TESTS OF $\Delta \mathit S$ = $\Delta \mathit Q$ RULE

Re(x$_{+}$)

INSPIRE   PDGID:
S011XRP
A non-zero value would violate the $\Delta \mathit S$ = $\Delta \mathit Q$ rule in $\mathit CPT$ conserving transitions. x$_{+}$ is defined above in the Re(x$_{-}$) section.

VALUE ($ 10^{-3} $) EVTS DOCUMENT ID TECN
$\bf{ -0.9 \pm3.0}$ OUR AVERAGE
$-2$ $\pm10$ 1
BATLEY
2007D
 NA48
$-0.5$ $\pm3.6$ 13k 2
AMBROSINO
2006E
 KLOE
$-1.8$ $\pm6.1$ 3
ANGELOPOULOS
1998D
 CPLR
1  Result obtained from the measurement $\Gamma $( ${{\mathit K}_S^0}$ $\rightarrow$ ${{\mathit \pi}}{{\mathit e}}{{\mathit \nu}}$ ) $/$ $\Gamma $( ${{\mathit K}_L^0}$ $\rightarrow$ ${{\mathit \pi}}{{\mathit e}}{{\mathit \nu}}$ ) = $0.993$ $\pm0.34$, neglecting possible $\mathit CPT$ non-invariance and using PDG 2006 values of B( ${{\mathit K}_L^0}$ $\rightarrow$ ${{\mathit \pi}}{{\mathit e}}{{\mathit \nu}}$ ) = $0.4053$ $\pm0.0015$, ${\mathit \tau}_{{{\mathit L}}}$ = ($5.114$ $\pm0.021$) $ \times 10^{-8}$ s and ${\mathit \tau}_{{{\mathit S}}}$ = ($0.8958$ $\pm0.0005$) $ \times 10^{-10}~$s.
2  Re(x$_{+}$) can be shown to be equal to the following combination of rates: Re(x$_{+}$) = ${1\over 2}{ \Gamma ( {{\mathit K}_S^0} \rightarrow {{\mathit \pi}} {{\mathit e}} {{\mathit \nu}} ) − \Gamma ( {{\mathit K}_L^0} \rightarrow {{\mathit \pi}} {{\mathit e}} {{\mathit \nu}} )\over \Gamma ( {{\mathit K}_S^0} \rightarrow {{\mathit \pi}} {{\mathit e}} {{\mathit \nu}} ) + \Gamma ( {{\mathit K}_L^0} \rightarrow {{\mathit \pi}} {{\mathit e}} {{\mathit \nu}} )}$ which is valid up to first order in terms violating $\mathit CPT$ and/or the $\Delta \mathit S$ = $\Delta \mathit Q$ rule.
3  Obtained neglecting $\mathit CPT$ violating amplitudes.
Conservation Laws:
$\Delta \mathit S$ = $\Delta \mathit Q$ RULE
References:
BATLEY 2007D
PL B653 145 Determination of the Relative Decay Rate ${{\mathit K}_S^0}$ $\rightarrow$ ${{\mathit \pi}}{{\mathit e}}{{\mathit \nu}}$ / ${{\mathit K}_L^0}$ $\rightarrow$ ${{\mathit \pi}}{{\mathit e}}{{\mathit \nu}}$
AMBROSINO 2006E
PL B636 173 Study of the Branching Ratio and Charge Asymmetry for the Decay ${{\mathit K}_S^0}$ $\rightarrow$ ${{\mathit \pi}}{{\mathit e}}{{\mathit \nu}}$ with the KLOE Detector
ANGELOPOULOS 1998D
PL B444 38 Measurement of the ${{\mathit K}_L^0}$ $−$ ${{\mathit K}_S^0}$ Mass Difference using Semileptonic Decays of Tagged Neutral Kaons
Also
EPJ C22 55 $\mathit T$-Violation and $\mathit CPT$-Invariance Measurements in the CPLEAR Experiment: a Detailed Description of the Analysis of Neutral-Kaon Decays to ${{\mathit e}}{{\mathit \pi}}{{\mathit \nu}}$