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 2019 FIT $0.65$ $\pm0.05$ $\pm0.06$ ESCHRICH 2001 CNTR ${{\mathit \pi}}$ ${{\mathit e}}$ $\rightarrow$ ${{\mathit \pi}}{{\mathit e}}$ $0.663$ $\pm0.006$ AMENDOLIA 1986 CNTR ${{\mathit \pi}}$ ${{\mathit e}}$ $\rightarrow$ ${{\mathit \pi}}{{\mathit e}}$ $0.663$ $\pm0.023$ DALLY 1982 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 2021 A FIT Fit existing data $0.655$ $\pm0.004$ 3 COLANGELO 2019 FIT Fit existing data $0.657$ $\pm0.003$ 4 ANANTHANARAYA.. 2017 FIT Fit existing data $0.6603$ $\pm0.0005$ $\pm0.0004$ 5 HANHART 2017 FIT Fit existing data $0.740$ $\pm0.031$ 6 LIESENFELD 1999 CNTR ${{\mathit e}}$ ${{\mathit p}}$ $\rightarrow$ ${{\mathit e}}{{\mathit \pi}^{+}}{{\mathit n}}$ $0.661$ $\pm0.012$ 7 BIJNENS 1998 CNTR ${{\mathit \chi}}$PT extraction $0.660$ $\pm0.024$ AMENDOLIA 1984 CNTR ${{\mathit \pi}}$ ${{\mathit e}}$ $\rightarrow$ ${{\mathit \pi}}{{\mathit e}}$ $0.711$ $\pm0.009$ $\pm0.016$ 6 BEBEK 1978 CNTR ${{\mathit e}}$ ${{\mathit N}}$ $\rightarrow$ ${{\mathit e}}{{\mathit \pi}}{{\mathit N}}$ $0.678$ $\pm0.004$ $\pm0.008$ 8 QUENZER 1978 CNTR ${{\mathit e}^{+}}$ ${{\mathit e}^{-}}$ $\rightarrow$ ${{\mathit \pi}^{+}}{{\mathit \pi}^{-}}$ $0.78$ ${}^{+0.09}_{-0.10}$ ADYLOV 1977 CNTR ${{\mathit \pi}}$ ${{\mathit e}}$ $\rightarrow$ ${{\mathit \pi}}{{\mathit e}}$ $0.74$ ${}^{+0.11}_{-0.13}$ BARDIN 1977 CNTR ${{\mathit e}}$ ${{\mathit p}}$ $\rightarrow$ ${{\mathit e}}{{\mathit \pi}^{+}}{{\mathit n}}$ $0.56$ $\pm0.04$ DALLY 1977 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:

CUI 2021A PL B822 136631 Pion charge radius from pion+electron elastic scattering data COLANGELO 2019 JHEP 1902 006 Two-pion contribution to hadronic vacuum polarization PDG 2019 RPP 2019 at pdg.lbl.gov Review of Particle Physics 2019 ANANTHANARAYAN 2017 PRL 119 132002 Electromagnetic Charge Radius of the Pion at High Precision HANHART 2017 EPJ C77 98 The Branching Ratio ${{\mathit \omega}}$ $\rightarrow$ ${{\mathit \pi}^{+}}{{\mathit \pi}^{-}}$ Revisited ESCHRICH 2001 PL B522 233 Measurement of the ${{\mathit \Sigma}^{-}}$ Charge Radius by ${{\mathit \Sigma}^{-}}$ Electron Elastic Scattering LIESENFELD 1999 PL B468 20 A Measurement of the Axial Form-factor of the Nucleon by the ${{\mathit p}}$ (${{\mathit e}}$, ${{\mathit e}^{\,'}}{{\mathit \pi}^{+}}$) ${{\mathit N}}$ Reaction at W = 1125 MeV BIJNENS 1998 JHEP 9805 014 The Vector and Scalar Formfactors of the Pion to Two Loops AMENDOLIA 1986 NP B277 168 A Measurement of the Space-like Pion Electromagnetic Formfactor AMENDOLIA 1984 PL 146B 116 A Measurement of the Pion Charge Radius DALLY 1982 PRL 48 375 Elastic Scattering Measurement of the Negative Pion Radius BEBEK 1978 PR D17 1693 Electroproduction of Single Pions at Low ${{\mathit \epsilon}}$ and a Measurement of the Pion Form-factor up to Q${}^{2}$ = 10 GeV${}^{2}$ QUENZER 1978 PL 76B 512 Pion Formfactor from 480 to 1100 MeV ADYLOV 1977 NP B128 461 A Measurement of the Electromagnetic Size of the Pion from Direct Elastic Pion Scattering Data at 50 ${\mathrm {GeV/}}\mathit c$ BARDIN 1977 NP B120 45 A Transverse and Longitudinal Cross Section Separation in a ${{\mathit \pi}^{+}}$ Electroproduction Coincidence Experiment and the Pion Radius DALLY 1977 PRL 39 1176 Direct Measurement of the ${{\mathit \pi}^{-}}$ Form-factor