${{\mathit \pi}^{\pm}}$ CHARGE RADIUS

INSPIRE   PDGID:
S008CR
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
2021A
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