We present the results of our observations of the early stages of the 2012--2013 outburst of the transient black hole X-ray binary (BHXRB), Swift J1745$-$26, with the VLA, SMA, and JCMT (SCUBA--2). Our data mark the first multiple-band mm & sub-mm observations of a BHXRB. During our observations the system was in the hard accretion state producing a steady, compact jet. The unique combination of radio and mm/sub-mm data allows us to directly measure the spectral indices in and between the radio and mm/sub-mm regimes, including the first mm/sub-mm spectral index measured for a BHXRB. Spectral fitting revealed that both the mm (230 GHz) and sub-mm (350 GHz) measurements are consistent with extrapolations of an inverted power-law from contemporaneous radio data (1--30 GHz). This indicates that, as standard jet models predict, a power-law extending up to mm/sub-mm frequencies can adequately describe the spectrum, and suggests that the mechanism driving spectral inversion could be responsible for the high mm/sub-mm fluxes (compared to radio fluxes) observed in outbursting BHXRBs. While this power-law is also consistent with contemporaneous optical data, the optical data could arise from either jet emission with a jet spectral break frequency of $ u_{{rm break}}gtrsim1times10^{14},{rm Hz}$ or the combination of jet emission with a lower jet spectral break frequency of $ u_{{rm break}}gtrsim2times10^{11},{rm Hz}$ and accretion disc emission. Our analysis solidifies the importance of the mm/sub-mm regime in bridging the crucial gap between radio and IR frequencies in the jet spectrum, and justifies the need to explore this regime further.
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