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

Understand

We calculate next-to-leading-order (NLO) corrections to exclusive processes in $k_T$ factorization theorem, taking $\pi\gamma^*\to\gamma$ as an example.

• Partons off-shell by $k_T^2$ are considered in both the quark diagrams from full QCD and the effective diagrams for the pion wave function.
• The gauge dependences in the above two sets of diagrams cancel, when deriving the $k_T$-dependent hard kernel as their difference.
• The gauge invariance of the hard kernel is then proven to all orders by induction.