Gauge Invariant Two-Photon-Exchange Contributions in $e^-\pi^+ \rightarrow e^-\pi^+$

The gauge invariant two-photon exchange (TPE) contributions in $e^-\pi^+ \rightarrow e^-\pi^+$ are discussed at hadronic level. The contact term is added to keep the full amplitude gauge invariant by two methods: one is to multiply form factors with the point-like particles scattering amplitude and another is to construct a gauge invariant Lagangian. The practical calculations show the TPE corrections by these two methods are almost the same, while the later method is favored when extending the discussion to processes including two charged finite-size particles like $ep \rightarrow en\pi^+$.