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 form factors, finite electron mass and recoil corrections prevent this simplified extraction from being an exact determination.