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${{\mathit K}_{{{{\mathit \ell}}4}}^{\pm}}$ FORM FACTORS

Based on the parametrizations of AMOROS 1999 , the ${{\mathit K}_{{{{\mathit \ell}}4}}^{\pm}}$ form factors can be expressed as

$\mathit F_{s}$ = $\mathit f_{s}$ + ${{\mathit f}_{{s}}^{\,'}}$ q${}^{2}$ + ${{\mathit f}_{{s}}^{''}}$ q${}^{4}$ + ${{\mathit f}_{{e}}^{\,'}}$ S$_{e}$ $/$ 4${{\mathit m}^{2}}_{{{\mathit \pi}}}$

$\mathit F_{p}$ = $\mathit f_{p}$

$\mathit G_{p}$ = $\mathit g_{p}$ + ${{\mathit g}_{{p}}^{\,'}}$ q${}^{2}$

$\mathit H_{p}$ = $\mathit h_{p}$

where q${}^{2}$ = (S$_{{{\mathit \pi}}}$ $/$ 4${{\mathit m}^{2}}_{{{\mathit \pi}}}$) $−$ 1, S$_{{{\mathit \pi}}}$ is the invariant mass squared of the dipion, and S$_{e}$ is the invariant mass squared of the dilepton.

${{\mathit f}_{{s}}^{\,'}}/{{\mathit f}_{{s}}}$ FOR ${{\mathit K}^{\pm}}$ $\rightarrow$ ${{\mathit \pi}^{+}}{{\mathit \pi}^{-}}{{\mathit e}^{\pm}}{{\mathit \nu}}$ DECAY

INSPIRE   PDGID:

S010RSP

VALUE ($ 10^{-2} $) EVTS DOCUMENT ID TECN CHG $15.2$ $\pm0.7$ $\pm0.5$ 1.13M 1 BATLEY 2010 C NA48 $\pm{}$ • • We do not use the following data for averages, fits, limits, etc. • • $17.2$ $\pm0.9$ $\pm0.6$ 670k 2 BATLEY 2008 A NA48 $\pm{}$ 1  Radiative corrections included. Using Roy equations and including isospin breaking, BATLEY 2010C obtains the following scattering lengths ${{\mathit a}_{{0}}^{0}}$ = $0.2220$ $\pm0.0128$ $\pm0.0050$ $\pm0.0037~$(theor.), ${{\mathit a}_{{0}}^{2}}$ = $-0.0432$ $\pm0.0086$ $\pm0.0034$ $\pm0.0028~$(theor.). The correlation with ${{\mathit f}_{{s}}^{''}}/{{\mathit f}_{{s}}}$ = $-0.954$ and with ${{\mathit f}_{{e}}^{\,'}}/{{\mathit f}_{{s}}}$ = 0.080. Supersedes BATLEY 2008A. 2  Radiative corrections included. Using Roy equations and not including isospin breaking, BATLEY 2008A obtains the following ${{\mathit \pi}}{{\mathit \pi}}$ scattering length $\mathit a{}^{0}_{0}$ = $0.233$ $\pm0.016$ $\pm0.007$ $\mathit a{}^{2}_{0}$ = $-0.0471$ $\pm0.011$ $\pm0.004$.

References:

BATLEY 2010C EPJ C70 635 Precise Tests of Low Energy QCD from ${{\mathit K}_{{e4}}}$ Decay Properties BATLEY 2008A EPJ C54 411 New High Statistics Measurement of ${{\mathit K}_{{e4}}}$ Decay Form Factors and ${{\mathit \pi}}{{\mathit \pi}}$ Scattering Phase Shifts

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