On the Use of Field RR Lyrae as Galactic Probes. III. The
  $\alpha$-element abundances
  We provide the largest and most homogeneous sample of $\alpha$-element (Mg,
Ca, Ti) and iron abundances for field RR Lyrae (RRLs, 162 variables) by using
high-resolution spectra. The current measurements were complemented with
similar abundances available in the literature for 46 field RRLs brought to our
metallicity scale. We ended up with a sample of old (t$\ge$ 10 Gyr), low-mass
stellar tracers (208 RRLs: 169 fundamental, 38 first overtone, 1 mixed mode)
covering three dex in iron abundance (-3.00$\le$[Fe/H]$\le$0.24). We found that
field RRLs are $\sim$0.3 dex more $\alpha$-poor than typical Halo tracers in
the metal-rich regime, ([Fe/H]$\ge$-1.2) while in the metal-poor regime
([Fe/H]$\le$-2.2) they seem to be on average $\sim$0.1 dex more
$\alpha$-enhanced. This is the first time that the depletion in
$\alpha$-elements for solar iron abundances is detected on the basis of a
large, homogeneous and coeval sample of old stellar tracers. Interestingly, we
also detected a close similarity in the [$\alpha$/Fe] trend between
$\alpha$-poor, metal-rich RRLs and red giants (RGs) in the Sagittarius dwarf
galaxy as well as between $\alpha$-enhanced, metal-poor RRLs and RGs in ultra
faint dwarf galaxies. These results are supported by similar elemental
abundances for 46 field Horizontal Branch (HB) stars. These stars share with
RRLs the same evolutionary phase and the same progenitors. This evidence
further supports the key role that old stellar tracers play in constraining the
early chemical enrichment of the Halo and, in particular, in investigating the
impact that dwarf galaxies have had in the mass assembly of the Galaxy.
