Parton distribution functions beyond leading twist from lattice QCD: The hL (x) case

We report the fast-ever calculation of the isovector flavor combination of the chiral-odd twist-3 parton distribution h(L) (x) for the proton from lattice QCD. We employ gauge configurations with two degenerate light, a strange and a charm quark (N-f = 2 + 1 + 1) of maximally twisted mass fermions with a clover improvement. The lattice has a spatial extent of 3 fm and lattice spacing of 0.093 fm. The values of the quark masses lead to a pion mass of 260 MeV. We use a source-sink time separation of 1.12 fm to control contamination from excited states. Our calculation is based on the quasi-distribution approach, with three values for the proton momentum: 0.83, 1.25, and 1.67 GeV. The lattice data are renormalized nonperturbatively using the RI' scheme, and the final result for h(L) (x) is presented in the (MS) over bar scheme at the scale of 2 GeV. Furthermore, we compute in the same setup the transversity distribution, h(1) (x), which allows us, in particular, to compare h(L) (x) to its Wandzura-Wilczek approximation. We also combine results for the isovector and isoscalar flavor combinations to disentangle the individual quark contributions for h(1) (x) and h(L) (x), and address the Wandzura-Wilczek approximation in that case as well.