  We present the first signature-based search for delayed photons using an
exclusive photon plus missing transverse energy final state. Events are
reconstructed in a data sample from the CDF II detector corresponding to $6.3
\text{fb}^{-1}$ of integrated luminosity from $\sqrt{s}=1.96$ TeV
proton-antiproton collisions. Candidate events are selected if they contain a
photon with an arrival time in the detector larger than expected from a
promptly-produced photon. The mean number of events from standard model sources
predicted by the data-driven background model based on the photon timing
distribution is $286 \pm 24$. A total of 322 events are observed. A $p$-value
of 12% is obtained, showing consistency of the data with standard model
predictions.
