ENERGY DEPENDENCE OF ${{\mathit K}_L^0}$ DALITZ PLOT

For discussion, see note on Dalitz plot parameters in the ${{\mathit K}^{\pm}}$ section of the Particle Listings above. For definitions of $\mathit a_{\mathit v}$, $\mathit a_{\mathit t}$, $\mathit a_{\mathit u}$, and $\mathit a_{\mathit y}$, see the earlier version of the same note in the 1982 edition of this $\mathit Review$ published in Physics Letters 111B 70 (1982).
$\vert $matrix element$\vert ^2$ = 1 + $\mathit g{}\mathit u$ + $\mathit h{}\mathit u{}^{2}$ + $\mathit j{}\mathit v$ + $\mathit k{}\mathit v{}^{2}$ + $\mathit f{}\mathit u{}\mathit v$ where $\mathit u$ = ($\mathit s_{3}$ $−$ $\mathit s_{0}$) $/$ ${{\mathit m}^{2}}_{{{\mathit \pi}}}$ and $\mathit v$ = ($\mathit s_{2}$ $−$ $\mathit s_{1}$) $/$ ${{\mathit m}^{2}}_{{{\mathit \pi}}}$

LINEAR COEFFICIENT $\mathit g$ FOR ${{\mathit K}_L^0}$ $\rightarrow$ ${{\mathit \pi}^{+}}{{\mathit \pi}^{-}}{{\mathit \pi}^{0}}$

INSPIRE   PDGID:
S013GT0
VALUE EVTS DOCUMENT ID TECN  COMMENT
$\bf{ 0.678 \pm0.008}$ OUR AVERAGE  Error includes scale factor of 1.5.  See the ideogram below.
$0.6823$ $\pm0.0044$ $\pm0.0044$ 500k
ANGELOPOULOS
1998C
 CPLR
$0.681$ $\pm0.024$ 6499
CHO
1977
HBC
$0.620$ $\pm0.023$ 4709
PEACH
1977
HBC
$0.677$ $\pm0.010$ 509k
MESSNER
1974
ASPK $\mathit a_{\mathit y}$ = $-0.917$ $\pm0.013$
• • We do not use the following data for averages, fits, limits, etc. • •
$0.69$ $\pm0.07$ 192 1
BALDO-CEOLIN
1975
HLBC
$0.590$ $\pm0.022$ 56k 1
BUCHANAN
1975
SPEC $\mathit a_{\mathit u}$ = $-0.277$ $\pm0.010$
$0.619$ $\pm0.027$ 20k 2, 1
BISI
1974
ASPK $\mathit a_{\mathit t}$ = $-0.282$ $\pm0.011$
$0.612$ $\pm0.032$ 1
ALEXANDER
1973B
 HBC
$0.73$ $\pm0.04$ 3200 1
BRANDENBURG
1973
HBC
$0.608$ $\pm0.043$ 1486 1
KRENZ
1972
HLBC $\mathit a_{\mathit t}$ = $-0.277$ $\pm0.018$
$0.650$ $\pm0.012$ 29k 1
ALBROW
1970
ASPK $\mathit a_{\mathit y}$ = $-0.858$ $\pm0.015$
$0.593$ $\pm0.022$ 36k 3, 1
BUCHANAN
1970
SPEC $\mathit a_{\mathit u}$ = $-0.278$ $\pm0.010$
$0.664$ $\pm0.056$ 4400 1
SMITH
1970
OSPK $\mathit a_{\mathit t}$ = $-0.306$ $\pm0.024$
$0.400$ $\pm0.045$ 2446 1
BASILE
1968B
 OSPK $\mathit a_{\mathit t}$ = $-0.188$ $\pm0.020$
$0.649$ $\pm0.044$ 1350 1
HOPKINS
1967
HBC $\mathit a_{\mathit t}$ = $-0.294$ $\pm0.018$
$0.428$ $\pm0.055$ 1198 1
NEFKENS
1967
OSPK $\mathit a_{\mathit u}$ = $-0.204$ $\pm0.025$
1  Quadratic dependence required by some experiments. (See sections on ``QUADRATIC COEFFICIENT $\mathit h$'' and ``QUADRATIC COEFFICIENT $\mathit k$'' below.) Correlations prevent us from averaging results of fits not including $\mathit g$, $\mathit h$, and $\mathit k$ terms.
2  BISI 1974 value comes from quadratic fit with quad. term consistent with zero. $\mathit g$ error is thus larger than if linear fit were used.
3  BUCHANAN 1970 result revised by BUCHANAN 1975 to include radiative correlations and to use more reliable ${{\mathit K}_L^0}$ momentum spectrum of second experiment (had same beam).

