The $Lambda_c^+K^-$ mass spectrum is studied with a data sample of $pp$ collisions at a centre-of-mass energy of 13 TeV corresponding to an integrated luminosity of 5.6 fb$^{-1}$ collected by the LHCb experiment. Three $Xi_c^0$ states are observed with a large significance and their masses and natural widths are measured to be begin{eqnarray*} m(Xi_c(2923)^0)&=& 2923.04 pm 0.25 pm 0.20 pm 0.14 ~mathrm{MeV}, Gamma(Xi_c(2923)^0) &=& 7.1 pm 0.8 pm 1.8 ~mathrm{MeV}, end{eqnarray*} begin{eqnarray*} m(Xi_c(2939)^0) &=& 2938.55 pm 0.21 pm 0.17 pm 0.14 ~mathrm{ MeV}, Gamma(Xi_c(2939)^0) &=& 10.2 pm 0.8 pm 1.1 ~mathrm{ MeV}, end{eqnarray*} begin{eqnarray*} m(Xi_c(2965)^0) &=& 2964.88 pm 0.26 pm 0.14 pm 0.14~mathrm{ MeV}, Gamma(Xi_c(2965)^0) &=& 14.1 pm 0.9 pm 1.3~mathrm{ MeV}, end{eqnarray*} where the uncertainties are statistical, systematic, and due to the limited knowledge of the $Lambda_c^+$ mass. The $Xi_c(2923)^0$ and $Xi_c(2939)^0$ baryons are new states. The $Xi_c(2965)^0$ state is in the vicinity of the known $Xi_c(2970)^0$ baryon; however, their masses and natural widths differ significantly.
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