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We discuss the impact on the study of gluon transverse momentum dependentdistributions (TMDs) of the associated production of a lepton pair and a$\Upsilon$ (or a $J/\psi$) in unpolarised proton-proton collisions, $pp\to{\cal Q} \, \ell \bar{\ell} X$, at LHC energies, where one can assume that suchfinal states are dominantly induced by gluon fusion. If the transverse momentumof the quarkonium-dilepton system - namely, the transverse momentum imbalanceof the quarkonium state and the lepton pair - is small, the corresponding crosssections can be calculated within the framework of TMD factorisation. Using thehelicity formalism, we show in detail how these cross sections are connected tothe moments of two independent TMDs: the distribution of unpolarised gluons,$f_1^g$, and the distribution of linearly polarised gluons, $h_1^{\perp g}$. Wecomplete our exhaustive derivation of these general relations with aphenomenological analysis of the feasibility of the TMD extraction, as well assome outlooks.

Associated production of a dilepton and a $\Upsilon(J/\psi)$ at the LHC as a probe of gluon transverse momentum dependent distributions