${{\mathit K}_{{{{{\mathit \ell}}3}}}^{\pm}}$ FORM FACTORS

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.
 $\mathit f_{0}$ = $\mathit f_{+}$ + $\mathit f_{−}$ $\mathit t/({{\mathit m}^{2}}_{{{\mathit K}^{+}}}$ $−$ ${{\mathit m}^{2}}_{{{\mathit \pi}^{0}}}$).
 $\mathit t$ = 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 $'$_{+}$.
  NA48/2 and OKA quadratic expansion coefficients are converted with
  $\lambda $'$_{+}{}^{PDG}$ = $\lambda $'$_{+}{}^{NA48/2}$ and $\lambda $''$_{+}{}^{PDG}$ = 2 $\lambda $''$_{+}{}^{NA48/2}$
  $\lambda $'$_{+}{}^{PDG}$ = (${{\mathit m}_{{{\mathit \pi}^{+}}}\over {\mathit m}_{{{\mathit \pi}^{0}}}}){}^{2}$ $\lambda $'$_{+}{}^{OKA}$ and
  $\lambda $''$_{+}{}^{PDG}$ = 2 (${{\mathit m}_{{{\mathit \pi}^{+}}}\over {\mathit m}_{{{\mathit \pi}^{0}}}}){}^{4}$ $\lambda $''$_{+}{}^{OKA}$
  OKA linear expansion coefficients are converted with
  $\lambda _{+}{}^{PDG}$ = (${{\mathit m}_{{{\mathit \pi}^{+}}}\over {\mathit m}_{{{\mathit \pi}^{0}}}}){}^{2}$ $\lambda _{+}{}^{OKA}$ and $\lambda _{0}{}^{PDG}$ = (${{\mathit m}_{{{\mathit \pi}^{+}}}\over {\mathit m}_{{{\mathit \pi}^{0}}}}){}^{2}$ $\lambda _{0}{}^{OKA}$
  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 following abbreviations are used:
 DP = Dalitz plot analysis.
 PI = ${{\mathit \pi}}$ spectrum analysis.
 MU = ${{\mathit \mu}}$ spectrum analysis.
 POL= ${{\mathit \mu}}$ polarization analysis.
 BR = ${{\mathit K}_{{{\mu3}}}^{\pm}}/{{\mathit K}_{{{e3}}}^{\pm}}$ branching ratio analysis.
 E = positron or electron spectrum analysis.
 RC = radiative corrections.

For previous $\lambda $'$_{+}$ and $\lambda $''$_{+}$ parametrizations used by NA48 (e.g. LAI 2007A) and ISTRA (e.g. YUSHCHENKO 2004B) see PDG 2018.

$\lambda $'$_{+}$ (LINEAR ${{\mathit K}_{{{e3}}}^{\pm}}$ FORM FACTOR FROM QUADRATIC FIT)

INSPIRE   JSON  (beta) PDGID:
S010LPE
VALUE ($ 10^{-2} $) EVTS DOCUMENT ID TECN CHG  COMMENT
$\bf{ 2.59 \pm0.04}$ OUR AVERAGE
$2.426$ $\pm0.078$ $\pm0.130$ 4.4M 1
BATLEY
2018
NA48 $\pm{}$
$2.611$ $\pm0.035$ $\pm0.028$ 5.25M
YUSHCHENKO
2018
OKA +
$2.485$ $\pm0.163$ $\pm0.034$ 919k 2, 3
YUSHCHENKO
2004B
 ISTR - DP
• • We do not use the following data for averages, fits, limits, etc. • •
$3.07$ $\pm0.21$ 550k 4, 2
AJINENKO
2003C
 ISTR - DP
1  Data collected in 2004 by NA48/2. Correlation coefficient with quadratic slope is $-0.929$. $\chi {}^{2}/\mathit NDF$ = 569.1/687. BATLEY 2018 also performed a combined ${{\mathit K}_{{{e3}}}^{\pm}}$ and ${{\mathit K}_{{{\mu3}}}^{\pm}}$ fit assuming ${{\mathit \mu}}−{{\mathit e}}$ universality and obtained ($24.24$ $\pm0.75$ $\pm1.3$) $ \times 10^{-3}$.
2  Rescaled to agree with our conventions as noted above.
3  YUSHCHENKO 2004B $\lambda $'$_{+}$ and $\lambda $''$_{+}$ are strongly correlated with coefficient $\rho (\lambda $'$_{+}$, $\lambda $''$_{+}$) = $−$0.95.
4  Superseded by YUSHCHENKO 2004B.
References