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longitudinal momentum fractions, we have made them explicit in the above formula. For a complete analysis, the above three terms have to be taken into account. The first term in the above equation, the contribution from the twist-three parton distributions φ (3) $_{i/A}$( x, x $^{′}$) from the polarized nucleon have been calculated in Refs. [17–19]. The second term from the twist- three effect in the unpolarized nucleon φ (3) $_{j/B}$( x, x $^{′}$) is found very small in the forward region of the polarized nucleon where the largest asymmetry has been found experimentally [28]. The third term is least known in the literature, which we will focus in this paper. There have been attempts to formulate this part of contribution [29]. However, a universality argument for the Collins fragmentation function [30, 31] would indicate that the contribution calcu- lated in Ref. [29] vanishes. This universality has also been extended to pp collisions [32]. Large azimuthal asymmetries coming from this Collins effect have been observed in semi- inclusive hadron production in deep inelastic scattering and di-hadron production in e $^{+}$e − annihilation processes by the HERMES [33] and COMPASS [34], and BELLE collabora- tions [35], respectively. In a recent publication, two of us have re-examined the twist-three fragmentation contribution to the single spin asymmetry [23]. In particular, we have identi- fied the twist-three fragmentation function corresponding to the TMD Collins fragmentation function, and shown that the TMD and collinear factorization approaches are consistent in the intermediate transverse momentum region in the SIDIS process. In this paper, we will extend this formalism to the twist-three fragmentation function contribution to the above process of (1), the single spin asymmetries in inclusive hadron production in p $^{↑}$p scattering. 

The twist-three fragmentation function which corresponds to the universal Collins TMD fragmentation function, is defined as 1 

where k + = P + h /z . This function can also be written as the transverse momentum moment of the Collins fragmentation function H ⊥ 1 ( z, p 2 $_{T}$), 

In the above definition Eq. (3), we have set the hadron momentum in the ˆ z direction. The index α is in the transverse direction perpendicular to the hadron momentum direction, and 

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