Nonperturbative renormalization of nonlocal quark bilinears for quasi-PDFs on the lattice using an auxiliary field

TL;DR

This approach shows how to understand the pattern of mixing that is allowed by chiral symmetry breaking and obtain a master formula for renormalizing the nonlocal operator that depends on three parameters and presents an approach for nonperturbatively determining these parameters and use perturbation theory to convert to the modified minimal subtraction scheme.

Abstract

Quasi-PDFs provide a path toward an ab initio calculation of parton distribution functions (PDFs) using lattice QCD. One of the problems faced in calculations of quasi-PDFs is the renormalization of a nonlocal operator. By introducing an auxiliary field, we can replace the nonlocal operator with a pair of local operators in an extended theory. On the lattice, this is closely related to the static quark theory. In this approach, we show how to understand the pattern of mixing that is allowed by chiral symmetry breaking, and obtain a master formula for renormalizing the nonlocal operator that depends on three parameters. We present an approach for nonperturbatively determining these parameters and use perturbation theory to convert to the MS-bar scheme. Renormalization parameters are obtained for two lattice spacings using Wilson twisted mass fermions and for different discretizations of the Wilson line in the nonlocal operator. Using these parameters we show the effect of renormalization on nucleon matrix elements with pion mass approximately 370 MeV, and compare renormalized results for the two lattice spacings. The renormalized matrix elements are consistent among the different Wilson line discretizations and lattice spacings.

Figures (6)

• Figure 1: Effective energy of the bare auxiliary field propagator, for two lattice spacings and three different link discretizations. Solid symbols show the finer lattice spacing and open symbols show the coarser one. The curve shows the three-loop perturbative result, shifted vertically by $-m$ to match it to the unsmeared data on the finer lattice spacing. Its error band indicates the size of the $O(\alpha_s^3)$ contribution.
• Figure 2: $Z_\phi$ for $\beta=2.10$, relative to the 5HYP case. Left: versus $\xi$, for $p\parallel n$ and $a^2p^2\approx 0.35$. Right: versus $p^2$, for $\xi=6a$ (unsmeared) and $\xi=10a$ (HYP). For each link discretization, we take the average of the two values at the smallest $p^2$.
• Figure 3: $Z_\phi$ for $\beta=2.10$, using unsmeared gauge links. Data are shown for a range of $p^2$ and $y\equiv |p|\xi$; the horizontal axis is $a^2p^2$, with a small displacement for different $y$ at the same $p^2$. The green open squares are given in our family of schemes, the blue filled diamonds show the result from conversion to $\overline{\text{MS}}$ at scale $|p|$ using Eq. \ref{['eq:conversion']}, and the orange filled triangles with black outlines show the $\overline{\text{MS}}$ results evolved to the scale 2 GeV, using the two-loop anomalous dimension of the static-light current Chetyrkin:2003viJi:1991prBroadhurst:1991fz.
• Figure 4: Matrix element for the helicity quasi-PDF versus $\xi$ on the $\beta=2.10$ ensemble, for three different link discretizations, bare (left) and renormalized (right). Only $\xi\geq 0$ is shown, since the real part is even in $\xi$ and the imaginary part is odd.
• Figure 5: Renormalized matrix element for the helicity quasi-PDF versus $\xi$ on the two ensembles, using five steps of HYP smearing.