CKM Matrix

The setup

Quarks come in two flavors per generation: up-type (u, c, t) and down-type (d, s, b). The Higgs mechanism gives each quark a mass; the resulting mass eigenstates are what propagate freely through spacetime. But the weak interaction — mediated by the W boson — couples quarks in a different basis, the weak eigenstates. The two bases are related by a unitary 3x3 rotation:

| d' |   | V_ud  V_us  V_ub | | d |
| s' | = | V_cd  V_cs  V_cb | | s |
| b' |   | V_td  V_ts  V_tb | | b |

The primed states on the left are what W sees; the unprimed on the right are mass eigenstates. The matrix V is CKM. Unitarity (V V† = I) leaves four physical parameters: three real mixing angles and one complex phase δ.

Measured magnitudes

From the 2024 Particle Data Group global fit (rounded):

	d	s	b
u	0.97373	0.2243	0.00382
c	0.221	0.975	0.0408
t	0.0086	0.0415	1.014

(Numbers are |V_ij|; central values with errors of order 0.01 or better on diagonals, 1–5% on off-diagonals.)

The diagonal is close to 1; entries grow smaller the more generations the transition crosses. |V_us| (one-step) is 0.22; |V_cb| (one-step in the lower-right) is 0.04; |V_ub| (two-step, the corner) is 0.004. The Wolfenstein expansion makes this hierarchy explicit: each off-diagonal is one extra power of λ = 0.225.

Worked example: K− semileptonic decay

Consider the decay K− → π° e− ν-bar_e. The kaon is a (s, u-bar) bound state. Inside the kaon a strange quark turns into an up quark by emitting a W−, which decays leptonically. The vertex amplitude is proportional to V_us.

amplitude ~ V_us * (form factor) * (lepton current)
rate ~ |V_us|^2 * G_F^2 * f+(0)^2 * m_K^5 / (192 pi^3) * (kinematic phase space)

Measured K− → π° e− ν-bar branching ratio: 5.07%. From this plus the lattice-QCD form factor f+(0) = 0.9698, one extracts |V_us| f+(0) = 0.2176, giving |V_us| = 0.2243. The Cabibbo angle is θ_C = arcsin(0.2243) = 12.97°.

CP violation

The single complex phase δ means some products of CKM elements are not real. Make a transition and its CP-conjugate (swap particles for antiparticles and mirror) and the rates can differ. The amount of CP violation in any process is bounded by the Jarlskog invariant:

J = Im(V_us V_cb V*_ub V*_cs) ≈ 3.08 × 10^-5

J is convention-independent: any product of four CKM elements with crossed indices has imaginary part ±J. Tiny in absolute terms — but it generates measurable asymmetries because B-meson lifetimes are long enough for interference to develop. The biggest CP asymmetry observed so far: A_CP(B° → J/ψ K_S) = sin(2β) = 0.706 ± 0.018.

The unitarity triangle

Unitarity gives six off-diagonal relations of the form V_i1 V*_j1 + V_i2 V*_j2 + V_i3 V*_j3 = 0. Each sum is three complex vectors closing to zero — a triangle in the complex plane. The most useful one (the "b-d triangle") is:

V_ud V*_ub + V_cd V*_cb + V_td V*_tb = 0

All three sides are O(λ³), so the triangle is not degenerate. Its three angles α, β, γ are CP-violation observables in B decays. Current measurements:

Angle	Measured	From decay
β	22.2° ± 0.7°	B° → J/ψ K_S
α	85.2° ± 4°	B° → ππ, ρρ, ρπ
γ	65.9° ± 3.5°	B → DK tree decays
Sum	173° ± 5°	Consistent with 180°

The triangle closes, so far. Any failure to close would indicate physics beyond the Standard Model.

