Interactions, mediators and gravity's exclusion

Demand that the Lagrangian survive a rotation U(x) = exp(−i α^a(x) T^a) that differs from point to point, and ∂_μ ψ picks up an unwanted (∂_μ U) ψ. The only cure is one vector field per generator, A^a_μ, transforming so that D_μ = ∂_μ − i g A^a_μ T^a rotates with ψ. The count is therefore the dimension of the Lie algebra: N² − 1 for SU(N) and 1 for U(1), so SU(3)_c × SU(2)_L × U(1)_Y supplies 8 + 3 + 1 = 12 gauge fields — eight gluons G^a_μ, three Wⁱ_μ and one B_μ. Their quantum numbers are fixed by the same algebra, because they sit in the adjoint representation: gluons carry colour–anticolour (an octet, not nine states, since the singlet combination is absent), the Wⁱ form a weak-isospin triplet, and B is a singlet with zero hypercharge. The structure constants decide who talks to whom: F^a_μν = ∂_μA^a_ν − ∂_νA^a_μ + g f^abc A^b_μ A^c_ν contains cubic and quartic self-couplings, so gluons scatter off gluons and W's off W's, while the abelian B has f = 0 and no self-coupling at all. And M² A_μ A^μ is not gauge invariant, so at this stage every one of the twelve is massless.

Symmetry breaking, not the group, sorts the twelve into massless and massive. The Higgs doublet's vacuum value v = 246 GeV is invariant under the combination Q = T₃ + Y/2 and under all of SU(3)_c, so the photon and the eight gluons keep zero mass, while the three broken generators hand mass to W⁺, W⁻ = (W¹ ∓ iW²)/√2 and to Z = cos θ_W W³ − sin θ_W B. The kinetic term |D_μ φ|² gives M_W = gv/2 = 0.653 × 246.2/2 = 80.4 GeV and M_Z = M_W/cos θ_W; with sin²θ_W = 0.231 that is 80.4/0.877 = 91.7 GeV against the measured 91.2 GeV, the half-percent gap being loop corrections. Read the bookkeeping honestly: v comes from G_F, g from M_W and G_F, θ_W from neutral-current data — the relations are predictions, the numbers are fits. The gluons are a warning in the other direction. They are exactly massless, yet the strong force between nucleons reaches only about 1 fm, because confinement lets only colour singlets propagate and the residual force is carried by the lightest of them, the pion, with range ℏ/m_π c = 197.3/139.6 = 1.41 fm. A massless mediator gives infinite range only when it can travel freely.

Exchange of a mediator of mass M between two static sources gives the momentum-space amplitude g²/(|q|² + M²) — the Feynman propagator 1/(q² − M²) with the energy transfer set to zero, so that q² = −|q|². Its Fourier transform is the potential: ∫ d³q/(2π)³ e^(iq⋅r)/(|q|² + M²) = e^(−Mr)/(4πr), a contour closed around the pole at |q| = iM, so |V(r)| = (g²ℏc/4πr) e^(−r/λ) with λ = ℏ/Mc once the constants are restored. M = 0 is Coulomb: the pole sits at |q| = 0 and the 1/r tail never ends. For the W, λ = ℏc/M_W c² = 197.3 MeV fm ÷ 80 400 MeV = 2.45 × 10⁻³ fm; for the Z, 197.3/91 200 = 2.16 × 10⁻³ fm. A proton's charge radius is 0.84 fm, so the weak force reaches under a three-hundredth of it, and at r = 0.01 fm the exponential has already cut the W's potential to e^(−4.1) = 0.017 of what a massless carrier would deliver. The range depends on M alone; the coupling g only scales the height of a curve whose shape M has already fixed.

For |q²| ≪ M² the W propagator loses its q-dependence: g²/(q² − M_W²) → −g²/M_W². Two currents joined by a W then look like four fermions meeting at a point, and matching the charged-current amplitude to Fermi's L_eff = −(G_F/√2) J^μ J_μ† gives G_F/√2 = g²/(8M_W²). Run it backwards with the measured G_F = 1.166 × 10⁻⁵ GeV⁻² and M_W = 80.4 GeV: g² = 8 × 6464 × 1.166 × 10⁻⁵/1.414 = 0.426, g = 0.653, and α_W = g²/4π = 0.034 = 1/29.5 — four and a half times the electromagnetic α = 1/137. The weak interaction is weak because its mediator is heavy, not because g is small. Compare the two amplitudes at a momentum transfer of 1 GeV: e²/|q²| = 0.092 GeV⁻² against g²/(|q²| + M_W²) = 6.6 × 10⁻⁵ GeV⁻², a ratio of 1.4 × 10³. Push q above M_W and the denominators become alike, the ratio tends to e²/g² = 0.22, and the weak amplitude is the larger. Fermi's constant also carries the warning that opens the last section: with mass dimension −2 it makes cross-sections grow as G_F² s, which cannot continue past √s of a few hundred GeV — and the W propagator is exactly what stops it.

A fair comparison of interactions uses dimensionless couplings, because a dimensionful one changes size with the units. Three of the four oblige: αₛ(M_Z) = 0.118, α_W = 1/29.5, α = 1/137 — strong, weak, electromagnetic, in that order, which upsets the folklore only for the weak one. Gravity does not oblige. G_N = 6.674 × 10⁻¹¹ m³ kg⁻¹ s⁻² has to be multiplied by a squared mass before it becomes a pure number, α_G = G_N m²/ℏc, so its strength depends on who is being asked. For two protons α_G = 6.674 × 10⁻¹¹ × (1.673 × 10⁻²⁷)²/(3.162 × 10⁻²⁶) = 5.9 × 10⁻³⁹; for two W bosons it is 4.3 × 10⁻³⁵; at the Planck mass Mₚₗ = (ℏc/G_N)¹⁄² = 1.22 × 10¹⁹ GeV it reaches 1. In natural units this is the statement [G_N] = −2: α_G = (E/Mₚₗ)² grows quadratically with energy, while α, α_W and αₛ move only logarithmically. That growth is what the twelve gauge couplings never do, and it is the technical reason gravity cannot be quantised the way they are.

Look for gravity's generator in SU(3)_c × SU(2)_L × U(1)_Y and there is none: each rotates an internal label, while gravity gauges spacetime itself, and its source is the energy-momentum tensor T^μν rather than the Noether charge of a phase. A rank-two source needs a rank-two carrier, so the graviton would be spin 2, not spin 1, and its exchange is universally attractive: no anticharge, no screening. Then the dimensions bite. In ℏ = c = 1 the Lagrangian has dimension 4 and the twelve gauge couplings dimension 0, so each loop adds only logarithms and a finite list of counterterms absorbs every divergence: the theory is renormalisable. The gravitational vertex carries √G_N with dimension −1, so the n-loop term grows as (E/Mₚₗ)⁽²ⁿ⁾ relative to tree level and every order demands new counterterms — Fermi's disease, which a W at 80 GeV cured. Nobody knows gravity's W. Below Mₚₗ the model is silent about it and gravity works as an effective theory; at 10¹⁹ GeV both fail together. Gravity is not left out for being weak but because its coupling has the wrong dimension for the machinery the other three share.