Measurement of Double Spin Asymmetry $A_{LT}$ in Semi-Inclusive Pion Electroproduction on a Transversely Polarized $^3$He Target

We recently measured the neutron double spin asymmetry $\mathcal{A}_{LT}$ in the semi-inclusive deep inelastic $^{3}He^{\uparrow}(\overrightarrow{e},e'\pi^{+/-})X$ reactions with polarized electron beam and a transversely polarized $^3$He target. The measurement was performed in Jefferson Lab Hall A, using 6 GeV polarized electron beam scattering on a 40 cm polarized $^3$He target. The produced pions were detected in the left high- resolution spectrometer in coincidence with the scattered electrons detected by the BigBite spectrometer. The kinematic coverage focused on the valence quark region, $x = 0.13 \sim 0.41$, at $Q^2=1.31 \sim 3.10~(\mathrm{GeV}/c)^2$. This data, when combined with the world data, will provide constraints on the $g^{q}_{1T}$ distribution functions, which describes longitudinal polarization of quarks in the transversely polarized target, for both $u$-quark and $d$- quark in the valence region. Experiment setup and preliminary data analysis progress will be presented in this talk.