Insights on star formation histories and physical properties of $1.2
  \leq z \lesssim 4 $ Herschel-detected galaxies
  We test the impact of using variable star forming histories (SFHs) and the
use of the IR luminosity (LIR) as a constrain on the physical parameters of
high redshift dusty star-forming galaxies. We explore in particular the stellar
properties of galaxies in relation with their location on the SFR-M* diagram.
We perform SED fitting of the UV-NIR and FIR emissions of a large sample of
GOODS-Herschel galaxies, for which rich multi-wavelength observations are
available. We test different SFHs and imposing energy conservation in the SED
fitting process, to face issues like the age-extinction degeneracy and produce
SEDs consistent with observations. Our models work well for the majority of the
sample, with the notable exception of the high LIR end, for which we have
indications that our simple energy conservation approach cannot hold true. We
find trends in the SFHs fitting our sources depending on stellar mass M* and z.
Trends also emerge in the characteristic timescales of the SED models depending
on the location on the SFR-M* diagram. We show that whilst using the same
available observational data, we can produce galaxies less star-forming than
usually inferred, if we allow declining SFHs, while properly reproducing their
observables. These sources can be post-starbursts undergoing quenching, and
their SFRs are potentially overestimated if inferred from their LIR. Fitting
without the IR constrain leads to a strong preference for declining SFHs, while
its inclusion increases the preference of rising SFHs, more so at high z, in
tentative agreement with the cosmic star formation history. Keeping in mind
that the sample is biased towards high LIR, the evolution shaped by our model
appears as both bursty (initially) and steady-lasting (later on). The global
SFH of the sample follows the cosmic SFH with a small scatter, and is
compatible with the "downsizing" scenario of galaxy evolution.
