Pion charge radius and the pion form factor

Abstract

The pion charge-form-factor data for positive and negative values of $k^2$, the $P$-wave $\pi -\pi$ phase-shift data, and the pion charge-radius data are analyzed using form factors previously developed by us and other authors. The peaked behavior near threshold for the $P$-wave $\pi -\pi$ phase shift suggested by Levin and Okubo is found not to be necessary for large charge-radius values, but larger slopes ($\frac{d{\delta }_{11}}{d{k}^2}$) at threshold do seem to be the important factor. We conclude that a more precise measurement of the pion charge radius is necessary to separate the different theoretical models. We also find that the form-factor data near (timelike) $k^2\sim -1.51\mathrm{}$ ${(\mathrm{G}\mathrm{e}\mathrm{V}/\mathit{c})}^2$ are sufficiently imprecise that nothing definitive can be said about the existence of the ${\rho }^{\prime }$ resonance. if such a resonance exists and if the pion has no hard core, the data are either consistent with a ${\rho }^{\prime }$ resonance of width ${\Gamma }_{{\rho }^{\prime }}\sim 2$ MeV or consistent with a ${\rho }^{\prime }$ resonance of large width if at least one additional ${\rho }^{\prime }$ resonance, ${\rho }^{\prime \prime }$, exists.