  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 matching
equation 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 broad range of $x$. This
work demonstrates the feasibility of accessing twist-3 parton distribution
functions from novel methods within lattice QCD and can provide essential
insights into the structure of hadrons.
