1. PHYS 745G Presentation Symmetries & Quarks
4/15/2017 7:15 PM
PHYS 745G Presentation Symmetries & Quarks
Shakil Mohammed
Department of Physics & Astronomy
Symmetries & Quarks

2. Overview A Brief Overview of Symmetries & Groups The SU(2) Group
Quark-Antiquark States: Mesons
Three Quark States: Baryons
Magnetic Moments

3. Symmetries in Physics
Isospin: Quantum number related to the Strong Interactions
For a two-nucleon system, the spin singlet and triplet states are:

4. Similarly, each nucleon has an isospin, I = ½, with I3=±
½ for protons and neutrons. Then the spin states are:

5. The Group SU(2) Generators Pauli Matrices The base states
Pauli Matrices are Hermitian
The 2×2 matrices known as U(2) and traceless 2×2 form a subgroup SU(2) in two dimension

6. Combining representations:
Composite system from 2 systems having angular momentum jA and jB
Combined operator
With a basis,
Where C = Clebsh-Gordan coefficients and M=mA+mB. The C’s are calculated by using
Symbolically,
For a third spin-1/2,

7. SU(2) of Isospin
The nucleon having an internal degree of freedom with two allowed states – Isospin
Isospin generators satisfy,
Generators are denoted as Ii = ½ τi, where
Isospin for Antiparticles
The antinucleaon states with operator C

8. Applying C to the state,
If we want to transform the antiparticle doublet the same way as particle doublet, then
A composite system of a nucleon-antinucleon pair

9. The Group SU(3)
The set of 3×3 matrices with detU = 1 for the group SU(3)
Fundamental representation of SU(3) is a triplet
The color charges of Quark R, G, B form a SU(3) symmetry group. They are denoted by λi, with i = 1,2,…,8.
The diagonal matrices are:
With eigenvalues:

10. Hypercharge: Y = B + S Charge Qe: Q = I3 + Y/2
Quark-Antiquark States: Mesons
Hypercharge: Y = B + S
Charge Qe: Q = I3 + Y/2

12. For 3 flavors of Quarks, q = u, d, s – 9 possible combinations of Quark-Antiquark
Among 9 combinations – 8 states are in SU(3) Octet and 1 state in SU(3) singlet
The 8 states transform among themselves, but do not mix with singlet state

13. The states uu*, dd*, ss* labeled A, B and C have I3 = Y = 0.

14. The singlet combination
C = √1/3(uu*+dd*+ss*)
State A, a member of the isospin triplet (du*,A,-ud*)
A= √1/2(uu*-dd*)
Isospin singlet state B (by requiring orthogonality to both A and C)
B= √1/6(uu*+dd*-2ss*)

15. The excited states of mesons correspond to the observed meson states
Parity of Meson, P = -(-1)L
The particle-antiparticle conjugation operator C is given by,
C = -(-1)S+1(-1)L = (-1)L+S
In each nonet of the meson, there are two isospin doublets

16. Mesons of Spin 0
Mesons of Spin 1

17. Three Quark States: Baryons
There are 27 possible qqq combinations involved in the SU(3) decomposition
First, the two qq combinations arrange themselves into two SU(3) multiplets having 6 symmetric and 3 anti-symmetric states

18. Next, we add the 3rd Quark triplet such that,

19. For the pA part,
pA = √1/2(ud-du)u
For the S part
Δ = √1/3[uud+(ud+du)u]
The remaining part requires orthogonality and thus,
pS = √1/6[(ud+du)u-2uud]

20. For the case of Spins,
Baryon spin multiplets with S = 3/2, 1/2, 1/2
Replacing u →↑ and d →↓we can have the spin multiplets,

21. Next, we combine the SU(3) flavor decomposition with SU(2) spin decomposition

22. The spin ½ baryon octet
The spin 3/2 baryon decuplet

23. The 3 possible values of color are R, G, B
For the case of Color
The 3 possible values of color are R, G, B
The quarks form fundamental triplet of an SU(3) color symmetry
The color wavefunction of a baryon is,
(qqq)col.singlet = √1/6(RGB-RBG+BRG-BGR+GBR-GRB)

24. Example: Wavefunction of spin-up proton

25. In ground state, l=l’=0 for the qqq
The parity in ground state = (-1)l+l’
In 1st excited state, l=1, l’=0 or l=0, l’=1
The first excited state contains (1+8+10) flavor multiplets of S = ½ baryons and octet of S = 3/2 baryons
The spins combine with L = 1 to give
Multiplets 1, 8, 10 with JP=1/2- and JP=3/2-
Three octets with JP=1/2-,3/2-,5/2-

26. Magnetic Moments The magnetic moment operator is given as
Where, the magnetic moment for Quark is
For proton (in the non-relativistic approx.)

27. Thank You!
Questions?