Quarks, leptons, and the Standard Model

Six quark flavours fall into two charge families: up-type u, c and t with Q = +2/3, and down-type d, s and b with Q = −1/3, in units of e. Every quark carries baryon number B = +1/3, the strange quark carries strangeness S = −1, and an antiquark reverses every sign. A baryon is three quarks and a meson a quark with an antiquark, and the hadron's additive numbers are simply sums. The proton uud has Q = 2/3 + 2/3 − 1/3 = +1 and B = 1; the Λ⁰ (uds) has Q = 0 and S = −1; the π⁺ (ud̄) has Q = 2/3 + 1/3 = +1 and B = 0; the K⁺ (us̄) has S = +1. Writing each up-type charge as 1 − 1/3 gives the shortcut Q = N_U − B. Mass is the exception: it does not add. The u and d quarks weigh about 2.2 and 4.7 MeV/c², so uud sums to 9 MeV/c², under 1% of the proton's 938.3 MeV/c². The rest is the energy of the gluon field and of the quarks' confined motion.

The Δ⁺⁺ is uuu with spin 3/2: three identical fermions in the spatial ground state with parallel spins, a state symmetric in every label we have named. Pauli exclusion forbids it unless the quarks differ in one more label. That label is colour, and it takes three values. Every hadron ever observed is colour neutral, one quark of each colour in a baryon and a colour with its anticolour in a meson, and no isolated quark has ever been seen. Colour neutrality needs N(q) − N(q̄) to be a multiple of 3, so B is an integer and, through Q = N_U − B, so is Q: that is why no confirmed free charge of e/3 has ever turned up. Here confinement is an observed rule, not a derived one; lattice calculations of the strong interaction reproduce it numerically, but no analytic proof exists. Its signature is a field between separating quarks that stores about 1 GeV per femtometre, a tension of 1.6 × 10⁵ N. At 1 fm that is already over seven pion rest energies, so well before 2 fm the field pays for new quark–antiquark pairs and snaps: a hard collision yields jets of hadrons, never a free quark.

The six leptons feel no strong force and come in three families, (e⁻, νₑ), (μ⁻, ν_μ) and (τ⁻, ν_τ), with charged-lepton rest energies of 0.511, 105.7 and 1777 MeV. Give Lₑ = +1 to e⁻ and νₑ, −1 to e⁺ and ν̄ₑ, and define L_μ and L_τ the same way; hadrons carry zero. In every decay in this unit each family total is conserved on its own. Muon decay, μ⁻ → e⁻ ν̄ₑ ν_μ, closes all three: L_μ goes 1 → 0 + 0 + 1 and Lₑ goes 0 → 1 − 1 + 0. The simpler-looking μ⁻ → e⁻ γ conserves charge, energy and total lepton number, yet changes both Lₑ and L_μ, and no search has ever seen it: fewer than one muon decay in 10¹² can go that way. The rule is empirical rather than fundamental. Neutrino oscillation shows that a νₑ can arrive as a ν_μ, so family number is not exactly conserved; but the violation this feeds into charged-lepton decays lies far below any measurement, and as a bookkeeping rule for decays it has never failed. Total lepton number, Lₑ + L_μ + L_τ, has shown no violation at all.

In hadrons, n → p + e⁻ + ν̄ₑ. In quarks it is udd → uud: one d quark becomes a u by emitting a W⁻, the W⁻ becomes the electron and the antineutrino, and the other two quarks are spectators. Check each vertex on its own. At the quark vertex, charge −1/3 → +2/3 + (−1) and B = 1/3 → 1/3; at the lepton vertex, −1 → −1 + 0 and Lₑ = 0 → +1 − 1. β⁺ decay runs u → d + W⁺ with W⁺ → e⁺ + νₑ. A free proton cannot do it, because the neutron and positron together outweigh it by 1.804 MeV/c², so it happens only inside nuclei whose binding supplies the energy. Electron capture, p + e⁻ → n + νₑ, is the same exchange with an electron absorbed instead of a positron emitted. With s in place of d the vertex drives strange decays, and only the W changes flavour: photon, gluon and Z leave every quark's flavour unchanged. So a change of strangeness marks a weak process, and the clock agrees. A strange hadron that can only decay weakly lives 10⁻¹⁰ to 10⁻⁸ s, against about 10⁻²³ s for a strong decay and 10⁻²⁰ to 10⁻¹⁶ s for an electromagnetic one.

The photon, eight gluons, the W± at 80.4 GeV and the Z⁰ at 91.2 GeV carry the electromagnetic, strong and weak interactions. Exchanging a carrier of mass M borrows energy Mc² for a time of order ħ/Mc², so it reaches R = ħc/Mc², and its potential takes the Yukawa form U(r) = −αħc e^(−r/R)/r. Let M → 0 and R → ∞: the Coulomb potential returns, which is the correspondence check. Yukawa's own case, the pion at 139.6 MeV, gives R = 1.41 fm, the reach of the force between nucleons; the W gives 2.45 × 10⁻³ fm. The weak coupling itself is not small: α_W ≈ 1/29.5, about 4.6 times α. What makes β decay slow is the range. A slow particle, whose wavelength far exceeds R, cannot resolve the potential's shape and feels only its volume integral, ∫U d³r = −4παħcR², so the effective strength goes as α_W/(M_W c²)². Multiply by π/√2 and you have Fermi's constant: G_F/(ħc)³ = 1.166 × 10⁻⁵ GeV⁻². Gluons are massless, yet the strong force does not reach far: there confinement, not mass, cuts it off near 1 fm.

The inventory is short: six quarks, six leptons, four kinds of carrier, and the Higgs boson at 125 GeV, whose field gives the W, the Z and the fermions their masses. Every ledger in this lesson says only whether a process may occur, not how fast. The Λ⁰ and the free neutron both pass theirs, yet the Λ⁰ lives 2.6 × 10⁻¹⁰ s and the neutron about 880 s, a factor of 3 × 10¹² that only the full field theory of rates can account for. Nor are the masses predicted. They are inputs, running from 0.511 MeV for the electron to 172.6 GeV for the top quark, a ratio of 3.4 × 10⁵, with the neutrino masses far smaller still. The number of families is measured, not derived; confinement is observed, not proved; and gravity is absent altogether. Use the model as a classification: a ledger that fails rules a process out, and one that balances only lets it in.