Collins-Soper Kernel and Reduced Soft Function in Lattice QCD

We evaluate the Collins-Soper kernel and the reduced soft function in lattice QCD, incorporating O(αₛ) matching corrections. The calculation relies on the evaluation of the quasi-transverse momentum-dependent wave function with asymmetric staple-shaped quark bilinear operators and four-point meson form factors. These quantities are computed non-perturbatively using two Nf=2+1+1 twisted-mass fermion ensembles with the same lattice spacing of $a=0.093$ fm: the first ensemble has a lattice size of 24³ × 48 and a pion mass of 346 MeV; the second one has a lattice size of 32³ × 64 and a pion mass of 261 MeV. The Collins-Soper kernel and the soft function are needed for the determination of the transverse momentum-dependent parton distribution functions.