  Short-period super-Earths and Neptunes are now known to be very frequent
around solar-type stars. Improving our understanding of these mysterious
planets requires the detection of a significant sample of objects suitable for
detailed characterization. Searching for the transits of the low-mass planets
detected by Doppler surveys is a straightforward way to achieve this goal.
Indeed, Doppler surveys target the most nearby main-sequence stars, they
regularly detect close-in low-mass planets with significant transit
probability, and their radial velocity data constrain strongly the ephemeris of
possible transits. In this context, we initiated in 2010 an ambitious Spitzer
multi-Cycle transit search project that targeted 25 low-mass planets detected
by radial velocity, focusing mainly on the shortest-period planets detected by
the HARPS spectrograph. We report here null results for 19 targets of the
project. For 16 planets out of 19, a transiting configuration is strongly
disfavored or firmly rejected by our data for most planetary compositions. We
derive a posterior probability of 83% that none of the probed 19 planets
transits (for a prior probability of 22%), which still leaves a significant
probability of 17% that at least one of them does transit. Globally, our
Spitzer project revealed or confirmed transits for three of its 25 targeted
planets, and discarded or disfavored the transiting nature of 20 of them. Our
light curves demonstrate for Warm Spitzer excellent photometric precisions: for
14 targets out of 19, we were able to reach standard deviations that were
better than 50ppm per 30 min intervals. Combined with its Earth-trailing orbit,
which makes it capable of pointing any star in the sky and to monitor it
continuously for days, this work confirms Spitzer as an optimal instrument to
detect sub-mmag-deep transits on the bright nearby stars targeted by Doppler
surveys.
