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.