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The NLO twist-3 contribution to the pion electromagnetic form factors in $k_{T}$ factorization

In this paper, by employing the $k_{T}$ factorization theorem, we calculate firstly the next-to-leading-order (NLO) twist-3 contributions to the pion electromagnetic form factors in the $πγ^* \to π$ process. From the analytical and numerical calculations we find the following points: (a) For the leading order (LO) twist-2, twist-3 and the NLO twist-2 contributions, our results agree very well with those obtained in previous works; (b) We extract out two factors $F^{(1)}_{\rm T3}(x_i,t,Q^2)$ and $\ov{F}^{(1)}_{\rm T3}(x_i,t,Q^2)$, which describe directly the NLO twist-3 contributions to the pion electromagnetic form factors $F^+(Q^2)$; (c) The NLO twist-3 contribution is negative in sign and cancel partially with the NLO twist-2 part, the total NLO contribution can therefore provide a roughly $\pm 20\%$ corrections to the total LO contribution in the considered ranges of $Q^2$; and (d) The theoretical predictions for $Q^2 F^+(Q^2)$ in the low-$Q^2$ region agree well with currently available data, this agreement can be improved by the inclusion of the NLO contributions.