Non-prompt $\mathrm{J}/\psi$ production in proton-proton collisions with ALICE

$\mathrm{J}/\psi$ production in high-energy hadronic collisions is sensitive to both perturbative and non-perturbative aspects of quantum chromodynamics (QCD) calculations. The production of a heavy-quark pair is well-described by perturbative QCD, whereas the formation of the bound state involves non-perturbative processes, treated in different ways by various available theoretical models. ALICE can measure inclusive $\mathrm{J}/\psi$ at both forward and midrapidity down to low ${p_{\rm T}}$ and the prompt and non-prompt $\mathrm{J}/\psi$ separation can be performed at midrapidity. The study of the production of non-prompt $\mathrm{J}/\psi$ originating from the decay of beauty hadrons, besides allowing to isolate the prompt $\mathrm{J}/\psi$ cross section from the inclusive $\mathrm{J}/\psi$ cross section, can be used to estimate open beauty-hadron production. Heavy-flavour particle production in pp collisions as a function of charged-particle multiplicity can provide insight into the processes occuring in the collision at the partonic level, as well as the interplay between the hard and soft mechanisms in particle production.