The quark model and hadron spectroscopy

Isospin came first: the proton and neutron differ by 1.3 MeV out of 939, so treat them as one particle carrying a two-valued label, I₃ = ±1/2. Strangeness came next, invented to explain why kaons and lambdas are always produced in pairs by the strong force and then decay slowly and singly — S is conserved by the strong interaction and violated by the weak. Combine the two into the hypercharge Y = B + S and plot every hadron at its point (I₃, Y). The spin-1/2 baryons land on eight sites: a hexagon with p and n along the top at Y = +1, the three Σ and the Λ across the middle at Y = 0, and the two Ξ below at Y = −1, the centre doubly occupied by Λ and Σ⁰. The spin-3/2 baryons land on a triangle of ten. Charge never needs an axis of its own, because Q = I₃ + Y/2 reads it off the two you already have.

Put u at (I₃, Y) = (+1/2, +1/3), d at (−1/2, +1/3) and s at (0, −2/3) — a small triangle — and Q = I₃ + Y/2 returns +2/3, −1/3 and −1/3 as required. Antiquarks sit at the reflected points. The patterns are then vector addition. A meson is qq̄, so its site is the sum of a quark site and an antiquark site, and the nine sums fall on a hexagon with a triply occupied centre: 3 ⊗ 3̄ = 8 ⊕ 1, the nonet. A baryon is qqq, giving 27 combinations that sort by exchange symmetry into 3 ⊗ 3 ⊗ 3 = 10 ⊕ 8 ⊕ 8 ⊕ 1 — a fully symmetric decuplet, two mixed octets and an antisymmetric singlet, and 10 + 8 + 8 + 1 = 27. That arithmetic is the content of the model. Nothing in it permits a light-quark baryon multiplet of six or twelve states, and none has ever been found.

In 1962 the 3/2⁺ decuplet had nine of its ten sites filled: four Δ at 1232 MeV with S = 0, three Σ* at 1385 with S = −1, two Ξ* at 1533 with S = −2. The apex, S = −3, was empty. Read the rows: 1385 − 1232 = 153 MeV and 1533 − 1385 = 148 MeV, a constant step near 150 MeV each time a light quark is swapped for a strange one. Extrapolating gives 1533 + 150 = 1683 MeV, and Q = I₃ + Y/2 with I₃ = 0 and Y = 1 − 3 = −2 gives Q = −1. Better still, at that mass every strangeness-conserving strong channel is shut — the lightest, Ξ⁰K⁻, needs 1809 MeV — so the state has to decay weakly and will leave a visible track. Brookhaven found it in 1964: mass 1672.5 MeV/c², charge −1, lifetime 8.2 × 10⁻¹¹ s. The octet obeys the same discipline through Gell-Mann–Okubo, (M_N + M_Ξ)/2 = (3M_Λ + M_Σ)/4, which gives 1128.6 against 1135.1 MeV.

Write the Δ⁺⁺ out. Its content is uuu, its spin is 3/2, and in the ground state there is no orbital angular momentum, so L = 0. Take the factors in turn. The spatial part with L = 0 is symmetric under swapping any two quarks. The flavour part, three identical u, is symmetric. The J_z = +3/2 state has all three spins up, so the spin part is symmetric too. Their product is symmetric — and three identical spin-1/2 fermions may not occupy a symmetric state. Either quarks are not fermions, or a factor is missing. The missing factor is colour: a label taking three values, combined as ε_abc q^a q^b q^c, which changes sign under any exchange and restores the antisymmetry. Δ⁻ = ddd and Ω⁻ = sss make the identical demand. Colour would be a cheap patch if this were its only evidence, but two independent measurements confirm the count of three: the ratio R of hadron to muon-pair production in e⁺e⁻ annihilation is three times the naive quark sum, reaching 10/3 above the charm threshold, and the π⁰ → γγ rate goes as N_c² and comes out a factor of nine above the colourless prediction.

Classification says nothing about mass, so the model adds one term: M = Σᵢmᵢ + A Σ_(i<j)(Sᵢ⋅Sⱼ)/(mᵢmⱼ), the colour analogue of a magnetic dipole–dipole interaction, which is where the 1/(mᵢmⱼ) comes from. Its power shows wherever flavour is degenerate. The ρ and the π have the same quark content and differ only in whether the pair's spins are parallel or antiparallel — yet they differ by 637 MeV, more than four pion masses. The K* and K, with one quark replaced by the heavier s, differ by only 396 MeV, and the ratio 637/396 = 1.61 is mₛ/mᵤ read straight off the spectrum. The Λ and the Σ⁰ are both uds and sit on the same lattice site, and the model still separates them: the Λ carries its ud pair in a spin singlet, the Σ⁰ in a triplet, so the Λ lies lower, 1115.7 against 1192.6 MeV. A 77 MeV gap between two states of identical quark content is spin doing the work.

Be clear about which half of this lesson is theorem and which is fit. The multiplet structure is group theory: given three flavours, 8 ⊕ 1 and 10 ⊕ 8 ⊕ 8 ⊕ 1 are forced, and Q = I₃ + Y/2 then holds site by site once Y is defined as B + S. The masses are another matter. The constituent values mᵤ ≈ 310 and mₛ ≈ 490 MeV/c² are fitted to the spectrum they are then used to describe, and they are not the masses in the QCD Lagrangian, which are roughly 2.2, 4.7 and 93 MeV/c² for u, d and s. The hyperfine constant A is fitted as well, and the η–η′ mixing angle is a third free number. The framework is also non-relativistic, which sits awkwardly with a 310 MeV/c² quark confined to about 1 fm, where the uncertainty principle alone gives it p ≈ ħ/r ≈ 200 MeV/c. The quark model organises the spectrum and predicts gaps inside it; computing a hadron mass from first principles waits for lattice QCD.