--- title: Quark Mixing and the CKM Matrix module: The Weak Interaction moduleNumber: 7 lessonNumber: 3 order: 703 summary: > The quark eigenstates the weak force acts on are not the mass eigenstates. Cabibbo captured this with one rotation angle; the GIM mechanism added a fourth quark to cancel dangerous flavour-changing neutral currents and predicted charm before its discovery. Three generations promote the rotation to the unitary Cabibbo–Kobayashi–Maskawa matrix — three angles and one irreducible complex phase, the sole source of Standard-Model CP violation. The Wolfenstein parametrization exposes its steep hierarchy, and unitarity closes into a triangle whose area measures the phase. topics: [The Weak Interaction] draft: false sources: - book: Griffiths ref: "Ch. 10 — Weak Interactions, §10.8 (Cabibbo angle, GIM, CKM)" - book: Thomson ref: "Ch. 14 — CP violation and the CKM matrix" - book: Tong ref: "The Standard Model (Cambridge Part III), §6.1–6.2 (diagonalising the Yukawas, the CKM matrix)" - book: PDG ref: "CKM matrix elements and the unitarity triangle — pdg.lbl.gov" --- 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** $\theta_c$: $$ d' = \cos\theta_c\, d + \sin\theta_c\, s , \qquad s' = -\sin\theta_c\, d + \cos\theta_c\, s . $$ The $W$ couples $u$ to $d'$ 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 $$ \sin\theta_c \approx 0.225, \qquad \theta_c \approx 13^\circ . $$ 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.[^tong-cabibbo] $$ % caption: The Cabibbo rotation. The weak eigenstate d-prime that the W couples to % is the mass eigenstate d rotated toward s through the Cabibbo angle. The rotation % mixes down and strange, so the same coupling reaches both same-generation and % cross-generation transitions. \begin{tikzpicture}[>=stealth, font=\footnotesize, scale=1.0] \definecolor{acc}{HTML}{4A6FA5} % mass axes \draw[->, black, thick] (0,0) -- (3.4,0) node[right, font=\scriptsize] {down}; \draw[->, black, thick] (0,0) -- (0,3.0) node[right, font=\scriptsize] {strange}; % weak axes rotated by ~13 deg \draw[->, acc, very thick] (0,0) -- (2.92,0.67) node[right, font=\scriptsize] {d-weak}; \draw[->, acc, very thick] (0,0) -- (-0.67,2.92) node[left, font=\scriptsize] {s-weak}; % angle arc \draw[acc, thick] (1.1,0) arc (0:13:1.1); \node[acc, font=\scriptsize, right] at (1.25,0.22) {Cabibbo angle}; \end{tikzpicture} $$ ## The GIM mechanism and the charm prediction Cabibbo's rotation has a dangerous side effect. Written out, the neutral current computed from the rotated $d'$ 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 $(c, s')$. 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: $$ \mathcal{M}(K^0\to\mu^+\mu^-) \propto \cos\theta_c\sin\theta_c - \sin\theta_c\cos\theta_c = 0 . $$ The cancellation is not perfect because $m_c \neq m_u$; 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.[^tong-gim] $$ % caption: The GIM cancellation. The flavour-changing neutral process (here a % neutral kaon to a muon pair) receives one contribution with an up quark in the % loop and one with a charm quark, carrying opposite signs from the Cabibbo factors. % They cancel exactly for equal masses, leaving a residue set by the up-charm mass % difference. \begin{tikzpicture}[>=stealth, font=\footnotesize, scale=1.0] \definecolor{acc}{HTML}{4A6FA5} % up loop box \draw[black, very thick, fill=black!8] (-3.6,-0.5) rectangle (-1.0,1.1); \node[black, font=\scriptsize] at (-2.3,0.6) {up in