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

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)

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)

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

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 

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

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,

Section VIII - Parity in Weak Interactions

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-

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

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”

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

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%

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

Section X - Weak Interactions of Quarks

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

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: + +

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)

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

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

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

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

Neutral weak current: observed not observed

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

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

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

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:

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