content_image/334215.jpg content_image/334222.jpg content_image/334221.jpg
and the hard factors H$_{ab}$$_{→}$$_{c}$ are defined as 

<img src='content_image/334215.jpg'>

where ˆ s , ˆ t , and ˆ u are the usual partonic Mandelstam variables. They are the same as the transverse spin transfer hard coefficients calculated in Ref. [36]. Substituting the expression of n$_{h}$ into Eq. (8), the factor in the bracket can also be written as, 

<img src='content_image/334222.jpg'>

In the forward rapidity region of the polarized nucleon, we have x ≫ x ′ and − ˆ u ≫ − ˆ t , and we can further simplify the transverse spin dependent differential cross section as 

<img src='content_image/334221.jpg'>

This term will be most phenomenological relevant for the single spin asymmetries of hadron production in the forward direction of the polarized nucleon. 

In order to demonstrate the twist-three fragmentation function contribution to the SSA in inclusive hadron production in p $^{↑}$p collisions, we need the unknown, but universal, twist- three fragmentation function ˆ H ( z ). We notice that ˆ H ( z ) can be related to the Collins fragmentation function H ⊥ 1 ( z, p 2 $_{⊥}$) as in Eq. (4), which has been studied from the available experimental data [37, 38]. However, we emphasize that the Collins function H ⊥ 1 ( z, p 2 $_{⊥}$) are fitted from small transverse momentum region where TMD factorization applies. To obtain the functional form for ˆ H ( z ), one has to assume a transverse momentum dependence in the Collins fragmentation functions. In principle, the twist-three fragmentation functions ˆ H ( z ) should be extracted from the experimental data of the SSAs at large transverse momentum 

