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

$\mathit f_{\mathit T}/\mathit f_{+}$ FOR ${{\mathit K}_{{\mu3}}^{\pm}}$ DECAY

INSPIRE   PDGID:
S010FTM
Ratio of tensor to $\mathit f_{+}$ couplings.
VALUE ($ 10^{-2} $) EVTS DOCUMENT ID TECN CHG  COMMENT
$-0.07$ $\pm0.71$ $\pm0.20$ 540k
YUSHCHENKO
2004
ISTR - DP
• • We do not use the following data for averages, fits, limits, etc. • •
$-2.1$ $\pm2.8$ $\pm1.4$ 112k 1
AJINENKO
2003
ISTR - DP
$2$ $\pm12$ 1585
BRAUN
1975
HLBC
1  The second error is the theoretical error from the uncertainty in the chiral perturbation theory prediction for~${{\mathit \lambda}}_{0}$. Superseded by YUSHCHENKO 2004 .
References:
YUSHCHENKO 2004
PL B581 31 High Statistic Study of the ${{\mathit K}^{-}}$ $\rightarrow$ ${{\mathit \pi}^{0}}{{\mathit \mu}^{-}}{{\mathit \nu}}$ Decay
AJINENKO 2003
PAN 66 105 Study of the ${{\mathit K}^{-}}$ $\rightarrow$ ${{\mathit \mu}^{-}}{{\overline{\mathit \nu}}}{{\mathit \pi}^{0}}$ Decay
BRAUN 1975
NP B89 210 A Study of the Semileptonic Decays of ${{\mathit K}^{+}}$ Mesons