  Low mass X-ray binaries (LMXBs) show evidence of a global correlation of
debated origin between X-ray and optical luminosity. We study for the first
time this correlation in two transient LMXBs, the black hole V404 Cyg and the
neutron star Cen X-4, over 6 orders of magnitude in X-ray luminosity, from
outburst to quiescence. After subtracting the contribution from the companion
star, the Cen X-4 data can be described by a single power law correlation of
the form $L_{opt}\propto\,L_{X}^{0.44}$, consistent with disk reprocessing. We
find a similar correlation slope for V404 Cyg in quiescence (0.46) and a
steeper one (0.56) in the outburst hard state of 1989. However, V404 Cyg is
about $160-280$ times optically brighter, at a given $3-9$ keV X-ray
luminosity, compared to Cen X-4. This ratio is a factor of 10 smaller in
quiescence, where the normalization of the V404 Cyg correlation also changes.
We show that once the bolometric X-ray emission is considered and the known
main differences between V404 Cyg and Cen X-4 are taken into account (a larger
compact object mass, accretion disk size, and the presence of a strong jet
contribution in the hard state for the black hole system) the two systems lie
on the same correlation. In V404 Cyg, the jet dominates spectrally at
optical-infrared frequencies during the hard state, but makes a negligible
contribution in quiescence, which may account for the change in its correlation
slope and normalization. These results provide a benchmark to compare with data
from the 2015 outburst of V404 Cyg and, potentially, other transient LMXBs as
well.
