An E$_6$ interpretation of an $e^+ e^- \gamma \gamma E\!\!\!/_T$ event

The lowest-dimensional representation of the group E$_6$ contains both the standard quarks and leptons and a set of exotic quarks and leptons whose decays can involve a series of chains ending in radiative decay of one light neutrino species to another. An example is given based on the decomposition E$_6 \to$ SU(2)$_I \times$ SU(6), where SU(2)$_I$ is an ``inert'' subgroup whose gauge bosons $W_I^{(\pm)}$ and $Z_I$ are all electromagnetically neutral, while SU(6) contains the conventional SU(5) grand-unified group. The possibility is explored that such a chain is responsible for an event observed by the Collider Detector at Fermilab (CDF) involving the production in proton-antiproton collisions at $E_{\rm c.m.} = 1.8$ TeV of an electron-positron pair, two photons, and missing energy ($e^+ e^- \gamma \gamma E \!\!\!/_T$).