We present the first far infrared (FIR) dust emission polarization map covering the full extent Milky Ways Central molecular zone (CMZ). The data, obtained with the PILOT balloon-borne experiment, covers the Galactic Center region $-2,^circ<l<2,^circ$, $-4,^circ<b<3,^circ$ at a wavelength of 240 $mu$m and an angular resolution $2.2,$. From our measured dust polarization angles, we infer a magnetic field orientation projected onto the plane of the sky that is remarkably ordered over the full extent of the CMZ, with an average tilt angle of $simeq 22,^circ$ clockwise with respect to the Galactic plane. Our results confirm previous claims that the field traced by dust polarized emission is oriented nearly orthogonal to the field traced by GHz radio synchrotron emission in the Galactic Center region. The observed field structure is globally compatible with the latest Planck polarization data at 353 GHz and 217 GHz. Upon subtraction of the extended emission in our data, the mean field orientation that we obtain shows good agreement with the mean field orientation measured at higher angular resolution by the JCMT within the 20 km/s and 50 km/s molecular clouds. We find no evidence that the magnetic field orientation is related to the 100 pc twisted ring structure within the CMZ. We propose that the low polarization fraction in the Galactic Center region and the highly ordered projected field orientation can be reconciled if the field is strong, with a 3D geometry that is is mostly oriented $simeq 15,^circ$ with respect to the line-of-sight towards the Galactic center. Assuming equipartition between the magnetic pressure and ram pressure, we obtain magnetic field strengths estimates as high as a few mG for several CMZ molecular clouds.
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