  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 GeV, 1.25 GeV, and 1.67 GeV.
The lattice data are renormalized non-perturbatively using the RI$'$ scheme,
and the final result for $h_L(x)$ is presented in the $\overline{\rm MS}$
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.
