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The cross-section of associated $J/\psi$-$\psi(2S)$ production inproton-proton collisions at a centre-of-mass energy of $\sqrt{s}=13$ TeV ismeasured using a data sample corresponding to an integrated luminosity of 4.2fb$^{-1}$, collected by the LHCb experiment. The measurement is performed forboth $J/\psi$ and $\psi(2S)$ mesons having transverse momentum$p_{\text{T}}<14$ GeV/$c$ and rapidity $2.0<y<4.5$, assuming negligiblepolarisation of the $J/\psi$ and $\psi(2S)$ mesons. The productioncross-section is measured to be $4.5\pm0.7\pm0.3$ nb, where the firstuncertainty is statistical and the second systematic. The differentialcross-sections are measured as functions of several kinematic variables of the$J/\psi$-$\psi(2S)$ candidates. The results are combined with a measurement of$J/\psi$-$J/\psi$ production, giving a cross-section ratio between$J/\psi$-$\psi(2S)$ and $J/\psi$-$J/\psi$ production of$0.274\pm0.044\pm0.008$, where the first uncertainty is statistical and thesecond systematic.

Measurement of associated $J/\psi$-$\psi(2S)$ production cross-section in $pp$ collisions at $\sqrt{s}=13$ TeV