Rapidity evolution of TMDs with running coupling

The scale of a coupling constant for rapidity-only evolution of transverse-momentum dependent (TMD) operators in the Sudakov kinematic region is calculated using the Brodsky-Lepage-Mackenzie optimal scale setting [S. J. Brodsky et al., Phys. Rev. D 28, 228 (1983).]. The effective argument of a coupling constant is halfway in the logarithmical scale between the transverse momentum and energy of TMD distribution. The resulting rapidity-only evolution equation is solved for quark and gluon TMDs.

Figure 1

One-loop diagrams for TMD operator (4) in the background quark field. The dashed lines denote gauge links.

Figure 2

Point-splitting regularization of rapidity divergence.

Figure 3

Quark loop correction to cut gluon propagator.

Figure 4

Quark loop correction to quark TMD evolution.

Figure 5

One-loop diagrams for gluon TMD operator (5) in the background gluon field.

Figure 6

Quark loop correction to gluon TMD evolution.

Figure 7

Quark loop correction to the gluon propagator in the background field.

Figure 8

Diagrams with the gauge-dependent part of the gluon propagator (denoted by wavy line).

Figure 9

Typical diagrams for one-loop evolution of the quark TMD operator.