content_image/329017.jpg content_image/329016.jpg
The TMD approach has also been used to calculate the SSA in the single inclusive hadron production in p $^{↑}$p collisions in a model dependent way [10, 24, 25]. However, there has been no factorization argument based on TMD parton distribution for this process [26]. In this paper, we follow the collinear factorization approach to study the single spin asymmetry coming from the twist-three fragmentation function contribution. In this process, there is only one large momentum scale, the transverse momentum P$_{h}$$_{⊥}$ of the final-state hadron, and the SSA is naturally suppressed by 1 /P$_{h}$$_{⊥}$ at large P$_{h}$$_{⊥}$ as a higher-twist effect, although how large this behavior should manifest is not clear. At low transverse momentum, it will ultimately fail, and a modification has to be made to account for the experimental data [3, 27]. 

In the twist-three collinear factorization approach, for the SSAs in hadron production in 

<img src='content_image/329017.jpg'>

the twist-three effects come from the distribution functions of the incoming polarized nucleon ( A ) with momentum P$_{A}$ or the unpolarized nucleon ( B ) with momentum P$_{B}$ , or the fragmen- tation function for the final state hadron ( h ) with momentum P$_{h}$ . Therefore, schematically, we can write down the single transverse spin dependent differential cross section in the following way [17, 18], 

<img src='content_image/329016.jpg'>

where ϵ ⊥ is defined as ϵ ⊥ = ϵ $^{µναβ}$P$_{Aµ}$ P$_{Bν}$/P$_{A}$ · P$_{B}$ with convention of ϵ = 1. Here we use ⊥ to label the transverse direction in the center of mass frame of P$_{A}$ and P$_{B}$ . In the above equation, ⊗ stands for the convolution in the longitudinal momentum fractions x , y , and z . The leading twist parton distributions are labeled by φ$_{i/A}$ ( x ) and φ$_{j/B}$ ( y ). Because hadron A is transversely polarized, we immediately see that φ$_{i/A}$ represents the leading-twist quark transversity distributions. However, for unpolarized hadron B , φ$_{j/B}$ represents both leading twist quark and gluon distributions. Leading twist parton fragmentation function is represented by D$_{h/c}$ ( z ) where parton c can be a quark or gluon. In the above equation, the superscript (3) represents the twist-three correlations for the distribution functions or the fragmentation functions. Since these twist-three functions normally involve two variables in 

