= Solution
Evidence for <quarks> comes from several kinds of observation that test different aspects of their interpretation. The light <meson> and <baryon> spectra organize into approximate flavour multiplets with the charges and strangeness expected for $q\bar q$ and $qqq$. In particular, the <baryon octet> and <baryon decuplet> encode the three light <quark flavours> and their <spin> couplings. This is an economical account of hadron quantum numbers, but spectroscopy alone would not establish that the constituents behave as localized scattering centers.
<Deep inelastic scattering> supplies that second test. In energetic <electron>–<proton> scattering the outgoing lepton measures the <energy> and <momentum> transferred through a virtual <photon>, while the broken-up target is summed over. Approximate <Bjorken scaling> means the structure functions depend primarily on $x=Q^2/(2P\cdot q)$ rather than varying rapidly with the resolving <momentum> scale $Q^2$. It indicates nearly pointlike constituents. In the leading spin-half <parton model>,
$$
F_2(x)=x\sum_q e_q^2[q(x)+\bar q(x)],\qquad F_2(x)=2xF_1(x).
$$
The charge-squared weights and the second relation test the constituents' fractional charges and spin-half nature. Scaling violations reflect <quark> and <gluon> radiation rather than a failure of the constituent picture.<Review of Particle Physics by Particle Data Group (2022)>[Structure Functions, §§18.1–18.3, deep-inelastic kinematics and leading parton expressions]{ref}
<Particle jets> give complementary final-state evidence. A high-energy <quark> produces a collimated spray of hadrons because <confinement> prevents it from emerging as an isolated particle. Back-to-back jets fit <quark>–antiquark production; three-jet events show hard <gluon> radiation. Their angular correlations test a spin-one radiated parton. This distinguishes constituents of hadrons from a literal observation of free fractional-charge particles.
Three independent arguments identify an additional threefold colour degree of freedom. First, the spin-aligned $uuu$ state of a ground-state $\Delta^{++}$ would have symmetric space, <spin> and flavour factors. The <Pauli exclusion principle> requires an antisymmetric additional factor. The <three-quark colour singlet> supplies it. The same construction accommodates the spin-aligned $sss$ state of the $\Omega^-$ without altering its measured quantum numbers.
Second, the inclusive <hadronic R ratio>, away from thresholds and narrow resonances, is at leading electromagnetic order
$$
R=\frac{\sigma(e^+e^-\to\text{hadrons})}{\sigma(e^+e^-\to\mu^+\mu^-)}
=N_c\sum_{q\ {
m accessible}}e_q^2.
$$
Each <quark flavour> has $N_c$ distinguishable colour final states. With $N_c=3$, the predicted plateaux for three, four and five accessible flavours are \b[$2$, $10/3$ and $11/3$]; without the factor three they would be a third as large. The measured inclusive normalization is therefore a colour-counting test, with radiative and threshold corrections included in precision comparisons.<Review of Particle Physics by Particle Data Group (2022)>[Plots of <Cross sections> and Related Quantities, inclusive <electron>–<Positron> hadronic <cross sections> and R plots]{ref}
\Image[/past-exam-of-the-particle-and-nuclear-physics-course-of-the-university-of-cambridge/2015/paper-1-colour-counting.png]
{title=Leading colour-counting predictions for the hadronic R ratio; these are model values, not plotted measurements}
Third, the neutral pion's two-photon amplitude contains a quark-loop colour sum, proportional to $N_c(e_u^2-e_d^2)$; its rate and the other evidence are consistent with three colours. <Quantum chromodynamics> identifies this degree of freedom as a gauge charge, with eight <gluons>. Observed hadrons are colour singlets; the evidence for colour is its quantitative and symmetry consequences, not visible colours or isolated coloured <quarks>.
Back to article page
