Quarks come in three colors (red, green, blue) and interact via gluons. Gluons carry one unit of color and one unit of anti-color. Why are there 8 gluons rather than 9 (3 colors times 3 anti-colors)?
AOne of the nine combinations is unphysical due to negative norm
BThe color-singlet combination (r r-bar + g g-bar + b b-bar)/sqrt(3) does not couple to color charge and must be excluded — it would mediate a long-range color force, which is not observed
CThree of the nine combinations are redundant due to symmetry
DThe ninth gluon has been observed but is too heavy to be relevant
The nine combinations of color-anticolor decompose under SU(3) as 3 x 3-bar = 8 + 1: an octet and a singlet. The octet states are the eight gluons — they carry net color charge and couple to quarks. The singlet (r r-bar + g g-bar + b b-bar)/sqrt(3) is color-neutral and would not be confined; if it existed as a physical gluon, it would mediate a long-range force between all hadrons. The absence of such a force is evidence that the gauge group is SU(3) (which has 8 generators) rather than U(3) (which has 9). The mathematical reason is that SU(3) is the group of 3x3 unitary matrices with determinant 1, excluding the U(1) phase.
Question 2 Multiple Choice
The QCD coupling constant alpha_s is approximately 0.12 at the Z boson mass (91 GeV). At the scale of a proton (approximately 1 GeV), alpha_s is approximately 0.5. Why does this large coupling make proton structure calculations fundamentally different from QED calculations of hydrogen?
ABecause the proton has three quarks while hydrogen has only one electron
BBecause alpha_s ~ 0.5 means the perturbative expansion in powers of alpha_s converges slowly or not at all — each higher-order correction is comparable to the previous one, so Feynman diagram perturbation theory is unreliable for low-energy QCD
CBecause quarks are heavier than electrons
DBecause gluons are massless like photons, so the calculations are equivalent
In QED, alpha ~ 1/137, so each loop adds a correction of order 1% — perturbation theory converges rapidly. In QCD at GeV scales, alpha_s ~ 0.5, so a one-loop correction is 50% of the tree level, a two-loop correction is 25%, etc. — the series does not converge reliably. This is why non-perturbative methods (lattice QCD, sum rules, chiral perturbation theory) are needed for hadron physics. At high energies (asymptotic freedom regime), alpha_s is small enough for perturbative QCD to work, which is why jet cross sections and deep inelastic scattering are calculable.
Question 3 True / False
QCD is an exact copy of QED with the replacement U(1) -> SU(3), and all differences between electromagnetism and the strong force follow from this single change.
TTrue
FFalse
Answer: True
This is correct at the level of the Lagrangian structure. QCD's Lagrangian is L = sum_f q-bar_f(i gamma^mu D_mu - m_f)q_f - (1/4)G^a_{mu nu}G^{a mu nu}, where D_mu = partial_mu - ig_s T^a A^a_mu is the covariant derivative with SU(3) generators T^a, and G^a_{mu nu} is the non-abelian field strength tensor. Replacing SU(3) -> U(1), T^a -> 1, and g_s -> e gives QED. All the differences — gluon self-interactions, asymptotic freedom, confinement, color neutrality of hadrons — follow from the non-abelian structure of SU(3) versus the abelian structure of U(1). The gauge principle and minimal coupling are identical.
Question 4 Short Answer
Explain what color confinement means physically and why it makes free quarks unobservable, despite quarks being confirmed as real constituents of protons and neutrons.
Think about your answer, then reveal below.
Model answer: Color confinement means that all observable particles must be color-neutral (color singlets). No free quark (which carries color charge) has ever been observed. When you try to separate two quarks, the energy stored in the color flux tube between them grows linearly with distance (unlike the Coulomb potential which falls off). At some point, the energy is sufficient to create a new quark-antiquark pair from the vacuum, producing two color-neutral hadrons rather than two free quarks. Quarks are real (they explain deep inelastic scattering, jet production, and the hadron spectrum) but are always confined inside hadrons — mesons (quark-antiquark) or baryons (three quarks). Confinement is a non-perturbative phenomenon that cannot be derived from Feynman diagrams.
Proving confinement rigorously from the QCD Lagrangian is one of the seven Millennium Prize Problems. Lattice QCD simulations confirm it numerically, and the linear potential has been verified, but an analytical proof remains elusive.