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Next-to-leading-order correction to pion form factor in $k_T$ factorization

We calculate next-to-leading-order (NLO) correction to the pion electromagnetic form factor at leading twist in the $k_T$ factorization theorem. Partons off-shell by $k_T^2$ are considered in both quark diagrams and effective diagrams for the transverse-momentum-dependent (TMD) pion wave function. The light-cone singularities in the TMD pion wave function are regularized by rotating the Wilson lines away from the light cone. The soft divergences from gluon exchanges among initial- and final-state partons cancel exactly. We derive the infrared-finite $k_T$-dependent NLO hard kernel for the pion electromagnetic form factor by taking the difference of the above two sets of diagrams. Varying the renormalization and factorization scales, we find that the NLO correction is smaller, when both the scales are set to the invariant masses of internal particles: it becomes lower than 40% of the leading-order (LO) contribution for momentum transfer squared $Q^2>7$ GeV$^2$. It is observed that the NLO leading-twist correction does not play an essential role in explaining the experimental data, but the LO higher-twist contribution does.