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

a$_{1}$/a$_{2}$ FORM FACTOR FOR M1 DIRECT EMISSION AMPLITUDE

INSPIRE   PDGID:
S013A12
Form factor = as described in ALAVI-HARATI 2000B.
VALUE (GeV${}^{2}$) EVTS DOCUMENT ID TECN  COMMENT
$\bf{ -0.737 \pm0.014}$ OUR AVERAGE
$-0.744$ $\pm0.027$ $\pm0.032$ 5241 1
ABOUZAID
2006
KTEV ${{\mathit \pi}^{+}}{{\mathit \pi}^{-}}{{\mathit e}^{+}}{{\mathit e}^{-}}$
$-0.738$ $\pm0.007$ $\pm0.018$ 111k 2
ABOUZAID
2006A
 KTEV ${{\mathit \pi}^{+}}{{\mathit \pi}^{+}}{{\mathit \gamma}}$
$-0.81$ ${}^{+0.07}_{-0.13}$ $\pm0.02$ 3
LAI
2003C
 NA48 ${{\mathit \pi}^{+}}{{\mathit \pi}^{-}}{{\mathit e}^{+}}{{\mathit e}^{-}}$
$-0.737$ $\pm0.026$ $\pm0.022$ 4
ALAVI-HARATI
2001B
 ${{\mathit \pi}^{+}}{{\mathit \pi}^{-}}{{\mathit \gamma}}$
$-0.720$ $\pm0.028$ $\pm0.009$ 1766 5
ALAVI-HARATI
2000B
 KTEV ${{\mathit \pi}^{+}}{{\mathit \pi}^{-}}{{\mathit e}^{+}}{{\mathit e}^{-}}$
1  ABOUZAID 2006 also measured $\vert {{\widetilde{\mathit g}}}_{M1}\vert $ = $1.11$ $\pm0.14$.
2  ABOUZAID 2006A also measured $\vert {{\widetilde{\mathit g}}}_{M1}\vert $ = $1.198$ $\pm0.035$ $\pm0.086$.
3  LAI 2003C also measured ${{\widetilde{\mathit g}}}_{M1}$ = $0.99$ ${}^{+0.28}_{-0.27}$ $\pm0.07$.
4  ALAVI-HARATI 2001B fit gives ${{\mathit \chi}^{2}}$/DOF = 38.8/27. Linear and quadratic fits give ${{\mathit \chi}^{2}}$/DOF = 43.2/27 and 37.6/26 respectively.
5  ALAVI-HARATI 2000B also measured $\vert {{\widetilde{\mathit g}}}_{M1}\vert $ = $1.35$ ${}^{+0.20}_{-0.17}$ $\pm0.04$.
References