and the hard factors H$_{ab}$$_{→}$$_{c}$ are defined as H$_{qq}$ $^{′}$→ qq ′ = H$_{q}$$_{¯}$ q $^{′}$→ q ¯ q ′ = N 2 c − 1 4 N 2 c 4ˆ s ˆ u − ˆ t 2 , H$_{qq}$$_{→}$$_{qq}$ = N 2 c − 1 4 N 2 c [ 4ˆ s ˆ u − ˆ t 2 − 1 N$_{c}$ 4ˆ s − ˆ t ] , H$_{q}$$_{¯}$ q → q ¯ q = H$_{¯}$ qq → ¯ qq = N 2 c − 1 4 N 2 c [ 4ˆ s ˆ u − ˆ t 2 + 1 N$_{c}$ 4ˆ u ˆ t ] , H$_{¯}$ qq → q ¯ q = H$_{q}$$_{¯}$ q → ¯ qq = − N 2 c − 1 N 3 c , H$_{qg}$$_{→}$$_{qg}$ = N 2 c − 1 N 2 c + 1 2 4ˆ s ˆ u − ˆ t 2 , (10) where ˆ s , ˆ t , and ˆ u are the usual partonic Mandelstam variables. They are the same as the transverse spin transfer hard coefficients calculated in Ref. [36]. Substituting the expression of n$_{h}$ into Eq. (8), the factor in the bracket can also be written as, In the forward rapidity region of the polarized nucleon, we have x ≫ x ′ and − ˆ u ≫ − ˆ t , and we can further simplify the transverse spin dependent differential cross section as This term will be most phenomenological relevant for the single spin asymmetries of hadron production in the forward direction of the polarized nucleon. In order to demonstrate the twist-three fragmentation function contribution to the SSA in inclusive hadron production in p $^{↑}$p collisions, we need the unknown, but universal, twist- three fragmentation function ˆ H ( z ). We notice that ˆ H ( z ) can be related to the Collins fragmentation function H ⊥ 1 ( z, p 2 $_{⊥}$) as in Eq. (4), which has been studied from the available experimental data [37, 38]. However, we emphasize that the Collins function H ⊥ 1 ( z, p 2 $_{⊥}$) are fitted from small transverse momentum region where TMD factorization applies. To obtain the functional form for ˆ H ( z ), one has to assume a transverse momentum dependence in the Collins fragmentation functions. In principle, the twist-three fragmentation functions ˆ H ( z ) should be extracted from the experimental data of the SSAs at large transverse momentum 8 E$_{h}$ d $^{3}$Δ σ ( s$_{⊥}$ ) d $^{3}$P$_{h}$ | forward = ϵ$_{⊥}$$_{αβ}$S α $_{⊥}$P β h ⊥ 2 α 2 s S ∑ a,b,c ∫ 1 x ′ min dx ′ x ′ f$_{b}$ ( x $^{′}$) 1 x h$_{a}$ ( x ) ∫ 1 z$_{min}$ dz z [ − z ∂ ∂z ( ˆ H ( z ) z 2 )] × 1 x $^{′}$S + T/z 1 − z ˆ u H$_{ab}$$_{→}$$_{c}$ (ˆ s, ˆ t, ˆ u ) . (12) { P β h + z ( p$_{2}$ · P$_{h}$n β h − p$_{2}$ · n$_{h}$P β $_{h}$) − z ˆ u } → ( P β h z ) x − x ′ x ( − ˆ u ) + x $^{′}$( − ˆ t ) . (11)