  [Abridged] In order to understand which process (e.g. galactic winds, cold
accretion) is responsible for the cool (T~10^4 K) halo gas around galaxies, we
embarked on a program to study the star-formation properties of galaxies
selected by their MgII absorption signature in quasar spectra. Specifically, we
searched for the H-alpha line emission from galaxies near very strong z=2 MgII
absorbers (with rest-frame equivalent width EW>2 \AA) because these could be
the sign-posts of outflows or inflows. Surprisingly, we detect H-alpha from
only 4 hosts out of 20 sight-lines (and 2 out of the 19 HI-selected
sight-lines), despite reaching a star-formation rate (SFR) sensitivity limit of
2.9 M/yr (5-sigma) for a Chabrier initial mass function. This low success rate
is in contrast with our z=1 survey where we detected 66%\ (14/21) of the MgII
hosts. Taking into account the difference in sensitivity between the two
surveys, we should have been able to detect >11.4 of the 20 z=2 hosts whereas
we found only 4 galaxies. Interestingly, all the z=2 detected hosts have
observed SFR greater than 9 M/yr, well above our sensitivity limit, while at
z=1 they all have SFR less than 9 M/yr, an evolution that is in good agreement
with the evolution of the SFR main sequence. Moreover, we show that the z=2
undetected hosts are not hidden under the quasar continuum after stacking our
data and that they also cannot be outside our surveyed area. Hence, strong MgII
absorbers could trace star-formation driven winds in low-mass halos (Mhalo <
10^{10.6} Msun). Alternatively, our results imply that z=2 galaxies traced by
strong MgII absorbers do not form stars at a rate expected (3--10 M/yr) for
their (halo or stellar) masses, supporting the existence of a transition in
accretion efficiency at Mhalo ~ 10^{11} Msun. This scenario can explain both
the detections and the non-detections.
