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We compute the contribution from the longitudinally polarized proton to thetwist-3 double-spin asymmetry $A_{LT}$ in inclusive (light) hadron productionfrom proton-proton collisions,i.e., $p^\uparrow \vec{p}\to h\,X$. We show thatusing the relevant QCD equation-of-motion relation and Lorentz invariancerelation allows one to eliminate the twist-3 quark-gluon correlator (associatedwith the longitudinally polarized proton) in favor of one-variable twist-3quark distributions and the (twist-2) transversity parton density. Includingthis result with the twist-3 pieces associated with the transversely polarizedproton and unpolarized final-state hadron (which have already been calculatedin the literature), we now have the complete leading-order cross section forthis process.

Twist-3 effect from the longitudinally polarized proton for $A_{LT}$ in hadron production from $pp$ collisions