A complete non-perturbative renormalization prescription for quasi-PDFs

TL;DR

The paper presents the first complete non-perturbative renormalization prescription for quasi-PDFs in the RI'$ scheme, addressing both logarithmic and linear divergences from Wilson lines and the mixing in the unpolarized case. It provides a framework to convert RI'$-renormalized results to ${\overline{\rm MS}}$ at $\bar{\mu}=2$ GeV and demonstrates this on a $N_f=2+1+1$ twisted-mass ensemble with $m_\pi\approx375$ MeV, including a detailed assessment of lattice artifacts and perturbative conversion uncertainties. The study yields renormalized helicity quasi-PDFs and shows that, after matching to light-front PDFs at one-loop, the results move toward phenomenology with reduced antiquark overestimation, while highlighting remaining systematic limitations. The work establishes a crucial renormalization groundwork for quasi-PDFs, enabling more reliable extractions of parton distributions from lattice QCD and guiding future improvements (e.g., two-loop conversions and artifact subtractions) to reach precision-level comparisons with experimental data.

Abstract

In this work we present, for the first time, the non-perturbative renormalization for the unpolarized, helicity and transversity quasi-PDFs, in an RI' scheme. The proposed prescription addresses simultaneously all aspects of renormalization: logarithmic divergences, finite renormalization as well as the linear divergence which is present in the matrix elements of fermion operators with Wilson lines. Furthermore, for the case of the unpolarized quasi-PDFs, we describe how to eliminate the unwanted mixing with the twist-3 scalar operator. We utilize perturbation theory for the one-loop conversion factor that brings the renormalization functions to the MS-scheme at a scale of 2 GeV. We also explain how to improve the estimates on the renormalization functions by eliminating lattice artifacts. The latter can be computed in one-loop perturbation theory and to all orders in the lattice spacing. We apply the methodology for the renormalization to an ensemble of twisted mass fermions with Nf=2+1+1 dynamical light quarks, and a pion mass of around 375 MeV.

Figures (10)

• Figure 1: One-loop conversion factor from the RI$'$ to the ${\overline{\rm MS}}$ scheme for $\bar{\mu}{=}\bar{\mu}_0$. In the left (right) plot we show the real (imaginary) part of the conversion factor as a function of the length of the Wilson line in lattice units. Two choices of the RI$'$ scale have been employed: $a\bar{\mu}_0{=}\frac{2\pi}{32}\, (\frac{7}{2}{+}\frac{1}{4}, 3,3,3)$ (blue circles) and $a\bar{\mu}_0{=}\frac{2\pi}{32}\, (\frac{4}{2}{+}\frac{1}{4}, 0,0,3)$ (red squares). We use the abbreviation (7,3,3,3) and (4,0,0,3) in the legends, respectively.
• Figure 2: The $z$-dependent renormalization function for the matrix element $\Delta h(P_3,z)$ with $a P_3=\frac{6\pi}{L}$. The "parallel" and "diagonal" choices for $\vec{\bar{\mu}}_0$ are shown in the left and right plots, respectively. Open (filled) symbols correspond to the RI$'$ (${\overline{\rm MS}}$) estimates.
• Figure 3: Left: Real part of $Z^{\overline{\rm MS}}_{\Delta h}$ for scales labeled by m1$-$m8 and $z/a{=}5,10,15$. Right: Same as left panel for the imaginary part of $Z^{\overline{\rm MS}}_{\Delta h}$.
• Figure 4: Extrapolated $Z$-factor for the helicity operator, $Z_{0, \,{\Delta h}}^{{\overline{\rm MS}}}$, for 5 steps of HYP smearing. The employed fit range for the RI$'$ scale is $(a\,\bar{\mu}_0)^2\,\epsilon\, [1.4 {-} 2.0]$.
• Figure 5: Left: multiplicative ($Z^{\rm RI'}_{VV}$) and mixing ($Z^{\rm RI'}_{VS}$) coefficients entering the renormalization of the unpolarized quasi-PDF, in the RI$'$ scheme for the "(7,3,3,3)" scale. Right: ratio of the real (red squares) and imaginary (blue circles) parts of $Z^{\rm RI'}_{VS}$ over $Z^{\rm RI'}_{VV}$.