= Solution
<Quark mixing> arises because the up-type and down-type <quark> <Yukawa matrices> need not be diagonal in the same <weak interaction> basis. In the <Standard Model> the three left-handed <quark> doublets $Q'_L$ transform as $(3,2,1/6)$ under $SU(3)_c\times SU(2)_L\times U(1)_Y$, whereas $u'_R$ and $d'_R$ transform as $(3,1,2/3)$ and $(3,1,-1/3)$. With a <Higgs doublet> $H$ and its conjugate $\widetilde H=i\sigma^2H^*$, the <gauge-invariant> <Yukawa interaction> is
$$
\mathcal L_Y=-\bar Q'_L Y_dH d'_R-\bar Q'_L Y_u\widetilde H u'_R+\text{h.c.}
$$
The generation indices are matrix-multiplied. <Electroweak symmetry breaking>, $\langle H\rangle=(0,v_H/\sqrt2)^T$, produces $M_u=v_HY_u/\sqrt2$ and $M_d=v_HY_d/\sqrt2$.
Each complex <fermion mass matrix> admits a <biunitary diagonalization>,
$$
U_{uL}^\dagger M_uU_{uR}=D_u,\qquad U_{dL}^\dagger M_dU_{dR}=D_d,
$$
where the diagonal entries are real nonnegative <quark> <masses>. Define $u'_L=U_{uL}u_L$ and $d'_L=U_{dL}d_L$, and similarly for right-handed fields. The <kinetic terms> retain their canonical form because the transformations are <unitary matrices>. However, the <weak charged current> becomes
$$
\mathcal L_{\rm cc}=-\frac{g_2}{\sqrt2}\bar u_L\gamma^\mu Vd_LW^+_\mu+\text{h.c.},\qquad
\boxed{V=U_{uL}^\dagger U_{dL}}.
$$
This <unitary matrix> is the <Cabibbo-Kobayashi-Maskawa matrix>. Its entries multiply transitions between up-type and down-type <mass eigenstates>. The minimal <Standard Model> has no analogous right-handed <charged weak current>.
<Quark mixing> does not introduce tree-level <flavor-changing neutral currents>. <Photon>, <gluon> and <weak neutral current> couplings are proportional to the identity within each fixed-charge <quark> sector, and $U^\dagger I U=I$. The single <Higgs doublet> also has diagonal neutral <scalar field> couplings after the same transformations that diagonalize the <masses>. At higher <loop orders>, flavor changes are possible. For example, an $s\to d$ amplitude contains $\sum_iV_{is}^*V_{id}f(m_i^2)$. <Unitarity> of the <CKM matrix> sets the sum of coefficients to zero, cancelling any <mass>-independent part. This is the <GIM mechanism>; <mass> splittings leave a suppressed, generally nonzero amplitude.
For $N$ generations a <unitary matrix> has $N^2$ real parameters: $N(N-1)/2$ mixing angles and $N(N+1)/2$ phases. Rephasing the $N$ up-type and $N$ down-type <quark> fields removes $2N-1$ phases; a common <baryon number> phase leaves every matrix entry unchanged. Hence the physical numbers are
$$
N_{\rm angles}=\frac{N(N-1)}2,\qquad
N_{\rm weak\ CP\ phases}=\frac{(N-1)(N-2)}2.
$$
Two generations therefore give a real Cabibbo rotation,
$$
V=\begin{pmatrix}\cos\theta_C&\sin\theta_C\\-\sin\theta_C&\cos\theta_C\end{pmatrix},
$$
with <strangeness>-changing charged-current transitions suppressed by $\sin\theta_C$. Three generations have three mixing angles and one irreducible weak <CP> phase. This counting assumes nondegenerate <masses>; degeneracies allow additional basis rotations.
Writing $s_{ij}=\sin\theta_{ij}$ and $c_{ij}=\cos\theta_{ij}$, a standard exact parametrization is
$$
V=\begin{pmatrix}
c_{12}c_{13}&s_{12}c_{13}&s_{13}e^{-i\delta}\\
-s_{12}c_{23}-c_{12}s_{23}s_{13}e^{i\delta}&c_{12}c_{23}-s_{12}s_{23}s_{13}e^{i\delta}&s_{23}c_{13}\\
s_{12}s_{23}-c_{12}c_{23}s_{13}e^{i\delta}&-c_{12}s_{23}-s_{12}c_{23}s_{13}e^{i\delta}&c_{23}c_{13}
\end{pmatrix}.
$$
Complex entries alone do not establish <CP violation>, because <quark> <quantum field rephasings> can change their phases. For nondegenerate <quark> <masses> the weak mixing sector conserves <CP> precisely when a basis makes $V$ real. A useful invariant test is the <Jarlskog invariant>,
$$
J=\operatorname{Im}(V_{ud}V_{cs}V_{us}^*V_{cd}^*)
=c_{12}c_{23}c_{13}^2s_{12}s_{23}s_{13}\sin\delta.
$$
A nonzero $J$ requires all three mixing angles and a nontrivial phase. If two same-charge <quark> <masses> coincide, the additional basis freedom removes the corresponding physical <CP> effect. The possible <QCD theta angle> is a separate <CP> parameter, not part of this weak-mixing count.
The <Wolfenstein parametrization> displays the hierarchy of <quark mixing> economically. Put $\lambda_W=s_{12}$, $s_{23}=A\lambda_W^2$, and $s_{13}e^{i\delta}=A\lambda_W^3(\rho+i\eta)$. Expanding the exact matrix gives
$$
V=\begin{pmatrix}
1-\lambda_W^2/2&\lambda_W&A\lambda_W^3(\rho-i\eta)\\
-\lambda_W&1-\lambda_W^2/2&A\lambda_W^2\\
A\lambda_W^3(1-\rho-i\eta)&-A\lambda_W^2&1
\end{pmatrix}+O(\lambda_W^4).
$$
The observed pattern is predominantly diagonal, with $1$-$2$ mixing largest and $2$-$3$, $1$-$3$ mixing successively suppressed. The expansion implies $J\simeq A^2\lambda_W^6\eta$. The truncated matrix must not be treated as an exactly <unitary matrix>.
Column <orthogonality> gives
$$
V_{ud}V_{ub}^*+V_{cd}V_{cb}^*+V_{td}V_{tb}^*=0.
$$
The three complex numbers form a <CKM unitarity triangle>, with area $|J|/2$. Dividing by $-V_{cd}V_{cb}^*$ puts its vertices at $0$, $1$ and $\bar\rho+i\bar\eta$, where $\bar\rho+i\bar\eta=-V_{ud}V_{ub}^*/(V_{cd}V_{cb}^*)$. The barred parameters include higher-order corrections to the Wolfenstein parameters. Comparing independent measurements of its sides and angles tests both <quark mixing> and the absence of additional flavor dynamics.
Semileptonic <decay rates> determine magnitudes of <CKM> entries through factors $|V_{ij}|^2$ once hadronic <matrix elements> are controlled. Neutral <kaon> and neutral B-<meson> mixing constrain products of entries through weak box <scattering amplitudes>. <CP> asymmetries probe their invariant relative phases, with <kaon> mixing providing indirect <CP violation> and <decay amplitudes> providing direct <CP violation>. These processes complement tests of the <unitarity> constraints on rows and columns. \b[The minimal three-generation <Standard Model> describes weak <quark mixing> by a <unitary matrix> with three angles and one physical <CP> phase; it does not predict the <Yukawa couplings> or their hierarchy.]
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