Double-spin asymmetry $A^{\cos\phi_S}_{LT}$ in semi-inclusive DIS at CLAS12 and EIC within the collinear framework

We study the longitudinal-transverse double-spin asymmetry with a $\cos\phi_S$ modulation in semi-inclusive deep inelastic scattering for charged and neutral pions production. We consider the particular case in which the transverse momentum of the final state hadron is integrated out. The corresponding asymmetry may be contributed by two parts: one is the convolution of the twist-3 distribution function $g_{T}(x)$ and the unpolarized fragmentation function $D_1(z)$, the other is related to the coupling of the transversity distribution function $h_1(x)$ and the collinear twist-3 fragmentation function $\tilde{E}(z)$. We take into account both contributions to predict the $\cos\phi_S$ asymmetry at the kinematics of CLAS12 and a future Electron Ion Collider. We find that the asymmetry of pion production at CLAS12 is sizable, and $\tilde{E}(z)$ can play an important role in the asymmetry in the large-$z$ region.