The measurement of the production of charm jets, identified by the presence of a ${\rm D^0}$ meson in the jet constituents, is presented in proton-proton collisions at centre-of-mass energies of $\sqrt{s}$ = 5.02 and 13 TeV with the ALICE detector at the CERN LHC. The ${\rm D^0}$ mesons were reconstructed from their hadronic decay ${\rm D^0} \rightarrow {\rm K^-}\pi^+$ and the respective charge conjugate. Jets were reconstructed from ${\rm D^0}$-meson candidates and charged particles using the anti-$k_{\rm T}$ algorithm, in the jet transverse momentum range $5<p_{\rm T;chjet}<50$ GeV/$c$, pseudorapidity $|\eta_{\rm jet}| <0.9-R$, and with the jet resolution parameters $R$ = 0.2, 0.4, 0.6. The distribution of the jet momentum fraction carried by a ${\rm D^0}$ meson along the jet axis ($z^{\rm ch}_{||}$) was measured in the range $0.4 < z^{\rm ch}_{||} < 1.0$ in four ranges of the jet transverse momentum. Comparisons of results for different collision energies and jet resolution parameters are also presented. The measurements are compared to predictions from Monte Carlo event generators based on leading-order and next-to-leading-order perturbative quantum chromodynamics calculations. A generally good description of the main features of the data is obtained in spite of a few discrepancies at low $p_{\rm T;chjet}$. Measurements were also done for $R = 0.3$ at $\sqrt{s}$ = 5.02 TeV and are shown along with their comparisons to theoretical predictions in an appendix to this paper.
The dependence of the ratio between the B$_\mathrm{s}^0$ and B$^+$ hadron production fractions, $f_\mathrm{s} / f_\mathrm{u}$, on the transverse momentum ($p_\mathrm{T}$) and rapidity of the B mesons is studied using the decay channels B$_\mathrm{s}^0$$\to$ J$/\psi\,\phi$ and B$^+$$\to$ J$/\psi$ K$^+$. The analysis uses a data sample of proton-proton collisions at a center-of-mass energy of 13 TeV, collected by the CMS experiment in 2018 and corresponding to an integrated luminosity of 61.6 fb$^{-1}$. The $f_\mathrm{s} / f_\mathrm{u}$ ratio is observed to depend on the B $p_\mathrm{T}$ and to be consistent with becoming asymptotically constant at large $p_\mathrm{T}$. No rapidity dependence is observed. The ratio of the B$^0$ to B$^+$ hadron production fractions, $f_\mathrm{d} / f_\mathrm{u}$, measured using the B$^0$$\to$ J$/\psi$ K$^{*0}$ decay channel, is found to be consistent with unity and independent of $p_\mathrm{T}$ and rapidity, as expected from isospin invariance.
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