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Published: 2014

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The $\eta$ meson provides a laboratory to study isospin violation and search for new flavor-conserving sources of C and CP violation with a sensitivity approaching $10{̂-6}$ of the isospin-conserving strong amplitude. Some of the most interesting rare $\eta$ decays are the neutral modes, yet the effective loss of photons from the relatively common decay $\eta \rightarrow 3\pi0̂ \rightarrow 6\gamma$ (33$\%$) has largely limited the sensitivity for decays producing 3-5$\gamma$'s. Particularly important relevant branches include the highly suppressed $\eta \rightarrow \pi0̂ 2\gamma \rightarrow 4\gamma$, which provides a rare window on testing models of $O(p6̂)$ contributions in ChPTh, and $\eta \rightarrow 3\gamma$ and $\eta \rightarrow 2\pi0̂ \gamma \rightarrow 5\gamma$ which provide direct constraints on C violation in flavor-conserving processes. The substitution of lead tungstate in the forward calorimeter of the GluEx setup in Jefferson Lab's new Hall D would allow dramatically improved measurements. The main niche of this facility, which we call the JLab Eta Factory (JEF), would be $\eta$ decay neutral modes. However, this could likely be expanded to rare $\eta'(958)$ decays for low energy QCD studies as well as $\eta$ decays involving muons for new physics searches.