  Understanding how galaxies cease to form stars represents an outstanding
challenge for galaxy evolution theories. This process of "star formation
quenching" has been related to various causes, including Active Galactic Nuclei
(AGN) activity, the influence of large-scale dynamics, and the environment in
which galaxies live. In this paper, we present the first results from a
follow-up of CALIFA survey galaxies with observations of molecular gas obtained
with the APEX telescope. Together with EDGE survey CARMA observations, we
collect $^{12}$CO observations that cover approximately one effective radius in
472 CALIFA galaxies. We observe that the deficit of galaxy star formation with
respect to the star formation main sequence (SFMS) increases with the absence
of molecular gas and with a reduced efficiency of conversion of molecular gas
into stars, in line with results of other integrated studies. However, by
dividing the sample into galaxies dominated by star formation and galaxies
quenched in their centres (as indicated by the average value of the H$\alpha$
equivalent width), we find that this deficit increases sharply once a certain
level of gas consumption is reached, indicating that different mechanisms drive
separation from the SFMS in star-forming and quenched galaxies. Our results
indicate that differences in the amount of molecular gas at a fixed stellar
mass are the primary driver for the dispersion in the SFMS, and the most likely
explanation for the start of star-formation quenching. However, once a galaxy
is quenched, changes in star formation efficiency drive how much a retired
galaxy separates in star formation rate from star-forming ones of similar
masses. In other words, once a paucity of molecular gas has significantly
reduced star formation, changes in the star formation efficiency are what
drives a galaxy deeper into the red cloud, retiring it.
