NNLO DGLAP splitting functions from collinear matching of TMDs

We report a complete computation of next-to-next-to-leading order (NNLO) helicity and transversity Dokshitzer-Gribov-Lipatov-Altarelli-Parisi (DGLAP) splitting functions, in both space-like and time-like kinematics. These results are obtained from the next-to-next-to-next-to-leading order (N$^3$LO) twist-2 matching of polarized transverse-momentum-dependent (TMD) parton distribution and fragmentation functions, including helicity, quark transversity, and linearly polarized gluons. We compare our results with existing calculations in the literature and discuss both agreements and discrepancies. Our results provide all perturbative ingredients required for the computation of N$^3$LO differential cross sections below the resolution scale $q_{T\mathrm{cut}}$ in transverse-momentum subtraction and enable next-to-next-to-next-to-next-to-leading logarithmic (N$^4$LL) resummation of $q_T$ observables in the Sudakov region. We further determine the small-$x$ structure of the polarized matching coefficients through N$^3$LO. These fixed-order results furnish the data for future small-$x$ resummation in polarized TMD factorization, where high-energy logarithms and Sudakov logarithms become simultaneously relevant. Establishing a consistent joint treatment of polarized small-$x$ evolution and transverse-momentum resummation remains an important open direction toward uniform precision in spin-dependent phenomenology. Our results provide essential theoretical input for precision spin physics at the forthcoming Electron-Ion Collider.