Cabibbo angle (${\theta }_C$)

The Cabibbo angle ${\theta }_C$ describes the real two-generation quark mixing rotationThus $V_{ud}=V_{cs}=\cos {\theta }_C$, $V_{us}=\sin {\theta }_C$ and $V_{cd}=-\sin {\theta }_C$. Nicola Cabibbo formulated the mixing of weak charged current couplings before the three-generation mechanism of CP violation; this foundational contribution is distinct from the later extension.^[ref]

If two semileptonic decays differ by $s\to u$ versus $d\to u$, an equal-hadronic-strength approximation and a fifth-power phase space ansatz give $r_s=K{\sin }^2{\theta }_C{\Delta }_s^5$ and $r_d=K{\cos }^2{\theta }_C{\Delta }_d^5$. Their partial decay rates are $r_i=B_i/{\tau }_i$, where $B_i$ is the appropriate branching fraction and ${\tau }_i$ the total mean lifetime. Taking the ratio gives the displayed expression. Different hadronic hadronic form factors, finite electron mass and recoil corrections prevent this simplified extraction from being an exact determination.

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.

In the charged-current spectator topology $c\to q{W}^{+}$, $W^{+}\to u{\bar{q}}^{\prime }$, the magnitudes of the CKM factors give favoured, singly suppressed and doubly suppressed amplitudes of orders $1$, $\tan {\theta }_C$ and ${\tan }^2{\theta }_C$. Squaring gives rate orders $1$, ${\tan }^2{\theta }_C$ and ${\tan }^4{\theta }_C$. Two singly suppressed channels are not thereby predicted to have equal rates: hadronic matrix elements, phase space and rescattering can differ.