Transverse single spin asymmetry in hadronic $\eta_{c,b}$ production

We study the transverse single spin asymmetry in $\eta_{c,b}$ production in polarized hadron collisions, employing the collinear twist-3 approach in combination with the color singlet model. Our main focus lies on the contribution from the twist-3 Efremov-Teryaev-Qiu-Sterman function. By extrapolating the derived spin dependent cross section to the small transverse momentum region, consistency between the collinear twist-3 approach and the transverse momentum dependent factorization approach is confirmed. As a byproduct of this work, we identify a term contributing to the scale evolution of tri-gluon correlations in the flavor-singlet case which was previously missed.