  We present a Markov-Chain Monte-Carlo (MCMC) forecast for the precision of
neutrino mass and cosmological parameter measurements with a Euclid-like galaxy
clustering survey. We use a complete perturbation theory model for the galaxy
one-loop power spectrum and tree-level bispectrum, which includes bias,
redshift space distortions, IR resummation for baryon acoustic oscillations and
UV counterterms. The latter encapsulate various effects of short-scale dynamics
which cannot be modeled within perturbation theory. Our MCMC procedure
consistently computes the non-linear power spectra and bispectra as we scan
over different cosmologies. The second ingredient of our approach is the
theoretical error covariance which captures uncertainties due to higher-order
non-linear corrections omitted in our model. Having specified characteristics
of a Euclid-like spectroscopic survey, we generate and fit mock galaxy power
spectrum and bispectrum likelihoods. Our results suggest that even under very
agnostic assumptions about non-linearities and short-scale physics a future
Euclid-like survey will be able to measure the sum of neutrino masses with a
standard deviation of 28 meV. When combined with the Planck cosmic microwave
background likelihood, this uncertainty decreases to 13 meV.
Over-optimistically reducing the theoretical error on the bispectrum down to
the two-loop level marginally tightens this bound to 11 meV. Moreover, we show
that the future large-scale structure (LSS) spectroscopic data will greatly
improve constraints on the other cosmological parameters, e.g. reaching a
percent (per mille) error on the Hubble constant with LSS alone (LSS + Planck).
