Diphotons at the $Z$-pole in Models of the 750 GeV Resonance Decaying to Axion-Like Particles

Models in which the 750 GeV resonance ($S$) decays to two light axion-like particles (ALPs $a$), which in turn decay to collimated photons mimicking the observed signal, are motivated by Hidden Valley scenarios and could also provide a mechanism by which a $S \to \gamma \gamma$ signal persists while $S \to Z \gamma,\; ZZ$ and $WW$ remain subdued in the near future. We point out that these Hidden Valley like models invoking $S \to aa \to 4 \gamma$ must also contend with $Z \to a (\to \gamma \gamma) \gamma$ constraints coming from CDF and ATLAS. Within an effective field theory framework, we work out the constraints on the couplings of $S$ to $a$ and gauge bosons coming from photonic $Z$ decays and ensuring that the ALPs decay inside the electromagnetic calorimeter, in two regimes - where $a$ decays primarily to photons, and where $a$ also has hadronic branchings. The analysis is done for both when $S$ has a large as well as a narrow width, and for different relative contributions to the signal coming from $S \to \gamma \gamma$ and $a \to \gamma \gamma$. Results for the particular case where $S$ and $a$ belong to the same complex field are also presented. A $\gamma\gamma$ resonance at the $Z$-pole coming from $Z \to a \gamma$ is expected in this class of models. Taking benchmark ALP masses below around 0.4 GeV and, assuming reasonable values for the fake jet rate and the identification efficiency of the photon-jet, we find the prospects for the discovery of diphotons at the $Z$-pole.