We study the temperature and baryon density dependence of the masses of the lightest charmed baryons $Lambda_c$, $Sigma_c$ and $Sigma^*_c$. We also look at the effects of the temperature and baryon density on the binding energies of the $Lambda_c N$ and $Lambda_c Lambda_c$ systems. Baryon masses and baryon-baryon interactions are evaluated within a chiral constituent quark model. Medium effects are incorporated in those parameters of the model related to the dynamical breaking of chiral symmetry, which are the masses of the constituent quarks, the $sigma$ and $pi$ meson masses, and quark-meson couplings. We find that while the in-medium $Lambda_c$ mass decreases monotonically with temperature, those of $Sigma_c$ and $Sigma^*_c$ have a nonmonotonic dependence. These features can be understood in terms of a simple group theory analysis regarding the one-gluon exchange interaction in those hadrons. The in-medium $Lambda_c N$ and $Lambda_c Lambda_c$ interactions are governed by a delicate balance involving a stronger attraction due to the decrease of the $sigma$ meson mass, suppression of coupled-channel effects and lower thresholds, leading to shallow bound states with binding energies of a few~MeV. The $Lambda_c$ baryon could possibly be bound to a large nucleus, in qualitative agreement with results based on relativistic mean field models or QCD sum rules. Ongoing experiments at RHIC or LHCb or the planned ones at FAIR and J-PARC may take advantage of the present results.
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