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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 .

$\lambda _{+}$ (LINEAR ENERGY DEPENDENCE OF $\mathit f_{+}$ IN ${{\mathit K}_{{e3}}^{\pm}}$ DECAY)

INSPIRE   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 2004 B 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 2003 C 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:

YUSHCHENKO 2018 JETPL 107 139 $K_{e3}$ decay studies in OKA experiment AKOPDZANOV 2009 PAN 71 2074 Study of ${{\mathit K}^{\pm}}$ $\rightarrow$ ${{\mathit e}^{\pm}}{{\mathit \nu}}{{\mathit \pi}^{0}}$ Decays at the KMN Setup 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 AJINENKO 2002 PAN 65 2064 Study of the ${{\mathit K}^{-}}$ $\rightarrow$ ${{\mathit \pi}^{0}}{{\mathit e}^{-}}{{\mathit \nu}}$ Decay SHIMIZU 2000 PL B495 33 Test of Exotic Scalar and Tensor Couplings in ${{\mathit K}^{+}}$ $\rightarrow$ ${{\mathit \pi}^{0}}{{\mathit e}^{+}}{{\mathit \nu}}$ Decay AKIMENKO 1991 PL B259 225 Measurement of the ${{\mathit K}^{+}}$ $\rightarrow$ ${{\mathit \pi}^{-}}{{\mathit e}^{+}}{{\mathit \nu}}$ Form Factors BOLOTOV 1988 JETPL 47 7 Measurement of Decay's Formfactor for ${{\mathit K}^{-}}$ $\rightarrow$ ${{\mathit \pi}^{0}}{{\mathit e}^{-}}{{\overline{\mathit \nu}}}$

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