14. Quark Flavour Mixing (CKM)¶

Note

A modelling chapter, not a pipeline stage: it shows how feynlag handles three-generation quark mixing without a symbolic diagonalization that does not exist in closed form.

Why insertion, not diagonalization¶

The CKM matrix is \(V = U_u^\dagger U_d\), the mismatch between the unitary rotations that diagonalize the up- and down-type Yukawas. For a generic three-generation complex Yukawa those rotations have no closed symbolic form — sympy’s Matrix.diagonalize returns unusable nested radicals (the same wall diagonalize_svd hits, flagged in examples/sm_scalar_gauge.py). So feynlag does what FeynRules’ Standard Model does: work in the quark mass basis with diagonal Yukawas and insert the CKM matrix directly into the charged current,

\[\mathcal L_W = \frac{g}{\sqrt2}\,W^+_\mu\;\bar u_i\,\gamma^\mu P_L\,V_{ij}\,d_j \;+\;\text{h.c.}\]

The neutral currents (\(Z\), \(\gamma\), gluon) are written flavour-diagonal in the mass basis, so they carry no CKM factor — the GIM mechanism, no flavour-changing neutral currents.

The unitarity trap¶

\(V\) must be unitary. A matrix of nine independent symbols is not: \(\sum_k V^*_{ki}V_{kj}\neq\delta_{ij}\) symbolically, which would fake a \(Z\)-current FCNC through the same rotation that produces the \(W\) mixing. There are two escapes, and feynlag uses both:

Exact standard parametrization — standard_ckm() builds \(V\) from three mixing angles and one CP phase in the PDG convention, unitary as a trigonometric identity, so V†V collapses to the identity under sympy.simplify.
Mass-basis insertion — putting \(V\) only in the \(W\) current, never in the neutral currents, so GIM holds by construction.

The two-generation Cabibbo case can be driven through the ordinary real mass-basis rotation machinery (7. Diagonalization and the Physical Basis), and there the GIM cancellation is an exact symbolic result: rotating the flavour-diagonal neutral current by rotation_2x2(θ_c) gives back a diagonal current (\(\cos^2+\sin^2=1\)), while the charged current picks up the off-diagonal \(\sin\theta_c\) term. This is the physics ground truth pinned in tests/test_ckm.py::TestTwoGenerationGIM.

Parameters and UFO export¶

CKM elements are exported as scalar parameters — never an IndexedBase/matrix, because ParameterSet is scalar-only and the UFO code printer would emit an invalid V[0,1] subscript. standard_ckm returns:

four real external parameters — the angles th12, th13, th23 and the phase deltaCP (PDG central values);
nine complex internal parameters Vud … Vtb, each defined by its standard-parametrization expression.

Internals are emitted type='complex' in dependency order, so the phase in Vub/Vtd survives; verify_ufo_numeric() evaluates the whole set at export.

Usage¶

from feynlag import standard_ckm

params, V = standard_ckm()          # V is a 3×3 sympy Matrix, exactly unitary
L_W = g / sqrt(2) * Wp * sum(
    V[a, b] * Bilinear(ubar[a], gamma_mu_PL, d[b])
    for a in range(3) for b in range(3))

See examples/sm_ckm.py for the full worked demo (charged-current vertices, GIM, UFO round-trip).

Verification¶

tests/test_ckm.py pins: the two-generation GIM cancellation (mixing in \(W\), none in \(Z\)); the three-generation V†V = 1 symbolically plus PDG magnitudes \(|V_{ud}|\approx0.974\), \(|V_{us}|\approx0.225\), \(|V_{cb}|\approx0.042\) and a non-zero CP phase in \(V_{ub}\); the inserted \(W\bar u d\) vertex \(= i\,g/\sqrt2\,V_{ud}\,\gamma^\mu P_L\); and the complex CKM internals round-tripping through UFO export.