We present X-ray and radio observations of what may be the closest type Iax supernova (SN) to date, SN 2014dt (d=12.3-19.3 Mpc) and provide tight constraints on the radio and X-ray emission. We infer a specific radio luminosity of < (1.0-2.4)E25 erg/s/Hz at a frequency of 7.5 GHz and a X-ray luminosity < 1.4E38 erg/s (0.3-10 keV) at ~38-48 days post-explosion. We interpret these limits in the context of Inverse Compton (IC) emission and synchrotron emission from a population of electrons accelerated at the forward shock of the explosion in a power-law distribution $N_e(gamma_e)propto gamma_e^{-p}$ with p=3. Our analysis constrains the progenitor system mass-loss rate to be smaller than 5E-6 solar masses per year at distances where r <= 1E16 cm for an assumed wind velocity v=100 km/s, and a fraction of post-shock energy into magnetic fields and relativistic electrons of epsilon_B=0.01 and epsilon_e=0.1, respectively. This result rules out some of the parameter space of symbiotic giant star companions, and it is consistent with the low mass-loss rates expected from He-star companions. Our calculations also show that the improved sensitivity of the next generation Very Large Array (ngVLA) is needed to probe the very low-density media characteristic of He stars that are the leading model for binary stellar companions of white dwarfs giving origin to type Iax SNe.
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