As part of the Dust, Ice, and Gas In Time (DIGIT) Herschel Open Time Key Program, we present Herschel photometry (at 70, 160, 250, 350 and 500 micron) of 31 Weak-Line T Tauri star (WTTS) candidates in order to investigate the evolutionary status of their circumstellar disks. Thirteen stars in our sample had circumstellar disks previously known from infrared observations at shorter wavelengths, while eighteen of them had no previous evidence for a disk. We detect a total of 15 disks as all previously known disks are detected at one or more Herschel wavelengths and two additional disks are identified for the first time. The spectral energy distributions (SEDs) of our targets seem to trace the dissipation of the primordial disk and the transition to the debris disk regime. Seven of the 15 disks appear to be optically thick primordial disks, including two objects with SEDs indistinguishable from those of typical Classical T Tauri stars, four objects that have significant deficit of excess emission at all IR wavelengths, and one pre-transitional object with a known gap in the disk. Despite their previous WTTS classification, we find that the seven targets in our sample with optically thick disks show evidence for accretion. The remaining eight disks have weaker IR excesses similar to those of optically thin debris disks. Six of them are warm and show significant 24 micron Spitzer excesses, while the last two are newly identified cold debris-like disks with photospheric 24 micron fluxes, but significant excess emission at longer wavelengths. The Herschel photometry also places strong constraints on the non-detections, where systems with F70/F70,star > 5 - 15 and L,disk/L,star > 1xE-3 to 1xE-4 can be ruled out. We present preliminary models for both the optically thick and optically thin disks and discuss our results in the context of the evolution and dissipation of circumstellar disks.
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