The properties of molecular gas, the fuel that forms stars, inside the cavity of the circumnuclear disk (CND) are not well constrained. We present results of a velocity-resolved submillimeter scan (~480 to 1250 GHz}) and [CII]158um line observations carried out with Herschel/HIFI toward Sgr A*; these results are complemented by a ~2x2 CO (J=3-2) map taken with the IRAM 30 m telescope at ~7 resolution. We report the presence of high positive-velocity emission (up to about +300 km/s) detected in the wings of CO J=5-4 to 10-9 lines. This wing component is also seen in H2O (1_{1,0}-1_{0,1}) a tracer of hot molecular gas; in [CII]158um, an unambiguous tracer of UV radiation; but not in [CI]492,806 GHz. This first measurement of the high-velocity CO rotational ladder toward Sgr A* adds more evidence that hot molecular gas exists inside the cavity of the CND, relatively close to the supermassive black hole (< 1 pc). Observed by ALMA, this velocity range appears as a collection of CO (J=3-2) cloudlets lying in a very harsh environment that is pervaded by intense UV radiation fields, shocks, and affected by strong gravitational shears. We constrain the physical conditions of the high positive-velocity CO gas component by comparing with non-LTE excitation and radiative transfer models. We infer T_k~400 K to 2000 K for n_H~(0.2-1.0)x10^5 cm^-3. These results point toward the important role of stellar UV radiation, but we show that radiative heating alone cannot explain the excitation of this ~10-60 M_Sun component of hot molecular gas inside the central cavity. Instead, strongly irradiated shocks are promising candidates.
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