  We report new Chandra hard X-ray ($>2\rm~keV$) 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 $\sim1.1\rm~kpc$. 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
$\sim1\rm~keV$, 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.
