Quark TMDPDFs from Large to Small $x$: A Unified Description of CSS, DGLAP, and Small-$x$ Double Logarithms

Abstract

We extend the full-kinematic background-field construction developed for gluon transverse-momentum dependent parton distribution functions (TMDPDFs) in Mukherjee et al. [Phys. Rev. D 109, 034035 (2024)] to quark TMDPDFs, providing a common framework for their large- and small-$x$ dynamics. We compute the leading-power quark TMD beam function at next-to-leading order using the background-field method. The background fields retain both longitudinal and transverse momenta, with neither collinear nor eikonal ordering imposed. We distinguish the renormalization scales that define the TMD operator from the factorization scales that separate its perturbative and non-perturbative modes. With standard TMD renormalization, the quark TMDPDF obeys Collins-Soper-Sterman (CSS) evolution, while the lower-scale separation is organized directly in terms of background-field TMD operators. We refer to this full-kinematic construction as the MSTT scheme. In the MSTT scheme, the leading term of the resulting operator expansion reproduces both the Dokshitzer-Gribov-Lipatov-Altarelli-Parisi (DGLAP) logarithm and the small-$x$ double logarithm in the appropriate limits, while retaining parts of the higher-twist and sub-eikonal corrections. We also formulate a rapidity-only scheme for quark TMDPDFs with distinct renormalization and factorization cutoffs. We derive its evolution equation, which reduces to DGLAP evolution in the collinear limit and to double-logarithmic quark evolution at small $x$, and establish its matching to the MSTT and standard TMD schemes.