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.
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