The N$^3$LO Twist-2 Matching of Helicity TMDs and SIDIS $q_\ast$ Spectrum

Abstract: We compute the twist-2 matching of transverse momentum dependent (TMD) helicity parton distribution and fragmentation functions at next-to-next-to-next-to-leading order (N$^3$LO) in QCD. This calculation entails the complete set of next-to-next-to-leading order (NNLO) Dokshitzer-Gribov-Lipatov-Altarelli-Parisi (DGLAP) splitting functions govering the evolution of helicity-dependent parton distribution functions (PDFs) and fragmentation functions (FFs). Within TMD factorization framework, we quantify the impact of radiative corrections by completing the next-to-next-to-next-to-leading logarithmic (N$^3$LL) prediction for lepton-hadron transverse momentum imbalance in semi-inclusive deep inelastic scattering (SIDIS). Our results provide the most precise theoretical input for probing the helicity structure and confined motion of quarks and gluons at future electron-ion collider (EIC).