We report on our exploratory study for the direct evaluation of the parton distribution functions from lattice QCD, based on a recently proposed new approach. We present encouraging results using N f ¼ 2 þ 1 þ 1 twisted mass fermions with a pion mass of about 370 MeV. The focus of this work is a detailed description of the computation, including the lattice calculation, the matching to an infinite momentum and the nucleon mass correction. In addition, we test the effect of gauge link smearing in the operator to estimate the influence of the Wilson line renormalization, which is yet to be done.

We present the results of our computation of the dimensionless chiral condensate r 0 Σ 1/3 with N f = 2 and N f = 2 + 1 + 1 flavours of maximally twisted mass fermions. The condensate is determined from the Dirac operator spectrum, applying the spectral projector method proposed by Giusti and Lüscher. We use 3 lattice spacings and several quark masses at each lattice spacing to perform the chiral and continuum extrapolations. We study the effect of the dynamical strange and charm quarks by comparing our results for N f = 2 and N f = 2 + 1 + 1 dynamical flavours.

In this paper, we show a comparison of different definitions of the topological charge on the lattice. We concentrate on one small-volume ensemble with 2 flavours of dynamical, maximally twisted mass fermions and use three more ensembles to analyze the approach to the continuum limit. We investigate several fermionic and gluonic definitions. The former include the index of the overlap Dirac operator, the spectral flow of the Wilson-Dirac operator and the spectral projectors. For the latter, we take into account different discretizations of the topological charge operator and various smoothing schemes to filter out ultraviolet fluctuations: the gradient flow, stout smearing, APE smearing, HYP smearing and cooling. We show that it is possible to perturbatively match different smoothing schemes and provide a well-defined smoothing scale. We relate the smoothing parameters for cooling, stout and APE smearing to the gradient flow time τ. In the case of hypercubic smearing the matching is performed numerically. We investigate which conditions have to be met to obtain a valid definition of the topological charge and susceptibility and we argue that all valid definitions are highly correlated and allow good control over topology on the lattice.