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.

Im($\epsilon {{}^\prime}/\epsilon $) = $−(\phi _{00}$ $−$ $\phi _{+−}$)/3

INSPIRE   PDGID:
S013EPI
For small $\vert {{\mathit \epsilon}^{\,'}}/{{\mathit \epsilon}}\vert $, Im(${{\mathit \epsilon}^{\,'}}/{{\mathit \epsilon}}$) is related to the phases of $\eta _{00}$ and $\eta _{+−}$ by the above expression.
VALUE ($^\circ{}$) DOCUMENT ID TECN  COMMENT
$\bf{ -0.11 \pm0.11}$ OUR FIT  Not assuming $\mathit CPT$
$\bf{ -0.002 \pm0.005}$ OUR FIT  Error includes scale factor of 1.7.  Assuming $\mathit CPT$
$-0.0985$ $\pm0.1157$ 1
ABOUZAID
2011
KTEV Not assuming $\mathit CPT$
1  ABOUZAID 2011 uses the full KTeV dataset collected in 1996, 1997, and 1999. The fit has $\Delta \mathit m$, ${{\mathit \tau}_{{s}}}$, ${{\mathit \phi}_{{\epsilon}}}$, Re(${{\mathit \epsilon}^{\,'}}/{{\mathit \epsilon}}$), and Im(${{\mathit \epsilon}^{\,'}}/{{\mathit \epsilon}}$) as free parameters. The reported value of Im(${{\mathit \epsilon}^{\,'}}/{{\mathit \epsilon}}$) = ($-17.20$ $\pm20.20$) $ \times 10^{-4}$ rad. The correlation coefficients are ${{\mathit \rho}}({{\mathit \phi}_{{\epsilon}}}$, $\Delta \mathit m$) = 0.828, ${{\mathit \rho}}({{\mathit \phi}_{{\epsilon}}}$, ${{\mathit \tau}_{{s}}}$) = $-0.765$, ${{\mathit \rho}}(\Delta \mathit m$, ${{\mathit \tau}_{{s}}}$) = $-0.858$, ${{\mathit \rho}}$(Im(${{\mathit \epsilon}^{\,'}}/{{\mathit \epsilon}}$), ${{\mathit \phi}_{{\epsilon}}}$) = $-0.041$, ${{\mathit \rho}}$(Im(${{\mathit \epsilon}^{\,'}}/{{\mathit \epsilon}}$), $\Delta \mathit m$) = 0.026, ${{\mathit \rho}}$(Im(${{\mathit \epsilon}^{\,'}}/{{\mathit \epsilon}}$), ${{\mathit \tau}_{{s}}}$) = $-0.010$.
References:
ABOUZAID 2011
PR D83 092001 Precise Measurements of Direct $\mathit CP$ Violation, $\mathit CPT$ Symmetry, and other Parameters in the Neutral Kaon System