We present arcsecond-resolution Submillimeter Array (SMA) polarimetric observations of the 880 um continuum emission from the protoplanetary disks around two nearby stars, HD 163296 and TW Hydrae. Although previous observations and theoretical work have suggested that a 2-3% polarization fraction should be common for the millimeter continuum emission from such disks, we detect no polarized continuum emission above a 3-sigma upper limit of 7 mJy in each arcsecond-scale beam, or <1% in integrated continuum emission. We compare the SMA upper limits with the predictions from the exploratory Cho & Lazarian (2007) model of polarized emission from T Tauri disks threaded by toroidal magnetic fields, and rule out their fiducial model at the ~10-sigma level. We explore some potential causes for this discrepancy, focusing on model parameters that describe the shape, magnetic field alignment, and size distribution of grains in the disk. We also investigate related effects like the magnetic field strength and geometry, scattering off of large grains, and the efficiency of grain alignment, including recent advances in grain alignment theory, which are not considered in the fiducial model. We discuss the impact each parameter would have on the data and determine that the suppression of polarized emission plausibly arises from rounding of large grains, reduced efficiency of grain alignment with the magnetic field, and/or some degree of magnetic field tangling (perhaps due to turbulence). A poloidal magnetic field geometry could also reduce the polarization signal, particularly for a face-on viewing geometry like the TW Hya disk. The data provided here offer the most stringent limits to date on the polarized millimeter-wavelength emission from disks around young stars.
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