loop}; \node[black, font=\scriptsize] at (-2.3,0.0) {sign plus}; % plus \node[black, font=\large] at (-0.3,0.3) {$+$}; % charm loop box \draw[black, very thick, fill=black!8] (0.4,-0.5) rectangle (3.0,1.1); \node[black, font=\scriptsize] at (1.7,0.6) {charm in loop}; \node[black, font=\scriptsize] at (1.7,0.0) {sign minus}; % equals \node[black, font=\large] at (3.7,0.3) {$=$}; % result \node[acc, font=\scriptsize] at (5.1,0.3) {nearly zero}; \node[black, font=\scriptsize] at (0.9,-1.1) {residue set by charm mass squared over W mass squared}; \end{tikzpicture} $$ ## Three generations: the CKM matrix With a third generation $(t, b)$, the rotation becomes a $3\times3$ 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 $V_{ij}$: $$ V_{\rm CKM} = \begin{pmatrix} V_{ud} & V_{us} & V_{ub} \\ V_{cd} & V_{cs} & V_{cb} \\ V_{td} & V_{ts} & V_{tb} \end{pmatrix} . $$ The matrix arises from the mismatch $V = (V^u)^\dagger V^d$ 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.[^tong-ckm] A general unitary $3\times3$ 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: $$ |V_{\rm CKM}| \approx \begin{pmatrix} 0.974 & 0.225 & 0.004 \\ 0.225 & 0.973 & 0.042 \\ 0.009 & 0.041 & 0.999 \end{pmatrix} . $$ Transitions within a generation are near unity; crossing one generation costs a factor $\sim0.2$ (the Cabibbo suppression, visible in $V_{us}$ and $V_{cd}$); crossing two generations costs far more.[^pdg-ckm] $$ % caption: The CKM matrix magnitudes as a hierarchy grid. Diagonal elements are near % one (darkest); the Cabibbo elements linking the first two generations are moderate; % elements crossing two generations are tiny (palest). Fill intensity tracks % magnitude; numerical values are given in the text. \begin{tikzpicture}[font=\footnotesize, scale=1.0] \definecolor{acc}{HTML}{4A6FA5} % rows top to bottom: up, charm, top; cols: down, strange, bottom % up row \fill[black!32] (0,2) rectangle (1,3); \node[black, font=\scriptsize] at (0.5,2.5) {Vud}; \fill[black!16] (1,2) rectangle (2,3); \node[black, font=\scriptsize] at (1.5,2.5) {Vus}; \fill[black!5] (2,2) rectangle (3,3); \node[black, font=\scriptsize] at (2.5,2.5) {Vub}; % charm row \fill[black!16] (0,1) rectangle (1,2); \node[black, font=\scriptsize] at (0.5,1.5) {Vcd}; \fill[black!32] (1,1) rectangle (2,2); \node[black, font=\scriptsize] at (1.5,1.5) {Vcs}; \fill[black!9] (2,1) rectangle (3,2); \node[black, font=\scriptsize] at (2.5,1.5) {Vcb}; % top row \fill[black!5] (0,0) rectangle (1,1); \node[black, font=\scriptsize] at (0.5,0.5) {Vtd}; \fill[black!9] (1,0) rectangle (2,1); \node[black, font=\scriptsize] at (1.5,0.5) {Vts}; \fill[black!33] (2,0) rectangle (3,1); \node[black, font=\scriptsize] at (2.5,0.5) {Vtb}; \draw[black] (0,0) grid (3,3); \node[black, font=\scriptsize, left] at (-0.15,2.5) {up}; \node[black, font=\scriptsize, left] at (-0.15,1.5) {charm}; \node[black, font=\scriptsize, left] at (-0.15,0.5) {top}; \node[black, font=\scriptsize, below] at (0.5,-0.1) {down}; \node[black, font=\scriptsize, below] at (1.5,-0.1) {strange}; \node[black, font=\scriptsize, below] at (2.5,-0.1) {bottom}; \end{tikzpicture} $$ ## 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 $$ V_{\rm CKM} \approx \begin{pmatrix} 1 - \tfrac12\lambda^2 & \lambda & A\lambda^3(\rho - i\eta) \\ -\lambda & 1 - \tfrac12\lambda^2 & A\lambda^2 \\ A\lambda^3(1 - \rho - i\eta) & -A\lambda^2 & 1 \end{pmatrix} , $$ with $A$, $\rho$, $\eta$ all of order unity. The measured values are $$ \lambda = 0.2250, \quad A = 0.826, \quad \bar\rho = 0.159, \quad \bar\eta = 0.348 . $$ 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.