Pion Beta Decay and CKM Unitarity

Abstract

Pion beta decay, ${\pi }^{+}\to {\pi }^0{e}^{+}\nu (\gamma )$, 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 $R_V=\Gamma \left(K\to \pi l\nu (\gamma )\right)/\Gamma \left({\pi }^{+}\to {\pi }^0{e}^{+}\nu (\gamma )\right)$. $R_V$ measures $f_{+}^K(0)|V_{\text{us}}|/f_{+}^{\pi }(0)|V_{\text{ud}}|$ 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 $f_{+}^K(0)/f_{+}^{\pi }(0)=0.970(2)$, we find $V_{\text{us}}/V_{\text{ud}}=0.22918(88)$ compared to $V_{\text{us}}/V_{\text{ud}}=0.23131(45)$ obtained from $R_A=\Gamma (K\to \mu \nu (\gamma ))/\Gamma (\pi \to \mu \nu (\gamma ))$. Those vector and axial-vector based $V_{\text{us}}/V_{\text{ud}}$ values exhibit a 2.2$\sigma$ discrepancy. That tension may be relieved by a shift in the lattice $f_{+}^K(0)/f_{+}^{\pi }(0)$ 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.