The first measurement at midrapidity ($|y| < 0.5$) of the production yield of the strange-charm baryons $Ξ_c^+$ and $Ξ_c^0$ as a function of charged-particle multiplicity in proton-proton collisions at $\sqrt{s} = 13$ TeV with the ALICE experiment at the LHC is reported. The $Ξ_c^+$ baryon is reconstructed via the $Ξ_c^+ \rightarrow Ξ^-π^+π^+$ decay channel in the range $4 < p_{\rm T} < 12$ GeV/$c$, while the $Ξ_c^0$ baryon is reconstructed via both the $Ξ_c^0 \rightarrow Ξ^-π^+$ and $Ξ_c^0 \rightarrow Ξ^-e^+ν_e$ decay channels in the range $2 < p_{\rm T} < 12$ GeV/$c$. The baryon-to-meson ($Ξ_c^{0,+}/D^0$) and the baryon-to-baryon ($Ξ_c^{0,+}/Λ_c^+$) production yield ratios show no significant dependence on multiplicity. In addition, the observed yield ratios are not described by theoretical predictions that model charm-quark fragmentation based on measurements at $e^+e^-$ and $e^-$p colliders, indicating differences in the charm-baryon production mechanism in pp collisions. A comparison with different event generators and tunings, including different modelling of the hadronisation process, is also discussed. Moreover, the branching-fraction ratio of BR($Ξ_c^0 \rightarrow Ξ^-e^+ν_e$)/BR($Ξ_c^0 \rightarrow Ξ^-π^+$) is measured as 0.825 $\pm$ 0.094 (stat.) $\pm$ 0.081 (syst.). This value supersedes the previous ALICE measurement, improving the statistical precision by a factor of 1.6.
$p_{\rm T}$-differential per-event yield of prompt $\Xi_c^0$ baryons measured in the different multiplicity classes.
$p_{\rm T}$-differential per-event yield of prompt $\Xi_c^+$ baryons measured in the different multiplicity classes.
Ratio between the prompt $\Xi_c^0$ baryons in a multiplicity class to the multiplicity-integrated (INEL $>$ 0) class.
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