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

Progress on Complex Langevin simulations of a finite density matrix model for QCD

We study the Stephanov model, which is an RMT model for QCD at finite density, using the Complex Langevin algorithm. Naive implementation of the algorithm shows convergence towards the phase quenched or quenched theory rather than to intended theory with dynamical quarks. A detailed analysis of this issue and a potential resolution of the failure of this algorithm are discussed. We study the effect of gauge cooling on the Dirac eigenvalue distribution and time evolution of the norm for various cooling norms, which were specifically designed to remove the pathologies of the complex Langevin evolution. The cooling is further supplemented with a shifted representation for the random matrices. Unfortunately, none of these modifications generate a substantial improvement on the complex Langevin evolution and the final results still do not agree with the analytical predictions.Comment: 8 pages, 7 figures, Proceedings of the 35th International Symposium on Lattice Field Theory, Granada, Spai

Random Matrix Models for Dirac Operators at finite Lattice Spacing

We study discretization effects of the Wilson and staggered Dirac operator with $N_{\rm c}>2$ using chiral random matrix theory (chRMT). We obtain analytical results for the joint probability density of Wilson-chRMT in terms of a determinantal expression over complex pairs of eigenvalues, and real eigenvalues corresponding to eigenvectors of positive or negative chirality as well as for the eigenvalue densities. The explicit dependence on the lattice spacing can be readily read off from our results which are compared to numerical simulations of Wilson-chRMT. For the staggered Dirac operator we have studied random matrices modeling the transition from non-degenerate eigenvalues at non-zero lattice spacing to degenerate ones in the continuum limit.Comment: 7 pages, 6 figures, Proceedings for the XXIX International Symposium on Lattice Field Theory, July 10 -- 16 2011, Squaw Valley, Lake Tahoe, California, PACS: 12.38.Gc, 05.50.+q, 02.10.Yn, 11.15.H

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