Insights on proton structure from lattice QCD: The twist-3 parton distribution function gT(x)

In this work, we present the first-ever calculation of the isovector flavor combination of the twist-3 parton distribution function $g_T(x)$ for the proton from lattice QCD. We use an ensemble of 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, lattice spacing of 0.093 fm, and reproduces a pion mass of $260$ MeV. We use the quasi-distribution approach and employ three values of the proton momentum boost, 0.83 GeV, 1.25 GeV and 1.67 GeV. We use a source-sink separation of 1.12 fm to suppress excited-states contamination. The lattice data are renormalized non-perturbatively. We calculate the appropriate matching within Large Momentum Effective Theory, which is applied to the lattice data in order to obtain $g_T$. The final distribution is presented in the $\overline{\rm MS}$ scheme at a scale of 2 GeV. We also calculate the helicity distribution $g_1$ to test the Wandzura-Wilczek approximation for $g_T$. We find that the approximation works well for a large range in $x$. This work demonstrates the feasibility of accessing twist-3 parton distribution functions from novel methods within lattice QCD, and can provide important insights on the structure of hadrons.