Weak Force

What the weak force does

Of the four known fundamental forces, three are familiar — gravity holds galaxies, electromagnetism runs chemistry and light, the strong force binds quarks into nucleons and nucleons into nuclei. The fourth, the weak force, is responsible for something the others cannot do: it changes the flavour of a quark or lepton. A down quark turns into an up quark; an electron pairs up with an electron neutrino; a muon decays into an electron and two neutrinos. None of these are possible without the weak interaction.

The defining process is nuclear beta decay. A neutron — which is otherwise a stable composite of two down quarks and one up — slowly converts into a proton, releasing an electron and an antineutrino. The conversion happens because one of the down quarks emits a virtual W− boson and becomes an up quark; the W− then materializes into an electron and an antineutrino. Beta decay is how the Sun produces neutrinos, how carbon-14 dating works, and how heavy nuclei in supernovae build their neutron-rich daughters.

The W and Z bosons

Boson	Mass (GeV)	Charge	Width Γ (GeV)	Lifetime (s)	Discovered
W+	80.379	+1	2.085	3.16 × 10⁻²⁵	CERN UA1/UA2 1983
W−	80.379	−1	2.085	3.16 × 10⁻²⁵	CERN UA1/UA2 1983
Z⁰	91.188	0	2.495	2.64 × 10⁻²⁵	CERN UA1 May 1983

These are immense by particle-physics standards — the W is ~85 times heavier than a proton, the Z ~97 times. Because they're so heavy, virtual W and Z exchange can travel only ~ℏ/(M c) ≈ 2.5 × 10⁻¹⁸ m before "going off shell" too far to matter. That's the range of the weak force: about a thousandth of a proton diameter. Outside that radius the force is undetectable; inside it, the force is comparable in strength to electromagnetism, as electroweak unification requires.

Beta decay at the quark level

The Feynman diagram for free neutron beta-minus decay:

n  =  u d d   →   u u d  =  p
                     |
                     d  →  u  +  W−
                                  W−  →  e−  +  ν̄_e

Total:  n  →  p  +  e−  +  ν̄_e

Only one of the three quarks changes; the other two are spectators. The W− is virtual — never observed directly in beta decay because the available energy (Q ≈ 0.78 MeV for free neutrons) is far below the W mass of 80 GeV. The matrix element scales as g²/M_W², which at low energies reduces to the Fermi four-fermion contact coupling G_F.

The free-neutron lifetime is 880.2 ± 1.0 s (or 611 s for the half-life). Inside a stable nucleus, the energy bookkeeping usually forbids the decay. Inside an unstable nucleus, beta decay drains protons or neutrons until the nuclide reaches the line of stability.

Worked example — comparing strengths

At low energies the four fundamental forces have wildly different strengths between two protons separated by 10⁻¹⁵ m (a femtometre, roughly nucleon size):

Force	Relative strength	Range	Mediator
Strong	1	~10⁻¹⁵ m	Gluon (8)
Electromagnetic	~10⁻²	Infinite	Photon
Weak	~10⁻⁶	~10⁻¹⁸ m	W±, Z
Gravitational	~10⁻³⁸	Infinite	(Graviton — hypothetical)

The weak force's low-energy weakness comes from its short range, not its intrinsic coupling. At energies near or above M_W, the running couplings of weak and EM converge — they merge into one electroweak coupling, mediated by a single SU(2) × U(1) gauge structure.

JavaScript — weak-force calculator

// Constants
const M_W_GeV = 80.379;
const M_Z_GeV = 91.188;
const G_F_GeV_minus2 = 1.166e-5;   // Fermi constant
const g_W = 0.6531;                // weak coupling (SU(2))
const sin2_theta_W = 0.23121;      // weak mixing angle, sin²θ_W
const e = 0.30282;                 // electromagnetic coupling
const hbar_GeVs = 6.582e-25;       // ℏ in GeV·s
const c_m_s = 2.998e8;
const hbarc_GeV_m = 1.973e-16;

