Interactions and their gauge bosons

Multiply every electron wavefunction by one constant phase, ψ → e^(iθ)ψ, and no probability, current or energy shifts; Noether returns a conserved charge and nothing more. Now let θ depend on position. The derivative stops commuting with the phase: ∂_μ(e^(iθ(x))ψ) = e^(iθ)(∂_μψ + i(∂_μθ)ψ), and that extra i(∂_μθ)ψ wrecks the equation of motion. There is exactly one cheap repair. Introduce a vector field A_μ that shifts as A_μ → A_μ − (ℏ/q)∂_μθ, and replace ∂_μ everywhere by D_μ = ∂_μ + i(q/ℏ)A_μ. The two changes cancel term by term, D_μψ now transforms exactly like ψ, and invariance is restored. What arrived is not a fudge: A_μ is the electromagnetic four-potential, its transformation law is the gauge freedom Maxwell's equations already had, and the qA_μ coupling that dropped out is minimal coupling — the reason the same e sits at every electron vertex in every process you will ever draw.

Repeat the construction with a larger symmetry and you get one mediator per independent generator: U(1) has one, SU(2) has three, SU(3) has eight, and SU(N) has N² − 1. That is where the photon, the W⁺, W⁻ and Z, and the eight gluons come from — counted, not fitted, once the U(1) boson B and the neutral W³ mix into the photon and the Z. For U(1) the generators commute, so the photon carries no electric charge and light does not scatter light at tree level. For SU(2) and SU(3) they do not commute: [T^a, T^b] = i f^(abc) T^c, and that commutator survives into the field strength as a term g f^(abc) A^b A^c, giving three- and four-boson vertices. The mediators therefore carry the charge they respond to — gluons carry colour, the W carries weak isospin and electric charge — and that self-coupling is the root of both asymptotic freedom and confinement. The counting also excludes something. Since 3 ⊗ 3* = 8 ⊕ 1, a ninth colour-singlet gluon is arithmetically available and physically absent; had it existed it would be unconfined, and the strong force would reach between colour-neutral atoms.

A quick argument gets the scale right. A quantum of mass M borrowed from the vacuum can persist for about Δt ≈ ℏ/(Mc²) before the energy-time bound closes on it, and in that time it travels at most cΔt = ℏ/(Mc). The careful version says the same thing without any borrowing: the static potential is the Fourier transform of the propagator 1/(q² + M²c²), and moving that pole off q = 0 turns the Coulomb 1/r into the Yukawa e^(−r/R)/r with R = ℏ/(Mc). Carry ℏc = 197.3 MeV⋅fm and the arithmetic is one division. The W: 197.3 ÷ 80400 = 2.45×10⁻³ fm. The Z: 197.3 ÷ 91190 = 2.16×10⁻³ fm — heavier boson, shorter reach, in strict inverse proportion. The photon: dividing by zero leaves the range unbounded, which is the same statement as saying Coulomb's law carries no exponential. Set against a proton's 0.84 fm charge radius, the W's reach is some 340 times smaller, which is why at nuclear energies the weak interaction behaves as though the two currents simply touch.

Feed M = 0 into R = ℏ/(Mc) for the gluon and you get an unbounded range, which is plainly wrong: no strong force acts between two atoms on a bench. The rule has not been misapplied, it simply does not govern here. Colour charge is confined. As two quarks separate the gluon field collapses into a flux tube of roughly 1 GeV per fermi — about 1.6×10⁵ N, the weight of sixteen tonnes — so pulling harder manufactures a new quark pair rather than a long-range field, and free colour is never seen. The strong interaction's reach is therefore set by hadronisation, not by any mediator mass. What survives between colour-singlet nucleons is a residual force, the QCD analogue of van der Waals between neutral atoms, carried by the lightest hadron light enough to be exchanged: R = 197.3 ÷ 139.6 = 1.41 fm for the pion. Yukawa ran this backwards in 1935 — from a nuclear range of about 2 fm he predicted a quantum of about 200 electron masses, near 100 MeV, and the pion turned up at 140 MeV in 1947.

A particle that can decay strongly does so as fast as its own size permits. The Δ(1232) has width Γ = 117 MeV, so τ = ℏ/Γ = 6.582×10⁻²² ÷ 117 = 5.6×10⁻²⁴ s — long enough for light to cross 1.7 fm and no longer. The Σ⁰ cannot reach Λπ⁰, since its 77 MeV mass gap will not make a pion, so it goes to Λγ in 7.4×10⁻²⁰ s, thirteen thousand times slower, and the real photon in the final state names the interaction outright. The Λ takes 2.63×10⁻¹⁰ s to reach pπ⁻, 4.7×10¹³ times the Δ's, and changes strangeness by one — which only the charged weak current can do. Its width is Γ = 6.582×10⁻²² ÷ 2.63×10⁻¹⁰ = 2.5 μeV, far below any spectrometer, which is why weak lifetimes are timed with a clock and strong ones are fitted from a line shape. Take the caution seriously though: inside the weak band the neutron lives 879 s and the muon 2.2 μs, a factor of 4×10⁸ apart, because phase space and matrix elements vary wildly. A lifetime narrows the candidates; the conservation ledger decides.

Strong, electromagnetic and weak are names awarded at the energy where each was first met, and they do not survive a change of scale. Near qc = 1 GeV the three effective strengths are αₛ ≈ 0.5, α = 1/137 = 0.0073, and — because the W propagator contributes q²/(M_W c)² — an effective weak strength near 0.034 ÷ 6464 = 5×10⁻⁶. Five decades separate the extremes, and the ordering matches the names. Now move to qc = 91.2 GeV, the Z mass. The strong coupling has run down to αₛ = 0.118, the electromagnetic has run up to about 1/128 = 0.0078, and the weak suppression has all but evaporated: the propagator factor is now 91.2²/(91.2² + 80.4²) = 0.56 rather than 1.5×10⁻⁴, so the effective weak strength is 0.034 × 0.56 = 0.019. The ordering is now 0.118 : 0.019 : 0.0078, a factor of fifteen across the lot, and the weak interaction is the stronger of the last two. Two different mechanisms did that, and they must not be blurred: αₛ and α run because of vacuum polarisation, while the weak's rise is pure kinematics, the propagator ceasing to suppress once q approaches M_W c.