${{\mathit K}_L^0}$ FORM FACTORS

For discussion, see note on form factors in the ${{\mathit K}^{\pm}}$ section of the Particle Listings above.
In the form factor comments, the following symbols are used.
 $\mathit f_{+}$ and $\mathit f_{−}$ are form factors for the vector matrix element.
 $\mathit f_{\mathit S}$ and $\mathit f_{\mathit T}$ refer to the scalar and tensor term.
 $\mathit f_{0}(\mathit t$) = $\mathit f_{+}(\mathit t$) + $\mathit f_{−}(\mathit t$) $\mathit t/({{\mathit m}^{2}}_{{{\mathit K}^{0}}}$ $−$ ${{\mathit m}^{2}}_{{{\mathit \pi}^{+}}}$).
 $\mathit t$ = momentum transfer to the ${{\mathit \pi}}$.
 $\lambda _{+}$ and $\lambda _{0}$ are the linear expansion coefficients of $\mathit f_{+}$ and $\mathit f_{0}$:
  $\mathit f_{+}(\mathit t$) = $\mathit f_{+}$(0) (1 + $\lambda _{+}\mathit t$ $/{{\mathit m}^{2}}_{{{\mathit \pi}^{+}}}$)
  For quadratic expansion
  $\mathit f_{+}(\mathit t$) = $\mathit f_{+}$(0) (1 + $\lambda $'$_{+}\mathit t$ $/{{\mathit m}^{2}}_{{{\mathit \pi}^{+}}}$ + ${\lambda ''_{+}\over 2}$ $\mathit t{}^{2}/\mathit m{}^{4}_{{{\mathit \pi}^{+}}}$ )
  as used by KTeV. If there is a non-vanishing quadratic term, then $\lambda _{+}$
  represents an average slope, which is then different from $\lambda $'$_{+}$.
  NA48 (${{\mathit K}_{{e3}}}$) and ISTRA quadratic expansion coefficients are converted with
  $\lambda $'$_{+}{}^{PDG}$ = $\lambda _{+}{}^{NA48}$ and $\lambda $''$_{+}{}^{PDG}$ = 2 $\lambda $'$_{+}{}^{NA48}$
  $\lambda $'$_{+}{}^{PDG}$ = (${{\mathit m}_{{{\mathit \pi}^{+}}}\over {\mathit m}_{{{\mathit \pi}^{0}}}}){}^{2}$ $\lambda _{+}{}^{ISTRA}$ and
  $\lambda $''$_{+}{}^{PDG}$ = 2 (${{\mathit m}_{{{\mathit \pi}^{+}}}\over {\mathit m}_{{{\mathit \pi}^{0}}}}){}^{4}$ $\lambda $'$_{+}{}^{ISTRA}$
  ISTRA linear expansion coefficients are converted with
  $\lambda _{+}{}^{PDG}$ = (${{\mathit m}_{{{\mathit \pi}^{+}}}\over {\mathit m}_{{{\mathit \pi}^{0}}}}){}^{2}$ $\lambda _{+}{}^{ISTRA}$ and $\lambda _{0}{}^{PDG}$ = (${{\mathit m}_{{{\mathit \pi}^{+}}}\over {\mathit m}_{{{\mathit \pi}^{0}}}}){}^{2}$ $\lambda _{0}{}^{ISTRA}$
  The pole parametrization is
  ${{\mathit f}_{{+}}}(\mathit t$) = ${{\mathit f}_{{+}}}$(0) (${{{\mathit M}_{{V}}^{2}}\over {{\mathit M}_{{V}}^{2}} − \mathit t }$ )
  ${{\mathit f}_{{0}}}(\mathit t$) = ${{\mathit f}_{{0}}}$(0) (${{{\mathit M}_{{S}}^{2}}\over {{\mathit M}_{{S}}^{2}} − \mathit t }$ )
  where ${{\mathit M}_{{V}}}$ and ${{\mathit M}_{{S}}}$ are the vector and scalar pole masses.
  The dispersive parametrization is
  ${{\mathit f}_{{+}}}(\mathit t$) = ${{\mathit f}_{{+}}}$(0) exp[ ${t\over m{}^{2}_{{{\mathit \pi}}} }$ (${{\mathit \Lambda}_{{+}}}$ + $\mathit H(\mathit t$)) ];
  ${{\mathit f}_{{0}}}(\mathit t$) = ${{\mathit f}_{{+}}}$(0) exp[ ${t\over m{}^{2}_{{{\mathit K}}} − m{}^{2}_{{{\mathit \pi}}} }$ (${\mathrm {ln}}[\mathit C$] $−$ $\mathit G(\mathit t$)) ],
  where ${{\mathit \Lambda}_{{+}}}$ is the slope parameter and ln[$\mathit C$ ] = ln[ ${{\mathit f}_{{0}}}$ ($\mathit m{}^{2}_{{{\mathit K}}}$ $−$ $\mathit m{}^{2}_{{{\mathit \pi}}}$ ) ]
  is the logarithm of the scalar form factor at the Callan-Treiman point.
  $\mathit H(t)$ and $\mathit G(t)$ are dispersive integrals.
  The following abbreviations are used:
 DP = Dalitz plot analysis.
 PI = ${{\mathit \pi}}$ spectrum analysis.
 MU = ${{\mathit \mu}}$ spectrum analysis.
 POL= ${{\mathit \mu}}$ polarization analysis.
 BR = ${{\mathit K}_{{\mu3}}^{0}}/{{\mathit K}_{{e3}}^{0}}$ branching ratio analysis.
 E = positron or electron spectrum analysis.
 RC = radiative corrections.

$\mathit L_{3}$ CHIRAL PERT. THEO. PARAM. FOR ${{\mathit K}_L^0}$ $\rightarrow$ ${{\mathit \pi}^{\pm}}{{\mathit \pi}^{0}}{{\mathit e}^{\mp}}{{\mathit \nu}_{{e}}}$

INSPIRE   PDGID:
S013L3
VALUE ($ 10^{-3} $) DOCUMENT ID TECN
$\bf{ -3.96 \pm0.28}$ OUR AVERAGE  Error includes scale factor of 1.6.
$-4.1$ $\pm0.2$
BATLEY
2004
NA48
$-3.4$ $\pm0.4$ 1
MAKOFF
1993
E731
1  MAKOFF 1993 sign has been changed to negative to agree with the sign convention used in BATLEY 2004 .
References:
BATLEY 2004
PL B595 75 Measurement of the Branching Ratio and Form Factors for the Decay ${{\mathit K}_L^0}$ $\rightarrow$ ${{\mathit \pi}^{\pm}}{{\mathit \pi}^{0}}{{\mathit e}^{\mp}}{{\mathit \nu}_{{e}}}$ (${{\overline{\mathit \nu}}_{{e}}}$)
MAKOFF 1993
PRL 70 1591 Study of the Decay ${{\mathit K}_L^0}$ $\rightarrow$ ${{\mathit \pi}^{\pm}}{{\mathit \pi}^{0}}{{\mathit e}^{\mp}}{{\overline{\mathit \nu}}}$( ${{\mathit \nu}}$)
Also
PRL 75 2069 (erratum) Study of the Decay ${{\mathit K}_L^0}$ $\rightarrow$ ${{\mathit \pi}^{\pm}}{{\mathit \pi}^{0}}{{\mathit e}^{\mp}}{{\overline{\mathit \nu}}}$( ${{\mathit \nu}}$)