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PHYS 745G Presentation
Symmetries & Quarks
Shakil Mohammed
Department of Physics & Astronomy
Overview
 A Brief Overview of Symmetries &
Groups
The SU(2) Group
The SU(3) Group
Quark-Antiquark States: Mesons
Three Quark States: Baryons
Magnetic Moments
Symmetries in Physics
Isospin: Quantum number related to the Strong
Interactions
For a two-nucleon system, the spin singlet and triplet
states are:
Similarly, each nucleon has an isospin, I = ½, with I3=±
½ for protons and neutrons. Then the spin states are:
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
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,
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
• 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 nucleonantinucleon pair
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:
Quark-Antiquark States: Mesons
Hypercharge: Y = B + S
Charge Qe: Q = I3 + Y/2
• 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
The states uu*, dd*, ss* labeled A, B and
C have I3 = Y = 0.
•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*)
• 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
Mesons of Spin 0
Mesons of Spin 1
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
• Next, we add the 3rd Quark triplet such that,
• 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]
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,
• Next, we combine the SU(3) flavor
decomposition with SU(2) spin decomposition
The spin ½ baryon octet
The spin 3/2 baryon decuplet
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)
Example: Wavefunction of spin-up proton
• 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=3/2Three octets with JP=1/2-,3/2-,5/2-
Magnetic Moments
• The magnetic moment operator is given as
• Where, the magnetic moment for Quark is
• For proton (in the non-relativistic approx.)