Quark Transverse Parton Distribution at the Next-to-Next-to-Next-to-Leading Order

We report a calculation of the perturbative matching coefficients for the transverse-momentum-dependent parton distribution functions for quark at the next-to-next-to-next-to-leading order in QCD, which involves calculation of nonstandard Feynman integrals with rapidity divergence. We introduce a set of generalized integration-by-parts equations, which allows an algorithmic evaluation of such integrals using the machinery of modern Feynman integral calculation.

Figure 1

Representative cut diagram for the color structure $d^{ABC}{d}_{ABC}=(N_c^2-1)(N_c^2-4)/N_c$ from VRR (left) and RRR (right).

Figure 2

$K$ factor at ${\mathrm{N}}^3\mathrm{LO}$ for the coefficient functions. Numerical evaluation of HPLs are made with the hpl package [86].

Figure 3

Integrated TMD PDFs at various perturbative order. We use nnpdf30_nnlo_as_0118 [87] throughout the numerical calculation.