Double Spin Asymmetries $A_{LT}^{\cos {\varphi }_S}$ and $A_{LT}^{\cos (2{\varphi }_h-{\varphi }_S)}$ 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 $\cos {\varphi }_S$ and $\cos (2{\varphi }_h-{\varphi }_S)$ 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,andthe$e_T^\perp H_1^\perp$terminthe$\cos\phi_S$asymmetry;andthe$g_T^\perp D_1$term,the$e_T H_1^\perp$term,andthe$e_T^\perp H_1^\perp$terminthe$\cos(2\phi_h -\phi_S)$ asymmetry. We calculate the four twist-3 distributions $g_T(x,{\mathit{k}}_T^2)$, $g_T^{\perp }(x,{\mathit{k}}_T^2)$, $e_T(x,{\mathit{k}}_T^2)$, and $e_T^{\perp }(x,{\mathit{k}}_T^2)$ 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 $\cos {\varphi }_S$ asymmetry has a strong dependence on the Bjorken $x$. Our predictions also show that the dominant contribution to the $\cos {\varphi }_S$ asymmetry comes from the $g_T{D}_1$ term, while the $g_T^{\perp }{D}_1$ term gives the main contribution to the $\cos (2{\varphi }_h-{\varphi }_S)$ asymmetry; the other two $T$-odd terms almost give negligible contributions. In particular, the $\cos (2{\varphi }_h-{\varphi }_S)$ asymmetry provides a unique opportunity to probe the distribution $g_T^{\perp }$.