University Physics IV · Particle Physics · 14.3
Interactions & Their Gauge Bosons
Four forces with four different-looking rulebooks — until you notice they are one construction run over three symmetry groups. This is where you learn to go from a symmetry to a mediator, from a mediator's mass to a range in fermis, and from a measured lifetime back to the interaction that caused it.
Build the model
Connect the measurement to the mechanism.
Start from a symmetry nobody can measure: multiplying every electron wavefunction by one constant phase changes no probability, current or energy. Now insist the phase be choosable independently at each point, and the free-particle equation breaks — differentiating a position-dependent phase leaves a stray term behind. Repairing it requires a new vector field whose own transformation cancels that debris exactly, and that field, quantised, is the photon.
Nothing was inserted by hand: the mediator, its spin-1 character and the way it attaches to charge all fall out of the demand. Run the argument on larger groups and even the head-count is fixed — one carrier for U(1), three for SU(2), eight for SU(3) — and for the non-abelian ones the carriers end up holding the very charge they respond to, so they couple to each other. The cost is specific.
A mass term for a gauge field is not gauge invariant, so the construction hands you massless mediators, and the W at 80.4 GeV and the Z at 91.2 GeV then need a separate mechanism to weigh anything. It fixes no coupling strength, no fermion mass, and no reason for these three groups rather than others. And the tidy rule that mass sets range, R = ℏc/Mc², works for the pion and the W and fails outright for the gluon, whose reach is ended by confinement instead.
- Simple definition
- A gauge boson is the spin-1 field that a local phase symmetry forces into existence, and the mass it carries — zero or otherwise — sets how far the interaction it mediates can reach.
- Example
- The W weighs 80.4 GeV/c², so its reach is R = ℏc/Mc² = 197.3 MeV⋅fm ÷ 80400 MeV = 2.45×10⁻³ fm, about 1/340 of a proton, while the massless photon's Coulomb tail never ends.
The gauge field is not added to the theory — it is whatever cancels the ∂μθ term, and that requirement also fixes how it couples.
θ(x) is dimensionless, Aμ the four-potential in V s m⁻¹, q the charge in C. Only Dμψ transforms the way ψ does.
The number of carriers is read off the symmetry rather than fitted: a ninth gluon would be colourless, unconfined, and would give a long-range strong force.
One B, three W, eight gluons. For colour, 3 ⊗ 3* = 8 ⊕ 1, and the singlet is the combination nature does not use.
Turns a mass measured in a collider into a distance measured across a nucleus, with no adjustable parameter anywhere in the conversion.
MW c² = 80.4 GeV → 2.45×10⁻³ fm; mπ c² = 139.6 MeV → 1.41 fm; M = 0 → unbounded. R lands in fm when Mc² is in MeV.
The exponential, not the 1/r, is the mediator's fingerprint: at r = R the interaction is already down to e⁻¹ = 0.368 of Coulomb strength.
g²/4π is the dimensionless coupling in natural units; r and R are lengths. Set M = 0 and the bare Coulomb 1/r returns.
At qc = 1 GeV the ratio is 6.7×10⁻⁴; at 100 GeV it is 2.6. The vertex never moved — only the propagator did.
In natural units g = 0.630 against e = 0.303; q is the momentum transfer and MW c = 80.4 GeV/c. These are amplitudes, not rates.
Strong near 10⁻²³ s, electromagnetic 10⁻²⁰ to 10⁻¹⁶ s, weak 10⁻¹⁰ s and longer — bands measured near 1 GeV, not universal laws.
Γ is the resonance width in MeV and τ the mean life in s. Δ(1232): Γ = 117 MeV gives τ = 5.6×10⁻²⁴ s.
A phase you cannot see, a field you cannot avoid
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 Aμ that shifts as Aμ → Aμ − (ℏ/q)∂μθ, and replace ∂μ everywhere by Dμ = ∂μ + i(q/ℏ)Aμ. The two changes cancel term by term, Dμψ now transforms exactly like ψ, and invariance is restored. What arrived is not a fudge: Aμ is the electromagnetic four-potential, its transformation law is the gauge freedom Maxwell's equations already had, and the qAμ coupling that dropped out is minimal coupling — the reason the same e sits at every electron vertex in every process you will ever draw.
The group counts the mediators — and charges some of them
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 W³, and the eight gluons come from — counted, not fitted. 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: [Ta, Tb] = i f(abc) Tc, and that commutator survives into the field strength as a term g f(abc) Ab Ac, 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 mediator mass turns 1/r into an exponential
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.
Where the range rule fails: massless gluons, 1.4 fm nuclei
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 near 1.4 fm he predicted a quantum near 140 MeV, and the pion turned up in 1947.
Reading the interaction off a measured lifetime
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.
The hierarchy is a snapshot taken at one energy
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/(MW c)² — an effective weak strength near 0.034 ÷ 6464 = 5×10⁻⁶. Four 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 evaporated: αW ≈ 0.034 outright. The ordering is now 0.118 : 0.034 : 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 MW c.
Change one variable at a time
Make the relationship visible.
Set the mass to 2.15 for a pion-like mediator: R = 1.40 fm, and half the Coulomb strength still survives at r = 1 fm. Slide to 4.90, the W, and R collapses to 0.0025 fm with only e⁻⁴⁰² left at 1 fm. The r = R marker always meets the 1/e line.
