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The ratio of production cross-sections of $\psi(2S)$ over $J/\psi$ mesons asa function of charged-particle multiplicity in proton-proton collisions at acentre-of-mass energy $\sqrt{s}=13$ TeV is measured with a data samplecollected by the LHCb detector, corresponding to an integrated luminosity of658 pb$^{-1}$. The ratio is measured for both prompt and non-prompt $\psi(2S)$and $J/\psi$ mesons. When there is an overlap between the rapidity ranges overwhich multiplicity and charmonia production are measured, amultiplicity-dependent modification of the ratio is observed for prompt mesons.No significant multiplicity dependence is found when the ranges do not overlap.For non-prompt production, the $\psi(2S)-to-J/\psi$ production ratio is roughlyindependent of multiplicity irrespective of the rapidity range over which themultiplicity is measured.  The results are compared to predictions of the co-mover model and agree wellexcept in the low multiplicity region. The ratio of production cross-sectionsof $\psi(2S)$ over $J/\psi$ mesons are cross-checked with other measurements indi-lepton channels and found to be compatible.

Multiplicity dependence of $\sigma_{\psi(2S)}/\sigma_{J/\psi}$ in $pp$ collisions at $\sqrt{s}=13$ TeV