In order to understand which process ( e.g . galactic winds , cold accretion ) is responsible for the cool ( T \sim 10 ^ { 4 } K ) halo gas around galaxies , we embarked on a program to study the star-formation properties of galaxies selected by their Mg ii absorption signature in quasar spectra . Specifically , we searched for the H \alpha line emission from galaxies near very strong z \simeq 2 Mg ii absorbers ( with rest-frame equivalent width \hbox { $W _ { r } ^ { \lambda 2796 } $ } \ga 2 Å ) 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 H i -selected sight-lines ) , despite reaching a star-formation rate ( SFR ) sensitivity limit of 2.9M _ { \odot } yr ^ { -1 } ( 5 \sigma ) for a Chabrier initial mass function . This low success rate ( 4/20 ) is in contrast with our z \simeq 1 survey where we detected 66 % ( 14/21 ) of the Mg ii hosts ( down to 0.6 M _ { \odot } yr ^ { -1 } , 5 \sigma ) . Taking into account the difference in sensitivity between the two surveys , we should have been able to detect \geq 11.4 ( \geq 7.6 ) of the 20 z \simeq 2 hosts —assuming that SFR evolves as \propto ( 1 + z ) ^ { \gamma } with \gamma = 2.5 ( or \gamma = 0 ) respectively— whereas we found only 4 galaxies . Interestingly , all the z = 2 detected hosts have observed SFRs \ga 9 M _ { \odot } yr ^ { -1 } , well above our sensitivity limit , while at z = 1 they all have SFR < 9 M _ { \odot } yr ^ { -1 } , an evolution that is in good agreement with the evolution of the SFR main sequence , i.e . with \gamma = 2.5 . Moreover , we show that the z = 2 undetected hosts are not hidden under the quasar continuum after stacking our data . They also can not be outside our surveyed area as this latter option runs against our sample selection criteria ( \hbox { $W _ { r } ^ { \lambda 2796 } $ } > 2 Å ) and the known W _ { r } ^ { \lambda 2796 } –impact parameter relation for low-ionization ions . Hence , strong Mg ii absorbers could trace star-formation driven winds in low-mass halos ( M _ { h } \leq 10 ^ { 10.6 } M _ { \odot } ) provided that the winds do not extend beyond 20 kpc in order not to violate the evolution of the absorber number density dN / dz ( \hbox { { Mg } { \sc ii } } ) . Alternatively , our results imply that z = 2 galaxies traced by strong Mg ii absorbers do not form stars at a rate expected ( 3–10M _ { \odot } yr ^ { -1 } ) for their ( halo or stellar ) masses , supporting the existence of a transition in accretion efficiency at M _ { h } \simeq 10 ^ { 11 } M _ { \odot } . This scenario can explain both the detections and the non-detections .