region where collinear factorization applies, similar to what has been done in [18], or from the transverse momentum weighted azimuthal asymmetry measurements in which the twist- three fragmentation function ˆ H ( z ) enters directly. For the purpose of estimating its contribution to the SSAs in the inclusive π 0 production in p $^{↑}$p collisions at RHIC energy, we parameterize the twist-three fragmentation function in terms of the leading-twist fragmentation function, We leave a more comprehensive parameterization for these fragmentation functions in a future study, including that for charged mesons as well. The additional z factor in the parameterization comes from the consideration that this novel fragmentation is mostly a valence behavior. Normally, we would also have a (1 − z ) suppression for the twist-three function. However, for scalar hadron production, the twist-three fragmentation function is not power suppressed in terms of (1 − z ), similar to the power counting of the Boer- Mulders function of π meson at large- x [39]. We have also adopted the quark transversity distributions from the parameterizations in Ref. [40] and the unpolarized fragmentation function in [41]. In Fig. 2, we show the predictions with the above parameterizations for the π 0 production with C$_{f}$ = − 0 . 4. The three curves from up to bottom correspond to a = 1 (solid), a = 2 (dashed), and a = 4 (dotted), respectively. With our parametrization of ˆ H ( z ), the twist-three fragmentation function can generate a sizable SSA in inclusive π 0 production at RHIC energy √$_{s}$ = 200 GeV. These contribution are comparable to that of the twist- three distribution from the polarized nucleon which have similar parameterizations [17, 18]. The single transverse spin asymmetry of η meson has also been studied by the STAR collaboration at RHIC recently [2]. A significantly larger asymmetry A$_{N}$ has been observed for η meson compared to π $^{0}$. As we discussed, in the twist-three collinear factorization, the dominate contribution comes from the twist-three distribution of the polarized nucleon, as shown in the first term of Eq. (2), and the twist-three fragmentation function, as shown in the third term of Eq. (2). The difference between η and π 0 from the first term will be mainly due to the difference in the unpolarized meson fragmentation function. However, from the measurements of spin-averaged cross section of η and π 0 by PHENIX collaboration at RHIC [42], one finds that for a large range of transverse momentum, η/π 0 ratio is a constant. This 9 ˆ H ( z ) = C$_{f}$z $^{a}$D ( z ) . (13)