Cabibbo-Kobayashi-Maskawa matrix ($V_{\mathrm{CKM}}$)

The Cabibbo-Kobayashi-Maskawa matrix is the unitary mismatch between the left-handed transformations that diagonalize the up- and down-quark Yukawa matrices. It appears in charged weak currents.

Orthogonality of two CKM rows or columns makes the three complex products form a closed triangle. Its area is $|J|/2$, where $J$ is the Jarlskog invariant. Normalizing the displayed column-orthogonality relation gives vertices $0,1,\bar{\rho }+i\bar{\eta }$, with $\bar{\rho }+i\bar{\eta }=-V_{ud}{V}_{ub}^{\ast }/(V_{cd}{V}_{cb}^{\ast })$. A collapsed triangle has zero weak rephasing-invariant CP violation for nondegenerate quark masses.

The Wolfenstein parametrization expands the three-generation CKM matrix in its small $1$-$2$ mixing parameter $\lambda =s_{12}$, defining $s_{23}=A{\lambda }^2$ and $s_{13}{e}^{i\delta }=A{\lambda }^3(\rho +i\eta )$. To cubic order its rows are $(1-{\lambda }^2/2,\lambda ,A{\lambda }^3(\rho -i\eta ))$, $(-\lambda ,1-{\lambda }^2/2,A{\lambda }^2)$, and $(A{\lambda }^3(1-\rho -i\eta ),-A{\lambda }^2,1)$. Higher orders restore exact unitarity. Rephasing-invariant barred parameters are defined from ratios of CKM entries rather than this truncated expansion.

For nondegenerate quark masses, independent phase changes of the up- and down-quark mass eigenfields alter the numerical CKM matrix by row and column phases without changing physics. Thus CP symmetry requires that some allowed phase convention makes the matrix real, rather than requiring real entries in every convention. With $N$ generic families, removing $2N-1$ phases from a unitary matrix leaves $(N-1)(N-2)/2$ physical CP-violating phases and $N(N-1)/2$ mixing angles.

Under charge conjugation, $\bar{u}{\gamma }^{\mu }(1-{\gamma }^5)d$ becomes $-\bar{d}{\gamma }^{\mu }(1+{\gamma }^5)u$. Parity reverses the chirality again and applies the Lorentz parity matrix. The minus sign cancels that of charge conjugation on the charged vector field. Thus each charged-current operator is exchanged with its Hermitian conjugate under CP symmetry, giving the displayed transformation for the CKM matrix $V$. For nondegenerate quark masses, physical CP conservation means that allowed quark rephasings can make $V$ real. A nonzero Jarlskog invariant obstructs this. A merely complex phase convention is not proof of physical CP violation, and $V=I$ is a CP-conserving counterexample to unconditional noninvariance.

An $N\times N$ CKM matrix starts with $N^2$ parameters of a unitary matrix. Independent quark rephasings remove $2N-1$ phases because the common phase has no effect. The remaining $(N-1{)}^2$ parameters consist of $N(N-1)/2$ angles and $(N-1)(N-2)/2$ CP-violating phases. Extra mass degeneracies may reduce this generic count.

The angle of the real two-generation quark mixing rotation, with $V_{ud}=V_{cs}=\cos {\theta }_C$, $V_{us}=\sin {\theta }_C$, and $V_{cd}=-\sin {\theta }_C$.

A small weak charged current amplitude factor associated with $\sin {\theta }_C$. A doubly suppressed amplitude contains two such factors, giving a rate suppression of order ${\tan }^4{\theta }_C$ relative to a favoured process when other matrix elements are comparable.

The Jarlskog invariant is a rephasing-invariant measure of CP violation in a three-generation quark mixing matrix. It is the common magnitude of imaginary quartic products such as $\mathrm{Im}(V_{ij}{V}_{k\ell }{V}_{i\ell }^{\ast }{V}_{kj}^{\ast })$ for distinct rows and columns.