${{\mathit K}_L^0}$ 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.

$\lambda _{0}$(LINEAR $\mathit f_{0}{{\mathit K}_{{\mu3}}^{0}}$ FORM FACTOR FROM QUADRATIC FIT)

INSPIRE   PDGID:
S013LZ
VALUE ($ 10^{-2} $) EVTS DOCUMENT ID TECN  COMMENT
$\bf{ 1.16 \pm0.09}$ OUR FIT  Error includes scale factor of 1.2.  Assuming ${{\mathit \mu}}-{{\mathit e}}$ universality
$\bf{ 1.07 \pm0.14}$ OUR FIT  Error includes scale factor of 1.3.  Not assuming ${{\mathit \mu}}-{{\mathit e}}$ universality
$0.91$ $\pm0.59$ $\pm0.26$ 1.8M 1
AMBROSINO
2007C
 KLOE no ${{\mathit \mu}}$ = ${{\mathit e}}$
$1.54$ $\pm0.18$ $\pm0.13$ 3.8M 2
AMBROSINO
2007C
 KLOE ${{\mathit \mu}}$ = ${{\mathit e}}$
$0.95$ $\pm0.11$ $\pm0.08$ 2.3M 3
LAI
2007A
 NA48 DP
$1.281$ $\pm0.136$ $\pm0.122$ 1.5M 4
ALEXOPOULOS
2004A
 KTEV DP, no ${{\mathit \mu}}$ = ${{\mathit e}}$
$1.372$ $\pm0.131$ 3.4M 5
ALEXOPOULOS
2004A
 KTEV PI, DP, ${{\mathit \mu}}$ = ${{\mathit e}}$
1  AMBROSINO 2007C, not assuming ${{\mathit \mu}}-{{\mathit e}}$ universality, gives a correlation matrix
  $\lambda $'$_{+}$ $\lambda $''$_{+}$
  $\lambda $''$_{+}$ $-0.97$ 1
  $\lambda _{0}$ 0.81 $-0.91$
2  AMBROSINO 2007C, assuming ${{\mathit \mu}}-{{\mathit e}}$ universality, gives a correlation matrix
  $\lambda $'$_{+}$ $\lambda $''$_{+}$
  $\lambda $''$_{+}$ $-0.95$ 1
  $\lambda _{0}$ 0.29 $-0.38$
3  LAI 2007A gives a correlation matrix
  $\lambda $'$_{+}$ $\lambda $''$_{+}$
  $\lambda $''$_{+}$ $\text{-}$0.96 1
  $\lambda _{0}$ 0.63 $-0.73$
4  ALEXOPOULOS 2004A, not assuming ${{\mathit \mu}}-{{\mathit e}}$ universality, gives a correlation matrix
  $\lambda $'$_{+}$ $\lambda $''$_{+}$ $\lambda _{0}$
  $\lambda $'$_{+}$ 1   
  $\lambda $''$_{+}$ $-0.96$ 1  
  $\lambda _{0}$ $0.65$ $-0.75$ 1
5  ALEXOPOULOS 2004A, assuming ${{\mathit \mu}}-{{\mathit e}}$ universality, gives a correlation matrix
  $\lambda $'$_{+}$ $\lambda $''$_{+}$ $\lambda _{0}$
  $\lambda $'$_{+}$ 1   
  $\lambda $''$_{+}$ $-0.97$ 1  
  $\lambda _{0}$ $0.34$ $-0.44$ 1
References:
AMBROSINO 2007C
JHEP 0712 105 Measurement of the ${{\mathit K}_L^0}$ $\rightarrow$ ${{\mathit \pi}}{{\mathit \mu}}{{\mathit \nu}}$ Form Factor Parameters with the KLOE Detector
LAI 2007A
PL B647 341 Measurement of ${{\mathit K}}{}^{0}_{{{\mathit \mu}}3}$ Form Factors
ALEXOPOULOS 2004A
PR D70 092007 Measurements of Semileptonic ${{\mathit K}_L^0}$ Decay Form Factors