We report new Chandra hard X-ray ($>2rm~keV$) and JVLA C-band observations of the nuclear superbubble of NGC 3079, an analog of the Fermi bubble in our Milky Way. We detect extended hard X-ray emission on the SW side of the galactic nucleus with coherent multi-wavelength features in radio, H$alpha$, and soft X-ray. The hard X-ray feature has a cone shape with possibly a weak cap, forming a bubble-like structure with a diameter of $sim1.1rm~kpc$. A similar extended feature, however, is not detected on the NE side, which is brighter in all other wavelengths such as radio, H$alpha$, and soft X-ray. Scattered photons from the nuclear region or other nearby point-like X-ray bright sources, inverse Compton emission from cosmic ray electrons via interaction with the cosmic microwave background, or any individually faint stellar X-ray source populations, cannot explain the extended hard X-ray emission on the SW side and the strongly NE/SW asymmetry. A synchrotron emission model, plus a thermal component accounting for the excess at $sim1rm~keV$, can well characterize the broadband radio/hard X-ray spectra. The broadband synchrotron spectra do not show any significant cutoff, and even possibly slightly flatten at higher energy. This rules out a loss-limited scenario in the acceleration of the cosmic ray electrons in or around this superbubble. As the first detection of kpc-scale extended hard X-ray emission associated with a galactic nuclear superbubble, the spatial and spectral properties of the multi-wavelength emissions indicate that the cosmic ray leptons responsible for the broad-band synchrotron emission from the SW bubble must be accelerated in situ, instead of transported from the nuclear region of the galaxy.
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