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

ln$\mathit (C)$ (DISPERSIVE SCALAR FORM FACTOR FOR ${{\mathit K}_{{{\mu3}}}^{0}}$ DECAY)

INSPIRE   JSON PDGID:
S013LCM
See the review on “${{\mathit K}_{{{{{\mathit \ell}}3}}}^{\pm}}$ and ${{\mathit K}_{{{{{\mathit \ell}}3}}}^{0}}$ Form Factors” for details of the dispersive parametrization.
VALUE ($ 10^{-1} $) EVTS DOCUMENT ID TECN  COMMENT
$\bf{ 1.75 \pm0.18}$ OUR AVERAGE  Error includes scale factor of 2.0.  See the ideogram below.
$1.915$ $\pm0.078$ $\pm0.094$ 3.4M 1
ABOUZAID
2010
 
KTEV ${{\mathit \mu}}$ = ${{\mathit e}}$
$2.04$ $\pm0.19$ $\pm0.15$ 3.8M 2
AMBROSINO
2007C
 
KLOE ${{\mathit \mu}}$ = ${{\mathit e}}$
$1.438$ $\pm0.080$ $\pm0.112$ 2.3M 3
LAI
2007A
 
NA48 DP
1  Obtained from a sample of 1.9 M ${{\mathit K}_{{{e3}}}}$ and 1.5 M ${{\mathit K}_{{{\mu3}}}}$. The correlation between ${{\mathit \Lambda}_{{{+}}}}$ and ln($\mathit C$) is $-0.269$.
2  AMBROSINO 2007C results include 2M ${{\mathit K}_{{{e3}}}}$ events from AMBROSINO 2006D. We convert (${{\mathit \Lambda}_{{{+}}}}$, ${{\mathit \Lambda}_{{{0}}}}$) to (${{\mathit \Lambda}_{{{+}}}}$, ln$\mathit (C)$) parametrization using ln$\mathit (C)$ = (${{\mathit \Lambda}_{{{0}}}}$ $\cdot{}$ 11.713 + 0.0398)$\pm0.0041$, where the error is due to theory parametrization of the form factor. The correlation between ${{\mathit \Lambda}_{{{+}}}}$ and ln$\mathit (C)$ is $-0.26$.
3  LAI 2007A gives a correlation $-0.44$ between their ${{\mathit \Lambda}_{{{+}}}}$ and ln$\mathit (C)$ measurements.

           ln$\mathit (C)$ (DISPERSIVE SCALAR FORM FACTOR FOR ${{\mathit K}_{{{\mu3}}}^{0}}$ DECAY) ($ 10^{-1} $)
References