Quark Mixing and the CKM Matrix

The charged weak current changes an up-type quark into a down-type quark, but it does not respect the quark generations cleanly: an up quark turns most often into a down, sometimes into a strange, rarely into a bottom. The reason is a mismatch between two bases. The quarks have definite mass in one basis and definite weak coupling in another, and the two are related by a rotation. For one generation the rotation is invisible; for three it becomes the Cabibbo–Kobayashi–Maskawa matrix, whose single complex phase is the only place the Standard Model can violate CP. This lesson builds that matrix from Cabibbo's angle through GIM's charm prediction to the unitarity triangle.

The Cabibbo rotation

Before charm was known, the weak decays of strange particles posed a puzzle: they proceeded, but suppressed relative to strangeness-conserving decays. Cabibbo's 1963 resolution was that the down-type quark the $W$ couples to is not the mass eigenstate $d$ but a rotated combination. With two generations, the weak eigenstate is the mass eigenstate turned through the Cabibbo angle ${\theta }_c$:

The $W$ couples $u$ to $d^{\prime }$ with full strength, which decomposes into a strangeness-conserving amplitude $u\to d$ weighted by $\cos {\theta }_c$ and a strangeness-changing amplitude $u\to s$ weighted by $\sin {\theta }_c$. Measuring the suppression of strange decays fixes

A kaon decays precisely because of the small $\sin {\theta }_c$ term: the strange quark has no lighter same-generation partner to decay into, so it must reach across generations to an up quark, paying a factor ${\sin }^2{\theta }_c\approx 0.05$ in rate. That factor is why strange particles live comparatively long.¹

The GIM mechanism and the charm prediction

Cabibbo's rotation has a dangerous side effect. Written out, the neutral current computed from the rotated $d^{\prime }$ contains a cross term $\propto \sin {\theta }_c\cos {\theta }_c\bar{d}s$ — a flavour-changing neutral current (FCNC) that would let a strange quark turn into a down quark by emitting a $Z$. Such transitions drive the decay $K^0\to {\mu }^{+}{\mu }^{-}$, and they were observed to be far rarer than this predicted.

In 1970 Glashow, Iliopoulos, and Maiani resolved this by adding a fourth quark, the charm, completing a second doublet $(c,s^{\prime })$. The charm's neutral-current contribution carries the opposite sign, $-\sin {\theta }_c\cos {\theta }_c\bar{d}s$, and the two cancel exactly when the up and charm masses are equal:

The cancellation is not perfect because $m_c\ne m_u$; the residue scales as $m_c^2/M_W^2$, which correctly gives the tiny observed rate. This GIM mechanism predicted both the existence of the charm quark and, roughly, its mass — four years before charm was discovered in the $J/\psi$ in 1974. Its modern statement is sharp: there are no tree-level flavour-changing neutral currents in the Standard Model; they arise only at loop level and are suppressed by the near-cancellation among the up-type quarks.²

Three generations: the CKM matrix

With a third generation $(t,b)$, the rotation becomes a $3\times 3$ unitary matrix acting on the down-type quarks. In the mass eigenbasis, the charged-current coupling of an up-type quark $u_i$ to a down-type quark $d_j$ carries the matrix element $V_{ij}$:

The matrix arises from the mismatch $V=(V^u{)}^{†}{V}^d$ between the unitary rotations that separately diagonalize the up-type and down-type Yukawa matrices — the leftover parameters that could not be absorbed into quark masses.³ A general unitary $3\times 3$ matrix has nine parameters; rephasing the quark fields removes five, leaving three real mixing angles and one irreducible complex phase. That surviving phase is decisive: it is the sole source of CP violation in the quark sector. A crucial counting fact is that two generations leave no physical phase — CP cannot be violated — so Kobayashi and Maskawa argued in 1973 that a third generation was required to accommodate the observed CP violation, before charm, let alone bottom or top, had been found.

The measured magnitudes reveal a steep hierarchy, nearly diagonal:

Transitions within a generation are near unity; crossing one generation costs a factor $\sim 0.2$ (the Cabibbo suppression, visible in $V_{us}$ and $V_{cd}$); crossing two generations costs far more.⁴

The Wolfenstein parametrization

The hierarchy invites an expansion. Wolfenstein observed that every element is a power of the single small number $\lambda =\sin {\theta }_c\approx 0.225$, and wrote

with $A$, $\rho$, $\eta$ all of order unity. The measured values are

The parametrization makes three things plain: the diagonal is $1$ to order ${\lambda }^2$; the Cabibbo mixing enters at order $\lambda$; and the complex phase$\eta$ appears only in the far-corner elements $V_{ub}$ and $V_{td}$, at order ${\lambda }^3$. CP violation is therefore intrinsically small — not because the phase is small, but because it rides on the most suppressed elements.⁵

The unitarity triangle

Unitarity, $V_{\mathrm{CKM}}^{†}{V}_{\mathrm{CKM}}=1$, forces each pair of distinct columns to be orthogonal. The orthogonality of the first and third down-type columns,

is the interesting one: all three terms are comparable, of order ${\lambda }^3$, so they form a genuine triangle in the complex plane rather than collapsing. Dividing through by $V_{cd}{V}_{cb}^{\ast }$ normalizes the base to unit length and places the apex at $(\bar{\rho },\bar{\eta })$. The three interior angles $\alpha$, $\beta$, $\gamma$ are measured independently in $B$-meson decays, and the requirement that the triangle close — that the three sides and three angles are mutually consistent — is among the most stringent tests of the Standard Model. So far every measurement lands on the same apex.⁶

The triangle's non-degeneracy is CP violation: its area equals half the Jarlskog invariant $J\approx 3\times 10^{-5}$, a basis-independent measure of how much the CKM matrix breaks CP. A flat triangle — a real matrix — would mean no CP violation. The apex sitting well above the real axis, $\bar{\eta }\ne 0$, is the geometric statement that the quark sector violates CP.

Summary

The weak eigenstates of the quarks are rotations of the mass eigenstates. Cabibbo's single angle ${\theta }_c\approx 13^{\circ }$ mixed down and strange, explaining suppressed strange decays; the GIM mechanism cancelled the resulting flavour-changing neutral currents by adding the charm quark, predicting it before its discovery. Three generations promote the rotation to the unitary CKM matrix, with three angles and one irreducible phase — the only source of CP violation in the Standard Model, and the reason Kobayashi and Maskawa demanded a third generation. The Wolfenstein parametrization exposes the steep $\lambda$-hierarchy and confines the phase to $V_{ub}$ and $V_{td}$; unitarity closes into a triangle whose area, the Jarlskog invariant, quantifies the CP violation. Where that violation is actually observed — in the neutral kaons and $B$ mesons — is the subject of the next lesson.