Pion beta decay and Cabibbo-Kobayashi-Maskawa unitarity

Abstract

Pion beta decay, ${\ensuremath{\pi}}^{+}\ensuremath{\rightarrow}{\ensuremath{\pi}}^{0}{e}^{+}\ensuremath{\nu}(\ensuremath{\gamma})$, provides a theoretically clean $\ifmmode\pm\else\textpm\fi{}0.3%$ determination of the Cabibbo-Kobayashi-Maskawa matrix element ${V}_{\mathrm{ud}}$. Although impressive, that result falls short of super-allowed nuclear beta decays where an order of magnitude better precision already exists. Here, we advocate a new strategy for utilizing pion beta decay, based on its utility in determining ${V}_{\mathrm{us}}/{V}_{\mathrm{ud}}$ via the ratio ${R}_{V}=\mathrm{\ensuremath{\Gamma}}(K\ensuremath{\rightarrow}\ensuremath{\pi}l\ensuremath{\nu}(\ensuremath{\gamma}))/\mathrm{\ensuremath{\Gamma}}({\ensuremath{\pi}}^{+}\ensuremath{\rightarrow}{\ensuremath{\pi}}^{0}{e}^{+}\ensuremath{\nu}(\ensuremath{\gamma}))$ which provides a measure of ${f}_{+}^{K}(0)|{V}_{\mathrm{us}}|/{f}_{+}^{\ensuremath{\pi}}(0)|{V}_{\mathrm{ud}}|$ independent of the Fermi constant and short-distance radiative corrections. Its dependence on the ratio of two hadronic vector current form factors provides an interesting computational goal for lattice gauge theory studies. Employing a recent lattice based value ${f}_{+}^{K}(0)/{f}_{+}^{\ensuremath{\pi}}(0)=0.970(2)$, we find ${V}_{\mathrm{us}}/{V}_{\mathrm{ud}}=0.22910(91)$ compared to ${V}_{\mathrm{us}}/{V}_{\mathrm{ud}}=0.23131(45)$ obtained from ${R}_{A}=\mathrm{\ensuremath{\Gamma}}(K\ensuremath{\rightarrow}\ensuremath{\mu}\ensuremath{\nu}(\ensuremath{\gamma}))/\mathrm{\ensuremath{\Gamma}}(\ensuremath{\pi}\ensuremath{\rightarrow}\ensuremath{\mu}\ensuremath{\nu}(\ensuremath{\gamma}))$. Those independent ${V}_{\mathrm{us}}/{V}_{\mathrm{ud}}$ determinations exhibit a $2.2\ensuremath{\sigma}$ discrepancy. That tension suggests a needed shift in the lattice ${f}_{+}^{K}(0)/{f}_{+}^{\ensuremath{\pi}}(0)$ value towards the consistency range 0.961(4), experimental/theory input changes or ``new physics'' effects. Other features and implications of ${R}_{V}$ and ${R}_{A}$ are also discussed.

Topics & Concepts