Transverse parton distribution and fragmentation functions at NNLO: the quark case

Abstract

We revisit the calculation of perturbative quark transverse momentum dependent parton distribution functions and fragmentation functions using the exponential regulator for rapidity divergences. We show that the exponential regulator provides a consistent framework for the calculation of various ingredients in transverse momentum dependent factorization. Compared to existing regulators in the literature, the exponential regulator has a couple of advantages which we explain in detail. As a result, the calculation is greatly simplified and we are able to obtain the next-to-next-to-leading order results up to O( 2 ) in dimensional regularization. These terms are necessary for a higher order calculation which is made possible with the simplification brought by the new regulator. As a by-product, we have obtained the two-loop quark jet function for the Energy-Energy Correlator in the back-to-back limit, which is the last missing ingredient for its N 3 LL resummation.

Transverse parton distribution and fragmentation functions at NNLO: the gluon case

“…We have also found similar discrepancy in the quark-to-quark coefficient as reported in Ref. [63]. Very recently, the two-loop results for the linearly polarized gluon TMDPDFs have been given in Ref.…”

“…The resulting master integrals have been solved in Ref. [63] using differential equation method [76,77]. For the details we refer to Ref.…”

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

“…The TMD beam function depends on the rapidity regulator being used. Rapidity-regulator-independent TMD PDFs can be simply obtained using [36,37] [44]. Therefore, quark TMD PDFs can also be extracted to this order.…”

“…where e −b0τ P ·K P + is the exponential regulator [27] in the collinear sector [36]. |Sp q←i (P, {l}, {k})| 2 is the spin and color averaged squared splitting amplitudes for i → q+X with n loops (counting at the squared amplitude level) and m real emission.…”

Renormalization and matching for the Collins-Soper kernel from lattice QCD

“…TMDPDFs measure the transverse momentum q T carried by the struck parton. For perturbative q T Λ QCD , they can be calculated in terms of collinear parton distribution functions, and the resulting matching formula is known to next-to-next-to-leading order (NNLO) [20][21][22][23][24][25][26][27]. In contrast, for nonperturbative q T Λ QCD , TMDPDFs become genuinely nonperturbative objects which so far have only been extracted from measurements by performing global fits to a variety of experimental data sets, see e.g.…”