Quark flavours and additive quantum numbers

Work in the six-dimensional flavour space spanned by |u⟩, |d⟩, |s⟩, |c⟩, |b⟩ and |t⟩, together with its conjugate spanned by the six antiquarks. For each flavour define the number operator N_f = n_f − n_f̄, diagonal in that basis with eigenvalue +1 on its own quark, −1 on its own antiquark and 0 on the other ten single-particle states. Six diagonal matrices commute pairwise, so a state can be a simultaneous eigenstate of all of them, and for a composite state each eigenvalue is the plain sum over its constituents. That is what additive means, and it is the exception rather than the rule: parity multiplies, C multiplies, and angular momenta need Clebsch-Gordan coefficients. Electric charge follows the same additive pattern, +2/3 for u, c and t against −1/3 for d, s and b, and so does baryon number at +1/3 per quark. The proton uud therefore carries B = 1 and Q = 2/3 + 2/3 − 1/3 = +1 by counting alone, with no dynamics anywhere in the argument.

The flavour labels predate the quarks. Strangeness was assigned in the early 1950s so that the K⁺ came out with S = +1, and when the quark model arrived the K⁺ turned out to be us̄ — which forces S = −1 on the s quark itself. The surviving rule is worth one line of memory: each flavour label takes the sign of that quark's electric charge. The down-type quarks s and b, at charge −1/3, carry S = −1 and B′ = −1; the up-type quarks c and t, at +2/3, carry C = +1 and T = +1. Antiquarks reverse every one of them. So D⁰ = cū has C = +1, B⁻ = bū has B′ = −1, and Λ⁰ = uds has S = −1. The convention looks perverse until you see what it buys: with these signs one formula covers every hadron in the tables, and with the signs flipped that same formula fails for every hadron carrying net strangeness, charm or bottomness.

Define Y = B + S + C + B′ + T, the sum of every additive label except the isospin component, and the claim is Q = I₃ + Y/2. This is not an experimental law but a linear identity among diagonal operators, and you can check it in three lines. Put B = (1/3)Σ_f N_f, I₃ = (1/2)(Nᵤ − N_d), S = −Nₛ, C = N_c, B′ = −N_b and T = Nₜ into the definition. Hypercharge becomes Y = (1/3)Nᵤ + (1/3)N_d − (2/3)Nₛ + (4/3)N_c − (2/3)N_b + (4/3)Nₜ. Adding I₃ to Y/2 gives (2/3)Nᵤ − (1/3)N_d − (1/3)Nₛ + (2/3)N_c − (1/3)N_b + (2/3)Nₜ, which is the charge operator itself. Test it on states: K⁺ = us̄ has B = 0 and S = +1, so Y = 1, and I₃ = +1/2, giving Q = +1. D⁰ = cū has B = 0, C = +1, so Y = 1, while the ū supplies I₃ = −1/2, giving Q = 0.

Why does the u–d pair get a doublet while every heavier flavour gets its own U(1)? Mass. At μ = 2 GeV in MS-bar the up and down masses are about 2.16 and 4.67 MeV, a difference far below Λ_QCD ≈ 200 MeV, so the strong Hamiltonian barely distinguishes them and SU(2) isospin is a good approximate symmetry. The strange quark near 93 MeV is comparable to Λ_QCD, so SU(3) flavour is broken but still useful: each unit of strangeness shifts the baryon decuplet by a nearly constant step, 1232 → 1385 → 1532 → 1672 MeV, spacings of 153, 147 and 140 MeV. The charm, bottom and top masses — 1.27, 4.18 and about 172.5 GeV — are hopeless, which is why nobody draws SU(4) multiplets and means it. What is left is the Cartan pair (I₃, Y): two commuting charges, hence the two-dimensional weight diagrams on which every SU(3) multiplet is plotted.

Every additive flavour number commutes with the strong and electromagnetic Hamiltonians, so those interactions conserve each one separately. The charged weak current does not: it turns one flavour into another, and at tree level it changes any one flavour number by at most one unit, so a first-order weak decay has |ΔS| ≤ 1 and |ΔS| = 2 requires second order — which is what makes K⁰–K̄⁰ mixing so slow. Reading a lifetime then reduces to checking the ledger. Δ⁺⁺ → pπ⁺ changes no flavour number and goes strong, in 5.6 × 10⁻²⁴ s. Σ⁰ → Λ⁰γ conserves strangeness but needs a photon, so it goes electromagnetic, in 7.4 × 10⁻²⁰ s. Λ⁰ → pπ⁻ has ΔS = +1 and must go weak, in 2.6 × 10⁻¹⁰ s: nearly fourteen orders of magnitude of lifetime read off one conservation law. Charge, baryon number and colour survive all three interactions; the individual flavour numbers do not.

Everything above is bookkeeping on labels that are integers or simple fractions. Mass is not like that. A quark is never isolated, so there is no rest frame to weigh it in; what appears in the Lagrangian is a renormalised parameter depending on both scheme and scale. The MS-bar values mᵤ ≈ 2.16 MeV, m_d ≈ 4.67 MeV and mₛ ≈ 93 MeV are quoted at μ = 2 GeV, and one-loop running carries mₛ to roughly 117 MeV at μ = 1 GeV — the same quark, a different number. Nor do they add up to anything you weigh: 2mᵤ + m_d = 8.99 MeV is under 1% of the proton's 938.3 MeV, the rest being the kinetic and field energy of the confined quarks and gluons, while the constituent masses near 336 MeV used in magnetic-moment fits are a different object again. The top makes the point sharpest: its width near 1.48 GeV gives τ ≈ 4.4 × 10⁻²⁵ s, some seven times shorter than the ħ/Λ_QCD ≈ 3.3 × 10⁻²⁴ s the strong interaction needs to dress it, so topness is a perfectly good quantum number that no hadron ever carries.