PARAMETERS FOR ${{\mathit K}_L^0}$ $\rightarrow$ 2 ${{\mathit \pi}}$ DECAY

$\eta _{+−}$ = A( ${{\mathit K}_L^0}$ $\rightarrow$ ${{\mathit \pi}^{+}}{{\mathit \pi}^{-}}$ ) $/$ A( ${{\mathit K}_S^0}$ $\rightarrow$ ${{\mathit \pi}^{+}}{{\mathit \pi}^{-}}$ ) $\eta _{00}$ = A( ${{\mathit K}_L^0}$ $\rightarrow$ ${{\mathit \pi}^{0}}{{\mathit \pi}^{0}}$ ) $/$ A( ${{\mathit K}_S^0}$ $\rightarrow$ ${{\mathit \pi}^{0}}{{\mathit \pi}^{0}}$ )
The fitted values of $\vert \eta _{+−}\vert $ and $\vert \eta _{00}\vert $ given below are the results of a fit to $\vert \eta _{+−}\vert $, $\vert \eta _{00}\vert $, $\vert \eta _{00}/\eta _{+−}\vert $, and Re($\epsilon {{}^\prime}/\epsilon $). Independent information on $\vert \eta _{+−}\vert $ and $\vert \eta _{00}\vert $ can be obtained from the fitted values of the ${{\mathit K}_L^0}$ $\rightarrow$ ${{\mathit \pi}}{{\mathit \pi}}$ and ${{\mathit K}_S^0}$ $\rightarrow$ ${{\mathit \pi}}{{\mathit \pi}}$ branching ratios and the ${{\mathit K}_L^0}$ and ${{\mathit K}_S^0}$ lifetimes. This information is included as data in the $\vert \eta _{+−}\vert $ and $\vert \eta _{00}\vert $ sections with a Document ID “BRFIT.” See the note “$\mathit CP$ violation in ${{\mathit K}_{{L}}}$ decays” above for details.

$\phi _{+−}$, PHASE of $\eta _{+−}$

INSPIRE   PDGID:
S013F+-
The dependence of the phase on $\Delta \mathit m$ and $\tau _{{{\mathit S}}}$ is given for each experiment in the comments below, where $\Delta \mathit m$ is the ${{\mathit K}_L^0}$ $–{{\mathit K}_S^0}$ mass difference in units $10^{10}$ $\hbar{}$s${}^{-1}$ and ${{\mathit \tau}_{{s}}}$ is the ${{\mathit K}_{{S}}}$ mean life in units $10^{-10}~$s. We also give the regeneration phase $\phi _{\mathit f}$ in the comments below.

OUR FIT is described in the note on “$\mathit CP$ violation in ${{\mathit K}_{{L}}}$ decays” in the ${{\mathit K}_L^0}$ Particle Listings. Most experiments in this section are included in both the “Not Assuming $\mathit CPT$” and “Assuming $\mathit CPT$” fits. In the latter fit, they have little direct influence on $\phi _{+−}$ because their errors are large compared to that assuming $\mathit CPT$, but they influence $\Delta \mathit m$ and $\tau _{\mathit s}$ through their dependencies on these parameters, which are given in the footnotes.
VALUE ($^\circ{}$) EVTS DOCUMENT ID TECN  COMMENT
$\bf{ 43.4 \pm0.5}$ OUR FIT  Error includes scale factor of 1.2.  Not assuming $\mathit CPT$
$\bf{ 43.51 \pm0.05}$ OUR FIT  Error includes scale factor of 1.2.  Assuming $\mathit CPT$
$42.9 \pm0.6 \pm0.3$ 70M 1
APOSTOLAKIS
1999C
 CPLR ${{\mathit K}^{0}}-{{\overline{\mathit K}}^{0}}$ asymmetry
$42.9 \pm0.8 \pm0.2$ 2, 3
SCHWINGENHEUE..
1995
E773 CH$_{1.1}$ regenerator
$41.4 \pm0.9 \pm0.2$ 4, 3
GIBBONS
1993
E731 B$_{4}$C regenerator
$44.5 \pm1.6 \pm0.6$ 5
CAROSI
1990
NA31 Vacuum regen.