[^pdg-wolfenstein] ## The unitarity triangle Unitarity, $V_{\rm CKM}^\dagger V_{\rm CKM} = 1$, forces each pair of distinct columns to be orthogonal. The orthogonality of the first and third down-type columns, $$ V_{ud}V_{ub}^* + V_{cd}V_{cb}^* + V_{td}V_{tb}^* = 0 , $$ 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 $V_{cd}V_{cb}^*$ 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.[^tong-triangle] The triangle's non-degeneracy _is_ CP violation: its area equals half the Jarlskog invariant $J \approx 3\times10^{-5}$, 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, $\bar\eta \neq 0$, is the geometric statement that the quark sector violates CP. $$ % caption: The unitarity triangle in the rho-eta plane. Unitarity of the CKM matrix % requires three complex numbers to sum to zero, forming a triangle with base % normalized to unit length. The apex at rho-bar, eta-bar sits above the real axis; % its height measures CP violation, and the interior angles are measured separately % in B decays. \begin{tikzpicture}[>=stealth, font=\footnotesize, scale=1.0] \definecolor{acc}{HTML}{4A6FA5} \draw[->, black] (-0.3,0) -- (4.0,0) node[right, font=\scriptsize] {rho}; \draw[->, black] (0,-0.3) -- (0,2.6) node[above, font=\scriptsize] {eta}; % triangle: base (0,0)-(3,0), apex (0.48,1.05) \coordinate (O) at (0,0); \coordinate (B) at (3,0); \coordinate (A) at (0.48,1.05); \draw[acc, very thick, fill=acc!8] (O) -- (B) -- (A) -- cycle; \fill[acc] (A) circle (2pt); \node[acc, above right, font=\scriptsize] at (A) {apex (rho, eta)}; \node[black, font=\scriptsize, below left] at (O) {gamma}; \node[black, font=\scriptsize, below right] at (B) {beta}; \node[black, font=\scriptsize, right] at (0.78,0.62) {alpha}; \node[black, font=\scriptsize, below] at (1.5,-0.05) {unit base}; \end{tikzpicture} $$ ## 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. [^tong-cabibbo]: Tong, _The Standard Model_ (Cambridge Part III), §6.2.1 — the Cabibbo angle, the rotated weak eigenstate, and the $\sin^2\theta_c$ suppression of strange-quark decays. [damtp.cam.ac.uk/user/tong/standardmodel.html](http://www.damtp.cam.ac.uk/user/tong/standardmodel.html) [^tong-gim]: Tong, _The Standard Model_ (Cambridge Part III), §6.3 — the GIM mechanism, the absence of tree-level FCNC, and the charm prediction from the $K^0\to\mu^+\mu^-$ cancellation; Griffiths, §10.8, gives the same argument. [^tong-ckm]: Tong, _The Standard Model_ (Cambridge Part III), §6.1–6.2 — the CKM matrix as the mismatch $(V^u)^\dagger V^d$, the parameter counting (three angles, one phase), and the Kobayashi–Maskawa third-generation argument. [^pdg-ckm]: CKM magnitudes and the two-generations-no-CP counting: Particle Data Group, _Review of Particle Physics_, CKM review, [pdg.lbl.gov](https://pdg.lbl.gov). [^pdg-wolfenstein]: Wolfenstein parameters $\lambda, A, \bar\rho, \bar\eta$: Particle Data Group, [pdg.lbl.gov](https://pdg.lbl.gov); Tong §6.2.3 derives the parametrization from the magnitude hierarchy. [^tong-triangle]: Tong, _The Standard Model_ (Cambridge Part III), §6.2.4 and §6.4.2 — the unitarity triangle, its apex at $(\bar\rho,\bar\eta)$, and the Jarlskog invariant $J = \mathrm{Im}(V_{ud}V_{ub}^*V_{tb}V_{td}^*)$ equal to twice the triangle area; measured values from the Particle Data Group, [pdg.lbl.gov](https://pdg.lbl.gov).