On the use of field RR Lyrae as Galactic probes: I. The Oosterhoff
  dichotomy based on fundamental variables
  We collected a large data set of field RR Lyrae stars (RRLs) by using
catalogues already available in the literature and Gaia DR2. We estimated the
iron abundances for a sub-sample of 2,382 fundamental RRLs ($\Delta$S method:
CaIIK, H$\beta$, H$\gamma$ and H$\delta$ lines) for which are publicly
available medium-resolution SDSS-SEGUE spectra. We also included similar
estimates available in the literature ending up with the largest and most
homogeneous spectroscopic data set ever collected for RRLs (2,903). The
metallicity scale was validated by using iron abundances based on high
resolution spectra for a fundamental field RRL (V~Ind), for which we collected
X-shooter spectra covering the entire pulsation cycle. The peak
([Fe/H]=-1.59$\pm$0.01) and the standard deviation ($\sigma$=0.43 dex) of the
metallicity distribution agree quite well with similar estimates available in
the literature. The current measurements disclose a well defined metal-rich
tail approaching Solar iron abundance. The spectroscopic sample plotted in the
Bailey diagram (period vs luminosity amplitude) shows a steady variation when
moving from the metal-poor ([Fe/H]=-3.0/-2.5) to the metal-rich
([Fe/H]=-0.5/0.0) regime. The smooth transition in the peak of the period
distribution as a function of the metallicity strongly indicates that the
long-standing problem of the Oosterhoff dichotomy among Galactic globulars is
the consequence of the lack of metal-intermediate clusters hosting RRLs. We
also found that the luminosity amplitude, in contrast with period, does not
show a solid correlation with metallicity. This suggests that
period-amplitude-metallicity relations should be cautiously treated.
