This paper reports a new way to calculate the Collins–Soper kernel, a key ingredient in the description of how quarks and gluons carry transverse momentum inside fast-moving protons. Instead of simulating moving protons, the authors compute a vacuum “soft function” made from long, space-like Wilson lines on a Euclidean lattice. They use complex directional vectors for those Wilson lines to control a quantity called rapidity, and extract the kernel with high statistical precision.

In plain terms, a soft function is a vacuum expectation value of certain gauge-field lines (Wilson lines) that capture low-energy, wide-angle gluon effects in scattering. The Collins–Soper (CS) kernel governs how transverse-momentum-dependent parton distributions (TMDPDFs) change with rapidity, which matters when comparing data taken in different experimental setups. The authors represent the Wilson lines by an auxiliary heavy-quark field and solve for its lattice propagator. By adjusting the complex direction vectors they can construct Wilson lines at arbitrary rapidity on the lattice, avoiding the need to simulate fast-moving hadrons.

On the numerical side, the team performed pure-gauge (quenched) calculations on three Wilson-gauge ensembles with physical box size about 2 fm and lattice spacings a = 0.048, 0.041, and 0.03 femtometers. They used 250–341–200 configurations and 2,048 stochastic sources per configuration, reaching sub-percent statistical errors for the soft function. To remove ultraviolet and other lattice divergences they form single and double ratios of lattice soft functions and then match the short-distance behavior to continuum perturbation theory in the MS (Modified Minimal Subtraction) scheme, using high-order perturbative input (N3LL). They choose cutoffs for the transverse separation (b) in the fit region: b_cut = 0.144 fm, b_th = 0.24 fm, and a matching point b_mch = 0.384 fm.