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We present a quantum theory of cooling of a mechanical resonator usingback-action with constant electron current. The resonator device is based on adoubly clamped nanotube, which mechanically vibrates and acts as a doublequantum dot for electron transport. Mechanical vibrations and electrons arecoupled electrostatically using an external gate. The fundamental eigenmode iscooled by absorbing phonons when electrons tunnel through the double quantumdot. We identify the regimes in which ground state cooling can be achieved forrealistic experimental parameters.

Cooling carbon nanotubes to the phononic ground state with constant electron current