The lightest mesons (pions, kaons, eta) form a nonet under SU(3) flavor symmetry. The nonet decomposes as 3 x 3-bar = 8 + 1: an octet plus a singlet. Why do we observe a nonet (nine particles) rather than separate octet and singlet multiplets?
ABecause the octet and singlet are exact eigenstates and all nine happen to have similar masses
BBecause SU(3) flavor symmetry is only approximate (broken by quark mass differences), so the physical eta and eta-prime are mixtures of the octet and singlet states — the mixing means all nine states are interrelated and best described together as a nonet
CBecause there are nine possible quark-antiquark combinations and each one is a separate particle
DBecause confinement forces all nine states to have the same mass
If SU(3) flavor were exact (m_u = m_d = m_s), the octet and singlet would be separate, non-mixing multiplets. But the strange quark is heavier (~95 MeV vs ~5 MeV for u,d), breaking SU(3) and allowing mixing between the I=0, S=0 members of the octet and singlet. For pseudoscalars, this produces the eta (~548 MeV, mostly octet) and eta-prime (~958 MeV, mostly singlet). The large eta-prime mass also receives a contribution from the axial anomaly. For vector mesons, the analogous mixing produces the omega and phi, where the phi is nearly pure s-sbar.
Question 2 True / False
The Delta++ baryon has charge +2, spin 3/2, and is composed of three up quarks (uuu). In the quark model, this state appears to violate the Pauli exclusion principle because all three quarks are in the same state.
TTrue
FFalse
Answer: False
This apparent paradox was one of the original motivations for introducing color charge. Three identical fermions in the same spatial and spin state would indeed violate the Pauli principle if there were no additional quantum number. But each quark carries one of three colors (red, green, blue), and the baryon wavefunction is totally antisymmetric in color (the color singlet epsilon_{ijk}). The full wavefunction is then antisymmetric: symmetric in space x spin x flavor (all u, all spin-up, ground state) times antisymmetric in color. The Pauli principle is satisfied, and the Delta++ is allowed.
Question 3 Short Answer
The quark model predicts that no hadrons exist with quantum numbers that cannot be made from qqbar (mesons) or qqq (baryons). What are 'exotic' hadrons, and what is their current experimental status?
Think about your answer, then reveal below.
Model answer: Exotic hadrons are states whose quantum numbers or quark content cannot be explained by the simple qqbar or qqq picture. These include tetraquarks (qqbar-qqbar), pentaquarks (qqqq-qbar), glueballs (bound states of gluons), and hybrid mesons (qqbar-g). QCD does not forbid these states — it only requires them to be color singlets. Since 2003, multiple exotic candidates have been observed: the X(3872), the Z_c and Z_b charged charmonium-like states, and the P_c pentaquark states discovered at LHCb. Their internal structure (compact multiquark state vs. loosely bound meson molecule) remains debated.
The existence of exotic hadrons does not contradict the quark model but extends it. The quark model's success in classifying the conventional hadron spectrum was so complete that exotics were long thought to be absent. Their discovery has reinvigorated hadron spectroscopy as a field.
Question 4 Multiple Choice
In the quark model, the proton (uud) and neutron (udd) both have spin 1/2. How do three spin-1/2 quarks combine to give spin 1/2 for the nucleon ground state, and why isn't the ground state spin 3/2?
AThe quarks have orbital angular momentum that cancels some of the spin
BTwo of the three quarks form a spin-0 pair, and the third quark carries the nucleon's spin 1/2 — this configuration is energetically favored over the fully aligned spin-3/2 state (which is the Delta baryon, approximately 300 MeV heavier)
COne of the quarks has its spin quantum number reduced by confinement
DThe nucleon is actually a mixture of spin-1/2 and spin-3/2 states
Three spin-1/2 quarks can couple to total spin S = 3/2 (all aligned) or S = 1/2 (one pair anti-aligned). In the ground state (L=0), the S=1/2 state is the nucleon (proton/neutron) and the S=3/2 state is the Delta. The mass difference (~300 MeV) is due to the color-magnetic interaction (the QCD analog of the spin-spin interaction), which is attractive for anti-aligned spins and repulsive for aligned spins. This hyperfine splitting is proportional to (sigma_i dot sigma_j)/(m_i m_j) and explains mass splittings throughout the hadron spectrum.