Metallicity calibrations for diffuse ionised gas and low ionisation
  emission regions
  Using integral field spectroscopic data of 24 nearby spiral galaxies obtained
with the Multi-Unit Spectroscopic Explorer (MUSE), we derive empirical
calibrations to determine the metallicity of the diffuse ionized gas (DIG)
and/or of the low-ionisation emission region (LI(N)ER) in passive regions of
galaxies. To do so, we identify a large number of HII--DIG/LIER pairs that are
close enough to be chemically homogeneous and we measure the metallicity
difference of each DIG/LIER region relative to its HII region companion when
applying the same strong line calibrations. The O3N2 diagnostic ($=$log [([O
III]/H$\beta$)/([N II]/H$\alpha$)]) shows a minimal offset (0.01--0.04 dex)
between DIG/LIER and HII regions and little dispersion of the metallicity
differences (0.05 dex), suggesting that the O3N2 metallicity calibration for
HII regions can be applied to DIG/LIER regions and that, when used on poorly
resolved galaxies, this diagnostic provides reliable results by suffering
little from DIG contamination. We also derive second-order corrections which
further reduce the scatter (0.03--0.04 dex) in the differential metallicity of
HII-DIG/LIER pairs. Similarly, we explore other metallicity diagnostics such as
O3S2 ($=$log([O III]/H$\beta$+[S II]/H$\alpha$)) and N2S2H$\alpha$ ($=$ log([N
II]/[S II]) + 0.264log([N II]/H$\alpha$)) and provide corrections for O3S2 to
measure the metallicity of DIG/LIER regions. We propose that the corrected O3N2
and O3S2 diagnostics are used to measure the gas-phase metallicity in quiescent
galaxies or in quiescent regions of star-forming galaxies.
