${{\mathit K}_{{{{{\mathit \ell}}3}}}^{\pm}}$ 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 ENERGY DEPENDENCE OF $\mathit f_{+}$ IN ${{\mathit K}_{{{e3}}}^{\pm}}$ DECAY)

INSPIRE   JSON  (beta) PDGID:
S010L+E
These results are for a linear expansion only. See the next section for fits including a quadratic term. For radiative correction of the ${{\mathit K}_{{{e3}}}^{\pm}}$ Dalitz plot, see GINSBERG 1967, BECHERRAWY 1970, CIRIGLIANO 2002, CIRIGLIANO 2004, and ANDRE 2007. Results labeled OUR FIT are discussed in the review “${{\mathit K}_{{{{{\mathit \ell}} 3}}}^{\pm}}$ and ${{\mathit K}_{{{{{\mathit \ell}} 3}}}^{0}}$ Form Factors” above. For earlier, lower statistics results, see the 2004 edition of this review, Physics Letters B592 1 (2004).
VALUE ($ 10^{-2} $) EVTS DOCUMENT ID TECN CHG  COMMENT
$\bf{ 2.959 \pm0.025}$ OUR FIT  Assuming ${{\mathit \mu}}-{{\mathit e}}$ universality
$\bf{ 2.956 \pm0.025}$ OUR AVERAGE
$2.95$ $\pm0.022$ $\pm0.018$ 5.25M
YUSHCHENKO
2018
OKA +
$3.044$ $\pm0.083$ $\pm0.074$ 1.1M
AKOPDZANOV
2009
TNF $\pm{}$
$2.966$ $\pm0.050$ $\pm0.034$ 919k 1
YUSHCHENKO
2004B
 ISTR - DP
$2.78$ $\pm0.26$ $\pm0.30$ 41k
SHIMIZU
2000
SPEC + DP
$2.84$ $\pm0.27$ $\pm0.20$ 32k 2
AKIMENKO
1991
SPEC PI, no RC
$2.9$ $\pm0.4$ 62k 3
BOLOTOV
1988
SPEC PI, no RC
• • We do not use the following data for averages, fits, limits, etc. • •
$3.06$ $\pm0.09$ $\pm0.06$ 550k 4, 1
AJINENKO
2003C
 ISTR - DP
$2.93$ $\pm0.15$ $\pm0.2$ 130k 4
AJINENKO
2002
SPEC DP
1  Rescaled to agree with our conventions as noted above.
2  AKIMENKO 1991 state that radiative corrections would raise $\lambda _{+}$ by $0.0013$.
3  BOLOTOV 1988 state radiative corrections of GINSBERG 1967 would raise $\lambda _{+}$ by $0.002$.
4  Superseded by YUSHCHENKO 2004B.
References