= Wolfenstein parametrization
{c}
{title2=$\lambda,\ A,\ \rho,\ \eta$}
The <Wolfenstein parametrization> organizes the <CKM matrix> in powers of a small parameter $\lambda=s_{12}$, defining $s_{23}=A\lambda^2$ and $s_{13}e^{i\delta}=A\lambda^3(\rho+i\eta)$. Substitution into the exact <unitary matrix> gives the cubic truncation
$$
V^{(3)}=\begin{pmatrix}
1-\lambda^2/2&\lambda&A\lambda^3(\rho-i\eta)\\
-\lambda&1-\lambda^2/2&A\lambda^2\\
A\lambda^3(1-\rho-i\eta)&-A\lambda^2&1
\end{pmatrix}.
$$
This is a useful truncation, with entrywise corrections starting at order $\lambda^4$, rather than an exactly <unitary matrix>. For example, its first-row squared norm is $1+\lambda^4/4+A^2\lambda^6(\rho^2+\eta^2)$, already different from one. The exact definitions of the sines can instead be inserted into the exact <CKM matrix>, retaining every cosine, to obtain exact <unitarity>.
For the illustrative parameters of the <CKM unitarity triangle>, the exact definitions give $\lambda=1/4$, $A=8/5$, $\rho=1/5$ and $\eta=\sqrt3/5$. The first-row norm error in $V^{(3)}$ is $0.0010765625$, whereas the exact <CKM matrix> has unit row norms. To leading order the <Jarlskog invariant> is $J\simeq A^2\lambda^6\eta$, but the exact expression includes cosine factors. The barred parameters $\bar\rho,\bar\eta$ are defined by the exact ratio in the <CKM unitarity triangle>; they differ from $\rho,\eta$ beyond leading order. This distinction matters when a geometrical closure test is used to test <unitarity>.
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