  With the existing and upcoming large multi-fibre low-resolution
spectrographs, the question arises how precise stellar parameters such as Teff
and [Fe/H] can be obtained from low-resolution K-band spectra with respect to
traditional photometric temperature measurements. Until now, most of the
effective temperatures in galactic Bulge studies come directly from photometric
techniques. Uncertainties in interstellar reddening and in the assumed
extinction law could lead to large systematic errors. We aim to obtain and
calibrate the relation between Teff and the $\rm ^{12}CO$ first overtone bands
for M giants in the galactic Bulge covering a wide range in metallicity. We use
low-resolution spectra for 20 M giants with well-studied parameters from
photometric measurements covering the temperature range 3200 < Teff < 4500 K
and a metallicity range from 0.5 dex down to -1.2 dex and study the behaviour
of Teff and [Fe/H] on the spectral indices. We find a tight relation between
Teff and the $\rm ^{12}CO(2-0)$ band with a dispersion of 95 K as well as
between Teff and the $\rm ^{12}CO(3-1)$ with a dispersion of 120 K. We do not
find any dependence of these relations on the metallicity of the star, making
them relation attractive for galactic Bulge studies. This relation is also not
sensitive to the spectral resolution allowing to apply this relation in a more
general way.
  We also found a correlation between the combination of the NaI, CaI and the
$\rm ^{12}CO$ band with the metallicity of the star. However this relation is
only valid for sub-solar metallicities. We show that low-resolution spectra
provide a powerful tool to obtain effective temperatures of M giants. We show
that this relation does not depend on the metallicity of the star within the
investigated range and is also applicable to different spectral resolution.
