${{\mathit K}_{{{{{\mathit \ell}}3}}}^{\pm}}$ 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.

$\vert \mathit f_{\mathit S}/\mathit f_{+}\vert $ FOR ${{\mathit K}_{{{e3}}}^{\pm}}$ DECAY

INSPIRE   PDGID:
S010FS
Ratio of scalar to $\mathit f_{+}$ couplings.
VALUE ($ 10^{-2} $) CL% EVTS DOCUMENT ID TECN CHG  COMMENT
$\bf{ (-80 {}^{+340}_{-400}) \times 10^{-3}}$ OUR AVERAGE
$0.013$ ${}^{+0.38}_{-0.46}$ 5.25M
YUSHCHENKO
2018
OKA + $\lambda $'$_{+}$, $\lambda $''$_{+}$, $\mathit f_{\mathit S}$ fit
$-0.37$ ${}^{+0.66}_{-0.56}$ $\pm0.41$ 919k
YUSHCHENKO
2004B
 ISTR - $\lambda $'$_{+}$, $\lambda $''$_{+}$, $\mathit f_{\mathit S}$ fit
$0.2$ $\pm2.6$ $\pm1.4$ 41k
SHIMIZU
2000
SPEC + $\lambda _{+}$, $\mathit f_{\mathit S}$, $\mathit f_{\mathit T}$ fit
• • We do not use the following data for averages, fits, limits, etc. • •
$0.2$ ${}^{+2.0}_{-2.2}$ $\pm0.3$ 550k 1
AJINENKO
2003C
 ISTR - $\lambda _{+}$, $\mathit f_{\mathit S}$, $\mathit f_{\mathit T}$ fit
$-1.9$ ${}^{+2.5}_{-1.6}$ 130k 1
AJINENKO
2002
SPEC $\lambda _{+}$, $\mathit f_{\mathit S}$ fit
$7.0$ $\pm1.6$ $\pm1.6$ 32k
AKIMENKO
1991
SPEC $\lambda _{+}$, $\mathit f_{\mathit S}$, $\mathit f_{\mathit T}$, ${{\mathit \phi}}$ fit
$0 \pm10$ 2827 2
BRAUN
1975
HLBC +
$\text{< 13}$ 90 4017
CHIANG
1972
OSPK +
$14 {}^{+3}_{-4}$ 2707 2
STEINER
1971
HLBC + $\lambda _{+}$, $\mathit f_{\mathit S}$, $\mathit f_{\mathit T}$, ${{\mathit \phi}}$ fit
$\text{< 23}$ 90
BOTTERILL
1968C
 ASPK
$\text{< 18}$ 90
BELLOTTI
1967B
 HLBC
$\text{< 30}$ 95
KALMUS
1967
HLBC +
1  Superseded by YUSHCHENKO 2004B.
2  Statistical errors only.
References