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this paper, we will report the derivative-term results, and carry out the numerical estimates for their contributions to the single spin asymmetries at RHIC. We will leave the detailed derivations, including the derivative and non-derivative terms in a future publication. 

We follow the technique developed in the last few years [17–23] to derive the contributions from the twist-three fragmentation functions. First, we calculate the associated scattering amplitudes in terms of various twist-three matrix elements of hadrons. From diagrammatic point of view, these individual contributions are not in a gauge invariant way. However, the final results will be gauge invariant when we sum up all the contributions. In terms of twist-three operators, we have the contributions from the ∂$_{T}$ ψ , A$_{T}$ , and ∂$_{T}$ A + ( ∂ $^{+}$A$_{T}$ ). These contributions indeed will form the gauge invariant results in terms of ˆ H ( z ), ˆ H$_{F}$ as defined above. In particular, ∂$_{T}$ ψ term corresponds to ˆ H ( z ) whereas A$_{T}$ term corresponds to ˆ H$_{F}$ . Since the derivative terms are associated with the function ˆ H ( z ), we will need to calculate the contributions from the ∂$_{T}$ ψ part. In order to carry out this calculation, we have to perform the collinear expansion for the hard partonic part associated with this matrix element. In Fig. 1, we illustrate the generic Feynman diagrams contributing to the derivative terms. In this diagram, p$_{1}$ and p$_{2}$ are the two incoming partons’ momenta from the polarized and unpolarized nucleons, respectively; k$_{1}$ and k$_{2}$ are the two outgoing partons’ momenta where k$_{1}$ fragments into the final-state hadron P$_{h}$ . The derivative terms come from the 

