Single-transverse spin asymmetries (SSAs) in hadronic processes, such as in the sin- gle inclusive hadron production in single transversely-polarized nucleon-nucleon scatter- ing, p $^{↑}$p → hX , have attracted much interests from both experimental and theoretical sides in the last few years, and great progress has been made in understanding the un- derlying physics [1]. Although it is a simple observable, defined as the spin asymme- try when one flips the transverse spin of one of the hadrons involved in the scattering: A$_{N}$ = ( dσ ( S$_{⊥}$ ) − dσ ( − S$_{⊥}$ )) / ( dσ ( S$_{⊥}$ ) + dσ ( − S$_{⊥}$ )), it is far more complicated to explain in the fundamental theory of strong interaction. It usually represents a correlation between the transverse polarization vector S$_{⊥}$ of one of the hadrons and the transverse momentum P$_{h}$$_{⊥}$ of the final-state hadron in the differential cross section. For example, in the process p $^{↑}$p → hX , it is the correlation between the polarization vector S$_{⊥}$ of the incoming nucleon and the transverse momentum P$_{h}$$_{⊥}$ of the final-state hadron that generates the asymmetry. In this paper, we will focus on the SSAs in these processes, especially for the neutral mesons, π 0 and η production, motivated by the recent striking experimental observations of large SSAs for them by the STAR collaboration at RHIC experiments [2, 3]. In particular, at forward direction of the polarized proton, it was found that the SSA for η meson is much larger than that for π 0 meson. This also confirmed previous experimental observations from the fixed target experiments [4]. In addition to these STAR results, both BRAHMS and PHENIX collaborations at RHIC have observed large single transverse spin asymmetries in charged and neutral meson production in the forward rapidity region [5, 6]. In the QCD framework, there have been mainly two approaches to study the SSAs in high energy scattering processes: the transverse momentum dependent (TMD) parton distribu- tion and fragmentation function approach [7–15] and the twist-three quark-gluon correlation approach in the collinear factorization [16–21]. In the TMD approach, it is required to have an additional hard momentum scale besides the transverse momentum P$_{h}$$_{⊥}$ of the hadron, such as Q $^{2}$, the momentum transfer square of the virtual photon in semi-inclusive deep in- elastic scattering (SIDIS) or Drell-Yan lepton pair production in pp collisions, to study the transverse momentum dependence in these processes. On the other hand, the twist-three approach is more appropriate in the processes that all momentum scales are much larger than the nonperturbative scale Λ$_{QCD}$ [17]. It has been shown that these two approaches are consistent with each other for the SIDIS and Drell-Yan processes in the intermediate transverse momentum region where they both apply [22, 23]. 2