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We calculate next-to-leading-order (NLO) corrections to exclusive processesin $k_T$ factorization theorem, taking $\pi\gamma^*\to\gamma$ as an example.Partons off-shell by $k_T^2$ are considered in both the quark diagrams fromfull QCD and the effective diagrams for the pion wave function. The gaugedependences in the above two sets of diagrams cancel, when deriving the$k_T$-dependent hard kernel as their difference. The gauge invariance of thehard kernel is then proven to all orders by induction. The light-conesingularities in the $k_T$-dependent pion wave function are regularized byrotating the Wilson lines away from the light cone. This regularizationintroduces a factorization-scheme dependence into the hard kernel, which can beminimized in the standard way. Both the large double logarithms $\ln^2k_T$ and$\ln^2 x$, $x$ being a parton momentum fraction, arise from the loop correctionto the virtual photon vertex, the former being absorbed into the pion wavefunction and organized by the $k_T$ resummation, and the latter absorbed into ajet function and organized by the threshold resummation. The NLO correctionsare found to be only few-percent for $\pi\gamma^*\to\gamma$, if setting thefactorization scale to the momentum transfer from the virtual photon.

Next-to-leading-order corrections to exclusive processes in $k_T$ factorization