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We investigate unpolarized and polarized gluon distributions and their applications to the Ioffe-time distributions, which are related to lattice QCD calculations of parton distribution functions. Guided by the counting rules based on the perturbative QCD at large momentum fraction $x$ and the color coherence of gluon couplings at small $x$, we parametrize gluon distributions in the helicity basis. By fitting the unpolarized gluon distribution, the inferred polarized gluon distribution from our parametrization agrees with the one from global analysis. A simultaneous fit to both unpolarized and polarized gluon distributions is also performed to explore the model uncertainty. The agreement with the global analysis supports the $(1-x)$ power suppression of the helicity-antialigned distribution relative to the helicity-aligned distribution. The corresponding Ioffe-time distributions and their asymptotic expansions are calculated from the gluon distributions. Our results of the Ioffe-time distributions can provide guidance to the extrapolation of lattice QCD data to the region lacking precise gluonic matrix elements. Therefore, they can help regulate the ill-posed inverse problem associated with extracting the gluon distributions from discrete data from first-principle calculations, which are available in a limited range of the nucleon momentum and the spatial separation between the gluonic currents. Given various limitations in obtaining lattice QCD data at large Ioffe time, phenomenological approaches can provide important complementary information to extract the gluon distributions in the entire $x$ region. The possibility of investigating higher-twist effects and other systematic uncertainties in the contemporary first-principle calculations of parton distributions from phenomenologically well-determined Ioffe-time distributions in the large Ioffe-time region is also discussed.
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We present a calculation of the pion valence quark distribution extracted using the formalism of reduced Ioffe time pseudo-distributions or more commonly known as pseudo-PDFs. Our calculation is carried out on two different 2+1 flavor QCD ensembles using the isotropic-clover fermion action, with lattice dimensions $24^3times 64$ and $32^3times 96$ at the lattice spacing of $a=0.127$ fm, and with the quark mass equivalent to a pion mass of $m_pi simeq 415$ MeV. We incorporate several combinations of smeared-point and smeared-smeared pion source-sink interpolation fields in obtaining the lattice QCD matrix elements using the summation method. After one-loop perturbative matching and combining the pseudo-distributions from these two ensembles, we extract the pion valence quark distribution using a phenomenological functional form motivated by the global fits of parton distribution functions. We also calculate the lowest four moments of the pion quark distribution through the OPE without OPE. We present a qualitative comparison between our lattice QCD extraction of the pion valence quark distribution with that obtained from global fits and previous lattice QCD calculations.
We present the results that are necessary in the ongoing lattice calculations of the gluon parton distribution functions (PDFs) within the pseudo-PDF approach. We identify the two-gluon correlator functions that contain the invariant amplitude determining the gluon PDF in the light-cone $z^2 to 0$ limit, and perform one-loop calculations in the coordinate representation in an explicitly gauge-invariant form. Ultraviolet (UV) terms, which contain $ln (-z^2)$-dependence cancel in the reduced Ioffe-time distribution (ITD), and we obtain the matching relation between the reduced ITD and the light-cone ITD. Using a kernel form, we get a direct connection between lattice data for the reduced ITD and the normalized gluon PDF.
We present the results that are necessary in the ongoing lattice calculations of the gluon parton distribution functions (PDFs) within the pseudo-PDF approach. We give a classification of possible two-gluon correlator functions and identify those that contain the invariant amplitude determining the gluon PDF in the light-cone $z^2 to 0$ limit. One-loop calculations have been performed in the coordinate representation and in an explicitly gauge-invariant form. We made an effort to separate ultraviolet (UV) and infrared (IR) sources of the $ln (-z^2)$-dependence at short distances $z^2$. The UV terms cancel in the reduced Ioffe-time distribution (ITD), and we obtain the matching relation between the reduced ITD and the light-cone ITD. Using a kernel form, we get a direct connection between lattice data for the reduced ITD and the normalized gluon PDF. We also show that our results may be used for a rather straightforward calculation of the one-loop matching relations for quasi-PDFs.
We discuss the structure of the parton quasi-distributions (quasi-PDFs) $Q(y, P_3)$ outside the canonical $-1 leq y leq 1$ support region of the usual parton distribution functions (PDFs). Writing the $y^n$ moments of $Q(y, P_3)$ in terms of the combined $x^{n-2l} k_perp^{2l}$-moments of the transverse momentum distribution (TMD) ${cal F} (x,k_perp^2)$, we establish a connection between the large-$|y|$ behavior of $Q(y,P_3)$ and large-$k_perp^2$ behavior of ${cal F} (x,k_perp^2)$. In particular, we show that the $1/k_perp^2$ hard tail of TMDs in QCD results in a slowly decreasing $sim 1/|y|$ behavior of quasi-PDFs for large $|y|$ that produces infinite $y^n$ moments of $Q(y,P_3)$. We also relate the $sim 1/|y|$ terms with the $ln z_3^2$-singulariies of the Ioffe-time pseudo-distributions $mathfrak{M} ( u, z_3^2)$. Converting the operator product expansion for $mathfrak{M} ( u, z_3^2)$ into a matching relation between the quasi-PDF $Q(y,P_3)$ and the light-cone PDF $f(x, mu^2)$, we demonstrate that there is no contradiction between the infinite values of the $y^n$ moments of $Q(y,P_3)$ and finite values of the $x^n$ moments of $f(x, mu^2)$.
We show that quasi-PDFs may be treated as hybrids of PDFs and primordial rest-frame momentum distributions of partons. This results in a complicated convolution nature of quasi-PDFs that necessitates using large $p_3 sim 3$ GeV momenta to get reasonably close to the PDF limit. As an alternative approach, we propose to use pseudo-PDFs $P(x, z_3^2)$ that generalize the light-front PDFs onto spacelike intervals and are related to Ioffe-time distributions $M ( u, z_3^2)$, the functions of the Ioffe time $ u = p_3 z_3$ and the distance parameter $z_3^2$ with respect to which it displays perturbative evolution for small $z_3$. In this form, one may divide out the $z_3^2$ dependence coming from the primordial rest-frame distribution and from the problematic factor due to lattice renormalization of the gauge link. The $ u$-dependence remains intact and determines the shape of PDFs.
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