  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.
