CSS Equation, etc, Follow from Structure of TMD Factorization

TL;DR

This paper demonstrates that the Collins–Soper–Sterman CS equation and the associated RG evolution for TMD PDFs arise from the intrinsic structure of TMD factorization rather than from operator-specific definitions. By exploiting a rapidity-based factorization framework and OPE-inspired RG arguments, it shows that the CS kernel tilde K depends only on b_T and μ, while the TMD anomalous dimensions gamma_f and gamma_K are universal, mass-independent (in MS), and yield a linear ln zeta dependence. The approach reconciles different treatments (BN, EIS) and clarifies why TMD evolution acquires a Q dependence via two Sudakov-type logarithms, while providing a perturbative small-b_T matching to integrated PDFs. The results set a unified foundation for phenomenology across the full small-q_T region, with caveats regarding heavy-quark masses.

Abstract

I show that the forms of the Collins-Soper-Sterman and renormalization-group equations for the evolution of transverse-momentum-dependent (TMD) parton densities in QCD follow from the structure of TMD factorization. A derivation does not need to directly use detailed properties of the operator definition of the TMD parton densities.