  We study the impact of neutrino masses on the shape and height of the BAO
peak of the matter correlation function, both in real and redshift space. In
order to describe the nonlinear evolution of the BAO peak we run N-body
simulations and compare them with simple analytic formulae. We show that the
evolution with redshift of the correlation function and its dependence on the
neutrino masses is well reproduced in a simplified version of the Zel'dovich
approximation, in which the mode-coupling contribution to the power spectrum is
neglected. While in linear theory the BAO peak decreases for increasing
neutrino masses, the effect of nonlinear structure formation goes in the
opposite direction, since the peak broadening by large scale flows is less
effective. As a result of this combined effect, the peak decreases by $\sim 0.6
\%$ for $ \sum m_\nu = 0.15$ eV and increases by $\sim1.2 \%$ for $ \sum m_\nu
= 0.3$ eV, with respect to a massless neutrino cosmology with equal value of
the other cosmological parameters. We extend our analysis to redshift space and
to halos, and confirm the agreement between simulations and the analytic
formulae. We argue that all analytical approaches having the Zel'dovich
propagator in their lowest order approximation should give comparable
performances, irrespectively to their formulation in Lagrangian or in Eulerian
space.
