Transverse momentum dependent parton quasidistributions

We investigate the transverse momentum dependent parton distributions (TMDs) in the parton quasidistributions framework. The longstanding hurdle of the so-called pinch pole singularity from the spacelike gauge links in the TMD definitions can be resolved by the finite length of the gauge link along the hadron moving direction. In addition, with the soft factor subtraction, the quasi-TMD is free of linear divergence. We further demonstrate that the energy evolution equation of the quasi-TMD a.k.a. the Collins-Soper evolution, only depends on the hadron momentum. This leads to a clear matching between the quasi-TMD and the standard TMDs.

Figure 1

Illustration of the gauge links in the unsubtracted quasi-TMD (a) and the soft factor (b).

Figure 2

Real diagram contributions to the Q-TMD quark distributions at one-loop order with a gluon exchange between quarks (a), between quark and gauge link (b) and between gauge links (c). The complex conjugate of Diagram (b) is implied. Diagram (c) would contain the pinch pole singularity with infinite length gauge links in the TMD definition. However, with a finite length $L$, this singularity is absent in Q-TMD.

Figure 3

Virtual diagram contributions to the Q-TMD quark distributions at one-loop order with quark self-energy (a), quark-gauge-link vertex correction (b) and gauge link self-energy (c). The complex conjugate is also implied. Diagram (c) requires the renormalization of the gauge link self-interaction. This follows recent examples in the collinear parton quasidistributions cases.