First-hand information for everyone
What the paper is about: The authors report the first nonperturbative information on the gluon Collins–Soper (CS) kernel. The CS kernel is a function that controls how transverse-momentum-dependent (TMD) distributions of gluons change when one shifts the rapidity scale. These TMDs describe how gluons inside a fast-moving hadron carry transverse momentum, and the kernel is an essential ingredient for predictions of many processes that probe gluon structure.
What the researchers did: They computed the kernel using lattice quantum chromodynamics (lattice QCD), a numerical method that puts the theory of quarks and gluons on a discrete space–time grid. The calculation used an ensemble of 1105 gauge-field configurations on a 323×48 lattice with spacing a = 0.15 fm and a pion mass close to the physical value, Mπ = 172(3) MeV. The authors extracted so-called quasi‑TMD beam functions from matrix elements of gluon operators in boosted pion states with momenta Pz between about 1.03 and 2.05 GeV. They matched those lattice observables to the physical lightlike TMDs using Large‑Momentum Effective Theory (LaMET) at next‑to‑next‑to‑leading logarithmic (NNLL) accuracy, and thereby obtained constraints on the gluon CS kernel for transverse momentum scales qT between 300 MeV and 1.3 GeV.
How it works at a high level: Direct TMDs live on the light cone and cannot be computed in Euclidean lattice QCD. LaMET provides a bridge: one computes related “quasi” distributions at large hadron momentum on the lattice, and then applies a perturbative matching formula to recover the lightlike quantities. The kernel itself is defined from the rapidity dependence of a gluon TMD beam function; in practice the authors compute Euclidean matrix elements built from gluon field-strength tensors connected by staple‑shaped Wilson lines, remove known ultraviolet divergences with a Wilson-loop renormalization factor, extrapolate to infinite staple length, and apply the NNLL matching to extract the CS kernel as a function of transverse separation (the Fourier partner of qT).
Why it matters: The gluon CS kernel is a universal ingredient in descriptions of many processes sensitive to gluon transverse motion, such as quarkonium and Higgs production or certain photoproduction measurements. Until now nonperturbative information on this kernel was lacking; phenomenology has relied on perturbative guidance and an assumed scaling relation with the quark kernel (Casimir scaling). These lattice results provide the first first‑principles constraints in the nonperturbative regime and are reported to be compatible within about one to two standard deviations with Casimir scaling. That compatibility, if confirmed, would support current phenomenological assumptions, while deviations could point to genuinely nonperturbative differences between quark and gluon evolution.
Important caveats and uncertainties: The calculation is only a first step. It uses a single lattice spacing (a = 0.15 fm) and a single lattice volume, so there is no continuum extrapolation yet. The hadron used on the lattice is the pion; while the CS kernel is universal in principle, the practical extraction used boosted pion matrix elements. The matching to lightlike TMDs is performed at NNLL accuracy, so truncation and power corrections in 1/(xPz) and 1/(bT xPz) remain and are controlled only approximately because the boosted momenta are finite. Statistical and systematic uncertainties are reported, and the authors note that future calculations at finer lattice spacings, higher momenta, and with additional checks will be needed to turn these constraints into precision determinations for phenomenology.