The quadratic scalar radius of the pion and the mixed $\pi-K$ radius

Abstract

We consider the quadratic scalar radius of the pion, $<r^2_{{\rm S},\pi}>$, and the mixed $K-\pi$ scalar radius, $<r^2_{{\rm S},K\pi}>$. With respect to the second, we point out that the more recent (post-1974) experimental results in $K_{l3}$ decays imply a value, $<r^2_{{\rm S},K\pi}>=0.31\pm0.06 {\rm fm}^2$, which is about $2 \sigma$ above estimates based on chiral perturbation theory. On the other hand, we show that this value of $<r^2_{{\rm S},K\pi}>$ suggests the existence of a low mass S$\tfrac{1}{2}$ $K\pi$ resonance. With respect to $<r^2_{{\rm S},\pi}>$, we contest the central value and accuracy of current evaluations, that give $<r^2_{{\rm S},\pi}>=0.61\pm0.04 {\rm fm}^2$. Based on experiment, we find a robust lower bound of $<r^2_{{\rm S},\pi}>\geq0.70\pm0.06 {\rm fm}^2$ and a reliable estimate, $<r^2_{{\rm S},\pi}>=0.75\pm0.07 {\rm fm}^2$, where the error bars are attainable. This implies, in particular, that the chiral result for $<r^2_{{\rm S},\pi}>$ is $1.4 \sigma$ away from experiment. We also comment on implications about the chiral parameter $\bar{l}_4$, very likely substantially larger (and with larger errors) than usually assumed.