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Double Spin Asymmetries $A_{LT}^{\cosϕ_S}$ and $A_{LT}^{\cos(2ϕ_h -ϕ_S)}$ in semi-inclusive DIS

We investigate the double spin asymmetries of pion production in semi-inclusive deep inelastic scattering with a longitudinal polarized beam off a transversely polarized proton target. Particularly, we consider the $\cosϕ_S$ and $\cos(2ϕ_h -ϕ_S)$ modulations, which can be interpreted by the convolution of the twist-3 transverse momentum dependent distributions and twist-2 fragmentation functions. Three different origins are taken into account simultaneously for each asymmetry: the $g_T D_1$ term, the $e_T H_1^\perp$ term, and the $e_T^\perp H_1^\perp$ term in the $\cosϕ_S$ asymmetry; and the $g_T^\perp D_1$ term, the $e_T H_1^\perp$ term, and the $e_T^\perp H_1^\perp$ term in the $\cos(2ϕ_h -ϕ_S)$ asymmetry. We calculate the four twist-3 distributions $g_T(x,\boldsymbol{k}_T^2)$, $g_T^\perp(x,\boldsymbol{k}_T^2)$, $e_T(x,\boldsymbol{k}_T^2)$, and $e_T^\perp(x,\boldsymbol{k}_T^2)$ in a spectator-diquark model including vector diquarks. Then we predict the two corresponding asymmetries for charged and neutral pions at the kinematics of HERMES, JLab, and COMPASS for the first time. The numerical estimates indicate that the two different angular-dependence asymmetries are sizable by several percent at HERMES and JLab, and the $\cosϕ_S$ asymmetry has a strong dependence on the Bjorken $x$. Our predictions also show that the dominant contribution to the $\cosϕ_S$ asymmetry comes from the $g_T D_1$ term, while the $g_T^\perp D_1$ term gives the main contribution to the $\cos(2ϕ_h -ϕ_S)$ asymmetry; the other two $T$-odd terms almost give negligible contributions. In particular, the $\cos(2ϕ_h -ϕ_S)$ asymmetry provides a unique opportunity to probe the distribution $g_T^\perp$.