// Range of the weak force
function rangeFromMass(M_GeV) {
  // R ≈ ℏc / (M c²)  with units in metres if M in GeV
  return hbarc_GeV_m / M_GeV;
}

console.log(`W range: ${rangeFromMass(M_W_GeV).toExponential(2)} m`);
// 2.5e-18 m
console.log(`Z range: ${rangeFromMass(M_Z_GeV).toExponential(2)} m`);
// 2.2e-18 m

// W boson lifetime from total width
function lifetimeFromWidth_s(Gamma_GeV) {
  return hbar_GeVs / Gamma_GeV;
}
console.log(`W lifetime: ${lifetimeFromWidth_s(2.085).toExponential(2)} s`);  // 3.2e-25 s
console.log(`Z lifetime: ${lifetimeFromWidth_s(2.495).toExponential(2)} s`);  // 2.6e-25 s

// Fermi vs. running weak coupling consistency
const GF_check = (g_W * g_W) / (4 * Math.sqrt(2) * M_W_GeV * M_W_GeV);
console.log(`G_F from g_W and M_W: ${GF_check.toExponential(3)} GeV^-2`);
// ≈ 1.16e-5 ✓

// Free-neutron beta decay half-life (Sargent / Fermi golden rule, schematic)
function neutronBetaHalflife_s() {
  // Empirical Q-value 0.7823 MeV; phase-space factor f ≈ 1.6887; matrix element |V_ud|² ≈ 0.9492² × g_A² adjustment
  // We just return the measured value here
  return 611.0;
}
console.log(`n half-life: ${neutronBetaHalflife_s()} s`);

// Weak mixing angle relations
const sin_thW = Math.sqrt(sin2_theta_W);
const cos_thW = Math.sqrt(1 - sin2_theta_W);
console.log(`sin θ_W = ${sin_thW.toFixed(4)} ; cos θ_W = ${cos_thW.toFixed(4)}`);
// M_W = M_Z cos θ_W
console.log(`M_W predicted: ${(M_Z_GeV * cos_thW).toFixed(2)} GeV vs measured ${M_W_GeV}`);
// 79.96 vs 80.38 — close; small electroweak radiative corrections explain the gap

// Cross-section for νe + e → νe + e at low energies (Fermi point coupling)
function nuESigma_cm2(E_GeV) {
  // σ ≈ (2 G_F² m_e E) / π  in natural units; convert to cm²
  const m_e_GeV = 0.000511;
  const sigma_GeV_minus2 = (2 * G_F_GeV_minus2 * G_F_GeV_minus2 * m_e_GeV * E_GeV) / Math.PI;
  return sigma_GeV_minus2 * 3.894e-28;  // 1 GeV^-2 = 3.894e-28 cm²
}
console.log(`σ(νe → νe) at 1 GeV: ${nuESigma_cm2(1).toExponential(2)} cm²`);
// ~1e-42 cm² — why neutrino detectors need kilotons of material

Parity and CP violation

Until 1956, every known interaction was symmetric under parity P — reflecting space coordinates produced a process that was just as physical. Lee and Yang noticed that no experiment had actually tested parity in the weak sector and proposed that the famous θ-τ puzzle (two particles with identical masses but opposite parities) could be resolved if the weak force violated P. Chien-Shiung Wu took up the challenge: she cooled cobalt-60 nuclei in a magnetic field and counted beta electrons in two hemispheres. The electrons came out preferentially in one direction — the mirror image of the experiment never occurred. Parity was maximally violated.

The mechanism: the weak force only couples to left-handed particles and right-handed antiparticles. Right-handed neutrinos and left-handed antineutrinos are sterile — they don't feel the weak force at all. This chiral structure is built into the V–A (vector minus axial-vector) form of the weak current.

In 1964, Cronin and Fitch observed that the combined CP symmetry (parity plus particle–antiparticle swap) is also violated, at the parts-per-thousand level in neutral kaon decay. CP violation is encoded in the CKM quark-mixing matrix as a single irreducible complex phase. It is a necessary precondition for the cosmological excess of matter over antimatter — though the Standard Model's amount of CP violation is too small by many orders of magnitude to account for that excess alone.

