Kobayashi and Maskawa: the CKM matrix and three quark generations

Cronin and Fitch’s CP violation had no accepted explanation until Kobayashi and Maskawa (1973) showed that a third generation of quarks – at the time, an unconfirmed extrapolation – forces the resulting three-generation quark-mixing matrix to admit exactly one physical complex phase, making CP violation an unavoidable feature of the weak interaction rather than an ad hoc addition. No dedicated CKM function exists in physicskit.particle (the package’s other CP-violation model, the neutral-kaon system, is phenomenological rather than built from an underlying mixing matrix); this example builds the standard three-generation unitary mixing matrix directly from Euler-like mixing angles and one phase – the same parametrization structure as the PMNS lepton-mixing matrix – checks its unitarity, and shows that the single complex phase is exactly what a two-generation (Cabibbo) matrix cannot admit.

import matplotlib.pyplot as plt
import numpy as np

The two-generation Cabibbo matrix: always real, no CP violation possible

A single mixing angle gives an orthogonal (real) 2x2 rotation – no room for a complex phase at all, since any phase on a 2x2 unitary matrix’s entries can be rotated away by redefining the quark fields’ overall phases.

theta_c = np.radians(13.02)  # the real Cabibbo angle
V_cabibbo = np.array([[np.cos(theta_c), np.sin(theta_c)], [-np.sin(theta_c), np.cos(theta_c)]])
print("Two-generation Cabibbo matrix (always real):")
print(np.round(V_cabibbo, 6))
print(f"is it unitary? max|V^dagger V - I| = {np.max(np.abs(V_cabibbo.T @ V_cabibbo - np.eye(2))):.2e}")

Two-generation Cabibbo matrix (always real):
[[ 0.974291  0.225291]
 [-0.225291  0.974291]]
is it unitary? max|V^dagger V - I| = 9.59e-19

The three-generation CKM matrix: one unavoidable complex phase

The standard parametrization: three mixing angles (theta_12, theta_23, theta_13) and one CP-violating phase delta, combined as a product of three complex rotations – with the phase entering only once a third generation exists at all.

theta12, theta23, theta13 = np.radians([13.04, 2.38, 0.201])  # close to the measured CKM angles
delta = np.radians(68.8)  # close to the measured CKM CP phase


def ckm_matrix(t12, t23, t13, delta):
    c12, s12 = np.cos(t12), np.sin(t12)
    c23, s23 = np.cos(t23), np.sin(t23)
    c13, s13 = np.cos(t13), np.sin(t13)
    e_idelta = np.exp(1j * delta)
    return np.array(
        [
            [c12 * c13, s12 * c13, s13 * np.conj(e_idelta)],
            [-s12 * c23 - c12 * s23 * s13 * e_idelta, c12 * c23 - s12 * s23 * s13 * e_idelta, s23 * c13],
            [s12 * s23 - c12 * c23 * s13 * e_idelta, -c12 * s23 - s12 * c23 * s13 * e_idelta, c23 * c13],
        ]
    )


V_ckm = ckm_matrix(theta12, theta23, theta13, delta)
print("\nThree-generation CKM matrix, |V_ij| (close to the measured values):")
print(np.round(np.abs(V_ckm), 6))

unitarity_error = np.max(np.abs(V_ckm.conj().T @ V_ckm - np.eye(3)))
print(f"\nis it unitary? max|V^dagger V - I| = {unitarity_error:.2e}")

Three-generation CKM matrix, |V_ij| (close to the measured values):
[[0.974207 0.22563  0.003508]
 [0.225488 0.973361 0.041527]
 [0.008736 0.040749 0.999131]]

is it unitary? max|V^dagger V - I| = 2.22e-16

The Jarlskog invariant: a basis-independent measure of the CP phase

A single number that vanishes if and only if there is no CP violation – for the real 2x2 Cabibbo matrix it is identically zero; for the CKM matrix with delta != 0 it is not.

J = np.imag(V_ckm[0, 0] * V_ckm[1, 1] * np.conj(V_ckm[0, 1]) * np.conj(V_ckm[1, 0]))
print(f"\nJarlskog invariant J = {J:.3e} (nonzero -- CP violation is unavoidable once three generations mix)")

V_ckm_no_phase = ckm_matrix(theta12, theta23, theta13, 0.0)
J_at_delta0 = np.imag(V_ckm_no_phase[0, 0] * V_ckm_no_phase[1, 1] * np.conj(V_ckm_no_phase[0, 1]) * np.conj(V_ckm_no_phase[1, 0]))
print(f"Jarlskog invariant at delta=0 (hypothetically no phase): {J_at_delta0:.3e} (vanishes -- delta is exactly what makes J nonzero)")

Jarlskog invariant J = 2.983e-05 (nonzero -- CP violation is unavoidable once three generations mix)
Jarlskog invariant at delta=0 (hypothetically no phase): -0.000e+00 (vanishes -- delta is exactly what makes J nonzero)

Visualizing the mixing strengths

fig, ax = plt.subplots(figsize=(5.5, 5))
im = ax.imshow(np.abs(V_ckm), cmap="viridis", vmin=0, vmax=1)
ax.set_xticks([0, 1, 2])
ax.set_xticklabels(["d", "s", "b"])
ax.set_yticks([0, 1, 2])
ax.set_yticklabels(["u", "c", "t"])
for i in range(3):
    for j in range(3):
        ax.text(j, i, f"{np.abs(V_ckm[i, j]):.3f}", ha="center", va="center", color="white" if np.abs(V_ckm[i, j]) < 0.6 else "black")
fig.colorbar(im, ax=ax, label="|V_ij|")
ax.set_title("CKM matrix magnitudes: strongly diagonal, small cross-generation mixing")
fig.tight_layout()

plt.show()