Measurement of the Branching Fractions of Semileptonic Kaon Decays and Extraction of the CKM Parameter $\lvert V_{us} \rvert$

Semileptonic kaon decays $(K_{l3} = K^+ → π^0 l^+ \nu$ with $l = e, \mu)$ provide a clean way to extract the parameter $\lvert V_{us} \rvert$ of the Cabibbo-Kobayashi-Maskawa (CKM) quark mixing matrix. It is one of the fundamental parameters of the Standard Model and is only accessible via experimental measurement. The NA62 collaboration collected $K^+$ decay data in 2007 which are analyzed in the presented work. In total $28 \cdot 10^6 K_{\mu 3}$ , $37 \cdot 10^6 K_{e3}$ , and $2\cdot 10^6 K_{2 \pi}$ candidates were extracted from the data set and used to compute the ratios of branching fractions $\frac{Br(K_{\mu 3})}{Br(K_{2 \pi})} = 0.16066(12) \text{stat.} (66) \text{syst.} = 0.1607(7)$, $\frac{Br(K_{e3})}{Br(K_{2 \pi})} = 0.24320(17) \text{stat.} (41) \text{syst.} = 0.2432(4)$, and $\frac{Br(K_{\mu 3})}{Br(K_{e3})} = 0.66061(17) \text{stat.} (245) \text{syst.} = 0.6606(25)$ which represent the most precise measurements available. The semileptonic branching fractions $Br(K_{\mu 3}) = 3.321(19) \%$ and $Br(K_{e3}) = 5.027(22) \%$ follow from the world average of $Br(K_{2 \pi})$ measurements. They are used to extract the CKM parameter $\lvert V_{us} \rvert = 0.2219(10)$. The new $\lvert V_{us} \rvert$ result leads to the unitarity relation $\lvert V_{ud} \rvert ^2 + \lvert V_{us} \rvert ^2 + \lvert V_{ub} \rvert ^2 = 0.9983(6)$ which disagrees by 2.8 standard deviations with the CKM unitarity prediction. The $e − \mu$ lepton universality is tested with the ratio of the semileptonic branching fractions $\frac{Br(K_{\mu 3})}{Br(K_{e3})}$ and agrees with the Standard Model prediction.