$43.3 \pm1.0 \pm0.5$ 6
GEWENIGER
1974B
 ASPK Vacuum regen.
• • We do not use the following data for averages, fits, limits, etc. • •
$43.76$ $\pm0.64$ 7
ABOUZAID
2011
KTEV Not assuming $\mathit CPT$
$44.12$ $\pm0.72$ $\pm1.20$ 8
ALAVI-HARATI
2003
KTEV Not assuming $\mathit CPT$
$42.5 \pm0.4 \pm0.3$ 9, 10
ADLER
1996C
 RVUE
$43.4 \pm1.1 \pm0.3$ 11
ADLER
1995B
 CPLR ${{\mathit K}^{0}}-{{\overline{\mathit K}}^{0}}$ asymmetry
$42.3$ $\pm4.4$ $\pm1.4$ 100k 12
ADLER
1992B
 CPLR ${{\mathit K}^{0}}-{{\overline{\mathit K}}^{0}}$ asymmetry
$47.7$ $\pm2.0$ $\pm0.9$ 13, 3
KARLSSON
1990
E731
$44.3 \pm2.8 \pm0.2$ 14
CARITHERS
1975
SPEC ${}^{}\mathrm {C}$ regenerator
1  APOSTOLAKIS 1999C measures $\phi _{+−}$ = ( $43.19$ $\pm0.53$ $\pm0.28$ ) + $300$ [ $\Delta \mathit m$$–$ $0.5301$ ] ($^\circ{}$) . We have adjusted the measurement to use our best values of ( $\Delta \mathit m$ = $0.5293$ $\pm0.0009$ ) . Our first error is their experiment's error and our second error is the systematic error from using our best values.
2  SCHWINGENHEUER 1995 measures $\phi _{+−}$ = ( $43.53$ $\pm0.76$ ) + $173$ [ $\Delta \mathit m$$–$ $0.5282$ ] $–$ $275$ [ ${{\mathit \tau}_{{s}}}$$–$ $0.8926$ ] ($^\circ{}$) . We have adjusted the measurement to use our best values of ( $\Delta \mathit m$ = $0.5293$ $\pm0.0009$ ) , ( ${{\mathit \tau}_{{s}}}$ = $0.8954$ $\pm0.0004$ ) . Our first error is their experiment's error and our second error is the systematic error from using our best values.
3  These experiments measure $\phi _{+−}-{{\mathit \phi}_{{f}}}$ and calculate the regeneration phase from the power law momentum dependence of the regeneration amplitude using analyticity and dispersion relations. SCHWINGENHEUER 1995 [GIBBONS 1993 ] includes a systematic error of $0.35^\circ{}$ [$0.5^\circ{}$] for uncertainties in their modeling of the regeneration amplitude.
4  GIBBONS 1993 measures $\phi _{+−}$ = ( $42.21$ $\pm0.9$ ) + $189$ [ $\Delta \mathit m$$–$ $0.5257$ ] $–$ $460$ [ ${{\mathit \tau}_{{s}}}$$–$ $0.8922$ ] ($^\circ{}$) . We have adjusted the measurement to use our best values of ( $\Delta \mathit m$ = $0.5293$ $\pm0.0009$ ) , ( ${{\mathit \tau}_{{s}}}$ = $0.8954$ $\pm0.0004$ ) . Our first error is their experiment's error and our second error is the systematic error from using our best values. This is actually reported in SCHWINGENHEUER 1995 , footnote$~$8. GIBBONS 1993 reports $\phi _{+−}$ ($42.2$ $\pm1.4)^\circ{}$. They measure $\phi _{+}-\phi _{\mathit f}$ and calculate the regeneration phase $\phi _{\mathit f}$ from the power law momentum dependence of the regeneration amplitude using analyticity. An error of $0.6^\circ{}$ is included for possible uncertainties in the regeneration phase.
