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${{\mathit K}_L^0}$ FORM FACTORS

For discussion, see note on form factors in the ${{\mathit K}^{\pm}}$ section of the Particle Listings above.
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.

$\mathit f_{0}(\mathit t$) = $\mathit f_{+}(\mathit t$) + $\mathit f_{−}(\mathit t$) $\mathit t/({{\mathit m}^{2}}_{{{\mathit K}^{0}}}$ $−$ ${{\mathit m}^{2}}_{{{\mathit \pi}^{+}}}$).

$\mathit t$ = 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 (${{\mathit K}_{{e3}}}$) and ISTRA quadratic expansion coefficients are converted with

$\lambda $'$_{+}{}^{PDG}$ = $\lambda _{+}{}^{NA48}$ and $\lambda $''$_{+}{}^{PDG}$ = 2 $\lambda $'$_{+}{}^{NA48}$

$\lambda $'$_{+}{}^{PDG}$ = (${{\mathit m}_{{{\mathit \pi}^{+}}}\over {\mathit m}_{{{\mathit \pi}^{0}}}}){}^{2}$ $\lambda _{+}{}^{ISTRA}$ and

$\lambda $''$_{+}{}^{PDG}$ = 2 (${{\mathit m}_{{{\mathit \pi}^{+}}}\over {\mathit m}_{{{\mathit \pi}^{0}}}}){}^{4}$ $\lambda $'$_{+}{}^{ISTRA}$

ISTRA linear expansion coefficients are converted with

$\lambda _{+}{}^{PDG}$ = (${{\mathit m}_{{{\mathit \pi}^{+}}}\over {\mathit m}_{{{\mathit \pi}^{0}}}}){}^{2}$ $\lambda _{+}{}^{ISTRA}$ and $\lambda _{0}{}^{PDG}$ = (${{\mathit m}_{{{\mathit \pi}^{+}}}\over {\mathit m}_{{{\mathit \pi}^{0}}}}){}^{2}$ $\lambda _{0}{}^{ISTRA}$

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 dispersive parametrization is

${{\mathit f}_{{+}}}(\mathit t$) = ${{\mathit f}_{{+}}}$(0) exp[ ${t\over m{}^{2}_{{{\mathit \pi}}} }$ (${{\mathit \Lambda}_{{+}}}$ + $\mathit H(\mathit t$)) ];

${{\mathit f}_{{0}}}(\mathit t$) = ${{\mathit f}_{{+}}}$(0) exp[ ${t\over m{}^{2}_{{{\mathit K}}} − m{}^{2}_{{{\mathit \pi}}} }$ (${\mathrm {ln}}[\mathit C$] $−$ $\mathit G(\mathit t$)) ],

where ${{\mathit \Lambda}_{{+}}}$ is the slope parameter and ln[$\mathit C$ ] = ln[ ${{\mathit f}_{{0}}}$ ($\mathit m{}^{2}_{{{\mathit K}}}$ $−$ $\mathit m{}^{2}_{{{\mathit \pi}}}$ ) ]

is the logarithm of the scalar form factor at the Callan-Treiman point.

$\mathit H(t)$ and $\mathit G(t)$ are dispersive integrals.

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 = ${{\mathit K}_{{\mu3}}^{0}}/{{\mathit K}_{{e3}}^{0}}$ branching ratio analysis.

E = positron or electron spectrum analysis.

RC = radiative corrections.

$\lambda _{+}$ (LINEAR ENERGY DEPENDENCE OF $\mathit f_{+}$ IN ${{\mathit K}_{{\mu3}}^{0}}$ DECAY)

INSPIRE   PDGID:

S013L+M

Results labeled OUR FIT are discussed in the review “${{\mathit K}_{{{{\mathit \ell}} 3}}^{\pm}}$ and ${{\mathit K}_{{{{\mathit \ell}} 3}}^{0}}$ Form Factors” in the ${{\mathit K}^{\pm}}$ Listings. For earlier, lower statistics results, see the 2004 edition of this review, Physics Letters B592 1 (2004).

VALUE ($ 10^{-2} $) EVTS DOCUMENT ID TECN  COMMENT $\bf{ 2.82 \pm0.04}$ OUR FIT  Error includes scale factor of 1.1.  Assuming ${{\mathit \mu}}-{{\mathit e}}$ universality $\bf{ 2.71 \pm0.10}$ OUR FIT  Error includes scale factor of 1.4.  Not assuming ${{\mathit \mu}}-{{\mathit e}}$ universality $2.67$ $\pm0.06$ $\pm0.08$ 2.3M 1 LAI 2007 A NA48 DP $2.745$ $\pm0.088$ $\pm0.063$ 1.5M ALEXOPOULOS 2004 A KTEV DP, no ${{\mathit \mu}}$ = ${{\mathit e}}$ $2.813$ $\pm0.051$ 3.4M ALEXOPOULOS 2004 A KTEV PI, DP, ${{\mathit \mu}}$ = ${{\mathit e}}$ $3.0$ $\pm0.3$ 1.6M DONALDSON 1974 B SPEC DP • • We do not use the following data for averages, fits, limits, etc. • • $4.27$ $\pm0.44$ 150k BIRULEV 1981 SPEC DP 1  LAI 2007A gives a correlation $-0.40$ between their ${{\mathit \lambda}_{{0}}}$ and ${{\mathit \lambda}_{{+}}}$ measurements.

References:

LAI 2007A PL B647 341 Measurement of ${{\mathit K}}{}^{0}_{{{\mathit \mu}}3}$ Form Factors ALEXOPOULOS 2004A PR D70 092007 Measurements of Semileptonic ${{\mathit K}_L^0}$ Decay Form Factors BIRULEV 1981 NP B182 1 A Study of the Semileptonic Decays of the Neutral Kaons Also SJNP 31 622 Study of Decays ${{\mathit K}_L^0}$ $\rightarrow$ ${{\mathit \pi}^{\pm}}{{\mathit \mu}^{\pm}}{{\mathit \nu}_{{\mu}}}$ DONALDSON 1974B PR D9 2960 Measurement of the Formfactors in the Decay ${{\mathit K}_L^0}$ $\rightarrow$ ${{\mathit \pi}}{{\mathit \mu}}{{\mathit \nu}}$ Also PRL 31 337 Measurement of the Formfactors in the Decay ${{\mathit K}_L^0}$ $\rightarrow$ ${{\mathit \pi}}{{\mathit \mu}}{{\mathit \nu}}$

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