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

Abstract

We consider the quadratic scalar radius of the pion, $<r_{\mathrm{S},\pi }^2>$, and the mixed $K-\pi$ scalar radius, $<r_{\mathrm{S},K\pi }^2>$. With respect to the second, we point out that the more recent (post-1974) experimental results in $K_{l3}$ decays imply a value, $<r_{\mathrm{S},K\pi }^2>=0.31\pm 0.06{\mathrm{fm}}^2$, which is about $2\sigma$ above estimates based on chiral perturbation theory. On the other hand, we show that this value of $<r_{\mathrm{S},K\pi }^2>$ suggests the existence of a low mass S$\frac{1}{2}$ $K\pi$ resonance. With respect to $<r_{\mathrm{S},\pi }^2>$, we contest the central value and accuracy of current evaluations, that give $<r_{\mathrm{S},\pi }^2>=0.61\pm 0.04{\mathrm{fm}}^2$. Based on experiment, we find a robust lower bound of $<r_{\mathrm{S},\pi }^2>\ge 0.70\pm 0.06{\mathrm{fm}}^2$ and a reliable estimate, $<r_{\mathrm{S},\pi }^2>=0.75\pm 0.07{\mathrm{fm}}^2$, where the error bars are attainable. This implies, in particular, that the chiral result for $<r_{\mathrm{S},\pi }^2>$ 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.

Comments: PlainTeX file. Corrected asymptotic phase; numerical results unaffected