The PAU Survey: Measurements of the 4000 {\AA} spectral break with
  narrow-band photometry
  The D4000 spectral break index is one of the most important features in the
visible spectrum, as it is a proxy for stellar ages and is also used in galaxy
classification. However, its direct measurement has always been reserved to
spectroscopic observations. In this work, we present a general method to
directly measure the D4000 spectral break index with narrow-band photometry:
this method has been validated using realistic simulations, and then evaluated
with PAUS NBs, cross-matched with VIPERS spectra ($i_{\rm AB} < 22.5$, $0.562 <
z < 0.967$). For comparison, we also determine the D4000 in this sample with
the SED-fitting code CIGALE (for both PAUS NBs and broad-band data from
CFHTLS). The direct D4000 measurement has significantly lower SNR than the
CIGALE D4000, however, we find that for $i_{\rm AB}<21$ (two magnitudes above
PAUS completeness limit) all direct D4000 measurements have $SNR>3$. Moreover,
CIGALE underestimates the error by $>$50\%, while the direct D4000 has proper
error estimation within $<$10\%. We study the D4000-stellar mass relationship
for red and blue galaxies, as well as the D4000-SFR dependence: all methods
show agreements with VIPERS within $1\sigma$, and the D4000-mass relation is
especially well reproduced for blue galaxies. We also evaluate how a D4000 cut
classifies galaxies into red/blue compared to machine learning: the CIGALE
D4000 with PAUS NBs has the best results while providing realistic cut values.
We conclude that the direct measurement of D4000 with narrow-band photometry is
a promising tool that allows individual measurements for objects measured with
$SNR > 3$, or averages by stacking, with results compatible with spectroscopy.