History — Fermi to CERN to LEP

Enrico Fermi proposed his beta-decay theory in 1933 — a contact four-fermion interaction with strength G_F. It correctly described the kinematics and spectrum of beta electrons but had no underlying gauge structure. In the 1950s the strength was measured precisely; by the 1960s a unified gauge theory of weak and electromagnetic interactions began to emerge: Glashow (1961), Salam (1968), Weinberg (1967). Renormalizability was proved by 't Hooft (1971). Neutral currents (Z exchange) were detected at CERN's Gargamelle bubble chamber in 1973. The W and Z bosons themselves were produced at CERN's SPS converted into a proton–antiproton collider in 1983, at the masses predicted ten years earlier. Final precision measurements came from LEP (1989–2000): M_Z to 2 ppm; M_W to 12 MeV. The Standard Model's electroweak sector is the most precisely tested theory in physics.

Common misconceptions

• "The weak force is intrinsically weak." No — at low energies it appears weak because of the W and Z masses, not because the coupling g_W is small. At energies near M_W the weak and electromagnetic couplings are comparable.
• "Beta decay shoots an electron straight out of the nucleus." No — there are no electrons inside the nucleus before the decay. The electron is created at the decay vertex when a W− converts; the original constituent was a down quark.
• "Neutrinos are weakly interacting therefore they don't matter." They permeate everything. About 65 billion solar neutrinos cross every square centimetre of Earth's surface per second. They drive supernova explosions and shape big-bang nucleosynthesis.
• "The W and Z were predicted from beta decay measurements." Partly. Beta decay constrains G_F = g²/(8M_W²). The individual values of g and M_W only emerge once you assume gauge unification with electromagnetism — Glashow–Salam–Weinberg.
• "Parity violation means right-handed particles don't exist." No — right-handed electrons, quarks, etc. exist and feel electromagnetism and gravity. They just don't couple to the W boson. Right-handed neutrinos may exist (and would explain neutrino mass via the seesaw mechanism) but if so they're sterile.
• "CP violation explains the matter–antimatter asymmetry." Not directly. The Standard Model's CP violation in the CKM is too small by ~10 orders of magnitude. Additional sources are needed — leptogenesis via Majorana neutrinos is the leading candidate.

Weak force vs other fundamental forces

Force	Mediator	Mass	Range	Coupling	Acts on
Strong	Gluon (8)	0	~1 fm (confined)	α_s ≈ 0.12 at 100 GeV	Colour-charged quarks, gluons
Electromagnetic	Photon	0	Infinite	α ≈ 1/137	Electric-charged particles
Weak	W+, W−, Z	80.4, 91.2 GeV	~10⁻¹⁸ m	α_W ≈ 1/30 (at M_W)	All fermions; flavour-changing
Gravity (not SM)	Graviton (hyp.)	0 (presumed)	Infinite	~10⁻³⁹ at fm scale	Everything with energy
Higgs Yukawa	Higgs boson	125.25 GeV	~10⁻¹⁸ m	Mass-dependent	All massive fermions, W, Z
Fermi (effective)	none (point)	—	0 (contact)	G_F ≈ 1.166 × 10⁻⁵ GeV⁻²	Low-E limit of weak

Where the weak force matters

• Stellar fusion. The first step of the proton–proton chain is p + p → d + e+ + ν_e — a weak conversion that's the bottleneck for solar burning.
• Big-bang nucleosynthesis. The freeze-out of weak interactions at ~1 second after the bang determines the neutron-to-proton ratio and thus primordial helium abundance.
• Radioactive dating. Carbon-14 β-decays with a 5730 yr half-life — same weak process, just inside a different nucleus.
• Neutrino observatories. Super-Kamiokande, IceCube, SNO+ — all detect neutrinos via charged-current (W exchange) or neutral-current (Z exchange) interactions.
• Reactor physics. Fission products are neutron-rich and β-decay back toward stability, releasing the residual heat that requires reactor cooling after shutdown.
• Supernova mechanism. Core-collapse supernovae release 99% of their energy as neutrinos via electron capture and pair processes; the next-door SN 1987A delivered the first detected extra-solar neutrino burst.
• Particle physics frontiers. The LHC's electroweak measurements (W mass, sin² θ_W, Higgs couplings) probe the SM at the per-mille level and search for new physics.