  A search for forbidden and exotic Z boson decays in the diphoton mass
spectrum is presented for the first time in hadron collisions, based on data
corresponding to 10.0 fb^-1 of integrated luminosity from proton-antiproton
collisions at sqrt(s) = 1.96 TeV collected by the CDF experiment. No evidence
of signal is observed, and 95% credibility level Bayesian upper limits are set
on the branching ratios of decays of the Z boson to a photon and neutral pion
(which is detected as a photon), a pair of photons, and a pair of neutral
pions. The observed branching ratio limits are 2.01 x 10^-5 for Z to
\pi^0\gamma, 1.46 x 10^-5 for Z to \gamma\gamma, and 1.52 x 10^-5 for Z to
\pi^0\pi^0. The Z to \pi^0\gamma\ and Z to \gamma\gamma\ limits improve by
factors of 2.6 and 3.6, respectively, the most stringent results from other
experiments. The Z to \pi^0\pi^0 branching ratio limit is the first
experimental result on this decay.