CKM vs PMNS (the lepton analogue)

	CKM (quarks)	PMNS (neutrinos)
Off-diagonal sizes	Small — hierarchical, < 0.23	Large — near maximal, > 0.5
Cabibbo-like angle	13°	θ_12 ≈ 33.4° (solar)
Atmospheric-like angle	θ_23 ≈ 2.4°	θ_23 ≈ 49°
Reactor-like angle	θ_13 ≈ 0.21°	θ_13 ≈ 8.6°
CP phase δ	About 65° (well-measured)	About 215° (hint, not yet established)
Majorana phases	None (Dirac fermions)	Two extra (if neutrinos are Majorana)

Why CKM is hierarchical and PMNS is anarchic remains an open question of flavor physics.

JavaScript — the Wolfenstein matrix

// Wolfenstein parameters (PDG 2024 central values)
const lambda = 0.22500;
const A      = 0.826;
const rho    = 0.159;
const eta    = 0.348;

// Build CKM matrix to O(lambda^4) accuracy
function makeCKM() {
  const l = lambda, l2 = l*l, l3 = l*l*l;
  const rebar = rho * (1 - l2/2);
  const etabar = eta * (1 - l2/2);
  return [
    // |V_ud|             V_us            V_ub
    [1 - l2/2,            l,              A*l3 * complex(rebar, -etabar)],
    [-l,                  1 - l2/2,       A*l2],
    [A*l3 * complex(1 - rebar, -etabar), -A*l2, 1]
  ];
}

function complex(re, im) { return { re, im, abs: () => Math.hypot(re, im) }; }

// Magnitude of each entry
const V = makeCKM();
console.log('|V_ud| =', V[0][0].toFixed(4));     // 0.9747
console.log('|V_us| =', V[0][1].toFixed(4));     // 0.2250
console.log('|V_ub| =', V[0][2].abs().toFixed(4)); // 0.00370
console.log('|V_cb| =', V[1][2].toFixed(4));     // 0.0418

// Jarlskog invariant
const J = A*A * Math.pow(lambda, 6) * eta * (1 - lambda*lambda/2);
console.log('J =', J.toExponential(2));  // 3.0e-5

// Cabibbo angle in degrees
const theta_C = Math.asin(lambda) * 180/Math.PI;
console.log('theta_C =', theta_C.toFixed(2) + ' deg');  // 13.00 deg

// CP phase
const delta = Math.atan2(eta, rho) * 180/Math.PI;
console.log('delta =', delta.toFixed(1) + ' deg');  // 65.4 deg

Where CKM matters

• Kaon physics. Kaon semileptonic and rare decays measure |V_us|, |V_td|, |V_ts|. NA62 measures K+ → pi+ nu nu-bar.
• B physics. LHCb, Belle II measure CP-violating asymmetries to over-constrain the unitarity triangle.
• Top physics. Single-top production at LHC directly measures |V_tb|.
• Nuclear beta decay. Superallowed 0+ → 0+ transitions give the most precise |V_ud|.
• Cosmology. CKM CP violation is one of three Sakharov conditions; quantifies how much SM contributes to baryogenesis.
• BSM searches. CKM unitarity tests are precision probes of new physics; the "Cabibbo angle anomaly" (tension at the 3σ level) is a current hot topic.

Common mistakes

• Calling CKM a probability matrix. CKM elements are amplitudes, not probabilities. |V_ij|^2 is the probability factor in a rate.
• Forgetting unitarity. The matrix is unitary by construction. Row and column sums of |V_ij|^2 must equal 1. This drastically reduces the independent parameters.
• Confusing weak and mass eigenstates. The CKM matrix relates the two bases. Mass eigenstates propagate; weak eigenstates appear in W vertices.
• Treating δ as a free knob. The CP phase is constrained by interference patterns in B-meson and kaon decays. Current value about 65 degrees with errors at the few-degree level.
• Assuming CKM explains baryogenesis. Jarlskog J = 3 x 10^-5 is far too small. SM CKM CP violation falls short of the observed baryon asymmetry by about ten orders of magnitude.
• Confusing with PMNS. Same mathematical structure (3x3 unitary) but the lepton matrix has large angles and possibly extra Majorana phases.