The relation between velocity dispersion and mass in simulated clusters
  of galaxies: dependence on the tracer and the baryonic physics
  [Abridged] We present an analysis of the relation between the masses of
cluster- and group-sized halos, extracted from $\Lambda$CDM cosmological N-body
and hydrodynamic simulations, and their velocity dispersions, at different
redshifts from $z=2$ to $z=0$. The main aim of this analysis is to understand
how the implementation of baryonic physics in simulations affects such
relation, i.e. to what extent the use of the velocity dispersion as a proxy for
cluster mass determination is hampered by the imperfect knowledge of the
baryonic physics. In our analysis we use several sets of simulations with
different physics implemented. Velocity dispersions are determined using three
different tracers, DM particles, subhalos, and galaxies.
  We confirm that DM particles trace a relation that is fully consistent with
the theoretical expectations based on the virial theorem and with previous
results presented in the literature. On the other hand, subhalos and galaxies
trace steeper relations, and with larger values of the normalization. Such
relations imply that galaxies and subhalos have a $\sim10$ per cent velocity
bias relative to the DM particles, which can be either positive or negative,
depending on halo mass, redshift and physics implemented in the simulation.
  We explain these differences as due to dynamical processes, namely dynamical
friction and tidal disruption, acting on substructures and galaxies, but not on
DM particles. These processes appear to be more or less effective, depending on
the halo masses and the importance of baryon cooling, and may create a
non-trivial dependence of the velocity bias and the $\soneD$--$\Mtwo$ relation
on the tracer, the halo mass and its redshift.
  These results are relevant in view of the application of velocity dispersion
as a proxy for cluster masses in ongoing and future large redshift surveys.
