We present a spectroscopic sample of z≈ 0.5 luminous red galaxies (LRGs) that are located within physical projected distances of a quasi-stellar object (QSO) sightline. Of the 37 LRGs in our sample, eight have associated Mg ii absorbers with rest-frame equivalent width Wr(2796)>0.3 Å and velocity separation km s−1 and 29 do not have associated Mg ii absorbers to a 2σ limit of Wr(2796)=0.3 Å. We perform a stellar population synthesis analysis using stacked spectra of the Mg ii absorbing and non-absorbing LRG subsamples. We find that LRGs with or without associated Mg ii absorbers share similar star formation histories and are best described by old stellar population models (≳1 Gyr). Younger stellar populations (≲1 Gyr) fail to reproduce their spectra. These findings are consistent with the lack of [O ii] emission features in the LRG spectra. The primarily old stellar populations in the LRGs indicate that starburst-driven outflows are unlikely to explain the observed Mg ii absorbers at large distances from the LRGs. In addition, the spectroscopic LRG sample allows us to derive a sensitive constraint for the cool gas covering fraction of 〈κ〉= 14 ± 6 per cent in the LRG haloes for absorbers of Wr(2796)>0.3 Å. Finally, we speculate on the origin of the observed Mg ii absorbers around the LRGs.