           Linear coeff. $\mathit g$ for ${{\mathit K}_L^0}$ $\rightarrow$ ${{\mathit \pi}^{+}}{{\mathit \pi}^{-}}{{\mathit \pi}^{0}}$ matrix element squared
References:
ANGELOPOULOS 1998C
EPJ C5 389 The Neutral Kaon Decays to ${{\mathit \pi}^{+}}{{\mathit \pi}^{-}}{{\mathit \pi}^{0}}$: a Detailed Analysis of the CPLEAR Data
CHO 1977
PR D15 587 A Measurement of ${{\mathit K}_L^0}$ $\rightarrow$ ${{\mathit \pi}^{+}}{{\mathit \pi}^{-}}{{\mathit \pi}^{0}}$ Decay Parameters
PEACH 1977
NP B127 399 A Study of the Dalitz Plot in the Decay ${{\mathit K}_L^0}$ $\rightarrow$ ${{\mathit \pi}^{+}}{{\mathit \pi}^{-}}{{\mathit \pi}^{0}}$
BALDO-CEOLIN 1975
NC 25A 688 Experimental Study of the ${{\mathit K}^{0}}$ $\rightarrow$ ${{\mathit \pi}^{+}}{{\mathit \pi}^{-}}{{\mathit \pi}^{0}}$ Decay
BUCHANAN 1975
PR D11 457 A Study of Three Body Charged Decays of ${{\mathit K}_L^0}$
BISI 1974
PL 50B 504 Measurement of the Slope Parameter for the Neutral Pion Spectrum in ${{\mathit K}_L^0}$ $\rightarrow$ ${{\mathit \pi}^{0}}{{\mathit \pi}^{+}}{{\mathit \pi}^{-}}$ Decay
MESSNER 1974
PRL 33 1458 Measurement of the ${{\mathit K}_L^0}$ $\rightarrow$ ${{\mathit \pi}^{+}}{{\mathit \pi}^{-}}{{\mathit \pi}^{0}}$ Dalitz Plot
ALEXANDER 1973B
NP B65 301 Measurement of the Branching Ratio and ${{\mathit \pi}^{0}}$ Energy Spectrum for ${{\mathit K}_L^0}$ $\rightarrow$ ${{\mathit \pi}^{+}}{{\mathit \pi}^{-}}{{\mathit \pi}^{0}}$
BRANDENBURG 1973
PR D8 1978 Measurement of the ${{\mathit K}_{{\mu3}}^{0}}/{{\mathit K}_{{e3}}^{0}}$ Branching Ratio, the ${{\mathit K}}{}^{0}_{{{\mathit \ell}}3}$ Form Factors, and the ${{\mathit K}}{}^{0}_{{{\mathit \pi}}3}$ Decay Parameters
KRENZ 1972
LNC 4 213 Yet Another Measurement of the Slope Parameter in the Decay ${{\mathit K}_L^0}$ $\rightarrow$ ${{\mathit \pi}^{+}}{{\mathit \pi}^{-}}{{\mathit \pi}^{0}}$
ALBROW 1970
PL 33B 516 An Experimental Study of the Decay ${{\mathit K}_L^0}$ $\rightarrow$ ${{\mathit \pi}^{+}}{{\mathit \pi}^{-}}$ ${{\mathit \pi}^{0}}$
BUCHANAN 1970
PL 33B 623 A Study of the Decay Mode ${{\mathit K}_L^0}$ $\rightarrow$ ${{\mathit \pi}^{+}}{{\mathit \pi}^{-}}{{\mathit \pi}^{0}}$
Also
Private Comm. Private Communication
SMITH 1970
PL 32B 133 Experimental Study of the Decay Spectrum of ${{\mathit K}_L^0}$ $\rightarrow$ ${{\mathit \pi}^{+}}{{\mathit \pi}^{-}}{{\mathit \pi}^{0}}$
BASILE 1968B
PL 28B 58 Experimental Study of the Decay Mode ${{\mathit K}_L^0}$ $\rightarrow$ ${{\mathit \pi}^{+}}{{\mathit \pi}^{-}}{{\mathit \pi}^{0}}$
HOPKINS 1967
PRL 19 185 Decay Rate and Spectra in ${{\mathit K}_L^0}$ $\rightarrow$ ${{\mathit \pi}^{+}}{{\mathit \pi}^{-}}{{\mathit \pi}^{0}}$
NEFKENS 1967
PR 157 1233 Measurement of Pion Spectra in the Decay ${{\mathit K}_{{2}}^{0}}$ $\rightarrow$ ${{\mathit \pi}^{+}}{{\mathit \pi}^{-}}{{\mathit \pi}^{0}}$