5  CAROSI 1990 measures $\phi _{+−}$ = ( $46.9$ $\pm1.4$ $\pm0.7$ ) + $579$ [ $\Delta \mathit m$$–$ $0.5351$ ] + $303$ [ ${{\mathit \tau}_{{s}}}$$–$ $0.8922$ ] ($^\circ{}$) . We have adjusted the measurement to use our best values of ( $\Delta \mathit m$ = $0.5293$ $\pm0.0009$ ) , ( ${{\mathit \tau}_{{s}}}$ = $0.8954$ $\pm0.0004$ ) . Our first error is their experiment's error and our second error is the systematic error from using our best values.
6  GEWENIGER 1974B measures $\phi _{+−}$ = ( $49.4$ $\pm1.0$ ) + $565$ [ $\Delta \mathit m$$–$ $0.540$ ] ($^\circ{}$) . We have adjusted the measurement to use our best values of ( $\Delta \mathit m$ = $0.5293$ $\pm0.0009$ ) . Our first error is their experiment's error and our second error is the systematic error from using our best values.
7  Not independent of other phase parameters reported in ABOUZAID 2011 .
8  ALAVI-HARATI 2003 $\phi _{+−}$ is correlated with their $\Delta \mathit m$ = ${\mathit m}_{{{\mathit K}_L^0} }–{\mathit m}_{{{\mathit K}_S^0} }$ and $\tau _{{{\mathit K}_{{S}}}}$ measurements in the ${{\mathit K}_L^0}$ and ${{\mathit K}_S^0}$ sections respectively. The correlation coefficients are $\rho\mathrm {(\phi _{+−},\Delta \mathit m)}=+0.955$, $\rho\mathrm {(\phi _{+−},\tau _{\mathit S})}=-0.871$, and $\rho\mathrm {(\tau _{\mathit S},\Delta \mathit m)}=-0.840$. $\mathit CPT$ is not assumed. Uses scintillator Pb regenerator. Superseded by ABOUZAID 2011 .
9  ADLER 1996C measures $\phi _{+−}$ = ( $43.82$ $\pm0.41$ ) + $339$ [ $\Delta \mathit m$$–$ $0.5307$ ] $–$ $252$ [ ${{\mathit \tau}_{{s}}}$$–$ $0.8922$ ] ($^\circ{}$) . We have adjusted the measurement to use our best values of ( $\Delta \mathit m$ = $0.5293$ $\pm0.0009$ ) , ( ${{\mathit \tau}_{{s}}}$ = $0.8954$ $\pm0.0004$ ) . Our first error is their experiment's error and our second error is the systematic error from using our best values.
10  ADLER 1996C is the result of a fit which includes nearly the same data as entered into the ``OUR$~$FIT'' value in the 1996 edition of this Review (Physical Review D54 1 (1996)).
11  ADLER 1995B measures $\phi _{+−}$ = ( $42.7$ $\pm0.9$ $\pm0.6$ ) + $316$ [ $\Delta \mathit m$$–$ $0.5274$ ] + $30$ [ ${{\mathit \tau}_{{s}}}$$–$ $0.8926$ ] ($^\circ{}$) . We have adjusted the measurement to use our best values of ( $\Delta \mathit m$ = $0.5293$ $\pm0.0009$ ) , ( ${{\mathit \tau}_{{s}}}$ = $0.8954$ $\pm0.0004$ ) . Our first error is their experiment's error and our second error is the systematic error from using our best values.
12  ADLER 1992B quote separately two systematic errors: $\pm0.4$ from their experiment and $\pm1.0$ degrees due to the uncertainty in the value of $\Delta \mathit m$.
13  KARLSSON 1990 systematic error does not include regeneration phase uncertainty.
14  CARITHERS 1975 measures $\phi _{+−}$ = ( $45.5$ $\pm2.8$ ) + $224$ [ $\Delta \mathit m$$–$ $0.5348$ ] ($^\circ{}$) . We have adjusted the measurement to use our best values of ( $\Delta \mathit m$ = $0.5293$ $\pm0.0009$ ) . Our first error is their experiment's error and our second error is the systematic error from using our best values. $\phi _{\mathit f}$ = $-40.9$ $\pm2.6^\circ{}$.
