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In this work, the two-photon-exchange (TPE) effects in $ep\rightarrowen\pi^+$ at small $-t$ are discussed within a hadronic model. Under the piondominance approximation the TPE contribution to the amplitude can be describedby a scalar function in the limit $m_e\rightarrow 0$. The TPE contributions tothe amplitude and the unpolarized differential cross section are both estimatedwhen only the elastic intermediate state is considered. We find that the TPEcorrections to the unpolarized differential cross section are about from $-4\%$to $-20\%$ at $Q^2=1$--$ 1.6$ GeV$^2$. After considering the TPE corrections tothe experimental data sets of unpolarized differential cross section, weanalyze the TPE corrections to the separated cross sections$\sigma_{\textrm{L,T,LT,TT}}$. We find that the TPE corrections (at$Q^2=1$--$1.6$ GeV$^2$) to $\sigma_{\textrm{L}}$ are about from $-10\%$ to$-30\%$, to $\sigma_{\textrm{T}}$ are about $20\%$, and to$\sigma_{\textrm{LT,TT}}$ are much larger. By these analysis, we conclude thatthe TPE contributions in $ep\rightarrow en\pi^+$ at small $-t$ are important toextract the separated cross sections $\sigma_{\textrm{L,T,LT,TT}}$ and theelectromagnetic magnetic form factor of $\pi^+$ in the experimental analysis.

Two-Photon-Exchange effects in $ep\rightarrow en\pi^+$ and their corrections to separated cross sections $\sigma_{\textrm{L,T,LT,TT}}$ at small $-t$