Parton distribution function with nonperturbative renormalization from lattice QCD

We present lattice results for the isovector unpolarized parton distribution with nonperturbative regularization-invariant momentum-subtraction scheme (RI/MOM) renormalization on the lattice. In the framework of large-momentum effective field theory (LaMET), the full Bjorken-$x$ dependence of a momentum-dependent quasidistribution is calculated on the lattice and matched to the ordinary light cone parton distribution at one-loop order, with power corrections included. The important step of RI/MOM renormalization that connects the lattice and continuum matrix elements is detailed in this paper. A few consequences of the results are also addressed here.

Figure 1

Comparison between the renormalization constants obtained with the point source and the momentum source for $z\le 2$, taking the $p_z=6\pi /L$ case as an example. The values are normalized by the central value of the renormalization constant at $z=0$ and the real parts are subtracted by unity for a better comparison. It is obvious that with the same configurations, the signal from the momentum source can be much better than that from the point source, while the values are consistent with each other.

Figure 2

The renormalization constant of the quasi-PDF operator $O_{{\gamma }_z}(z)$ (red boxes) and the mixing with the scalar quasi-PDF operator $O_{\mathcal{I}}(z)$ (blue dots) with the momentum along the Wilson link being $6\pi /L=1.29\mathrm{GeV}$ and ${\mu }_R^2=p^2=5.74{\mathrm{GeV}}^2$. The size of the mixing coefficient is about an order of magnitude smaller than the renormalization factor in the large-$z$ region.

Figure 3

The bare ${\tilde{h}}_{{\gamma }_z}(z,P_z,{\mu }_R)$ (blue) and renormalized ${\tilde{h}}_R(z,P_z,{\mu }_R)$ (red) for $P_z=4\pi /L$ (upper row) and $6\pi /L$ (lower row) with the renormalization scale ${\mu }_R=2.4\mathrm{GeV}$. The left and right panels show the real and imaginary parts, respectively.

Figure 4

The renormalized unpolarized isovector quark distribution after one-loop matching and mass correction at the renormalization scale ${\mu }^2=5.76{\mathrm{GeV}}^2$ in the $\overline{\mathrm{MS}}$ scheme, with $P_z=6\pi /L$ and three different RI/MOM renormalization scales ${\mu }_R$. Three distributions agree with each other within the statistical uncertainties, it shows the ${\mu }_R$ dependence is almost cancelled numerically. The negative-$x$ part is related to the antiquark distribution via $\overline{u}(x)-\overline{d}(x)=-u(-x)+d(-x)$ for $x>0$.

Figure 5

The renormalized unpolarized isovector quark distribution after one-loop matching and mass correction at the renormalization scale ${\mu }^2=5.76{\mathrm{GeV}}^2$ in the $\overline{\mathrm{MS}}$ scheme. The negative-$x$ part is related to the antiquark distribution via $\overline{u}(x)-\overline{d}(x)=-u(-x)+d(-x)$ for $x>0$.

Figure 6

Comparison of the renormalized function ${\tilde{h}}_R(z)$ of this work (dashed lines) and from a Fourier transform of phenomenological PDFs into the function of the Ioffe time $z{P}_z$. All the values are converted to ${\mu }^2=5.76{\mathrm{GeV}}^2$ in the $\overline{\mathrm{MS}}$ scheme. The solid lines are the Fourier transform of the corresponding CJ15 PDF (blue), after matching and mass corrections (green and red). The left and right panels show the real and imaginary parts, respectively.