Connecting Different TMD Factorization Formalisms in QCD

TL;DR

The paper provides a systematic bridge between the original CSS1 and updated CSS2 TMD factorization formalisms, showing that the non-perturbative $g$-functions are scheme-invariant and that perturbative inputs can be derived from collinear-factorization calculations and quark form-factor results. By deriving explicit mappings between $A$, $B$, $ ilde{K}$, $ ilde{C}$, and $H^{ m DY}$ across CSS1 and CSS2—and relating them to SCET frameworks—the authors unify multiple TMD approaches and establish universal evolution for TMD parton densities and fragmentation functions via $ ilde{K}$ and $oldsymbol{ ilde{eta}}$. They compute high-order perturbative ingredients up to $O(a_s^3)$, including the DY hard factor $H^{ m DY}$, the evolution kernel $ ilde{K}$, and the anomalous dimensions $oldsymbol{ ilde{eta}}$ and $oldsymbol{ ilde{eta}}$, while providing explicit matching coefficients $ ilde{C}$ for TMD quark densities. The results enable practical, scheme-independent phenomenology for DY and SIDIS and demonstrate deep consistency between different factorization formalisms, including SCET treatments, ultimately advancing precision QCD predictions in transverse momentum dependent processes.

Abstract

In the original Collins-Soper-Sterman (CSS) presentation of the results of transverse-momentum-dependent (TMD) factorization for the Drell-Yan process, results for perturbative coefficients can be obtained from calculations for collinear factorization. Here we show how to use these results, plus known results for the quark form factor, to obtain coefficients for TMD factorization in more recent formulations, e.g., that due to Collins, and apply them to known results at order $α_s^2$ and $α_s^3$. We also show that the "non-perturbative" functions as obtained from fits to data are equal in the two schemes. We compile the higher-order perturbative inputs needed for the updated CSS scheme by appealing to results obtained in a variety of different formalisms. In addition, we derive the connection between both versions of the CSS formalism and several formalisms based in soft-collinear effective theory (SCET). Our work uses some important new results for factorization for the quark form factor, which we derive.

Figures (1)

• Figure 1: Graphs for the space-like (a) and time-like (b) quark form factor.