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Quarks
 Experience the strong,
weak, and EM
interactions
 There are anti-quarks as
well
 Quark masses are not welldefined
 Quarks carry color (RGB)
 Color is the charge of the
strong interaction (SI)
 Free quarks do not exist?
 Quarks form bound states
through the SI to produce
the hadron spectrum of
several hundred observed
particles
 These bound states are
colorless
Phys 450
 Structureless and pointlike
Spring 2003
Quarks
 Experience the strong,
weak, and EM
interactions
 There are anti-quarks as
well
 Quark masses are not welldefined
 Quarks carry color (RGB)
 Color is the charge of the
strong interaction (SI)
 Free quarks do not exist?
 Quarks form bound states
through the SI to produce
the hadron spectrum of
several hundred observed
particles
 These bound states are
colorless
Phys 450
 Structureless and pointlike
Spring 2003
Quark Content
 Here are some particles for which you should know
the quark content









p = uud, n = udd
Δ’s = uuu, uud, udd, ddd
π = ud, (uu + dd)/√2, du
K0 = ds, K0 = sd, there are also K+, KΛ = uds, Ω- = sss
J/ψ = cc, Υ = bb (the “oops Leon”)
D0 = cu, D0 = uc, there are also D+, DB0 = db, B0 = bd, there are also B+, BNote there are no bound states of the top quark
 This is because the top quark decays before it
hadronizes
Phys 450
Spring 2003
Hadrons
 Hadrons == particles that have strong
interactions


Baryons (fermions)
Mesons (bosons)
 Baryons == 3 quarks (or antiquarks)

p = uud, n = ddu, Λ = uds, Ω- = sss
 Mesons == quark plus antiquark


π+ = u(d-bar), π- = d(u-bar),
π0 = (u(u-bar)+d(d-bar))/√2)
 Hadrons can decay via the strong, weak, or
electromagnetic interaction
Phys 450
Spring 2003
Quark Model
 By the 1960’s scores of “elementary
particles” had been discovered suggesting a
periodic table

“The discoverer of a new particles used to be
awarded the Nobel Prize; now, he should be
fined $10000” – Lamb
 Underlying structure to this spectrum was
suggested by Gell-Mann in the 1960’s

First through the “Eightfold Way” and later
through the quark model
 It took approximately a decade for
physicists to accept quarks as being “real”

Discovery of J/ψ and deep inelastic scattering Phys 450
Spring 2003
experiments gave evidence that partons = quarks
Quark Model
 One of the early successes of the quark
model (Eightfold Way) was the prediction of
the existence of the Ω- before its discovery
Phys 450
Spring 2003
A Little More (review)
on Spin
 Physics should be unchanged under symmetry
operations


Rotations form a symmetry group
So do infinitesimal rotations
 The angular momentum operators are the
generators of the infinitesimal rotation group


An infinitesimal rotation ε about z is
 U ψ(x,y,z) = ψ (R-1r) ~ ψ (x+εy,y-εx,z)
 = ψ(x,y,z) + ε(y∂ψ/∂x - x∂ψ/∂y)
 = (1 - iε(xpy – ypx))ψ = (1 – iεJ3)ψ
And the generators (angular momentum operators) satisfy
commutation relations and have eigenvalues shown on the
previous page
Phys 450
Spring 2003
SU(2) Group (Jargon)
 SU(2) group is the set of all traceless unitary 2x2
matrices (detU = 1)




U(2) group is the set of all unitary 2x2 matrices
U† U = 1
U(θi) = exp(-iθiσi/2)
σi are the Pauli matrices and Ji = σi/2
 The generators of this group are the Ji
 The SU(2) algebra is just the algebra of the
generators Ji
 The lowest, nontrivial representation of the group
are the Pauli matrices
 The basis for this representation are the column
vectors
Phys 450
Spring 2003
SU(2) Group Representations
 Higher order representations (higher order
spin states) can be built from the
fundamental representation (by adding spin
states via the CG coefficients)



A composite system is described in terms of the
basis |jAjBJM> == |jAmA>|jBmB>
The J’s and M’s follow the normal rules for
addition of angular momentum
|jAjBJM> = ∑ CG(mAmB;JM>|jAjBmAmB> where the
CG are the Clebsch-Gordon coefficients we
talked about earlier in the course
Phys 450
Spring 2003
SU(2) Representations
 The product of 2 irreducible representations
of dimension 2jA+1 and 2jB+1 may be
decomposed into the sum of irreducible
representations of dimension 2J+1 where J
= jA+jB, …, |jA+jB|

Irreducible means …
 What is he talking about???
Phys 450
Spring 2003
SU(3) Group (Jargon)
 SU(3) group is the set of all traceless
unitary 3x3 matrices (detU = 1)
 The generators of this group are the Fa
2
 There are 3 -1 = 8 generators Fa
 They satisfy the algebra [Fa,Fb] = ifabcFc
 fabc== structure constants
 The generators Fa = 1/2λa where λa are
the Gell-Mann matrices (see next page)
 The basis for this representation are the
column vectors
Phys 450
Spring 2003
SU(3) Group
 Note F3 and F8 are diagonal




F3 == Isospin operator
F8 == Hypercharge operator
Later we’ll define Y = B+S and
Experimentally we find Q = I3 + Y/2
Phys 450
Spring 2003
SU(3) Represenations
 Combining 2 SU(3) objects
 3 x 3 = 6 + 3

It’s a 3 because in Y, I3 space the u, d, s
triangle looks like the ud, us, ds triangle
Phys 450
Spring 2003
SU(3) Representations





Combining 3 SU(3) objects
3 x 3 x 3 = 3 x (6 + 3) = 10 + 8 + 8 + 1
Note the 8’s!
Note the symmetry is S, MS, MA, A
The mixed symmetry representations are
given on the next page
Phys 450
Spring 2003
Quark Model
 Hopefully you’ve caught on to what we’ve
done
 Let u, d, s be the SU(3) basis states
 Define isospin Ii = λi/2
 Define hypercharge Y = λ8/√3 = B+S


Since λ3 and λ8 are diagonal, I3 and Y are
conserved and represent additive quantum
numbers
Note I2, S, Q = I3 + Y/2 are also diagonal and
hence are conserved and represent additive
quantum numbers
Phys 450
Spring 2003
Quark Model
u
d
s
u
d
s
I
1/2
1/2
0
I
½
1/2
0
I3
1/2
-1/2
0
I3
-1/2
½
0
Y
1/3
1/3
-2/3
Y
-1/3
-1/3
2/3
Q
2/3
-1/3
-1/3
Q
-2/3
1/3
1/3
B
1/3
1/3
1/3
B
-1/3
-1/3
-1/3
S
0
0
-1
S
0
0
1
Spin
1/2
1/2
1/2
Spin
1/2
1/2
1/2
P
+
+
+
P
-
-
Phys 450
Spring 2003
Quark Model
 A convenient way to display the multiplet is
to show its elements on a weight diagram in
Y-I3 space
 Note that the combinations ud, us, ds would
appear in the same triangle as s, d, u
Phys 450
Spring 2003
Mesons
 3 x 3 = 8 + 1
 One can determine the multiplet by explicit calculation of the
representation or by the following trick
Phys 450
Spring 2003