Cabibbo-angle extraction from semileptonic decay rates (${\tan }^2{\theta }_C=(B_s/{\tau }_s)({\tau }_d/B_d)({\Delta }_d/{\Delta }_s{)}^5$)

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.