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${{\mathit K}_{{{{\mathit \ell}}3}}^{\pm}}$ 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 $''$_{+}$(QUADRATIC ${{\mathit K}_{{e3}}^{\pm}}$ FORM FACTOR)

INSPIRE   PDGID:

S010LQE

VALUE ($ 10^{-2} $) EVTS DOCUMENT ID TECN CHG  COMMENT $\bf{ 0.186 \pm0.021}$ OUR AVERAGE $0.164$ $\pm0.030$ $\pm0.039$ 4.4M 1 BATLEY 2018 NA48 $\pm{}$ $0.191$ $\pm0.019$ $\pm0.014$ 5.25M YUSHCHENKO 2018 OKA + $0.192$ $\pm0.062$ $\pm0.071$ 919k 2, 3 YUSHCHENKO 2004 B ISTR - DP • • We do not use the following data for averages, fits, limits, etc. • • $-0.5$ $\pm0.7$ $\pm1.5$ 550k 4, 2 AJINENKO 2003 C 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 ($1.67$ $\pm0.29$ $\pm0.41$) $ \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:

BATLEY 2018 JHEP 1810 150 Measurement of the form factors of charged kaon semileptonic decays YUSHCHENKO 2018 JETPL 107 139 $K_{e3}$ decay studies in OKA experiment YUSHCHENKO 2004B PL B589 111 High Statistic Measurement of the ${{\mathit K}^{-}}$ $\rightarrow$ ${{\mathit \pi}^{0}}{{\mathit e}^{-}}{{\mathit \nu}}$ Decay Form Factor AJINENKO 2003C PL B574 14 High Statistics Study of the ${{\mathit K}^{-}}$ $\rightarrow$ ${{\mathit \pi}^{0}}{{\mathit e}^{-}}{{\mathit \nu}}$ Decay

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