the quark model and su(3) - welcome to...
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The Quark Model and SU(3)
Peyton Rose
10 June 2013Physics 251
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Scattering Experiments at SLAC in the 1960s suggested that pointlike
particles exist in the proton
Atom
Proton
ddddddddd
uuuuuuuuu
uuuuuuuuu
+
+ +++
-
-
- -
-
Pre-Experiment Post-Experiment
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Six quarks and six anti-quarks have been discovered
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An additional quantum number – color – restricts combinations of
quarks. In nature, we observe qq and qqq
q q
q (q)
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The free quark Lagrangian density has a SU(n) symmetry for n quarks*
* with identical masses
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First, let's consider a model with only up and down quarks.
In this case, the relevant quantum number is isospin
with an SU(2) symmetry.
Generators of the 2-dimensional representation of SU(2):
Commutation Relations: Raising/Lowering Operators
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There are two eigenstates of s3
In the 2 representation, u corresponds to and d corresponds to In the 2* representation,-d corresponds to and u corresponds to
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Meson states are formed from 2 2*
2 2* = 3 1 → We expect 4 meson states
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With appropriate linear combinations and normalization, we recover the pions ...
...which fit into a 3-dim representation of SU(2)
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In the quark model, we consider three quarks – up, down, and strange
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There are 8 matrices in the defining representation of SU(3)
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Two of these matrices are diagonal
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SU(3) contains pieces of SU(2)
Y I
U V
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We can form raising and lowering operators within each subgroup
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Using the eigenvalues of and , we can draw weight diagrams
3 3*
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Meson states are 3 3*
3 3* 1 8
=
→ We expect 9 states!
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To find states, overlay 3* on 3 in the weight diagram
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States on the hexagon vertices have a unique quark content
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One (0,0) state is found using raising/lowering operators
Other (0,0) states are orthogonal
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Baryon states are 3 3 3
3 3 6 3*
=
3 3 3 10 8 8 1
=
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In this case, the overlaid weight diagram is more difficult
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Using our knowledge of SU(3), we construct each N-let
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In practice, it is helpful to consider other physical properties
● Combining three spin-1/2 particles can only result in spin-3/2 or spin-1/2
● Particles of half integer spin must have completely anti-symmetric wave functions
● The color state is anti-symmetric
→ The combination of flavor and spin must be symmetric
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For a spin-up proton:
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References
● An Introduction to Quarks and Partons by F. E. Close
● A Modern Introduction to Particle Physics by Fayyazuddin and Riazuddin
● Particle Physics: A Comprehensive Introduction by Abraham Seiden
● Class Notes, Physics 221A, Bruce Schumm
● Class Notes, Physics 251, Howie Haber