  Since the discovery of cosmic rays (CRs) over a century ago, their origin
remains an open question. Galactic CRs with energy up to the knee ($10^{15}$
eV) are considered to originate from supernova remnants, but this scenario has
recently been questioned due to lack of TeV $\gamma$-ray counterparts in many
cases. Extragalactic CRs on the other hand, are thought to be associated with
accelerated particles in the relativistic jets launched by supermassive
accreting black holes at the center of galaxies. Scaled down versions of such
jets have been detected in X-ray binaries hosting a stellar black hole (BHXBs).
In this work, we investigate the possibility that the smaller-scale jets in
transient outbursts of low-mass BHXBs could be sources of Galactic CRs. To
better test this scenario, we model the entire electromagnetic spectrum of such
sources focusing on the potential TeV regime, using the `canonical' low-mass
BHXB GX 339-4 as a benchmark. Taking into account both the leptonic radiative
processes and the $\gamma$-rays produced via neutral pion decay from inelastic
hadronic interactions, we predict the GeV and TeV $\gamma$-ray spectrum of GX
339-4 using lower-frequency emission as constraints. Based on this test-case of
GX 339-4 we investigate whether other, nearby low-mass BHXBs could be detected
by the next-generation very-high-energy $\gamma$-ray facility the Cherenkov
Telescope Array, which would establish them as additional and numerous
potential sources of CRs in the Galaxy.
