On the Renormalizability of Quasi Parton Distribution Functions

TL;DR

The paper analyzes the ultraviolet structure of coordinate-space quasi-PDFs and proves they renormalize multiplicatively to all orders, with divergences localized to the region where all loop momenta are large. A power-counting framework identifies finite sets of potentially divergent topologies, and gauge-link-irreducible analysis shows 3-D integrations do not generate real UV divergences. The renormalization consists of an exponential power-divergence factor and standard wavefunction/vertex renormalization, with no operator mixing, making coordinate-space quasi-PDFs good candidates for lattice-based PDF extraction after addressing nonperturbative power divergences. It also highlights the need for consistent subtraction schemes when Fourier transforming to momentum-space quasi-PDFs at large $\tilde{x}$.

Abstract

Quasi-parton distribution functions have received a lot of attentions in both perturbative QCD and lattice QCD communities in recent years because they not only carry good information on the parton distribution functions, but also could be evaluated by lattice QCD simulations. However, unlike the parton distribution functions, the quasi-parton distribution functions have perturbative ultraviolet power divergences because they are not defined by twist-2 operators. In this paper, we identify all sources of ultraviolet divergences for the quasi-parton distribution functions in coordinate-space, and demonstrate that power divergences, as well as all logarithmic divergences can be renormalized multiplicatively to all orders in QCD perturbation theory.

Figures (9)

• Figure 1: Feynman diagrams for quasi-quark PDFs of an asymptotic quark of momentum $p$ at one-loop order.
• Figure 2: Feynman diagram for quasi-quark PDFs of an asymptotic gluon of momentum $p$ at one-loop order.
• Figure 3: Some loop diagrams, with an additional gluon (denoted as dotted curves) attachments, that could contribute to the quasi-PDFs at higher loop orders.
• Figure 4: A general "gauge-link-irreducible" (GLI) diagram, where $l_0=q-l_1-\cdots-l_n$.
• Figure 5: A diagram made of two "gauge-link-irreducible" (GLI) sub-diagrams.