Parton distribution functions beyond leading twist from lattice QCD: The <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:msub><mml:mi>h</mml:mi><mml:mi>L</mml:mi></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mi>x</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:math> case
Shohini Bhattacharya, Krzysztof Cichy, Martha Constantinou, Andreas Metz, Aurora Scapellato, Fernanda Steffens
Abstract
We report the first-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 ${\mathrm{RI}}^{\ensuremath{'}}$ scheme, and the final result for ${h}_{L}(x)$ is presented in the $\overline{\mathrm{MS}}$ 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.