Conservation Laws:
$\mathit CP$ VIOLATION OBSERVED
References:
ABOUZAID 2011
PR D83 092001 Precise Measurements of Direct $\mathit CP$ Violation, $\mathit CPT$ Symmetry, and other Parameters in the Neutral Kaon System
ALAVI-HARATI 2003
PR D67 012005 Measurements of Direct $\mathit CP$ Violation, $\mathit CPT$ Symmetry, and other Parameters in the Neutral Kaon System
Also
PR D70 079904 (errat.) Erratum to ALAVI-HARATI 2003 : “Measurements of Direct $\mathit CP$-Violation, $\mathit CPT$ Symmetry, and other Parameters in the Neutral Kaon System''
APOSTOLAKIS 1999C
PL B458 545 A Determination of the $\mathit CP$ Violation Parameter $\eta _{+−}$ from the Decay of Strangeness Tagged Neutral Kaons
Also
EPJ C18 41 A Detailed Description of the Analysis of the Decay of Neutral Kaons to ${{\mathit \pi}^{+}}{{\mathit \pi}^{-}}$ in the CPLEAR Experiment
ADLER 1996C
PL B369 367 Evaluation of the Phase of the $\mathit CP$ Violation Parameter $\eta _{+−}$ and ${{\mathit K}_L^0}$ $−$ ${{\mathit K}_S^0}$ Mass Difference from a Correlation Analysis of Different Experiments
ADLER 1995B
PL B363 243 Measurement of the $\mathit CP$ Violation Parameter $\eta _{+−}$ using Tagged ${{\mathit K}^{0}}$ and ${{\overline{\mathit K}}^{0}}$
SCHWINGENHEUER 1995
PRL 74 4376 $\mathit CPT$ Tests in the Neutral Kaon System
GIBBONS 1993
PRL 70 1199 New Measurements of the Neutral Kaon Parameters $\Delta _{m}$, $\tau _{S}$, $\Phi _{00}$ $−$ $\Phi _{+-}$, and $\Phi _{+-}$
Also
PR D55 6625 $\mathit CP$ and $\mathit CPT$ Symmetry Test from the Two Pion Decays of the Neutral Kaon with the FNAL E731 Detector
ADLER 1992B
PL B286 180 First Determination of $\mathit CP$ Violation Parameter from ${{\mathit K}^{0}}$ $−$ ${{\overline{\mathit K}}^{0}}$ Decay Asymmetry
Also
SJNP 55 840 First Results of the CPLEAR Experiment at CERN: Study of $\mathit CP$ Violation and $\mathit CPT$ Test using ${{\mathit K}^{0}}$ $−$ ${{\overline{\mathit K}}^{0}}$ Interferometry
CAROSI 1990
PL B237 303 A Measurement of the Phases of the $\mathit CP$ Violating Amplitudes in ${{\mathit K}^{0}}$ $\rightarrow$ 2 ${{\mathit \pi}}$ Decays and a Test of $\mathit CPT$ Invariance
KARLSSON 1990
PRL 64 2976 A Test of $\mathit CPT$ Symmetry Through a Determination of the Difference in the Phases of $\eta _{00}$ and $\eta _{+−}$ in ${{\mathit K}}$ $\rightarrow$ 2 ${{\mathit \pi}}$ Decays
CARITHERS 1975
PRL 34 1244 Measurement of the Phase of the $\mathit CP$ Nonconservation Parameter $\eta _{+−}$ and ${{\mathit K}_S^0}$ Total Decay Rate
GEWENIGER 1974B
PL 48B 487 A New Determination of the ${{\mathit K}^{0}}$ $\rightarrow$ ${{\mathit \pi}^{+}}{{\mathit \pi}^{-}}$ Decay Parameters
Also
PL 52B 119 The Phase $\phi _{+−}$ of $\mathit CP$ Violation in the ${{\mathit K}^{0}}$ $\rightarrow$ ${{\mathit \pi}^{+}}{{\mathit \pi}^{-}}$ Decay