This paper reports a global fit showing that a widely used theory for heavy quark bound states can describe how certain charmonium particles are produced and decay at high transverse momentum in proton collisions. The authors determine six nonperturbative parameters that appear in non‑relativistic QCD (NRQCD). These parameters, called long‑distance matrix elements (LDMEs), control how a charm–anticharm quark pair turns into observable particles like J/ψ, ψ(2S) and χ_c.

NRQCD separates the short‑distance creation of a heavy quark pair, which can be calculated with standard methods, from the long‑distance process of forming a bound state, which is encoded in the LDMEs. The notation O(v²) means the authors included terms up to second order in the typical relative speed v between the charm quarks inside the bound state. Including these relativistic corrections makes the description more complete than a lowest‑order treatment.

To extract the LDMEs the team used a leading‑power fragmentation framework and advanced perturbative calculations labeled NLO+NLL (next‑to‑leading order with next‑to‑leading logarithm resummation). They fit LHC measurements from ATLAS and CMS at a collision energy of 13 TeV. The fit used ψ(2S)/J/ψ yield ratios and ψ(2S) polarization data, and it accounted both for direct production and for “feed‑down” — the J/ψ particles that come from decays of heavier states such as χ_c and ψ(2S). They also used one decay width per charmonium state as an extra input to constrain the fit.

The fitted LDMEs give very good agreement with CMS and ATLAS cross‑section data for J/ψ up to transverse momentum P_T = 360 GeV. The same LDME set also matches CMS measurements of J/ψ polarization. From these results the authors conclude that NRQCD, when extended to O(v²) and combined with the fragmentation approach, is sufficient to describe hadroproduction of J/ψ, ψ(2S) and χ_c at large P_T.