Probing the gluon Transverse Momentum-Dependent distributions inside the proton through quarkonium-pair production at the LHC

Transverse momentum-dependent factorisation is used to describe hadronic collisions while taking into account the intrinsic transverse momentum of partons inside hadrons. This requires the use of Transverse Momentum-Dependent Parton Distribution Functions (TMDPDFs or simply TMDs) in order to parameterize the parton correlator. Such distributions need to be extracted from experimental data. Quark TMDs are relatively well known thanks to processes such as semi-inclusive deep inelastic scattering (SIDIS) and Drell-Yan for which numerous data exist. Gluon TMDs remain poorly known, since there is no ideal process to probe them in the operating colliders. The future Electron-Ion Collider (EIC) will offer a much better access to them, but its first run remains at least 10 years from now. It is also important to study TMDs in various kinds of processes in order to check their universality which is not as trivial as that of collinear PDFs. We propose to use quarkonium-pair production to study the two leading-twist gluon TMDs accessible through unpolarized proton collisions at the Large Hadron Collider (LHC). Quarkonia are mesons, i.e. bound states of a quark-antiquark pair. In the case of a quarkonium, the pair is made of heavy flavours: charmonium combines a charm with an anti charm, while bottomonium combines a bottom with an anti-bottom. J/ψ mesons are the lowest lying vector state of charmonium and are produced in large amounts at the LHC. J/ψ pairs originate from gluon fusion in vast majority, which is important in order to focus on gluon TMDs. Studying two-particle final states also allows one to tune the hard scale of the process commensurate to the pair mass, which in turn allows one to study TMD evolution. We first use a model of Gaussian-based TMDs to compute observables in J/ψ-pair production that are sensitive to the TMDs. These observables are the transverse-momentum spectrum of the pair, mostly sensitive to the unpolarized gluon TMD, and azimuthal asymmetries, whose existence requires the linearly-polarised gluon TMD. We see that J/ψ pair production is an ideal process to probe the linearly-polarised gluon distribution through one azimuthal asymmetry that is maximal at large hard scales. We also use the LHCb data on the J/ψ pair transverse momentum to fit the average gluon transverse momentum using our Gaussian-based model. The large value that is obtained is interpreted as a consequence of TMD evolution that perturbatively enhances the intrinsic transverse momentum of the gluon at such large hard scales. We then improve our predictions by including TMD evolution in the formalism used to describe the gluon TMDs in our calculations. In this picture, the unpolarized gluon distribution is a leading contribution in an expansion of the strong coupling, while the linearly-polarised distribution is sub-leading. The remaining non-perturbative component is modelled using a Gaussian. We observe that the computed magnitude of the azimuthal asymmetries in J/ψ-pair production are lower than when using the purely Gaussian model. However, we observe that these asymmetries remain sizeable and could be detected at the LHC. We also provide predictions for Υ-pair production (the Υ is the bottomonium equivalent of the J/ψ). We finally study the helicity structure of the quarkonium-pair production amplitude. It can be written as a sum of sub-amplitudes corresponding to various helicity states of the initial-state gluons and final-state quarkonia. In the high-mass limit of the pair, the amplitudes greatly simplify and explain how the hard-scattering coefficients of J/ψ-pair production maximise the size of one azimuthal asymmetry, as previously observed. Moreover, it is shown that the amplitude zero for longitudinally polarised pairs predicted at leading order in the collinear regime exists as well in TMD factorisation. It should survive for intermediate masses as hard gluon emissions are suppressed in the TMD regime. (author)