  We present an extension of the scale invariance formalism by Heinz & Sunyaev
(2003) which includes the effects of radiative cooling on the X-ray synchrotron
emission from core dominated jets and outflows. We derive the scaling relations
between the radio luminosity L_r, the synchrotron X-ray luminosity L_x, the
mass M, and the accretion rate of an accreting black hole. We argue that the
inclusion of synchrotron losses into the scaling relations is required by the
data used in Merloni, Heinz, & DiMatteo (2003) to define the "fundamental
plane'' correlation between L_r, L_x, and M in AGNs and Galactic X-ray
binaries. We then fit the new scaling relations to the "fundamental plane"
correlation. This allows us to derive statistical constraints on the
contribution from jet synchrotron emission to the X-rays from black holes, or,
alternatively, on the particle acceleration scheme at work in these jets. We
find that, in order for jet synchrotron X-rays to be consistent with the
observed "fundamental plane" correlation, the global particle spectrum must be
both steep: f(gamma) ~ gamma^(-(3.4 (+0.6-0.5)), and unbroken down to energies
well below the radio synchrotron regime. Such a steep particle distribution
would imply steeper X-ray spectra than observed in most sources contributing to
the correlation with avalailable X-ray spectral index measurements. We suggest
that, unless some assumptions of the scale invariance hypothesis are broken,
another emission mechanism (likely either disk radiation and/or inverse Compton
scattering in the jet) contributes significantly to the X-rays of a sizeable
fraction of the sources.
