#### ${{\mathit \pi}^{+}}{{\mathit \pi}^{-}}{{\mathit e}^{+}}{{\mathit e}^{-}}$ DECAY-PLANE ASYMMETRY PARAMETER ${{\mathit A}_{{{\phi}}}}$

In the ${{\mathit \eta}}$ rest frame, the total momentum of the ${{\mathit e}^{+}}{{\mathit e}^{-}}$ pair is equal and opposite to that of the ${{\mathit \pi}^{+}}{{\mathit \pi}^{-}}$ pair. Let $\hat{{\mathit z}}$ be the unit vector along the momentum of the ${{\mathit e}^{+}}{{\mathit e}^{-}}$ pair; let $\hat{{\mathit n}}_{ee}$ and $\hat{{\mathit n}}_{ {{\mathit \pi}} {{\mathit \pi}} }$ be the unit vectors normal to the ${{\mathit e}^{+}}{{\mathit e}^{-}}$ and ${{\mathit \pi}^{+}}{{\mathit \pi}^{-}}$ planes; and let $\phi$ be the angle between the two normals. Then
 sin$\phi$ cos $\phi$ = [($\hat{{\mathit n}}_{ee}{\times }\hat{{\mathit n}}_{ {{\mathit \pi}} {{\mathit \pi}} }$) $\cdot{}$ $\hat{{\mathit z}}$] ($\hat{{\mathit n}}_{ee}\cdot{}\hat{{\mathit n}}_{ {{\mathit \pi}} {{\mathit \pi}} }$) ,
 and
 ${{\mathit A}_{{{\phi}}}}{}\equiv$ ${{{\mathit N}_{{{{sin{{\mathit \phi}}~cos{{\mathit \phi}}>0}}}}}~−~{{\mathit N}_{{{{sin{{\mathit \phi}}~cos{{\mathit \phi}}<0}}}}}\over {{\mathit N}_{{{{sin{{\mathit \phi}}~cos{{\mathit \phi}}>0}}}}}~+~{{\mathit N}_{{{{sin{{\mathit \phi}}~cos{{\mathit \phi}}<0}}}}} }$ .

VALUE ($10^{-2}$) EVTS DOCUMENT ID TECN  COMMENT
$-0.6$ $\pm2.5$ $\pm1.8$ $1555$ $\pm52$
 2009 B
KLOE ${{\mathit e}^{+}}$ ${{\mathit e}^{-}}$ $\rightarrow$ ${{\mathit \phi}}$ $\rightarrow$ ${{\mathit \eta}}{{\mathit \gamma}}$
Conservation Laws:
$\mathit CP$ INVARIANCE
References:
 AMBROSINO 2009B
PL B675 283 Measurement of the Branching Ratio and Search for a $\mathit CP$ Violating Asymmetry in the ${{\mathit \eta}}$ $\rightarrow$ ${{\mathit \pi}^{+}}{{\mathit \pi}^{-}}{{\mathit e}^{+}}{{\mathit e}^{-}}$( ${{\mathit \gamma}}$) Decay at KLOE