Renormalizability of quasiparton distribution functions

Quasiparton distribution functions have received a lot of attention in both the 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 quasiparton 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 quasiparton 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.

Figure 1

Feynman diagrams for quasiquark PDFs of an asymptotic quark of momentum $p$ at one-loop order.

Figure 2

Feynman diagram for quasiquark 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 GLI diagram, where $l_0=q-l_1-\cdots -l_n$.

Figure 5

A diagram made of two GLI subdiagrams.

Figure 6

A two-loop diagram that might give a UV divergent contribution to the quasiquark PDF based on the power counting rules and building blocks in Figs. 1 and 2.

Figure 7

Three topologies of diagrams that could give UV divergent contributions to the quasiquark PDFs.

Figure 8

Sample topologies of diagrams that give UV finite contributions to the quasiquark PDFs.

Figure 9

Contributions to the general diagrams in Fig. 7 with all loop diagrams reorganized in terms of 1PI diagrams.