The charge radius of the pion $\sqrt {\langle r{}^{2}_{{{\mathit \pi}}}\rangle }$ is defined in relation to the form factor of the pion electromagnetic vertex, called vector form factor VFF, F${}^{V}_{{{\mathit \pi}}}$. The VFF is a function of the squared four-momentum transfer $\mathit t$, or of the squared c.m. energy $\mathit s$, depending on the channel in which the photon exchange takes place. In both cases, it is related to the slope of the VFF at zero, namely

$\langle $r${}^{2}_{{{\mathit \pi}}}\rangle $ = 6 ${d F{}^{V}_{{{\mathit \pi}}}(\mathit q)\over d\mathit q}(\mathit q$=0) where $\mathit q$ = $\mathit t$, $\mathit s$.

The quantity cannot be measured directly. It can be extracted from the cross sections of three processes: pion electroproduction, ${{\mathit e}}$ ${{\mathit N}}$ $\rightarrow$ ${{\mathit e}}{{\mathit N}}{{\mathit \pi}}$, and pion electron scattering ${{\mathit e}}$ ${{\mathit \pi}}$ $\rightarrow$ ${{\mathit e}}{{\mathit \pi}}$, for the $\mathit t$ channel, and positron electron annihilation into two charged pions, ${{\mathit e}^{+}}$ ${{\mathit e}^{-}}$ $\rightarrow$ ${{\mathit \pi}^{+}}{{\mathit \pi}^{-}}$, for the $\mathit s$ channel. We encode all measurements, but we do not use electroproduction data in averaging because the extraction of the pion radius involves, in this case, theoretical uncertainties that cannot be controlled at the needed level of accuracy. In case of analyses based on the same data set, as ANANTHANARAYAN 2017 and COLANGELO 2019, which cannot be averaged, we combine the results into a common value, with the uncertainty range chosen to cover both analyses. Note that for consistency the form factor needs to be defined in both channels with the vacuum polarisation removed. For details see COLANGELO 2019 or Appendix B of ANANTHANARAYAN 2016A.

VALUE (fm) DOCUMENT ID TECN  COMMENT $\bf{ 0.659 \pm0.004}$ OUR AVERAGE $0.656$ $\pm0.005$ 1 PDG 01   FIT $0.65$ $\pm0.05$ $\pm0.06$ ESCHRICH 00   CNTR ${{\mathit \pi}}$ ${{\mathit e}}$ $\rightarrow$ ${{\mathit \pi}}{{\mathit e}}$ $0.663$ $\pm0.006$ AMENDOLIA 98   CNTR ${{\mathit \pi}}$ ${{\mathit e}}$ $\rightarrow$ ${{\mathit \pi}}{{\mathit e}}$ $0.663$ $\pm0.023$ DALLY 98   CNTR ${{\mathit \pi}}$ ${{\mathit e}}$ $\rightarrow$ ${{\mathit \pi}}{{\mathit e}}$ • • We do not use the following data for averages, fits, limits, etc. • • $0.640$ $\pm0.007$ 2 CUI 02 A   FIT Fit existing data $0.655$ $\pm0.004$ 3 COLANGELO 01   FIT Fit existing data $0.657$ $\pm0.003$ 4 ANANTHANARAYA.. 01   FIT Fit existing data $0.6603$ $\pm0.0005$ $\pm0.0004$ 5 HANHART 01   FIT Fit existing data $0.740$ $\pm0.031$ 6 LIESENFELD 99   CNTR ${{\mathit e}}$ ${{\mathit p}}$ $\rightarrow$ ${{\mathit e}}{{\mathit \pi}^{+}}{{\mathit n}}$ $0.661$ $\pm0.012$ 7 BIJNENS 99   CNTR ${{\mathit \chi}}$PT extraction $0.660$ $\pm0.024$ AMENDOLIA 98   CNTR ${{\mathit \pi}}$ ${{\mathit e}}$ $\rightarrow$ ${{\mathit \pi}}{{\mathit e}}$ $0.711$ $\pm0.009$ $\pm0.016$ 6 BEBEK 97   CNTR ${{\mathit e}}$ ${{\mathit N}}$ $\rightarrow$ ${{\mathit e}}{{\mathit \pi}}{{\mathit N}}$ $0.678$ $\pm0.004$ $\pm0.008$ 8 QUENZER 97   CNTR ${{\mathit e}^{+}}$ ${{\mathit e}^{-}}$ $\rightarrow$ ${{\mathit \pi}^{+}}{{\mathit \pi}^{-}}$ $0.78$ ${}^{+0.09}_{-0.10}$ ADYLOV 97   CNTR ${{\mathit \pi}}$ ${{\mathit e}}$ $\rightarrow$ ${{\mathit \pi}}{{\mathit e}}$ $0.74$ ${}^{+0.11}_{-0.13}$ BARDIN 97   CNTR ${{\mathit e}}$ ${{\mathit p}}$ $\rightarrow$ ${{\mathit e}}{{\mathit \pi}^{+}}{{\mathit n}}$ $0.56$ $\pm0.04$ DALLY 97   CNTR ${{\mathit \pi}}$ ${{\mathit e}}$ $\rightarrow$ ${{\mathit \pi}}{{\mathit e}}$ 1  This value combines the measurements of ANANTHANARAYAN 2017 and COLANGELO 2019 which are based on the same data set. The uncertainty range is chosen to cover both results. 2  CUI 2021A perform a fit including spacelike data only. Employ a new mathematical procedure based on interpolation via continued fractions augmented by statistical sampling. Also do not impose the charge conserving normalization condition F(0) = 1. 3  COLANGELO 2019 fit existing F$_{V}$ data, using an extended Omnes dispersive representation. This analysis is based on the same data set of ANANTHANARAYAN 2017 . Accordingly, they cannot be averaged. We combine the results into a common value, with the uncertainty range chosen to cover the uncertainty ranges of both analyses. 4  ANANTHANARAYAN 2017 fit existing F$_{V}$ data, using a mixed phase-modulus dispersive representation. This analysis is based on the same data set of COLANGELO 2019. Accordingly, they cannot be averaged. We combine the results into a common value, with the uncertainty range chosen to cover the uncertainty ranges of both analyses. 5  According to the authors the uncertainty could be underestimated. The value quoted omits the BaBar data AUBERT 2009. 6  The extractions could contain an additional theoretical uncertainty which cannot be sufficiently quantified. 7  BIJNENS 1998 fits existing data. 8  The extraction is based on a parametrization that does not have correct analytic properties.

References