Pion Beta Decay and CKM Unitarity

Pion beta decay, $π^+\to π^0 e^+ν(γ)$, provides a theoretically clean $\pm 0.3\%$ determination of the CKM matrix element $V_{\text{ud}}$. That value falls short of super-allowed nuclear beta decays where an order of magnitude better precision already exists. We advocate a new strategy for utilizing pion beta decay, based on its role in determining ${V_{\text{us}} / V_{\text{ud}}}$ via the ratio $R_V=Γ\left(K\to πl ν(γ)\right)/Γ\left(π^+\to π^0 e^+ν(γ)\right)$. $R_V$ measures ${ f_+^K(0) |V_{\text{us}}| / f_+^π(0) |V_{\text{ud}}| }$ and is insensitive to the Fermi constant and some radiative corrections. Its dependence on the ratio of form factors may also prove useful for lattice gauge theory calculations. Employing a lattice based value ${ f_+^K(0) / f_+^π(0) } = 0.970(2)$, we find $V_{\text{us}}/V_{\text{ud}}=0.22918(88)$ compared to $V_{\text{us}}/V_{\text{ud}}=0.23131(45)$ obtained from $R_A=Γ(K\to μν(γ))/Γ(π\to μν(γ))$. Those vector and axial-vector based $V_{\text{us}}/V_{\text{ud}}$ values exhibit a 2.2$σ$ discrepancy. That tension may be relieved by a shift in the lattice ${ f_+^K(0) / f_+^π(0) }$ towards the consistency range 0.961(4), changes in experimental input or new physics. Other implications of $R_V$ and $R_A$ are also discussed.

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