DAW

1

• DAW 3 is due this afternoon-ish.
• I will be away on Friday.
• Lecture will be given by Dr. Ben Speakman and Dr.
  Alex Smith Introduction to Neutrinos.
• Test 2 take-home. Out on Friday, April 18,
  back on Monday, April 21.
• 4 problems, potentially covering anything since
  Test 1.

2
SU(2) ? SU(3)
Isospin
Quarks

• n 2, n2 1 3 generators (2?2 linearly
  independent, traceless Hermitian matrices)

• n 3, n2 1 8 generators (3?3 linearly
  independent, traceless Hermitian matrices)

Pauli Matrices
Gell-Mann Matrices
One is diagonal ? one quantum number I3
Two are diagonal ? two quantum numbers I3, Y B
S
More quarks? SU(3) ? SU(4) ? SU(5) ?
3
Assembling Particles from Quarks
Meson quark antiquark
u and d form an isodoublet
4
Constructing the meson multiplet in SU(3)
Expect an octet and a singlet. If SU(3) were a
perfect symmetry, all particles in a multiplet
would have the same mass. (SU(3) is a badly
broken symmetry.)
5
Building mesons

• Quark-antiquark bound state (think H or
  positronium) with the strong interaction doing
  the binding.
• Different sets of mesons are assembled from the
  basic quark-antiquark combinations with specific
  values of the spin and orbital angular momenta.
• L 0 gives the two lowest-mass multiplets

Pseudoscalar Mesons (??)
Vector Mesons (??)
6
Quark-composition assignment is very
straightforward for most states, except for the
neutral octet and singlet members, which can mix
in hard-to-predict ways. More on this later, as
well as how the quark model can be used to
predict/understand meson properties like masses.

• First observations
• The s quark is clearly more massive than u and d.
• V mesons more massive than PS hyperfine
  splitting (spin-spin effect).

7
Add the fourth flavor, charm
SU(3) ? SU(4)
(Anyone care to sketch the five-flavor
multiplets?)
8
Baryon quark quark quark
Assembling Particles from Quarks
Total of combinations is 27 3 quarks, 3
possible flavors for each
9
Baryon Decuplet JP (3/2)
The 10 are grouped together because of symmetry
in both flavor and spin under interchange of two
quarks.
Corner states are ddd, uuu and sss, with
automatic symmetry others are constructed, e.g.
the udd (?0) state is
10
Baryon Octet JP (1/2)
The 8 are grouped together because of symmetry
under simultaneous exchange of flavor and spin
for any quark pair (mixed symmetry).
11
The Need for Color

• The lowest-mass J 3/2 baryons are assumed (and
  later shown) to be states of zero orbital angular
  momentum. This suggests a spatially symmetric
  ground state.
• J 3/2 assembly of three spin-1/2 quarks ?
  symmetric spin state (? u?u?u?).
• ? the baryon decuplet wave functions are
  symmetric in space, spin and flavor even though,
  as fermions (Pauli principle) they must be
  antisymmetric overall identical fermions cannot
  exist in the same quantum state.
• Another factor in the wave function? Perhaps
  there is a hidden degree of freedom.
• Other evidence?

12
The Need for Color Problems!
1

• Theory predicts that (except near flavor
  threshold) R should equal the sum of the squares
  of all accessible quark charges. Too small by ?3!

2

• Very similarly, theory predicts a total decay
  rate for ?0 meson that is too small by ?3.

13
Quarks with Color

• Assign a three-valued quantum number to the
  quarks color.
• Solves all three problems! Baryon wave functions
  symmetric in all other variables are
  antisymmetric in color. In QED calculations each
  quark flavor counts as three types of quark - one
  for each color.
• Color is the charge of the strong interaction
  quantum chromodynamics (QCD) is the theory.
• Uses the metaphor of color theory. The color
  charges of the quarks are hidden - observable
  hadrons are colorless combinations

Baryon
Meson
Antibaryon
14
Why do leptons and quarks decay?
Leptons
Quarks

• The weak interaction provides a flavor-changing
  current

• All decays within doublets.
• Conservation of lepton is absolute (SM!).

• Decays within and btw doublets.
• Weak interaction does not conserve flavor quantum
  s.