Parton distribution functions on the lattice and in the continuum

Ioffe-time distributions, which are functions of the Ioffe-time $\nu$, are the Fourier transforms of parton distribution functions with respect to the momentum fraction variable $x$. These distributions can be obtained from suitable equal time, quark bilinear hadronic matrix elements which can be calculated from first principles in lattice QCD, as it has been recently argued. In this talk I present the first numerical calculation of the Ioffe-time distributions of the nucleon in the quenched approximation.Comment: 8 pages, 10 figures. arXiv admin note: text overlap with arXiv:1706.0537

Moments of Ioffe time parton distribution functions from non-local matrix elements

We examine the relation of moments of parton distribution functions to matrix elements of non-local operators computed in lattice quantum chromodynamics. We argue that after the continuum limit is taken, these non-local matrix elements give access to moments that are finite and can be matched to those defined in the $\overline{MS}$ scheme. We demonstrate this fact with a numerical computation of moments through non-local matrix elements in the quenched approximation and we find that these moments are in excellent agreement with the moments obtained from direct computations of local twist-2 matrix elements in the quenched approximation.Comment: 1+11 pages, 1 figure, version to appear in JHE

Reconstructing parton distribution functions from Ioffe time data: from Bayesian methods to Neural Networks

The computation of the parton distribution functions (PDF) or distribution amplitudes (DA) of hadrons from first principles lattice QCD constitutes a central open problem. In this study, we present and evaluate the efficiency of a selection of methods for inverse problems to reconstruct the full $x$-dependence of PDFs. Our starting point are the so called Ioffe time PDFs, which are accessible from Euclidean time calculations in conjunction with a matching procedure. Using realistic mock data tests, we find that the ill-posed incomplete Fourier transform underlying the reconstruction requires careful regularization, for which both the Bayesian approach as well as neural networks are efficient and flexible choices.Comment: 1+41 pages, 20 figure

Lattice QCD Exploration of Parton Pseudo-Distribution Functions

We demonstrate a new method of extracting parton distributions from lattice calculations. The starting idea is to treat the generic equal-time matrix element M(Pz(3), z(3)(2)) as a function of the Ioffe time nu = Pz(3) and the distance z(3). The next step is to divide M(Pz(3), z(3)(2)) by the rest-frame density M(0, z(3)(2)). Our lattice calculation shows a linear exponential z(3)-dependence in the rest-frame function, expected from the Z(z(3)(2)) factor generated by the gauge link. Still, we observe that the ratio M (Pz(3), z(3)(2))/M(Pz(3), z(3)(2)) has a Gaussian-type behavior with respect to z(3) for 6 values of P used in the calculation. This means that Z (z(3)(2))factor was canceled in the ratio. When plotted as a function of nu and z(3), the data are very close to z(3)-independent functions. This phenomenon corresponds to factorization of the x-and k(perpendicular to)-dependence for the TMD F(x, k(perpendicular to)(2)) . For small z(3) \u3c= 4a, the residual z(3)-dependence is explained by perturbative evolution, with alpha(s)/pi = 0.1

The Continuum and Leading Twist Limits of Parton Distribution Functions in Lattice QCD

In this study, we present continuum limit results for the unpolarized parton distribution function of the nucleon computed in lattice QCD. This study is the first continuum limit using the pseudo-PDF approach with Short Distance Factorization for factorizing lattice QCD calculable matrix elements. Our findings are also compared with the pertinent phenomenological determinations. Inter alia, we are employing the summation Generalized Eigenvalue Problem (sGEVP) technique in order to optimize our control over the excited state contamination which can be one of the most serious systematic errors in this type of calculations. A crucial novel ingredient of our analysis is the parameterization of systematic errors using Jacobi polynomials to characterize and remove both lattice spacing and higher twist contaminations, as well as the leading twist distribution. This method can be expanded in further studies to remove all other systematic errors.Comment: 56 pages, 29 figure

Long-distance contribution to ϵK\epsilon_K from lattice QCD

A lattice QCD approach to the calculation of the long-distance contributions to $\epsilon_K$ is presented. This parameter describes indirect CP violation in $K\to\pi\pi$ decay. While the short-distance contribution to $\epsilon_K$ can be accurately calculated in terms of standard model parameters and a single hadronic matrix element, $B_K$, there is a long-distance part which is estimated to be approximately $5\%$ of the total and is more difficult to determine. A method for determining this small but phenomenologically important contribution to $\epsilon_K$ using lattice QCD is proposed and a complete exploratory calculation of the contribution is presented. This exploratory calculation uses an unphysical light quark mass corresponding to a 339 MeV pion mass and an unphysical charm quark mass of 968 MeV, expressed in the $\overline{\mathrm{MS}}$ scheme at 2 GeV. This calculation demonstrates that future work should be able to determine this long-distance contribution from first principles with a controlled error of 10\% or less

Pion Valence Structure from Ioffe-Time Parton Pseudodistribution Functions

We present a calculation of the pion valence quark distribution extracted using the formalism of reduced Ioffe-time pseudodistributions 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 243 × 64 and 323 × 96 at the lattice spacing of a = 0.127 fm, and with the quark mass equivalent to a pion mass of mπ ≃ 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 pseudodistributions 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 “operator product expansion without operator product expansion.” 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