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

${{\mathit A}_{{\phi}}}{}\equiv$ ${{{\mathit N}_{{{#def{sfcf}#def{>}0}}}}~−~{{\mathit N}_{{{#def{sfcf}#def{<}0}}}}\over {{\mathit N}_{{{#def{sfcf}#def{>}0}}}}~+~{{\mathit N}_{{{#def{sfcf}#def{<}0}}}} }$ .

VALUE ($ 10^{-2} $) EVTS DOCUMENT ID TECN  COMMENT $-0.6$ $\pm2.5$ $\pm1.8$ $1555$ $\pm52$ AMBROSINO 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