
Kaons:
hyperons:
Hence we can evaluate relationships amongst different strangeness production processes, in same manner as pion-nucleon scattering
(BUT do NOT use isospin for EM or Weak processes - e.g. strange particle decays !)
I=1/2 doublets
I=0 singlet
I=1 triplet
I=1/2 doublet

5. Charge Conjugation
C is operator that interchanges particles with antiparticles, but with no change of spin or partial coordinates
C is another discrete symmetry (like parity)  another multiplicative quantum number, C
Almost all of physics is invariant under C, with the exception of weak interaction
Only neutral particles are eigenstates of charge conjugation 

6. Charge Conjugation From QED In EM decay C conservation in EM decay:
Experimental limit:
forbidden

7. Charge Conjugation
Under charge conjugation fermion and antifermion can be considered as two states of same f
Overall wavefunction must be antisymmetric as if they are a pair
of identical fermions
has symmetry
To make wavefunction totally antisymmetric,

8. Section VIII - Parity in Weak Interactions

9. Parity Violation in Beta Decay
Parity violation was first observed in the b decay of 60Co nuclei (C.S.Wu et. al. Phys. Rev. 105 (1957) 1413)
Align 60Co nuclei with field and look at direction of emission of electrons
Under parity:
If PARITY is CONSERVED, expect equal numbers of electrons parallel and antiparallel to
J=5 J=4 e- e-

10. Most electrons emitted opposite to direction of field
 PARITY VIOLATION in b DECAY
Polarized
Unpolarized
T = 0.01 K
As 60Co heats up,
thermal excitation
randomises the spins

11. Origin of Parity Violation
SPIN and HELICITY
Consider a free particle of constant momentum, .
Total angular momentum, , is ALWAYS conserved.
The orbital angular momentum, , is perpendicular to
The spin angular momentum, , can be in any direction relative to
Define the sign of the component of spin along the direction of
motion as the
HELICITY
“RIGHT-HANDED”
“LEFT-HANDED”

12. The WEAK interaction distinguishes between LEFT and RIGHT-HANDED states.
The weak interaction couples preferentially to
LEFT-HANDED PARTICLES
and
RIGHT-HANDED ANTIPARTICLES
In the ultra-relativistic (massless) limit, the coupling to RIGHT-HANDED particles vanishes.
i.e. even if RIGHT-HANDED n’s exist – they are unobservable !
60Co experiment: e
60Co
60Ni +
J=5 J=4

13. PARITY is ALWAYS conserved in the STRONG/EM interactions
Parity Violation
The WEAK interaction treats LH and RH states differently and therefore can violate PARITY (i.e. the interaction Hamiltonian does not commute with P ).
PARITY is ALWAYS conserved in the STRONG/EM interactions
Example
PARITY CONSERVED PARITY VIOLATED
Branching fraction = 32% Branching fraction < 0.001%

14. PARITY is USUALLY violated in the WEAK interaction
but NOT ALWAYS !
Example
PARITY VIOLATED PARITY CONSERVED
Branching fraction ~ 21% Branching fraction ~ 6%

15. Section X - Weak Interactions of Quarks

16. Weak Interactions of Quarks
In the Standard Model, the leptonic weak couplings take place within a particular generation:
Natural to expect same pattern for QUARKS, i.e.
Unfortunately, not that simple !!
Example:
The decay suggests a coupling

17. Cabibbo Mixing Angle + + Four-Flavour Quark Mixing
The states which take part in the WEAK interaction are ORTHOGONAL combinations of the states of definite flavour (d, s)
For 4 flavours, {d, u, s and c}, the mixing can be described by a single parameter
 CABIBBO ANGLE (from experiment)
Weak Eigenstates Flavour Eigenstates
Couplings become: + +

18. u u n d d p d u e Example: Nuclear b decay Recall
strength of ud coupling
Hence, expect
It works,
Cabibbo Favoured
Cabibbo Suppressed n d d p d u e
(compare muon decay and
nuclear decay)

19. Example:
coupling  Cabibbo suppressed
Expect
Measure
is DOUBLY Cabibbo suppressed

20. GIM mechanism W- W+ u nm m+ m- d s K0 c nm d s m+ m- W- K0 W+ cosqC
Historical puzzle: had a much lower BR than expected
cosqC W- W+ u nm m+ m-
sinqC d s K0
Glashow, Iliopoulis, Maiani postulated existence of extra quark before the
charm quark was discovered (because of the absence of FCNC) c nm d s m+ m-
cosqC
- sinqC W- K0 W+
Vertex factors enter with opposite overall sign : two diagrams approximately cancel
Known as GIM mechanism

21. With neutral as well as charged currents, one would expect
to have similar rate to m- m+ d s K0 Z0 m+ u s K+ W+ nm
whereas experiments give
Conclude that Flavour Changing Neutral Currents do not exist:
cannot change the quark flavour

22. The Weak NC Vertex
All weak neutral current interactions of quarks can be described by the Z0 boson propagator and the weak vertices:
Z0 NEVER changes type of particle
Z0 NEVER changes quark flavour
Z0 couplings are a MIXTURE of EM and WEAK couplings and therefore depend on Z0
STANDARD MODEL
WEAK NC QUARK
VERTEX
+antiparticles

23. Neutral weak current:
observed
not observed

24. CKM Matrix Cabibbo-Kobayashi-Maskawa Matrix Extend to 3 generations
Weak Eigenstates Flavour Eigenstates
Giving couplings

25. The Weak CC Quark Vertex
All weak charged current quark interactions can be described by the W boson propagator and the weak vertex:
W bosons CHANGE quark flavour
W likes to couple to quarks in the SAME generation, but quark state mixing means that CROSS-GENERATION coupling can occur.
W-Lepton coupling constant gW
W-Quark coupling constant gW VCKM
STANDARD MODEL
WEAK CC
QUARK VERTEX
+antiparticles

26. CKM Matrix VCKM is rotation matrix in 3 dimensions
Unitary 3x3 matrix requires 4 parameters, 3 mixing angles and 1 phase
Elements are complex numbers, not predicted by Standard Model
In practice c2≈1 and c3≈1, hence 2x2 sub-matrix unitary

27. CKM Matrix Alternative (approximate) parameterization (Wolfenstein):
 4 free real parameters
3rd generation almost decoupled:
b lifetime surprisingly high given large mass
bu decays highly suppressed, usual chain bcsu
Parameterization of mixing among charge = -1/3
quarks purely conventional. Could redefine VCKM
and have same physics:

28. Example: gw Vud, Vcs, Vtb O(1) gw Vcd, Vus O(l) gw Vcb, Vts O(l2)
(Log scale)
t d, b u O(l3)
very small