Perturbative Renormalization of quasi-PDFs

TL;DR

The paper addresses the renormalization of gauge-invariant nonlocal Wilson-line operators used to access quasi-PDFs in lattice QCD, revealing both linear and logarithmic ultraviolet divergences as well as finite operator mixing at one loop.Using Wilson/clover fermions and Symanzik-improved gluon actions, the authors perform a detailed one-loop analysis in dimensional regularization and lattice regularization, computing renormalization constants, mixing coefficients, and conversion factors between RI′ and MS̄ schemes.A key result is the identification of a linear divergence proportional to |z|/a arising from tadpole diagrams on the lattice, along with a finite, symmetry-allowed mixing pattern that can be tuned away with appropriate clover parameter choices.The work also outlines nonperturbative strategies to extract and subtract the linear divergence, and proposes a nonperturbative renormalization program (including RI′/MS̄ matching) for unpolarized, helicity, and transversity quasi-PDFs, enabling more reliable lattice determinations of parton distributions.

Abstract

In this paper we present results for the renormalization of gauge invariant nonlocal fermion operators which contain a Wilson line, to one-loop level in lattice perturbation theory. Our calculations have been performed for Wilson/clover fermions and a wide class of Symanzik improved gluon actions. The extended nature of such long-link operators results in a nontrivial renormalization, including contributions which diverge linearly as well as logarithmically with the lattice spacing, along with additional finite factors. On the lattice there is also mixing among certain subsets of these nonlocal operators; we calculate the corresponding finite mixing coefficients, which are necessary in order to disentangle individual matrix elements for each operator from lattice simulation data. Finally, extending our perturbative setup, we present non-perturbative prescriptions to extract the linearly divergent contributions.

Figures (4)

• Figure 1: Feynman diagrams contributing to the one-loop calculation of the Green's functions of operator $\mathcal{O}_\Gamma$. The straight (wavy) lines represent fermions (gluons). The operator insertion is denoted by a filled rectangle.
• Figure 2: Real (left panel) and imaginary (right panel) parts of the conversion factors for the operators $V_1$ and $T$ as a function of $z/a$ in the Landau gauge. The RI$'$ momentum scale employed is $a\bar{q}=\frac{2\pi}{32}\left(4{+}\frac{1}{4},0,0,4\right)$.
• Figure 3: Real (left panel) and imaginary (right panel) parts for the function $F_{V_1(A_1)}$ versus $qz$. Solid and dashed lines correspond to the Landau and Feynman gauge, respectively.
• Figure 4: The imaginary part of the ratio of Eq. (\ref{['RR']}) for simulation data at $m_\pi{=}375$ MeV using Twisted Mass fermions from Ref. Alexandrou:2016jqi (open symbols), as well as preliminary data at $m_\pi{=}130$ MeV using Twisted Mass clover-improved fermions (filled symbols). The unpolarized (blue circles, filled orange diamonds), polarized (red squares) and transversity (green triangles) quasi-PDFs are presented.