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We present an improved determination of the $\pi\pi$ continuum contributionto the isovector spectral functions of the nucleon electromagnetic formfactors. Our analysis includes the most up-to-date results for the$\pi\pi\to\bar N N$ partial waves extracted from Roy-Steiner equations,consistent input for the pion vector form factor, and a thorough discussion ofisospin-violating effects and uncertainty estimates. As an application, weconsider the $\pi\pi$ contribution to the isovector electric and magnetic radiiby means of sum rules, which, in combination with the accurately known neutronelectric radius, are found to slightly prefer a small proton charge radius.

On the pi pi continuum in the nucleon form factors and the proton radius puzzle