Scalar radius of the pion in the Kroll-Lee-Zumino renormalizable theory

The Kroll-Lee-Zumino renormalizable Abelian quantum field theory of pions and a massive $\ensuremath{\rho}$-meson is used to calculate the scalar radius of the pion at next-to-leading (one-loop) order in perturbation theory. Because of renormalizability, this determination involves no free parameters. The result is $⟨{r}_{\ensuremath{\pi}}^{2}{⟩}_{s}=0.40\text{ }\text{ }{\mathrm{fm}}^{2}$. This value gives for ${\overline{\ensuremath{\ell}}}_{4}$, the low energy constant of chiral perturbation theory, ${\overline{\ensuremath{\ell}}}_{4}=3.4$, and ${F}_{\ensuremath{\pi}}/F=1.05$, where $F$ is the pion decay constant in the chiral limit. Given the level of accuracy in the masses and the $\ensuremath{\rho}\ensuremath{\pi}\ensuremath{\pi}$ coupling, the only sizable uncertainty in this result is due to the (uncalculated) next-to-next-to-leading order contribution.

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