CP violation in $\eta \to \pi \pi$ by Higgs - $\eta$ mixing through two-loop quantum effects

Strictly within the standard electro-weak interaction, CP violation in the flavour conserving process $\eta \ra \pi + \pi$ could originate from the mixing of the $\eta$ meson with the virtual scalar Higgs $H^{0}$ via $W^{+}+W^{-}$ and $Z^{0}+Z^{0}$ exchange. The parity-violation carried by these weak gauge bosons makes the mixing possible at two-loop level. Nowhere the Kobayashi-Maskawa (KM) phase mechanism is needed. For the Higgs mass between 100-600 GeV, the $\eta \ra \pi +\pi$ branching ratio is found to be $3\cdot 10^{-26} - 2\cdot 10^{-29}$, hence unconventional CP violation mechanisms are the only ones that could give rise to its observation at the existing or near future $\eta$ factories, unless the Higgs mass is improbably as light as 10 MeV.