MEDIATOR MASS Mc²141 MeV
RANGE R = ℏc/Mc²1.3970 fm
SEPARATION r1.000 fm
STRENGTH ÷ COULOMB0.4888
Live interpretationMEDIATOR MASS Mc²: 141 MeV. RANGE R = ℏc/Mc²: 1.3970 fm. SEPARATION r: 1.000 fm. STRENGTH ÷ COULOMB: 0.4888
Catch the common trap
Explain before calculating.
Gluons are massless, so R = ℏ/(Mc) returns an unbounded range — yet the nuclear force dies away past about 1.4 fm. What accounts for this?
Choose an answer to test the model.
Practice & worked examples
Reason from the model, then test the result.
EasyTake ℏc = 197.3 MeV⋅fm. Find the Yukawa range of the Z boson (MZ c² = 91.19 GeV) and of the W (MW c² = 80.4 GeV), compare the two, and say what the same formula returns for the photon.
- The range is the mediator's reduced Compton wavelength: R = ℏ/(Mc) = ℏc/(Mc²). Working in MeV and fm keeps it to a single division, because ℏc = 197.3 MeV⋅fm.
- Z: MZ c² = 91.19 GeV = 9.119×10⁴ MeV, so RZ = 197.3 ÷ 91190 = 2.16×10⁻³ fm.
- W: MW c² = 8.04×10⁴ MeV, so RW = 197.3 ÷ 80400 = 2.45×10⁻³ fm. Their ratio RW/RZ = 2.45/2.16 = 1.13 is exactly MZ/MW = 91.19/80.4 = 1.13 — range and mass are inversely proportional, with nothing else entering.
- Photon: M = 0, so ℏc/0 is unbounded. The Yukawa factor e(−r/R) becomes e⁰ = 1 at every r and the potential reverts to bare Coulomb 1/r, which is why the electrostatic force has no range at all.
AnswerRZ = 2.16×10⁻³ fm and RW = 2.45×10⁻³ fm — about 1/390 and 1/340 of a proton's 0.84 fm charge radius — while the photon's range is unbounded.
MediumThe Δ(1232) resonance has measured width Γ = 117 MeV; the Σ⁰ has mean life 7.4×10⁻²⁰ s and the Λ 2.63×10⁻¹⁰ s. Using Γτ = ℏ with ℏ = 6.582×10⁻²² MeV⋅s, find the Δ's lifetime, name the interaction behind each decay, and give the Λ's natural width.
- Δ: τ = ℏ/Γ = 6.582×10⁻²² ÷ 117 = 5.63×10⁻²⁴ s.
- Sanity-check that as a distance: cτ = (3.00×10²³ fm s⁻¹)(5.63×10⁻²⁴ s) = 1.69 fm, about one hadron across. Something that decays in the time light needs to cross it can only be decaying strongly.
- Σ⁰ → Λγ at 7.4×10⁻²⁰ s is 7.4×10⁻²⁰ ÷ 5.63×10⁻²⁴ = 1.3×10⁴ times slower. A real photon in the final state, plus a lifetime in the 10⁻²⁰ to 10⁻¹⁶ s band, both say electromagnetic.
- Λ → pπ⁻ at 2.63×10⁻¹⁰ s is 2.63×10⁻¹⁰ ÷ 5.63×10⁻²⁴ = 4.7×10¹³ times slower than the Δ, and it changes strangeness by one, which neither the strong nor the electromagnetic interaction can do. Charged weak current.
- Λ width: Γ = ℏ/τ = 6.582×10⁻²² ÷ 2.63×10⁻¹⁰ = 2.5×10⁻¹² MeV = 2.5 μeV — orders below any spectrometer's resolution, which is why weak decays are timed rather than fitted as line shapes.
AnswerτΔ = 5.6×10⁻²⁴ s (strong); the Σ⁰ is 1.3×10⁴ times slower (electromagnetic) and the Λ 4.7×10¹³ times slower (weak); ΓΛ = 2.5 μeV.
HardW exchange and photon exchange between the same two currents give amplitudes in the ratio (g²/e²) q²/(q² + MW²c²), with g²/e² = 1/sin²θW = 4.33 and MW c² = 80.4 GeV. Evaluate the ratio at momentum transfer qc = 1 GeV and at qc = 100 GeV, convert each to a ratio of rates, and say what actually changed between the two.
- Fix the vertex factor first. In natural units e = √(4πα) = √(4π/137.0) = 0.303, and g = e/sin θW = 0.303/0.481 = 0.630, so g²/e² = 1/0.2312 = 4.33. The weak vertex is the stronger of the two, by 2.08 in amplitude.
- The propagator supplies everything else: (MW c²)² = 80.4² = 6464 GeV², against which q²c² = 1 GeV² is negligible.
- At qc = 1 GeV: ratio = 4.33 × 1/(1 + 6464) = 4.33/6465 = 6.70×10⁻⁴. Rates go as amplitude squared, so W exchange runs at (6.70×10⁻⁴)² = 4.5×10⁻⁷ of the electromagnetic rate — the suppression that earned the interaction its name.
- At qc = 100 GeV: q²c² = 1.00×10⁴ GeV², so the ratio is 4.33 × 10⁴/(10⁴ + 6464) = 4.33 × 0.607 = 2.63, and the rate ratio is 2.63² = 6.9.
- Nothing about the vertex moved: g is 0.630 at both energies. The whole hierarchy was the factor q²/(q² + MW²c²), worth 1.5×10⁻⁴ at 1 GeV and 0.607 at 100 GeV. Above the W mass the weak interaction is the stronger one.
Answer6.70×10⁻⁴ in amplitude (4.5×10⁻⁷ in rate) at qc = 1 GeV, rising to 2.63 (6.9 in rate) at 100 GeV. The coupling never changed; the propagator did.